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# Symmetric Ground States Solutions of m-Coupled nonlinear Schrödinger
equations
Hichem Hajaiej Justus-Liebig-Universität Giessen
Mathematisches Institut
Arnd Str 2, 35392 Giessen
Germany hichem.hajaiej@gmail.com
###### Abstract.
We prove the existence of radial and radially decreasing ground states of an
m-coupled nonlinear Schrödinger equation with a general nonlinearity.
## 1\. Introduction
The following Cauchy problem of an m-coupled nonlinear Schrödinger equations:
(1.1)
$\begin{cases}i\partial_{t}\Phi_{1}+\Delta\Phi_{1}+g_{1}\left(|x|,|\Phi_{1}|^{2},\ldots,|\Phi_{m}|^{2}\right)\Phi_{1}&=0,\\\
\qquad\qquad\vdots&\\\
i\partial_{t}\Phi_{m}+\Delta\Phi_{m}+g_{m}\left(|x|,|\Phi_{1}|^{2},\ldots,|\Phi_{m}|^{2}\right)\Phi_{m}&=0,\\\
\hfill\Phi_{i}(0,x)&=\Phi_{i}^{0}(x)\,\quad\mbox{for}\,1\leq i\leq
m.\end{cases}$
For $1\leq i\leq m:\Phi_{i}^{0}:\mathbb{R}^{N}\rightarrow\mathbb{C}$ and
$g_{i}:\mathbb{R}_{+}^{*}\times\mathbb{R}_{+}^{m}\rightarrow\mathbb{R}$,
$\Phi_{i}:\mathbb{R}_{+}\times\mathbb{R}^{N}\rightarrow\mathbb{C}$, has
numerous applications in physical problems. It appears in the study of spatial
solitons in nonlinear waveguides [30], the theory of Bose-Einstein condensates
[12], interactions of m-wave packets [5], optical pulse propagation in
birefringent fibers [25, 26], wavelength division multiplexed optical systems.
Physically, the solution $\Phi_{i}$ is the $i$th component of the beam in
Kerr-like photorefractive media [1]. In the most relevant cases, it is
possible to write (1.1) in a vectorial form as follows:
(1.2) $\begin{cases}i\frac{\partial\Phi}{\partial t}=E^{\prime}(\Phi)&\\\
\Phi(0,x)=\Phi^{0}=(\Phi^{0}_{1},\ldots,\Phi^{0}_{m})&\end{cases}$
where
(1.3)
$E(\Phi)=\frac{1}{2}\|\nabla\Phi\|_{2}^{2}-\int{G\left(|x|,\Phi_{1},\ldots,\Phi_{m}\right)}\,dx.$
$G:(0,\infty)\times\mathbb{R}^{m}\rightarrow\mathbb{R}$ satisfies the
following system:
(1.4) $\begin{cases}\frac{\partial G}{\partial
u_{1}}=g_{1}\left(|x|,u_{1}^{2},\ldots,u_{m}^{2}\right)u_{1},&\\\
\qquad\vdots&\\\ \frac{\partial G}{\partial
u_{m}}=g_{m}\left(|x|,u_{1}^{2},\ldots,u_{m}^{2}\right)u_{m}.&\end{cases}$
When $m=1$, $G$ can be easily given by the explicit expression:
$G(r,s)=\frac{1}{2}\int_{0}^{s^{2}}g(r,t)\,dt$.
In the general case:
(1.5) $\displaystyle G(r,u_{1},\ldots,u_{m})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{u_{1}^{2}}g_{1}(r,t,u_{2}^{2},\ldots,u_{m}^{2})\,dt+K_{1}(u_{2},\ldots,u_{m})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{u_{i}^{2}}g_{i}(r,u_{1}^{2},\ldots,t_{i},\ldots,u_{m}^{2})\,dt_{i}+K_{i}(u_{1},\ldots,u_{i-1},u_{i+1},\ldots,u_{m})$
$\displaystyle=$ $\displaystyle\quad\ldots$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{u_{m}^{2}}g_{m}(r,u_{1}^{2},\ldots,t)\,dt+K_{m}(u_{1},\ldots,u_{m-1}).$
A soliton or standing wave of (1.1) is a solution of the form:
$\Phi(t,x)=\left(\Phi_{1}(t,x),\ldots,\Phi_{m}(t,x)\right)$, where for $1\leq
j\leq m:\Phi_{j}(t,x)=u_{j}(x)e^{-i\lambda_{j}t}$, $\lambda_{j}$ are real
numbers. Therefore $\mathcal{U}=(u_{1},\ldots,u_{m})$ is a solution of the
following $m\times m$ elliptic eigenvalue problem:
(1.6) $\begin{cases}\Delta
u_{1}+\lambda_{1}u_{1}+g_{1}\left(|x|,u_{1}^{2},\ldots,u_{m}^{2}\right)u_{1}=0,&\\\
\qquad\vdots&\\\ \Delta
u_{m}+\lambda_{m}u_{m}+g_{m}\left(|x|,u_{1}^{2},\ldots,u_{m}^{2}\right)u_{m}=0.&\end{cases}$
Among all the standing waves, let us mention the ground states which
correspond to the least energy solutions $\mathcal{U}$ of (1.6), defined by:
(1.7) $E(\mathcal{U})=\frac{1}{2}\sum\limits_{i=1}^{m}|\nabla
u_{i}|_{2}^{2}-\int_{\mathbb{R}^{N}}G\left(|x|,u_{1}(x),\ldots,u_{m}(x)\right)\,dx$
under constraints
(1.8)
$S_{c}=\left\\{\mathcal{U}=(u_{1},\ldots,u_{m})\in\mathrm{H}^{1}(\mathbb{R}^{N})\times\ldots\mathrm{H}^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}u_{i}^{2}=c_{i}\right\\}$
where $c_{i}>0$ are $m$ prescribed numbers.
Ground states are solutions of the minimization problem:
(1.9) $\mbox{For given }\hfill c_{i}>0,M_{c}=\inf_{\mathcal{U}\in
S_{c}}E(\mathcal{U}).\hfill$
Profiles of stable electromagnetic waves traveling along a medium are given by
(1.9). Note that in (1.7), $|x|$ is the position relative to the optical axis,
$G$ is related to the index of refraction of the medium. In the most relevant
cases, $G$ has jumps at interfaces between layers of different media (core and
claddings). Therefore, $G$ is not continuous with respect to the first
variable in many practical cases.
The existence of ground states has been investigated by many authors following
different methods. In [2, 14, 17, 21, 15, 16, 27, 31, 32, 34, 33] by numerical
arguments; in [3, 4, 22, 23, 24, 28], the mathematical analysis using the
variational approach has been pursued to prove the existence of ground states.
These works addressed the special case $m=2$ and
(1.10)
$\begin{cases}g_{1}(|x|,u_{1}^{2},u_{2}^{2})=\left(|u_{1}|^{2p-2}+\beta|u_{1}|^{p-2}|u_{2}|^{p}\right),&\\\
g_{2}(|x|,u_{1}^{2},u_{2}^{2})=\left(|u_{2}|^{2p-2}+\beta|u_{2}|^{p-2}|u_{1}|^{p}\right).&\end{cases}$
This is a very interesting case where we can easily determine $G$, indeed
using (1.5) it is obvious that
$G(r,s_{1},s_{2})=\frac{1}{2p}u_{1}^{2p}+\frac{\beta}{p}u_{1}^{p}u_{2}^{p}+K_{1}(u_{2})=\frac{1}{2p}u_{2}^{2p}+\frac{\beta}{p}u_{1}^{p}u_{2}^{p}+K_{2}(u_{1})$.
A straightforward computation implies:
$G(r,s_{1},s_{2})=\frac{1}{2p}s_{1}^{2p}+\frac{1}{2p}s_{2}^{2p}+\frac{\beta}{p}s_{1}^{p}s_{2}^{p}$.
In [3, 24], not only the existence of ground states has been established, for
(1.1) with $g_{i}$ given by (1.10), but also the orbital stability has been
discussed. Of course, we are interested in the orbital stability of ground
states of (1.1) with general non-linearities. However, an inescapable step
consists in the establishment of suitable assumptions of $g_{i}$ under which
(1.1) admits a unique solution. This is a very challenging open question under
investigation.
Following a self-contained approach, we establish the existence of radial and
radially decreasing ground states [Theorem 3.1]. Our main assumptions are that
$G$ satisfies a growth condition and it is a supermodular function, that is to
say:
(1.11) $G(r,y+he_{i}+ke_{j})+G(r,y)\geq G(r,y+he_{i})+G(r,y+ke_{j})$ (1.12)
$G(r_{1},y+he_{i})+G(r_{0},y)\leq G(r_{1},y)+G(r_{0},y+he_{i})$
for every $i\neq j,h,k>0;y=(y_{1},\ldots,y_{m})$ and $e_{i}$ denotes the $i$th
standard basis vector in $\mathbb{R}^{m},r>0$ and $0<r_{0}<r_{1}$.
These inequalities are connected to the cooperativity of (1.6). When
$\lambda_{i}\equiv 0$, W.C. Troy proved in [35] the necessity of this
hypothesis. Contrary to previous works, we will not use minimization under the
so-called Nehari Manifold; neither results involving the Palais-Smale
condition. Instead, we take advantage of some recent results of symmetrization
inequalities. More precisely, in [13], it has been proved that if $G$
satisfies (1.11) and (1.12), then:
(1.13)
$\int_{\mathbb{R}^{N}}G\left(|x|,u_{1}(x),\ldots,u_{m}(x)\right)\,dx\leq\int_{\mathbb{R}^{N}}G\left(|x|,u^{*}_{1}(x),\ldots,u^{*}_{m}(x)\right)\,dx.$
Here $u^{*}$ denotes the Schwarz symmetrization of a function $u$ vanishing at
infinity. It is well known that the norm of the gradient does not increase
under Schwarz symmetrization in L2. Moreover rearrangements preserve the L2
norm:
$\int|\nabla u^{*}|^{2}\leq\int|\nabla u|^{2}$ $\int u^{2}=\int{(u^{*})}^{2}.$
Finally let us point out that, as mentioned in [11], in many valuable papers
the study of (1.9) with $m=1$ relied on the fact that one could look for
minima in the class of radial functions using rearrangement inequalities. The
compact embedding of such functions in Lp enables us to conclude [6, 7, 8, 9,
10, 36, 37]. H. Brezis and E.H. Lieb [11] concluded this remark saying “It is
not known whether the minimum action lies in the class of radial solutions for
$m>1$ because rearrangement inequalities are not applicable.” In this paper we
build on a method enabling us to use such vectorial inequalities to solve
(1.9).
Thanks to these inequalities, we first prove that: Given
$c_{1},\ldots,c_{m}>0$:
1. (1)
(1.9) always admits a minimizing sequence
$\mathcal{U}_{n}=(u_{n,1},\ldots,u_{n,m})$ such that each component $u_{n,i}$
is radial and radially decreasing.
2. (2)
Noticing that any minimizing sequence of (1.9) is bounded, we will prove that
if $\mathcal{U}_{n}=\mathcal{U}^{*}_{n}\rightharpoonup\mathcal{U}$ then
$\lim\limits_{n\rightarrow+\infty}\int_{\mathbb{R}^{N}}G\left(|x|,u_{n,1},\ldots,u_{n,m}\right)\,dx=\int_{\mathbb{R}^{N}}G\left(|x|,u_{1}(x),\ldots,u_{m}(x)\right)\,dx$
which implies that $\mathcal{U}=(u_{1},\ldots,u_{m})$ is such that
$E(\mathcal{U})\leq M_{c}$.
3. (3)
To conclude, it is sufficient to prove that $\mathcal{U}\in S_{c}$.
This paper contains four more sections. In the next section, we introduce the
notation and definitions. In the third section, we state our main result and
give a detailed proof. The fourth part is dedicated to a variant of our
approach. The last section is dedicated to some challenging open problems.
## 2\. Preliminaries and Notation
* •
In the sequel, $m,N\in\mathbb{N}^{*}$.
* •
For $1\leq p<\infty$, $|\cdot|_{p}$ denotes the norm in
L${}^{p}(\mathbb{R}^{N})$.
* •
If $V=(v_{1},\ldots,v_{m})$ with
$v_{i}\in$L${}^{2}(\mathbb{R}^{N}):\|V\|_{2}^{2}=|v_{1}|_{2}^{2}+\ldots+|v_{m}|_{2}^{2}$.
* •
If $V=(v_{1},\ldots,v_{m})$ with $v_{i}\in$H${}^{1}(\mathbb{R}^{N}):\|\nabla
V\|_{2}^{2}=|\nabla v_{1}|_{2}^{2}+\ldots+|\nabla v_{m}|_{2}^{2}$.
* $[$H${}^{1}(\mathbb{R}^{N})]^{m}=$H${}^{1}(\mathbb{R}^{N})\times\ldots\times$H${}^{1}(\mathbb{R}^{N})$.
* •
All statements about measurability refer to the Lebesgue measure, $\mu$, on
$\mathbb{R}^{N}$ or $(0,\infty)$. When no domain of integration is indicated,
the integral extends over $\mathbb{R}^{N}$.
* •
M$(\mathbb{R}^{N})$ is the set of measurable functions on $\mathbb{R}^{N}$.
* •
F$(\mathbb{R}^{N})$ is the set of symmetrizable functions:
$\left\\{u\in\textrm{M}(\mathbb{R}^{N}):u\geq 0\mbox{ and
}\mu\\{x\in\mathbb{R}^{N}:u(x)>t\\}<\infty\quad\forall t>0\right\\}.$
* •
For $u\in$F$(R^{N})$, $u^{*}$ denotes the Schwarz symmetrization of $u$. For
more details, see [13].
* •
We say that $u$ is Schwarz symmetric if $u\equiv u^{*}$.
* •
For
$V\in\mathrm{F}(\mathbb{R}^{N})\times\ldots\times\mathrm{F}(\mathbb{R}^{N})$,
$V$ is Schwarz symmetric if each of its components has its property.
* •
For the convenience of the reader, let us recall some important symmetrization
inequalities [18]:
(2.1) $\displaystyle\forall u\in\mathrm{H}^{1}(\mathbb{R}^{N}):|\nabla
u|_{2}^{2}$ $\displaystyle=$
$\displaystyle\Big{|}\nabla|u|\Big{|}^{2}_{2}\geq\Big{|}\nabla|u|^{*}\Big{|}^{2}_{2}$
(2.2) $\displaystyle\forall u\in\mathrm{L}^{2}(\mathbb{R}^{N}):|u|_{2}^{2}$
$\displaystyle=$ $\displaystyle|u^{*}|_{2}^{2}.$
###### Definition 2.1.
A function $G:(0,\infty)\times\mathbb{R}^{m}\rightarrow\mathbb{R}$ is an
m-Carathéodory function if
1. (1)
$G(\cdot,s_{1},\ldots,s_{m}):(0,\infty)\rightarrow\mathbb{R}$ is measurable on
$(0,\infty)\setminus\Gamma$, where $\Gamma$ is a subset of $(0,\infty)$ having
one dimensional measure zero, for all $s_{1},\ldots,s_{m}\geq 0$,
2. (2)
For all $1\leq n\leq m$, every $(m-1)$ tuple $s_{i}\geq 0$ and
$r\in(0,\infty)\setminus\Gamma$, the function:
$\displaystyle\mathbb{R}$ $\displaystyle\rightarrow$ $\displaystyle\mathbb{R}$
$\displaystyle s_{n}$ $\displaystyle\mapsto$ $\displaystyle
G(r,\ldots,s_{n},\ldots)$
is continuous on $\mathbb{R}$.
This definition establishes the standard context for handling the
measurability of the composite functions
$G\left(|x|,u_{1}(x),\ldots,u_{m}(x)\right),u_{i}\in M(\mathbb{R}^{N})$. An
important property of an m-Carathéodory function is that $x\mapsto
G\left(|x|,u_{1}(x),\ldots,u_{m}(x)\right)$ is measurable on $\mathbb{R}^{N}$
for every $u_{1},\ldots,u_{m}\in M(\mathbb{R}^{N})$
* •
For the convenience of the reader, let us also recall that for an
m-Carathéodory function satisfying (1.11) and (1.12), we have (1.13); [13].
* •
For $r>0:B(0,r)=\\{x\in\mathbb{R}^{N}:|x|<r\\},|x|$ is the euclidean norm in
$\mathbb{R}^{N}$, there is a constant $V_{N}$ such that
$\mu(B(0,r))=V_{N}r^{N}$ for all $r>0$.
## 3\. Main result
###### Theorem 3.1.
Let $G:(0,\infty)\times\mathbb{R}^{m}\rightarrow\mathbb{R}$ be such that:
* (G0)
$G$ is an m-Carathéodory function such that
$G(r,s_{1},\ldots,s_{m})\leq G(r,|s_{1}|,\ldots,|s_{m}|)$
for every $r>0$ and $s_{1},\ldots,s_{m}\in\mathbb{R}$,
* (G1)
For all $r>0;s_{1},\ldots,s_{m}\geq 0$, we have
$0\leq G(r,s_{1},\ldots,s_{m})\leq
K\left(|s|^{2}+\sum\limits_{i=1}^{m}s_{i}^{\ell_{i}+2}\right):\newline
s=(s_{1},\ldots,s_{m});K>0\textrm{ and }0<\ell_{i}<\frac{4}{N},$
* (G2)
$G$ satisfies (1.11) and (1.12),
* (G3)
$\forall\varepsilon>0,\exists R_{0}>0$ and $S_{0}>0$ such that
$G(r,s_{1},\ldots,s_{m})\leq\varepsilon|s|^{2}$ for all $r>R_{0}$,
$s_{1},\ldots,s_{m}<S_{0};s=(s_{1},\ldots,s_{m})$,
* (G4)
$G(r,t_{1}s_{1},\ldots,t_{m}s_{m})\geq t^{2}_{\max}G(r,s_{1},\ldots,s_{m})$
for any $t_{1},\ldots,t_{m}\geq 1;r>0;s_{1},\ldots,s_{m}\geq 0$ where
$t_{\max}=\max\limits_{1\leq i\leq m}t_{i}$.
Suppose additionally that $M_{c}<0$, then:
$\forall c_{1},\ldots,c_{m}>0$ there exist
$V_{c}=\left(v_{1}^{c_{1}},\ldots,v_{m}^{c_{m}}\right)$ such that $V_{c}\in
S_{c}$ and $E(V_{c})=M_{c}$.
The proof of the result is divided in three parts: (step 1 $\rightarrow$ step
3):
###### Lemma 3.2.
Suppose that $G$ satisfies (G0) and (G1), then all the minimizing sequences of
(1.9) are bounded in $[\mathrm{H}^{1}(\mathbb{R}^{N})]^{m}$.
Proof: Let $\mathcal{U}=(u_{1},\ldots,u_{m})\in S_{c}$, (G0) and (G1) imply
that
$\int G(|x|,\mathcal{U}(x))\,dx\leq
Kc+K\int\sum\limits_{i=1}^{m}|u_{i}(x)|^{\ell_{i}+2}\,dx.$
For $1\leq i\leq m$, the Gagliardo-Nirenberg inequality tells us that:
$|u_{i}|_{\ell_{i}+2}\leq C|u_{i}|_{2}^{1-\sigma_{i}}\cdot|\nabla
u_{i}|_{2}^{\sigma_{i}};\sigma_{i}=\frac{N}{2}\frac{\ell_{i}}{\ell_{i}+2}.$
Now let $\varepsilon>0,p_{i}=\frac{4}{N\ell_{i}},q_{i}$ is such that
$\frac{1}{p_{i}}+\frac{1}{q_{i}}=1$. Applying Young’s inequality, we obtain:
$|u_{i}|_{\ell_{i}+2}\leq\left\\{\frac{C^{\ell_{i}+2}}{\varepsilon}|u_{i}|_{2}^{(1-\sigma_{i})(\ell_{i}+2)}\right\\}^{q_{i}}\frac{1}{q_{i}}+\frac{N\ell_{i}}{4}\left\\{\varepsilon^{\frac{4}{N\ell_{i}}}|\nabla
u_{i}|^{2}_{2}\right\\}.$
Consequently:
$E(\mathcal{U})\geq\left\\{\frac{1}{2}-Km\sum_{i=1}^{m}\frac{N\ell_{i}}{4}\varepsilon^{\frac{4}{N\ell_{i}}}\right\\}\|\nabla\mathcal{U}\|_{2}^{2}-Kc-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}C^{\ell_{i}+2}c^{\frac{(1-\sigma_{i})(\ell_{i}+2)}{2}}.$
Taking $\varepsilon$ such that
$\frac{1}{2}-Km\sum_{i=1}^{m}\frac{N\ell_{i}}{4}\varepsilon^{\frac{4}{N\ell_{i}}}\geq
0$, we prove that $E$ is bounded from below. To show that any minimizing
sequence of (1.9) is bounded in $[\mathrm{H}^{1}(\mathbb{R}^{N})]^{m}$, it is
enough to take the latter inequality with the strict sign.
###### Remark 3.3.
* •
The lemma remains true if we replace (G1) by the more general growth
condition:
$G(r,s_{1},\ldots,s_{m})\leq
K\left(|s|^{2}+\sum\limits_{k=0}^{\alpha}\left(\xi_{1,k}s_{1}+\ldots+\xi_{m,k}s_{m}\right)^{\ell_{k}+2}\right),$
for all $r>0$ and $s_{1},\ldots,s_{m}\geq 0$, where $K$ is a positive
constant, $\alpha\in\mathbb{N}^{*}$ and for $0\leq
k\leq\alpha,0<\ell_{k}<\frac{4}{N}$. For $0\leq k\leq\alpha,1\leq j\leq
m:\xi_{j,k}$ can take arbitrarily the value 0 or 1.
* •
The growth condition stated in our lemma is optimal, in the sense that if
$\ell>\frac{4}{N}$, we can prove that $M_{c}=-\infty$.
Under the hypotheses of Theorem 3.1, we will first prove that:
Step 1:
(3.1) $\mbox{For any
}\mathcal{U}=(u_{1},\ldots,u_{m})\in\left[\mathrm{H}^{1}(\mathbb{R}^{N})\right]^{m}:E(\mathcal{U})\geq
E(\mathcal{U}^{*}).$
This inequality enables us to assert that for any m-tuple
$c_{1},\ldots,c_{m}>0$, (1.9) always admits a Schwarz symmetric minimizing
sequence. For such minimizing sequence, we have the following compactness
property:
Step 2: If $\mathcal{U}_{n}=U_{n}^{*}\rightharpoonup\mathcal{U}$ in
$\left[\mathrm{H}^{1}(\mathbb{R}^{N})\right]^{m}:E[\mathcal{U}]\leq\lim\inf
E(\mathcal{U}_{n})$.
Finally we will show that this $\mathcal{U}$ belongs to the constraint when
$M_{c}<0$.
Step 1:
###### Lemma 3.4.
Suppose that $G$ satisfies (G0), (G1) and (G2). If $(\mathcal{U}_{n})$ is a
minimizing sequence of (1.9), $\left(|\mathcal{U}_{n}|^{*}\right)$ also has
this property.
Proof: Let
$\mathcal{U}=(u_{1},\ldots,u_{m})\in\left[\mathrm{H}^{1}(\mathbb{R}^{N})\right]^{m}$.
First note that for any $u_{i}\in\mathrm{H}^{1}(\mathbb{R}^{N})$ and $|\nabla
u_{i}|_{2}=\Big{|}\nabla|u_{i}|\Big{|}_{2}$, thus using (G0);
$E(|\mathcal{U}|)=E(|u_{1}|,\ldots,|u_{m}|)\leq E(u_{1},\ldots,u_{m})$.
To achieve the proof, it is sufficient to show that for any
$V=(v_{1},\ldots,v_{m})$ with $v_{i}\geq 0$,
$E(v_{1}^{*},\ldots,v_{m}^{*})\leq E(v_{1},\ldots,v_{m})$, which follows
immediately from (2.1) and (1.13). Note finally that by (2.2): if $\int
v_{i}^{2}=c_{i}$ then $\int(v_{i}^{*})^{2}=c_{i}$, this completes the proof.
From now on:
(3.2) $\mathcal{U}_{n}=(u_{n,1},\ldots,u_{n,m})\textrm{ is a minimizing
sequence of~{}(\ref{eq1.9}), which is Schwarz symmetric. }$
By Lemma 3.2, it is bounded in $[\mathrm{H}^{1}(\mathbb{R}^{N})]^{m}$. We know
that (up to a subsequence) there exists $\mathcal{U}=(u_{1},\ldots,u_{m})$
such that
(3.3) $u_{n,j}\rightharpoonup u_{j}\quad\forall 1\leq j\leq m.$
Step 2:
###### Lemma 3.5.
Let $G$ be a function satisfying (G0), (G1) and (G3). ($\mathcal{U}_{n}$) be a
minimizing sequence satisfying (3.2) and (3.3) then
$E(\mathcal{U})\leq\lim\inf E(\mathcal{U}_{n})$.
Proof: $\forall 1\leq i\leq m$, we know that $|\nabla
u_{i}|_{2}^{2}\leq|\nabla u_{n,i}|_{2}^{2}$. Let us prove that
$\lim\limits_{n\rightarrow+\infty}\int
G(|x|,u_{n,1}(x),\ldots,u_{n,m}(x))\,dx=\int
G(|x|,u_{1}(x),\ldots,u_{m}(x))\,dx.$
Let $R>0$, we first show that
$\lim\limits_{n\rightarrow+\infty}\int_{|x|\leq
R}G(|x|,u_{n,1}(x),\ldots,u_{n,m}(x))\,dx=\int_{|x|\leq
R}G(|x|,u_{1}(x),\ldots,u_{m}(x))\,dx.$
For $1\leq i\leq m$, $(u_{n,i})$ converges weakly to $u_{i}$ in
H${}^{1}(\mathbb{R}^{N})$, it then converges to $u_{i}$ in L${}^{\ell_{i}+2}$
$(|x|\leq R)$. Therefore, up to a subsequence (which we also denote by
$u_{n,i}$), $u_{n,i}\rightarrow u_{i}$ for almost every $|x|\leq R$,
$|u_{n,i}|<h_{i}$ where $h_{i}\in\mathrm{L}^{\ell_{i}+2}(|x|\leq R)$.
Now using (G1):
$G(|x|,u_{n,1}(x),\ldots,u_{n,m}(x))\leq
K\left(\sum\limits_{i=1}^{m}h^{2}_{i}(x)+\sum\limits_{i=1}^{m}h_{i}^{\ell_{i}+2}(x)\right).$
All functions involved in this sum are in L${}^{1}(|x|\leq R)$. By the
dominated convergence theorem, it follows that
(3.4) $\lim\limits_{n\rightarrow+\infty}\int_{|x|\leq
R}G(|x|,u_{n,1}(x),\ldots,u_{n,m}(x))\,dx=\int_{|x|\leq
R}G(|x|,u_{1}(x),\ldots,u_{m}(x))\,dx.$
Now fix $n\in\mathbb{N}$ and $1\leq i\leq n$. Since $u_{n,i}$ is Schwarz
symmetric:
$V_{N}|x|^{N}u_{n,i}^{2}(x)\leq\int_{|y|\leq|x|}u^{2}_{n,i}(y)\,dy\leq c_{i}.$
Consequently
$u_{n,i}(x)\leq\frac{c_{i}^{1/2}}{V_{N}^{1/2}|x|^{N/2}}\leq\frac{c_{i}^{1/2}}{V_{N}^{1/2}R^{N/2}}$
for all $|x|>R$.
Let $\varepsilon>0$, choose $R$ large enough, (G3) implies that
$\int_{|x|>R}G(|x|,u_{n,1}(x),\ldots,u_{n,m}(x))\,dx\leq\varepsilon\sum\limits_{i=1}^{m}\int_{|x|>R}u_{n,i}^{2}(x)\,dx\leq\varepsilon
c,$
where $c=\sum\limits_{i=1}^{m}c_{i}$.
Proving that:
(3.5)
$\lim\limits_{R\rightarrow\infty}\lim\limits_{n\rightarrow\infty}\int\limits_{|x|>R}G(|x|,u_{n,1}(x),\ldots,u_{n,m}(x))\,dx=0.$
The two properties we need to prove (3.5) are: $\int u_{n,i}^{2}(x)\leq c_{i}$
and $(u_{n,i})$ is Schwarz symmetric $\forall 1\leq i\leq m$.
Clearly $\int u_{i}^{2}\leq c_{i}$. The second property is inherited by
$u_{i}$ almost everywhere. Indeed for $R>0$, there exists $n_{k}(R)$ such that
$(u_{n_{k},i})$ converges to $u_{i}$ almost everywhere and we obtain:
$\lim\limits_{R\rightarrow\infty}\int\limits_{|x|>R}G(|x|,u_{1}(x),\ldots,u_{m}(x))\,dx=0.$
Consequently
$\lim\limits_{n\rightarrow\infty}\int
G(|x|,u_{n,1}(x),\ldots,u_{n,m}(x))\,dx=\int
G(|x|,u_{1}(x),\ldots,u_{m}(x))\,dx.$
Thanks to our lemmas, we know that $E(\mathcal{U})\leq M_{c}$;
($\mathcal{U}=(u_{1},\ldots,u_{m})$ is given by (3.3)):
(3.6) $|u_{i}|^{2}_{2}\leq c_{i}\quad\forall 1\leq i\leq m.$
Step 3: To conclude that the infinum is achieved, we have to prove that
$\mathcal{U}\in S_{c}$. Suppose that $M_{c}<0$, set
$t_{i}=\frac{c_{i}^{1/2}}{|u_{i}|_{2}}$, by (3.6):
(3.7) $t_{i}\geq 1\mbox{ and }(t_{1}u_{1},\ldots,t_{m}u_{m})\in S_{c}\qquad
t_{\max}=\max\limits_{1\leq i\leq m}t_{i}\geq 1.$
$E\left(t_{1}u_{1},\ldots,t_{m}u_{m}\right)=\frac{1}{2}\sum\limits_{i=1}^{m}|t_{i}\nabla
u_{i}|^{2}_{2}-\int G(|x|,t_{1}u_{1}(x),\ldots,t_{m}u_{m}(x))\,dx.$
By (G4):
$\displaystyle E(t_{1}u_{1},\ldots t_{m}u_{m})$ $\displaystyle\leq$
$\displaystyle t_{\max}^{2}E(u_{1},\ldots,u_{m}).$ $\displaystyle M_{c}\leq
E(t_{1}u_{1},\ldots t_{m}u_{m})$ $\displaystyle\leq$ $\displaystyle
t_{\max}^{2}E(u_{1},\ldots,u_{m})\leq t_{\max}^{2}M_{c},$
since $t_{i}\geq 1$ by Lemma 3.5, it follows that $M_{c}\leq
t^{2}_{\max}M_{c}\Rightarrow t^{2}_{\max}\leq 1$, hence $t_{i}=1$ for any
$1\leq i\leq m$. This ends the proof of Theorem 3.1.
On the hypothesis $\mathbf{M_{c}<0}$:
Inspired by [29] and closely following the approach therein, we prove that if
$G$ satisfies:
* (G5)
There exist $R_{1}>0,S_{1}>0$. For any $1\leq i\leq m$, there exist
$A_{i}>0,t_{i}\in[0,2)$ and $0\leq\sigma_{i}\leq\frac{2(2-t_{i})}{N}$ such
that
$G(r,s_{1},\ldots,s_{m})\geq\sum\limits_{i=1}^{m}A_{i}r^{-t_{i}}s_{i}^{\sigma_{i}+2}\mbox{
for all }r>R_{1},0<s<s_{1}$
then $M_{c}<0$.
Set $d(N)=\int
e^{-2|y|^{2}}\,dy,D(N)=\frac{4}{d^{2}(N)}\int|y|^{2}e^{-2|y|^{2}}\,dy$. For
$\alpha\in(0,1]$, we set $w_{\alpha}:\mathbb{R}^{N}\rightarrow\mathbb{R}$
defined by $w_{\alpha}(x)=\frac{\alpha^{N/4}e^{-\alpha|x|^{2}}}{d(N)}$. A
straightforward computation shows that $|w_{\alpha}|_{2}=1$ and $|\nabla
w_{\alpha}|_{2}^{2}=\alpha D(N)$.
On the other hand, there exists $B>R_{1}$ such that for any $|x|>B$,
$w_{\alpha}(x)\leq S_{1}$.
$\int G(|x|,w_{\alpha}(x),\ldots,w_{\alpha}(x))\geq\int_{|x|\geq
B}\sum\limits_{i=1}^{m}\frac{A_{i}}{[d(N)]^{\sigma_{i}+2}}|x|^{-t_{i}}e^{-\alpha(\sigma_{i}+2)|x|^{2}}\alpha^{\frac{N}{4}(\sigma_{i}+2)}\,dx.$
By the change of variable $y=\alpha^{\frac{1}{2}}x$, we obtain:
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}\frac{A_{i}}{[d(N)]^{\sigma_{i}+2}}\alpha^{\frac{N\sigma_{i}}{4}+\frac{t_{i}}{2}}\int_{|y|\geq
B\alpha^{\frac{1}{2}}}|y|^{-t_{i}}e^{-(\sigma_{i}+2)|y|^{2}}\,dy$
$\displaystyle\geq$
$\displaystyle\sum\limits_{i=1}^{m}\frac{A_{i}}{[d(N)]^{\sigma_{i}+2}}\alpha^{\frac{N\sigma_{i}}{4}+\frac{t_{i}}{2}}\int_{|y|\geq
B}|y|^{-t_{i}}e^{-(\sigma_{i}+2)|y|^{2}}\,dy$
Set $I_{i}=\int_{|y|\geq B}|y|^{-t_{i}}e^{-(\sigma_{i}+2)|y|^{2}}\,dy$, it
follows that:
$E(w_{\alpha},\ldots,w_{\alpha})\leq\alpha\left\\{mD(N)-\sum\limits_{i=1}^{m}\frac{A_{i}}{[d(N)]^{\sigma_{i}+2}}I_{i}\alpha^{\frac{N\sigma_{i}}{4}+\frac{t_{i}}{2}-1}\right\\}.$
The fact that $\sigma_{i}<2(2-t_{i})/N$ enables us to conclude that
$E(w_{\alpha},\ldots,w_{\alpha})<0$ for $\alpha$ sufficiently small. Taking
$u_{i}=\frac{c_{i}^{1/2}w_{\alpha}}{|w_{\alpha}|_{2}}$, we can easily see that
$E(u_{1},\ldots,u_{m})<0$ with $(u_{1},\ldots,u_{m})\in S_{c}$, thus
$M_{c}<0$.
## 4\. Variant of our result
Our approach also applies to the following variational problem:
$\tilde{M_{c}}=\inf_{\mathcal{U}\in\tilde{S_{c}}}\tilde{E}(\mathcal{U}),\mbox{
for
}\mathcal{U}=(u_{1},\ldots,u_{m})\in\left[\mathrm{H}^{1}(\mathbb{R}^{N})\right]^{m},$
$\tilde{E}(\mathcal{U})=\frac{1}{2}\sum\limits_{i=1}^{m}|\nabla
u_{i}|_{2}^{2}-\frac{1}{2}\int p(|x|)\sum\limits_{i=1}^{m}u_{i}^{2}(x)-\int
G(|x|,u_{1}(x),\ldots,u_{m}(x)).$
For a prescribed $c>0$:
$\tilde{S_{c}}=\left\\{\mathcal{U}=(u_{1},\ldots,u_{m}):\|\mathcal{U}\|^{2}_{2}=c\right\\}$.
Then we have the following result:
###### Theorem 4.1.
Suppose that $p:(0,\infty)\rightarrow\mathbb{R}$ satisfies
* (P1)
$p$ is non-negative, non-increasing and
$\lim\limits_{r\rightarrow\infty}p(r)=0$;
* (P2)
* –
If $N=1,2$, there exists $a\in(0,1]$ such that $p(a)>0$;
* –
If $N\geq 3$, there exists $R>0$ such that
$p(r)>\frac{j^{2}_{N/2-1,1}}{R^{2}}$ where $j^{2}_{N/2-1,1}$ is the first zero
of the Bessel function $J_{N/2-1}$.
Suppose that $G$ satisfies $(G0)\rightarrow(G4)$ in which each $t_{i}$ is
replaced by $t$, then, for any $c>0$, there exists
$\mathcal{U}_{c}=(u_{c}^{1},\ldots,u_{c}^{m})$ Schwarz symmetric such that
$\tilde{E}(\mathcal{U}_{c})=\tilde{M_{c}}$.
Proof: Following the same approach as in the previous Theorem, step 1, step 2
and step 3 can be proven under minor modifications. Therefore we are done if
$\tilde{M_{c}}<0$. Since $G$ is non-negative, it is sufficient to prove that
we can construct $v\in\mathrm{H}^{1}(\mathbb{R}^{N})$ such that
(4.1) $\frac{1}{2}|\nabla v|_{2}^{2}-\frac{1}{2}\int p(|x|)v^{2}<0.$
For the convenience of the reader, we will mention all the details. These test
functions were constructed in [19] and used in [20].
* •
Case $N=1$:
Take $w(x)=e^{-|x|}$, $\alpha\in(0,1],0<d\leq a$ and $w_{\alpha}(x)=w(\alpha
x)$
(4.2) $\frac{1}{2}\int|\nabla
w_{\alpha}|^{2}-p(|x|)w^{2}_{\alpha}(x)\,dx=\frac{1}{2}\int\alpha^{2}|\nabla
w(\alpha x)|^{2}-p(|x|)w^{2}(\alpha x)\,dx.$
By the change of variables $y=\alpha x$, we obtain:
$\displaystyle(\ref{eq4.2})$ $\displaystyle\leq$
$\displaystyle\frac{1}{2\alpha}\left\\{\alpha^{2}|\nabla w|_{2}^{2}-\int
p\left(\frac{|y|}{\alpha}\right)w^{2}(y)\,dy\right\\}\leq\frac{1}{2\alpha}\left\\{\alpha^{2}|\nabla
w|_{2}^{2}-w^{2}(d)\int_{|y|\leq
d}p\left(\frac{|y|}{\alpha}\right)\,dy\right\\}$ $\displaystyle(\ref{eq4.2})$
$\displaystyle\leq$ $\displaystyle\frac{\alpha}{2}\left\\{|\nabla
w|_{2}^{2}-\frac{w^{2}(d)p(d)2d}{\alpha}\right\\}.$
In the last inequality, we have used the change of variables
$z=\frac{y}{\alpha}$, then used the monotonicity of $p$.
Therefore for $\alpha$ small enough, (4.2)$<0$. Now for $c>0$ and $\alpha$
small enough take: $v_{i}=\frac{c^{1/2}w_{\alpha}}{m^{1/2}|w_{\alpha}|_{2}}$,
then $\frac{1}{2}\int|\nabla v_{i}|_{2}^{2}-\frac{1}{2}\int
p(|x|)v_{i}^{2}<0$, $v=(v_{1},\ldots,v_{m})\in\tilde{S_{c}}$ and
$\tilde{E}(v_{1},\ldots,v_{m})<0$.
* •
Case $N=2$:
Let $u(x)=\begin{cases}\left(\log{\frac{1}{|x|}}\right)^{1/3}&\mbox{if
}|x|<1,\\\ 0&\mbox{otherwise.}\end{cases}$
$u\in\mathrm{H}^{1}(\mathbb{R}^{2})$ but it is an unbounded function because
of its singularity in $0$. Let $K=\left(\int_{|x|\leq
1}p(|x|)\,dx\right)^{-1}$, there exists $d\in\mathbb{R}^{2}$ such that
(4.3) $u^{2}(d)>K|\nabla u|_{2}^{2}.$
Set $w_{d}(x)=u(|d|x),w_{d}\in\mathrm{H}^{1}(\mathbb{R}^{2})$ and:
$\displaystyle\frac{1}{2}|\nabla w_{d}|^{2}_{2}-\frac{1}{2}\int
p(|x|)w_{d}^{2}(x)\,dx\leq\frac{1}{2}\int|d|^{2}\Big{|}\nabla
u(|d|x)\Big{|}^{2}-p(|x|)u^{2}(|d|x)\,dx$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int|\nabla
u(y)|^{2}-\frac{1}{|d|^{2}}p\left(\frac{|y|}{|d|}\right)u^{2}(y)\,dy\leq\frac{1}{2}\int|\nabla
u(y)|^{2}-\frac{1}{2|d|^{2}}\int_{|y|\leq
d}p\left(\frac{|y|}{|d|}\right)u^{2}(y)\,dy$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\left\\{|\nabla u|^{2}_{2}-u^{2}(d)\int_{|z|\leq
1}p(|z|)\,dz\right\\}<0\mbox{ by (\ref{eq4.3})}.$
The proof goes as previously setting
$v_{i}=\frac{c^{1/2}w_{d}}{m^{1/2}|w_{d}|_{2}}$ for $1\leq i\leq n$.
* •
Case $N\geq 3$: Let $x\in\mathcal{B}(0,1)$, set
$\varphi_{1}(x)=|x|^{-\left(\frac{N}{2}-1\right)}J_{N/2-1}\left(j_{N/2-1,1}|x|\right)$.
It is easy to check that $\varphi_{0}\in\mathrm{H}_{0}^{1}(|x|<1)$ and
$-\Delta\varphi_{1}=j^{2}_{N/2-1,1}\varphi_{1}$. For $R$ given by (P2), set
$\varphi_{R}(x)=\varphi_{1}\left(\frac{x}{R}\right)$ then
$\varphi_{R}\in\mathrm{H}_{0}^{1}(|x|<R)$ and
$-\nabla\varphi_{R}=\frac{j^{2}_{N/2-1,1}}{R^{2}}\varphi_{R}$.
Now set $w_{R}=\begin{cases}\varphi_{R}&\mbox{if }|x|<R\\\
0&\mbox{otherwise.}\end{cases}$
$w_{R}\in\mathrm{H}^{1}(\mathbb{R}^{N})$ and $\frac{1}{2}\int|\nabla
w_{R}|-\frac{1}{2}\int p(|x|)w^{2}_{R}(x)\,dx\leq\frac{1}{2}\int_{|x|\leq
R}\left\\{\frac{j^{2}_{N/2-1,1}}{R^{2}}-p(|x|)\right\\}w_{R}^{2}(x)\,dx<0$ by
(P2).
We conclude in the same way as in the previous cases.
###### Remark 4.2.
Theorem 4.1 holds true when (P2) is replaced by (G5).
Examples of functions $\mathbf{G}$ satisfying $\mathbf{(G0)\rightarrow(G5)}$:
Let $m=2,k\in\mathbb{N}^{*}$:
(R)
$G(r,s)=b(r)|s|^{2}+a(r)\sum\limits_{j=1}^{k}|s_{1}|^{\ell_{1,j}+1}|s_{2}|^{\ell_{2,j}+1}$
* (R1)
$\ell_{1,j}$ and $\ell_{2,j}>1$ with $\ell_{1,j}+\ell_{2,j}<\frac{4}{N}$ for
$1\leq j\leq k$.
* (R2)
$a(r)$ is a non-negative, non-increasing function bounded from above and below
by two positive constants.
* (R3)
$b(r)$ is a non-negative, non-increasing bounded function tending to zero as
$r$ goes to infinity.
Then $G$ satisfies $(G0)\rightarrow(G5)$.
Remarks:
* •
For $m>2$, functions $G$ satisfying $(G0)$ to $(G5)$ are given in a similar
way as (R) with a sum involving products of all $s_{i},1\leq i\leq m$. This
ensures $(G4)$.
* •
Note that in (R), $|s|^{2}$ can be replaced by $|s|^{\sigma+2}$ with
$0<\sigma<\frac{4}{N}$. In this case $b(r)$ can be taken as a positive
constant: (R’)
* •
Finally when one deals with functions $G$ that are not necessarily sums of
products involving all $s_{i}$ with $1\leq i\leq m$, we should apply Theorem
4.1, from which we can easily see that (1.10) is a particular case of this
result. More precisely, take $a\equiv\frac{\beta}{p}$, $b=\frac{1}{2p}$,
$\ell_{1}=\ell_{2}=\frac{\sigma}{2}=p-1$ with $1<p<\frac{2}{N}$ in (R’).
## 5\. Concluding remarks
In this paper, we have determined suitable assumptions of the operator $G$,
involved in the m-coupled nonlinear Schrödinger equations such that (1.1)
admits a radial and radially decreasing ground state with respect to each
component. Moreover, if (1.11) and (1.12) hold true with strict inequality
[21, Theorem 2], it follows that $E(\mathcal{U}^{*})<E(\mathcal{U})$ for any
$\mathcal{U}\in[H^{1}(\mathbb{R}^{N})]^{m}$. Consequently all the ground
states of (1.1) are Schwarz symmetric. A challenging question is the
establishment of the uniqueness of these least energy solutions. Until now, we
are not aware of any result in this direction when $N>1$ and $m>1$. Another
very interesting question is the study of the orbital stability of these
standing waves. We expect that for $\ell_{i}<4/N$, the ground states are
stable. A crucial step to establish such a result is to prove the uniqueness
of the solutions of (1.1). For more general nonlinearities $g_{i}$, this open
problem, under investigation, seems to be extremely complicated.
## Acknowledgment
The author is extremely grateful to Dr. Yvan Pointurier for his precious help.
The author is also grateful to the referees, Stefan Le Coz and Louis Jeanjean
for their valuable comments.
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|
arxiv-papers
| 2009-03-16T20:51:59 |
2024-09-04T02:49:01.188801
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hichem Hajaiej",
"submitter": "Hichem Hajaiej",
"url": "https://arxiv.org/abs/0903.2854"
}
|
0903.2900
|
# Time evolution of Wigner function in laser process derived by entangled
state representation
Li-yun Hu1,2∗and Hong-yi Fan2 1College of Physics & Communication Electronics,
Jiangxi Normal University, Nanchang 330022, China 2Department of Physics,
Shanghai Jiao Tong University, Shanghai, 200030, P.R. China *Corresponding
author. E-mail addresses: hlyun2008@126.com
###### Abstract
Evaluating the Wigner function of quantum states in the entangled state
representation is introduced, based on which we present a new approach for
deriving time evolution formula of Wigner function in laser process.
Application of this fomula to calculating time evolution of photon number is
also presented, as an example, the case when the initial state is photon-added
coherent state is discussed.
## I Introduction
One of the major topics in Quantum Statistical Mechanics is the evolution of
pure states into mixed states 1 ; 2 . Such evolution usually happens when a
system is immersed in a thermal environment, or a signal (a quantum state)
passes through a quantum channel, and is described by a master equation.
Alternately, description of evolution of density matrices $\rho$ can be
replaced by its Wigner function’s evolution in phase space 3 ; 4 . The partial
negativity of Wigner function can be considered as an indicator of
nonclassicality of quantum state. On the basis of the entangled state
representation and the thermo field dynamics we present a new approach for
deriving time evolution formula of Wigner function in amplitude-damping
channel and laser process. Application of this fomula to calculating time
evolution of photon number is also presented, as an example, the case when the
initial state is photon-added coherent state is discussed.
## II Wigner function formula in thermo entangled state representation
We begin with briefly reviewing the thermo entangled state representation
(TESR). On the basis of Umezawa-Takahash thermo field dynamcs (TFD) 5 ; 6 ; 7
we have constructed the TESR in doubled Fock space 8 ; 9 ,
$\left|\eta\right\rangle=\exp\left[-\frac{1}{2}|\eta|^{2}+\eta
a^{\dagger}-\eta^{\ast}\tilde{a}^{\dagger}+a^{\dagger}\tilde{a}^{\dagger}\right]\left|0,\tilde{0}\right\rangle,$
(1)
or
$\left|\eta\right\rangle=D\left(\eta\right)\left|\eta=0\right\rangle,\text{ \
}D\left(\eta\right)=e^{\eta a^{\dagger}-\eta^{\ast}a},$ (2)
where $D\left(\eta\right)$ is the displacement operator, $\tilde{a}^{\dagger}$
is a fictitious mode accompanying the real photon creation operator
$a^{\dagger},$
$\left|0,\tilde{0}\right\rangle=\left|0\right\rangle\left|\tilde{0}\right\rangle,$
and $\left|\tilde{0}\right\rangle$ is annihilated by $\tilde{a},$
$\left[\tilde{a},\tilde{a}^{\dagger}\right]=1$. Operating $a$ and $\tilde{a}$
on $\left|\eta\right\rangle$ in Eq.(1) we obtain the eigen-equations of
$\left|\eta\right\rangle$,
$\displaystyle(a-\tilde{a}^{\dagger})\left|\eta\right\rangle$
$\displaystyle=\eta\left|\eta\right\rangle,\;(a^{\dagger}-\tilde{a})\left|\eta\right\rangle=\eta^{\ast}\left|\eta\right\rangle,$
$\displaystyle\left\langle\eta\right|(a^{\dagger}-\tilde{a})$
$\displaystyle=\eta^{\ast}\left\langle\eta\right|,\
\left\langle\eta\right|(a-\tilde{a}^{\dagger})=\eta\left\langle\eta\right|.$
(3)
Note that $\left[(a-\tilde{a}^{\dagger}),(a^{\dagger}-\tilde{a})\right]=0,$
thus $\left|\eta\right\rangle$ is the common eigenvector of
$(a-\tilde{a}^{\dagger})$ and $(\tilde{a}-a^{\dagger}).$ Using the normally
ordered form of vacuum projector $\left|0,\tilde{0}\right\rangle\left\langle
0,\tilde{0}\right|=\colon\exp\left(-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}\right)\colon,$
and the technique of integration within an ordered product (IWOP) of operators
10 ; 11 ; 12 , we can easily prove that $\left|\eta\right\rangle$ is complete
and orthonormal,
$\int\frac{d^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=1,\text{
}\left\langle\eta^{\prime}\right.\left|\eta\right\rangle=\pi\delta\left(\eta^{\prime}-\eta\right)\delta\left(\eta^{\prime\ast}-\eta^{\ast}\right).$
(4)
It is easily seen that $\left|\eta=0\right\rangle$ has the properties
$\text{ \
}\left|\eta=0\right\rangle=e^{a^{\dagger}\tilde{a}^{\dagger}}\left|0,\tilde{0}\right\rangle=\sum_{n=0}^{\infty}\left|n,\tilde{n}\right\rangle,$
(5)
and
$\displaystyle a\text{\ }\left|\eta=0\right\rangle$
$\displaystyle=\tilde{a}^{\dagger}\left|\eta=0\right\rangle,$ $\displaystyle
a^{\dagger}\left|\eta=0\right\rangle$
$\displaystyle=\tilde{a}\left|\eta=0\right\rangle,$ (6)
$\displaystyle\left(a^{\dagger}a\right)^{n}\left|\eta=0\right\rangle$
$\displaystyle=\left(\tilde{a}^{\dagger}\tilde{a}\right)^{n}\left|\eta=0\right\rangle.$
Note that density operators $\rho$($a^{\dagger}$,$a)$ are defined in the real
space which are commutative with operators ($\tilde{a}^{\dagger}$,$\tilde{a})$
in the tilde space.
Next, we shall derive a new expression of Wigner function in the TESR.
According to the definition of Wigner function 13 ; 14 of density operator
$\rho,$
$W\left(\alpha\right)=\text{Tr}\left[\Delta\left(\alpha\right)\rho\right],$
(7)
where $\Delta\left(\alpha\right)$ is the single-mode Wigner operator 13 ,
whose explicit form is
$\Delta\left(\alpha\right)=\frac{1}{\pi}\colon
e^{-2\left(a^{\dagger}-\alpha^{\ast}\right)\left(a-\alpha\right)}\colon=\frac{1}{\pi}D\left(2\alpha\right)(-1)^{a^{\dagger}a}.$
(8)
By using
$\left\langle\tilde{n}\right|\left.\tilde{m}\right\rangle=\delta_{n,m}$ and
introducing $\left|\rho\right\rangle\equiv\rho\left|I\right\rangle$ we can
reform Eq.(7) as
$\displaystyle W\left(\alpha\right)$
$\displaystyle=\sum_{m,n}^{\infty}\left\langle
n,\tilde{n}\right|\Delta\left(\alpha\right)\rho\left|m,\tilde{m}\right\rangle$
$\displaystyle=\frac{1}{\pi}\left\langle\eta=0\right|D\left(2\alpha\right)(-1)^{a^{\dagger}a}\left|\rho\right\rangle$
$\displaystyle=\frac{1}{\pi}\left\langle\eta=-2\alpha\right|(-1)^{a^{\dagger}a}\left|\rho\right\rangle$
$\displaystyle=\frac{1}{\pi}\left\langle\xi=2\alpha\right|\left.\rho\right\rangle,$
(9)
where $\left|\xi\right\rangle$ is defined as
$\displaystyle\left|\xi\right\rangle_{\xi=\eta}$
$\displaystyle=(-1)^{a^{\dagger}a}\left|\eta=-\xi\right\rangle$
$\displaystyle=\exp\left(-\frac{1}{2}|\xi|^{2}+\xi
a^{\dagger}+\xi^{\ast}\tilde{a}^{\dagger}-a^{\dagger}\tilde{a}^{\dagger}\right)\left|0,\tilde{0}\right\rangle$
$\displaystyle=D\left(\xi\right)e^{-a^{\dagger}\tilde{a}^{\dagger}}\left|0,\tilde{0}\right\rangle.$
(10)
It can be proved that
$\left\langle\eta\right|\left.\xi\right\rangle=\frac{1}{2}\exp\left(\frac{\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right),$
(11)
a Fourier transformation kernel, so $\left|\xi\right\rangle$ can be considered
the conjugate state of $\left|\eta\right\rangle,$ which also possess
orthonormal and complete properties
$\int\frac{d^{2}\xi}{\pi}\left|\xi\right\rangle\left\langle\xi\right|=1,\text{
}\left\langle\xi^{\prime}\right.\left|\xi\right\rangle=\pi\delta\left(\xi^{\prime}-\xi\right)\delta\left(\xi^{\prime\ast}-\xi^{\ast}\right).$
(12)
Eq.(9) is just a new formula for evaluating the Wigner function of quantum
states: by calculating the overlap between two “pure states” in enlarged Fock
space rather than using the ensemble average in real mode space.
For example, for number state $\left|n\right\rangle\left\langle n\right|,$
noticing $\left|n\right\rangle\left\langle
n\right|\left.I\right\rangle=\left|n,\tilde{n}\right\rangle$, and the
generating function of two-variable Hermite polynomial 15 ; 16
$H_{m,n}\left(x,y\right)$,
$\sum_{m,n}^{\infty}\frac{t^{m}t^{\prime
n}}{m!n!}H_{m,n}\left(x,y\right)=\exp\left[-tt^{\prime}+tx+t^{\prime}y\right],$
(13)
we see
$\displaystyle W_{\left|n\right\rangle\left\langle
n\right|}\left(\alpha\right)$ $\displaystyle=\frac{1}{\pi}\left\langle
n,\tilde{n}\right|\left.\xi_{=2\alpha}\right\rangle=\frac{1}{n!\pi}e^{-\frac{1}{2}|\xi|^{2}}H_{n,n}\left(\xi,\xi^{\ast}\right)$
$\displaystyle=\frac{\left(-1\right)^{n}}{\pi}e^{-2|\alpha|^{2}}L_{n}\left(4|\alpha|^{2}\right),$
(14)
in the last step in Eq.(14) we have used the relation between
$H_{m,n}\left(x,y\right)$ and Laguerre polynomial $L_{m}\left(x\right)$ 17 ,
$L_{n}\left(xy\right)=\frac{\left(-1\right)^{n}}{n!}H_{n,n}\left(x,y\right).$
(15)
Similarly, for coherent state $\left|z\right\rangle\left\langle z\right|$
($\left|z\right\rangle=\exp(-\left|z\right|^{2}/2+za^{\dagger})\left|0\right\rangle$)
18 ; 19 , due to $\left|z\right\rangle\left\langle
z\right|\left.I\right\rangle=D\left(z\right)\tilde{D}\left(z^{\ast}\right)\left|0\tilde{0}\right\rangle=\left|z,\tilde{z}^{\ast}\right\rangle,$
we have
$\displaystyle W_{\left|z\right\rangle\left\langle
z\right|}\left(\alpha\right)$ $\displaystyle=\frac{1}{\pi}\left\langle
0,\tilde{0}\right|\exp\left(-2|\alpha|^{2}+2\alpha^{\ast}a^{\dagger}+2\alpha\tilde{a}-a\tilde{a}\right)\left|z,\tilde{z}^{\ast}\right\rangle$
$\displaystyle=\frac{1}{\pi}\exp\left[-2\left|\alpha-z\right|^{2}\right].$
(16)
Further, using Eq.(11) and the completeness of $\left\langle\eta\right|$ in
Eq.(4), we can reform Eq.(9) as
$W\left(\alpha\right)=\int\frac{d^{2}\eta}{\pi^{2}}\left\langle\xi=2\alpha\right|\left.\eta\right\rangle\left\langle\eta\right|\left.\rho\right\rangle=\int\frac{d^{2}\eta}{2\pi^{2}}e^{\alpha^{\ast}\eta-\alpha\eta^{\ast}}\left\langle\eta\right|\left.\rho\right\rangle.$
(17)
Once $\left\langle\eta\right|\left.\rho\right\rangle$ is known, one can
calculate the Wigner function by taking the Fourier transform of
$\left\langle\eta\right|\left.\rho\right\rangle$. Eqs. (9) and (13) are two
ways accessing to Wigner function, we can use either one to derive Wigner
functions.
## III Evolution formula of Wigner function for amplitude damping channel
In this section, we consider Wigner function’s time evolution in the amplitude
decay channel (dissipation in a lossy cavity) described by the following
master equation 20
$\frac{d\rho}{dt}=\kappa\left(2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho
a^{\dagger}a\right),$ (18)
where $\kappa$ is the rate of decay. In Ref. 21 we have reformed (18) as
$\frac{d}{dt}\left|\rho\right\rangle=\kappa\left(2a\tilde{a}-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}\right)\left|\rho\right\rangle,$
(19)
thus the formal solution of Eq.(19) is
$\left|\rho\left(t\right)\right\rangle=e^{\kappa
t\left(a\tilde{a}-\tilde{a}^{\dagger}a^{\dagger}+1\right)}e^{\left(1-e^{2\kappa
t}\right)\left(a^{\dagger}-\tilde{a}\right)\left(a-\tilde{a}^{\dagger}\right)/2}\left|\rho_{0}\right\rangle.$
(20)
Then projecting Eq.(20) on $\left\langle\eta\right|$, and noticing
$\exp\left[\kappa
t\left(a\tilde{a}-\tilde{a}^{\dagger}a^{\dagger}\right)\right]$ being the two-
mode squeezing operator,
$\left\langle\eta\right|\exp\left[\kappa
t\left(a\tilde{a}-\tilde{a}^{\dagger}a^{\dagger}\right)\right]=e^{-\kappa
t}\left\langle\eta e^{-\kappa t}\right|,$ (21)
as well as Eq.(3), we obtain
$\left\langle\eta\right.\left|\rho\left(t\right)\right\rangle=e^{-\frac{1}{2}T\left|\eta\right|^{2}}\left\langle\eta
e^{-\kappa t}\right|\left.\rho_{0}\right\rangle,$ (22)
where $T=1-e^{-2\kappa t}.$ Substituting Eq.(22) into Eq.(17), we derive the
Wigner function at time $t$
$W\left(\alpha,t\right)=\int\frac{d^{2}\eta}{2\pi^{2}}e^{\alpha^{\ast}\eta-\alpha\eta^{\ast}-\frac{1}{2}T\left|\eta\right|^{2}}\left\langle\eta
e^{-\kappa t}\right|\left.\rho_{0}\right\rangle.$ (23)
Inserting the completeness relation (12) into Eq.(23) and noticing Eqs.(9) as
well as (11), we can reform Eq.(23) as
$\displaystyle W\left(\alpha,t\right)$
$\displaystyle=\int\frac{d^{2}\xi^{\prime}}{\pi}\int\frac{d^{2}\eta}{2\pi^{2}}e^{-\frac{1}{2}T\left|\eta\right|^{2}}\left\langle\eta
e^{-\kappa
t}\right.\left|\xi^{\prime}\right\rangle\left\langle\xi^{\prime}\right|\left.\rho_{0}\right\rangle$
$\displaystyle=\int\frac{d^{2}\beta
d^{2}\eta}{\pi^{2}}e^{-\frac{T}{2}\left|\eta\right|^{2}+\eta\left(\alpha^{\ast}-\beta^{\ast}e^{-\kappa
t}\right)+\eta^{\ast}\left(\beta e^{-\kappa
t}-\alpha\right)}W\left(\beta,0\right)$
$\displaystyle=\frac{2}{T}\int\frac{d^{2}\beta}{\pi}\exp\left[-\frac{2}{T}\left|\alpha-\beta
e^{-\kappa t}\right|^{2}\right]W\left(\beta,0\right)$ (24)
where $W\left(\beta,0\right)$ is the Wigner function at initial time, and we
have used the following integral formula 17
$\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}\right)=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0.$
(25)
Eq.(24) is the expression of time evolution of Wigner function for amplitude
damping channel.
For example, for the photon-added coherent state
$C_{m}a^{{\dagger}m}\left|z\right\rangle$, where
$C_{m}=[m!L_{m}(-\left|z\right|^{2})]^{-1}$ is the normalization factor, the
initial Wigner function $W\left(\beta,0\right)$ is given by 22
$W\left(\beta,0\right)=\frac{\left(-1\right)^{m}e^{-2\left|\beta-z\right|^{2}}}{\pi
L_{m}(-\left|z\right|^{2})}L_{m}(\left|2\beta-z\right|^{2}).$ (26)
Substituting Eq.(26) into Eq.(24) and using Eq.(15) as well as the another
generating function of $H_{m,n}\left(x,y\right),$
$H_{m,n}\left(x,y\right)=\left.\frac{\partial^{m+n}}{\partial\tau^{m}\partial\tau^{\prime
n}}\exp\left[-\tau\tau^{\prime}+\tau
x+\tau^{\prime}y\right]\right|_{\tau=\tau^{\prime}=0},$ (27)
we have
$\displaystyle W\left(\alpha,t\right)$
$\displaystyle=\frac{2}{T}\frac{e^{-2\left(\left|z\right|^{2}+\frac{1}{T}\left|\alpha\right|^{2}\right)}}{\pi
m!L_{m}(-\left|z\right|^{2})}\frac{\partial^{2m}}{\partial\tau^{m}\partial\tau^{\prime
m}}e^{-\tau\tau^{\prime}-\tau z-z^{\ast}\tau^{\prime}}$
$\displaystyle\int\frac{d^{2}\beta}{\pi}\exp\left[-\frac{\allowbreak
2}{T}\left|\beta\right|^{2}+2\beta\left(z^{\ast}+\allowbreak\frac{\alpha^{\ast}}{T}e^{-t\kappa}+\tau\right)\right.$
$\displaystyle\left.+2\beta^{\ast}\left(z+\frac{\alpha}{T}e^{-t\kappa}+\tau^{\prime}\right)\right]_{\tau=\tau^{\prime}=0}$
$\displaystyle=\frac{e^{-2\allowbreak\left|\alpha-ze^{-\kappa
t}\right|^{2}}}{\pi
m!L_{m}(-\left|z\right|^{2})}\frac{\partial^{2m}}{\partial\tau^{m}\partial\tau^{\prime
m}}\exp\left[\left(1-2e^{-2t\kappa}\right)\tau\tau^{\prime}\right.$
$\displaystyle+\left[\left(1-2e^{-2t\kappa}\right)z+2\alpha
e^{-t\kappa}\right]\tau$
$\displaystyle+\left.\left[\allowbreak\left(1-2e^{-2t\kappa}\right)z^{\ast}+2\alpha^{\ast}e^{-t\kappa}\right]\tau^{\prime}\right]_{\tau=\tau^{\prime}=0}.$
(28)
With use of a scaled transformation in the right-hand part of Eq.(28) we
finally get
$\displaystyle W\left(\alpha,t\right)$
$\displaystyle=\frac{\left(1-2e^{-2\kappa t}\right)^{m}}{\pi
L_{m}(-\left|z\right|^{2})}e^{-2\allowbreak\left|\alpha-ze^{-\kappa
t}\right|^{2}}$ $\displaystyle\times L_{m}\left[-\frac{\left|2\alpha
e^{-\kappa t}+z\left(1-2e^{-2\kappa t}\right)\right|^{2}}{1-2e^{-2\kappa
t}}\right],$ (29)
which is the analytical expression of the time evolution of Wigner function
for any number ($m$) photon-added coherent state in photon loss channel 23 .
In particular, when $t=0,$ Eq.(29) just reduce to Eq.(26).
## IV Evolution formula of Wigner function for Laser process
We now generalize the master equation to the case of Laser theory. The
mechanism of laser is described by the following master equation
$\displaystyle\frac{d\rho\left(t\right)}{dt}$
$\displaystyle=g\left[2a^{\dagger}\rho\left(t\right)a-aa^{\dagger}\rho\left(t\right)-\rho\left(t\right)aa^{\dagger}\right]$
$\displaystyle+\kappa\left[2a\rho\left(t\right)a^{\dagger}-a^{\dagger}a\rho\left(t\right)-\rho\left(t\right)a^{\dagger}a\right],$
(30)
where $g$ and $\kappa$ are the cavity gain and the loss, respectively. Eq.(30)
reduces to Eq.(18) when $g=0;$ while for $g\rightarrow\kappa\bar{n}$ and
$\kappa\rightarrow\kappa\left(\bar{n}+1\right),$ Eq.(30) becomes
$\displaystyle\frac{d\rho}{dt}$
$\displaystyle=\kappa\left(\bar{n}+1\right)\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right)$
$\displaystyle+\kappa\bar{n}\left(2a^{\dagger}\rho a-aa^{\dagger}\rho-\rho
aa^{\dagger}\right),$ (31)
which corresponds to the master equation in thermal environment 20 .
Similar to the way of deriving Eq.(22), we have derived in Ref. 21
$\displaystyle\left|\rho\left(t\right)\right\rangle$
$\displaystyle=\exp\left[\left(a\tilde{a}-\tilde{a}^{\dagger}a^{\dagger}+1\right)\left(\kappa-g\right)t\right]$
$\displaystyle\times\exp\left[\frac{\left(\kappa+g\right)\left(1-e^{2\left(\kappa-g\right)t}\right)}{2\left(\kappa-g\right)}\left(a^{\dagger}-\tilde{a}\right)\left(a-\tilde{a}^{\dagger}\right)\right]\left|\rho_{0}\right\rangle.$
(32)
Thus the matrix element
$\left\langle\eta\right|\left.\rho\left(t\right)\right\rangle$ is given by
$\left\langle\eta\right|\left.\rho\left(t\right)\right\rangle=\exp\left[-\frac{A}{2}|\eta|^{2}\right]\left\langle\eta
e^{-\left(\kappa-g\right)t}\right|\left.\rho_{0}\right\rangle,$ (33)
where
$A=\frac{\kappa+g}{\kappa-g}\left(1-e^{-2\left(\kappa-g\right)t}\right).$ (34)
According to Eq.(13) the Wigner function’s evolution for Laser process is
given by
$\displaystyle W\left(\alpha,t\right)$
$\displaystyle=\int\frac{d^{2}\eta}{2\pi^{2}}e^{-\frac{A}{2}|\eta|^{2}+\alpha^{\ast}\eta-\alpha\eta^{\ast}}\left\langle\eta
e^{-\left(\kappa-g\right)t}\right|\left.\rho_{0}\right\rangle$
$\displaystyle=\int\frac{d^{2}\xi
d^{2}\eta}{2\pi^{2}}e^{-\frac{A}{2}|\eta|^{2}+\alpha^{\ast}\eta-\alpha\eta^{\ast}}\left\langle\eta
e^{-\left(\kappa-g\right)t}\right.\left|\xi_{=2\beta}\right\rangle
W\left(\beta,0\right)$ $\displaystyle=\int\frac{d^{2}\xi
d^{2}\eta}{\pi^{2}}e^{-\frac{A}{2}|\eta|^{2}+\eta\left(\alpha^{\ast}-\beta^{\ast}e^{-\left(\kappa-g\right)t}\right)+\eta^{\ast}\left(\beta
e^{-\left(\kappa-g\right)t}-\alpha\right)}W\left(\beta,0\right)$
$\displaystyle=\frac{2}{A}\int\frac{d^{2}\beta}{\pi}\exp\left[-\frac{2}{A}\left|\alpha-\beta
e^{-\left(\kappa-g\right)t}\right|^{2}\right]W\left(\beta,0\right),$ (35)
where we have used Eq.(25). In particular, when $g=0,$ Eq.(35) reduces to
Eq.(24). For $g\rightarrow\kappa\bar{n}$ and
$\kappa\rightarrow\kappa\left(\bar{n}+1\right)$, leading to
$A=\left(2\bar{n}+1\right)T,$ Eq.(35) becomes
$W\left(\alpha,t\right)=\frac{2}{\left(2\bar{n}+1\right)T}\int\frac{d^{2}\beta}{\pi}W\left(\beta,0\right)e^{-2\frac{\allowbreak\left|\alpha-\beta
e^{-\kappa t}\right|^{2}}{\left(2\allowbreak\bar{n}+1\right)T}},$ (36)
or
$W\left(\alpha,t\right)=2e^{2\kappa t}\int d^{2}\beta
W_{T}\left(\beta\right)W\left(e^{\kappa t}(\alpha-\sqrt{T}\beta),0\right),$
(37)
where
$W_{T}\left(\beta\right)=\frac{1}{\pi\left(2\bar{n}+1\right)}e^{-\frac{2\allowbreak\left|\beta\right|^{2}}{2\allowbreak\bar{n}+1}}$
is the Wigner function of the thermal state with mean photon number $\bar{n}$.
Similar to the way of deriving Eq.(29), when the initial state is
$C_{m}a^{{\dagger}m}\left|z\right\rangle,$ substituting Eq.(26) into Eq.(35)
we have
$\displaystyle W\left(\beta,\beta^{\ast},t\right)$
$\displaystyle=\frac{e^{-C-2\left|\beta\right|^{2}}}{\pi
L_{m}(-\left|z\right|^{2})}\frac{A^{m}}{\left(2\bar{n}T+1\right)}$
$\displaystyle\times\frac{\left[\left(\bar{n}+1\right)T\right]^{m}}{\left(\bar{n}T+1\right)^{m}}L_{m}\left(-\frac{\left|B\right|^{2}}{A}\right),$
(38)
where
$\displaystyle A$ $\displaystyle=1-\allowbreak\frac{e^{-2\kappa
t}/T}{\left(2T\bar{n}+1\right)\left(\bar{n}+1\right)},$ $\displaystyle B$
$\displaystyle=\sqrt{\frac{\left(\bar{n}+1\right)T}{\bar{n}T+1}}z^{\ast}+\frac{\sqrt{\bar{n}T+1}e^{-\kappa
t}\left(2\beta^{\ast}-\frac{z^{\ast}e^{-\kappa
t}}{\bar{n}T+1}\right)}{\left(2\bar{n}T+1\right)\sqrt{\left(\bar{n}+1\right)T}},$
$\displaystyle C$
$\displaystyle=\allowbreak\frac{1}{2\bar{n}T+1}\left(\frac{3\bar{n}T+2}{T\bar{n}+1}\left|ze^{-\kappa
t}\right|^{2}+4T^{2}\bar{n}^{2}\left|\beta\right|^{2}\right)$
$\displaystyle\text{ \ \ }-\allowbreak\frac{2e^{-\kappa
t}\left(T\bar{n}+1\right)}{2\bar{n}T+1}\allowbreak\left(z\beta^{\ast}+\beta
z^{\ast}\right).$ (39)
In particular, when $\bar{n}=0,$ leading to
$A=\allowbreak\frac{1-2e^{-2t\kappa}}{T},B=\frac{1}{\sqrt{T}}\left(\left(1-2e^{-2\kappa
t}\right)z^{\ast}+2e^{-\kappa t}\beta^{\ast}\right),$ and
$-C-2\left|\beta\right|^{2}=-2\left|\beta-ze^{-t\kappa}\right|^{2},$ thus
Eq.(39) reduces to Eq.(29).
Eq.(38) manifestly shows that the Wigner function of
$C_{m}a^{{\dagger}m}\left|z\right\rangle$ in thermal environment is closely
related to the Laguerre polynomials. In addition, due to
$L_{m}\left(-\left|x\right|^{2}\right)>0,$ so $C_{m}>0,$ thus it is easily
seen that when $A>0,$ which means the condition
$\kappa t\geqslant\kappa
t_{c}=\frac{1}{2}\ln\frac{2\left(\bar{n}+1\right)}{2\bar{n}+1},$ (40)
the Wigner function (38) is always positive-definite. Thus we emphasize that
for any values of $m$, when the condition (40) is satisfied, the Wigner
function has no chance to be negative.
## V Time evolution of photon number for the laser process
Next we consider the photon number (PN) of density operator $\rho$ for the
laser process. According to the TFD, we can reform the PN
$p\left(n\right)=\left\langle n\right|\rho\left|n\right\rangle$ as
$\displaystyle p\left(n\right)$ $\displaystyle=\left\langle
n\right|\rho\left|n\right\rangle=\sum_{m=0}^{\infty}\left\langle
n,\tilde{n}\right|\rho\left|m,\tilde{m}\right\rangle$
$\displaystyle=\left\langle
n,\tilde{n}\right|\rho\left|I\right\rangle=\left\langle
n,\tilde{n}\right|\left.\rho\right\rangle,$ (41)
thus the PN is converted to the matrix element $\left\langle
n,\tilde{n}\right|\left.\rho\right\rangle$ in thermo dynamics frame. Then
using the completeness of $\left\langle\xi\right|$ and Eq.(9) as well as
Eq.(14), we see
$\displaystyle p\left(n\right)$
$\displaystyle=\int\frac{d^{2}\xi}{\pi}\left\langle
n,\tilde{n}\right|\left.\xi\right\rangle\left\langle\xi\right|\left.\rho\right\rangle$
$\displaystyle=\int d^{2}\xi\left\langle
n,\tilde{n}\right|\left.\xi\right\rangle W\left(\alpha=\xi/2\right)$
$\displaystyle=4\pi\int d^{2}\alpha W_{\left|n\right\rangle\left\langle
n\right|}\left(\alpha\right)W\left(\alpha\right),$ (42)
one can see this formula also in 1 ; 24 . Thus one can calculate the PN by
combining Eq.(35) and (42).
Now we evaluate the PN of the above decoherence model in Eq.(30). Substituting
Eq.(35) into Eq.(42), we see
$p\left(n\right)=\frac{8}{A}\int d^{2}\beta
W\left(\beta,0\right)G\left(\beta\right),$ (43)
where
$\displaystyle G\left(\beta\right)$ $\displaystyle\equiv\int d^{2}\alpha
W_{\left|n\right\rangle\left\langle
n\right|}\left(\alpha\right)e^{-\frac{2}{A}\left|\alpha-\beta
e^{-\left(\kappa-g\right)t}\right|^{2}-2|\alpha|^{2}}$
$\displaystyle=(-1)^{n}\int\frac{d^{2}\alpha}{\pi}L_{n}\left(4\left|\alpha\right|^{2}\right)e^{-\frac{2}{A}\left|\alpha-\beta
e^{-\left(\kappa-g\right)t}\right|^{2}-2|\alpha|^{2}}.$ (44)
Using Eqs.(25) and (27) we can evaluate Eq.(44) as
$\displaystyle G\left(\beta\right)$
$\displaystyle=\frac{1}{n!}\frac{\partial^{n+n}}{\partial\tau^{n}\partial\tau^{\prime
n}}e^{-\tau\tau^{\prime}-\frac{2}{A}\left|\beta\right|^{2}e^{-2\left(\kappa-g\right)t}}\int\frac{d^{2}\alpha}{\pi}$
$\displaystyle\times\exp\left[-2\frac{A+1}{A}|\alpha|^{2}+2\alpha\left(\tau+\frac{\beta^{\ast}}{Ae^{\left(\kappa-g\right)t}}\right)\right.$
$\displaystyle\left.+2\alpha^{\ast}\left(\tau^{\prime}+\frac{\beta}{Ae^{\left(\kappa-g\right)t}}\right)\right]_{\tau=\tau^{\prime}=0}$
$\displaystyle=\frac{Ae^{-\frac{2e^{-2\left(\kappa-g\right)t}}{A+1}\left|\beta\right|^{2}}}{2\left(A+1\right)n!}\frac{\partial^{n+n}}{\partial\tau^{n}\partial\tau^{\prime
n}}\exp\left[-\frac{1-A}{1+A}\tau^{\prime}\tau\right.$
$\displaystyle+\left.\frac{2\beta^{\ast}e^{-\left(\kappa-g\right)t}}{A+1}\tau^{\prime}+\tau\frac{2\beta
e^{-\left(\kappa-g\right)t}}{A+1}\right]_{\tau=\tau^{\prime}=0}.$ (45)
After making some scaled transformations, we finally obtain
$\displaystyle G\left(\beta\right)$
$\displaystyle=\frac{A\left(A-1\right)^{n}}{2\left(1+A\right)^{n+1}n!}e^{-\frac{2e^{-2\left(\kappa-g\right)t}}{A+1}\left|\beta\right|^{2}}$
$\displaystyle\times\frac{\partial^{n+n}}{\partial\tau^{n}\partial\tau^{\prime
n}}\left.e^{-\tau^{\prime}\tau+\frac{2\beta^{\ast}e^{-\left(\kappa-g\right)t}}{\sqrt{1-A^{2}}}\tau^{\prime}+\tau\frac{2\beta
e^{-\left(\kappa-g\right)t}}{\sqrt{1-A^{2}}}}\right|_{\tau=\tau^{\prime}=0}$
$\displaystyle=\frac{A\left(A-1\right)^{n}}{2\left(1+A\right)^{n+1}}e^{-\frac{2e^{-2\left(\kappa-g\right)t}}{A+1}\left|\beta\right|^{2}}L_{n}\left(\frac{4\left|\beta\right|^{2}e^{-2\left(\kappa-g\right)t}}{1-A^{2}}\right).$
(46)
Substituting Eq.(46) into Eq.(43) yields
$\displaystyle p\left(n\right)$
$\displaystyle=\frac{4\left(A-1\right)^{n}}{\left(A+1\right)^{n+1}}\int
d^{2}\beta
e^{-\frac{2e^{-2\left(\kappa-g\right)t}}{A+1}\left|\beta\right|^{2}}$
$\displaystyle\times
L_{n}\left\\{\frac{4e^{-2\left(\kappa-g\right)t}}{1-A^{2}}\left|\beta\right|^{2}\right\\}W\left(\beta,0\right),$
(47)
which is a new formula for calculating the photon number distribution of the
open system in enviornment. From Eq.(47) it is easily seen that once the
Wigner function of initial state is known, one can obtain its photon number
distribution by performing the integration in Eq.(47).
In particular, when $g=0,$ $A=1-e^{-2\kappa t}=T,$ Eq.(47) reduces to
$\displaystyle p\left(n\right)$ $\displaystyle=\frac{4(-1)^{n}e^{2\kappa
t}}{\left(2e^{2\kappa t}-1\right)^{n+1}}\int d^{2}\beta
e^{-\frac{2}{2e^{2\kappa t}-1}\left|\beta\right|^{2}}$ $\displaystyle\times
L_{n}\left\\{\frac{4e^{2\kappa t}}{2e^{2\kappa
t}-1}\left|\beta\right|^{2}\right\\}W\left(\beta,0\right),$ (48)
which corresponds to the photon number of density operator in the amplitude
damping quantum channel.
While for $g\rightarrow\kappa\bar{n}$ and
$\kappa\rightarrow\kappa\left(\bar{n}+1\right)$, , Eq.(47) becomes to
$\displaystyle p\left(n\right)$
$\displaystyle=\frac{4\left(\mathcal{A}-1\right)^{n}}{\left(\mathcal{A}+1\right)^{n+1}}\int
d^{2}\beta e^{-\frac{2e^{-2\kappa t}}{\mathcal{A}+1}\left|\beta\right|^{2}}$
$\displaystyle\times L_{n}\left\\{\frac{4e^{-2\kappa
t}}{1-\mathcal{A}^{2}}\left|\beta\right|^{2}\right\\}W\left(\beta,0\right),$
(49)
where
$\mathcal{A}=\left(2\bar{n}+1\right)T=\left(2\bar{n}+1\right)\left(1-e^{-2\kappa
t}\right).$ Eq.(49) corresponds to the photon number of system interacting
with thermal bath.
For example, we still consider the photon-added coherent state field.
Substituting Eq.(26) into Eq.(47) and uisng Eqs.(25) and (27) yields
$\displaystyle p\left(n\right)$
$\displaystyle=Ne^{-2\left|z\right|^{2}}\int\frac{d^{2}\beta}{\pi}L_{m}(\left|2\beta-z\right|^{2})L_{n}\left\\{\frac{4e^{-2\left(\kappa-g\right)t}}{1-A^{2}}\left|\beta\right|^{2}\right\\}$
$\displaystyle\times\exp\left[\allowbreak 2\left(z\beta^{\ast}+\beta
z^{\ast}\right)-2\left(\allowbreak
1+\frac{e^{-2\left(\kappa-g\right)t}}{A+1}\right)\left|\beta\right|^{2}\right]$
$\displaystyle=\frac{Ne^{-2\left|z\right|^{2}}\left(-1\right)^{m+n}}{m!n!}\frac{\partial^{2m}}{\partial\upsilon^{m}\partial\upsilon^{\prime
m}}\frac{\partial^{2n}}{\partial\tau^{n}\partial\tau^{\prime
n}}e^{-\upsilon\upsilon^{\prime}-z^{\ast}\upsilon^{\prime}}$
$\displaystyle\times
e^{-z\upsilon-\tau\tau^{\prime}}\int\frac{d^{2}\beta}{\pi}\exp\left[-2\mu\left|\beta\right|^{2}+2\left(\sigma\tau+z^{\ast}+\upsilon\right)\beta\right.$
$\displaystyle\left.+2\left(\sigma\tau^{\prime}+z+\upsilon^{\prime}\right)\beta^{\ast}\right]_{\upsilon=\upsilon^{\prime}=\tau=\tau^{\prime}=0}$
$\displaystyle=\frac{\allowbreak N\left(-1\right)^{m+n}}{2\mu
m!n!}e^{\frac{2-2\mu}{\mu}\left|z\right|^{2}}\frac{\partial^{2m}}{\partial\upsilon^{m}\partial\upsilon^{\prime
m}}\frac{\partial^{2n}}{\partial\tau^{n}\partial\tau^{\prime n}}$
$\displaystyle\times\exp\left[\omega\upsilon\upsilon^{\prime}+\allowbreak\left(\lambda\sigma-1\right)\allowbreak\tau\tau^{\prime}+\lambda\left(\tau\upsilon^{\prime}+\upsilon\tau^{\prime}\right)\right.$
$\displaystyle\left.+\omega\left(z^{\ast}\upsilon^{\prime}+z\upsilon\right)+\lambda\left(z\tau+z^{\ast}\tau^{\prime}\right)\right]_{\upsilon=\upsilon^{\prime}=\tau=\tau^{\prime}=0},$
(50)
where we have set
$\omega=\frac{2-\mu}{\mu},\lambda=\frac{2\sigma}{\mu},\text{
}\sigma=\frac{e^{-\left(\kappa-g\right)t}}{\sqrt{1-A^{2}}},$ (51)
and
$N=\frac{4\left(A-1\right)^{n}}{\left(A+1\right)^{n+1}}\frac{\left(-1\right)^{m}}{L_{m}(-\left|z\right|^{2})},\mu=\allowbreak
1+\frac{e^{-2\left(\kappa-g\right)t}}{A+1}.$ (52)
Further expanding the exponential item
$\exp\left[\omega\upsilon\upsilon^{\prime}+\allowbreak\left(\lambda\sigma-1\right)\tau\tau^{\prime}\right],$
we finally obtain
$\displaystyle p\left(n\right)$ $\displaystyle=\frac{\allowbreak
N\lambda^{2n}e^{\frac{2-2\mu}{\mu}\left|z\right|^{2}}}{2\mu\left(-\omega\right)^{n-m}}\sum_{l,k=0}^{m,n}\frac{m!n!\left[\omega\left(\lambda\sigma-1\right)/\lambda^{2}\right]^{k}}{l!k!\left[\left(m-l\right)!\left(n-k\right)!\right]^{2}}$
$\displaystyle\times\left|H_{m-l,n-k}\left(i\sqrt{\omega}z,i\sqrt{\omega}z^{\ast}\right)\right|^{2}.$
(53)
In particular, when $g=0,$ leading to $A=\omega=T,\sigma=\frac{e^{-\kappa
t}}{\sqrt{1-T^{2}}},\mu=\allowbreak\frac{2}{2-e^{-2\kappa
t}},\lambda\sigma=1,$ and $\lambda=\sqrt{1+T}$, thus
$\displaystyle p\left(n\right)$
$\displaystyle=\frac{m!}{n!}\frac{\left(1-\omega\right)^{n}}{L_{m}(-\left|z\right|^{2})}\sum_{l=0}^{m}\frac{\omega^{m-n}e^{-e^{-2t\kappa}\left|z\right|^{2}}}{l!\left[\left(m-l\right)!\right]^{2}}$
$\displaystyle\times\left|H_{m-l,n}\left(i\sqrt{\omega}z,i\sqrt{\omega}z^{\ast}\right)\right|^{2},$
(54)
which concides with Eq.(43) with idea detection efficiency in Ref. 23 .
In sum, by virtue of the thermo entangled state representation that has a
fictitious mode as a counterpart mode of the system mode, we have derived the
relation between the Wigner functions at $t$ time and the initial time when
quantum system interacts with envoirnment, such as decoherence, damping and
amplification. As another quantity describing quantum system, the formula of
photon number distribution has also been derived, which can be evaluated by
performing an integration for the initial Wigner function. Our deriviations
seem more concise.
ACKNOWLEDGEMENT: Work supported by the National Natural Science Foundation of
China under grants: 10775097 and 10874174.
## References
* (1) W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
* (2) H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Springer-Verlag, Berlin, 1999; H. J. Carmichael, Statistical Methods in Quantum Optics 2: Non-Classical Fields, (Springer-Verlag, Berlin, 2008).
* (3) Wolfgang P. Schleich, Quantum Optics in Phase Space, (Wiley-VCH, Birlin, 2001).
* (4) M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, Phys. Rep. 106, (1984) 121.
* (5) Memorial Issue for H. Umezawa, Int. J. Mod. Phys. B 10, (1996) 1695 memorial issue and references therein.
* (6) H. Umezawa, Advanced Field Theory – Micro, Macro, and Thermal Physics (AIP 1993)
* (7) Y. Takahashi and H. Umezawa, Collecive Phenomena 2, (1975) 55.
* (8) Hong-yi Fan and Yue Fan, Phys. Lett. A 246, (1998) 242; ibid, 282, (2001) 269.
* (9) Hong-yi Fan and Yue Fan, J. Phys. A 35, (2002) 6873; Hong-yi Fan and Hai-liang Lu, Mod. Phys. Lett. B, 21, (2007) 183.
* (10) Hong-yi Fan, Hai-liang Lu and Yue Fan, Ann. Phys _._ 321, (2006) 480.
* (11) Hong-yi Fan, H. R. Zaidi and J. R. Klauder, Phys. Rev. D 35, (1987) 1831.
* (12) A. Wünsche, J. Opt. B: Quantum Semiclass. Opt. 1, (1999) R11.
* (13) E. P. Wigner, Phys. Rev. 40, (1932) 749
* (14) G. S. Agarwal and E. Wolf, Phys. Rev. D 2, (1970) 2161; R. F. O’Connell and E. P. Wigner, Phys. Lett. A 83, (1981) 145.
* (15) A. Wünsche, J. Computational and Appl. Math. 133 (2001) 665.
* (16) A. Wünsche, J . Phys. A: Math. and Gen. 33 (2000) 1603.
* (17) R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, Berlin, 2001), Appendix A.
* (18) R. J. Glauber, Phys. Rev. 130, (1963) 2529; Phys. Rev. 131, (1963) 2766.
* (19) J. R. Klauder and B. S. Skargerstam, Coherent States, (World Scientific, Singapore, 1985).
* (20) C. Gardiner and P. Zoller, Quantum Noise (Springer Berlin, 2000).
* (21) Hong-yi Fan and Li-yun Hu, Opt. Commun. 282, (2009) 932; 281, (2008) 5571.
* (22) G. S. Agarwal and K. Tara, Phys. Rev. A 43, (1991) 492.
* (23) Li-yun Hu and Hong-yi Fan, Phys. Scr. 79, (2009) 035004.
* (24) Hong-yi Fan and Li-yun Hu, Opt. Lett. 33, (2008) 443.
|
arxiv-papers
| 2009-03-17T05:15:48 |
2024-09-04T02:49:01.197668
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu and Hong-yi Fan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/0903.2900"
}
|
0903.2932
|
# Experimental investigation of electric field distributions in a chaotic 3D
microwave rough billiard
Oleg Tymoshchuk, Nazar Savytskyy, Oleh Hul, Szymon Bauch, and Leszek Sirko
Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668
Warszawa, Poland
(January 28, 2007)
###### Abstract
We present the first experimental study of the electric field distributions
$E_{N}$ of a three-dimensional (3D) microwave chaotic rough billiard with the
translational symmetry. The translational symmetry means that the cross-
section of the billiard is invariant under translation along $z$ direction.
The 3D electric field distributions were measured up to the level number
$N=489$. In this way the experimental spatial correlation functions
$C_{N,p}({\bf x,s})\propto\langle E_{N,p}({\bf x}+\frac{1}{2}{\bf
s})E_{N,p}^{\ast}({\bf x}-\frac{1}{2}{\bf s})\rangle$ were found and compared
with the theoretical ones. The experimental results for higher two-dimensional
level number $N_{\bot}$ appeared to be in good agreement with the theoretical
predictions.
###### pacs:
05.45.Mt,05.45.Jn
In this paper we present the first experimental investigation of electric
field distributions of the chaotic 3D microwave rough billiard with the
translational symmetry. Due to experimental difficulties there are very few
experimental studies devoted to 3D chaotic microwave cavities Sirko1995 ;
Alt1997 ; Dorr1998 ; Eckhardt1999 ; Dembowski2002 . In a pioneering experiment
Deus et al. Sirko1995 have been measured eigenfrequencies of the 3D chaotic
(irregular) microwave cavity in order to confirm that their distribution
displays behavior characteristic for classically chaotic quantum systems,
viz., the Wigner distribution. Three-dimensional chaotic cavities as well as
properties of random electromagnetic vector field have been also scarcely
studied theoretically Primack2000 ; Prosen1997 ; Arnaut2006 .
In general, there is no analogy between quantum billiards and electromagnetic
cavities in three dimensions. However, for 3D cavities with the translational
symmetry the classification of the modes into transverse electric (TE) and
transverse magnetic (TM) is possible. The TM modes are especially important
because they allow for the simulation of 2D quantum billiards on cross-
sectional planes of 3D cavities. Furthermore, we show in this paper that the
distributions of the electric field of TM modes of the 3D chaotic rough cavity
can be experimentally measured.
Figure 1: Upper panel: Sketch of the chaotic half-circular 3D microwave rough
billiard in the $xy$ plane. Dimensions are given in cm. The cavity sidewalls
are marked by 1 and 2 (see text). Squared wave functions
$|\psi_{N,p}(R_{c},\theta)|^{2}$ were evaluated on a half-circle of fixed
radius $R_{c}=9.25$ cm. Billiard’s rough boundary $\Gamma$ is marked with the
bold line. Lower panel: White circle marks the position of the hole drilled in
the upper wall of the cavity. This hole was used to introduce the perturber
inside the cavity in order to measure the $z$-component of the electric field
distributions $E_{N,p}({\bf x})$.
In the experiment we used 3D cavity with the translational symmetry in the
shape of a rough half-circle (Fig. 1) with the height $h=60$ mm. The cavity
was made of polished aluminium.
We suppose that the direction of the translational symmetry of the cavity is
along the $z$-axis. The boundary conditions at $z=0$ and $z=h$ demand that the
$z$ dependence of the $z$-component of the electric and magnetic fields
$E_{N,p}({\bf x})$ and $B_{N,p}({\bf x})$ of TM modes be in the form
$E_{N,p}({\bf x})\equiv E_{N,p}(x,y,z)=A_{N,p}\psi_{N,p}(x,y)f_{p}(z)$, where
$f_{p}(z)=\cos(p\pi z/h)$, $p=0,1,2\ldots$, $A_{N,p}$ is the normalization
constant and $B_{N,p}({\bf x})=0$. The functional dependence of $E_{N,p}({\bf
x})$ on the plane cross section coordinates is denoted by the amplitude
$\psi_{N,p}(x,y)\equiv E_{N,p}(x,y)$. The amplitude $\psi_{N,p}(x,y)$
satisfies the Helmholtz equation
$(\bigtriangleup_{\bot}+k_{N,p}^{2})\psi_{N,p}(x,y)=0,$ $None$
where $\bigtriangleup_{\bot}$ is two-dimensional Laplacian operator and
$k_{N,p}=(k_{N}^{2}-(p\pi/h)^{2})^{1/2}$ is the effective wave vector. The
wave vector $k_{N}=2\pi\nu_{N}/c$, where $\nu_{N}$ is the resonance frequency
of the level $N$ and $c$ is the speed of light in the vacuum. The equation (1)
is equivalent to the Schrödinger equation (in units $\hbar=1$) describing a
particle of mass $m=1/2$ with the kinetic energy $k_{N}^{2}$ in an external
potential $V=(p\pi/h)^{2}$ Kim2005 . Therefore, microwave 3D cavities with the
translational symmetry simulate on the cross-sectional planes quantum
billiards with the external potential $(p\pi/h)^{2}$. In this way microwave
cavities can be effectively used beyond the standard 2D frequency limit (the
case $p=0$) Hans in simulation of quantum systems. The amplitude
$\psi_{N,p}(x,y)$ fulfills Dirichlet boundary conditions on the sidewalls of
the billiard. Therefore, throughout the text the amplitudes $\psi_{N}(x,y)$
are also often called the wave functions $\psi_{N}(x,y)$. It is important to
note that the full electric field $E_{N,p}({\bf x})$ satisfies additionally
Neumann boundary conditions at the top and the bottom of the cavity.
Because of the relatively low quality factor of the cavity ($Q\simeq 4000$)
the value of the level number $N$ was evaluated from the Balian–Bloch formula
Balian
$N(k)=\frac{1}{3\pi^{2}}Vk^{3}-\frac{2}{3\pi^{2}}\int_{S}\frac{d\sigma_{\omega}}{R_{\omega}}k,$
$None$
where k is the wave vector, $V=(9.43\pm 0.01)\cdot 10^{-4}$ m3 is the volume
of the cavity and $\int_{S}\frac{d\sigma_{\omega}}{R_{\omega}}=0.932$ m $\pm
0.005$ m is the surface curvature averaged over the surface of the cavity.
The measurements allowed us for the first time to evaluate the spatial
correlation function Kaufman88
$C_{N,p}({\bf x},{\bf s})=\frac{1}{\langle|E_{N,p}({\bf
x})|^{2}\rangle}\langle E_{N,p}({\bf x}+\frac{1}{2}{\bf s})E_{N,p}^{\ast}({\bf
x}-\frac{1}{2}{\bf s})\rangle,$ $None$
where the local average $\langle\cdots\rangle$ is defined as follows
$\langle|E_{N,p}({\bf
x})|^{2}\rangle=\frac{1}{\Delta^{n}}\int_{-\Delta/2}^{\Delta/2}|E_{N,p}({\bf
x}+{\bf s})|^{2}d^{n}s.$ $None$
The 3D cavity sidewalls are made of 2 segments (see Fig. 1). The rough segment
1 is described on the cross-sectional planes by the radius function
$R(\theta)=R_{0}+\sum_{m=2}^{M}{a_{m}\sin(m\theta+\phi_{m})}$, where the mean
radius $R_{0}$=10.0 cm, $M=20$, $a_{m}$ and $\phi_{m}$ are uniformly
distributed on [0.084,0.091] cm and [0,2$\pi$], respectively, and
$0\leq\theta<{\pi}$. (Here, for the convenience, the polar coordinates $r$ and
$\theta$ are used instead of the Cartesian ones $x$ and $y$.)
The surface roughness of a billiard on the cross-sectional planes is
characterized by the function $k(\theta)=(dR/d\theta)/R_{0}$. For our billiard
we have the angle average
$\tilde{k}=(\left<k^{2}(\theta)\right>_{\theta})^{1/2}\simeq 0.400$. In such a
billiard the classical dynamics is diffusive in orbital momentum due to
collisions with the rough boundary because $\tilde{k}$ is much above the chaos
border $k_{c}=M^{-5/2}=0.00056$ Frahm97 . The roughness parameter $\tilde{k}$
determines also other properties of the billiard Frahm on the cross-sectional
planes. The amplitudes $\psi_{N,p}(r,\theta)$ are localized for the two-
dimensional level number $N_{\bot}<N_{e}=1/128\tilde{k}^{4}$, where
$N_{\bot}=\frac{A}{4\pi}k_{N,p}^{2}-\frac{P}{4\pi}k_{N,p}$. $A=(1.572\pm
0.002)\cdot 10^{-2}$ m2 and $P=0.537$ m $\pm 0.001$ m are the cross-sectional
plane area and its perimeter, respectively. Because of a large value of the
roughness parameter $\tilde{k}$ the localization border lies very low,
$N_{e}\simeq 1$. The border of Breit-Wigner regime is
$N_{W}=M^{2}/48\tilde{k}^{2}\simeq 52$. It means that between
$N_{e}<N_{\bot}<N_{W}$ Wigner ergodicity Frahm ought to be observed and for
$N_{\bot}>N_{W}$ Shnirelman ergodicity should emerge.
To measure the amplitudes $\psi_{N,p}(r,\theta)$ of the 3D electric field
distributions we used a very effective method described in Savytskyy2003 . It
is based on the perturbation technique Slater52 and preparation of the “trial
functions” Savytskyy2004 ; Hul2005 ; Hul2006 . In the perturbation method a
small perturber is introduced inside the cavity to alter its resonant
frequencies and in this way to evaluate the squared wave functions
$|\psi_{N,p}(R_{c},\theta)|^{2}$ (see Fig. 1). The perturber (4.0 mm in length
and 0.3 mm in diameter, oriented in $z$-direction) was moved by the stepper
motor via the Kevlar line hidden in the groove (0.4 mm wide, 1.0 mm deep) made
in the cavity’s bottom wall along the half-circle $R_{c}$. The measurements
were performed at 0.36 mm steps along a half-circle with fixed radius
$R_{c}=9.25$ cm.
In order to find the dependence of the electric field distributions
$E_{N,p}({\bf x})$ on the $z$ coordinate and to estimate the wave vector
$k_{3}=p\pi/h$ we measured the electric field inside the 3D cavity along the
$z$-axis. Also in this case the perturber (4.5 mm in length and 0.3 mm in
diameter) was attached to the Kevlar line and moved by the stepper motor. The
perturber entered and exited the cavity by small holes (0.4 mm) drilled in the
upper and the bottom walls of the cavity. The both holes were located at the
position: $r=9.11$ cm, $\theta=0.47$ radians.
Using the method of the “trial wave function” we were able to reconstruct 75
experimental wave functions $\psi_{N,p}(r,\theta)$, which belonged to TM modes
of the rough half-circular 3D billiard with the level number $N$ between 2 and
489. The range of corresponding eigenfrequencies was from $\nu_{2}\simeq 2.47$
GHz to $\nu_{489}\simeq 11.99$ GHz. The remaining wave functions belonging to
TM modes, from the range $N=2-489$, were not reconstructed because of near-
degeneration of the neighboring eigenfrequencies or due to the problems with
the measurements of $|\psi_{N,p}(R_{c},\theta)|^{2}$ along a half-circle
coinciding for its significant part with one of the nodal lines of
$\psi_{N,p}(r,\theta)$.
Figure 2: The reconstructed wave function $\psi_{460,0}(r,\theta)$ of the
chaotic half-circular microwave rough billiard. The amplitudes have been
converted into a grey scale with white corresponding to large positive and
black corresponding to large negative values, respectively. Dimensions of the
billiard are given in cm. In the figure the $z$ dependence of the electric
field distribution
$E_{460,0}(r,\theta,z)|_{x=0}\propto\psi_{460,0}(r,\theta)|_{x=0}f_{0}(z)$ is
also shown.
In Fig. 2 and Fig. 3 we show two examples of reconstructed wave functions
$\psi_{460,0}(r,\theta)$ and $\psi_{463,4}(r,\theta)$, respectively. The
character of the wave functions predominantly depends on the effective wave
vector $k_{N,p}$. It is seen that the wave function $\psi_{463,4}(r,\theta)$
in Fig. 3 is more regular than the one presented in Fig. 2 in spite of having
the larger level number $N$.
Figure 3: The reconstructed wave function $\psi_{463,4}(r,\theta)$ of the
chaotic half-circular microwave rough billiard. The $z$ dependence of the
electric field distribution
$E_{463,4}(r,\theta,z)|_{x=0}\propto\psi_{463,4}(r,\theta)|_{x=0}f_{4}(z)$ is
also shown.
In order to check ergodicity of the billiard’s wave functions
$\psi_{N,p}(R_{c},\theta)$, especially close to the ergodicity borders, one
should use some additional measures such as e.g., calculation of the
structures of their energy surfaces Frahm97 . For this reason we extracted
wave function amplitudes $C^{(N,p)}_{nl}=\left<n,l|N,p\right>$ in the basis
$n,l$ of a half-circular billiard with radius $r_{max}$, where $n=1,2,3\ldots$
enumerates the zeros of the Bessel functions and $l=1,2,3\ldots$ is the
angular quantum number. As expected, close to the border of the regimes of
Breit-Wigner and Shnirelman ergodicity the wave function
$\psi_{460,0}(r,\theta)$ ($N_{\bot}=65$) was found to be extended
homogeneously over the whole energy surface Hlushchuk01 (figure not shown
here). In contrary, the wave function $\psi_{463,4}(r,\theta)$, $N_{\bot}=16$,
which lies closer to the localization boarder, displays the tendency to
localization in $n,l$ basis (figure not shown here).
The measurement of 3D electric field distributions $E_{N,p}({\bf x})$ allowed
us for the first time to find the experimental spatial correlation function
$C_{N,p}({\bf x,s})$. It is easy to show Berry77 that for the 3D chaotic
cavity with the translational symmetry the spatial correlation function should
have the following form:
$C_{N,p}({\bf x},|{\bf s}|)\equiv C_{N,p}(|{\bf
s}|)=J_{0}(k_{N,p}s_{xy})\cos(p\pi s_{z}/h),$ $None$
where $|{\bf s}|=(s_{xy}^{2}+s_{z}^{2})^{1/2}$. For the cross-sectional planes
$z=const$ the correlation function $C_{N,p}(|{\bf s}|)\sim
J_{0}(k_{N,p}s_{xy})$ is reduced to the well known result of Berry Berry77
for chaotic 2D wave functions described by a random superposition of plane
waves.
Figure 4: Panels (a)-(c) show the experimental correlation function
$C_{460,0}({\bf x},|{\bf s}|)$ calculated at ${\bf x}=$ (-2.75 cm, 4.35 cm, 0
cm) for the three projection angles $\phi=0$, $\pi/4$, and $\pi/2$,
respectively. Experimental correlation function $C_{460,0}({\bf x},|{\bf s}|)$
(full line) is compared with the theoretical one (dashed line).
In Fig. 4(a)-(c) we show a representative example of the experimental
correlation function $C_{460,0}({\bf x},|{\bf s}|)$ ($N_{\bot}=65$) calculated
at ${\bf x}=$ (-2.75 cm, 4.35 cm, 0 cm) for the three different projection
angles $\phi=0$, $\pi/4$, and $\pi/2$, respectively, where
$\phi=\arcsin(s_{z}/|{\bf s}|)$. The local average $\langle\cdots\rangle$
required for the evaluation of $C_{N,p}({\bf x},|{\bf s}|)$ (see the formulas
(3) and (4)) was calculated on the cross-sectional plane $xy$ in the range
$\Delta/2=2\pi/k_{N,p}$. The experimental correlation functions
$C_{460,0}({\bf x},|{\bf s}|)$ are compared in Fig. 4 with the theoretical
ones. In all cases we find good agreement with the theoretical predictions
given by the formula (5). Small discrepancies observed in Fig. 4(a) for $|{\bf
s}|>1$ can be connected with the finiteness of the system and were
theoretically studied in Baecker02 .
Figure 5: Panels (a)-(c) show the experimental correlation function
$C_{463,4}({\bf x},|{\bf s}|)$ calculated at ${\bf x}=$ (-2.75 cm, 4.35 cm, 0
cm) for the three projection angles $\phi=0$, $\pi/4$, and $\pi/2$,
respectively. Experimental correlation function $C_{463,4}({\bf x},|{\bf s}|)$
(full line) is compared with the theoretical one (dashed line).
Fig. 5(a)-(c) shows the experimental correlation function $C_{463,4}({\bf
x},|{\bf s}|)$ ($N_{\bot}=16$) calculated at ${\bf x}=$ (-2.75 cm, 4.35 cm, 0
cm) for the three different projection angles $\phi=0$, $\pi/4$, and $\pi/2$,
respectively. The experimental correlation functions $C_{463,4}(|{\bf s}|)$
are compared in Fig. 5 with the theoretical ones. In Fig. 5 (a), even for
small $|{\bf s}|$, we find a significant departure of the experimental
correlation function from the theoretical prediction, which clearly suggests
that the wave function $\psi_{463,4}(r,\theta)$ is not chaotic. Also in Fig. 5
(b)-(c) the experimental correlation functions $C_{463,4}(|{\bf s}|)$ show for
larger $|{\bf s}|$ significant deviations from the theoretical ones. The
discrepancies between the correlation function $C_{463,4}({\bf x},|{\bf
s}|=z)$ for the $z$-component of the electric field distribution and the
theoretical prediction in Fig. 5(c) arise mainly due to the procedure of
averaging of the correlation function $C_{N,p}({\bf x},|{\bf s}|)$ in
$z$-direction, which was taken over the period of the cosine function.
In summary, we measured the wave functions of the chaotic 3D rough microwave
billiard with the translational symmetry up to the level number $N=489$. For
the first time the experimental correlation function $C_{N,p}({\bf x},{\bf
s})$ was estimated and compared with the theoretical prediction. For the
states with higher $N_{\bot}$ we find, especially for small values of the
parameter $|{\bf s}|$, good agreement with the theoretical predictions, which
show that the wave functions are chaotic. For the states with lower $N_{\bot}$
significant discrepancies between experimental and theoretical results are
observed.
Acknowledgments. This work was partially supported by the Ministry of Science
and Higher Education grants No. N202 099 31/0746 and 2 P03B 047 24.
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|
arxiv-papers
| 2009-03-17T09:54:32 |
2024-09-04T02:49:01.205757
|
{
"license": "Public Domain",
"authors": "Oleg Tymoshchuk, Nazar Savytskyy, Oleh Hul, Szymon Bauch and Leszek\n Sirko",
"submitter": "Oleh Hul",
"url": "https://arxiv.org/abs/0903.2932"
}
|
0903.2939
|
# Investigation of nodal domains in a chaotic three-dimensional microwave
rough billiard with the translational symmetry
Nazar Savytskyy, Oleg Tymoshchuk, Oleh Hul, Szymon Bauch and Leszek Sirko
Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668
Warszawa, Poland
(March 20, 2007)
###### Abstract
We show that using the concept of the two-dimensional level number $N_{\bot}$
one can experimentally study of the nodal domains in a three-dimensional (3D)
microwave chaotic rough billiard with the translational symmetry. Nodal
domains are regions where a wave function has a definite sign. We found the
dependence of the number of nodal domains $\aleph_{N_{\bot}}$ lying on the
cross-sectional planes of the cavity on the two-dimensional level number
$N_{\bot}$. We demonstrate that in the limit $N_{\bot}\rightarrow\infty$ the
least squares fit of the experimental data reveals the asymptotic ratio
$\aleph_{N_{\bot}}/N_{\bot}\simeq 0.059\pm 0.029$ that is close to the
theoretical prediction $\aleph_{N_{\bot}}/N_{\bot}\simeq 0.062$. This result
is in good agreement with the predictions of percolation theory.
###### pacs:
05.45.Mt,05.45.Df
In this paper we show that measuring the distributions of the electric field
of TM modes of a 3D chaotic rough cavity with the translational symmetry one
can find the dependence of the number of nodal domains $\aleph_{N_{\bot}}$
lying on the cross-sectional planes of the cavity on the two-dimensional level
number $N_{\bot}$. The translational symmetry means that the cross-section of
the billiard is invariant under translation along $z$ direction.
In the seminal papers Blum et al. Blum2002 and Bogomolny and Schmit
Bogomolny2002 showed that the distributions of the number of nodal domains in
two-dimensional (2D) systems can be used to distinguish between the systems
with integrable and chaotic underlying classical dynamics. The theoretical
findings have been tested in a series of experiments with chaotic microwave 2D
rough billiards Savytskyy2004 ; Hul2005 ; Hul2006 .
Due to severe experimental problems there are very few experimental studies
devoted to 3D chaotic microwave cavities Sirko1995 ; Alt1997 ; Dorr1998 ;
Eckhardt1999 ; Dembowski2002 . In a pioneering experiment Deus et al.
Sirko1995 have been measured eigenfrequencies of the 3D chaotic (irregular)
microwave cavity in order to confirm that their distribution displays behavior
characteristic for classically chaotic quantum systems, viz., the Wigner
distribution. In other important experiments the periodic orbits Alt1997 , the
distributions and the correlation function of the frequency shifts caused by
the external perturbation Dorr1998 ; Eckhardt1999 and a trace formula for
chaotic 3D cavities Dembowski2002 have been respectively studied. Quite
recently the spatial correlation functions of the 3D experimental microwave
chaotic rough billiard with the translational symmetry have been studied by
Tymoshchuk et al Tymoshchuk2007 . Three-dimensional chaotic cavities and
properties of random electromagnetic vector field have been also studied in
several theoretical papers Primack2000 ; Prosen1997 ; Arnaut2006 .
The important feature of 3D cavities with the translational symmetry is
connected with the fact that their modes can be classified into transverse
electric (TE) and transverse magnetic (TM). Although, there is no analogy
between quantum billiards and electromagnetic cavities in three dimensions,
the TM modes are especially important because they allow for the simulation of
2D quantum billiards on cross-sectional planes of 3D cavities.
Figure 1: Sketch of the chaotic half-circular 3D microwave rough billiard in
the $xy$ plane. Dimensions are given in cm. The cavity sidewalls are marked by
1 and 2 (see text). Squared wave functions $|\psi_{N,p}(R_{c},\theta)|^{2}$
were evaluated on a half-circle of fixed radius $R_{c}=9.25$ cm. Billiard’s
rough boundary $\Gamma$ is marked with the bold line. The white circle
centered at $x=8.12$ cm and $y=4.13$ cm marks the position of the hole drilled
in the upper wall of the cavity. The hole was used to introduce the perturber
inside the cavity in order to measure the $z$-component of the electric field
distributions $E_{N,p}({\bf x})$.
In the experiment we used 3D cavity with the translational symmetry in the
shape of a rough half-circle (Fig. 1) with the height $h=60$ mm. The cavity
was made of polished aluminium. The upper and bottom walls of the cavity were
attached to the sidewalls with 48 screws in order to make good electrical
contact.
Assuming that the direction of the translational symmetry of the cavity is
along the $z$-axis the boundary conditions at $z=0$ and $z=h$ demand that the
$z$ dependence of the $z$-component of the electric and magnetic fields
$E_{N,p}({\bf x})$ and $B_{N,p}({\bf x})$ of TM modes be in the form
$E_{N,p}({\bf x})\equiv E_{N,p}(x,y,z)=A_{N,p}\psi_{N,p}(x,y)f_{p}(z)$, where
$f_{p}(z)=\cos(p\pi z/h)$, $p=0,1,2\ldots$, $A_{N,p}$ is the normalization
constant and $B_{N,p}({\bf x})=0$. The dependence of $E_{N,p}({\bf x})$ on the
plane cross section coordinates we denote by the amplitude
$\psi_{N,p}(x,y)\equiv E_{N,p}(x,y)$. Then, the amplitude $\psi_{N,p}(x,y)$
satisfies the Helmholtz equation
$(\bigtriangleup_{\bot}+k_{N,p}^{2})\psi_{N,p}(x,y)=0,$ $None$
where $\bigtriangleup_{\bot}$ is two-dimensional Laplacian operator and
$k_{N,p}=(k_{N}^{2}-(p\pi/h)^{2})^{1/2}$ is the effective wave vector. The
wave vector $k_{N}=2\pi\nu_{N}/c$, where $\nu_{N}$ is the resonance frequency
of the level $N$ and $c$ is the speed of light in the vacuum. One can easily
see that the equation (1) is equivalent to the Schrödinger equation (in units
$\hbar=1$) describing a particle of mass $m=1/2$ with the kinetic energy
$k_{N}^{2}$ in an external potential $V=(p\pi/h)^{2}$ Kim2005 . Therefore, 3D
microwave cavities can be effectively used beyond the standard 2D frequency
limit (the case $p=0$) Hans in simulation of quantum systems. The amplitude
$\psi_{N,p}(x,y)$ fulfills Dirichlet boundary conditions on the sidewalls of
the billiard and therefore, throughout the text it is also called the wave
functions $\psi_{N}(x,y)$. It is worth noting that the full electric field
$E_{N,p}({\bf x})$ satisfies Neumann boundary conditions at the top and the
bottom of the cavity.
The measurements of $E_{N,p}({\bf x})$ of a 3D microwave cavity allowed us to
test experimentally an important finding of the papers by Blum et al. Blum2002
and Bogomolny and Schmit Bogomolny2002 which connects the number of nodal
domains of 2D billiards with the level number $N$. We will show that for the
3D cavities with the translational symmetry the number of nodal domains
$\aleph_{N_{\bot}}$ lying on the cross-sectional planes of the cavity is
connected with the two-dimensional level number $N_{\bot}$. The condition
$E_{N,p}({\bf x})|_{z=const}=0$ on the cross-sectional planes of the cavity
determines a set of nodal lines which separate regions (nodal domains) with
opposite signs of the electric field distribution $E_{N,p}({\bf
x})|_{z=const}$.
The value of the level number $N$ of the 3D cavity was evaluated from the
Balian–Bloch formula Balian .
$N(k)=\frac{1}{3\pi^{2}}Vk^{3}-\frac{2}{3\pi^{2}}\int_{S}\frac{d\sigma_{\omega}}{R_{\omega}}k,$
$None$
where k is the wave vector, $V=(9.43\pm 0.01)\cdot 10^{-4}$ m3 is the volume
of the cavity and $\int_{S}\frac{d\sigma_{\omega}}{R_{\omega}}=0.932$ m $\pm
0.005$ m is the surface curvature averaged over the surface of the cavity. We
used this formula because of the relatively low quality factor of the cavity
($Q\simeq 4000$) some resonances overlapped.
The two-dimensional level number $N_{\bot}$ is defined by the standard
Weyl–Bloch formula $N_{\bot}=\frac{A}{4\pi}k_{N,p}^{2}-\frac{P}{4\pi}k_{N,p}$,
where $A=(1.572\pm 0.002)\cdot 10^{-2}$ m2 and $P=0.537$ m $\pm 0.001$ m are
the cross-sectional plane area of the cavity and its perimeter, respectively.
The cavity sidewalls consist of two segments (see Fig. 1). The rough segment 1
is described on the cross-sectional planes by the radius function
$R(\theta)=R_{0}+\sum_{m=2}^{M}{a_{m}\sin(m\theta+\phi_{m})}$, where the mean
radius $R_{0}$=10.0 cm, $M=20$, $a_{m}$ and $\phi_{m}$ are uniformly
distributed on [0.084,0.091] cm and [0,2$\pi$], respectively, and
$0\leq\theta<{\pi}$. (For convenience, the polar coordinates $r$ and $\theta$
are used instead of the Cartesian ones $x$ and $y$.) It is worth noting that
following our earlier experience Hlushchuk01b ; Hlushchuk01 we decided to use
a rough desymmetrized half-circular cavity instead of a rough circular cavity,
because the first one lowers the number of nearly degenerated
eigenfrequencies. Additionally, a half-circular geometry of the cavity was
suitable in the procedure of accurate measurements of the electric field
distributions inside the billiard.
The roughness of a billiard on the cross-sectional planes can be characterized
by the function $k(\theta)=(dR/d\theta)/R_{0}$. For our microwave billiard we
have the angle average
$\tilde{k}=(\left<k^{2}(\theta)\right>_{\theta})^{1/2}\simeq 0.400$. The value
of $\tilde{k}$ is much above the chaos border $k_{c}=M^{-5/2}=0.00056$ Frahm97
which indicates that in such a billiard the classical dynamics is diffusive in
orbital momentum due to collisions with the rough boundary.
The other properties of the billiard Frahm are also determined by the
roughness parameter $\tilde{k}$. The amplitudes $\psi_{N,p}(r,\theta)$ are
localized for the two-dimensional level number
$N_{\bot}<N_{e}=1/128\tilde{k}^{4}$. Because of a large value of the roughness
parameter $\tilde{k}$ the localization border lies very low, $N_{e}\simeq 1$.
The border of Breit-Wigner regime is $N_{W}=M^{2}/48\tilde{k}^{2}\simeq 52$.
It means that between $N_{e}<N_{\bot}<N_{W}$ Wigner ergodicity Frahm ought to
be observed and for $N_{\bot}>N_{W}$ Shnirelman ergodicity should emerge.
In order to measure the amplitudes $\psi_{N,p}(r,\theta)$ of the 3D electric
field distributions we used an effective method described in Savytskyy2003 .
It is based on the perturbation technique and preparation of the “trial
functions”. In this method the amplitudes $\psi_{N}(r,\theta)$ (electric field
distribution $E_{N}(r,\theta)$ inside the cavity) are determined from the form
of electric field $E_{N,p}(R_{c},\theta)$ evaluated on a half-circle of fixed
radius $R_{c}$ (see Fig. 1). The first step in evaluation of
$E_{N,p}(R_{c},\theta)$ is measurement of $|E_{N,p}(R_{c},\theta)|^{2}$. The
perturbation technique developed in Slater52 and used successfully in
Slater52 ; Sridhar91 ; Richter00 ; Anlage98 was implemented for this purpose.
In this method a small perturber is introduced inside the cavity to alter its
resonant frequency.
The perturber (4.0 mm in length and 0.3 mm in diameter, oriented in
$z$-direction) was moved by the stepper motor via the Kevlar line hidden in
the groove (0.4 mm wide, 1.0 mm deep) made in the cavity’s bottom wall along
the half-circle $R_{c}$. Before closing the cavity we carefully inspected
whether the pin moves smoothly, oriented in vertical position. Using such a
perturber we had no positive frequency shifts that would exceed the
uncertainty of frequency shift measurements (15 kHz).
In order to determine the dependence of the electric field distributions
$E_{N,p}({\bf x})$ on the $z$ coordinate and to estimate the wave vector
$k_{3}=p\pi/h$ we measured the electric field inside the 3D cavity along the
$z$-axis. The perturber (4.5 mm in length and 0.3 mm in diameter) was attached
to the Kevlar line and moved by the stepper motor. It entered and exited the
cavity by small holes (0.4 mm) drilled in the upper and the bottom walls of
the cavity. The both holes were located at the position: $r=9.11$ cm,
$\theta=0.47$ radians.
To eliminate the variation of resonant frequencies connected with the thermal
expansion of the aluminium cavity the temperature of the cavity was stabilized
with the accuracy of 0.05 $\deg$.
Using a field perturbation technique we were able to measure squared wave
functions $|\psi_{N,p}(R_{c},\theta)|^{2}$ for 80 TM modes within the region
$2\leq N\leq 489$. The range of corresponding eigenfrequencies was from
$\nu_{2}\simeq 2.47$ GHz to $\nu_{489}\simeq 11.99$ GHz. The measurements were
performed at 0.36 mm steps along a half-circle with fixed radius $R_{c}=9.25$
cm. This step was small enough to reveal in details the space structure of
high-lying levels.
Figure 2: Panel (a): Squared wave function $|\psi_{430,2}(R_{c},\theta)|^{2}$
(in arbitrary units) measured on a half-circle with radius $R_{c}=9.25$ cm
($\nu_{430}\simeq 11.50$ GHz). Panel (b): Squared $z$-component of the
electric field distribution $|f_{2}(z)|^{2}$ measured at $r=9.11$ cm and
$\theta=0.47$ radians.
In Fig. 2 (a) and Fig. 2 (b) we show the examples of the squared amplitude
$|\psi_{N,p}(R_{c},\theta)|^{2}$ and the squared $z$-component of the field,
respectively, evaluated for the level number $N=430$.
The perturbation method used in our measurements allows us to extract
information about the modulus of the wave function amplitude
$|\psi_{N,p}(R_{c},\theta)|$ at any given point of the cross-sectional plane
$z=0$ but it doesn’t allow to determine the sign of
$\psi_{N,p}(R_{c},\theta)$. In order to obtain information about the sign of
$\psi_{N,p}(R_{c},\theta)$ we used the method of the “trial wave function”
precisely described in Savytskyy2003 ; Savytskyy2004 ; Hul2005 .
The amplitudes $\psi_{N,p}(r,\theta)$ of the electric field distributions of a
rough half-circular 3D billiard may be expanded in terms of circular waves
(here only odd states in expansion are considered)
$\psi_{N,p}(r,\theta)=\sum_{s=1}^{L}a_{s}J_{s}(k_{N,p}r)\sin(s\theta),$ $None$
where $J_{s}$ is the Bessel function of order $s$.
In Eq. (5) the number of basis functions is limited to
$L=k_{N,p}r_{max}=l_{N}^{max}$, where $r_{max}=10.64$ cm is the maximum radius
of the cavity. $l_{N}^{max}=k_{N,p}r_{max}$ is a semiclassical estimate for
the maximum possible angular momentum for a given $k_{N}$. Circular waves with
angular momentum $s>L$ correspond to evanescent waves and can be neglected.
Coefficients $a_{s}$ may be extracted from the “trial wave function”
$\psi_{N,p}(R_{c},\theta)$ via
$a_{s}=[\frac{\pi}{2}J_{s}(k_{N,p}R_{c})]^{-1}\int_{0}^{\pi}\psi_{N,p}(R_{c},\theta)\sin(s\theta)d\theta.$
$None$
Due to experimental uncertainties and the finite step size in the measurements
of $|\psi_{N,p}(R_{c},\theta)|^{2}$ the wave functions $\psi_{N,p}(r,\theta)$
are not exactly zero at the boundary $\Gamma$. As the quantitative measure of
the sign assignment quality we chose the integral
$I=\gamma\int_{\Gamma}|\psi_{N,p}(r,\theta)|^{2}dl$ calculated along the
billiard’s rough boundary $\Gamma$, where $\gamma$ is length of $\Gamma$. For
correctly reconstructed wave functions the integral $I$ was several times
smaller than in the case of not correctly reconstructed ones.
It is worth noting that since the pin is attached to the line it cannot be
stuck. However, one may assume that during the movement the pin may be
accidentally, from time to time, slightly slanted, adding small ”a noise-like
component” to the measured electric field. The formula (5) shows that each
wave function is expanded in terms of $L$ circular waves which filters out
noise-like higher frequency Fourier components from the reconstructed wave
function. The same filtering removes out the influence of the experimental
uncertainties of frequency shifts on the reconstructed wave functions.
Figure 3: The “trial wave function” $\psi_{430,2}(R_{c},\theta)$ (in arbitrary
units) with the correctly assigned signs, which was used in the reconstruction
of the wave function $E_{430,2}(r,\theta,z)$ of the billiard (see Fig. 4).
In Fig. 3 we show the “trial wave function” $\psi_{430,2}(R_{c},\theta)$ with
the correctly assigned signs, which was used in the reconstruction of the wave
function $\psi_{430,2}(r,\theta)$ of the billiard (see Fig. 4). In Fig. 4
different nodal domains are separated by the bold full lines.
Figure 4: The reconstructed wave function $\psi_{430,2}(r,\theta)$ of the
chaotic half-circular microwave rough billiard. The amplitudes have been
converted into a grey scale with white corresponding to large positive and
black corresponding to large negative values, respectively. The structure of
the nodal lines are shown by the bold full lines. Dimensions of the billiard
are given in cm. In the figure the $z$ dependence of the electric field
distribution
$E_{430,2}(r,\theta,z)|_{x=0}\propto\psi_{430,2}(r,\theta)|_{x=0}f_{2}(z)$ is
also shown.
Using the method of the “trial wave function” we were able to reconstruct 75
experimental wave functions $\psi_{N,p}(r,\theta)$, which belonged to TM modes
of the rough half-circular 3D billiard with the level number $N$ between 2 and
489. The remaining wave functions belonging to TM modes, from the range
$N=2-489$, were not reconstructed because of near-degeneration of the
neighboring eigenfrequencies or due to the problems with the measurements of
$|\psi_{N,p}(R_{c},\theta)|^{2}$ along a half-circle coinciding for its
significant part with one of the nodal lines of $\psi_{N,p}(r,\theta)$.
Figure 5: Structure of the energy surface of the wave functions lying close to
the boarder of the regimes of Breit-Wigner and Shnirelman ergodicity
($N_{W}=52$), panels (a) and (b), and for the low wave function in the regime
of Breit-Wigner ergodicity, panel (c). Panel (a): The moduli of amplitudes
$|C^{(460,0)}_{nl}|$ for the wave function $\psi_{460,0}(r,\theta)$,
$N_{\bot}=65$, lying in the regime of Shnirelman ergodicity. Panel (b): The
moduli of amplitudes $|C^{(430,2)}_{nl}|$ for the wave function
$\psi_{430,2}(r,\theta)$, $N_{\bot}=50$, in the regime of Breit-Wigner
ergodicity . Panel (c): The moduli of amplitudes $|C^{(463,4)}_{nl}|$ for the
wave function $\psi_{463,4}(r,\theta)$ lying in the regime of Breit-Wigner
ergodicity close to the localization boarder. Full lines show the
semiclassical estimation of the energy surface (see text).
The borders of Breit-Wigner and Shnirelman ergodicities are not sharp.
Therefore, to check ergodicity of the billiard’s wave functions
$\psi_{N,p}(r,\theta)$, especially close to the borders, one should use some
additional measures such as e.g., calculation of the structures of their
energy surfaces Frahm97 . For this reason we extracted wave function
amplitudes $C^{(N,p)}_{nl}=\left<n,l|N,p\right>$ in the basis $n,l$ of a half-
circular billiard with radius $r_{max}$, where $n=1,2,3\ldots$ enumerates the
zeros of the Bessel functions and $l=1,2,3\ldots$ is the angular quantum
number. The moduli of amplitudes $|C^{(N,p)}_{nl}|$ and their projections into
the energy surface for the experimental wave functions
$\psi_{460,0}(r,\theta)$, $\psi_{430,2}(r,\theta)$ and
$\psi_{463,4}(r,\theta)$ are shown in Fig. 5(a-c). As expected, on the border
of the regimes of Breit-Wigner and Shnirelman ergodicity the wave functions
$\psi_{460,0}(r,\theta)$ ($N_{\bot}=65$) and $\psi_{430,2}(r,\theta)$
($N_{\bot}=50$) are extended homogeneously over the whole energy surface
Hlushchuk01 . The wave function $\psi_{463,4}(r,\theta)$, $N_{\bot}=16$, which
lies closer to the localization boarder, is also extended along the energy
surface, however it displays the tendency to localization in $n,l$ basis (see
Fig. 5(c)). The full lines on the projection planes in Fig. 5(a-c) mark the
energy surface of a half-circular billiard $H(n,l)=k^{2}_{N,p}$ estimated from
the semiclassical formula Hlushchuk01b :
$\sqrt{(l^{max}_{N})^{2}-l^{2}}-l\arctan(l^{-1}\sqrt{(l^{max}_{N})^{2}-l^{2}})+\pi/4=\pi
n$. It is clearly visible that the peaks $|C^{(N,p)}_{nl}|$ are spread almost
perfectly along the lines marking the energy surface.
Figure 6: The number of nodal domains $\aleph_{N_{\bot}}$ (full circles) on
the cross-section planes of the chaotic half-circular 3D microwave rough
billiard. Full line shows the least squares fit
$\aleph_{N_{\bot}}=a_{1}N_{\bot}+b_{1}\sqrt{N_{\bot}}$ to the experimental
data (see text), where $a_{1}=0.059\pm 0.029$, $b_{1}=0.991\pm 0.190$. The
prediction of the theory of Bogomolny and Schmit Bogomolny2002 $a_{1}=0.062$.
The number of nodal domains $\aleph_{N_{\bot}}$ on the cross-sectional plane
$z=0$ vs. the level number $N_{\bot}$ in the chaotic 3D microwave rough
billiard is plotted in Fig. 6. The full line in Fig. 6 shows the least squares
fit $\aleph_{N_{\bot}}=a_{1}N_{\bot}+b_{1}\sqrt{N_{\bot}}$ of the experimental
data, where $a_{1}=0.059\pm 0.029$, $b_{1}=0.991\pm 0.190$. The coefficient
$a_{1}=0.059\pm 0.029$ coincides with the prediction of the percolation model
of Bogomolny and Schmit Bogomolny2002 $\aleph_{N_{\bot}}/N_{\bot}\simeq
0.062$ within the error limits. The relatively large uncertainty of the
coefficient $a_{1}$ is connected with the fact that in the least squares fit
procedure we used only 27 higher states with $N_{\bot}>20$. The states with
lower $N_{\bot}$ were not taken into account because they were not fully
chaotic (see Fig. 5(c)). The second term in the least squares fit corresponds
to a contribution of boundary domains, i.e. domains, which include the
billiard boundary. Numerical calculations of Blum et al. Blum2002 performed
for the Sinai and stadium billiards showed that the number of boundary domains
scales as the number of the boundary intersections, that is as
$\sqrt{N_{\bot}}$. Our results clearly suggest that in the rough billiard, at
the level numbers $20<N_{\bot}\leq 65$, the boundary domains also
significantly influence the scaling of the number of nodal domains
$\aleph_{N_{\bot}}$, leading to the departure from the predicted scaling
$\aleph_{N_{\bot}}\sim N_{\bot}$.
In summary, we measured the wave functions of the chaotic 3D rough microwave
billiard with the translational symmetry up to the level number $N=489$. We
showed that for the two-dimensional level numbers $20<N_{\bot}\leq 65$ the
scaling of the number of nodal domains $\aleph_{N_{\bot}}$ significantly
departures from the predicted scaling $\aleph_{N_{\bot}}\sim N_{\bot}$, which
suggests that the boundary domains influence the scaling Bogomolny2002 . In
the limit $N_{\bot}\rightarrow\infty$ the least squares fit of the
experimental data yields the asymptotic number of nodal domains
$\aleph_{N_{\bot}}/N_{\bot}\simeq a_{1}=0.059\pm 0.029$ that is close to the
theoretical prediction $\aleph_{N_{\bot}}/N_{\bot}\simeq 0.062$. Finally, our
results show that 3D microwave cavities with the translational symmetry can be
effectively used beyond the standard 2D frequency limit in simulation of
quantum systems.
Acknowledgments. This work was partially supported by the Ministry of
Education and Science grant No. N202 099 31/0746.
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|
arxiv-papers
| 2009-03-17T11:02:14 |
2024-09-04T02:49:01.210948
|
{
"license": "Public Domain",
"authors": "Nazar Savytskyy, Oleg Tymoshchuk, Oleh Hul, Szymon Bauch and Leszek\n Sirko",
"submitter": "Oleh Hul",
"url": "https://arxiv.org/abs/0903.2939"
}
|
0903.3024
|
# A Vector Generalization of Costa’s Entropy-Power
Inequality with Applications
Ruoheng Liu, Tie Liu, H. Vincent Poor, and Shlomo Shamai (Shitz) This research
was supported by the United States National Science Foundation under Grants
CNS-06-25637 and CCF-07-28208, the European Commission in the framework of the
FP7 Network of Excellence in Wireless Communications NEWCOM++, and the Israel
Science Foundation. The material in this paper was presented in part at the
New Result Session of the 2008 IEEE International Symposium on Information
Theory, Toronto, Ontario, Canada, July 2008.Ruoheng Liu and H. Vincent Poor
are with the Department of Electrical Engineering, Princeton University,
Princeton, NJ 08544, USA. Email: {rliu,poor}@princeton.eduTie Liu is with the
Department of Electrical and Computer Engineering, Texas A&M University,
College Station, TX 77843, USA. Email: tieliu@tamu.eduShlomo Shamai (Shitz) is
with the Department of Electrical Engineering, Technion-Israel Institute of
Technology, Technion City, Haifa 32000, Israel. Email:
sshlomo@ee.technion.ac.il
###### Abstract
This paper considers an entropy-power inequality (EPI) of Costa and presents a
natural vector generalization with a real positive semidefinite matrix
parameter. This new inequality is proved using a perturbation approach via a
fundamental relationship between the derivative of mutual information and the
minimum mean-square error (MMSE) estimate in linear vector Gaussian channels.
As an application, a new extremal entropy inequality is derived from the
generalized Costa EPI and then used to establish the secrecy capacity regions
of the degraded vector Gaussian broadcast channel with layered confidential
messages.
###### Index Terms:
Entropy-power inequality (EPI), extremal entropy inequality, information-
theoretic security, mutual information and minimum mean-square error (MMSE)
estimate, vector Gaussian broadcast channel
## I Introduction
In information theory, the entropy-power inequality (EPI) of Shannon [1] and
Stam [2] has played key roles in the solution of several canonical network
communication problems. Celebrated examples include Bergmans’s solution [3] to
the Gaussian broadcast channel problem, Leung-Yan-Cheong and Hellman’s
solution [4] to the Gaussian wire-tap channel problem, Ozarow’s solution [5]
to the Gaussian two-description problem, Oohama’s solution [6] to the
quadratic Gaussian CEO problem, and more recently Weingarten, Steinberg and
Shamai’s solution [7] to the multiple-input multiple-output Gaussian broadcast
channel problem.
Let $\mathbf{X}$ and $\mathbf{Z}$ be two independent random $n$-vectors with
densities in ${\mathbb{R}}^{n}$, where ${\mathbb{R}}$ denotes the set of real
numbers. The classical EPI of Shannon [1] and Stam [2] can be written as
$\displaystyle\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{Z})\right]\geq\exp\left[\frac{2}{n}h(\mathbf{X})\right]+\exp\left[\frac{2}{n}h(\mathbf{Z})\right]$
(1)
where $h(\mathbf{X})$ denotes the differential entropy of $\mathbf{X}$. The
equality holds if and only if $\mathbf{X}$ and $\mathbf{Z}$ are Gaussian and
with proportional covariance matrices.
In network information theory, most applications focus on the special case of
(1) where one of the random vectors is fixed to be Gaussian. In this setting,
the classical EPI of Shannon and Stam can be further strengthened as shown by
Costa [8]. Let $\mathbf{Z}$ be a Gaussian random $n$-vector with a positive
definite covariance matrix, and let $a$ be a real scalar such that
$a\in[0,1]$. Costa’s EPI [8] can be written as
$\displaystyle\exp\left[\frac{2}{n}h(\mathbf{X}+\sqrt{a}\mathbf{Z})\right]$
$\displaystyle\geq(1-a)\exp\left[\frac{2}{n}h(\mathbf{X})\right]+a\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{Z})\right]$
(2)
for any random $n$-vector $\mathbf{X}$ independent of $\mathbf{Z}$. The
equality holds if and only if $\mathbf{X}$ is also Gaussian and with a
covariance matrix proportional to that of $\mathbf{Z}$’s.
Though not as widely known as the classical EPI of Shannon and Stam, Costa’s
EPI has found useful applications in deriving capacity bounds for the Gaussian
interference channel [9] and the multiantenna flat-fading channel [10]. The
original proof of Costa’s EPI provided in [8] was based on rather detailed
calculations. Simplified proofs based on a Fisher information inequality [11]
and a fundamental relationship between the derivative of mutual information
and minimum mean-square error (MMSE) in linear Gaussian channels [12] can be
found in [13] and [14], respectively.
Note that Costa’s EPI (2) provides a strong relationship among the
differential entropies of three random vectors: $\mathbf{X}$,
$\mathbf{X}+\sqrt{a}\mathbf{Z}$ and $\mathbf{X}+\mathbf{Z}$. To apply, the
increments of $\mathbf{X}+\sqrt{a}\mathbf{Z}$ and $\mathbf{X}+\mathbf{Z}$ over
$\mathbf{X}$ need to be Gaussian and have _proportional_ covariance matrices.
For some applications in network information theory (as we will see shortly),
the proportionality requirement may turn out to be overly restrictive. A main
contribution of this paper is to prove a natural generalization of Costa’s EPI
(2) by replacing the real scalar $a$ with a positive semidefinite _matrix_
parameter. The result is summarized in the following theorem.
###### Theorem 1 (Generalized Costa’s EPI)
Let $\mathbf{Z}$ be a Gaussian random $n$-vector with a positive definite
covariance matrix $\mathbf{N}$, and let $\mathbf{A}$ be an $n\times n$ real
symmetric matrix such that $0\preceq\mathbf{A}\preceq\mathbf{I}$. Here,
$\mathbf{I}$ denotes the $n\times n$ identity matrix, and “$\preceq$” denotes
“less or equal to” in the positive semidefinite partial ordering between real
symmetric matrices. Then,
$\displaystyle\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z})\right]$
$\displaystyle\geq|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X})\right]+|\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{Z})\right]$
(3)
for any random $n$-vector $\mathbf{X}$ independent of $\mathbf{Z}$. The
equality holds if $\mathbf{Z}$ is Gaussian and with a covariance matrix
$\mathbf{B}$ such that $\mathbf{B}-\mathbf{A}\mathbf{B}$ and
$\mathbf{B}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}$ are
proportional.
Note that when $\mathbf{A}=a\mathbf{I}$, the generalized Costa EPI (3) reduces
to the original Costa EPI (2). On the other hand, when $\mathbf{A}$ is not a
scaled identity, the covariance matrices of increments of
$\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z}$ and $\mathbf{X}+\mathbf{Z}$
over $\mathbf{X}$ do not need to be proportional. As we will see, the ability
to cope with a _general_ matrix parameter makes the generalized Costa EPI more
flexible and powerful than the original Costa EPI.
A different but related generalization of Costa’s EPI was considered by Payaró
and Palomar [15], where they examined the concavity of the entropy-power
$\exp\left[\frac{2}{n}h(\mathbf{A}^{\frac{1}{2}}\mathbf{X}+\mathbf{Z})\right]$
with respect to the matrix parameter $\mathbf{A}$. This line of research was
motivated by the observation that the original Costa EPI (2) is equivalent to
the concavity of the entropy power
$\exp\left[\frac{2}{n}h(\sqrt{a}\mathbf{X}+\mathbf{Z})\right]$ with respect to
the scalar parameter $a$. Unlike the scalar case, Payaró and Palomar [15]
showed that the entropy-power
$\exp\left[\frac{2}{n}h(\mathbf{A}^{\frac{1}{2}}\mathbf{X}+\mathbf{Z})\right]$
is in general _not_ concave with respect to the matrix parameter $\mathbf{A}$.
However, the concavity does hold when $\mathbf{A}$ is restricted to be
_diagonal_ [15].
In information theory, a main application of the EPI is to derive extremal
entropy inequalities, which can then be used to solve network communication
problems. In their work [16], Liu and Viswanath derived an extremal entropy
inequality based on the classical EPI of Shannon [1] and Stam [2] and used it
to establish the private message capacity region of the vector Gaussian
broadcast channel via the Marton outer bound [17, Theorem 5]. In this paper,
we will derive a new extremal entropy inequality based on the generalized
Costa EPI and use it to characterize the secrecy capacity regions of the
degraded vector Gaussian broadcast channel with layered confidential messages.
The rest of the paper is organized as follows. In Section II, we summarize the
main results of the paper, including a new extremal entropy inequality and its
applications on the degraded vector Gaussian broadcast channel with layered
confidential messages. In Section III, we prove the generalized Costa EPI,
following a perturbation approach via a fundamental relationship between the
derivative of mutual information and MMSE estimate in linear vector Gaussian
channels [18, Theorem 2]. In Section IV, we derive the new extremal entropy
inequality from the generalized Costa EPI. The coding theorems for the
degraded vector Gaussian broadcast channel with layered confidential messages
are proved in Section V and Section VI. Finally, in Section VII, we conclude
the paper with some remarks.
## II Summary of Main Results
The following notation will be used throughout the paper. A random vector is
denoted with an upper-case letter (e.g., $\mathbf{X}$), its realization is
denoted with the corresponding lower-case letter (e.g., $\mathbf{x}$), and its
probability density function is denoted with
$p(\mathbf{x})=p_{\mathbf{X}}(\mathbf{x})$. We use ${\sf E}[\mathbf{X}]$ to
denote the expectation of $\mathbf{X}$. Thus, the covariance matrix of
$\mathbf{X}$ is given by
$\displaystyle{\sf Cov}(\mathbf{X})={\sf E}\left[(\mathbf{X}-{\sf
E}[\mathbf{X}])(\mathbf{X}-{\sf E}[\mathbf{X}])^{\textsf{T}}\right].$
Given any jointly distributed random vectors $(\mathbf{X},\mathbf{Y})$, the
MMSE estimate of $\mathbf{X}$ from the observation $\mathbf{Y}$ is the
conditional mean ${\sf E}[\mathbf{X}|\mathbf{Y}]$. The MMSE (matrix) is given
by:
$\displaystyle{\sf Cov}(\mathbf{X}|\mathbf{Y})={\sf E}\left[(\mathbf{X}-{\sf
E}[\mathbf{X}|\mathbf{Y}])(\mathbf{X}-{\sf
E}[\mathbf{X}|\mathbf{Y}])^{\textsf{T}}\right].$
### II-A A New Extremal Entropy Inequality
The following extremal entropy inequality is a consequence of the generalized
Costa EPI.
###### Theorem 2
Let $\mathbf{Z}_{k}$, $k=0,\ldots,K$, be a total of $K+1$ Gaussian random
$n$-vectors with positive definite covariance matrices $\mathbf{N}_{k}$,
respectively. Assume that $\mathbf{N}_{1}\preceq\ldots\preceq\mathbf{N}_{K}$.
If there exists an $n\times n$ positive semidefinite matrix $\mathbf{B}^{*}$
such that
$\displaystyle\sum_{k=1}^{K}\mu_{k}(\mathbf{B}^{*}+\mathbf{N}_{k})^{-1}+\mathbf{M}_{1}=(\mathbf{B}^{*}+\mathbf{N}_{0})^{-1}+\mathbf{M}_{2}$
(4)
for some $n\times n$ positive semidefinite matrices $\mathbf{M}_{1}$,
$\mathbf{M}_{2}$ and $\mathbf{S}$ with
$\displaystyle\mathbf{B}^{*}\mathbf{M}_{1}$ $\displaystyle=0$ (5)
$\displaystyle\mbox{and}\quad\quad(\mathbf{S}-\mathbf{B}^{*})\mathbf{M}_{2}$
$\displaystyle=0$ (6)
and real scalars $\mu_{k}\geq 0$ with $\sum_{k=1}^{K}\mu_{k}=1$, then
$\displaystyle\sum_{k=1}^{K}\mu_{k}h(\mathbf{X}+\mathbf{Z}_{k}|U)-h(\mathbf{X}+\mathbf{Z}_{0}|U)$
$\displaystyle\leq\sum_{k=1}^{K}\frac{\mu_{k}}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{k}|-\frac{1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{0}|$
(7)
for any $(\mathbf{X},U)$ independent of
$(\mathbf{Z}_{0},\ldots,\mathbf{Z}_{K})$ such that ${\sf
E}[\mathbf{X}\mathbf{X}^{\textsf{T}}]\preceq\mathbf{S}$.
Note that (4)–(6) are precisely the Karush-Kuhn-Tucker (KKT) conditions (see
[7, Appendix IV] and [19, Section 5.2]) for the optimization program:
$\displaystyle\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left[\sum_{k=1}^{K}\frac{\mu_{k}}{2}\log|\mathbf{B}+\mathbf{N}_{k}|-\frac{1}{2}\log|\mathbf{B}+\mathbf{N}_{0}|\right].$
Therefore, (7) implies that a jointly _Gaussian_ $(U,\mathbf{X})$ such that
for each $U=u$, $\mathbf{X}$ has the _same_ covariance matrix is an optimal
solution to the optimization program:
$\displaystyle\max_{(U,\mathbf{X})}\left[\sum_{k=1}^{K}\mu_{k}h(\mathbf{X}+\mathbf{Z}_{k}|U)-h(\mathbf{X}+\mathbf{Z}_{0}|U)\right]$
where the maximization is over all $(U,\mathbf{X})$ independent of
$(\mathbf{Z}_{0},\ldots,\mathbf{Z}_{K})$ such that ${\sf
E}[\mathbf{X}\mathbf{X}^{\textsf{T}}]\preceq\mathbf{S}$. Note that when $K=1$,
this is a special case of [16, Theorem 8] with $\mu=1$.
### II-B Applications on the Degraded Vector Gaussian Broadcast Channel with
Layered Confidential Messages
(a) Communication scenario 1
(b) Communication scenario 2
Figure 1: Degraded vector Gaussian broadcast channel with layered confidential
messages
Consider the following vector Gaussian broadcast channel with three receivers:
$\displaystyle\mathbf{Y}_{k}[t]=\mathbf{X}[t]+\mathbf{Z}_{k}[t],\quad k=1,2,3$
(8)
where $\\{\mathbf{Z}_{k}[t]\\}_{t}$, $k=1,2,3$, are independent and
identically distributed additive vector Gaussian noise processes with zero
means and positive definite covariance matrices $\mathbf{N}_{k}$,
respectively. The channel input $\\{\mathbf{X}[t]\\}_{t}$ is subject to a
matrix constraint:
$\frac{1}{n}\sum_{t=1}^{n}\mathbf{X}[t]\mathbf{X}^{\textsf{T}}[t]\preceq\mathbf{S}$
(9)
where $\mathbf{S}$ is a positive semidefinite matrix, and $n$ is the block
length. We assume that the noise covariance matrices are ordered as
$\mathbf{N}_{1}\preceq\mathbf{N}_{2}\preceq\mathbf{N}_{3},$ (10)
i.e., the received signal $\mathbf{Y}_{3}[t]$ is (stochastically) degraded
with respect to $\mathbf{Y}_{2}[t]$, which is further degraded with respect to
$\mathbf{Y}_{1}[t]$.
We consider two different communication scenarios, both with two independent
messages $W_{1}$ and $W_{2}$. In the first scenario (see Fig. 1-(a)), message
$W_{1}$ is intended for receiver 1 but needs to be kept secret from receivers
2 and 3, and message $W_{2}$ is intended for receivers 1 and 2 but needs to be
kept confidential from receiver 3. In the second scenario (see Fig. 1-(b)),
message $W_{1}$ is intended for receivers 1 but needs to be kept secret from
receiver receiver 3, and message $W_{2}$ is intended for receivers 1 but needs
to be kept secret from receiver 3. The confidentiality of the messages at the
unintended receivers is measured using the normalized information-theoretic
criteria [20, 21]:
$\displaystyle\frac{1}{n}I(W_{1};\mathbf{Y}_{2}^{n})\rightarrow
0,\quad\frac{1}{n}I(W_{1};\mathbf{Y}_{3}^{n})\rightarrow
0,\quad\text{and}\quad\frac{1}{n}I(W_{2};\mathbf{Y}_{3}^{n})\rightarrow 0$
(11)
for the first scenario and
$\displaystyle\frac{1}{n}I(W_{1};\mathbf{Y}_{3}^{n})\rightarrow
0,\quad\text{and}\quad\frac{1}{n}I(W_{2};\mathbf{Y}_{3}^{n})\rightarrow 0$
(12)
for the second scenario. Here, the limits are taken as the block length
$n\rightarrow\infty$. The goal is to characterize the entire secrecy rate
region $\mathcal{C}_{s}=\\{(R_{1},R_{2})\\}$ that can be achieved by any
coding scheme.
To characterize the secrecy capacity regions, we will first consider the
discrete memoryless version of the problem with transition probability
$p(y_{1},y_{2},y_{3}|x)$ and degradedness order
$X\rightarrow Y_{1}\rightarrow Y_{2}\rightarrow Y_{3}.$ (13)
We have the following single-letter characterizations of the secrecy capacity
regions.
###### Theorem 3
The secrecy capacity region of the discrete memoryless broadcast channel
$p(y_{1},y_{2},y_{3}|x)$ with confidential messages $W_{1}$ (intended for
receiver 1 but needs to be kept secret from receivers 2 and 3) and $W_{2}$
(intended for receivers 1 and 2 but needs to be kept secret from receiver 3)
under the degradedness order (13) is given by the set of nonnegative rate
pairs $(R_{1},R_{2})$ such that
$\displaystyle R_{1}\leq$ $\displaystyle\;I(X;Y_{1}|U)-I(X;Y_{2}|U)$ and
$\displaystyle R_{2}\leq$ $\displaystyle\;I(U;Y_{2})-I(U;Y_{3})$ (14)
for some jointly distributed $(U,X)$ satisfying the Markov relation
$U\rightarrow X\rightarrow(Y_{1},Y_{2},Y_{3}).$
###### Theorem 4 ([22, Theorem 2])
The secrecy capacity region of the discrete memoryless broadcast channel
$p(y_{1},y_{2},y_{3}|x)$ with confidential messages $W_{1}$ (intended for
receiver 1 but needs to be kept secret from receiver 3) and $W_{2}$ (intended
for receivers 1 and 2 but needs to be kept secret from receiver 3) under the
degradedness order (13) is given by the set of nonnegative rate pairs
$(R_{1},R_{2})$ such that
$\displaystyle R_{1}\leq$ $\displaystyle\;I(X;Y_{1}|U)-I(X;Y_{3}|U)$ and
$\displaystyle R_{2}\leq$ $\displaystyle\;I(U;Y_{2})-I(U;Y_{3})$ (15)
for some jointly distributed $(U,X)$ satisfying the Markov relation
$U\rightarrow X\rightarrow(Y_{1},Y_{2},Y_{3}).$
A proof of Theorem 4 can be found in [22]. Theorem 3 can be proved in a
similar fashion; for completeness, a proof is included in Appendix A. For the
vector Gaussian broadcast channel (8) under the degradedness order (10), the
single-letter expressions (14) and (15) can be further evaluated using the
extremal entropy inequality (7). The results are summarized in the following
theorems.
###### Theorem 5
The secrecy capacity region of the vector Gaussian broadcast channel (8) with
confidential messages $W_{1}$ (intended for receiver 1 but needs to be kept
secret from receivers 2 and 3) and $W_{2}$ (intended for receivers 1 and 2 but
needs to be kept secret from receiver 3) and degradedness order (10) under the
matrix constraint (9) is given by the set of nonnegative secrecy rate pairs
$(R_{1},R_{2})$ such that
$\displaystyle R_{1}$
$\displaystyle\leq\;\frac{1}{2}\log\left|\frac{\mathbf{B}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|$
and $\displaystyle R_{2}$
$\displaystyle\leq\;\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}+\mathbf{N}_{3}}\right|$
(16)
for some $0\preceq\mathbf{B}\preceq\mathbf{S}$.
###### Theorem 6
The secrecy capacity region of the vector Gaussian broadcast channel (8) with
confidential messages $W_{1}$ (intended for receiver 1 but needs to be kept
secret from receiver 3) and $W_{2}$ (intended for receivers 1 and 2 but needs
to be kept secret from receiver 3) and degradedness order (10) under the
matrix constraint (9) is given by the set of nonnegative secrecy rate pairs
$(R_{1},R_{2})$ such that
$\displaystyle R_{1}$
$\displaystyle\leq\;\frac{1}{2}\log\left|\frac{\mathbf{B}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}+\mathbf{N}_{3}}{\mathbf{N}_{3}}\right|$
and $\displaystyle R_{2}$
$\displaystyle\leq\;\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}+\mathbf{N}_{3}}\right|$
(17)
for some $0\preceq\mathbf{B}\preceq\mathbf{S}$.
## III Proof of Theorem 1
In this section, we prove the generalized Costa EPI (3) as stated in Theorem
1. We first examine the equality condition. Note that when $\mathbf{X}$ is
Gaussian, the generalized Costa EPI (3) becomes the matrix inequality:
$\displaystyle|\mathbf{B}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}|^{\frac{1}{n}}$
$\displaystyle\geq|\mathbf{B}-\mathbf{A}\mathbf{B}|^{\frac{1}{n}}+|\mathbf{A}\mathbf{B}+\mathbf{A}\mathbf{N}|^{\frac{1}{n}}.$
Suppose that $\mathbf{B}-\mathbf{A}\mathbf{B}$ and
$\mathbf{B}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}$ are
proportional, i.e., there exists a real scalar $c$ such that
$\displaystyle\mathbf{B}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}=c(\mathbf{B}-\mathbf{A}\mathbf{B}).$
Since both matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric, this implies
that $\mathbf{A}\mathbf{B}$ is also symmetric, i.e.,
$\mathbf{A}\mathbf{B}=\mathbf{B}^{\textsf{T}}\mathbf{A}^{\textsf{T}}=\mathbf{B}\mathbf{A}.$
Therefore, $\mathbf{A}$ and $\mathbf{B}$ must have the _same_ eigenvector
matrix [23] and hence
$\displaystyle\mathbf{A}\mathbf{B}$
$\displaystyle=\mathbf{A}^{\frac{1}{2}}\mathbf{B}\mathbf{A}^{\frac{1}{2}}.$
It follows that
$\displaystyle\mathbf{A}^{\frac{1}{2}}\mathbf{B}\mathbf{A}^{\frac{1}{2}}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}$
$\displaystyle=\mathbf{B}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}-(\mathbf{B}-\mathbf{A}\mathbf{B})$
$\displaystyle=(c-1)(\mathbf{B}-\mathbf{A}\mathbf{B})$
i.e.,
$\mathbf{A}^{\frac{1}{2}}\mathbf{B}\mathbf{A}^{\frac{1}{2}}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}$
and $\mathbf{B}-\mathbf{A}\mathbf{B}$ are proportional. Therefore,
$\displaystyle|\mathbf{B}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}|^{\frac{1}{n}}$
$\displaystyle=|\mathbf{B}-\mathbf{A}\mathbf{B}+(\mathbf{A}^{\frac{1}{2}}\mathbf{B}\mathbf{A}^{\frac{1}{2}}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}})|^{\frac{1}{n}}$
$\displaystyle=|\mathbf{B}-\mathbf{A}\mathbf{B}|^{\frac{1}{n}}+|\mathbf{A}^{\frac{1}{2}}\mathbf{B}\mathbf{A}^{\frac{1}{2}}+\mathbf{A}^{\frac{1}{2}}\mathbf{N}\mathbf{A}^{\frac{1}{2}}|^{\frac{1}{n}}$
$\displaystyle=|\mathbf{B}-\mathbf{A}\mathbf{B}|^{\frac{1}{n}}+|\mathbf{A}\mathbf{B}+\mathbf{A}\mathbf{N}|^{\frac{1}{n}}.$
This proved the desired equality condition.
We now turn to the proof of the inequality. First consider the special case
when $|\mathbf{A}|=0$. Since
$\displaystyle
h(\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z})-h(\mathbf{X})=I(\mathbf{A}^{\frac{1}{2}}\mathbf{Z};\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z})\geq
0,$
we have
$\displaystyle\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z})\right]$
$\displaystyle\geq\exp\left[\frac{2}{n}h(\mathbf{X})\right]$
$\displaystyle\geq|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X})\right]$
where the last inequality follows from the assumption that
$0\preceq\mathbf{A}\preceq\mathbf{I}$ and hence
$0\leq|\mathbf{I}-\mathbf{A}|\leq 1$.
Next, consider the general case when $|\mathbf{A}|>0$. The proof is rather
long so we divide it into several steps.
_Step 1–Constructing a monotone path._ To prove the generalized Costa EPI (3),
we can equivalently show that
$\displaystyle\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{Z})\right]$
$\displaystyle\leq|\mathbf{A}|^{-\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z})\right]-\left(\frac{|\mathbf{I}-\mathbf{A}|}{|\mathbf{A}|}\right)^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X})\right].$
(18)
Since $\mathbf{X}$ and $\mathbf{Z}$ are independent, we have
$\displaystyle h(\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z})-h(\mathbf{X})$
$\displaystyle=h(\mathbf{A}^{-\frac{1}{2}}\mathbf{X}+\mathbf{Z})-h(\mathbf{A}^{-\frac{1}{2}}\mathbf{X})$
$\displaystyle=h(\mathbf{A}^{-\frac{1}{2}}\mathbf{X}+\mathbf{Z})-h(\mathbf{A}^{-\frac{1}{2}}\mathbf{X}|\mathbf{Z})$
$\displaystyle=I(\mathbf{Z};\mathbf{A}^{-\frac{1}{2}}\mathbf{X}+\mathbf{Z})$
(19)
and
$h(\mathbf{X}+\mathbf{Z})-h(\mathbf{X})=I(\mathbf{Z};\mathbf{X}+\mathbf{Z}).$
(20)
Divide both sides of (18) by $\exp\left[\frac{2}{n}h(\mathbf{X})\right]$ and
use (19) and (20). Then, (18) can be equivalently written as
$\displaystyle\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{X}+\mathbf{Z})\right]$
$\displaystyle\leq|\mathbf{A}|^{-\frac{1}{n}}\left\\{\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{A}^{-\frac{1}{2}}\mathbf{X}+\mathbf{Z})\right]-|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\right\\}.$
(21)
Let
$\displaystyle F(\mathbf{D})$
$\displaystyle:=|\mathbf{D}|^{\frac{2}{n}}\Biggl{\\{}\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]-|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}\Biggr{\\}}.$
(22)
With this definition, (21) can be equivalently written as
$\displaystyle F(\mathbf{I})\leq F(\mathbf{A}^{-\frac{1}{2}}).$ (23)
To show the inequality (23), it is sufficient to construct a family of
$n\times n$ positive definite matrices $\\{\mathbf{D}(\gamma)\\}_{\gamma}$
connecting $\mathbf{I}$ and $\mathbf{A}^{-\frac{1}{2}}$ such that
$F(\mathbf{D}(\gamma))$ is monotone along the path. Unlike the scalar case
where there is only one path connecting $1$ to $1/\sqrt{a}$, in the matrix
case there are infinitely many paths connecting $\mathbf{I}$ and
$\mathbf{A}^{-\frac{1}{2}}$. Here, we consider the special choice
$\mathbf{D}(\gamma):=\left[\mathbf{I}+\gamma(\mathbf{A}^{-1}-\mathbf{I})\right]^{\frac{1}{2}}$
(24)
and show that
$\frac{\partial F}{\partial\gamma}\geq 0,\quad\forall\gamma\in[0,1].$ (25)
along this particular path.
_Step 2–Calculating the derivative $\frac{\partial F}{\partial\gamma}$._
Following [14, Theorem 5], we have
$I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})=I(\mathbf{X};\mathbf{D}\mathbf{X}+\mathbf{Z})+h(\mathbf{Z})-h(\mathbf{X})-\log|\mathbf{D}|$
and
${\sf Cov}(\mathbf{X}|\mathbf{D}\mathbf{X}+\mathbf{Z})=\mathbf{D}^{-1}\,{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})\mathbf{D}^{-\textsf{T}}.$
Let $\mathbf{N}:={\sf Cov}(\mathbf{Z})$ and note that $\mathbf{D}$ is
symmetric. We have
$\displaystyle\frac{\partial}{\partial\mathbf{D}}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})$
$\displaystyle=\frac{\partial}{\partial\mathbf{D}}I(\mathbf{X};\mathbf{D}\mathbf{X}+\mathbf{Z})-\mathbf{D}^{-1}$
$\displaystyle=\mathbf{N}^{-1}\mathbf{D}\,{\sf
Cov}(\mathbf{X}|\mathbf{D}\mathbf{X}+\mathbf{Z})-\mathbf{D}^{-1}$
$\displaystyle=\left(\mathbf{N}^{-1}{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})-\mathbf{I}\right)\mathbf{D}^{-1}$
(26)
where the second equality follows from the fundamental relationship between
the derivative of mutual information and MMSE estimate in linear vector
Gaussian channels as stated in [18, Theorem 2].
From (26), the derivative $\frac{\partial F}{\partial\mathbf{D}}$ can be
calculated as
$\displaystyle\frac{\partial F}{\partial\mathbf{D}}=$
$\displaystyle\frac{2}{n}|\mathbf{D}|^{\frac{2}{n}}\mathbf{D}^{-1}\Biggl{\\{}\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]-|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}\Bigg{\\}}+$
$\displaystyle|\mathbf{D}|^{\frac{2}{n}}\Biggl{\\{}\frac{2}{n}\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]\frac{\partial
I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})}{\partial\mathbf{D}}-\frac{2}{n}|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}(\mathbf{I}-\mathbf{D}^{-2})^{-1}\mathbf{D}^{-3}\Biggr{\\}}$
$\displaystyle=$
$\displaystyle\frac{2}{n}|\mathbf{D}|^{\frac{2}{n}}\Biggl{\\{}\left\\{\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]-|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}\right\\}\mathbf{I}+$
$\displaystyle\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right](\mathbf{N}^{-1}{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})-\mathbf{I})-|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}(\mathbf{D}^{2}-\mathbf{I})^{-1}\Biggr{\\}}\mathbf{D}^{-1}$
$\displaystyle=$
$\displaystyle\frac{2}{n}|\mathbf{D}|^{\frac{2}{n}}\Biggl{\\{}\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]\mathbf{N}^{-1}{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})-|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}\left[\mathbf{I}+(\mathbf{D}^{2}-\mathbf{I})^{-1}\right]\Biggr{\\}}\mathbf{D}^{-1}.$
(27)
The derivative $\frac{\partial\mathbf{D}}{\partial\gamma}$ can be calculated
as
$\displaystyle\frac{\partial\mathbf{D}}{\partial\gamma}$
$\displaystyle=\frac{1}{2}\left[\mathbf{I}+\gamma(\mathbf{A}^{-1}-\mathbf{I})\right]^{-\frac{1}{2}}(\mathbf{A}^{-1}-\mathbf{I})$
$\displaystyle=\frac{1}{2\gamma}\mathbf{D}^{-1}(\mathbf{D}^{2}-\mathbf{I})$
$\displaystyle=\frac{1}{2\gamma}\mathbf{D}(\mathbf{I}-\mathbf{D}^{-2}).$ (28)
By (27), (28) and the chain rule of differentiation [24, Chapter 17.5],
$\displaystyle\frac{\partial F}{\partial\gamma}$ $\displaystyle={\sf
Tr}\left\\{\frac{\partial
F}{\partial\mathbf{D}}\,\frac{\partial\mathbf{D}}{\partial\gamma}\right\\}$
$\displaystyle=\frac{|\mathbf{D}|^{\frac{2}{n}}}{n}{\sf
Tr}\Biggl{\\{}\left[\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]\mathbf{N}^{-1}{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})-|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}\left[\mathbf{I}+(\mathbf{D}^{2}-\mathbf{I})^{-1}\right]\right]\frac{\mathbf{I}-\mathbf{D}^{-2}}{\gamma}\Biggr{\\}}$
$\displaystyle=\frac{|\mathbf{D}|^{\frac{2}{n}}}{n\gamma}{\sf
Tr}\left\\{\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]\mathbf{N}^{-1}{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})(\mathbf{I}-\mathbf{D}^{-2})-|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}\mathbf{I}\right\\}$
$\displaystyle=\frac{|\mathbf{D}|^{\frac{2}{n}}}{n\gamma}\left\\{\exp\left[\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]{\sf
Tr}\left\\{\mathbf{N}^{-1}{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})(\mathbf{I}-\mathbf{D}^{-2})\right\\}-n|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}\right\\}.$
(29)
_Step 3–Proving $\frac{\partial F}{\partial\gamma}\geq 0$._ The mutual
information $I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})$ can be bounded
from below as follows:
$\displaystyle I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})$
$\displaystyle\geq I(\mathbf{Z};{\sf
E}[\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z}])$
$\displaystyle=h(\mathbf{Z})-h(\mathbf{Z}|{\sf
E}[\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z}])$
$\displaystyle=\frac{1}{2}\log(2\pi e)^{n}|\mathbf{N}|-h(\mathbf{Z}-{\sf
E}[\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z}]|{\sf
E}[\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z}])$
$\displaystyle\geq\frac{1}{2}\log(2\pi e)^{n}|\mathbf{N}|-h(\mathbf{Z}-{\sf
E}[\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z}])$
$\displaystyle\geq\frac{1}{2}\log(2\pi e)^{n}|\mathbf{N}|-\frac{1}{2}\log(2\pi
e)^{n}\bigl{|}{\sf Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})\bigr{|}$
$\displaystyle=\frac{1}{2}\log\frac{|\mathbf{N}|}{|{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})|}.$ (30)
Here, the first inequality follows from the Markov relation
$\mathbf{Z}\rightarrow\mathbf{D}\mathbf{X}+\mathbf{Z}\rightarrow{\sf
E}[\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z}]$
and the chain rule of mutual information [25, Chapter 2.8]; the second
inequality follows from the fact that conditioning reduces differential
entropy [25, Chapter 9.6]; and the third inequality follows from the well-
known fact that Gaussian maximizes differential entropy for a given covariance
matrix [25, Chapter 9.6]. By (30),
$\displaystyle|\mathbf{I}-\mathbf{D}^{-2}|^{\frac{1}{n}}\exp\left[-\frac{2}{n}I(\mathbf{Z};\mathbf{D}\mathbf{X}+\mathbf{Z})\right]$
$\displaystyle\leq\;|\mathbf{N}^{-1}{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})(\mathbf{I}-\mathbf{D}^{-2})|^{\frac{1}{n}}$
$\displaystyle\leq\;\frac{1}{n}{\sf Tr}\left\\{\mathbf{N}^{-1}{\sf
Cov}(\mathbf{Z}|\mathbf{D}\mathbf{X}+\mathbf{Z})(\mathbf{I}-\mathbf{D}^{-2})\right\\}$
(31)
where the last inequality follows from the well-known inequality of arithmetic
and geometric means [26, p. 136].
Finally, substituting (31) into (29) establishes the fact that $\frac{\partial
F}{\partial\gamma}\geq 0$ for all $\gamma\in[0,1]$. In particular, we have
$F(\mathbf{D}(1))\geq F(\mathbf{D}(0))$. This proved the desired inequality
(21) and hence the generalized Costa EPI (3).
## IV Proof of Theorem 2
In this section, we prove the extremal entropy inequality (7) as stated in
Theorem 2. We will first state a series of corollaries of Theorem 1, as
intermediate results leading to Theorem 2. Based on the final corollary, we
will prove Theorem 2 using an _enhancement_ argument.
###### Corollary 1
Let $\mathbf{Z}$ be a Gaussian random $n$-vector with a positive definite
covariance matrix, and let $\mathbf{A}$ be an $n\times n$ positive real
symmetric matrix such that $0\preceq\mathbf{A}\preceq\mathbf{I}$. Then
$\displaystyle\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z}|U)\right]$
$\displaystyle\geq|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}|U)\right]+|\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{Z}|U)\right]$
(32)
for any $(\mathbf{X},U)$ independent of $\mathbf{Z}$.
###### Corollary 2
Let $\mathbf{Z}_{1}$, $\mathbf{Z}_{2}$ and $\mathbf{Z}_{3}$ be Gaussian random
$n$-vectors with positive definite covariance matrices $\mathbf{N}_{1}$,
$\mathbf{N}_{2}$ and $\mathbf{N}_{3}$, respectively. Assume that
$\mathbf{N}_{1}\preceq\mathbf{N}_{3}$. If there exists an $n\times n$ positive
semidefinite matrix $\mathbf{B}^{*}$ such that
$\displaystyle(\mathbf{B}^{*}+\mathbf{N}_{1})^{-1}+\mu(\mathbf{B}^{*}+\mathbf{N}_{3})^{-1}=(1+\mu)(\mathbf{B}^{*}+\mathbf{N}_{2})^{-1}$
(33)
for some real scalar $\mu\geq 0$, then
$\displaystyle h(\mathbf{X}+\mathbf{Z}_{1}|U)+\mu h$
$\displaystyle(\mathbf{X}+\mathbf{Z}_{3}|U)-(1+\mu)h(\mathbf{X}+\mathbf{Z}_{2}|U)$
$\displaystyle\leq\frac{1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{1}|+\frac{\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{3}|-\frac{1+\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{2}|$
(34)
for any $(\mathbf{X},U)$ independent of
$(\mathbf{Z}_{1},\mathbf{Z}_{2},\mathbf{Z}_{3})$.
###### Corollary 3
Let $\mathbf{Z}_{k}$, $k=0,\ldots,K$, be a collection of $K+1$ Gaussian random
$n$-vectors with respective positive definite covariance matrices
$\mathbf{N}_{k}$. Assume that
$\mathbf{N}_{1}\preceq\ldots\preceq\mathbf{N}_{K}$. If there exists an
$n\times n$ positive semidefinite matrix $\mathbf{B}^{*}$ such that
$\displaystyle\sum_{k=1}^{K}\mu_{k}(\mathbf{B}^{*}+\mathbf{N}_{k})^{-1}=(\mathbf{B}^{*}+\mathbf{N}_{0})^{-1}$
(35)
for some $\mu_{k}\geq 0$ with $\sum_{k=1}^{K}\mu_{k}=1$, then
$\displaystyle\sum_{k=1}^{K}\mu_{k}h(\mathbf{X}+\mathbf{Z}_{k}|U)-h(\mathbf{X}+\mathbf{Z}_{0}|U)$
$\displaystyle\leq\sum_{k=1}^{K}\frac{\mu_{k}}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{k}|-\frac{1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{0}|$
(36)
for any $(\mathbf{X},U)$ independent of
$(\mathbf{Z}_{0},\ldots,\mathbf{Z}_{K})$.
A proof of Corollaries 1, 2 and 3 can be found in Appendices B, C and D,
respectively. We are now ready to prove Theorem 2. Note that the special case
with $\mathbf{M}_{1}=\mathbf{M}_{2}=0$ was proved in Corollary 3. To extend
the result of Corollary 3 to nonzero $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$, we
will consider an enhancement argument, which was first introduced by
Weingarten, Steinberg and Shamai in [7].
Let $\widetilde{\mathbf{N}}_{1}$ and $\widetilde{\mathbf{N}}_{0}$ be $n\times
n$ real symmetric matrices such that:
$\displaystyle\mu_{1}(\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{1})^{-1}$
$\displaystyle=\mu_{1}(\mathbf{B}^{*}+\mathbf{N}_{1})^{-1}+\mathbf{M}_{1}$
(37)
$\displaystyle\mbox{and}\quad\quad(\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{0})^{-1}$
$\displaystyle=(\mathbf{B}^{*}+\mathbf{N}_{0})^{-1}+\mathbf{M}_{2}.$ (38)
As shown in [7, Lemma 11 and 12], $\widetilde{\mathbf{N}}_{1}$ and
$\widetilde{\mathbf{N}}_{0}$ satisfy the following properties:
$\displaystyle 0\prec\widetilde{\mathbf{N}}_{1}$
$\displaystyle=\left(\mathbf{N}_{1}^{-1}+\mu_{1}^{-1}\mathbf{M}_{1}\right)^{-1}\preceq\mathbf{N}_{1},$
(39) $\displaystyle\widetilde{\mathbf{N}}_{1}$
$\displaystyle\preceq\widetilde{\mathbf{N}}_{0}\preceq\mathbf{N}_{0},$ (40)
$\displaystyle\left|\frac{\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{1}}{\widetilde{\mathbf{N}}_{1}}\right|$
$\displaystyle=\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|$
(41)
and
$\displaystyle\left|\frac{\mathbf{S}+\widetilde{\mathbf{N}}_{0}}{\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{0}}\right|$
$\displaystyle=\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{*}+\mathbf{N}_{2}}\right|.$
(42)
Let $\widetilde{\mathbf{Z}}_{0}$ and $\widetilde{\mathbf{Z}}_{1}$ be two
Gaussian $n$-vectors with covariance matrices $\widetilde{\mathbf{N}}_{0}$ and
$\widetilde{\mathbf{N}}_{1}$, respectively. Note from (39) that
$\widetilde{\mathbf{N}}_{1}\preceq\mathbf{N}_{1}\preceq\mathbf{N}_{2}\preceq\ldots\preceq\mathbf{N}_{K}$.
Moreover, substitute (37) and (38) into (4) and we have
$\displaystyle\mu_{1}(\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{1})^{-1}+\sum_{k=2}^{K}\mu_{k}(\mathbf{B}^{*}+\mathbf{N}_{k})^{-1}$
$\displaystyle=(\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{0})^{-1}.$ (43)
Thus, by Corollary 3
$\displaystyle\mu_{1}h(\mathbf{X}+\widetilde{\mathbf{Z}}_{1}|U)+$
$\displaystyle\sum_{k=2}^{K}\mu_{k}h(\mathbf{X}+\mathbf{Z}_{k}|U)-h(\mathbf{X}+\widetilde{\mathbf{Z}}_{0}|U)$
$\displaystyle\leq\frac{\mu_{1}}{2}(\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{1})^{-1}+\sum_{k=2}^{K}\frac{\mu_{k}}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{k}|-\frac{1}{2}\log|\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{0}|$
(44)
for any $(\mathbf{X},U)$ independent of
$(\widetilde{\mathbf{Z}}_{0},\widetilde{\mathbf{Z}}_{1},\mathbf{Z}_{2},\ldots,\mathbf{Z}_{K})$.
On the other hand, note from (39) that
$\widetilde{\mathbf{N}}_{1}\preceq\mathbf{N}_{1}$. We have
$\displaystyle I(\mathbf{X};\mathbf{X}+\mathbf{Z}_{1}|U)\leq
I(\mathbf{X};\mathbf{X}+\widetilde{\mathbf{Z}}_{1}|U)$
for any $(\mathbf{X},U)$ independent of
$(\mathbf{Z}_{1},\widetilde{\mathbf{Z}}_{1})$. Thus,
$\displaystyle
h(\mathbf{X}+\widetilde{\mathbf{Z}}_{1}|U)-h(\mathbf{X}+\mathbf{Z}_{1}|U)$
$\displaystyle\geq h(\widetilde{\mathbf{Z}}_{1})-h(\mathbf{Z}_{1})$
$\displaystyle=\frac{1}{2}\log\left|\frac{\widetilde{\mathbf{N}}_{1}}{\mathbf{N}_{1}}\right|$
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{1}}{\mathbf{B}^{*}+\mathbf{N}_{1}}\right|$
(45)
where the last equality follows from (41).
Also note from (40) that $\widetilde{\mathbf{N}}_{0}\preceq\mathbf{N}_{0}$.
Let $\hat{\mathbf{Z}}_{0}$ be a Gaussian $n$-vector with covariance matrix
$\mathbf{N}_{0}-\widetilde{\mathbf{N}}_{0}$ and independent of
$(\widetilde{\mathbf{Z}}_{0},\mathbf{X},U)$. We have
$\displaystyle
h(\mathbf{X}+\mathbf{Z}_{0}|U)-h(\mathbf{X}+\widetilde{\mathbf{Z}}_{0}|U)$
$\displaystyle=h(\mathbf{X}+\widetilde{\mathbf{Z}}_{0}+\hat{\mathbf{Z}}_{0}|U)-h(\mathbf{X}+\widetilde{\mathbf{Z}}_{0}|U)$
$\displaystyle=I(\hat{\mathbf{Z}}_{0};\mathbf{X}+\widetilde{\mathbf{Z}}_{0}+\hat{\mathbf{Z}}_{0}|U)$
$\displaystyle\geq
I(\hat{\mathbf{Z}}_{0};\mathbf{X}+\widetilde{\mathbf{Z}}_{0}+\hat{\mathbf{Z}}_{0})$
$\displaystyle\geq\frac{1}{2}\log\left|\frac{{\sf
Cov}(\mathbf{X})+\mathbf{N}_{0}}{{\sf
Cov}(\mathbf{X})+\widetilde{\mathbf{N}}_{0}}\right|$
$\displaystyle\geq\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{0}}{\mathbf{S}+\widetilde{\mathbf{N}}_{0}}\right|$
(46)
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{0}}{\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{0}}\right|$
(47)
for any $(\mathbf{X},U)$ independent of
$(\mathbf{Z}_{0},\widetilde{\mathbf{Z}}_{0})$ such that ${\sf
E}[\mathbf{X}\mathbf{X}^{\textsf{T}}]\preceq\mathbf{S}$. Here, the first
inequality follows from the independence of $\hat{\mathbf{Z}}_{0}$ and $U$;
the second inequality follows from the worst noise result [27, Lemma II.2];
the third inequality follows from the fact that
$\widetilde{\mathbf{N}}_{0}\preceq\mathbf{N}_{0}$ and ${\sf
Cov}(\mathbf{X})\preceq{\sf
E}[\mathbf{X}\mathbf{X}^{\textsf{T}}]\preceq\mathbf{S}$; and the last
inequality follows from (42).
Finally, put together (44), (45) and (47) and we may obtain
$\displaystyle\sum_{k=1}^{K}\mu_{k}$ $\displaystyle
h(\mathbf{X}+\mathbf{Z}_{k}|U)-h(\mathbf{X}+\mathbf{Z}_{0}|U)$
$\displaystyle=\left[\mu_{1}h(\mathbf{X}+\widetilde{\mathbf{Z}}_{1}|U)+\sum_{k=2}^{K}\mu_{k}h(\mathbf{X}+\mathbf{Z}_{k}|U)-h(\mathbf{X}+\widetilde{\mathbf{Z}}_{0}|U)\right]-$
$\displaystyle\hskip
16.0pt\mu_{1}\left[h(\mathbf{X}+\widetilde{\mathbf{Z}}_{1}|U)-h(\mathbf{X}+\mathbf{Z}_{1}|U)\right]-\left[h(\mathbf{X}+\mathbf{Z}_{0}|U)-h(\mathbf{X}+\widetilde{\mathbf{Z}}_{0}|U)\right]$
$\displaystyle\leq\left[\frac{\mu_{1}}{2}(\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{1})^{-1}+\sum_{k=2}^{K}\frac{\mu_{k}}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{k}|-\frac{1}{2}\log|\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{0}|\right]-$
$\displaystyle\hskip
16.0pt\frac{\mu_{1}}{2}\log\left|\frac{\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{1}}{\mathbf{B}^{*}+\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{0}}{\mathbf{B}^{*}+\widetilde{\mathbf{N}}_{0}}\right|$
$\displaystyle=\sum_{k=1}^{K}\frac{\mu_{k}}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{k}|-\frac{1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{0}|$
for any $(\mathbf{X},U)$ independent of
$(\mathbf{Z}_{0},\mathbf{Z}_{1},\ldots,\mathbf{Z}_{K})$ such that ${\sf
E}[\mathbf{X}\mathbf{X}^{\textsf{T}}]\preceq\mathbf{S}$. This completes the
proof of Theorem 2.
## V Proof of Theorem 5
In this section, we prove Theorem 5. Note that the achievability of the
secrecy rate region (16) can be obtained from the secrecy rate region (14) by
letting $\mathbf{U}$ and $\mathbf{V}$ be two independent Gaussian vectors with
zero means and covariance matrices $\mathbf{S}-\mathbf{B}$ and $\mathbf{B}$,
respectively and $\mathbf{X}=\mathbf{U}+\mathbf{V}$. We therefore concentrate
on the converse part of the theorem.
To show that (16) is indeed the secrecy capacity region of the vector Gaussian
broadcast channel (8), we will consider proof by contradiction. Assume that
$(R_{1}^{o},R_{2}^{o})$ is an achievable secrecy rate pair that lies _outside_
the secrecy rate region (16). Note that $\mathbf{N}_{1}\preceq\mathbf{N}_{2}$.
From [28, Theorem 1], we can bound $R_{1}^{o}$ by
$\displaystyle
R_{1}^{o}\leq\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|=R_{1}^{max}.$
Note that when $R_{2}^{o}=0$, $R_{1}^{max}$ is achievable by letting
$\mathbf{B}=\mathbf{S}$ in (14). Thus, we may assume that $R_{2}^{o}>0$ and
write $R_{1}^{o}=R_{1}^{*}+\delta$ for some $\delta>0$ where $R_{1}^{*}$ is
given by
$\displaystyle\max_{\mathbf{B}}$
$\displaystyle\quad\left[\frac{1}{2}\log\left|\frac{\mathbf{B}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|\right]$
subject to: $\displaystyle\quad 0\preceq\mathbf{B}\preceq\mathbf{S}$
$\displaystyle\quad\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}+\mathbf{N}_{3}}\right|\geq
R_{2}^{o}.$
Let $\mathbf{B}^{*}$ be an optimal solution to the above optimization program.
Then, $\mathbf{B}^{*}$ must satisfy the following KKT conditions111As this
optimization program is not convex, a set of constraint qualifications (CQs)
should be checked to make sure that the KKT conditions indeed hold. The CQs
stated in Appendix IV of [7] hold in a trivial manner for this program.:
$\displaystyle(\mathbf{B}^{*}+\mathbf{N}_{1})^{-1}+\mu(\mathbf{B}^{*}+\mathbf{N}_{3})^{-1}+\mathbf{M}_{1}$
$\displaystyle=(1+\mu)(\mathbf{B}^{*}+\mathbf{N}_{2})^{-1}+\mathbf{M}_{2}$
(48) $\displaystyle\mathbf{B}^{*}\mathbf{M}_{1}$ $\displaystyle=0$ (49)
$\displaystyle\mbox{and}\quad\quad(\mathbf{S}-\mathbf{B}^{*})\mathbf{M}_{2}$
$\displaystyle=0$ (50)
where $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ are $n\times n$ positive
semidefinite matrices, and $\mu$ is a nonnegative real scalar such that
$\mu>0$ if and only if
$\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{*}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}^{*}+\mathbf{N}_{3}}\right|=R_{2}^{o}.$
Thus,
$\displaystyle R_{1}^{o}+\mu
R_{2}^{o}=\left[\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|\right]+\mu\left[\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{*}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}^{*}+\mathbf{N}_{3}}\right|\right]+\delta.$
(51)
On the other hand, by the converse part of Theorem 3
$\displaystyle R_{1}^{o}+\mu R_{2}^{o}\leq\;$
$\displaystyle[I(\mathbf{X};\mathbf{X}+\mathbf{Z}_{1}|U)-I(\mathbf{X};\mathbf{X}+\mathbf{Z}_{2}|U)]+\mu[I(U;\mathbf{X}+\mathbf{Z}_{2})-I(U;\mathbf{X}+\mathbf{Z}_{3})]$
$\displaystyle=\;$
$\displaystyle[h(\mathbf{Z}_{2})-h(\mathbf{Z}_{1})]-\mu[h(\mathbf{X}+\mathbf{Z}_{3})-h(\mathbf{X}+\mathbf{Z}_{2})]+$
$\displaystyle[h(\mathbf{X}+\mathbf{Z}_{1}|U)+\mu
h(\mathbf{X}+\mathbf{Z}_{3}|U)-(1+\mu)h(\mathbf{X}+\mathbf{Z}_{2}|U)]$
$\displaystyle=\;$
$\displaystyle\frac{1}{2}\log\left|\frac{\mathbf{N}_{2}}{\mathbf{N}_{1}}\right|-\mu[h(\mathbf{X}+\mathbf{Z}_{3})-h(\mathbf{X}+\mathbf{Z}_{2})]+$
$\displaystyle[h(\mathbf{X}+\mathbf{Z}_{1}|U)+\mu
h(\mathbf{X}+\mathbf{Z}_{3}|U)-(1+\mu)h(\mathbf{X}+\mathbf{Z}_{2}|U)]$ (52)
for some jointly distributed $(U,\mathbf{X})$ independent of
$(\mathbf{Z}_{1},\mathbf{Z}_{2},\mathbf{Z}_{3})$. Note that
$\mathbf{N}_{2}\preceq\mathbf{N}_{3}$. Similar to (46), we may obtain
$\displaystyle h(\mathbf{X}+\mathbf{Z}_{3})-h(\mathbf{X}+\mathbf{Z}_{2})$
$\displaystyle\geq\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{S}+\mathbf{N}_{2}}\right|.$
(53)
Moreover, by letting
$\displaystyle\mu_{1}=\frac{1}{1+\mu},\quad\mu_{3}=\frac{\mu}{1+\mu},\quad\tilde{\mathbf{M}}_{1}=\frac{\mathbf{M}_{1}}{1+\mu},\quad\mbox{and}\;\;\tilde{\mathbf{M}}_{2}=\frac{\mathbf{M}_{2}}{1+\mu}$
we can rewrite the KKT conditions (48)–(50) as
$\displaystyle\mu_{1}(\mathbf{B}^{*}+\mathbf{N}_{1})^{-1}+\mu_{3}(\mathbf{B}^{*}+\mathbf{N}_{3})^{-1}+\tilde{\mathbf{M}}_{1}$
$\displaystyle=(\mathbf{B}^{*}+\mathbf{N}_{2})^{-1}+\tilde{\mathbf{M}}_{2}$
$\displaystyle\mathbf{B}^{*}\tilde{\mathbf{M}}_{1}$ $\displaystyle=0$
$\displaystyle\mbox{and}\quad\quad(\mathbf{S}-\mathbf{B}^{*})\tilde{\mathbf{M}}_{2}$
$\displaystyle=0.$
Thus, by Theorem 2
$\displaystyle h(\mathbf{X}+\mathbf{Z}_{1}|U)+\mu h$
$\displaystyle(\mathbf{X}+\mathbf{Z}_{3}|U)-(1+\mu)h(\mathbf{X}+\mathbf{Z}_{2}|U)$
$\displaystyle\leq\frac{1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{1}|+\frac{\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{3}|-\frac{1+\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{2}|.$
(54)
Substituting (53) and (54) into (52), we have
$\displaystyle R_{1}^{o}+\mu R_{2}^{o}\leq$
$\displaystyle\;\frac{1}{2}\log\left|\frac{\mathbf{N}_{2}}{\mathbf{N}_{1}}\right|-\frac{\mu}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{S}+\mathbf{N}_{2}}\right|+$
$\displaystyle\;\left[\frac{1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{1}|+\frac{\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{3}|-\frac{1+\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{2}|\right]$
$\displaystyle=$
$\displaystyle\;\left[\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{2}}{\mathbf{N}_{2}}\right|\right]+\mu\left[\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{*}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}^{*}+\mathbf{N}_{3}}\right|\right].$
(55)
Thus, we have obtained a contradiction between (51) and (55). As a result, all
the achievable rate pairs must be inside the secrecy rate region (16). This
completes the proof of the theorem.
## VI Proof of Theorem 6
In this section, we prove Theorem 6 following similar steps as those used in
the proof for Theorem 5. The achievability of the secrecy rate region (17) can
be obtained from the secrecy rate region (15) by letting $\mathbf{U}$ and
$\mathbf{V}$ be two independent Gaussian vectors with zero means and
covariance matrices $\mathbf{S}-\mathbf{B}$ and $\mathbf{B}$, respectively and
$\mathbf{X}=\mathbf{U}+\mathbf{V}$. We therefore concentrate on the converse
part of the theorem.
To show that (17) is indeed the secrecy capacity region of the vector Gaussian
broadcast channel (8), we will use proof by contradiction. Assume that
$(R_{1}^{o},R_{2}^{o})$ is an achievable secrecy rate pair that lies _outside_
the secrecy rate region (17). Note that $\mathbf{N}_{1}\preceq\mathbf{N}_{3}$.
From [28, Theorem 1], we can bound $R_{1}^{o}$ by
$\displaystyle
R_{1}^{o}\leq\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{N}_{3}}\right|=R_{1}^{max}.$
Note that when $R_{2}^{o}=0$, $R_{1}^{max}$ is achievable by letting
$\mathbf{B}=\mathbf{S}$ in (15). Thus, we may assume that $R_{2}^{o}>0$ and
write $R_{1}^{o}=R_{1}^{*}+\delta$ for some $\delta>0$ where $R_{1}^{*}$ is
given by
$\displaystyle\max_{\mathbf{B}}$
$\displaystyle\quad\left[\frac{1}{2}\log\left|\frac{\mathbf{B}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}+\mathbf{N}_{3}}{\mathbf{N}_{3}}\right|\right]$
subject to: $\displaystyle\quad 0\preceq\mathbf{B}\preceq\mathbf{S}$
$\displaystyle\quad\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}+\mathbf{N}_{3}}\right|\geq
R_{2}^{o}.$
Let $\mathbf{B}^{*}$ be an optimal solution to the above optimization program.
Then, $\mathbf{B}^{*}$ must satisfy the following KKT conditions:
$\displaystyle(\mathbf{B}^{*}+\mathbf{N}_{1})^{-1}+(\mu-1)(\mathbf{B}^{*}+\mathbf{N}_{3})^{-1}+\mathbf{M}_{1}$
$\displaystyle=\mu(\mathbf{B}^{*}+\mathbf{N}_{2})^{-1}+\mathbf{M}_{2}$ (56)
$\displaystyle\mathbf{B}^{*}\mathbf{M}_{1}$ $\displaystyle=0$ (57)
$\displaystyle\mbox{and}\quad\quad(\mathbf{S}-\mathbf{B}^{*})\mathbf{M}_{2}$
$\displaystyle=0$ (58)
where $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ are $n\times n$ positive
semidefinite matrices, and $\mu$ is a nonnegative real scalar such that
$\mu\geq 1$.222If $\mu<1$, it is easy to see that $\mathbf{B}^{*}=\mathbf{S}$
is an optimal solution and hence contradicts the assumption that
$R_{2}^{o}>0$. Therefore,
$R_{2}^{o}=\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{*}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}^{*}+\mathbf{N}_{3}}\right|$
and
$\displaystyle R_{1}^{o}+\mu
R_{2}^{o}=\left[\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{3}}{\mathbf{N}_{3}}\right|\right]+\mu\left[\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{*}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}^{*}+\mathbf{N}_{3}}\right|\right]+\delta.$
(59)
On the other hand, by the converse part of Theorem 4
$\displaystyle R_{1}^{o}+\mu R_{2}^{o}\leq\;$
$\displaystyle[I(\mathbf{X};\mathbf{X}+\mathbf{Z}_{1}|U)-I(\mathbf{X};\mathbf{X}+\mathbf{Z}_{3}|U)]+\mu[I(U;\mathbf{X}+\mathbf{Z}_{2})-I(U;\mathbf{X}+\mathbf{Z}_{3})]$
$\displaystyle=\;$
$\displaystyle[h(\mathbf{Z}_{3})-h(\mathbf{Z}_{1})]-\mu[h(\mathbf{X}+\mathbf{Z}_{3})-h(\mathbf{X}+\mathbf{Z}_{2})]+$
$\displaystyle[h(\mathbf{X}+\mathbf{Z}_{1}|U)+(\mu-1)h(\mathbf{X}+\mathbf{Z}_{3}|U)-\mu
h(\mathbf{X}+\mathbf{Z}_{2}|U)]$ $\displaystyle\leq\;$
$\displaystyle\frac{1}{2}\log\left|\frac{\mathbf{N}_{3}}{\mathbf{N}_{1}}\right|-\frac{\mu}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{S}+\mathbf{N}_{2}}\right|+$
$\displaystyle[h(\mathbf{X}+\mathbf{Z}_{1}|U)+(\mu-1)h(\mathbf{X}+\mathbf{Z}_{3}|U)-\mu
h(\mathbf{X}+\mathbf{Z}_{2}|U)]$ (60)
for some jointly distributed $(U,\mathbf{X})$ independent of
$(\mathbf{Z}_{1},\mathbf{Z}_{2},\mathbf{Z}_{3})$, where the last inequality
follows from (53).
Since $\mu\geq 1$, by letting
$\displaystyle\mu_{1}=\frac{1}{\mu},\quad\mu_{3}=\frac{\mu-1}{\mu},\quad\tilde{\mathbf{M}}_{1}=\frac{\mathbf{M}_{1}}{\mu},\quad\mbox{and}\;\;\tilde{\mathbf{M}}_{2}=\frac{\mathbf{M}_{2}}{\mu}$
we can rewrite the KKT conditions (56)–(58) as
$\displaystyle\mu_{1}(\mathbf{B}^{*}+\mathbf{N}_{1})^{-1}+\mu_{3}(\mathbf{B}^{*}+\mathbf{N}_{3})^{-1}+\tilde{\mathbf{M}}_{1}$
$\displaystyle=(\mathbf{B}^{*}+\mathbf{N}_{2})^{-1}+\tilde{\mathbf{M}}_{2}$
$\displaystyle\mathbf{B}^{*}\tilde{\mathbf{M}}_{1}$ $\displaystyle=0$
$\displaystyle\mbox{and}\quad\quad(\mathbf{S}-\mathbf{B}^{*})\tilde{\mathbf{M}}_{2}$
$\displaystyle=0.$
Thus, by Theorem 2
$\displaystyle h(\mathbf{X}+\mathbf{Z}_{1}|U)+(\mu-1)h$
$\displaystyle(\mathbf{X}+\mathbf{Z}_{3}|U)-\mu
h(\mathbf{X}+\mathbf{Z}_{2}|U)$
$\displaystyle\leq\frac{1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{1}|+\frac{1-\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{3}|-\frac{\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{2}|.$
(61)
Substituting (54) into (60), we have
$\displaystyle R_{1}^{o}+\mu R_{2}^{o}\leq$
$\displaystyle\;\frac{1}{2}\log\left|\frac{\mathbf{N}_{3}}{\mathbf{N}_{1}}\right|-\frac{\mu}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{S}+\mathbf{N}_{2}}\right|+$
$\displaystyle\;\left[\frac{1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{1}|+\frac{\mu-1}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{3}|-\frac{\mu}{2}\log|\mathbf{B}^{*}+\mathbf{N}_{2}|\right]$
$\displaystyle=$
$\displaystyle\;\left[\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{B}^{*}+\mathbf{N}_{3}}{\mathbf{N}_{3}}\right|\right]+\mu\left[\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{*}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{3}}{\mathbf{B}^{*}+\mathbf{N}_{3}}\right|\right].$
(62)
Thus, we have obtained a contradiction between (59) and (62). As a result, all
the achievable rate pairs must be inside the secrecy rate region (17). This
completes the proof of the theorem.
## VII Conclusions
This paper has considered an EPI of Costa and has established a natural
generalization by replacing the scalar parameter in the original Costa EPI
with a matrix one. The generalized Costa EPI has been proven using a
perturbation approach via a fundamental relationship between the derivative of
mutual information and the MMSE in linear vector Gaussian channels. This is an
example of how the connections between information theory and statistics can
be explored to provide new mathematical tools for information theory.
As an application, a new extremal entropy inequality has been derived from the
generalized Costa EPI and then used to characterize the secrecy capacity
regions of the degraded vector Gaussian broadcast channel problem with layered
confidential messages. We expect that the generalized Costa EPI will also play
important roles in solving some other Gaussian network communication problems.
## Appendix A Proof of Theorem 3
### A-A Achievability
We first show that the secrecy rate region (14) is achievable. Following the
idea of superposition coding for the degraded broadcast channel [3], we
introduce an auxiliary codebook which can be distinguished by both receiver 1
and receiver 2. The codebook is generated using random binning [20, 21].
Fix $p(u)$ and $p(x|u)$ and let
$\displaystyle R^{\prime}_{1}$ $\displaystyle=I(X;Y_{2}|U)-\epsilon_{1}$ (63a)
and $\displaystyle R^{\prime}_{2}$ $\displaystyle=I(U;Y_{3})-\epsilon_{1}$
(63b)
for some $\epsilon_{1}>0$. Let
$\displaystyle L_{k}$ $\displaystyle=2^{nR_{k}},\quad
J_{k}=2^{nR^{\prime}_{k}}\quad\text{and},\quad T_{k}=L_{k}J_{k}\quad k=1,2.$
Without loss of generality, $L_{k}$, $L^{\prime}_{k}$ and $J_{k}$ are assumed
to be integers.
#### Codebook generation
Generate $T_{2}$ independent codewords $u^{n}$ of length $n$ according to
$\prod_{i=1}^{n}p(u_{i})$ and label them as
$u^{n}(w_{2},j_{2}),\quad w_{2}\in\\{1,\dots,L_{2}\\},\quad
j_{2}\in\\{1,\dots,J_{2}\\}.$
For each codeword $u^{n}(w_{2},j_{2})$, generate $T_{1}$ independent codewords
$x^{n}$ according to $\prod_{i=1}^{n}p(x_{i}|u_{i})$ and label them as
$x^{n}(w_{1},j_{1},w_{2},j_{2})=x^{n}\bigl{(}w_{1},j_{1},u^{n}(w_{2},j_{2})\bigr{)},\quad
w_{k}\in\\{1,\dots,L_{k}\\}\quad\mbox{and}\quad j_{k}\in\\{1,\dots,J_{k}\\}.$
#### Encoding
To send a message pair $(w_{1},w_{2})$, the transmitter randomly chooses a
pair $(j_{1},j_{2})$ and sends the corresponding codeword
$x^{n}(w_{1},j_{1},w_{2},j_{2})$ through the channel.
#### Decoding
Receiver 2 determines the unique $w_{2}$ such that
$\bigl{(}u^{n}(w_{2},j_{2}),y_{2}^{n}\bigr{)}\in\mathcal{A}_{\epsilon}^{(n)}(p_{U,Y_{2}})$
where $\mathcal{A}_{\epsilon}^{(n)}(p_{U,Y_{2}})$ denotes the set of jointly
typical sequences $u^{n}$ and $y_{2}^{n}$ with respect to $p(u,y_{2})$. If
there are none such or more than one such, an error is declared. Receiver 1
looks for the unique $(w_{1},w_{2})$ such that
$\bigl{(}u^{n}(w_{2},j_{2}),x^{n}(w_{1},j_{1},w_{2},j_{2}),y_{1}^{n}\bigr{)}\in\mathcal{A}_{\epsilon}^{(n)}(p_{U,X,Y_{1}})$
where $\mathcal{A}_{\epsilon}^{(n)}(p_{U,X,Y_{1}})$ denotes the set of jointly
typical sequences $u^{n}$, $x^{n}$ and $y_{1}^{n}$ with respect to
$p(u,x,y_{1})$. Otherwise, an error is declared.
#### Error probability analysis
By the symmetry of the codebook generation, the probability error does not
depend on which codeword was sent. Hence, without loss of generality, we may
assume that the transmitter sends the message pair $(w_{1},w_{2})=(1,1)$
associated with the codeword $x^{n}(1,1,1,1)$ and define the corresponding
event
$\mathcal{K}_{1}:=\\{x^{n}(1,1,1,1)\;\text{was sent}\\}.$
First consider the decoding at receiver 2, for which we will show that
receiver 2 is able to decode $u^{n}(w_{2},j_{2})$ with small probability of
error if $R_{2}+R^{\prime}_{2}<I(U;Y_{2})$. To prove this, define the event
$\displaystyle\mathcal{E}_{2}(w_{2},j_{2}):=\left\\{\bigl{(}u^{n}(w_{2},j_{2}),y_{2}^{n}\bigr{)}\in\mathcal{A}_{\epsilon}^{(n)}(p_{U,Y_{2}})\right\\}.$
Then, the probability of error at receiver 2 can be bounded from above as
$\displaystyle P^{(n)}_{e,2}$
$\displaystyle\leq\Pr\left\\{\bigcap_{j_{2}}\mathcal{E}_{2}^{c}(1,j_{2})\Big{|}\mathcal{K}_{1}\right\\}+\sum_{w_{2}\neq
1,\,j_{2}}\Pr\\{\mathcal{E}_{2}(w_{2},j_{2})|\mathcal{K}_{1}\\}$
$\displaystyle\leq\Pr\\{\mathcal{E}_{2}^{c}(1,1)|\mathcal{K}_{1}\\}+\sum_{w_{2}\neq
1,\,j_{2}}\Pr\\{\mathcal{E}_{2}(w_{2},j_{2})|\mathcal{K}_{1}\\}$
where
$\mathcal{E}_{2}^{c}(1,j_{2}):=\left\\{\bigl{(}u^{n}(1,j_{2}),y_{2}^{n}\bigr{)}\notin\mathcal{A}_{\epsilon}^{(n)}(p_{U,Y_{2}})\right\\}.$
For large enough $n$ and $R_{2}+R_{2}^{\prime}<I(U;Y_{2})$, the joint
asymptotic equipartition property (AEP) [25, Chapter 14.2] implies
$\displaystyle P_{e,2}^{(n)}$
$\displaystyle\leq\epsilon+T_{2}2^{-n[I(U;Y_{2})-\epsilon]}$
$\displaystyle=\epsilon+2^{n(R_{2}+R^{\prime}_{2})}\,2^{-n[I(U;Y_{2})-\epsilon]}$
$\displaystyle\leq 2\epsilon.$ (64)
Next, we will show that receiver 1 can successfully decode both $u^{n}$ and
$x^{n}$ if
$\displaystyle R_{1}+R^{\prime}_{1}$ $\displaystyle<I(X;Y_{1}|U)$ and
$\displaystyle R_{2}+R^{\prime}_{2}$ $\displaystyle<I(U;Y_{2}).$ (65)
Define the events
$\displaystyle\mathcal{E}_{1,1}(w_{1},j_{1},w_{2},j_{2})$
$\displaystyle:=\left\\{\bigl{(}u^{n}(w_{2},j_{2}),x^{n}(w_{1},j_{1},w_{2},j_{2}),y_{1}^{n}\bigr{)}\in\mathcal{A}_{\epsilon}^{(n)}(p_{U,X,Y_{1}})\right\\}.$
and $\displaystyle\mathcal{E}_{1}(w_{2},j_{2})$
$\displaystyle:=\left\\{\bigl{(}u^{n}(w_{2},j_{2}),y_{1}^{n}\bigr{)}\in\mathcal{A}_{\epsilon}^{(n)}(p_{U,Y_{1}})\right\\}$
where $\mathcal{A}_{\epsilon}^{(n)}(p_{U,Y_{1}})$ denotes the set of jointly
typical sequences $u^{n}$ and $y_{1}^{n}$ with respect to $p(u,y_{1})$. Then,
the probability of error
$\displaystyle P^{(n)}_{e,1}$
$\displaystyle\leq\Pr\\{\mathcal{E}_{1}^{c}(1,1)|\mathcal{K}_{1}\\}+\sum_{w_{2}\neq
1,\,j_{2}}\Pr\\{\mathcal{E}_{1}(w_{2},j_{2})|\mathcal{K}_{1}\\}+\sum_{w_{1}\neq
1,j_{1},}\Pr\\{\mathcal{E}_{1,1}(w_{1},j_{1},1,1)|\mathcal{K}_{1}\\}$
where
$\mathcal{E}_{1}^{c}(1,1):=\left\\{\bigl{(}u^{n}(1,1),y_{1}^{n}\bigr{)}\notin\mathcal{A}_{\epsilon}^{(n)}(p_{U,Y_{1}})\right\\}.$
By the AEP [25, Chapter 14.2],
$\displaystyle\Pr\\{\mathcal{E}_{1}^{c}(1,1)|\mathcal{K}_{1}\\}$
$\displaystyle\leq\epsilon,$
$\displaystyle\Pr\\{\mathcal{E}_{1}(w_{2},j_{2})|\mathcal{K}_{1}\\}$
$\displaystyle\leq 2^{-n[I(U;Y_{1})-\epsilon]},\quad\text{for}~{}w_{2}\neq 1,$
and $\displaystyle\Pr\\{\mathcal{E}_{1,1}(w_{1},j_{1},1,1)|\mathcal{K}_{1}\\}$
$\displaystyle\leq 2^{-n[I(X;Y_{1}|U)-\epsilon]},\quad\text{for}~{}w_{1}\neq
1.$
Since the channel is degraded, we have $I(U;Y_{1})\geq I(U;Y_{2})$. Hence, if
$n$ is large enough and the condition (65) holds, the probability of error at
receiver 1 can be bounded from above as
$\displaystyle P_{e,1}^{(n)}$
$\displaystyle\leq\epsilon+T_{2}2^{-n[I(U;Y_{1})-\epsilon]}+T_{1}2^{-n[I(X;Y_{1}|U)-\epsilon]}$
$\displaystyle\leq\epsilon+2^{n(R_{2}+R^{\prime}_{2})}2^{-n[I(U;Y_{2})-\epsilon]}+2^{n(R_{1}+R^{\prime}_{1})}2^{-n[I(X;Y_{1}|U)-\epsilon]}$
$\displaystyle\leq 3\epsilon.$ (66)
Together, (64) and (66) illustrate that messages $(w_{1},w_{2})$ can be
decoded at receiver 1 with a total probability of error that goes to $0$ as
long as the rate pair $(R_{1},R_{2})$ satisfies (14).
#### Equivocation calculation
To show that (11) holds, we consider the following lower bound on the
equivocation:
$\displaystyle H(W_{1}|Y_{2}^{n})$ $\displaystyle\geq
H(W_{1}|Y_{2}^{n},U^{n})$
$\displaystyle=H(W_{1},Y_{2}^{n}|U^{n})-H(Y_{2}^{n}|U^{n})$
$\displaystyle=H(X^{n},Y_{2}^{n}|U^{n})-H(X^{n}|W_{1},Y_{2}^{n},U^{n})-H(Y_{2}^{n}|U^{n})$
$\displaystyle=H(X^{n}|U^{n})+H(Y_{2}^{n}|X^{n},U^{n})-H(X^{n}|W_{1},Y_{2}^{n},U^{n})-H(Y_{2}^{n}|U^{n})$
$\displaystyle=H(X^{n}|U^{n})-H(X^{n}|W_{1},Y_{2}^{n},U^{n})-I(X^{n};Y_{2}^{n}|U^{n})$
(67)
where the second equality is due to the fact that $W_{1}$ is independent of
everything else given $X^{n}$.
According to the codebook generation, for a given $U^{n}=u^{n}$, $X^{n}$ has
$T_{1}$ possible values with equal probabilities. Hence,
$\displaystyle H(X^{n}|U^{n})$ $\displaystyle=n(R_{1}+R^{\prime}_{1})$
$\displaystyle=n[R_{1}+I(X;Y_{2}|U)-\epsilon_{1}]$ (68)
where (68) follows from the definition of $R^{\prime}_{1}$ in (63a).
Next, we show that for any given $\epsilon_{2}>0$,
$H(X^{n}|W_{1},Y_{2}^{n},U^{n})\leq n\epsilon_{2}$ for large enough $n$. To
calculate $H(X^{n}|W_{1},Y_{2}^{n},U^{n})$, consider the following
hypothetical scenario. Fix $W_{1}=w_{1}$, and assume that the transmitter
sends a codeword $x^{n}\bigl{(}w_{1},j_{1},u^{n}(w_{2},j_{2})\bigr{)}$,
$j_{1}\in\\{1,\dots,J_{1}\\}$. Assume that receiver 2 knows the sequence
$U^{n}=u^{n}(w_{2},j_{2})$. Given index $W_{1}=w_{1}$, receiver 2 decodes the
codeword $x^{n}(w_{1},j_{1},u^{n})$ (i.e., looks for the index $j_{1}$) based
on the received sequence $y_{2}$. Let $\lambda(w_{1})$ denote the average
probability of error of decoding the index $j_{1}$ at receiver 2. By the AEP
[25, Chapter 14.2], we have $\lambda(w_{1})\leq\epsilon$ for sufficiently
large $n$. By Fano’s inequality [25, Chapter 2.11],
$\displaystyle\frac{1}{n}H(X^{n}|W_{1}=w_{1},Y_{2}^{n},U^{n})$
$\displaystyle\leq\frac{1}{n}+\lambda(w_{1})\frac{\log_{2}J_{1}}{n}$
$\displaystyle\leq\frac{1}{n}+\epsilon R^{\prime}_{1}$
$\displaystyle:=\epsilon_{2}.$
Consequently,
$\displaystyle\frac{1}{n}H(X^{n}|W_{1},Y_{2}^{n},U^{n})$
$\displaystyle=\frac{1}{n}\sum_{w_{1}=1}^{L_{1}}\Pr(W_{1}=w_{1})H(X^{n}|W_{1}=w_{1},Y_{2}^{n},U^{n})$
$\displaystyle\leq\epsilon_{2}.$ (69)
By the AEP [25, Chapter 14.2], for any $\epsilon_{3}$
$\displaystyle I(X^{n};Y_{2}^{n}|U^{n})\leq nI(X;Y_{2}|U)+n\epsilon_{3}$ (70)
for sufficiently large $n$. Substituting (68), (69) and (70) into (67), we
have
$\displaystyle\frac{1}{n}H(W_{1}|Y_{2}^{n})$ $\displaystyle\geq
R_{1}-(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}).$
Similarly, we can show that
$\displaystyle H(W_{2}|Y_{3}^{n})\geq
H(U^{n})-H(U^{n}|W_{2},Y_{3}^{n})-I(U^{n};Y_{3}^{n})$
where
$\displaystyle H(U^{n})=n[R_{2}+I(U;Y_{3})-\epsilon_{1}]$ $\displaystyle
H(U^{n}|W_{2},Y_{3}^{n})\leq n\epsilon^{\prime}_{2}$ and $\displaystyle
I(U^{n};Y_{3}^{n})\leq n[I(U;Y_{3})+\epsilon^{\prime}_{3}],$
where $\epsilon^{\prime}_{2}$ and $\epsilon^{\prime}_{3}$ vanishes in the
limit as $n\rightarrow\infty$. Hence,
$\displaystyle\frac{1}{n}H(W_{2}|Y_{3}^{n})\geq
R_{2}-(\epsilon_{1}+\epsilon^{\prime}_{2}+\epsilon^{\prime}_{3}).$
Note that $Y_{3}$ is degraded with respect to $Y_{2}$. Therefore,
$\displaystyle H(W_{1}|Y_{3}^{n})$ $\displaystyle\geq$ $\displaystyle
H(W_{1}|Y_{2}^{n},Y_{3}^{n})$ $\displaystyle=$ $\displaystyle
H(W_{1}|Y_{2}^{n})$ $\displaystyle\geq$ $\displaystyle
R_{1}-(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}).$
This proves the security condition (11) and hence the achievability part of
the theorem.
### A-B The Converse
We first bound from above the secrecy rate $R_{1}$. The perfect secrecy
condition (11) implies that for all $\epsilon>0$,
$\displaystyle H(W_{1}|Y_{2}^{n})$ $\displaystyle\geq H(W_{1})-n\epsilon$
(71a) and $\displaystyle H(W_{2}|Y_{3}^{n})$ $\displaystyle\geq
H(W_{2})-n\epsilon.$ (71b)
On the other hand, Fano’s inequality [25, Chapter 2.11] implies that for any
$\epsilon_{0}>0$,
$\displaystyle H(W_{1}|Y_{1}^{n})$
$\displaystyle\leq\epsilon_{0}\log\left(2^{nR_{1}}-1\right)+h(\epsilon_{0}):=n\delta_{1}$
(72a) and $\displaystyle H(W_{2}|Y_{2}^{n})$
$\displaystyle\leq\epsilon_{0}\log\left(2^{nR_{2}}-1\right)+h(\epsilon_{0}):=n\delta_{2}.$
(72b)
Thus,
$\displaystyle nR_{1}$ $\displaystyle=H(W_{1})$
$\displaystyle\leq\bigl{[}H(W_{1}|Y_{2}^{n})+n\epsilon\bigr{]}+\bigl{[}n\delta_{1}-H(W_{1}|Y_{1}^{n})\bigr{]}$
$\displaystyle\leq
H(W_{1},W_{2}|Y_{2}^{n})-H(W_{1}|Y_{1}^{n},W_{2})+n(\epsilon+\delta_{1})$
$\displaystyle\leq
H(W_{1}|Y_{2}^{n},W_{2})-H(W_{1}|Y_{1}^{n},W_{2})+n(\epsilon+\delta_{1}+\delta_{2})$
(73)
where the first inequality follows from (71a) and (72a), and the last
inequality follows from (72b). Let $\delta=\epsilon+\delta_{1}+\delta_{2}$. By
the chain rule of the mutual information [25, Chapter 2.5],
$\displaystyle n(R_{1}-\delta)$ $\displaystyle\leq
I(W_{1};Y_{1}^{n}|W_{2})-I(W_{1};Y_{2}^{n}|W_{2})$
$\displaystyle=\sum_{i=1}^{n}\left[I(W_{1};Y_{1,i}|W_{2},Y_{1,i+1}^{n})-I(W_{1};Y_{2,i}|W_{2},Y_{2}^{i-1})\right]$
$\displaystyle=\sum_{i=1}^{n}\left[I(W_{1};Y_{1,i}|W_{2},Y_{1,i+1}^{n},Y_{2}^{i-1})-I(W_{1};Y_{2,i}|W_{2},Y_{1,i+1}^{n},Y_{2}^{i-1})\right]$
(74)
where the last equality follows from [21, Lemma 7]. Let
$\displaystyle V_{i}:=\left(Y_{1,i+1}^{n},Y_{2}^{i-1}\right).$ (75)
We can further bound (74) from above as
$\displaystyle n(R_{1}-\delta)$
$\displaystyle\leq\sum_{i=1}^{n}\left[I(W_{1},X_{i};Y_{1,i}|W_{2},V_{i})-I(W_{1},X_{i};Y_{2,i}|W_{2},V_{i})\right]$
$\displaystyle\qquad-\sum_{i=1}^{n}\left[I(X_{i};Y_{1,i}|W_{1},W_{2},V_{i})-I(X_{i};Y_{2,i}|W_{1},W_{2},V_{i})\right]$
$\displaystyle\leq\sum_{i=1}^{n}\left[I(W_{1},X_{i};Y_{1,i}|W_{2},V_{i})-I(W_{1},X_{i};Y_{2,i}|W_{2},V_{i})\right]$
$\displaystyle=\sum_{i=1}^{n}\left[I(X_{i};Y_{1,i}|W_{2},V_{i})-I(X_{i};Y_{2,i}|W_{2},V_{i})\right]$
(76)
where the second inequality follows from the Markov relation
$(W_{1},W_{2},V_{i})\rightarrow X_{i}\rightarrow Y_{1,i}\rightarrow Y_{2,i},$
and the last equality is due to the fact that $Y_{1,i}$ and $Y_{2,i}$ are
conditionally independent of everything else given $X_{i}$.
Next, we bound from above the secrecy rate $R_{2}$. By (71b) and (72b),
$\displaystyle nR_{2}$ $\displaystyle=H(W_{2})$
$\displaystyle\leq\bigl{[}H(W_{2}|Y_{3}^{n})+n\epsilon\bigr{]}+\bigl{[}n\delta_{2}-H(W_{2}|Y_{2}^{n})\bigr{]}$
$\displaystyle=I(W_{2};Y_{2}^{n})-I(W_{2};Y_{3}^{n})+n(\epsilon+\delta_{2})$
$\displaystyle=\sum_{i=1}^{n}\left[I(W_{2};Y_{2,i}|Y_{2,i+1}^{n})-I(W_{2};Y_{3,i}|Y_{3}^{i-1})\right]+n(\epsilon+\delta_{2}).$
(77)
Let $\delta^{\prime}:=\epsilon+\delta_{2}$ and
$\displaystyle V^{\prime}_{i}$
$\displaystyle:=\left(Y_{2,i+1}^{n},Y_{3}^{i-1}\right).$ (78)
Applying [21, Lemma 7] again, we may obtain
$\displaystyle n(R_{2}-\delta^{\prime})$
$\displaystyle\leq\sum_{i=1}^{n}\left[I(W_{2};Y_{2,i}|V^{\prime}_{i})-I(W_{2};Y_{3,i}|V^{\prime}_{i})\right]$
$\displaystyle=\sum_{i=1}^{n}\left[I(W_{2},V^{\prime}_{i};Y_{2,i})-I(W_{2},V^{\prime}_{i};Y_{3,i})\right]-\sum_{i=1}^{n}\left[I(V^{\prime}_{i};Y_{2,i})-I(V^{\prime}_{i};Y_{3,i})\right]$
$\displaystyle\leq\sum_{i=1}^{n}\left[I(W_{2},V^{\prime}_{i};Y_{2,i})-I(W_{2},V^{\prime}_{i};Y_{3,i})\right]$
(79)
where the last inequality follows from the Markov relation
$V^{\prime}_{i}\rightarrow Y_{1,i}\rightarrow Y_{2,i}$. Furthermore, by the
definitions of $V_{i}$ and $V_{i}^{\prime}$ in (75) and (78) respectively,
$\displaystyle
V^{\prime}_{i}\rightarrow(W_{2},V_{i})\rightarrow(Y_{2,i},Y_{3,i}).$ (80)
By (79) and (80),
$\displaystyle n(R_{2}-\delta^{\prime})$
$\displaystyle\leq\sum_{i=1}^{n}\left[I(W_{2},V^{\prime}_{i},V_{i};Y_{2,i})-I(W_{2},V^{\prime}_{i},V_{i};Y_{3,i})\right]-\sum_{i=1}^{n}\left[I(V_{i};Y_{2,i}|W_{2},V^{\prime}_{i})-I(V_{i};Y_{3,i}|W_{2},V^{\prime}_{i})\right]$
$\displaystyle=\sum_{i=1}^{n}\left[I(W_{2},V_{i};Y_{2,i})-I(W_{2},V_{i};Y_{3,i})\right]-\sum_{i=1}^{n}\left[I(V_{i};Y_{2,i}|W_{2},V^{\prime}_{i})-I(V_{i};Y_{3,i}|W_{2},V^{\prime}_{i})\right].$
(81)
Note that $Y_{3,i}$ is conditionally independent of everything else given
$Y_{2,i}$. Hence,
$\displaystyle I(V_{i};Y_{3,i}|W_{2},V^{\prime}_{i})$ $\displaystyle\leq
I(V_{i};Y_{2,i},Y_{3,i}|W_{2},V^{\prime}_{i})$
$\displaystyle=I(V_{i};Y_{2,i}|W_{2},V^{\prime}_{i})+I(V_{i};Y_{3,i}|Y_{2,i},W_{2},V^{\prime}_{i})$
$\displaystyle=I(V_{i};Y_{2,i}|W_{2},V^{\prime}_{i}).$ (82)
Substituting (82) into (81), we have
$\displaystyle R_{2}$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\left[I(W_{2},V_{i};Y_{2,i})-I(W_{2},V_{i};Y_{3,i})\right]+\delta^{\prime}.$
(83)
Finally, let
$\displaystyle U_{i}:=(W_{2},V_{i}).$ (84)
With this definition, (76) and (83) can be rewritten as
$\displaystyle R_{1}$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\left[I(X_{i};Y_{1,i}|U_{i})-I(X_{i};Y_{2,i}|U_{i})\right]+\delta.$
$\displaystyle\mbox{and}\quad\quad R_{2}$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\left[I(U_{i};Y_{2,i})-I(U_{i};Y_{3,i})\right]+\delta^{\prime}.$
(85)
Following the standard single-letterization process (e.g., see [25, Chapter
14.3]), we have the desired converse result.
## Appendix B Proof of Corollary 1
Fix $U=u$. By the generalized Costa EPI (3), we have
$\displaystyle h(\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z}|U=u)$
$\displaystyle\geq\frac{n}{2}\log\left\\{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}|U=u)\right]+|\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{Z}|U=u)\right]\right\\}.$
(86)
Taking expectation over $U$ on both sides of (86), we may obtain
$\displaystyle h(\mathbf{X}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z}|U)$
$\displaystyle\geq\frac{n}{2}{\sf
E}\left[\log\left\\{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}|U=u)\right]+|\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{Z}|U=u)\right]\right\\}\right]$
$\displaystyle\geq\frac{n}{2}\log\left\\{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}{\sf
E}\left[h(\mathbf{X}|U=u)\right]\right]+|\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}{\sf
E}\left[h(\mathbf{X}+\mathbf{Z}|U=u)\right]\right]\right\\}$
$\displaystyle=\frac{n}{2}\log\left\\{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}|U)\right]+|\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{X}+\mathbf{Z}|U)\right]\right\\}$
(87)
where the second inequality follows from Jensen’s inequality [25, Chapter 2.6]
and the convexity of $\log\left(a_{1}e^{x_{1}}+a_{2}e^{x_{2}}\right)$ in
$(x_{1},x_{2})$ for $a_{1},a_{2}\geq 0$. Taking logarithm on both sides of
(87) proves the desired inequality (32).
## Appendix C Proof of Corollary 2
Note that when $\mu=0$, (33) implies that $\mathbf{N}_{1}=\mathbf{N}_{2}$.
Thus, both sides of (34) are equal to zero and the inequality holds trivially
with an equality. For the rest of the proof, we will assume that $\mu>0$. The
proof is rather long so we divide it into several steps.
_Step 1–Generalized eigenvalue decomposition._ We start by applying
generalized eigenvalue decomposition [23] to the positive define matrices
$\mathbf{B}^{*}+\mathbf{N}_{1}$ and $\mathbf{B}^{*}+\mathbf{N}_{2}$. There
exists an _invertible_ generalized eigenvector matrix $\mathbf{V}$ such that
$\displaystyle\mathbf{V}^{\textsf{T}}(\mathbf{B}^{*}+\mathbf{N}_{1})\mathbf{V}=\mathbf{\Lambda}_{1}$
(88) and
$\displaystyle\mathbf{V}^{\textsf{T}}(\mathbf{B}^{*}+\mathbf{N}_{2})\mathbf{V}=\mathbf{\Lambda}_{2}$
(89)
where $\mathbf{\Lambda}_{1}$ and $\mathbf{\Lambda}_{2}$ are positive definite
_diagonal_ matrices. Let
$\displaystyle\mathbf{\Lambda}_{3}:=\mathbf{V}^{\textsf{T}}(\mathbf{B}^{*}+\mathbf{N}_{3})\mathbf{V}$
(90)
be an $n\times n$ positive definite matrix. By (33),
$\displaystyle\mathbf{\Lambda}_{1}^{-1}+\mu\mathbf{\Lambda}_{3}^{-1}=(1+\mu)\mathbf{\Lambda}_{2}^{-1}.$
(91)
Thus, $\mathbf{\Lambda}_{3}$ is also diagonal. Moreover, since
$\mathbf{N}_{1}\preceq\mathbf{N}_{3}$,
$\displaystyle\mathbf{\Lambda}_{1}-\mathbf{\Lambda}_{3}=\mathbf{V}^{\textsf{T}}(\mathbf{N}_{1}-\mathbf{N}_{3})\mathbf{V}\preceq
0.$
and hence
$\displaystyle\mathbf{\Lambda}_{1}\preceq\mathbf{\Lambda}_{3}.$ (92)
_Step 2–Choosing matrix parameter $\mathbf{A}$._ Let
$\tilde{\mathbf{\Lambda}}_{3}=\mathbf{\Lambda}_{3}+\epsilon\mathbf{I}$ for
some $\epsilon>0$, and let $\tilde{\mathbf{\Lambda}}_{2}$ be an $n\times n$
matrix such that
$\displaystyle\mathbf{\Lambda}_{1}^{-1}+\mu\tilde{\mathbf{\Lambda}}_{3}^{-1}=(1+\mu)\tilde{\mathbf{\Lambda}}_{2}^{-1}.$
(93)
Clearly, $\tilde{\mathbf{\Lambda}}_{2}$ is diagonal. Moreover, by (92)
$\displaystyle\mathbf{\Lambda}_{1}\prec\tilde{\mathbf{\Lambda}}_{3}.$ (94)
Note that $\mu>0$ so by (93) and (94)
$\displaystyle\mathbf{\Lambda}_{1}\prec\tilde{\mathbf{\Lambda}}_{2}\prec\tilde{\mathbf{\Lambda}}_{3}.$
(95)
Comparing (91) and (93) and using the fact that
$\mathbf{\Lambda}_{3}\prec\tilde{\mathbf{\Lambda}}_{3}$, we have
$\displaystyle\mathbf{\Lambda}_{2}\prec\tilde{\mathbf{\Lambda}}_{2}.$ (96)
Now let
$\displaystyle\mathbf{Y}_{1}:=\mathbf{V}^{\textsf{T}}(\mathbf{X}+\mathbf{Z}_{1})$
$\displaystyle\mathbf{Y}_{2}:=\mathbf{V}^{\textsf{T}}(\mathbf{X}+\widetilde{\mathbf{Z}}_{2})$
and
$\displaystyle\mathbf{Y}_{3}:=\mathbf{V}^{\textsf{T}}(\mathbf{X}+\widetilde{\mathbf{Z}}_{3})$
where $\widetilde{\mathbf{Z}}_{2}$ and $\widetilde{\mathbf{Z}}_{3}$ are
Gaussian $n$-vectors with covariance matrices
$\displaystyle\widetilde{\mathbf{N}}_{2}$
$\displaystyle=\mathbf{V}^{-\textsf{T}}\tilde{\mathbf{\Lambda}}_{2}\mathbf{V}^{-1}-\mathbf{B}^{*}$
$\displaystyle\succ\mathbf{V}^{-\textsf{T}}\mathbf{\Lambda}_{2}\mathbf{V}^{-1}-\mathbf{B}^{*}$
$\displaystyle=(\mathbf{B}^{*}+\mathbf{N}_{2})-\mathbf{B}^{*}$
$\displaystyle=\mathbf{N}_{2}$
and
$\displaystyle\widetilde{\mathbf{N}}_{3}$
$\displaystyle=\mathbf{V}^{-\textsf{T}}\tilde{\mathbf{\Lambda}}_{3}\mathbf{V}^{-1}-\mathbf{B}^{*}$
$\displaystyle=\mathbf{V}^{-\textsf{T}}(\mathbf{\Lambda}_{3}+\epsilon\mathbf{I})\mathbf{V}^{-1}-\mathbf{B}^{*}$
$\displaystyle=(\mathbf{B}^{*}+\mathbf{N}_{3}+\epsilon\mathbf{V}^{-\textsf{T}}\mathbf{V}^{-1})-\mathbf{B}^{*}$
$\displaystyle=\mathbf{N}_{3}+\epsilon\mathbf{V}^{-\textsf{T}}\mathbf{V}^{-1}$
respectively and are independent of $\mathbf{X}$. The covariance matrices of
$\mathbf{Y}_{k}$, $k=1,2,3$, can be calculated as
$\mathbf{V}^{\textsf{T}}[{\sf
Cov}(\mathbf{X})-\mathbf{B}^{*}]\mathbf{V}+\mathbf{\Lambda}_{1}$,
$\mathbf{V}^{\textsf{T}}[{\sf
Cov}(\mathbf{X})-\mathbf{B}^{*}]\mathbf{V}+\tilde{\mathbf{\Lambda}}_{2}$ and
$\mathbf{V}^{\textsf{T}}[{\sf
Cov}(\mathbf{X})-\mathbf{B}^{*}]\mathbf{V}+\tilde{\mathbf{\Lambda}}_{3}$,
respectively. Thus, $\mathbf{Y}_{2}$ and $\mathbf{Y}_{3}$ can be equivalently
written as
$\displaystyle\mathbf{Y}_{3}=\mathbf{Y}_{1}+\mathbf{Z}$ and
$\displaystyle\mathbf{Y}_{2}=\mathbf{Y}_{1}+\mathbf{A}^{\frac{1}{2}}\mathbf{Z}$
where $\mathbf{Z}$ is a Gaussian $n$-vector with covariance matrix
$\tilde{\mathbf{\Lambda}}_{3}-\mathbf{\Lambda}_{1}\succ 0$ and is independent
of $\mathbf{Y}_{1}$, and
$\displaystyle\mathbf{A}$
$\displaystyle:=(\tilde{\mathbf{\Lambda}}_{2}-\mathbf{\Lambda}_{1})(\tilde{\mathbf{\Lambda}}_{3}-\mathbf{\Lambda}_{1})^{-1}.$
(97)
Clearly, $\mathbf{A}$ is diagonal. Moreover, by (95)
$0\prec\mathbf{A}\prec\mathbf{I}$.
_Step 3–Applying generalized Costa’s EPI._ By the generalized Costa EPI (3),
$\displaystyle
h(\mathbf{Y}_{2}|U)\geq\frac{n}{2}\log\left\\{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{Y}_{1}|U)\right]+|\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{Y}_{3}|U)\right]\right\\}.$
Thus,
$\displaystyle h($ $\displaystyle\mathbf{Y}_{1}|U)+\mu
h(\mathbf{Y}_{3}|U)-(1+\mu)h(\mathbf{Y}_{2}|U)$ $\displaystyle\leq
h(\mathbf{Y}_{1}|U)+\mu
h(\mathbf{Y}_{3}|U)-\frac{(1+\mu)n}{2}\log\left\\{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{Y}_{1}|U)\right]+|\mathbf{A}|^{\frac{1}{n}}\exp\left[\frac{2}{n}h(\mathbf{Y}_{3}|U)\right]\right\\}.$
(98)
Now we consider the function
$\displaystyle f(b,c)=b+\mu
c-\frac{(1+\mu)n}{2}\log\left[|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp\left(\frac{2b}{n}\right)+|\mathbf{A}|^{\frac{1}{n}}\exp\left(\frac{2c}{n}\right)\right].$
Note that
$\displaystyle\nabla f(b,c)$
$\displaystyle=\left[\begin{matrix}\displaystyle{1-(1+\mu)\frac{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp(2b/n)}{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp(2b/n)+|\mathbf{A}|^{\frac{1}{n}}\exp(2c/n)}}\\\\[8.53581pt]
\displaystyle{\mu-(1+\mu)\frac{|\mathbf{A}|^{\frac{1}{n}}\exp(2c/n)}{|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp(2b/n)+|\mathbf{A}|^{\frac{1}{n}}\exp(2c/n)}}\end{matrix}\right]$
and
$\displaystyle\nabla^{2}f(b,c)=-\frac{2(1+\mu)}{n}\frac{|\mathbf{A}|^{\frac{1}{n}}|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp[(2b+2c)/n]}{\left[|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\exp(2b/n)+|\mathbf{A}|^{\frac{1}{n}}\exp(2c/n)\right]^{2}}\left[\begin{matrix}1&-1\\\
-1&1\end{matrix}\right]\preceq 0.$
So $f(b,c)$ is concave in $(b,c)$. By setting $\nabla f(b,c)=0$, the global
maximum is achieved when
$\displaystyle
c=b+\frac{n}{2}\log\left[\mu\left(\frac{|\mathbf{I}-\mathbf{A}|}{|\mathbf{A}|}\right)^{\frac{1}{n}}\right]$
and the maximum is given by
$\displaystyle\frac{\mu
n}{2}\log\left[\mu\left(\frac{|\mathbf{I}-\mathbf{A}|}{|\mathbf{A}|}\right)^{\frac{1}{n}}\right]-\frac{(1+\mu)n}{2}\log\left[(1+\mu)|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\right].$
Hence,
$\displaystyle h(\mathbf{Y}_{1}|U)+$ $\displaystyle\mu
h(\mathbf{Y}_{3}|U)-(1+\mu)h(\mathbf{Y}_{2}|U)$ $\displaystyle\leq\frac{\mu
n}{2}\log\left[\mu\left(\frac{|\mathbf{I}-\mathbf{A}|}{|\mathbf{A}|}\right)^{\frac{1}{n}}\right]-\frac{(1+\mu)n}{2}\log\left[(1+\mu)|\mathbf{I}-\mathbf{A}|^{\frac{1}{n}}\right].$
(99)
_Step 4–Calculating $\log|\mathbf{A}|$ and $\log|\mathbf{I}-\mathbf{A}|$._
Note that (93) can be rewritten as
$\displaystyle\mu(\mathbf{\Lambda}_{1}^{-1}-\tilde{\mathbf{\Lambda}}_{3}^{-1})=(1+\mu)(\mathbf{\Lambda}_{1}^{-1}-\tilde{\mathbf{\Lambda}}_{2}^{-1})$
which gives
$\displaystyle\left|\frac{\tilde{\mathbf{\Lambda}}_{2}-\mathbf{\Lambda}_{1}}{\tilde{\mathbf{\Lambda}}_{3}-\mathbf{\Lambda}_{1}}\right|$
$\displaystyle=\left(\frac{\mu}{1+\mu}\right)^{n}\left|\frac{\tilde{\mathbf{\Lambda}}_{2}}{\tilde{\mathbf{\Lambda}}_{3}}\right|.$
(100)
Similarly, we have
$\displaystyle(\mathbf{\Lambda}_{1}^{-1}-\tilde{\mathbf{\Lambda}}_{3}^{-1})=(1+\mu)(\tilde{\mathbf{\Lambda}}_{2}^{-1}-\tilde{\mathbf{\Lambda}}_{3}^{-1})$
and hence
$\displaystyle\left|\frac{\tilde{\mathbf{\Lambda}}_{3}-\tilde{\mathbf{\Lambda}}_{2}}{\tilde{\mathbf{\Lambda}}_{3}-\mathbf{\Lambda}_{1}}\right|$
$\displaystyle=\left(\frac{1}{1+\mu}\right)^{n}\left|\frac{\tilde{\mathbf{\Lambda}}_{2}}{\tilde{\mathbf{\Lambda}}_{1}}\right|.$
(101)
According to the definition of $\mathbf{A}$ in (97),
$\displaystyle\log|\mathbf{A}|$
$\displaystyle=\log\left|\frac{\tilde{\mathbf{\Lambda}}_{2}-\mathbf{\Lambda}_{1}}{\tilde{\mathbf{\Lambda}}_{3}-\mathbf{\Lambda}_{1}}\right|$
$\displaystyle=\log\left[\left(\frac{\mu}{1+\mu}\right)^{n}\left|\frac{\tilde{\mathbf{\Lambda}}_{2}}{\tilde{\mathbf{\Lambda}}_{3}}\right|\right]$
(102)
and
$\displaystyle\log|\mathbf{I}-\mathbf{A}|$
$\displaystyle=\log\left|\frac{\tilde{\mathbf{\Lambda}}_{3}-\tilde{\mathbf{\Lambda}}_{2}}{\tilde{\mathbf{\Lambda}}_{3}-\mathbf{\Lambda}_{1}}\right|$
$\displaystyle=\log\left[\left(\frac{1}{1+\mu}\right)^{n}\left|\frac{\tilde{\mathbf{\Lambda}}_{2}}{\mathbf{\Lambda}_{1}}\right|\right]$
(103)
where (102) and (103) follow (100) and (101), respectively. Substituting (102)
and (103) into (99), we have
$\displaystyle h(\mathbf{Y}_{1}|U)+\mu
h(\mathbf{Y}_{3}|U)-(1+\mu)h(\mathbf{Y}_{2}|U)$
$\displaystyle\leq\frac{1}{2}\log|\mathbf{\Lambda}_{1}|+\frac{\mu}{2}\log|\tilde{\mathbf{\Lambda}}_{3}|-\frac{1+\mu}{2}\log|\tilde{\mathbf{\Lambda}}_{2}|.$
(104)
_Step 5–Letting $\epsilon\downarrow 0$._ Note that
$\tilde{\mathbf{\Lambda}}_{3}=\mathbf{\Lambda}_{3}+\epsilon\mathbf{I}\rightarrow\mathbf{\Lambda}_{3}$
and
$\widetilde{\mathbf{N}}_{3}=\mathbf{N}_{3}+\epsilon\mathbf{V}^{-\textsf{T}}\mathbf{V}^{-1}\rightarrow\mathbf{N}_{3}$
in the limit as $\epsilon\downarrow 0$. Moreover, by (93) we have
$\tilde{\mathbf{\Lambda}}_{2}\rightarrow\mathbf{\Lambda}_{2}$ and hence
$\displaystyle\widetilde{\mathbf{N}}_{2}$
$\displaystyle=\mathbf{V}^{-\textsf{T}}\tilde{\mathbf{\Lambda}}_{2}\mathbf{V}^{-1}-\mathbf{B}^{*}$
$\displaystyle\rightarrow\mathbf{V}^{-\textsf{T}}\mathbf{\Lambda}_{2}\mathbf{V}^{-1}-\mathbf{B}^{*}$
$\displaystyle=(\mathbf{B}^{*}+\mathbf{N}_{2})-\mathbf{B}^{*}$
$\displaystyle=\mathbf{N}_{2}.$
Letting $\epsilon\downarrow 0$ on both sides of (104), we have
$\displaystyle h(\mathbf{V}^{\textsf{T}}(\mathbf{X}+\mathbf{N}_{1})|U)+\mu
h(\mathbf{V}^{\textsf{T}}(\mathbf{X}+\mathbf{N}_{3})|U)-$
$\displaystyle(1+\mu)h(\mathbf{V}^{\textsf{T}}(\mathbf{X}+\mathbf{N}_{2})|U)$
$\displaystyle\leq\frac{1}{2}\log|\mathbf{\Lambda}_{1}|+\frac{\mu}{2}\log|\mathbf{\Lambda}_{3}|-\frac{1+\mu}{2}\log|\mathbf{\Lambda}_{2}|.$
(105)
Using the fact that
$\displaystyle
h(\mathbf{V}^{\textsf{T}}(\mathbf{X}+\mathbf{N}_{1})|U)=h(\mathbf{X}+\mathbf{N}_{1}|U)+\log|\mathbf{V}|$
and
$\displaystyle\log|\mathbf{\Lambda}_{k}|$
$\displaystyle=\log|\mathbf{V}^{\textsf{T}}(\mathbf{B}^{*}+\mathbf{N}_{k})\mathbf{V}|$
$\displaystyle=\log|\mathbf{B}^{*}+\mathbf{N}_{k}|+2\log|\mathbf{V}|$
for $k=1,2,3$, the desired inequality (34) can be obtained from (105). This
completes the proof of the corollary.
## Appendix D Proof of Corollary 3
Here, we prove Corollary 3 using mathematical induction. Note that when $K=1$,
(35) implies that $\mathbf{N}_{1}=\mathbf{N}_{0}$. Thus, the inequality (36)
holds trivially with equality for any $(U,\mathbf{X})$ independent of
$(\mathbf{Z}_{0},\mathbf{Z}_{1})$.
Assume that the inequality (36) holds for $K=Q-1$. Let $\mathbf{N}$ be an
$n\times n$ symmetric matrix such that
$\displaystyle(\mathbf{B}^{*}+\mathbf{N})^{-1}=\sum_{k=1}^{Q-1}\mu_{k}^{\prime}(\mathbf{B}^{*}+\mathbf{N}_{k})^{-1}$
(106)
where
$\displaystyle\mu_{k}^{\prime}:=\frac{\mu_{k}}{\sum_{j=1}^{Q-1}\mu_{j}},\quad
j=1,\ldots,Q.$
By the assumption $\mathbf{N}_{1}\preceq\ldots\preceq\mathbf{N}_{Q-1}$, we
have from (106)
$\displaystyle\mathbf{N}_{1}\preceq\mathbf{N}\preceq\mathbf{N}_{Q-1}.$ (107)
Let $\mathbf{Z}$ be a Gaussian random $n$-vector with covariance matrix
$\mathbf{N}$ and independent of $(U,\mathbf{X})$. By the induction assumption
and (106),
$\displaystyle\sum_{k=1}^{Q-1}\mu_{k}^{\prime}h(\mathbf{X}+\mathbf{Z}_{k}|U)-h(\mathbf{X}+\mathbf{Z}|U)$
$\displaystyle\leq\sum_{k=1}^{Q-1}\frac{\mu_{k}^{\prime}}{2}\log|\mathbf{B}+\mathbf{N}_{k}|-\frac{1}{2}\log|\mathbf{B}+\mathbf{N}|.$
(108)
On the other hand, substitute (106) into (35) and we have
$\displaystyle(\mathbf{B}+\mathbf{N})^{-1}+\mu_{Q}^{\prime}(\mathbf{B}+\mathbf{N}_{Q})^{-1}=(1+\mu_{Q}^{\prime})(\mathbf{B}+\mathbf{N}_{0})^{-1}.$
Note from (107) that $\mathbf{N}\preceq\mathbf{N}_{Q-1}\preceq\mathbf{N}_{Q}$.
Thus, by Corollary 2
$\displaystyle h(\mathbf{X}+\mathbf{Z}|U)+\mu_{Q}^{\prime}h$
$\displaystyle(\mathbf{X}+\mathbf{Z}_{Q}|U)-(1+\mu_{Q}^{\prime})h(\mathbf{X}+\mathbf{Z}_{0}|U)$
$\displaystyle\leq\frac{1}{2}\log|\mathbf{B}+\mathbf{N}|+\frac{\mu_{Q}^{\prime}}{2}\log|\mathbf{B}+\mathbf{N}_{Q}|-\frac{1+\mu_{Q}^{\prime}}{2}\log|\mathbf{B}+\mathbf{N}_{0}|.$
(109)
Putting together (108) and (109), we have
$\displaystyle\sum_{j=1}^{Q}\mu_{j}h(\mathbf{X}+\mathbf{Z}_{j}|U)-h(\mathbf{X}+\mathbf{Z}_{0}|U)$
$\displaystyle\leq\sum_{j=1}^{Q}\frac{\mu_{j}}{2}\log|\mathbf{B}+\mathbf{N}_{j}|-\frac{1}{2}\log|\mathbf{B}+\mathbf{N}_{0}|.$
This proved the induction step and hence the corollary.
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|
arxiv-papers
| 2009-03-17T19:05:55 |
2024-09-04T02:49:01.219248
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruoheng Liu, Tie Liu, H. Vincent Poor and Shlomo Shamai (Shitz)",
"submitter": "Ruoheng Liu",
"url": "https://arxiv.org/abs/0903.3024"
}
|
0903.3119
|
11institutetext: Department of Astronomy, Nanjing University, Nanjing 210093,
China
# On the afterglow from the receding jet of gamma-ray burst
Xin Wang Y. F. Huang hyf@nju.edu.cn S. W. Kong
(Received month day 2009 / Accepted month day 2009)
According to popular progenitor models of gamma-ray bursts, twin jets should
be launched by the central engine, with a forward jet moving toward the
observer and a receding jet (or the counter jet) moving backwardly. However,
in calculating the afterglows, usually only the emission from the forward jet
is considered. Here we present a detailed numerical study on the afterglow
from the receding jet. Our calculation is based on a generic dynamical
description, and includes some delicate ingredients such as the effect of the
equal arrival time surface. It is found that the emission from the receding
jet is generally rather weak. In radio bands, it usually peaks at a time of
$t\geq 1000$ d, with the peak flux nearly 4 orders of magnitude lower than the
peak flux of the forward jet. Also, it usually manifests as a short plateau in
the total afterglow light curve, but not as an obvious rebrightening as once
expected. In optical bands, the contribution from the receding jet is even
weaker, with the peak flux being $\sim 23$ magnitudes lower than the peak flux
of the forward jet. We thus argue that the emission from the receding jet is
very difficult to detect. However, in some special cases, i.e., when the
circum-burst medium density is very high, or if the parameters of the receding
jet is quite different from those of the forward jet, the emission from the
receding jet can be significantly enhanced and may still emerge as a marked
rebrightening. We suggest that the search for receding jet emission should
mostly concentrate on nearby gamma-ray bursts, and the observation campaign
should last for at least several hundred days for each event.
###### Key Words.:
gamma rays: bursts — ISM: jets and outflows — stars: neutron
## 1 Introduction
Thanks to the discovery of X-ray, optical and radio afterglows of gamma-ray
bursts (GRBs), it is now clear that most GRBs are situated at cosmological
distances (Costa et al. 1997; van Paradijs et al. 1997; Frail et al. 1997). A
lot of progresses have been achieved during the past decade (Piran 2004;
Mészáros 2006). Especially, through the detection of GRB 030329, the
association of long GRBs with supernovae is firmly established (Hjorth et al.
2003), which strongly supports the collapsar model as the energy mechanism for
long GRBs (Woosley 1993; MacFadyen & Woosley 1999). Theoretically, the
collapse of a massive star will most likely give birth to a black hole,
surrounded by a temporal accretion disk. It is a common sense that the
accretion system will produce double-sided jets (MacFadyen & Woosley 1999;
Aloy et al. 2000; Rhoads 1999; Mészáros 2002). The GRB can be observed only
when our line of sight is right on one of the two jets. The collimation of GRB
ejecta can be tested observationally, through various beaming effects, such as
the achromatic break in GRB afterglow light curves (Sari et al. 1999; Liang et
al. 2008), the polarization in both the main burst phase and the afterglow
phase (Lazzati 2006), the predicted existence of orphan afterglows (Rhoads
1997; Huang et al. 2002; Granot & Loeb 2003), and the energy crisis already
noted in some GRBs (Frail et al. 2001). In fact, more and more observational
evidences have been accumulated today, supporting the idea that many GRB
ejecta might be highly collimated.
Current studies on the beaming effects are mostly concentrated on the emission
from the forward jet, i.e., the jet moving toward the observer. The emission
from the receding jet (or the counter jet) is generally omitted. It is
interesting to note that this ingredient recently has been studied by a few
authors (Granot & Loeb 2003; Li & Song 2004). By some simple analytical
derivations, Li & Song (2004) argued that the emission from the receding jet
can be detected in a few cases in the non-relativistic phase of GRB
afterglows. However, previous studies did not consider some important effects,
such as the action of the equal arrival time surface (EATS). Recently, Zhang &
MacFadyen (2009) presented a two-dimensional simulation of GRB outflow. The
emission from the receding jet has also been included in their calculations,
but they did not investigate the effects of various parameters on the receding
jet component.
In this paper, we will present our detailed numerical investigation on the
emission from the receding jet of GRB in the deep Newtonian stage. Although
the GRB jet may be complicatedly structured (Mészáros et al. 1998; Kumar &
Granot 2003; Huang et al. 2004), and the circum-burst environment may be wind
medium and even associated with some complex density variations (Mészáros et
al. 1998; Chevalier & Li 2000; Gou et al. 2001; Wu et al. 2004), here we will
only consider the simplest situation, i.e, the homogeneous double-sided jets
expanding into a homogeneous interstellar medium, which is favored by some
recent fits (Huang et al. 2000a; Yost et al. 2003).
The structure of our paper is organized as follows. §2 is mainly a review of
the dynamics and radiation model we used in our calculations. In §3 we present
the numerical results, together with our tentative explanations. §4 is our
conclusion and discussion.
## 2 Model Description
In the afterglow phase, the GRB ejecta expands into the interstellar medium
(ISM) and is decelerated continuously, giving rise to a strong external shock.
The swept-up electrons are accelerated by the blastwave, producing the
afterglow mainly through synchrotron radiation. In radio bands, the shell is
no longer optically thin, so that the synchrotron self-absorption should be
considered. In our study, we will use the simplified dynamical equations
suggested by Huang et al. (1999, 2000b), which is consistent with the self-
similar solution of Blandford & McKee (1976) in the ultra-relativistic phase,
and is consistent with the Sedov solution (Sedov 1969) in the non-relativistic
phase. The beaming effects (Rhoads 1997, 1999) can also be conveniently
simulated in this way. Here, for completeness, we first describe the dynamics
and the radiation process briefly.
### 2.1 Hydrodynamical Evolution
In our description, $t$ is the photon arrival time measured in the lab frame;
$R$ is the radial coordinate measured in the burst frame relative to the
initiation point; $m$ is the rest mass of the swept-up medium; $\theta$ is the
half-opening angle of the ejecta; $\gamma$ is the bulk Lorentz factor of the
moving material; $p$ is the electron distribution index which is typically
between 2 and 3; $n$ is the number density of ISM; $\xi_{\rm e}$ and
$\xi^{2}_{\rm B}$ are the energy equipartition factors for electrons and the
comoving magnetic field. We further denote the initial values of the rest
mass, the isotropic equivalent energy, the Lorentz factor and the half-opening
angle of the ejecta as $M_{\rm ej},E_{\rm 0,iso},\gamma_{0},\theta_{\rm j}$,
respectively.
The overall dynamical evolution of the GRB ejecta can be depicted by
$\displaystyle\frac{\mathrm{d}R}{\mathrm{d}t}$ $\displaystyle=$
$\displaystyle\beta c\gamma\left({\gamma+\sqrt{\gamma^{2}-1}}\right),$ (1)
$\displaystyle\frac{\mathrm{d}m}{\mathrm{d}R}$ $\displaystyle=$ $\displaystyle
2\pi\left({1-\cos\theta}\right)R^{2}nm_{\rm p},$ (2)
$\displaystyle\frac{\mathrm{d}\theta}{\mathrm{d}t}$ $\displaystyle=$
$\displaystyle\frac{c_{\rm s}}{R}\left({\gamma+\sqrt{\gamma^{2}-1}}\right),$
(3) $\displaystyle\frac{\mathrm{d}\gamma}{\mathrm{d}m}$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{2}-1}{M_{\rm ej}+\varepsilon
m+2(1-\varepsilon)\gamma m},$ (4)
where $\beta=\sqrt{1-1/\gamma^{2}}$, $c$ is the speed of light, and $c_{\rm
s}$ is the comoving sound speed, which can be calculated by $c_{\rm
s}^{2}=\hat{\gamma}(\hat{\gamma}-1)(\gamma-1)c^{2}/\left[1+\hat{\gamma}(\gamma-1)\right]$
with $\hat{\gamma}\approx(4\gamma+1)/({3\gamma})$ being a reasonable
approximation for the adiabatic index. In Equation (4), $\varepsilon$ is the
radiative efficiency. In the extreme case, $\varepsilon=0$ means adiabatic
condition and $\varepsilon=1$ refers to highly radiative situation. Note that
in realistic case, $\varepsilon$ should evolve gradually from 1 to 0, in about
several hours.
Equations (1) — (4) is a convenient description of GRB afterglow dynamics that
is applicable in both the initial ultra-relativistic phase and the late
Newtonian phase.
### 2.2 Radiation Process
Basically, we assume that the shock-accelerated electrons follow a power-law
distribution according to their energies, $\mathrm{d}N^{\prime}_{\rm
e}/{\mathrm{d}\gamma_{\rm e}}\propto\gamma_{\rm e}^{-p}$, However, to ensure
that the calculation in the deep Newtonian phase is correct, we need to modify
the basic distribution function as $\mathrm{d}N^{\prime}_{\rm
e}/{\mathrm{d}\gamma_{\rm e}}\propto\left(\gamma_{\rm e}-1\right)^{-p}$ (Huang
& Cheng 2003). The minimum and maximum Lorentz factors of electrons can be
calculated as $\gamma_{\rm e,\min}=\xi_{e}(\gamma-1)m_{\rm p}(p-2)/[m_{\rm
e}(p-1)]+1$ and $\gamma_{\rm e,\max}=\sqrt{6\pi e/\left(\sigma_{\rm
T}B^{\prime}\right)}\approx 10^{8}(B^{\prime}/1{\rm G})^{-1/2}$, where
$B^{\prime}$ is the comoving magnetic field strength, $m_{\rm p}$ and $m_{\rm
e}$ are masses of proton and electron, respectively. As usual, we assume that
the energy ratio of magnetic field with respect to internal energy is
$\xi^{2}_{\rm B}$, so that the energy density of magnetic field is $B^{\prime
2}/(8\pi)=\xi^{2}_{\rm
B}\left(\hat{\gamma}-1\right)^{-1}(\hat{\gamma}\gamma+1)(\gamma-1)nm_{\rm
p}c^{2}$.
The cooling of electrons due to synchrotron radiation will lead to a steep
distribution function above a critical Lorentz factor, $\gamma_{\rm c}$. The
expression for $\gamma_{\rm c}$ can be derived as $\gamma_{\rm c}=6\pi m_{\rm
e}c/\left(\sigma_{\rm T}\gamma B^{\prime 2}t\right)$, where $\sigma_{\rm T}$
is the Thompson scattering cross section (Sari et al. 1998). Considering all
the above ingredients, we finally use the following electron distribution
function in our calculations (Huang & Cheng 2003):
1\. $\gamma_{\rm c}\leq\gamma_{\rm e,\min}$,
$\frac{\mathrm{d}N^{\prime}_{\rm e}}{\mathrm{d}\gamma_{\rm
e}}\propto\left\\{\begin{array}[]{l}\left(\gamma_{\rm e}-1\right)^{-2}\hskip
22.76228pt(\gamma_{\rm c}\leq\gamma_{\rm e}<\gamma_{\rm e,\min}),\\\
\left(\gamma_{\rm e}-1\right)^{-(p+1)}\hskip 9.53186pt(\gamma_{\rm
e,\min}\leq\gamma_{\rm e}\leq\gamma_{\rm e,\max});\\\ \end{array}\right.$ (5)
2\. $\gamma_{\rm e,\min}<\gamma_{\rm c}\leq\gamma_{\rm e,\max}$,
$\frac{\mathrm{d}N^{\prime}_{\rm e}}{\mathrm{d}\gamma_{\rm
e}}\propto\left\\{\begin{array}[]{l}\left(\gamma_{\rm e}-1\right)^{-p}\hskip
22.1931pt(\gamma_{\rm e,\min}\leq\gamma_{\rm e}\leq\gamma_{\rm c}),\\\
\left(\gamma_{\rm e}-1\right)^{-(p+1)}\hskip 9.67383pt(\gamma_{\rm
c}<\gamma_{\rm e}\leq\gamma_{\rm e,\max});\\\ \end{array}\right.$ (6)
3\. $\gamma_{\rm c}>\gamma_{\rm e,\max}$,
$\frac{\mathrm{d}N^{\prime}_{\rm e}}{\mathrm{d}\gamma_{\rm
e}}\propto\left(\gamma_{\rm e}-1\right)^{-p}\hskip 15.6491pt(\gamma_{\rm
e,\min}\leq\gamma_{\rm e}\leq\gamma_{\rm e,\max}).$ (7)
In the comoving frame, the synchrotron radiation power at $\nu^{\prime}$ is
(Rybicki & Lightman 1979)
$P^{\prime}(\nu^{\prime})=\frac{\sqrt{3}e^{3}B^{\prime}}{m_{\rm
e}c^{2}}\int_{\min\left(\gamma_{\rm e,\min}\ ,\ \gamma_{\rm
c}\right)}^{\gamma_{\rm e,\max}}{\left(\frac{\mathrm{d}N^{\prime}_{\rm
e}}{\mathrm{d}\gamma_{\rm e}}\right)}\
F\left(\frac{\nu^{\prime}}{\nu^{\prime}_{\rm e}}\right)d\gamma_{\rm e},$ (8)
with $F(x)=x\int_{x}^{+\infty}{K_{5/3}(k)\,\mathrm{d}k}$ being the Bessel
function and $\nu^{\prime}_{\rm e}=3\gamma^{2}_{\rm e}eB^{\prime}/\left(4\pi
m_{\rm e}c\right)$ being the characteristic emission frequency (Shu 1991;
Longair 1992). To calculate the radio afterglows, we must consider the
synchrotron self-absorption. The optical depth of synchrotron self-absorption
can be obtained as
$\tau_{\nu^{\prime}}=\frac{\sqrt{3}e^{3}B^{\prime}}{8\pi m^{2}_{\rm
e}c^{2}\nu^{\prime 2}}\int_{\min\left(\gamma_{\rm e,\min}\ ,\ \gamma_{\rm
c}\right)}^{\gamma_{\rm e,\max}}{(q+2)\left(\frac{\mathrm{d}n^{\prime}_{\rm
e}}{\mathrm{d}\gamma_{\rm e}}\right)\frac{1}{\gamma_{\rm e}}}\
F\left(\frac{\nu^{\prime}}{\nu^{\prime}_{\rm e}}\right)d\gamma_{\rm e},$ (9)
where $\mathrm{d}n^{\prime}_{\rm e}/\mathrm{d}\gamma_{\rm e}$ denotes the
column density distribution of electrons measured in the comoving frame on the
line of sight; $q$ is the electron power-law distribution index which varies
from $2$ to $p+1$ for fast-cooling and from $p$ to $p+1$ for slow-cooling. The
synchrotron self-absorption will affect the radiation by a reduction-factor
(Waxman et al. 1998)
$f(\tau)=\frac{1-e^{-\tau_{\nu^{\prime}}}}{\tau_{\nu^{\prime}}}.$ (10)
Let us define the Doppler-factor as
$D=\left[\gamma\left(1-\beta\mu\right)\right]^{-1}$ (Mészáros 2006), where
$\mu=\cos\Theta$ and $\Theta$ is the angle between the velocity of the
emitting material and the line of sight. Also we denote the viewing angle as
$\theta_{\rm obs}$. Then the observed frequency is $\nu=D\nu^{\prime}/(1+z)$,
and the observed flux density from a point-like source is
$F_{\nu}=\frac{(1+z)D^{3}}{4\pi d_{\rm
L}^{2}}f(\tau)P^{\prime}\left[(1+z)D^{-1}\nu\right],$ (11)
where $d_{\rm L}$ is the luminosity distance. Finally, we can integrate the
flux density over the EATS (Waxman 1997; Sari 1998) determined by
$t_{\rm obs}=(1+z)\int\frac{\mathrm{d}R}{\beta\gamma cD}\equiv\rm{const}.$
(12)
## 3 Numerical Results
In this section, we present our numerical results concerning the emission from
the receding jet. First of all, for simplicity, we assume that the twin jets
have the same characteristics, i.e., the same initial energy, opening angle,
initial Lorentz factor, and the circum-burst ISM density. We also assume that
the microphysics shock parameters ($p$, $\xi_{\rm e}$, $\xi^{2}_{\rm B}$) are
the same for the receding and forward blastwaves. For convenience, we define a
set of parameter values as the “standard” condition: $n=1/{\rm cm}^{3}$,
$E_{\rm 0,iso}=10^{53}{\rm ergs}$, $\theta_{\rm j}=0.1$, $\varepsilon=0$,
$\xi_{\rm e}=0.1$, $\xi^{2}_{\rm B}=0.01$, $p=2.5$, $\theta_{\rm obs}=0$, and
$\gamma_{0}=300$. These values are typical in the study of GRB afterglows. For
redshift, we adopt the value of $z=0.1$ (which corresponds to $d_{\rm L}=454$
Mpc according to the popular cosmology model, Wright 2006).
Firstly we illustrate the evolution of the Lorentz factors of the twin jets in
Fig. 1. Note that the X-axis is observers’ time. For the observer, the
dynamical evolution of the receding jet is quite different from that of the
forward jet, especially in the relativistic phase. We see that in a rather
long time ($t\sim 50$ d), $\gamma$ of the receding jet remains almost
constant. This is due to the time delay induced by the long distance between
the twin jets. It also implies that the emission from the receding jet will be
very weak in this period, since it is highly beamed backwardly. At the
observers’ time of $t\sim 340$ d, the Lorentz factor of the receding jet is
still more than 10, while the forward jet’s Lorentz factor has already
decreased to less than 1.1 .
In Fig. 2, we show some examples of the equal arrival time surfaces (EATSes)
at three moments. As expected, at any particular moment, the typical radius of
the surface is much larger for the forward jet branch as compared with that
for the receding jet branch. Also, we notice that the curvature of the two
branches is quite different. Generally, the EATS is much flatter on the
receding jet. Another interesting feature is that the area of the EATS on the
forward branch is much larger than that of the corresponding receding branch.
Fig. 3 shows the radio and optical afterglow light curves under the “standard”
condition (thick lines). Here, the thick dotted line corresponds to emission
from the forward jet, the thick dashed line corresponds to emission from the
receding jet, and the thick sold line is the total light curve. Under the
“standard” condition, for the forward jet, the afterglow light curve (the
dotted line) becomes slightly flattened in the non-relativistic phase. It is
consistent with previous results in the deep Newtonian phase (Huang & Cheng
2003). Also it can be seen that the receding jet really can contribute a
significant portion in the total emission at very late stage. The role played
by the receding jet is reasonably more important in the radio band than in the
optical band. However, the dashed component is generally not very strong, so
that it can only lead to a plateau in the total light curve, but not an
obvious rebrightening or a marked peak as expected by Li & Song (2004).
Interestingly, our result is consistent with the simulation of Zhang &
Macfadyen (2009). We believe that the discrepancy between our numerical result
and Li & Song’s analytical result mainly comes from the effect of the EATS.
Below, we will give some detailed analyses on this point. Additionally, it
should be noted that in the radio band, the peak flux of the receding
component is about 4 orders of magnitude weaker than that of the forward
component. It essentially means that the receding component is very weak, and
is very difficult to detect. In the optical band, the condition is even more
awkward. The peak flux of the receding component is about 23 magnitudes dimmer
than that of the forward component in R band. Even comparing with the flux of
the forward jet at the jet break time, it is still 16 — 17 magnitudes weaker.
So, in optical band, it is even much more difficult to observe the receding
jet component.
According to Li & Song (2004), the time when the receding jet becomes notably
visible ($t_{\rm NR}^{\rm RJ}$) is relevant to the time when the forward jet
enters the non-relativistic phase ($t_{\rm NR}$), i.e.,
$t_{\rm NR}^{\rm RJ}=t_{\rm NR}+\frac{2r_{\rm NR}}{c},$ (13)
where $r_{\rm NR}$ is the radius of the forward jet at $t_{\rm NR}$. In the
standard frame work (Blandford & Mckee 1976; Rhoads 1999), the sphere-like
phase of a highly collimated GRB ejecta ends at the so called jet break time
determined by $\gamma_{\rm j}=1/\theta_{\rm j}$, with the shock radius being
$r_{\rm j}=\left(3E_{\rm 0,iso}\theta_{\rm j}^{2}/\left[{4\pi nm_{\rm
p}c^{2}}\right]\right)^{1/3}$. After the sphere-like phase, the jet spreads
laterally at the co-moving sound speed $c_{\rm s}$ so that we have
$\gamma\propto t^{-1/2}$ and $r_{\rm NR}\approx r_{\rm j}$ (Rhoads 1999). Then
finally we obtain
$t_{\rm NR}=\frac{1}{2c}\left(\frac{3E_{\rm 0,iso}\theta_{\rm j}^{2}}{4\pi
nm_{\rm p}c^{2}}\right)^{1/3},\\\ \qquad t_{\rm NR}^{\rm RJ}=5t_{\rm NR}.$
(14)
Adopting the standard values of our parameters, Equation (14) yields $t_{\rm
NR}\approx 104$ d and $t_{\rm NR}^{\rm RJ}\approx 520$ d. After correcting for
the cosmological time dilation ($z=0.1$), we get the corresponding observers’
time of $t_{1}=(1+z)\ t_{\rm NR}\approx 114$ d and $t_{3}=(1+z)\ t_{\rm
NR}^{\rm RJ}\approx 572$ d. In fact, in Fig. 2, the EATSes for these two
moments have been displayed. So, according to Li & Song’s suggestion, the
contribution from the receding jet should peak at $t_{3}\approx 572$ d. In our
Fig. 3, for the “standard” condition, the peak is postponed to $t_{\rm
peak}\sim 1140$ d for 8.46 GHz, and to $t_{\rm peak}\sim 1700$ d in R band.
So, the EATS effect and the deceleration of the external shock can lead to
some subtle difference between the analytical results and the numerical
results. Actually, Zhang & MacFadyen’s numerical results have clearly shown
that the observers’ time does not equal to the burst frame time at $t_{\rm
NR}$ (Zhang & MacFadyen 2009). Unfortunately, in previous analysises it is
usually assumed that these two times are equal.
Another reason that suppresses the rebrightening of the receding jet is as
follows. According to Li & Song’s analysis, at the observers’ time $t_{3}$,
the receding jet should be at the radius of $r_{\rm NR}$. However, from our
Fig. 2, we see that the typical radius of the EATS at $t_{3}$ on the receding
jet is much smaller than the radius of the forward jet at $t_{1}$. The reason
is again due to the EATS effect. It means that the receding jet still does not
decelerate enough at $t_{3}$ (actually, the bulk Lorentz factor is still
3.95), and its emission is still mainly directed forwardly (not backwardly
toward the observer). Additionally, Fig. 2 shows clearly that the area of the
receding jet at $t_{3}$ (corresponding to $t_{\rm NR}^{\rm RJ}$ ) is much
smaller than that of the forward jet at $t_{1}$ (corresponding to $t_{\rm
NR}$). So, the number of electrons involved in the radiation process is
typically much smaller on the receding jet at $t_{\rm NR}^{\rm RJ}$, as
compared to that on the forward jet at $t_{\rm NR}$. Due to the above reasons,
the contribution from the receding jet is naturally much weaker than that
deduced from $L_{\rm\nu}^{\rm RJ}(t)\approx L_{\rm\nu}(t-4t_{\rm NR}),\ (t\geq
t_{\rm NR}^{\rm RJ})$ (Equation (7) in Li & Song (2004) ).
However, although the receding jet emission is generally very weak in our
“standard” condition, we guess that in some special cases it still can be
enhanced. Obviously, a denser environment will help to decelerate the jet more
quickly, thus lead to a smaller $t_{\rm peak}$ and a higher intensity. In Fig.
3, we have also plotted in thin lines our numerical results for a double-sided
jet that locates in a dense circum-burst medium ($n=1000/{\rm cm}^{3}$). Note
that other parameters involved here are the same as the “standard” case.
Encouragingly, in Fig. 3(a) we see that the peak time of the receding jet can
be as early as $t_{\rm peak}\sim 150$ d, with the peak flux as large as a few
mJy in radio band (i.e., only several times less than the peak level of the
forward jet). In Fig. 3(b), the optical contribution from the receding jet is
still very weak, with the peak flux being about 28m.
In Fig. 4, we plot the afterglow light curves in more radio and
optical/infrared bands. Generally speaking, $t_{\rm peak}$ is about 1140 d in
radio bands and is about 1700 d in optical bands. Such a difference in the
peak time is insignificant, considering that the frequency difference between
radio and optical wavelengths is really huge. We notice that $t_{\rm peak}$
almost remains the same from radio to X-ray bands in Fig. 7 of Zhang &
MacFadyen (2009). Thus our results are roughly consistent with Zhang &
MacFadyen’s. Another interesting conclusion that can be drawn from our Figs. 3
and 4 is that at lower frequency, the relative intensity of the receding jet
component (its peak flux), as compared with the peak of the forward jet
component, becomes stronger. Such a tendency can also be roughly seen in Fig.
7 of Zhang & MacFadyen (2009).
Fig. 5 illustrates the effects of some parameters ($n$, $E_{\rm 0,iso}$,
$\theta_{\rm j}$, and $\varepsilon$) on the receding jet component in the
afterglow light curve. Fig. 5(a) shows that the circum-burst medium density
($n$) affects the peak time ($t_{\rm peak}$) of receding jet dramatically. A
larger number density usually leads to a smaller $t_{\rm peak}$. The strength
of the receding jet component is also obviously enhanced. It again hints that
the receding jet component is most likely detectable in a dense environment.
Similarly, the initial kinetic energy ($E_{\rm 0,iso}$) also affects $t_{\rm
peak}$ significantly, with larger $E_{\rm 0,iso}$ corresponding to a larger
$t_{\rm peak}$ (Fig. 5(b)). The effect of the initial jet opening angle
($\theta_{\rm j}$) on $t_{\rm peak}$ can also be clearly seen in Fig. 5(c). It
should be further noted that the receding jet component is more marked when
the opening angle is smaller. In Fig. 5(d), we can observe an obvious
rebrightening when the radiation efficiency ($\varepsilon$) is large. However,
in realistic case, $\varepsilon$ is unlikely to be so large. Actually, at such
late stages, the external shock should be adiabatic, so that $\varepsilon$
should be nearly zero.
In Fig. 5(d), we also plot the radio afterglow light curves for double-sided
jets under some special physical assumptions. The dash-dotted line is plotted
by assuming that both the forward jet and the receding jet do not experience
any lateral expansion. Since the deceleration of the jets is much slower in
this case, we see that the receding jet component emerges much later and is
also much less obvious as compared with our “standard” case. The dotted line
is plotted by assuming a much smaller initial Lorentz factor
($\gamma_{0}=30$), which may correspond to the so called failed GRBs (Huang et
al. 2002). The receding jet component emerges slightly earlier as compared
with the solid line, but its role becomes less significant correspondingly.
Fig. 6 illustrates the effects of other four parameters ($\xi_{\rm e}$,
$\xi_{\rm B}^{2}$, $p$, and $\theta_{\rm obs}$) on the receding jet component.
Generally speaking, a larger $\xi_{\rm e}$ and/or $\xi_{\rm B}^{2}$ can
enhance the receding jet component markedly. On the other hand, although $p$
has an important influence on the overall afterglow light curve, its impact on
the relative strength of the receding jet component is not significant. Again,
note that in all the cases, the contribution from the receding jet only
emerges as a plateau, but not as any obvious rebrightening. In Fig. 6(d), when
the observing angle ($\theta_{\rm obs}$) increases, the forward jet component
becomes weaker, while the receding jet component becomes stronger. It is in
good accord with our expectation (also see Granot & Loeb 2003). However, the
contribution from the receding jet still generally plays a minor role in the
total afterglow light curve. Additionally, for off-axis twin jets, the GRB
from the forward jet is un-observable, so that even the afterglow from the
forward jet itself (i.e., the orphan afterglow) is difficult to observe. Note
that in Fig. 6(d), when $\theta_{\rm obs}=\pi/2$ (i.e., the thick solid line),
the contribution from the receding jet and the forward jet are actually equal.
Equation (14) tells us that the peak time of the receding component should be
relevant to the 3 parameters of $n$, $E_{\rm 0,iso}$, $\theta_{\rm j}$; on the
other hand, other parameters such as $\xi_{\rm e}$, $\xi_{\rm B}^{2}$, $p$ do
not affect the peak time. These tendency can be clearly seen in Figs. 5 and 6.
In all the above calculations, we have assumed that the conditions and
parameters of the twin jets are the same. However, this may not be the case
for realistic GRBs. The circum-burst environment and the micro-physics
parameters may actually be different for the twin jets, as that may happen in
the two component jet structure (Huang et al. 2004; Jin et al. 2007; Racusin
et al. 2008). In Fig. 7, we have plotted the overall afterglow light curves by
assuming different parameters for the forward jet and the receding jet. In
each panel of Fig. 7, we first plot a common light curve (the solid line) by
adopting the standard parameter set, but change $\xi_{\rm e}$ to 0.01 and
change $\xi^{2}_{\rm B}$ to $10^{-4}$. We then increase the values of
$\xi_{\rm e}$, $\xi_{\rm B}^{2}$, and $n$ for the receding jet to see their
effects on the afterglow light curve. It is encouraging to see that the
emission from the receding jet really can be greatly enhanced, so that it can
manifest as an obvious rebrightening in the overall light curve. In Fig. 7(a),
7(b) and 7(d), the peak flux of the rebrightening can be nearly 100 times
larger than the “background” level in the best cases. It is imaginable that in
the most favorable cases, when all $\xi_{\rm e}$, $\xi_{\rm B}^{2}$ and $n$
are larger for the receding jet at the same time, the rebrightening will be
even more remarkable. However, note that the contrary condition may also exist
in realistic GRBs, i.e., these parameters may also be smaller for the receding
jet. Then the emission from the receding jet will be completely unnoticeable.
## 4 Conclusion and Discussion
We have studied the emission of the receding jet numerically. The effect of
the EATS is included in our calculations. Clearly, this effect plays an
important role in the process. It is found that the contribution from the
receding jet is generally quite weak. In most cases, it only manifests as a
short plateau in the overall afterglow light curve, but not a marked
rebrightening. The flux density of the plateau is usually much less than 100
$\mu$Jy in radio bands even at a small redshift of $z=0.1$ . If we place the
GRB at a more typical redshift of $z=1$, then the flux density of the plateau
will be less than 0.1 $\mu$Jy at 8.46 GHz. We noticed that the observed radio
afterglow emission is generally on the level of 0.1 — 1 mJy at about the peak
time. After several months, the radio afterglow usually decreases to a very
low level, and is submersed by the emission from the host galaxy, whose
strength can be 40 — 70 $\mu$Jy (Berger et al. 2001). Additionally, the error
bar of radio observations is usually $\sim$ 30 — 50 $\mu$Jy at very late
stages (Frail et al. 2003). Thus the contribution from the receding jet, i.e.
the plateau, is actually very difficult to detect currently, especially for
those GRBs at $z\sim 1$. Our results are consistent with a recent
observational report by van der Horst et al. (2008), who failed to detect any
clear clues of the receding jet emission.
However, as shown in our Fig. 7, if the micro-physics parameters of the
receding jet were different from the forward jet, or if the receding jet were
in a much denser environment, then it is still possible that the contribution
from the receding jet can be greatly enhanced. For example, if $\xi_{\rm e}$
and/or $\xi_{\rm B}^{2}$ of the receding jet is much larger than that of the
forward jet, then the receding jet can really manifest as an obvious
rebrightening.
Also, our Fig. 5(a) shows that a dense circum-burst environment can suppress
the emission of the forward jet, and enhance the contribution from the
receding jet. If the GRB occurs in a very dense molecular cloud with
$n>10^{3}/{\rm cm}^{3}$ (Dai & Lu 1999), the contribution from the receding
jet may be much easier to detect. Additionally, if the GRB is very near to us
at the same time, then the possibility of successfully detecting the receding
jet is very high (see the thin lines in Fig. 3(a)).
In short, we believe that the effort of trying to search for the afterglow
contribution from the receding jet is still meaningful. If observed, it would
provide useful clues to study the circum-burst environment and the micro-
physics of external shocks. We suggest that nearby GRBs (with redshift $z\leq
0.1$) should be good candidates for such studies.
###### Acknowledgements.
We would like to thank the anonymous referee for constructive suggestions that
lead to an overall improvement of this study. We also thank Z. Li for
stimulating discussion. This research was supported by the National Natural
Science Foundation of China (grant 10625313), and by the National Basic
Research Program of China (grant 2009CB824800). Xin Wang is also supported by
2008’ National Undergraduate Innovation Program of China (grant 081028441).
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Figure 1: The evolution of the Lorentz factors of the twin jets. The solid
line corresponds to the receding jet and the dashed line is plotted for the
forward jet. The twin jets are in “standard” condition as defined in Sect. 3.
The observers’ time has been corrected for the cosmological effect ($z=0.1$).
Figure 2: Schematic illustration of the EATSes at three moments, $t_{1}\approx
114$ d, $t_{2}\approx 286$ d and $t_{3}\approx 572$ d. In this calculation, we
have used the “standard” parameter set as defined in Sect. 3. “O” is the
position of the central engine, and the observer is on the far right side with
Y=0. The dotted lines indicate the jet boundary. For the receding jet, the
EATSes are plotted in thick solid lines, while for the forward jet the
surfaces are plotted in thin solid lines. Note that on the forward jet
branches, the bulk Lorentz factors of the material at the peak of the EATSes
are 1.17, 1.07, and 1.03 for $t_{1}$, $t_{2}$, and $t_{3}$, respectively. On
the receding jet branches, the bulk Lorentz factors of the material at the
peak of the EATSes are 56.07, 11.79, and 3.95 for $t_{1}$, $t_{2}$, and
$t_{3}$, respectively.
Figure 3: 8.46 GHz radio afterglow (a) and R-band optical afterglow (b) from
the forward jet and the receding jet. The thick lines are plotted for a
“standard” double-sided jet as defined in Sect. 3. The thin lines are plotted
for the double-sided jet with only one parameter altered as compared with the
“standard” condition, i.e. $n=1000/{\rm cm}^{3}$. In each group, the dotted
line reflects the emission from the forward jet, the dashed line reflects the
contribution from the receding jet, and the solid line is the total light
curve.
Figure 4: Multiwavelength afterglow light curves of a double-sided jet. Radio
afterglows are illustrated in panel (a), and optical/IR afterglows are plotted
in panel (b). In this calculation, we have used the “standard” parameter set
as defined in Sect. 3.
Figure 5: The effects of various parameters ($n$, $E_{\rm 0,iso}$,
$\theta_{\rm j}$, and $\varepsilon$) on the 8.46 GHz radio afterglow light
curves of double-sided jets. In each panel, “(s)” corresponds to the
“standard” condition as defined in Sect. 3, and other lines are drawn with
only one certain parameter altered or one condition changed. In panel (d), the
dash-dotted line is plotted for a double-sided jet without lateral expansion;
and the dotted line is plotted for a double-sided jet with a low initial
Lorentz factor ($\gamma_{\rm 0}=30$), which may correspond to the so called
failed GRBs.
Figure 6: The effects of various parameters ($\xi_{\rm e}$, $\xi_{\rm B}^{2}$,
$p$, and $\theta_{\rm obs}$) on the 8.46 GHz radio afterglow light curves of
double-sided jets. In each panel, “(s)” corresponds to the “standard”
condition as defined in Sect. 3, and other lines are drawn with only one
certain parameter altered.
Figure 7: 8.46 GHz radio afterglow light curves of double-sided jets. In this
figure, we assume that the parameters of the receding jet can be different
from those of the forward jet. In each panel, the solid line is plotted under
the “standard” condition, i.e., the parameters are completely the same for the
twin jets (but note that we have evaluated $\xi_{\rm e}$ as 0.01 and $\xi_{\rm
B}^{2}$ as $10^{-4}$ here). For other light curves, one or two parameters are
changed for the receding jet, to see its effect on the afterglows.
|
arxiv-papers
| 2009-03-18T10:00:18 |
2024-09-04T02:49:01.233691
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Wang, Y. F. Huang, Si-Wei Kong",
"submitter": "Y. F. Huang",
"url": "https://arxiv.org/abs/0903.3119"
}
|
0903.3136
|
# Experimental and numerical investigation of the reflection coefficient and
the distributions of Wigner’s reaction matrix for irregular graphs with
absorption
Michał Ławniczak1, Oleh Hul1, Szymon Bauch1, Petr Šeba2,3, and Leszek Sirko1
1Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46,
02-668 Warszawa, Poland
2University of Hradec Králové, Hradec Králové, Czech Republic
3Institute of Physics, Academy of Sciences of the Czech Republic,
Cukrovarnická 10, 162 53 Praha, Czech Republic
(April 16, 2008)
###### Abstract
We present the results of experimental and numerical study of the distribution
of the reflection coefficient $P(R)$ and the distributions of the imaginary
$P(v)$ and the real $P(u)$ parts of the Wigner’s reaction $K$ matrix for
irregular fully connected hexagon networks (graphs) in the presence of strong
absorption. In the experiment we used microwave networks, which were built of
coaxial cables and attenuators connected by joints. In the numerical
calculations experimental networks were described by quantum fully connected
hexagon graphs. The presence of absorption introduced by attenuators was
modelled by optical potentials. The distribution of the reflection coefficient
$P(R)$ and the distributions of the reaction $K$ matrix were obtained from the
measurements and numerical calculations of the scattering matrix $S$ of the
networks and graphs, respectively. We show that the experimental and numerical
results are in good agreement with the exact analytic ones obtained within the
framework of random matrix theory (RMT).
###### pacs:
05.45.Mt,03.65.Nk
Quantum graphs of connected one-dimensional wires were introduced more than
sixty years ago by Pauling Pauling . Next the same idea was used by Kuhn Kuhn
to describe organic molecules by free electron models. Quantum graphs can be
considered as idealizations of physical networks in the limit where the
lengths of the wires are much bigger than their widths, i.e. assuming that the
propagating waves remain in a single transversal mode. Among the systems
modelled by quantum graphs one can find e.g., electromagnetic optical
waveguides Flesia ; Mitra , mesoscopic systems Imry ; Kowal , quantum wires
Ivchenko ; Sanchez and excitation of fractons in fractal structures Avishai ;
Nakayama . Recently it has been shown that quantum graphs are excellent
paradigms of quantum chaos Kottossmilansky ; Kottos ; Prlkottos ; Zyczkowski ;
Kus ; Tanner ; Kottosphyse ; Kottosphysa ; Gaspard ; Blumel ; Hul2004 . More
complicated and thus more realistic systems - microwave networks with moderate
absorption strength $\gamma=2\pi\Gamma/\Delta\leq 7.1$, where $\Gamma$ is the
absorption width and $\Delta$ is the mean level spacing, have been
experimentally investigated in Hul2005 ; Hul2007 . Other interesting open
objects - quantum graphs with leads - have been analyzed in details in
Kottosphyse ; Kottosphysa . However, the properties of networks and graphs
with strong absorption have not been studied experimentally neither
numerically so far. Therefore, in this paper we study experimentally and
numerically the distribution of the reflection coefficient $P(R)$ and the
distributions of the Wigner’s reaction matrix Akguc2001 (in the literature
often called $K$ matrix Fyodorov2004 ) for networks (graphs) with time
reversal symmetry ($\beta=1$) in the presence of strong absorption.
In the case of a single channel antenna experiment the $K$ matrix is related
to the scattering matrix $S$ by the following relation
$S=\frac{1-iK}{1+iK}.$ (1)
Eq. (1) holds for the systems with absorption but without direct processes
Fyodorov2004 . It is important to mention that the function $Z=iK$ has a
direct physical meaning of the electric impedance that has been recently
measured in the microwave cavity experiment Anlage2005 . In the one channel
case the $S$ matrix can be parameterized as
$S=\sqrt{R}e^{i\theta},$ (2)
where $R$ is the reflection coefficient and $\theta$ the phase.
Properties of the statistical distributions of the $S$ matrix with direct
processes and imperfect coupling have been studied theoretically in several
important papers Lopez1981 ; Doron1992 ; Brouwer1995 ; Savin2001 ;
Fyodorov2003 ; Fyodorov2005 . Recently the distribution of the $S$ matrix has
been also measured experimentally for chaotic microwave cavities with
absorption Kuhl2005 . The distribution $P(R)$ of the reflection coefficient
$R$ and the distributions of the imaginary $P(v)$ and the real $P(u)$ parts of
the Wigner’s reaction $K$ matrix are theoretically known for any dimensionless
absorption strength $\gamma$ Fyodorov2004 ; Savin2005 . In the case of time
reversal systems (symmetry index $\beta=1$) $P(R)$ has been studied
experimentally by Méndez-Sánchez et al. Sanchez2003 . The distributions $P(v)$
and $P(u)$ have been studied for chaotic microwave cavities in Anlage2005 ;
Anlage2006 and for microwave networks for moderate absorption strength
$\gamma\leq 7.1$ in Hul2005 ; Hul2007 . For systems without time reversal
symmetry ($\beta=2$) and a single perfectly coupled channel $P(R)$ was
calculated by Beenakker and Brouwer Beenakker2001 while the exact formulas
for the distributions $P(v)$ and $P(u)$ were given by Fyodorov and Savin
Fyodorov2004 .
In the experiment quantum graphs can be simulated by microwave networks. The
analogy between quantum graphs and microwave networks is based upon the
equivalency of the Schrödinger equation describing the quantum system and the
telegraph equation describing the microwave circuit Hul2004 .
A general microwave network consists of $N$ vertices connected by bonds e.g.,
coaxial cables. A coaxial cable consists of an inner conductor of radius
$r_{1}$ surrounded by a concentric conductor of inner radius $r_{2}$. The
space between the inner and the outer conductors is filled with a homogeneous
material having a dielectric constant $\varepsilon$. For a frequency $\nu$
below the onset of the next TE11 mode only the fundamental TEM mode can
propagate inside a coaxial cable. (This mode is in the literature often called
a Lecher wave.) The cut-off frequency of the TE11 mode is
$\nu_{c}\simeq\frac{c}{\pi(r_{1}+r_{2})\sqrt{\varepsilon}}=32.9$ GHz Jones ,
where $r_{1}$ = 0.05 cm is the inner wire radius of the coaxial cable (SMA-
RG402), while $r_{2}$ = 0.15 cm is the inner radius of the surrounding
conductor, and $\varepsilon\simeq 2.08$ is the Teflon dielectric constant
Breeden1967 ; Savytskyy2001 .
From the experimental point of view absorption of the networks can be changed
by the change of the bonds’ (cables’) lengths Hul2004 or more effectively by
the application of microwave attenuators Hul2005 ; Hul2007 . In the numerical
calculations weak absorption inside the cables can be described with the help
of complex wave vector Hul2004 . We will show that strong absorption inside an
attenuator can be described by a simple optical potential. The corresponding
mathematical theory has been developed in Ex1 .
The distribution $P(R)$ of the reflection coefficient $R$ and the
distributions of the imaginary and real parts of the Wigner’s reaction matrix
$K$ for microwave networks with absorption were found using the impedance
approach Anlage2005 ; Anlage2006 ; Hul2007 . In this approach the real and
imaginary parts of the normalized impedance $Z$
$Z=\frac{\textrm{Re }Z_{n}+i(\textrm{Im }Z_{n}-\textrm{Im }Z_{r})}{\textrm{Re
}Z_{r}}$ (3)
of a chaotic microwave system are measured, with
$Z_{n(r)}=Z_{0}(1+S_{n(r)})/(1-S_{n(r)})$ being the network (radiation)
impedance expressed by the network (radiation) scattering matrix $S_{n(r)}$
and $Z_{0}$ is the characteristic impedance of the transmission line. The
radiation impedance $Z_{r}$ is the impedance seen at the output of the
coupling structure for the same coupling geometry, but with the vertices of
the network removed to infinity. The Wigner’s reaction matrix $K$ can be
expressed by the normalized impedance as $K=-iZ$. The scattering matrix $S$ of
a network for the perfect coupling case (no direct processes present) required
for the calculation of the reflection coefficient $R$ (see Eq. (2)) can be
finally extracted from the formula $S=(1-Z)/(1+Z)$.
Figure 1: (a) The scheme of the experimental set-up for measurements of the
scattering matrix $S_{n}$ of the microwave fully connected networks with
absorption. Absorption in the networks was varied by the change of the
attenuators. (b) The scheme of the setup used to measure the radiation
scattering matrix $S_{r}$. Instead of a network five 50 $\Omega$ loads were
connected to the 6-joint.
Figure 1(a) shows the experimental setup for measuring the single-channel
scattering matrix $S_{n}$ of fully connected hexagon microwave networks
necessary for finding of the impedance $Z_{n}$. We used Hewlett-Packard 8720A
microwave vector network analyzer to measure the scattering matrix $S_{n}$ of
the networks in the frequency window: 7.5–11.5 GHz. The networks were
connected to the vector network analyzer through a lead - a HP 85131-60012
flexible microwave cable - connected to a 6-joint vertex. The other five
vertices of the networks were connected by 5-joints. Each bond of the network
presented in Fig. 1(a) contains a microwave attenuator.
The radiation impedance $Z_{r}$ was found experimentally by measuring the
scattering matrix $S_{r}$ of the 6-joint connector with five joints terminated
by 50 $\Omega$ loads (see Figure 1(b)).
The experimentally measured fully connected hexagon networks were described in
numerical calculations by quantum fully connected hexagon graphs with one lead
attached to the 6-joint vertex. In the calculations attenuators (absorbers)
were modelled by optical potentials Ex1 . To be explicit we suppose that the
fully connected hexagon graph $\Upsilon$ with one coupled antenna is described
in the Hilbert space
$L^{2}(\Upsilon):=\bigoplus_{(j,n)}L^{2}(0,\ell_{jn})\bigoplus
L^{2}(0,\infty)$, where $\ell_{jn}$ stays for the lengths of the bond
connecting the vertices $j$ and $n$ and the halfline $(0,\infty)$ describes
the attached antenna.
We define the Schrödinger operator $H$ by
$H{\psi_{jn}:=\,-\psi^{\prime\prime}_{jn}+U_{jn}\psi_{jn}},$ (4)
with $\psi_{jn}\in L^{2}(0,\ell_{jn})$ for the bonds and
$H{\psi_{0n}:=\,-\psi_{0n}^{\prime\prime}},$ (5)
with $\psi_{0n}\in L^{2}(0,\infty)$ describing the wave function of the
antenna connected to the vertex $n$ (note that the ”infinite” vertex of the
antenna has index 0) .
At the vertices the wave functions are linked together with the boundary
values
$\psi_{jn}(j):=\lim_{x\to
0+}\psi_{jn}(x)\,,\quad\psi^{\prime}_{jn}(j):\lim_{x\to
0+}\psi^{\prime}_{jn}(x)\,,$ (6)
satisfying boundary conditions $\,\psi_{jn}(j)=\psi_{jm}(j)=:\psi_{j}$ for all
$n,m$ describing connected vertices, and
$\sum_{n\in\nu(j)}\psi^{\prime}_{jn}(j)=0.$ (7)
The optical potentials $U_{jn}$ are purely imaginary and describe the absorber
inserted between the vertices $(j,n)$.
Since the graph $\Upsilon$ is infinite (due to the attached antenna) we can
look for solutions of the equation
$H\psi=k^{2}\psi,$ (8)
referring to the continuous spectrum, where $k$ is the wave vector. For
microwaves propagating inside a lossless bond with a dielectric constant
$\varepsilon$ the wave vector $k=2\pi\varepsilon\nu/c$, where $\nu$ and $c$
denote the frequency of a microwave field and the speed of light in the
vacuum, respectively. To solve this equation we used the graph duality
principle Ex1 . According to this principle we need to solve the equation
$-f^{\prime\prime}+U_{jn}f=k^{2}f$ on $[0,\ell_{jn}]$ satisfying the
normalized Dirichlet boundary conditions
$u_{jn}(\ell_{jn})=1\\!-\\!(u_{jn})^{\prime}(\ell_{jn})=0\,,\;\;v_{jn}(0)1\\!-\\!(v_{jn})^{\prime}(0)=0\,.$
(9)
The Wronskian of this solution is naturally equal to
$W_{jn}-v_{jn}(\ell_{jn})=u_{jn}(0)$. Then according to Ex1 the corresponding
boundary values (6) satisfy the equation
$\sum_{n}{\psi_{n}\over
W_{jn}}\,-\,\left(\,\sum_{n\in\nu(j)}{(v_{jn})^{\prime}(\ell_{jn})\over
W_{jn}}\,\right)\psi_{j}\,=\,0\,.$ (10)
Conversely, any solution $\psi_{j}$ of the system (10) determines a solution
of (8) by
$\displaystyle\psi_{jn}(x)={\psi_{n}\over W_{jn}}\,u_{jn}(x)-\,{\psi_{j}\over
W_{jn}}\,v_{jn}(x)\;\;$ $\displaystyle{\rm if}$ $\displaystyle n=1,..,6\,,$
(11) $\displaystyle\psi_{jn}(x)=-\,{\psi_{j}\over W_{jn}}\,v_{jn}(x)\;\;$
$\displaystyle{\rm if}$ $\displaystyle n=0\,.$ (12)
As already mentioned the microwave attenuators are modelled by optical
potentials localized inside the inserted component. It is well known that any
smooth and localized potential can be easily approximated by a sequence of
delta potentials inside the support of the potential - see demkov for
details. We will use this fact and express the optical potential as a sum of
$N$ delta-potentials with imaginary coupling constants:
$U(x)=ib\sum_{r=1}^{N}\delta(x-(r-1)l_{b}/(N-1))$. The delta-potentials were
equally spaced inside the length $l_{b}$ of the absorbing element
(attenuator). By changing the number $N$ and the strength $b$ of delta-
potentials we were able to vary absorbing properties as well as reflective
properties of attenuators. We used $N=10$ delta-potentials with $b=0.028$
$m^{-1}$ for simulation of the 1 dB attenuators and $N=12$ delta-potentials
with $b=0.045$ $m^{-1}$ for the 2 dB attenuators. In both cases the length of
the attenuator was $l_{b}=2.65$ cm. Furthermore, in the numerical calculations
of the scattering matrices $S_{n}$ of the graphs the weak absorption inside
the microwave cables was taken into account by replacing the real wave vector
$k$ by the complex vector $k+ia\sqrt{k}$ Goubau , where the absorption
coefficient was assumed to be $a=0.009$ $m^{-1/2}$ Hul2004 .
Figure 2: In the panels (a) and (b) the modulus $|S_{n}|$ and the phase
$\theta$ of the scattering matrix $S$ measured for the network with
$\gamma=19.9$ are plotted in the frequency range 7.5 - 9 GHz. In (c) and (d)
$|S_{n}|$ and $\theta$ of the scattering matrix $S_{n}$ are plotted for the
network with $\gamma=47.9$ in the same frequency range. The measurements have
been done for the two networks which in each bond contained: 1 dB attenuator
((a) and (b)) and 2 dB attenuator ((c) and (d)), respectively. The total
“optical” length of the microwave networks including joints and attenuators
were 574 cm and 554 cm, respectively.
In order to find the distribution $P(R)$ of the reflection coefficient $R$ and
the distributions of the imaginary and real parts of the K matrix we measured
the scattering matrix $S_{n}$ of $88$ and $74$ network configurations
containing in each bond a single 1 dB and 2 dB microwave SMA attenuator,
respectively. The total optical lengths of the microwave networks containing 1
dB attenuators, including joints and attenuators, varied from 574 cm to 656
cm. For the networks with 2 dB attenuators the optical lengths varied from 554
cm to 636 cm. To avoid degeneracy of eigenvalues of the networks the lengths
of the bonds were chosen as incommensurable.
In Figure 2 the modulus $|S_{n}|$ and the phase $\theta$ of the scattering
matrix $S_{n}$ of the microwave networks with $\gamma=19.9\textrm{ and }47.9$,
respectively, are presented in the frequency range 7.5 - 9 GHz. The
measurements were done for two networks containing 1 dB and 2 dB attenuators,
respectively. Their total “optical” lengths including joints and attenuators
were 574 cm and 554 cm, respectively.
For systems with time reversal symmetry ($\beta=1$), the explicit analytic
expression for the distribution $P(R)$ of the reflection coefficient $R$ is
given by Savin2005
$P(R)=\frac{2}{(1-R)^{2}}P_{0}\Bigl{(}\frac{1+R}{1-R}\Bigr{)}.$ (13)
The probability distribution $P_{0}(x)$ is given by the expression
$P_{0}(x)=-\frac{dW(x)}{dx},$ (14)
where the integrated probability distribution $W(x)$ is expressed by the
formula Savin2005
$W(x)=\frac{x+1}{4\pi}\Bigl{[}f_{1}(w)g_{2}(w)+f_{2}(w)g_{1}(w)+h_{1}(w)j_{2}(w)+h_{2}(w)j_{1}(w)\Bigr{]}_{w=(x-1)/2}.$
(15)
The functions $f_{1},g_{1},h_{1},j_{1}$ are defined as follows
$f_{1}(w)=\int_{w}^{\infty}dt\frac{\sqrt{t\mid t-w\mid}e^{-\gamma
t/2}}{(1+t)^{3/2}}\Bigl{[}1-e^{-\gamma}+\frac{1}{t}\Bigr{]},$ (16)
$g_{1}(w)=\int_{w}^{\infty}dt\frac{e^{-\gamma t/2}}{\sqrt{t\mid
t-w\mid}(1+t)^{3/2}},$ (17)
$h_{1}(w)=\int_{w}^{\infty}dt\frac{\sqrt{\mid t-w\mid}e^{-\gamma
t/2}}{\sqrt{t(1+t)}}\Bigl{[}\gamma+(1-e^{-\gamma})(\gamma t-2)\Bigr{]},$ (18)
$j_{1}(w)=\int_{w}^{\infty}dt\frac{e^{-\gamma t/2}}{\sqrt{t\mid
t-w\mid}(1+t)^{1/2}}.$ (19)
Their counterparts with the index 2 are given by the same expressions but the
integration is performed in the interval $t\in[0,w]$ instead of $[w,\infty)$.
Figure 3: Experimental distribution $P(R)$ of the reflection coefficient $R$
for the microwave fully connected hexagon networks at $\bar{\gamma}=19.3$
(open squares) and $\bar{\gamma}=47.7$ (full squares). The corresponding
theoretical distribution $P(R)$ evaluated from the Eq. (13) is marked by the
solid line ($\gamma=19.3$) and dashed line ($\gamma=47.7$), respectively.
Figure 4: Numerical distribution $P(R)$ of the reflection coefficient $R$ for
fully connected hexagon quantum graphs at $\bar{\gamma}=19.3$ (open circles)
and $\bar{\gamma}=47.7$ (full circles). The corresponding theoretical
distribution $P(R)$ evaluated from the Eq. (13) is marked by the solid line
($\gamma=19.3$) and dashed line ($\gamma=47.7$), respectively.
The distributions of the imaginary and the real parts $P(v)$ and $P(u)$ of the
$K$ matrix Fyodorov2004 can be also expressed by the probability distribution
$P_{0}(x)$:
$P(v)=\frac{\sqrt{2}}{\pi
v^{3/2}}\int^{\infty}_{0}dqP_{0}\Bigl{[}q^{2}+\frac{1}{2}\Bigl{(}v+\frac{1}{v}\Bigr{)}\Bigr{]},$
(20)
and
$P(u)=\frac{1}{2\pi\sqrt{u^{2}+1}}\int^{\infty}_{0}dqP_{0}\Bigl{[}\frac{\sqrt{u^{2}+1}}{2}\Bigl{(}q+\frac{1}{q}\Bigr{)}\Bigr{]},$
(21)
where $-v=\textrm{Im}\,K<0$ and $u=\textrm{Re}\,K$ are, respectively, the
imaginary and real parts of the $K$ matrix.
Figure 3 shows the experimental distributions $P(R)$ (squares) of the
reflection coefficient $R$ for two mean values of the parameter
$\bar{\gamma}$, viz., 19.3 and 47.7. The distribution for $\bar{\gamma}=19.3$
is obtained by averaging over 88 realizations of the microwave networks
containing 1 dB attenuators. The distribution for $\bar{\gamma}=47.7$ is
obtained by averaging over 74 realizations of the microwave networks
containing 2 dB attenuators. The experimental values of the $\gamma$ parameter
were estimated for each realization of the network by adjusting the
theoretical mean reflection coefficient $\langle R\rangle_{th}$ to the
experimental one $\langle R\rangle=\langle SS^{{\dagger}}\rangle$, where
$\langle R\rangle_{th}=\int_{0}^{1}dRRP(R).$ (22)
Figure 3 also presents the corresponding distributions $P(R)$ (solid and
dashed lines, respectively) evaluated from Eq. (13). A good overall agreement
of the experimental distributions $P(R)$ with their theoretical counterparts
is seen.
Figure 4 shows the numerically evaluated distributions $P(R)$ (circles) of the
reflection coefficient $R$ for the graphs at $\bar{\gamma}=19.3\textrm{ and
}47.7$ compared to the theoretical ones evaluated from the formula Eq. (13).
The numerical distributions are the result of averaging over 162 and 214
realizations of the graphs with optical potentials simulating 1 dB and 2 dB
attenuators, respectively. The numerical values of $\gamma$ parameter were
also estimated by adjusting the theoretical mean reflection coefficient to the
numerical one. The agreement between the numerical results for
$\bar{\gamma}=47.7$ and the theoretical ones (dashed line) is good. However,
for $\bar{\gamma}=19.3$ for $R<0.15$ some discrepancies between the numerical
results and the theoretical ones (solid line) are visible.
Figure 5: Experimental distribution $P(v)$ of the imaginary part of the $K$
matrix for the two values of the mean absorption parameter:
$\bar{\gamma}=19.3$ (open squares) and $\bar{\gamma}=47.7$ (full squares),
respectively. The corresponding theoretical distribution $P(v)$ evaluated from
the Eq. (20) is marked by the solid line ($\gamma=19.3$) and dashed line
($\gamma=47.7$), respectively.
Figure 6: Numerical distribution $P(v)$ of the imaginary part of the $K$
matrix for the two values of the mean absorption parameter:
$\bar{\gamma}=19.3$ (open circles) and $\bar{\gamma}=47.7$ (full circles),
respectively. The corresponding theoretical distribution $P(v)$ evaluated from
the Eq. (20) is marked by the solid line ($\gamma=19.3$) and dashed line
($\gamma=47.7$), respectively.
In Figure 5 the experimental distribution $P(v)$ of the imaginary part of the
$K$ matrix is shown for the two mean values of the parameter
$\bar{\gamma}=19.3\textrm{ and }47.7$, respectively. The distribution is the
result of averaging over 88 and 74 realizations of the networks with the
attenuators 1 dB and 2 dB, respectively. The experimental results in Figure 5
are in general in good agreement with the theoretical ones. However, both
experimental distributions are slightly higher than the theoretical ones in
the vicinity of their maxima.
Figure 7: Experimental distribution $P(u)$ of the real part of the $K$ matrix
for the two values of the mean absorption parameter: $\bar{\gamma}=19.3$ (open
squares) and $\bar{\gamma}=47.7$ (full squares), respectively. The experiment
is compared to the theoretical distribution $P(u)$ evaluated from the Eq.
(21): solid line ($\gamma=19.3$) and dashed line ($\gamma=47.7$).
Figure 8: Numerical distribution $P(u)$ of the real part of the $K$ matrix for
the two values of the mean absorption parameter: $\bar{\gamma}=19.3$ (open
circles) and $\bar{\gamma}=47.7$ (full circles), respectively. The numerical
results are compared to the theoretical distributions $P(u)$ evaluated from
the Eq. (21): solid line ($\gamma=19.3$) and dashed line ($\gamma=47.7$).
Results of the numerical calculations of the distributions $P(v)$ are shown in
Figure 6 for two mean values of the parameter $\bar{\gamma}=19.3\textrm{ and
}47.7$, respectively. They are compared to the theoretical ones evaluated from
the formula Eq. (20). Figure 6 shows also a good agreement between the
numerical and theoretical results, which confirms usefulness of the optical
potential approach in describing the microwave attenuators.
Measurements of the distribution $P(u)$ of the real part of the Wigner’s
reaction matrix give an additional test of the consistency of the $\gamma$
evaluation. In Figure 7 we show this distribution obtained for two values of
$\bar{\gamma}=19.3\textrm{ and }47.7$, respectively, compared to the
theoretical ones evaluated from the formula Eq. (21). Also here we observe
good overall agreement between the experimental and theoretical results.
However, Figure 7 shows that for the networks with 2 dB attenuators the
theoretical distribution is in the middle ($-0.1<u<0.1$) slightly higher than
the experimental one. According to the definition of the $K$ matrix (see Eq.
(1)) such a behavior of the experimental distribution $P(u)$ suggests
deficiency of small values of $|\textrm{Im}S|$, whose origin is not known.
Moreover, the experimental distribution $P(u)$ obtained for the networks
assembled with 1 dB attenuators is slightly asymmetric for $|u|>0.5$.
In Figure 8 the comparison of the numerical distribution $P(u)$ obtained for
two values of $\bar{\gamma}=19.3\textrm{ and }47.7$, respectively, to the
theoretical one evaluated from the formula Eq. (21) is presented. In this case
we see a good overall agreement between the numerical and theoretical results.
In spite of the above mentioned discrepancies which appeared mainly in the
case of the experimental distribution $P(u)$ the overall good agreement
between the experimental and theoretical results justifies a posteriori the
chosen procedure of calculating the experimental $\gamma$. The same is true
also for the numerical simulations.
In summary, we measured and calculated numerically the distribution of the
reflection coefficient $P(R)$ and the distributions of the imaginary $P(v)$
and the real $P(u)$ parts of the Wigner’s reaction $K$ matrix for irregular
fully connected hexagon networks and graphs in the presence of strong
absorption. In the case of the microwave networks consisting of SMA cables and
attenuators the application of attenuators allowed for effective change of
absorption in the graphs. In the numerical calculations absorption in an
attenuator was modelled by an optical potential. We showed that in the case of
the time reversal symmetry ($\beta=1$) the experimental and numerical results
for $P(R)$, $P(v)$ and $P(u)$ are in good overall agreement with the
theoretical predictions. The agreement of the numerical and theoretical
results strongly confirms the usefulness of the optical potential approach in
the description of the microwave attenuators.
Acknowledgments. This work was partially supported by Polish Ministry of
Science and Higher Education grant No. N202 099 31/0746 and by the Ministry of
Education, Youth and Sports of the Czech Republic within the project LC06002.
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|
arxiv-papers
| 2009-03-18T11:21:06 |
2024-09-04T02:49:01.240936
|
{
"license": "Public Domain",
"authors": "Michal Lawniczak, Oleh Hul, Szymon Bauch, Petr Seba and Leszek Sirko",
"submitter": "Oleh Hul",
"url": "https://arxiv.org/abs/0903.3136"
}
|
0903.3355
|
# Two-Stream Instability Model With Electrons Trapped in Quadrupoles
Paul J. Channell
Los Alamos National Laboratory, Los Alamos, NM 87545
111This work was supported by the US Department of Energy under Contract
Number DE-AC52-06NA25396.
###### Abstract
We formulate the theory of the two-stream instability (e-cloud instability)
with electrons trapped in quadrupole magnets. We show that a linear
instability theory can be sensibly formulated and analyzed. The growth rates
are considerably smaller than the linear growth rates for the two-stream
instability in drift spaces and are close to those actually observed.
## 1 Introduction
The Proton Storage Ring (PSR) at Los Alamos has been troubled for some time,
[1], [2], [3], [4], [5], [6], [7], [8], [9], by an instability that is
probably a two-stream instability of the proton beam with background
electrons, i.e. an electron-cloud instability. We have previously considered
the possibility that the instability for a bunched beam occurred because of
electrons in drift regions that were renewed from turn to turn, [10]; in this
case the phase memory of the coherent motion has to reside in the proton beam
which excited the fresh electrons on each turn which then drove the tail of
the proton bunch to larger amplitudes. In this note we will consider instead
the possibility that the instability is due to electrons that survive from
turn to turn. The most likely place in the ring where the electrons can
survive with coherent phase information from turn to turn is in the
quadrupoles, where they are trapped in the magnetic mirrors formed by the
cusp-shaped fields and can drive the e-p instability in a similar way to free
electrons. In this note we will present a simple model of this two-stream
instability with electrons trapped in quadrupoles.
### 1.1 Electron Trapping and Dynamics
A major assumption of this note is that there are abundant electrons in the
PSR; experimentally this has been observed, though the source is not
completely clear. It is likely that some form of beam induced multipactor
gives rise to the electrons, perhaps initiated by a very small number of lost
beam particles, though other explanations are possible. Normally, one would
expect that with a bunched beam electrons would be expelled during the beam
gap and that one could not have an e-p instability; however, the electrons,
however they are produced, cannot be driven quickly to the walls in the
quadrupoles which act in the transverse direction as very effective magnetic
mirrors. It thus seems possible that electrons in the quadrupoles could drive
the e-p instability. To investigate this possibility further in this section
we will make simple estimates of the electron motion in quadrupoles to
establish that electrons can be trapped there for multiple turns and thus
carry coherent phase information to drive the instabillity. A more accurate
investigation of the electron motion in the complex geometry can and should be
done using computer codes, [11].
The dominant aspect of electron motion in the quads is the rapid rotation
about the magnetic field lines; the cyclotron frequency is
$f_{c}={eB\over 2\pi mc},$ (1)
where $e$ is the charge, $B$ is the magnetic field, $m$ is the mass, and $c$
is the speed of light. For electrons we have
$f_{c}=2.8{\cal B}\,\,{\rm GHz},$ (2)
where ${\cal B}$ is the magnetic field measured in kilogauss. Thus, even very
low fields near the axis give rise to cyclotron frequencies that are hundreds
of MHz; most electrons will have cyclotron frequencies that are multiple GHz.
The radius of this rotational motion is, for electrons,
$\rho=3.37*10^{-3}{\sqrt{\cal E}\over{\cal B}}\,\,{\rm cm},$ (3)
where ${\cal E}$ is the transverse electron energy in eV. Only very energetic
electrons in low field regions will have gyroradii approaching $1$ cm; most
will have gyroradii that are much less than $1$ mm. Electrons are thus
confined transversely to the magnetic field on cyclotron orbits of small radii
and many are confined longitudinally (for electrons) along the magnetic field
by the increasing magnetic field with radius, i.e. by ‘mirror’ confinement.
(Note that longitudinal for the electrons is transverse to the beam
direction.)
Of course, particles with large components of velocity parallel to the
magnetic field, i.e. those in the ‘loss cone’, are not confined; presumably
these give rise to the electron ‘tracking’ that has been observed in the
quadrupoles. We will ignore the rapid electron cyclotron motion in the quads
and concentrate on the longitudinal electron mirror motion and transverse
drifts due to electric fields and to magnetic field non-uniformity, i.e. a
‘guiding center’ description of the trapped electrons.
In the transverse direction (for electrons) there are three components of
electron drift, that due to the gradient in the magnetic field, the so-called
${\nabla B\;}$drift, that due to the field line curvature, and that due to any
electric fields that are present, the ${E\times B\;}$drift. These drifts give
rise to electron velocities perpendicular to the magnetic field; in fact, in
the quads, the drifts are along the direction of the beam axis and thus can
lead to electron loss out the ends of the quads.
The ${\nabla B\;}$drift and curvature velocities are given by
$V_{\nabla B}={m(v_{\perp}^{2}+2v_{\parallel}^{2})\over
2eB}{\hat{b}\times\nabla B\over B}c,$ (4)
where $\hat{b}$ is a unit vector in the direction of the magnetic field, the
$v_{\perp}^{2}$ term is due to the gradient drift and the $v_{\parallel}^{2}$
term is due to the curvature. If we assume the parallel and perpendicular
electron velocities to be roughly the same and adopt the usual model of
quadrupole magnetic fields in which a component is linear in transverse
displacement from the axis, i.e.
$B=B^{\prime}r,$ (5)
then, defining the ${\nabla B\;}$confinement time, $T_{\nabla B}$ to be the
time for an electron to drift half the length of a quad, $L_{Q}$, we get
$T_{\nabla B}={eB^{\prime}r^{2}L_{Q}\over 4Ec},$ (6)
where $E$ is the thermal energy of the electron. This becomes
$T_{\nabla B}={\bar{B}^{\prime}\bar{r}^{2}\bar{L}_{Q}\over 4{\cal
E}}\,\,\mu{\rm sec},$ (7)
where $\bar{B}^{\prime}$ is the field gradient in T/m, $\bar{r}$ is the radius
in cm, ${\cal E}$ is the energy in eV, and $\bar{L}_{Q}$ is the quad length in
cm. As an example typical of the PSR, if we take $\bar{B}^{\prime}=3.7$,
$\bar{r}=2.5$, and $\bar{L}_{Q}=47$, then
$T_{\nabla B}={272\over{\cal E}}\,\,\mu{\rm sec}.$ (8)
Note that this is an overestimate of the drifts since the actual drift
reverses sign as the electrons move out along the magnetic field lines toward
the poles. If the electrons only have energies that are a few hundred eV then
the confinement time is tens to hundreds of turns and is probably longer than
the growth time for the e-p instability.
The ${E\times B\;}$drift velocity is given by
$V_{E\times\\!B}={E_{\perp}\times B\over|B|^{2}}c$ (9)
The electric field is due to the proton beam and to any electrons that are
present. The electric potential due to the proton beam alone is given by
$e\phi={2eI\over\beta c},$ (10)
where $\beta$ is the beam velocity scaled by the speed of light and $I$ is the
(time-dependent) beam current. The beam current varies by $100\%$ in one
revolution period (the beam is bunched), but we will estimate drifts using the
average current and resulting field. Note that electrons spend a lot of time
near the magnetic mirror points where we expect that the $E$ field will mostly
be parallel to the $B$ field and will give rise to only small drifts.
Nevertheless, the ${E\times B\;}$drift velocity due to this term alone,
assuming it acts all the time, would give an electron confinement time of
$T_{E\times B}=8.34{\beta\bar{B}^{\prime}\bar{r}^{2}\bar{L}_{Q}\over{\cal
I}}\,\,{\rm nsec},$ (11)
where the current, ${\cal I}$, is measured in amps. if we again take
$\bar{B}^{\prime}=3.7$, $\bar{r}=2.5$, ${\cal I}=10$, and $\bar{L}_{Q}=47$,
then $T_{E\times B}=761$ nsec, i.e. electrons would be confined for several
turns, even with this overestimate of the ${E\times B\;}$drift. With a more
realistic calculation, including the full orbit dynamics of the electrons and
the reverse drifts that occur when only the electrons are present, it is
likely that the electrons will be confined for many turns.
Electrons to the left and right of the beam, horizontally, are free to move
vertically (initially) until they move out radially along the field line to a
region of greater field strength. Electrons above and below the beam,
vertically, are free to move horizontally (initially) until they move out
radially along a field line to a region of greater field strength. A complete
model of the electron motion is very complicated, but a simple model will
suffice to treat the motion of the center of mass of the electrons for
oscillations near the beam axis. Let us note that for electrons that can move
vertically, i.e. those to the left and right of the beam, the restoring mirror
force exactly vanishes at zero vertical position and the restoring force
reverses sign there. For electrons that can move horizontally, i.e. those
above and below the beam, the restoring mirror force exactly vanishes at zero
horizontal position and the restoring force reverses sign there. Thus, in both
transverse directions we should expect the restoring potential for an electron
to be approximately a harmonic oscillator potential near the axis. To see this
in more detail, let us begin with the equation from Krall and Trivelpiece,
[12], for the equation of motion along a field line of a particle in a
magnetic field
${d^{2}s\over dt^{2}}\approx-{v_{\perp 0}^{2}\over 2B_{0}}{\partial
B\over\partial s},$ (12)
where $s$ is the distance along the field line, $v_{\perp 0}$ is the initial
value of the transverse velocity, and $B_{0}$ is the initial value of the
magnitude of the magnetic field. The components of the quadrupole field are
$B_{x}=B_{0}^{\prime}y,$ (13)
$B_{y}=B_{0}^{\prime}x.$ (14)
We thus see that
${\partial B\over\partial s}={2B_{0}^{\prime}xy\over x^{2}+y^{2}}.$ (15)
From this we see that a particle that starts at $x=x_{0}$, $y=0$ satisfies the
approximate equation
${d^{2}y\over dt^{2}}\approx-({B_{0}^{\prime}v_{\perp 0}^{2}\over
B_{0}x_{0}})y,$ (16)
i.e. it is approximately a harmonic oscillator with a squared angular
frequency of
$\omega_{m}^{2}={B_{0}^{\prime}v_{\perp 0}^{2}\over B_{0}x_{0}}.$ (17)
But $B_{0}\approx B_{0}^{\prime}x_{0}$, so
$\omega_{m}^{2}\approx{v_{\perp 0}^{2}\over x_{0}^{2}}.$ (18)
It thus appears that modeling the mirror trapping of the electrons by a
harmonic oscillator potential, but with a large spread in oscillation
frequencies, should be a fairly good approximation.
## 2 Dipole Model of the e-p Instability
In this section, in order to find thresholds and growth rates, we will present
a simple theory of the _e-p_ instability. The model for the linear theory of
the instability in this section that we use is similar to the theory of Keil
and Zotter, [13]. We model the proton beam by the beam centroid at each
azimuthal position around the ring. The background electrons have a complex
distribution both in physical and in velocity space determined by their
formation, capture in the quadrupoles, interaction with the proton beam, and
loss, as discussed in the previous section. We cannot hope to accurately model
all of these effects in an analytic theory; we will simply assume that the
electrons have a distribution in the squared magnetic bounce frequency,
$g_{m}=\omega_{m}^{2}$, and that at each bounce frequency those electrons are
described by their centroid position, with electrons at a different bounce
frequency having a different centroid. We assume the proton beam moves at a
constant azimuthal velocity around the ring and is subject to a constant
transverse focusing force that produces betatron oscillations at the betatron
frequency, i.e. we make the smooth approximation, [14]. We only model proton
beam and electron motion in one transverse direction. The protons and
electrons are assumed to interact with each other via a force that is linear
in the relative displacement of the centroids of the protons and electrons.
The equations of motion for the centroids are thus given by
$(\frac{\partial}{\partial
t}+\omega_{0}\frac{\partial}{\partial\theta})^{2}y_{p}+\Gamma_{d}(\frac{\partial}{\partial
t}+\omega_{0}\frac{\partial}{\partial\theta})y_{p}=-\omega^{2}_{\beta}y_{p}+\omega_{p}^{2}(Y_{e}-y_{p})$
(19)
$\frac{\partial^{2}y_{em}}{\partial
t^{2}}+\omega^{2}_{m}y_{em}=\omega^{2}_{e}(y_{p}-y_{em})$ (20)
where $y_{p}(\theta,t)$ is the proton centroid position at an azimuth,
$\theta$, around the machine and time, $t$, $\omega_{0}$ is the proton beam
angular revolution frequency in the machine, and $\omega_{\beta}$ is the
angular betatron frequency of the protons. The proton beam centroid only
responds to the net electron centroid position, $Y_{e}$, which is given by
$Y_{e}=\int f(g_{m})y_{em}(\theta,t)dg_{m},$ (21)
where $f(g_{m})$ is the equilibrium distribution function of electrons in the
squared bounce frequency and $y_{em}(\theta,t)$ is the centroid of electrons
with a particular bounce frequency. The coupling frequencies $\omega_{p}$ and
$\omega_{e}$ are given by
$\omega^{2}_{e}=\frac{2N_{p}r_{e}c^{2}}{\pi b(a+b)R}$ (22)
$\omega^{2}_{p}=(\frac{Fm_{e}}{\gamma m_{p}})\omega^{2}_{e}$ (23)
with $N_{p}$ the number of protons in the machine, $r_{e}$ the classical
electron radius, $c$ the velocity of light, $\gamma$ the relativistic factor
of the proton beam, $a$ and $b$ the sizes of the proton beam, $F$ the
neutralization fraction of electrons, and $R$ the effective radius of the
ring. Note that the inter-species force is assumed to depend linearly on the
distance between the beam centroids; this is approximately correct for small
amplitudes of oscillation, but clearly fails at larger oscillation amplitudes.
Also note that we have inserted a linear damping term with coefficient
$\Gamma_{d}$ into the proton equation to account for the chromatic spread in
proton revolution frequencies; the different revolution frequencies will give
different longitudinal velocities which will Landau damp the transverse
oscillations. A more extensive model would have the proton beam described by a
distribution function in the azimuthal direction and take into account the
Landau damping due to the spread in azimuthal velocities. The approximation we
have adopted mimics this damping and has the same functional dependence as the
result of this more extensive model (see below), i.e. the damping depends on
1) the energy spread, 2) the momentum compaction factor, and 3) the mode
number (through the derivative in the damping term). Thus, this damping term
will give rise to the correct qualitative behavior with the correct functional
dependencies, i.e. damping of off-axis oscillations as they phase-mix away. We
can estimate this damping rate of transverse oscillations due to this spread
to be the chromatic fractional tune spread times the betatron frequency. Note
that the chromatic fractional tune spread is just the chromaticity times the
energy spread, i.e. it measures the longitudinal velocity spread and its
influence on the transverse oscillations. We do not include the transverse
tune spread due to space charge and machine nonlinearities because we are
using a dipole model and the centroid motion of the protons does not depend on
these terms.
$\Gamma_{d}\sim({\Delta\nu\over\nu}){\omega_{\beta}\over 2\pi}.$ (24)
Because we are using an unbunched beam model, i.e. the smooth approximation,
the average neutralization around the ring will be smaller than the
neutralization in the quadrupoles by roughly the ratio of the ratio of total
quadrupole length to the ring circumference; thus the neutralization fraction
in a quadrupole will be about $20$ times $F$ since quadrupoles are about $10$%
of the circumference and only about half the electrons can move vertically.
We have seen in the context of the drift space instability model, [10], that
bunching doesn’t have a large effect on the instability, and we assume the
same to be true here. There seems to be no simple way to incorporate bunching;
a moderately realistic model would result in a dispersion equation which would
be an infinite matrix eqation with all unbunched beam modes coupled. The
unbunched beam model of this paper would then be just the diagonal
approximation to this matrix equation. It is likely that an extensive
numerical investigation would be required to resolve the behavior.
The above model is overly simplified, but contains most of the important
physics. It will break down, of course, if the electron loss rate is too high.
Of couse, we are also assuming that the background electron density, on
average, is constant so that if electron generation and loss rates fluctuate
rapidly our model should fail.
The model of Bosch, [15], for the effect of beam gaps on the trapped ion
instability in an electron ring also considers the effect of a large spread of
(ion) oscillation frequencies on the instability, and his formulation is
similar to ours.
If we assume that the perturbations have a dependence on time and angle
proportional to $e^{-i\omega t+in\theta}$, then the equations become
$(-(\omega-n\omega_{0})^{2}-i\Gamma_{d}(\omega-n\omega_{0})+\omega^{2}_{\beta}+\omega^{2}_{p})y_{p}=\omega^{2}_{p}Y_{e}$
(25)
$(\omega^{2}_{e}+\omega^{2}_{m}-\omega^{2})y_{e}=\omega^{2}_{e}y_{p}.$ (26)
Solving equation 26 for $y_{e}$ and using equations 21 and 25 we find
$((\omega-n\omega_{0})^{2}+i\Gamma_{d}(\omega-n\omega_{0})-\omega^{2}_{\beta}-\omega^{2}_{p})=-\omega^{2}_{e}\omega^{2}_{p}\int{f(g_{m})\over
g_{m}+\omega^{2}_{e}-\omega^{2}}dg_{m},$ (27)
where we have used the definition of $g_{m}=\omega^{2}_{m}$. We have to deal
with the singularity in the integral on the right hand side of this equation.
We adopt the Landau prescription, see [12], where the integral is replaced by
the principal value plus $\pi i$ times the residue at the pole;
$\int{f(g_{m})\over
g_{m}+\omega^{2}_{e}-\omega^{2}}dg_{m}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int{f(g_{m})\over
g_{m}+\omega^{2}_{e}-\omega^{2}}dg_{m}+\pi if(\omega^{2}-\omega^{2}_{e}).$
(28)
Equation 27 thus becomes
$\displaystyle((\omega-n\omega_{0})^{2}+i\Gamma_{d}(\omega-n\omega_{0})-\omega^{2}_{\beta}-\omega^{2}_{p})$
$\displaystyle=$
$\displaystyle-\omega^{2}_{e}\omega^{2}_{p}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int{f(g_{m})\over
g_{m}+\omega^{2}_{e}-\omega^{2}}dg_{m}$ (29) $\displaystyle-\pi
i\omega^{2}_{e}\omega^{2}_{p}f(\omega^{2}-\omega^{2}_{e}),$
where the bar through the integral sign indicates principal value. This is the
dispersion relation for the two-stream mode. To solve it we have to specify
the distribution function of electron bounce frequencies, $f$. Of course,
there should be no electrons in the ‘loss-cone’, i.e. at zero
$\omega^{2}_{m}$, but otherwise the detailed distribution depends on their
formation, capture in the quadrupoles, interaction with the proton beam, and
loss. We will simply take one distribution as an example, one in which the
distribution is constant between a minimum squared bounce frequency and a
maximum squared bounce frequency; i.e.
$\displaystyle f(g_{m})$ $\displaystyle=$ $\displaystyle{1\over
g_{max}-g_{min}}\;\;\;g_{min}\leq g_{m}\leq g_{max},$ (30) $\displaystyle
0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm otherwise}$
With this distribution the dispersion equation, 29, becomes
$\displaystyle(\omega-n\omega_{0})^{2}$ $\displaystyle=$
$\displaystyle\omega^{2}_{\beta}+\omega^{2}_{p}-i\Gamma_{d}(\omega-n\omega_{0})$
(31) $\displaystyle-{\omega^{2}_{e}\omega^{2}_{p}\over
g_{max}-g_{min}}(\ln({g_{max}-\omega^{2}+\omega^{2}_{e}\over\omega^{2}-\omega^{2}_{e}-g_{min}})+\pi
i)$
Though this is a transcendental equation and can’t be solved exactly, we note
that the coefficient of the logarithmic term is small and the logarithm varies
slowly, so we can simply solve iteratively. The remainder of the equation is a
quadratic for $\omega-n\omega_{0}$ and the resulting approximate solution is
$\displaystyle\omega$ $\displaystyle\approx$ $\displaystyle
n\omega_{0}-{i\Gamma_{d}\over 2}$ (32) $\displaystyle\pm{1\over
2}\bigl{[}4(\omega^{2}_{\beta}+\omega^{2}_{p})-\Gamma_{d}^{2}$
$\displaystyle-{4\omega^{2}_{e}\omega^{2}_{p}\over
g_{max}-g_{min}}(\ln({g_{max}-(n\omega_{0})^{2}+\omega^{2}_{e}\over(n\omega_{0})^{2}-\omega^{2}_{e}-g_{min}})+\pi
i)\bigr{]}^{1\over 2}$
We note the damping due to the phase mixing term, as expected, and the usual
upper and lower sidebands. Note that we have taken $\omega\approx n\omega_{0}$
inside the logarithm because the mode numbers are usually rather high $30-50$
and this is a good (few percent) approximation for the real part of the
frequency. Let us expand just the imaginary term under the square root to find
the damping and growth rates. For convenience define the real frequency shift
to be
$\omega^{2}_{s}\equiv{1\over
4}\bigl{[}4(\omega^{2}_{\beta}+\omega^{2}_{p})-\Gamma_{d}^{2}\\\
-{4\omega^{2}_{e}\omega^{2}_{p}\over
g_{max}-g_{min}}(\ln({g_{max}-(n\omega_{0})^{2}+\omega^{2}_{e}\over(n\omega_{0})^{2}-\omega^{2}_{e}-g_{min}}))\bigr{]}$
Note that to a good approximation $\omega_{s}\approx\omega_{\beta}$. Expanding
the imaginary term in the square root we get
$\omega\approx n\omega_{0}-{i\Gamma_{d}\over 2}\pm\omega_{s}(1-{\pi
i\omega^{2}_{e}\omega^{2}_{p}\over 2\omega^{2}_{s}(g_{max}-g_{min}))})$ (33)
Note that the upper side band (plus sign) is always damped, but that the lower
side band can be unstable if
${\pi\omega^{2}_{e}\omega^{2}_{p}\over\omega_{s}(g_{max}-g_{min}))}>\Gamma_{d},$
(34)
with growth rate given by
$\gamma_{\rm growth}={\pi\omega^{2}_{e}\omega^{2}_{p}\over
2\omega_{s}(g_{max}-g_{min}))}-{\Gamma_{d}\over 2}.$ (35)
A number of modes in lower side bands can be unstable, limited only by the
condition $\omega_{e}^{2}+g_{min}<\omega^{2}<\omega_{e}^{2}+g_{max}$, with
roughly equal growth rates (there is some weak dependence on mode number in
$\omega_{s}$) and this is consistent with experiments where multiple modes are
usually observered, [16].
### 2.1 Example
Let us look at an example typical of the PSR; let us take
$a=b=1.8\;{\rm cm},$ $\omega_{0}=2\pi*2.8\;{\rm MHz},$
$\omega_{\beta}=2.2*\omega_{0},$ $2\pi R=89\;{\rm m}$
If we express the number of particles in the ring as
$N_{p}={\cal N}\times 10^{13},$ (36)
then we can compute
$\omega^{2}_{e}=1.76*{\cal N}*10^{17}\;{\rm sec}^{-2},$ (37)
and
$\omega^{2}_{p}=0.5182*F*{\cal N}*10^{14}\;{\rm sec}^{-2}.$ (38)
In the PSR the measured vertical chromaticity is about $-1.68$ and the energy
spread (typical conditions) is about $0.5\%$ so we take the chromatic tune
spread to be about $0.009$, i.e. a fractional tune spread of $0.43$%, then
$\Gamma_{d}\approx 0.0252*10^{6}\;{\rm sec}^{-1}.$ (39)
We take the frequency shift to be
$\omega_{s}\approx\omega_{\beta}=3.9*10^{7}\;{\rm sec}^{-1}.$ (40)
To estimate $g_{max}$ we use equation 18, setting the maximum transverse
energy to the beam potential; the result is
$g_{max}\approx 3.48*{\cal N}*10^{17}\;{\rm sec}^{-2},$ (41)
where we used $x_{0}\approx a=1.8$ cm. Note that we simply ignore $g_{min}$,
i.e. assume it is zero; it only modifies our results by a small factor.
If we evaluate the threshold condition, equation 34, using equations 37, 38,
39, 40, and 41 we find the criterion for instability to be
$F*{\cal N}>0.188;$ (42)
in other words, once the product of the particle number (times $10^{13}$) and
percent neutralization is about $19.0$, we can expect instability. Recall that
the neutralization fraction in quadrupoles will be about $20$ times higher
than $F$ since quadrupoles are only about $10$% of the ring and only half the
electrons can move vertically. At threshold the growth time is infinite, but
if, for simplicity, we assume that we are a factor of $2$ above the threshold,
$F*{\cal N}=9.4*10^{-2}$, then using 37, 38, 39, 40, and 41 in equation 35 we
find
$\gamma_{\rm growth}\approx 12.6\;{\rm KHz},$ (43)
i.e. a growth time of about $222$ turns. These estimates are only intended to
show that the results seem to be within a factor of two or three of the
observations and that the theory is thus a possible explanation of the
observed instability.
## 3 Discussion
Our results show that electrons trapped in quadrupoles are a plausible
explanation of the two-stream instability observed in the PSR. The growth
times found are considerably closer to the observed values than the linear
growth times derived from the instability treatment for electrons in drift
spaces, [10]. The reason for this is that the electrons confined in
quadrupoles have a very large frequency spread due to the wide variation in
magnetic bounce frequencies as compared to those in drift spaces which have
only a very small spread in space charge confinement bounce frequencies. Thus,
many fewer electrons are resonant at a particular frequency.
In addition, if the instability is due to electrons trapped in quadrupoles,
then the transverse momentum kick given to the protons is easily explained;
the momentum is transferred from the quadrupoles via the electrons, rather
than having to be transferred only from electrons, as in the drift space
theory.
Clearly a great deal more work can be done to refine this model. A kinetic
description of the proton beam could be used, and would give a more sensitive
dependence of the phase-mixing damping that depends on the detailed proton
distribution. An investigation of different electron distribution functions,
perhaps motivated by detailed simulation of electron formation and capture
dynamics, would give threshold and growth rate estimates that are better
founded than those in this note. The formulation of a bunched beam model would
be considerably more difficult, but might be worthwhile. Finally, a composite
model with both drift space electrons and quadrupole trapped electrons would
be very difficult to analyze but might be necessary to fit all observations in
real machines.
## References
* [1] George P. Lawrence, Proceedings of the 1987 Particle Accelerator Conference, Washington, DC (IEEE, Piscataway, NJ, 1987), p. 825.
* [2] D. Neuffer, E. Colton, D. Fitzgerald, T. Hardek, R. Hutson, R. Macek, M. Plum, H. Thiessen, and T.-S Wang, Nucl. Instrum. Methods Phys. Res., Sect. A 321, 1 (1992).
* [3] R. Macek, A. Browman, D. Fitzgerald, R. McCrady, F. Merrill, M. Plum, T. Spickermann, T. S. Wang, J. Griffin, K. Y. Ng, D. Wildman, K. Harkay, R. Custom, and R. Rosenberg, Proceedings of the 2001 Particle Accelerator Conference, Chicago, IL (IEEE, Piscataway, NJ, 2001), p. 688.
* [4] R. J. Macek, M. Borden, A. Browman, D. Fitzgerald, T. S. Wang, T. Zaugg, K. Harkay, and R. A. Rosenberg, Proceedings of the 2003 Particle Accelerator Conference, Portland, OR (IEEE, Piscataway, NJ, 2003), p. 508.
* [5] M. Plum, J. Allen, M. Borden, D. Fitzgerald, R. Macek, and T. S. Wang, Proceedings of 1995 Particle Accelerator Conference, Dallas, Texas (IEEE, Piscataway, NJ, 1996), p. 3406.
* [6] M. A. Plum, D. H. Fitzgerald, D. Johnson, J. Langenbrunner, R. J. Macek, F. Merrill, P. Morton, B. Prichard, O. Sander, M. Shulze, H. A. Thiessen, T. S. Wang, and C. A. Wilkinson, Proceedings of the 1997 Particle Accelerator Conference, Vancouver, Canada (IEEE, Piscataway, NJ, 1998), p. 1611.
* [7] R. J. Macek, Proceedings of ECLOUD’02 Workshop, Geneva, edited by G. Rumolo, p. 259 (CERN-2002-001).
* [8] R. J. Macek, A. A. Browman, M. J. Borden, D. H. Fitzgerald, R. C. McCrady, T. Spickermann, and T. J. Zaugg, Proceedings of ECLOUD’04, Napa, California, 2004, edited by M. Furman, p. 63 (CERN-2005-001).
* [9] R. J. Macek and A. A. Browman, Proceedings of the 2005 Particle Accelerator Conference, Knoxville, TN, 2005 (IEEE, Piscataway, NJ, 2005), p. 2047.
* [10] Paul J. Channell, ‘Phenomenological two-stream instability model in the nonlinear electron regime’ Phys. Rev. ST Accel. Beams 5, 114401 (2002)
* [11] M. T. F. Pivi and M. A. Furman, Phys. Rev. ST Accel. Beams 6, 034201 (2003).
* [12] N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, New York, (1973).
* [13] E. Keil and B. Zotter, ‘Landau-Damping of Coupled Electron-Proton Oscillations’, CERN Internal Note CERN-ISR-TH/71-58, December 1971.
* [14] Paul J. Channell, ‘Systematic solution of the Vlasov-Poisson equations for charged particle beams’, Phys. Plasmas 6, 982 (1999)
* [15] R.A. Bosch, Nucl. Instrum. and Meth. A 450,(2000) p 223.
* [16] R.J. Macek, private communication (2008).
|
arxiv-papers
| 2009-03-19T16:02:26 |
2024-09-04T02:49:01.251212
|
{
"license": "Public Domain",
"authors": "Paul J. Channell",
"submitter": "Paul Channell",
"url": "https://arxiv.org/abs/0903.3355"
}
|
0903.3357
|
# Equivariant Yamabe problem and Hebey–Vaugon conjecture
Farid Madani
Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie
Équipe: d’Analyse Complexe et Géométrie
175, rue Chevaleret
75013 Paris, France. madani@math.jussieu.fr
###### Abstract.
In their study of the Yamabe problem in the presence of isometry group, E.
Hebey and M. Vaugon announced a conjecture. This conjecture generalizes T.
Aubin’s conjecture, which has already been proven and is sufficient to solve
the Yamabe problem. In this paper, we generalize Aubin’s theorem and we prove
the Hebey–Vaugon conjecture in some new cases.
###### Key words and phrases:
Conformal metric; Isometry group; Scalar curvature; Yamabe problem.
###### 1991 Mathematics Subject Classification:
53A30, 53C21, 35J20.
## 1\. Introduction
Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Denote by
$I(M,g)$, $C(M,g)$ and $R_{g}$ the isometry group, the conformal
transformations group and the scalar curvature, respectively. Let $G$ be a
subgroup of the isometry group $I(M,g)$. E. Hebey and M. Vaugon[5] considered
the following problem:
### Hebey–Vaugon problem
_Is there some $G-$invariant metric $g_{0}$ which minimizes the functional_
$J(g^{\prime})=\frac{\int_{M}R_{g^{\prime}}\mathrm{d}v(g^{\prime})}{(\int_{M}\mathrm{d}v(g^{\prime}))^{\frac{n-2}{n}}}$
_where $g^{\prime}$ belongs to the $G-$invariant conformal class of metrics
$g$ defined by:_
$[g]^{G}:=\\{\tilde{g}=e^{f}g/f\in
C^{\infty}(M),\;\sigma^{*}\tilde{g}=\tilde{g}\quad\forall\sigma\in G\\}$
The positive answer would have two consequences. The first is that there
exists an $I(M,g)-$invariant metric $g_{0}$ conformal to $g$ such that the
scalar curvature $R_{g_{0}}$ is constant. The second is that the A.
Lichnerowicz’s conjecture [7], stated below, is true. By the works of J.
Lelong-Ferrand[6] and M. Obata[9], we know that if $(M,g)$ is not conformal to
$(S_{n},g_{can})$ (the unit sphere endowed with its standard metric
$g_{can}$), then $C(M,g)$ is compact and there exists a conformal metric
$g^{\prime}$ to $g$ such that $I(M,g^{\prime})=C(M,g)$. This implies that the
first consequence is equivalent to the
### A. Lichnerowicz conjecture
_For every compact Riemannian manifold $(M,g)$ which is not conformal to the
unit sphere $S_{n}$ endowed with its standard metric, there exists a metric
$\tilde{g}$ conformal to $g$ for which $I(M,\tilde{g})=C(M,g)$, and the scalar
curvature $R_{\tilde{g}}$ is constant._
To such metrics correspond functions which are necessarily solutions of the
Yamabe equation. In other words, if $\tilde{g}=\psi^{\frac{4}{n-2}}g$, $\psi$
is a $G-$invariant smooth positive function then $\psi$ satisfies
$\frac{4(n-1)}{n-2}\Delta_{g}\psi+R_{g}\psi=R_{\tilde{g}}\psi^{\frac{n+2}{n-2}}.$
The classical Yamabe problem, which consists to find a conformal metric with
constant scalar curvature on a compact Riemannian manifold, is the particular
case of the problem above when $G=\\{\mathrm{id}\\}$. Denote by $O_{G}(P)$ the
orbit of $P\in M$ under $G$, $W_{g}$ the Weyl tensor associated to the
manifold $(M,g)$ and $\omega_{n}$ the volume of the unit sphere $S_{n}$. We
define the integer $\omega(P)$ at the point $P$ as
$\omega(P)=\inf\\{i\in\mathbb{N}/\|\nabla^{i}W_{g}(P)\|\neq
0\\}\;(\omega(P)=+\infty\text{ if }\forall
i\in\mathbb{N},\;\|\nabla^{i}W_{g}(P)\|=0)$
### Hebey–Vaugon conjecture
_Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$ and $G$
be a subgroup of $I(M,g)$. If $(M,g)$ is not conformal to $(S_{n},g_{can})$ or
if the action of $G$ has no fixed point, then the following inequality holds _
(1)
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})<n(n-1)\omega_{n}^{2/n}(\inf_{Q\in
M}\mathrm{card}O_{G}(Q))^{2/n}$
###### Remarks 1.1.
1. (1)
This conjecture is the generalization of the former T. Aubin’s conjecture [1]
for the Yamabe problem corresponding to $G=\\{\mathrm{id}\\}$, where the
constant in the right side of the inequality is equal to
$\inf_{g^{\prime}\in[g_{can}]}J(g^{\prime})$ for $S_{n}$. In this case, the
conjecture is completely proved.
2. (2)
The inequality is obvious if $\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})$ is
nonpositive, it is the case when there exists a Yamabe metric with nonpositive
scalar curvature.
3. (3)
If for any $Q\in M$, $\mathrm{card}O_{G}(Q)=+\infty$ then this conjecture is
also obvious.
The only results known about this conjecture are given in the following
theorem:
###### Theorem 1.1 (E. Hebey and M. Vaugon).
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ and
$G$ be a subgroup of $I(M,g)$. We always have :
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})\leq
n(n-1)\omega_{n}^{2/n}(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
and inequality (1) holds if one of the following items is satisfied.
1. (1)
The action of $G$ on $M$ is free
2. (2)
$3\leq\dim M\leq 11$
3. (3)
There exists a point $P$ with minimal orbit (finite) under $G$ such that
$\omega(P)>(n-6)/2$ or $\omega(P)\in\\{0,1,2\\}$.
The case $\omega=3$ was studied by A. Rauzy (private communications).
In this prove we prove the following results:
### Main theorem
_The Hebey–Vaugon conjecture holds if there exists a point $P\in M$ with
minimal orbit (finite) for which $\omega(P)\leq 15$ or if the degree of the
leading part of $R_{g}$ is greater or equal to $\omega(P)+1$, in the
neighborhood of this point $P$._
###### Corollary 1.1.
Hebey–Vaugon conjecture holds for every smooth compact Riemannian manifold
$(M,g)$ of dimension $n\in[3,37]$.
To prove the main theorem, we need to construct a $G-$invariant test function
$\phi$ such that
$I_{g}(\phi)<n(n-1)\omega_{n}^{2/n}(\inf_{Q\in M}\mathrm{card}O_{G}(Q))^{2/n}$
Thus, all the difficulties are in the construction of a such function. For
some cases, we can use the test functions constructed by T. Aubin [1] and R.
Schoen [10] in the case of Yamabe problem. They have been already proven by E.
Hebey and M. Vaugon [5]. But the item 3, presented in Theorem 1.1, uses test
functions different than T. Aubin and R. Schoen ones.
We multiply T. Aubin’s test function $u_{\varepsilon,P}$ by a function as
follows:
(2) $\varphi_{\varepsilon}(Q)=(1-r^{\omega+2}f(\xi))u_{\varepsilon,P}(Q)$ (3)
$u_{\varepsilon,P}(Q)=\begin{cases}\biggl{(}\displaystyle\frac{\varepsilon}{r^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}-\biggl{(}\frac{\varepsilon}{\delta^{2}+\varepsilon^{2}}\biggr{)}^{\frac{n-2}{2}}&\mbox{
if }Q\in B_{P}(\delta)\\\ 0&\mbox{ if }Q\in M-B_{P}(\delta)\end{cases}$
for all $Q\in M$, where $r=d(Q,P)$ is the distance between $P$ and $Q$.
$(r,\xi^{j})$ is a geodesic coordinates system in the neighborhood of $P$ and
$B_{P}(\delta)$ is the geodesic ball of center $P$ with radius $\delta$ fixed
sufficiently small. $f$ is a function depending only on $\xi$, chosen such
that $\int_{S_{n-1}}fd\sigma=0$. Without loss of generality, we suppose that
in the coordinates system $(r,\xi^{j})$ we have $\det g=1+o(r^{m})$ for $m\gg
1$. In fact, E. Hebey and M. Vaugon proved that there exists
$\tilde{g}\in[g]^{G}$ for which $\det\tilde{g}=1+o(r^{m})$ and
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})$ does not depend on the conformal
$G-$invariant metric.
## 2\. Computation of $\int_{M}R_{g}\varphi_{\varepsilon}^{2}dv$
Let be
$I_{a}^{b}(\varepsilon)=\int_{0}^{\delta/\varepsilon}\frac{t^{b}}{(1+t^{2})^{a}}dt\text{
and }I_{a}^{b}=\lim_{\varepsilon\to 0}I_{a}^{b}(\varepsilon)$
then $I_{a}^{2a-1}(\varepsilon)=\log\varepsilon^{-1}+O(1)$. If $2a-b>1$ then
$I_{a}^{b}(\varepsilon)=I_{a}^{b}+O(\varepsilon^{2a-b-1})$ and by integration
by parts, we establish the following relationships :
(4)
$I_{a}^{b}=\frac{b-1}{2a-b-1}I_{a}^{b-2}=\frac{b-1}{2a-2}I_{a-1}^{b-2}=\frac{2a-b-3}{2a-2}I_{a-1}^{b},\quad\frac{4(n-2)I_{n}^{n+1}}{(I^{n-2}_{n})^{(n-2)/n}}=n$
Using the inequality $(a-b)^{\beta}\geq a^{\beta}-\beta a^{\beta-1}b$ for
$0<b<a$, we have for $\beta\geq 2$, $0\leq\alpha<(n-2)(\beta-1)-n$
(5)
$\int_{M}r^{\alpha}u_{\varepsilon,P}^{\beta}\mathrm{d}v=\omega_{n-1}I_{(n-2)\beta/2}^{\alpha+n-1}\varepsilon^{\alpha+n-\beta(n-2)/2}+O(\varepsilon^{n-2})$
This integral appears frequently in the following computations, and it allows
us to neglect the constant term in the expression of $u_{\varepsilon}$, when
we choose $\delta$ sufficiently small and $\varepsilon$ smaller than $\delta$.
Denote by $I_{g}$ the Yamabe functional defined for all $\psi\in H^{1}(M)$ by
(6)
$I_{g}(\psi)=\biggr{(}\int_{M}|\nabla_{g}\psi|^{2}\mathrm{d}v+\frac{(n-2)}{4(n-1)}\int_{M}R_{g}\psi^{2}\mathrm{d}v\biggl{)}\|\psi\|_{N}^{-2}$
where $N=2n/(n-2)$ and $\nabla_{g}$ is the gradient of the metric $g$.
The second integral of the functional $I_{g}$ with the scalar curvature term
needs a special consideration. Let $\mu(P)$ be an integer defined as follows :
$|\nabla_{\beta}R_{g}(P)|=0$ for all $|\beta|<\mu(P)$ and there exists
$\gamma\in\mathbb{N}^{\mu(P)}$ such that $|\nabla_{\gamma}R_{g}(P)|\neq 0$
then
$R_{g}(Q)=\bar{R}+O(r^{\mu(P)+1})$
where $\bar{R}=r^{\mu(P)}\sum_{|\beta|=\mu}\nabla_{\beta}R_{g}(P)\xi^{\beta}$
is a homogeneous polynomial of degree $\mu(P)$, the $\beta$ are multi-indices.
For simplicity, we drop the letter $P$ in $\omega(P)$ and $\mu(P)$.
By E. Hebey and M. Vaugon [5] results:
###### Lemma 2.1.
$\mu\geq\omega$, $g_{ij}=\delta_{ij}+O(r^{\omega+2})$ and
$\bar{\int}_{S(r)}R_{g}=O(r^{2\omega+2})$ which implies that
$\int_{S(r)}\bar{R}d\sigma=\leavevmode\nobreak\ 0$ when $\mu<2\omega+2$
$\bar{\int}$ denotes the average. Then
(7)
$\begin{split}\int_{M}R_{g}\varphi_{\varepsilon}^{2}\mathrm{d}v&=\int_{M}R_{g}u_{\varepsilon,P}^{2}\mathrm{d}v-2\int_{M}fu_{\varepsilon,P}^{2}R_{g}r^{\omega+2}\mathrm{d}v+\int_{M}f^{2}u_{\varepsilon,P}^{2}R_{g}r^{2\omega+4}\mathrm{d}v\\\
&=\varepsilon^{2\omega+4}\omega_{n-1}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma
I_{n-2}^{n+2\omega+1}(\varepsilon)-\\\
&2\varepsilon^{\omega+\mu+4}I_{n-2}^{\omega+\mu+n+1}(\varepsilon)\omega_{n-1}\bar{\int}_{S(r)}r^{-\mu}f(\xi)\bar{R}\mathrm{d}\sigma(\xi)+O(\varepsilon^{n-2})\\\
\end{split}$
Moreover T. Aubin [2] proved that:
###### Theorem 2.1.
If $\mu\geq\omega+1$ then there exists $C(n,\omega)>0$ such that
$\bar{\int}_{S_{n-1}(r)}R\mathrm{d}\sigma=C(n,\omega)(-\Delta_{g})^{\omega+1}R(P)r^{2\omega+2}+o(r^{2\omega+2})$
$(-\Delta_{g})^{\omega+1}R(P)$ is negative. Then
$I_{g}(u_{\varepsilon,P})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$.
From now until the end of this section, we make the assumption that
$\mu=\omega$. Now, we recall some results obtained by T. Aubin in his papers
[3, 4]:
$\bar{R}$ is homogeneous polynomial of degree $\omega$ then
$\Delta_{\mathcal{E}}\bar{R}$ is homogeneous of degree $\omega-2$ and
$\Delta_{\mathcal{E}}\bar{R}=r^{-2}(\Delta_{s}\bar{R}-\omega(n+\omega-2)\bar{R})$
where $\Delta_{\mathcal{E}}$ is the Euclidean Laplacian and $\Delta_{s}$ is
the Laplacian on the sphere $S_{n-1}$. $\Delta_{\mathcal{E}}^{k-1}\bar{R}$ is
homogeneous of degree $\omega-2k+2$ and
$\Delta^{k}_{\mathcal{E}}\bar{R}=r^{-2}(\Delta_{s}-\nu_{k}\mathrm{id})\Delta^{k-1}_{\mathcal{E}}\bar{R}=r^{-2k}\prod_{p=1}^{k}(\Delta_{S}-\nu_{p}\mathrm{id})\bar{R}$
with
(8) $\nu_{k}=(\omega-2k+2)(n+\omega-2k)$
The sequence of integers $(\nu_{k})_{\\{1\leq k\leq[\omega/2]\\}}$ is
decreasing. It will play the role of the eigenvalues of the Laplacian on the
sphere $S_{n-1}$. It is known that the eigenvalues of the geometric Laplacian
are non-negative and increasing. Our $\nu_{k}$ are in the opposite order.
We know by T. Aubin’s paper [2] that
$\Delta^{[\omega/2]}_{\mathcal{E}}\bar{R}=0$ and
$\int_{S(r)}\bar{R}\mathrm{d}\sigma=0$, then
$q=\min\\{k\in\mathbb{N}/\Delta_{\mathcal{E}}^{k}\bar{R}=0\\}$
is well defined and $r^{-\omega}\bar{R}\in\bigoplus_{k=1}^{q}E_{k}$, with
$E_{k}$ the eigenspace associated to the positive eigenvalues $\nu_{k}$ of the
Laplacian $\Delta_{s}$ on the sphere $S_{n-1}$. If $j\neq k$, then $E_{k}$ is
orthogonal to $E_{j}$, for the standard scalar product in
$H_{1}^{2}(S_{n-1})$. Moreover, since $\int\bar{R}d\sigma=0$ there exist
$\varphi_{k}\in E_{k}$ (eigenfunctions of $\Delta_{s}$) such that
(9)
$\bar{R}=r^{\omega}\Delta_{s}\sum_{k=1}^{q}\varphi_{k}=r^{\omega}\sum_{k=1}^{q}\nu_{k}\varphi_{k}$
According to Lemma 2.1, we can split the metric $g$ in the following way:
(10) $g=\mathcal{E}+h$
where $\mathcal{E}$ is the Euclidean metric and $h$ is a symmetric 2-tensor
defined in our geodesic coordinates system by
(11)
$h_{ij}=r^{\omega+2}\bar{g}_{ij}+r^{2(\omega+2)}\hat{g}_{ij}+\tilde{h}_{ij}\text{
and }h_{ir}=h_{rr}=0$
where $\bar{g}$, $\hat{g}$ and $\tilde{h}$ are symmetric 2-tensors defined on
the sphere $S_{n-1}$. We denote by $s$ the standard metric on the sphere,
$\nabla$, $\Delta$ are the associated gradient and Laplacian on $S_{n-1}$. By
straightforward computations, Aubin [3] proved that:
###### Lemma 2.2.
$\bar{R}=\nabla^{ij}\bar{g}_{ij}r^{\omega}\text{ and }$
$\bar{\int}_{S_{n-1}(r)}R\mathrm{d}\sigma=[B/2-C/4-(1+\omega/2)^{2}Q]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
where
$B=\bar{\int}_{S_{n-1}}\nabla^{i}\bar{g}^{jk}\nabla_{j}\bar{g}_{ik}\mathrm{d}\sigma$,
$C=\bar{\int}_{S_{n-1}}\nabla^{i}\bar{g}^{jk}\nabla_{i}\bar{g}_{jk}\mathrm{d}\sigma$
and $Q=\bar{\int}_{S_{n-1}}\bar{g}_{ij}\bar{g}^{ij}\mathrm{d}\sigma$
For further details refer to [8].
The integrals $Q$, $B$ and $C$ are given in terms of the tensor $\bar{g}$. Our
goal is to compute them using the eigenfunctions $\varphi_{k}$ above. Let us
define
$b_{ij}=\sum_{k=1}^{q}\frac{1}{(n-2)(\nu_{k}+1-n)}[(n-1)\nabla_{ij}\varphi_{k}+\nu_{k}\varphi_{k}s_{ij}]$
and $a_{ij}$ such that $\bar{g}_{ij}=a_{ij}+b_{ij}$ then, according to (9), we
check that
(12) $\bar{R}=\bar{R}_{b}=\nabla^{ij}b_{ij}r^{\omega}\quad\text{and
}\bar{R}_{a}=\nabla^{ij}a_{ij}r^{\omega}=0$
If $\bar{g}_{ij}=a_{ij}$ then $\bar{R}=\bar{R}_{a}=0$ and $\mu\geq\omega+1$.
By Theorem 2.1
$\bar{\int}_{S_{n-1}(r)}R\mathrm{d}\sigma=\bar{\int}_{S_{n-1}(r)}R_{a}\mathrm{d}\sigma<0$
If $\bar{g}_{ij}=b_{ij}$ then
$\bar{\int}_{S_{n-1}(r)}R\mathrm{d}\sigma=\bar{\int}_{S_{n-1}(r)}R_{b}\mathrm{d}\sigma=[B_{b}/2-C_{b}/4-(1+\omega/2)^{2}Q_{b}]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
where $B_{b}$, $C_{b}$ and $Q_{b}$ are the same integrals defined in Lemma 2.2
when the considered tensor $\bar{g}_{ij}=b_{ij}$. We compute them in terms of
$\varphi_{k}$
$\displaystyle
Q_{b}=\bar{\int}_{S_{n-1}}\bar{b}_{ij}\bar{b}^{ij}\mathrm{d}\sigma=\frac{n-1}{n-2}\sum_{k=1}^{q}\frac{\nu_{k}}{\nu_{k}-n+1}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
$\displaystyle
B_{b}=-(n-1)Q_{b}+\sum_{k=1}^{q}\nu_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
$\displaystyle
C_{b}=-(n-1)Q_{b}+\frac{n-1}{n-2}\sum_{k=1}^{q}\nu_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
To find these expressions, we used several times the identity
$\nabla^{i}b_{ij}=-\sum_{k=1}^{q}\nabla_{j}\varphi_{k}$ and Stokes formula
(more details are given in [3, 4] and [8]). In the general case, we deduce
that
###### Lemma 2.3.
If $\mu=\omega$ and $\bar{g}_{ij}=a_{ij}+b_{ij}$, where $b_{ij}$ is defined
above,
(13)
$\bar{\int}_{S_{n-1}(r)}R\mathrm{d}\sigma=\bar{\int}_{S_{n-1}(r)}R_{a}+R_{b}\mathrm{d}\sigma\leq[B_{b}/2-C_{b}/4-(1+\omega/2)^{2}Q_{b}]r^{2(\omega+1)}+o(r^{2(\omega+1)})$
and
(14)
$B_{b}/2-C_{b}/4-(1+\omega/2)^{2}Q_{b}=\sum_{k=1}^{q}u_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
with
(15)
$u_{k}=\biggl{(}\frac{n-3}{4(n-2)}-\frac{(n-1)^{2}+(n-1)(\omega+2)^{2}}{4(n-2)(\nu_{k}-n+1)}\biggr{)}\nu_{k}$
$u_{k}$ is obtained using the expressions of $Q_{b}$, $B_{b}$ and $C_{b}$
above.
## 3\. Generalization of T. Aubin’s theorem
###### Theorem 3.1.
If there exists $P\in M$ such that $\omega(P)\leq(n-6)/2$ then there exists
$f\in C^{\infty}(S_{n-1})$ with vanishing mean integral such that
$I_{g}(\varphi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$
The case $\omega=0$ of the this theorem has already been proven by T. Aubin
[1]. He also proved the theorem when $\mu\geq\omega+1$ (see Theorem 2.1).
From now until the end of this paper, we drop the letter $P$ in $\omega(P)$
and $\mu(P)$.
###### Proof.
If $\mu\geq\omega+1$ then the inequality holds by Theorem 2.1. So we suppose
that $\mu=\omega$ until the end of the proof. We start by computing the first
integral of the Yamabe functional (6) with $\psi=\varphi_{\varepsilon}$. Using
formula
$|\nabla_{g}\varphi_{\varepsilon}|^{2}=(\partial_{r}\varphi_{\varepsilon})^{2}+r^{-2}|\nabla_{s}\varphi_{\varepsilon}|^{2}$,
we obtain:
$\int_{M}|\nabla_{g}\varphi_{\varepsilon}|^{2}\mathrm{d}v=\int_{M}|\nabla_{g}u_{\varepsilon,P}|^{2}\mathrm{d}v+\int_{0}^{\delta}[\partial_{r}(r^{(\omega+2)}u_{\varepsilon,P})]^{2}r^{n-1}\mathrm{d}r\int_{S_{n-1}}f^{2}\mathrm{d}\sigma+\\\
\int_{0}^{\delta}u^{2}_{\varepsilon,P}r^{n+2\omega+1}\mathrm{d}r\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma$
The substitution $t=r/\varepsilon$ gives
(16)
$\int_{M}|\nabla_{g}\varphi_{\varepsilon}|^{2}\mathrm{d}v=(n-2)^{2}\omega_{n-1}I_{n}^{n+1}(\varepsilon)+\varepsilon^{2\omega+4}\biggl{\\{}\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma I_{n-2}^{2\omega+n+1}(\varepsilon)+\\\
\int_{S_{n-1}}f^{2}\mathrm{d}\sigma[(\omega-n+4)^{2}I_{n}^{2\omega+n+5}(\varepsilon)+2(\omega+2)(\omega-n+4)I_{n}^{2\omega+n+3}(\varepsilon)+(\omega+2)^{2}I_{n}^{2\omega+n+1}(\varepsilon)]\biggr{\\}}$
For $\|\varphi_{\varepsilon}\|_{N}^{-2}$, we need to compute the Taylor
expansion of :
$\varphi_{\varepsilon}^{N}(Q)=[1-Nr^{\omega+2}f(\xi)+\frac{N(N-1)}{2}r^{2\omega+4}f^{2}(\xi)+o(r^{2\omega+4})]u_{\varepsilon,P}^{N}$
Using the fact that $\int_{S_{n-1}}f\mathrm{d}\sigma(\xi)=0$ and formula (5),
we conclude that
$\begin{split}\|\varphi_{\varepsilon}\|_{N}^{N}&=\int_{0}^{\delta}\int_{S_{n-1}}[1+\frac{N(N-1)}{2}r^{2(\omega+2)}f^{2}(\xi)+o(r^{2\omega+4})]r^{n-1}u^{N}_{\varepsilon,P}\mathrm{d}r\mathrm{d}\sigma(\xi)\\\
&=\omega_{n-1}I^{n-1}_{n}+\frac{N(N-1)}{2}\varepsilon^{2(\omega+2)}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma
I_{n}^{2\omega+n+3}+o(\varepsilon^{2\omega+4})\end{split}$
then
(17)
$\|\varphi_{\varepsilon}\|_{N}^{-2}=(\omega_{n-1}I^{n-1}_{n})^{-2/N}\bigl{\\{}1\\\
-(N-1)\varepsilon^{2(\omega+2)}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma
I_{n}^{2\omega+n+3}/(\omega_{n-1}I^{n-1}_{n})\bigr{\\}}+o(\varepsilon^{2\omega+4})$
By Eqs (16), (17), (7) and the relationship (4), if $n>2\omega+6$ then :
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+(\omega_{n-1}I^{n-1}_{n})^{-2/N}I_{n-2}^{n+2\omega+1}\varepsilon^{2\omega+4}\times\\\
\biggl{\\{}\frac{(n-2)\omega_{n-1}}{4(n-1)}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma-\frac{n-2}{2(n-1)}\int_{S_{n-1}}f(\xi)\bar{R}\mathrm{d}\sigma+\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma+\\\
-\frac{n(n-2)^{2}-(\omega+2)^{2}(n^{2}+n+2)}{(n-1)(n-2)}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma\biggr{\\}}+o(\varepsilon^{2\omega+4})$
If $n=2\omega+6$ then
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+(\omega_{n-1}I^{n-1}_{n})^{-2/N}\varepsilon^{2\omega+4}\log\varepsilon^{-1}\times\\\
\biggl{\\{}\frac{(n-2)\omega_{n-1}}{4(n-1)}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma-\frac{n-2}{2(n-1)}\int_{S_{n-1}}f(\xi)\bar{R}\mathrm{d}\sigma+\\\
\int_{S_{n-1}}|\nabla
f|^{2}\mathrm{d}\sigma+(\omega+2)^{2}\int_{S_{n-1}}f^{2}\mathrm{d}\sigma\biggr{\\}}+O(\varepsilon^{2\omega+4})$
For further details refer to [8].
Let $I_{S}$ be the functional defined for a function $f$ on the sphere
$S_{n-1}$, with zero mean integral , by
$I_{S}(f)=\bar{\int}_{S_{n-1}}4(n-1)(n-2)|\nabla
f|^{2}-[4n(n-2)^{2}-4(\omega+2)^{2}(n^{2}+n+2)]f^{2}+\\\
-2(n-2)^{2}f\bar{R}\mathrm{d}\sigma$
This implies that if $n>2\omega+6$
(18)
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+\frac{\omega_{n-1}^{2/n}I_{n-2}^{n+2\omega+1}\varepsilon^{2\omega+4}}{4(n-1)(n-2)(I^{n-1}_{n})^{2/N}}\times\\\
\\{(n-2)^{2}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma+I_{S}(f)\\}+o(\varepsilon^{2\omega+4})$
and if $n=2\omega+6$
(19)
$I_{g}(\varphi_{\varepsilon})=\frac{n(n-2)}{4}\omega_{n-1}^{2/n}+\frac{\omega_{n-1}^{2/n}I_{n-2}^{n+2\omega+1}\varepsilon^{2\omega+4}\log\varepsilon^{-1}}{4(n-1)(n-2)(I^{n-1}_{n})^{2/N}}\times\\\
\\{(n-2)^{2}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma+I_{S}(f)\\}+O(\varepsilon^{2\omega+4})$
Notice that if $k\neq j$ then
$I_{S}(\varphi_{k}+\varphi_{j})=I_{S}(\varphi_{k})+I_{S}(\varphi_{j})$.
Indeed, $\varphi_{k}$ and $\varphi_{j}$ are orthogonal for the standard scalar
product in $H_{1}^{2}(S_{n-1})$.
$\begin{split}I_{S}(c_{k}\nu_{k}\varphi_{k})&=\bigl{\\{}d_{k}c_{k}^{2}-2(n-2)^{2}c_{k}\bigr{\\}}\nu_{k}^{2}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma\\\
&=-\frac{(n-2)^{4}}{d_{k}}\nu_{k}^{2}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma\end{split}$
where
$d_{k}=4[(n-1)(n-2)\nu_{k}-n(n-2)^{2}+(\omega+2)^{2}(n^{2}+n+2)]\text{ and
}c_{k}=\frac{(n-2)^{2}}{d_{k}}$
Using (8), we can check easily that $d_{k}$ is positive for any $1\leq
k\leq[\omega/2]$. Now, let us consider
$f=\sum_{1}^{q}c_{k}\nu_{k}\varphi_{k}$. Then
$I_{S}(f)=-\sum_{1}^{q}\frac{(n-2)^{4}}{d_{k}}\nu_{k}^{2}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
and by Lemma 2.3
$(n-2)^{2}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma+I_{S}(f)\leq\sum_{1}^{q}(u_{k}(n-2)^{2}-\frac{(n-2)^{4}}{d_{k}}\nu_{k}^{2})\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma+o(1)$
The following lemma implies that
$I_{g}(\varphi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}$ ∎
###### Lemma 3.1.
For any $k\leq q\leq[\omega/2]$ the following inequality holds
$u_{k}-\frac{(n-2)^{2}}{d_{k}}\nu_{k}^{2}<0$
###### Proof.
Recall the expression of $\nu_{k}$ given in (8). The sequence $(U_{k})$
defined by
$U_{k}:=(\nu_{k}-n+1)d_{k}\\{(n-2)\frac{u_{k}}{\nu_{k}}-\frac{(n-2)^{3}}{d_{k}}\nu_{k}\\}$
is polynomial decreasing in $\nu_{k}$ when $\nu_{k}\geq 0$. In fact,
$U_{k}=P(\nu_{k})$ with $P$ the decreasing polynomial in $\mathbb{R}_{+}$,
defined by
$P(x)=[(n-1)(n-2)x-n(n-2)^{2}+(\omega+2)^{2}(n^{2}+n+2)]\times\\\
[(n-3)(x-n+1)-(n-1)^{2}-(n-1)(\omega+2)^{2}]-(n-2)^{3}(x^{2}-(n-1)x)$
The derivative of $P$ is
$P^{\prime}(x)=-2(n-2)x-2n(n-2)^{3}+2(n^{2}-3n-2)(\omega+2)^{2}$
By assumption $\omega+2\leq(n-2)/2$ then $P$ is decreasing in
$\mathbb{R}_{+}$. Hence
$U_{k}=P(\nu_{k})\leq P(\nu_{\omega/2})=U_{\omega/2}$
for all $k\leq\omega/2$. It easy to check that $u_{\omega/2}$ is negative so
$U_{k}\leq U_{\omega/2}<0$. ∎
## 4\. Proof of the main theorem
By Remarks 1.1, we consider only the positive case (i.e.,
$\inf_{g^{\prime}\in[g]^{G}}J(g^{\prime})>0$) and the case when there exists
$P\in M$ such that
$O_{G}(P)=\\{P_{i}\\}_{1\leq i\leq m},\;\;m=\mathrm{card}O_{G}(P)=\inf_{Q\in
M}\mathrm{card}O_{G}(Q),\;\omega\leq\frac{n-6}{2}\text{ and }P_{1}=P$
Let $\tilde{\varphi}_{\varepsilon,i}$ be a function defined as follows:
(20)
$\tilde{\varphi}_{\varepsilon,i}(Q)=(1-r_{i}^{\omega+2}f_{i}(\xi))u_{\varepsilon,P_{i}}(Q)$
where $r_{i}=d(Q,P_{i})$, the function $u_{\varepsilon,P_{i}}$ is defined as
in (3) and $f_{i}$ is defined by:
(21)
$f_{i}(Q)=cr_{i}^{-\omega}\nabla_{g}^{\omega}R_{(P_{i})}(\exp_{P_{i}}^{-1}Q,\cdots,\exp_{P_{i}}^{-1}Q)$
$\exp_{P_{i}}$ is the exponential map. In a geodesic coordinates system
$\\{r,\xi^{j}\\}$ with origin $P$, induced by the exponential map
$f_{1}=cr^{-\omega}\bar{R}=c\sum_{k=1}^{q}\nu_{k}\varphi_{k}$
where $\bar{R}$, $\varphi_{k}$ and $\nu_{k}$ are defined in Section 2. Thus
the functions $f_{i}$ are defined on the sphere $S_{n-1}$. The choice of the
constant $c$ is important.
###### Lemma 4.1.
Suppose that $\omega\leq(n-6)/2$. If $\omega\in[3,15]$ or if
$\mathrm{deg}\bar{R}\geq\omega+1$ then there exists $c\in\mathbb{R}$ such that
the corresponding functions $\tilde{\varphi}_{\varepsilon,i}$ satisfy :
(22)
$I_{g}(\tilde{\varphi}_{\varepsilon,i})<\frac{1}{4}n(n-2)\omega_{n}^{2/n}$
###### Remarks 4.1.
1. (1)
We proved inequality of this lemma for any $\omega\leq(n-6)/2$, using test
function $\varphi_{\varepsilon}$ (see Theorem 3.1). We notice that the
difference between $\varphi_{\varepsilon}$ and
$\tilde{\varphi}_{\varepsilon,i}$ is on the construction of the corresponding
functions $f$ and $f_{i}$ respectively. From $\tilde{\varphi}_{\varepsilon,i}$
we define a $G-$invariant function (see proof of the main theorem below), this
property is not possible with the function $\varphi_{\varepsilon}$.
2. (2)
For $\omega=16$ and $n$ sufficiently big, we can check that for any
$c\in\mathbb{R}$, inequality (22) is false.
###### Proof.
1\. If $\mathrm{deg}\bar{R}\geq\omega+1$, then by Theorem 2.1
$I_{g}(u_{\varepsilon,P_{i}})<\frac{n(n-2)}{4}\omega_{n}^{2/n}$
It is sufficient to take $c=0$, hence
$\tilde{\varphi}_{\varepsilon,i}=u_{\varepsilon,P_{i}}$.
2\. If $\mathrm{deg}\bar{R}=\omega$. Using estimates given in the proof of
Theorem 3.1 (see (18), (19)), it is sufficient to show that there exists
$c\in\mathbb{R}$ such that
(23)
$I_{S}(f_{1})+(n-2)^{2}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}<0$
We keep the notations used in the proof of Theorem 3.1. Thus
$I_{S}(f_{1})=\sum_{k=1}^{q}I_{S}(c\nu_{k}\varphi_{k})=\bigl{\\{}d_{k}c^{2}-2(n-2)^{2}c\bigr{\\}}\nu_{k}^{2}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
$\text{and
}\bar{\int}_{S(r)}r^{-2\omega-2}R_{g}\mathrm{d}\sigma_{r}=\sum_{k=1}^{q}u_{k}\bar{\int}_{S_{n-1}}\varphi_{k}^{2}\mathrm{d}\sigma$
To prove inequality (23), it is sufficient to prove that
(24) $\forall k\leq
q\quad\frac{d_{k}}{2(n-2)}c^{2}-(n-2)c+(n-2)\frac{u_{k}}{2\nu_{k}^{2}}<0$
The left side of the inequality above is a second degree polynomial with
variable $c$, his discriminant is:
(25) $\Delta_{k}=(n-2)^{2}-\frac{d_{k}u_{k}}{\nu_{k}^{2}}$
Using Lemma 3.1, we deduce that for any $k\leq q$, $\Delta_{k}>0$. Hence, the
polynomial above admits two different roots denoted $x_{k}<y_{k}$ and given by
$x_{k}=\frac{(n-2)^{2}-(n-2)\sqrt{\Delta_{k}}}{d_{k}},\qquad
y_{k}=\frac{(n-2)^{2}+(n-2)\sqrt{\Delta_{k}}}{d_{k}}$
Inequality (24) holds if and only if
(26) $\bigcap_{k=1}^{q}(x_{k},y_{k})\neq\varnothing$
The sequence $(d_{k})_{k\leq[\omega/2]}$ decreases. It is easy to check that
(27) $\forall k<j\leq[\frac{\omega}{2}]\qquad x_{k}<y_{j}$
Hence intersection (26) is not empty if
(28) $\forall k<j\leq[\frac{\omega}{2}]\qquad x_{j}<y_{k}$
We also check that if $\omega$ is even, $u_{\omega/2}<0$, which implies
$x_{\omega/2}<0$.
* $i.$
If $\omega=3$ then $q=1$, intersection above is not empty. It is sufficient to
take $c=(x_{1}+y_{2})/2$.
* $ii.$
If $\omega=4$ then $k\in\\{1,2\\}$, $x_{2}<0$ (because $u_{2}<0$) and
$0<x_{1}<y_{2}$. Hence intersection $]x_{1},y_{1}[\cap]x_{2},y_{2}[$ is not
empty.
* $iii.$
If $5\leq\omega\leq 15$, it is sufficient to prove (28) which is equivalent to
prove that
(29) $\forall
k<j\leq[\frac{\omega}{2}]\quad(n-2)(d_{j}-d_{k})+d_{k}\sqrt{\Delta_{j}}+d_{j}\sqrt{\Delta_{k}}>0$
Notice that $\Delta_{k}$ given by (25) is a rational fraction in $n$. By
straightforward computations, we check that there exists reel numbers
$a_{k},\;b_{k},\;e_{k},\;h_{k}$ and $s_{k}$ which depend on $k$ and $\omega$
such that
(30)
$\displaystyle\Delta_{k}=a_{k}n^{2}+b_{k}n+e_{k}+\frac{h_{k}}{n-2}+\frac{s_{k}}{\nu_{k}+1-n}$
(31) $\displaystyle\sqrt{\Delta_{k}}>\sqrt{a_{k}}(n+\frac{b_{k}}{2a_{k}})$
Inequality (29) holds if we use (31).
The expressions of the reel numbers above are known explicitly (we used the
software Maple to compute them, see [8]). For simplicity, we omit to give
these expressions.
∎
###### Proof of the main theorem.
The orbit of $P$ under the action of $G$ is supposed to be minimal (i.e.
$\mathrm{card}O_{G}(P)=\inf_{Q\in M}\mathrm{card}O_{G}(Q)$). Without loss of
generality, we suppose that $3\leq\omega\leq(n-6)/2$, because if
$\omega>(n-6)/2$ or $\omega\leq 2$, we conclude using Theorem 1.1. From
functions $\tilde{\varphi}_{\varepsilon,i}$ defined by (20), we define the
function $\phi_{\varepsilon}$ as follows:
$\phi_{\varepsilon}=\sum_{k=1}^{m}\tilde{\varphi}_{\varepsilon,i}$
$\phi_{\varepsilon}$ is $G-$invariant. In fact, for any $\sigma\in G$, such
that $\sigma(P_{i})=P_{j}$
$u_{\varepsilon,P_{i}}=u_{\varepsilon,P_{j}}\circ\sigma\text{ and
}f_{i}=f_{j}\circ\sigma$
$f_{i}$ are defined by (21), we deduce that
$\tilde{\varphi}_{\varepsilon,i}=\tilde{\varphi}_{\varepsilon,j}\circ\sigma$
The support of $\tilde{\varphi}_{\varepsilon,i}$ is included in the ball
$B_{P_{i}}(\delta)$. We choose $\delta$ sufficiently small such that for all
integers $i\neq j$ in $[1,m]$, intersection $B_{P_{j}}(\delta)\cap
B_{P_{i}}(\delta)=\varnothing$. Thus
$I_{g}(\phi_{\varepsilon})=(\mathrm{card}O_{G}(P))^{2/n}I_{g}(\varphi_{\varepsilon})$
By Lemma 4.1, we conclude that
$I_{g}(\phi_{\varepsilon})<\frac{n(n-2)}{4}\omega_{n-1}^{2/n}(\mathrm{card}O_{G}(P))^{2/n}$
It remains to notice that if $\tilde{g}=\phi_{\varepsilon}^{4/(n-2)}g$ then
$J(\tilde{g})=4\frac{n-1}{n-2}I_{g}(\phi_{\varepsilon})<n(n-1)\omega_{n-1}^{2/n}(\mathrm{card}O_{G}(P))^{2/n}$
where $\varepsilon$ is sufficiently smaller than $\delta$. ∎
###### Proof of the Corollary 1.1.
Suppose that the orbit of $P$ under the action of $G$ is minimal (otherwise
the conjecture is obvious).
If $\omega=\omega(P)>[(n-6)/2]$, we conclude using Theorem 1.1.
If $\omega\leq[(n-6)/2]\leq 15$, we conclude using main theorem. ∎
## References
* [1] T. Aubin, _Équations différentielles non linéaires et problème de Yamabe_ , J. Math. Pures et appl 55 (1976), 269–296.
* [2] by same author, _Sur quelques problèmes de courbure scalaire_ , J. Funct. Anal 240 (2006), 269–289.
* [3] by same author, _Solution complète de la ${C}^{0}$ compacité de l’ensemble des solutions de l’équation de Yamabe_, J. Funct. Anal. 244 (2007), 579–589.
* [4] by same author, _On the ${C}^{0}$ compactness of the set of the solutions of the Yamabe equation_, Bull. Sci. Math (2008).
* [5] E. Hebey and M. Vaugon, _Le problème de Yamabe équivariant_ , Bull. Sci. Math 117 (1993), 241–286.
* [6] J. Lelong-Ferrand, Mém. Acad. Royale Belgique, Classe des Sciences 39 (1971).
* [7] A. Lichnerowicz, _Sur les transformations conformes d’une variété riemannienne compacte_ , C. R. Acad. Sci. Paris 259 (1964).
* [8] F. Madani, _Le problème de Yamabe avec singularités et la conjecture de Hebey–Vaugon_ , Ph.D. thesis, Université Pierre et Marie Curie, 2009. ArXiv: 0910.0562.
* [9] M. Obata, _The conjectures on conformal transformations of riemannian manifolds_ , J. Diff. Geom. 6 (1971), 247–258.
* [10] R. Schoen, _Conformal deformation of a riemannian metric to constant scalar curvature_ , J. Differ. Geom 20 (1984), 479–495.
|
arxiv-papers
| 2009-03-19T16:13:17 |
2024-09-04T02:49:01.257694
|
{
"license": "Public Domain",
"authors": "Farid Madani",
"submitter": "Farid Madani",
"url": "https://arxiv.org/abs/0903.3357"
}
|
0903.3448
|
# Comment on “Quantum Key Distribution with Classical Bob”
Yong-gang Tan ygtan@lynu.edu.cn Physics and Information Engineering College,
Luoyang Normal College, Luoyang 471022, Henan, People’s Republic of China Hua
Lu Department of Physics, Hubei University of Technology, Wuhan 430068,
People’s Republic of China Qing-yu Cai qycai@wipm.ac.cn State Key Lab of
Magnetics Resonance and Atom and Molecular Physics, Wuhan Institute of Physics
and Mathematics Wuhan 430071, People’s Republic of China
###### pacs:
03.67.Hk
M. Boyer _et al._ Boyer07 recently proposed an interesting quantum key
distribution scheme (BKM07). It claimed that Bob doesn’t need quantum capacity
to ensure the protocol’s security. That is to say, a ”classical” Bob can
ensure the security of the key. This work is conceptually novel and
interesting. However, in this comment, we will show that classical Bob is not
good enough for detecting a powerful Eve’s eavesdropping.
In BKM07, when Alice’s photons flying into Bob’s Lab, Bob measures about half
of the incoming photons to generate key and reflects back the others. Among
the registered photons on Bob’s detectors, Alice and Bob drop the results
prepared on the $X$ basis and keep the left as their raw key. Then Alice and
Bob’s photons can be classified into two categories: the CTRL photons which
are reflected back to Alice and the SIFT photons which are used to generate
key. If Eve has tagged all of Alice’s photons before they enter Bob’s realm,
she can differentiate Bob’s SIFT photons from CTRL photons: Bob consumed all
the SIFT photons during the course of his measurement, so he has to send fresh
photons which are not tagged in the SIFT mode. Therefore, Eve can distinguish
the SIFT photons from the CTRL photons in the return line and she can thus
obtain the information of the INFO bits by using the method in the mock
protocol presented in Boyer07 .
In fact, Eve’s tag can be finished with practical technology. Suppose Eve has
an optical wavelength converter which can provide a very small wavelength
change to the aim photons interpretation1 . In practice, the information may
be encoded on the photon’s polarization (If the phase-encoding was used, Eve
can select to tag the polarization of the travel photons.). Since the
polarization is communicative with the wavelength, Eve’s operation does not
affect the information encoded on the photons. A practical eavesdropping
scheme can hence be depicted as following.
1\. Alice prepares a string of photons randomly in the $X$ basis or in the $Z$
basis. Let the wavelength of Alice’s photons be $\lambda$.
2\. Eve performs a CNOT from the incoming photons into a $|0\rangle_{blank}$
ancilla before they entering the wavelength converter. The wavelength becomes
$\lambda+\delta\lambda$ after the photons passed through Eve’s Lab. Eve
forwards the tagged photons to Bob.
3\. Bob randomly operates the incoming photons in the CTRL mode or in the SIFT
mode. In the former case, Bob just reflects Alice’s photons back to Alice. In
the latter case, he measures the photons in the $Z$ basis to read out the
information, then he makes a copy of the information he obtained on a string
of fresh photons and sends them to Alice.
4a. Eve operates a CNOT conversely on the tagged photons as that in step2
which can reset her ancilla and erase the interaction on the initial photon
prepared by Alice.
4b. If Bob’s photons are not tagged, Eve measure her ancilla on the $Z$ basis
to extract its information. It is the same as the information Bob read from
Alice’s photons.
5\. Alice and Bob declare which mode the photons are operated in. If the
quantum bit error rate (QBER) is below a tolerant threshold, they will use the
information obtained from the $Z$-SIFT mode as their raw key. Or else, they
discard the protocol.
In Eve’s eavesdropping, if $\delta\lambda$ is chosen appropriately, the tagged
photons can register on Bob’s detectors correctly. With the above
eavesdropping scheme, Eve may obtain all Alice and Bob’s information without
being detected.
Thus we have showed the classical Bob is not good enough to discover a
powerful Eve. Furthermore, the practical apparatuses of Alice and Bob can not
be the same. The photons prepared by Bob may have different characters with
that of Alice’s and then a powerful Eve can distinguish Alice’s photons from
that of Bob’s. In this case, Eve even doesn’t need to tag Alice’s photon but
Bob himself tags the travel photons.
This work is funded by NSFC under Grant No. 10504039 and Wuhan Chenguang
Project. Y.-G. Tan also thank the youth fund of Luoyang Normal College.
## References
* (1) Michel Boyer, Dan Kenigsberg, and Tal Mor, Phys. Rev. Lett. 99, 140501 (2007).
* (2) S. Preble and M. Lipson, _Dynamic Wavelength Converterus_ , WO/2008/024458 (US Patent).
|
arxiv-papers
| 2009-03-20T03:48:38 |
2024-09-04T02:49:01.265698
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yong-gang Tan, Hua Lu and Qing-yu Cai",
"submitter": "Yonggang Tan",
"url": "https://arxiv.org/abs/0903.3448"
}
|
0903.3460
|
# The best bound of the area–length ratio in Ahlfors Covering surface theory
(I)
Guang Yuan Zhang Department of Mathematical Sciences, Tsinghua University,
Beijing 100084, P. R. China. Email:gyzhang@math.tsinghua.edu.cn
###### Abstract.
In Ahlfors’ covering surface theory, it is well known that there exists a
positive constant $h$ such that for any nonconstant holomorphic mapping
$f:\overline{\Delta}\rightarrow S,$ if
$f(\Delta)\cap\\{0,1,\infty\\}=\emptyset,$ then
$A(f,\Delta)\leq hL(f,\partial\Delta),$
where $\Delta$ is the disk $|z|<1$ in $\mathbb{C},$ $S$ is the unit Riemann
sphere, $A(f,\Delta)$ is the area of the image of $\Delta$ and
$L(f,\partial\Delta)$ is the length of the image of $\partial\Delta$, both
counting multiplicities.
In this paper, we will show that the best lower bound for $h$ is the number
$h_{0}=\max_{\tau\in[0,1]}\left[\frac{\sqrt{1+\tau^{2}}\left(\pi+\arcsin\tau\right)}{\mathrm{{arccot}\frac{\sqrt{1-\tau^{2}}}{\sqrt{1+\tau^{2}}}}}-\tau\right]=4.\,\allowbreak
034\,159\,790\,\allowbreak 51\dots,$
and this is the exact estimation, i.e. there exists a sequence of holomorphic
mappings $f_{n}:\overline{\Delta}\rightarrow S$ such that
$f_{n}(\Delta)\cap\\{0,1,\infty\\}=\emptyset$ and
$\lim_{n\rightarrow\infty}A(f_{n},\Delta)/L(f_{n},\partial\Delta)=h_{0}.$
###### 2000 Mathematics Subject Classification:
30D35, 30D45, 52B60
Project 10271063 and 10571009 supported by NSFC
## 1\. Introduction
In this paper, the Riemann sphere $S$ is the unit sphere
$S=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3};\;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\\}$
endowed with the stereographic projection
$P:\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}\rightarrow S$
with $P(0)=(0,0,-1)$, $P(\infty)=(0,0,1).$ The lengths of curves and the areas
of domains in $S$ are defined in the usual way. Thus, $P$ induces the
spherical metric $ds=\rho(z)|dz|=\frac{2}{1+|z|^{2}}|dz|,z\in\mathbb{C}.$ For
a set $V$ in $\overline{\mathbb{C}},$ we denote by $\partial V$ its boundary
and $\overline{V}$ its closure.
We will identify the extended plane
$\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ with $S,$ via the
stereographic projection $P$. So for any set $D\subset\mathbb{C}$, we will
also write $D\subset S,$ but in the later relation, $D$ in fact means the set
$P(D).$ When we write $0\in S$, for example, $0$ indicates the point
$P(0)=(0,0,-1)$ in $S.$ In this way, some notations in $\mathbb{C}$ will be
used in $S$: we use the interval notation $[-1,1],[0,+\infty]$ to denote the
line segment $P([-1,1]),P([0,+\infty])$ in $S,$ etc.
For a Jordan domain $U$ in $\mathbb{C}$ and a holomorphic mapping
$g:\overline{U}\rightarrow S,$ we denote by $A(g,U)$ the spherical area of the
image of $U,$ counted with multiplicities, and denote by $L(g,\partial U)$ the
spherical length of the image of $\partial U,$ counted with multiplicities. If
we regard $g$ as a mapping from from $\overline{U}$ into
$\overline{\mathbb{C}}=\mathbb{C\cup\\{\infty\\}}$, via the stereographic
projection $P$, we have
$A(g,U)=\iint_{U}(\rho(g(z))|g(z)|)^{2}dxdy,\ z=x+iy;$ $L(g,\partial
U)=\int_{\partial U}\rho(g(z))\left|g(z)\right||dz|.$
In Ahlfors’ covering surface theory ([1], [4]), it is well known that there
exists a positive constant $h$ such that for any holomorphic mapping
$f:\overline{\Delta}\rightarrow S,$ if $f(z)\neq 0,1,\infty$ for any
$z\in\Delta,$ then
(1.1) $A(f,\Delta)\leq hL(f,\partial\Delta).$
The goal of this paper is to give the best lower bound for $h,$ and our main
result is the following theorem.
###### Theorem 1.1.
Let $f:\overline{\Delta}\rightarrow S$ be a nonconstant holomorphic mapping
such that $f(z)\neq 0,1,\infty$ for any $z\in\Delta$. Then
(1.2) $A(f,\Delta)<h_{0}L(f,\partial\Delta),$
where
(1.3)
$h_{0}=\max_{\tau\in[0,1]}\left[\frac{\sqrt{1+\tau^{2}}\left(\pi+\arcsin\tau\right)}{\mathrm{arccot}\frac{\sqrt{1-\tau^{2}}}{\sqrt{1+\tau^{2}}}}-\tau\right]=4.03415979051\dots,$
and $h_{0}$ is the best lower bound in the sense that there exists a sequences
of holomorphic mappings $f_{n}:\overline{\Delta}\rightarrow S$ such that
$f_{n}(\Delta)\cap\\{0,1,\infty\\}=\emptyset$ and
$\lim_{n\rightarrow\infty}\frac{A(f_{n},\Delta)}{L(f_{n},\partial\Delta)}=h_{0}.$
Consider the function
(1.4)
$h(\tau)=\frac{\sqrt{1+\tau^{2}}\left(\pi+\arcsin\tau\right)}{\mathrm{arccot}\frac{\sqrt{1-\tau^{2}}}{\sqrt{1+\tau^{2}}}}-\tau,\tau\in[0,1].$
It is clear that
$h(0)=4,h(1)=3\sqrt{2}-1<4,$
and
$h^{\prime}(0)=\frac{4}{\pi}-1>0.$
Thus, $h$ takes its maximum $h_{0}$ at some point $\tau_{0}\in(0,1)$ and
$h_{0}>4.$
For a domain $U$ in $S,$ we denote by $A(U)$ the area of $U.$ If
$U\subset\mathbb{C},$ we still use the notation $A(U)$ to denote the spherical
area of $U,$ which is the area of $P(U)$ given by
$A(U)=\iint_{U}(\rho(x+iy))^{2}dxdy.$
For a curve $\Gamma=\Gamma(t),t\in[0,1],$ in $S,$ we denote by $L(\Gamma)$ the
length of the set $\Gamma=\\{\Gamma(t);t\in[0,1]\\}.$ If
$\Gamma=\Gamma(t),t\in[0,1],$ is a curve in $\mathbb{C}$, we still denote by
$L(\Gamma)$ the spherical length of $\Gamma,$ which is the length of the set
$P(\Gamma),$ and in the case that $\Gamma$ is simple we have
$L(\Gamma)=\int_{\Gamma}\rho(z)|dz|.$
Now we explain the geometric meaning of the function $h(\tau)$ given by (1.4).
Let $D$ be the disk in $S$ with diameter $\overline{1,\infty},$ the shortest
path from $1$ to $\infty$ in $S.$ Let $l\in[\pi,\sqrt{2}\pi]$ and let $D_{l}$
be a domain inside $D$ whose boundary is composed of the two congruent
circular arcs, each of which has endpoints $\\{1,\infty\\}$ and spherical
length $\frac{l}{2}.$ Then we have $L(\partial D_{l})=l.$ It is clear that
$D_{l},$ regarded as a domain in $\mathbb{C},$ is an angular domain whose
vertex is $1$ and bisector is the ray $[1,+\infty)$ in $\mathbb{C}$. We denote
by $2\theta_{l}$ the value of the angle of this angular domain. Then it is
clear that $\theta_{l}<\frac{\pi}{2}.$
It is proved in Section 4 that the area $A(D_{l})$ and the length $L(\partial
D_{l})=l$ are real analytic functions of
$\tau=\sin\theta_{l},\theta_{l}\in[0,\frac{\pi}{2}],$ and when we understand
$A(D_{l})$ and $l,$ in the ratio $\frac{4\pi+A(D_{l})}{l},$ as functions of
$\tau=\sin\theta_{l},$ we obtain the function $h(\tau)$ given by (1.4):
(1.5) $h(\tau)=\frac{A(S)+A(D_{l})}{l}=\frac{4\pi+A(D_{l})}{l},\tau\in[0,1].$
This is the geometrical meaning of the function $h(\tau).$
Considering that $l\geq\pi$ and $A(D_{l})\leq
A(D)=2\pi(1-\frac{\sqrt{2}}{2}),$ we have
$h(\tau)\leq\frac{4\pi+2\pi(1-\frac{\sqrt{2}}{2})}{\pi}<4.6,$
and then
$4<h_{0}<4.6.$
A numerical computation shows that
$h_{0}=4.034\,159\,790\,51\dots$
The inequality (1.1) directly follows from the fundamental theorem of L.
Ahlfors’ covering surface theory ([1], [4]) for a finite number of points
$a_{1},\dots,a_{q}$:
###### Theorem 1.2 (Ahlfors).
Let $a_{1},\dots,a_{q}$ be distinct $q$ points in $S.$ Then there exists a
positive constant $h=h(a_{1},\dots,a_{q})$ such that for any meromorphic
function defined on $\overline{\Delta}$
(1.6)
$(q-2)A(f,\Delta)/4\pi\leq\sum_{m=1}^{q}n(f,a_{m})+hL(f,\partial\Delta),$
where $n(f,a_{m})$ is the number of solutions of the equation
$f(z)=a_{m},z\in\Delta,$ ignoring multiplicities.
###### Remark 1.1.
J. Dufresnoy’s work [3] may be the first literature estimating the number $h$
in (1.6) explicitly, in which it is shown that the number $h$ in (1.6) can be
taken to be $h=h_{1}=\frac{3}{2\delta_{0}},$ where $\delta_{0}$ is the
smallest spherical distance between the points $a_{m},m=1,\dots,q$. When
$f(z)\neq 0,1,\infty,z\in\Delta,$ Dufresnoy’s result is that
$A(f,\Delta)\leq 12L(f,\partial\Delta).$
###### Remark 1.2.
J. Dufresnoy’s work [3] also studied the relationship between the constant in
(1.1) and some other classical constants, such as Landau’s, Bloch’s and
Schotkii’s constants. This is also introduced in the book [4] by Haymann.
To prove the main theorem, the difficulty lies in the inequality (1.2). It
seems hard to estimate the best lower bound for the constant $h$ by following
Ahlfors’ method in his covering surface theory. Fortunately, we managed to re-
understand Ahlfors’s theory via the classical isoperimetric inequality of the
unit hemisphere which is obtained by F. Bernstein [2] in 1905 (see Section 4).
The following is the outline of the proof of the main theorem.
(A). Observation for certain class of open mappings. We have been able to find
that the area-length ratio is relative easy to figure out for a special family
$F$ of mappings from $\overline{\Delta}$ into $S$ such that for each $f\in F,$
$f$ satisfies the following conditions (a)–(e):
(a) $f$ is open, discrete111The term _discrete_ means that for each $q\in
f(\overline{\Delta}),$ $f^{-1}(q)$ is a finite set. and continuous, the
boundary curve $\Gamma_{f}=f(z),z\in\partial\Delta,$ is a polygonal curve in
$S$ and $f(\Delta)\cap\\{0,1,\infty\\}=\emptyset.$
(b) Each natural edge222See Definition 2.2 (2) and (3). of $\Gamma$ has
spherical length strictly less than $\pi.$
(c) $\Gamma_{f}$ is locally convex everywhere except at $0,1,\infty.$
(d) All branched points of $f$ are located in $\\{0,1,\infty\\}.$
(e) $\Gamma_{f}\cap[0,+\infty]$ contains at most finitely many points. Here
$[0,+\infty]$ denotes the line segment in $S$ from $0$ to $\infty$ passing
through $1.$
It is clear that normal mappings defined in Section 3 satisfy condition (a).
Conversely, any mapping satisfying (a) that is orientation preserved is a
normal mapping333We will not introduce the proof for this conclusion, since it
is not used in this paper.. It is relatively easy to estimate the area–length
ratio for mappings in the family $F:$ for each $f\in F\ $one can obtain the
following inequality by Lemmas 14.1 and 14.2,
$A(f,\Delta)\leq h_{0}L(g,\partial\Delta)-\min\\{A(f,\Delta),4\pi\\},$
where $h_{0}$ is given by (1.3).
On the other hand, it is fortunate that we are able to prove that, for any
holomorphic mapping $f:\overline{\Delta}\rightarrow S$ with
$f(\Delta)\cap\\{0,1,\infty\\}=\emptyset,$ and for sufficiently small
$\varepsilon>0,$ there exist a finite number of mappings
$\\{g_{1},\dots,g_{n}\\}$ in the family $F,$ such that
(1.7) $\sum_{j=1}^{n}A(g_{j},\Delta)\geq A(f,\Delta)-\varepsilon,\
\mathrm{and\ }\sum_{j=1}^{n}L(g_{j},\partial\Delta)\leq
L(f,\partial\Delta)+\varepsilon.$
Summarizing the above two aspects, we obtain (1.2).
The existence of the family $\\{g_{1},\dots,g_{n}\\},$ which is given by
Theorem 12.1, is the first key step to prove the main theorem. Sections 8–11
is prepared for proving Theorem 12.1: we first prove Theorems 10.1 and 11.1,
and then we apply these two results to deduce Theorem 12.1 in Section 12. The
ingredients of Sections 8 and 9 are Theorem 8.1, Lemma 9.2 and Lemma 9.3,
which are just used to prove Theorem 10.1 and Theorem 11.1. We will give the
outline for the proof of Theorem 12.1 in the following part (B).
The content of Sections 4–7 and 13 is for proving Lemmas 14.1 and 14.2, which,
with the existence of the family $\\{g_{1},\dots,g_{n}\\},$ deduce the main
theorem in the last section, Section 14. In Section 4, we introduce two
classical results, the Bernstein’s isoperimetric inequality of the unit
hemisphere and the Ladó’s theorem, from which we prove Theorems 4.3 and 4.4
that is used in Section 14 for proving Lemmas 14.1 and 14.2. Sections 5 and 6
are prepared for Section 7, and the ingredient of Section 7 is Theorem 7.1,
which is the second key step to prove the main theorem: with Theorems 4.3, 4.4
and 13.1, it deduces Lemmas 14.1 and 14.2. Theorem 13.1, which is proved just
based on Lemma 6.3 and Corollary 7.1, is the third key step to prove the main
theorem.
(B). The existence of $\\{g_{1},\dots,g_{n}\\}$ in (A). Now, we introduce the
outline to prove the existence of $\\{g_{1},\dots,g_{n}\\}.$ Let
$f:\overline{\Delta}\rightarrow S$ be a holomorphic mapping with
$f(\Delta)\cap\\{0,1,\infty\\}=\emptyset.$ To show the existence of the family
$\\{g_{1},\dots,g_{n}\\},$ for any $\varepsilon>0,$ we first approximate $f$
by an open mapping $f_{1}$ such that $f_{1}$ satisfies (a) and (b) in (A) and
$A(f_{1},\Delta)>A(f,\Delta)-\frac{\varepsilon}{2}\ \mathrm{and\
}L(f_{1},\partial\Delta)<L(f,\partial\Delta)-\frac{\varepsilon}{2}.$
Then we are able to first show that there exist a finite number of mappings
$\\{G_{1},\dots,G_{n}\\}$ that satisfy (1.7) and (a)–(d) as follows.
Operation 1: (a)(b)$\rightarrow$(a)(b)(c). We can apply Theorem 11.1 several
times to obtain a mapping $f_{2}$ such that $f_{2}$ satisfies (a)–(c) and
$A(f_{2},\Delta)\geq A(f_{1},\Delta)\ \mathrm{and\
}L(f_{2},\partial\Delta)\leq L(f_{1},\partial\Delta).$
If $f_{2}$ satisfies (d), then $\\{G_{1}\\}=\\{f_{2}\\}$ is the desired
family. Otherwise we turn to next operation.
Operation 2: (a)(b)(c)$\rightarrow$(a)(b)(d). If $f_{2}$ does not satisfies
(d), then we can apply Theorem 10.1 a finite number of times to decompose
$f_{2}$ into a finite444By Theorem 10.1 (iv) we may assume
$\sum_{j=1}^{m}V(f_{2j})\leq V(f_{2})+2(m-1),$ where $V(f_{2})$ is the number
of natural vertices (see Definition 2.2) of the polygonal curve
$\Gamma_{f_{2}}=f_{2}(z),z\in\partial\Delta.$ Then by Lemma 12.1 we have
$3m\leq V(f_{2})+2(m-1),$ which implies $m\leq V(f_{2})-2,$ and then the
finiteness follows. number of mappings $f_{2j},j=1,\dots,m,$ that satisfy (a),
(b), (d) and
$\sum_{j=1}^{m}A(f_{2j},\Delta)\geq A(f_{2},\Delta)\ \mathrm{and\
}\sum_{j=1}^{m}L(f_{2j},\partial\Delta)\leq L(f_{2},\partial\Delta).$
Operation 2 may destroy condition (c)! We try to repair this by applying
Operation 1 to all the mappings $f_{2j}$ and obtain mappings
$f_{12j},j=1,\dots,m,$ that satisfy (a)–(c) and
$A(f_{12j},\Delta)\geq A(f_{2j},\Delta)\ \mathrm{and\
}L(f_{12j},\partial\Delta)\leq L(f_{2j},\partial\Delta),j=1,\dots,m.$
But Operation 1 may destroy condition (d)! We try to repair this by applying
Operation 2 to each $f_{12j}$ that has ramification points in $\Delta$ and
obtain more mappings. But then condition (c) may again be destroyed for the
mappings obtained from Operation 2.
It seems we are arguing in a circle! Luckily, we are able to prove that
Operations 1 and 2 can not be applied infinitely many times! This is the
ingredient of Theorem 12.1. Thus, we can execute Operations 1 and 2
alternatively with in a finite number of steps to obtain the desired mappings
$G_{j},j=1,\dots,n.$
From the mappings $G_{j}$ we can easily obtain the mappings
$g_{j},j=1,2,\dots,n,$ by slightly perturb each $G_{j}$.
###### Remark 1.3.
The method in this paper can also be used to estimate the best bound of the
constant $h$ in Ahlfors’s fundamental theorem for any number $(\geq 3)$ of
points. We will discuss this in another paper.
## 2\. Some notations and definitions related to curves in $S$
In this section we introduce some notations, definitions and make some
conventions. _Locally convex polygonal paths_ and _locally convex polygonal
curves_ in the Riemann sphere $S$ defined in this section play a central role
in this paper.
Let $\Gamma=\Gamma(t),t\in[\alpha,\beta],$ be a curve in $\mathbb{C}$ or $S$.
Then the orientation of the curve $\Gamma$ will be regarded as the orientation
as $t$ increases. Therefore, if $\Gamma$ is not closed, the orientation of
$\Gamma$ is from $\Gamma(\alpha)$ to $\Gamma(\beta),$ and we will denote by
$-\Gamma=\Gamma(t_{2}+t_{1}-t),t\in[t_{1},t_{2}],$
the same curve with opposite orientation.
If $\Gamma_{j}=\Gamma_{j}(t),t\in[t_{j1},t_{j2}],$ are two curves in
$\mathbb{C}$ (or $S$) and $\Gamma_{1}(t_{12})=\Gamma_{2}(t_{21}),$ we will
denote by $\Gamma_{1}+\Gamma_{2}$ the curve
$\Gamma(t)=\left\\{\begin{array}[]{ll}\Gamma_{1}(t),&t\in[t_{11},t_{12}],\\\
\Gamma_{2}(t+t_{21}-t_{12}),&t\in(t_{12},t_{12}+t_{22}-t_{21}].\end{array}\right.$
When $\Gamma_{1}+(-\Gamma_{2})$ makes sense, we will write it by
$\Gamma_{1}-\Gamma_{2}.$
Curves in this paper are always oriented and continuous curves. Some times a
curve $\Gamma$ will be regard as a set in $S.$ But this is only in the case
that the curve is involved in some set operations.
For a Jordan domain $D$ in $\mathbb{C},$ the boundary $\partial D$ of $D$ is
always regarded as an oriented curve with the anticlockwise orientation. If
$D$ is a Jordan domain in $\mathbb{C}$ and $f:\overline{D}\rightarrow S$ is a
continuous mapping, then the the boundary curve
(2.1) $\Gamma_{f}=\Gamma_{f}(z),z\in\partial D,$
of $f$ is always regarded as an oriented curve with the oreintation induced by
$\partial D$.
The notation $\Gamma_{f}$ will be used through out this paper, which alway
denotes the curve given by (2.1) for any given Jordan domain $D$ of
$\mathbb{C}$ and any mapping $f:\overline{D}\rightarrow S$.
An oriented great circle $C$ in $S$ divides the sphere into two hemispheres.
We will call the hemisphere that is on the left hand side of $C$ _inside, or
enclosed by,_ $C$, in the sense that we are standing on the sphere with our
heads pointing to the center of $S,$ and going along $C$ in the orientation of
$C.$ For example, when $\Delta$ is regarded as a disk in $S,$ $\Delta$ is the
lower hemisphere of $S$ and $\Delta$ is inside the oriented circle
$\partial\Delta,$ i.e. $P(\Delta)$ is inside the great circle
$P(\partial\Delta);$ and the upper hemisphere
$\overline{\mathbb{C}}\backslash\overline{\Delta}$ in $S$ is inside the
oriented circle $-\partial\Delta.$
If $\Gamma$ is a Jordan curve in $S$, then the domain in $S$ that is bounded
by $\Gamma$ and is inside $\Gamma$ is also called the domain inside, or
enclosed by, $\Gamma.$ Of course, here “inside” means “on the left hand side
of”.
A section of a great circle in $S$ is called a _line segment_. To emphasize
this, we also call it _straight line segment_ or _geodesic line segment._
The spherical distance of two points $p$ and $q$ in $S$ will be denoted by
$d(p,q).$ In the case that $p$ and $q$ are not antipodal, we denote by
$\overline{pq}$ the shortest (simple) path in $S$ from $p$ to $q,$ which is
unique and is in fact the shorter of the two arcs with end points $p$ and $q$
of the great circle of $S$ passing through $p$ and $q.$ We will write
$\overline{q_{1}q_{2}\dots
q_{n}}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{n-1}q_{n}},$
if each term of the right hand side makes sense, where $q_{1},\dots,q_{n}$ are
points in $S.$
We write $\overline{pq}$ by $\overline{p,q},$ if $p,$ or $q,$ or both, is
replaced by explicit complex numbers. For example, we denote by the shortest
path from $p=1$ to $q=2$ by $\overline{pq}=\overline{1,2}.$ Note that we
identify $\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ with $S,$ via the
stereographic projection $P.$
When $\overline{pq}$ makes sense, we will denote by $\overline{pq}^{\circ}$
the interior of the path.
###### Definition 2.1.
A closed curve
$\Gamma=f(z),z\in\partial\Delta,$
in $S$ is called a _polygonal closed curve_ if and only if there exist a
finite number of points $p_{j}\in\partial\Delta,j=1,\dots,n,$ with
(2.2) $\arg p_{1}<\arg p_{2}<\dots<\arg p_{n}<\arg p_{1}+2\pi$
such that for each section555A section of a curve always inherits the
orientation of the curve. $\alpha_{j}$ of $\partial\Delta$ from $p_{j}$ to
$p_{j+1}$ $(p_{n+1}=p_{1}),$ the section $\Gamma_{j}$ of $\Gamma$ restricted
to $\alpha_{j}$ is a locally simple and locally straight path, and in this
case
$\Gamma=\Gamma_{1}+\dots+\Gamma_{n}$
is called a _partition_ of $\Gamma$.
Note that the term _partition_ emphasizes that each term $\Gamma_{j}$ is
locally simple and locally straight. A locally simple and locally straight
curve in $S$ must be contained in some great circle of $S.$ So, for each
$\alpha_{j}$ in the above definition, each $p_{0}\in\alpha_{j}$ has a
neighborhood $L_{p_{0}}$ in $\alpha_{j}$ such that $\Gamma$ restricted to
$L_{p_{0}}$ is a homeomorphism onto a line segment in $S.$
Through out this paper, we denote by $E$ the set $\\{0,1,\infty\\}$ in $S.$
###### Definition 2.2.
Let $\Gamma=f(z),z\in\partial\Delta,$ be a polygonal closed curve in $S$.
(1) A point $p_{0}\in\partial D$ is called a _natural vertex_ of $\Gamma$ if
and only if one of the following conditions holds:
(a) $f(p_{0})\in E=\\{0,1,\infty\\}.$
(b) $f(p_{0})\notin E$ and for any neighborhood $I_{p_{0}}$ of $p_{0}$ in
$\partial\Delta,$ the restriction $\Gamma|_{I_{p_{0}}}=f(z),z\in I_{p_{0}},$
can not be a straight and simple path.
(2) In the case that $\Gamma$ has at least two natural vertices, a closed
interval $I$ in $\partial\Delta$ is called a _natural edge_ of $\Gamma$ if and
only if the endpoints of $I$ are both natural vertices of $\Gamma$ but the
interior of $I$ does not contain any natural vertex of $\Gamma.$
(3) If $I$ is a natural edge of $\Gamma,$ then the restriction
$\Gamma|_{I}=f(z),z\in I,$ is also called a _natural edge_ of $\Gamma.$
For the above definition (2), the reader should be aware that a natural edge
can not contain any point of
$f^{-1}\left(E\right)=f^{-1}\left(\\{0,1,\infty\\}\right)$ in its interior (in
$\partial\Delta$), because by definition each point in $f^{-1}\left(E\right)$
is a natural vertex. Thus, one can not understand any natural edge to be a
maximal interval on which $\Gamma$ is locally simple and locally straight. If
we regard the great circle $C$ determined666This is in the sense that $C$
contains $\overline{0,1}$ and is oriented by $\overline{0,1}.$ by
$\overline{0,1}$ as a simple closed curve, it has three natural edges
$\overline{0,1},\overline{1,\infty}$ and
$\overline{\infty,-1,1}=\overline{\infty,-1}+\overline{-1,1}$, but the whole
curve $C$ is simple and straight.
If $\Gamma$ has no any natural vertex, $\Gamma$ must be a closed curve
contained in some great circle $C_{1}$ of $S$ with
$C_{1}\cap\\{0,1,\infty\\}=\emptyset$ and $\Gamma$ is locally simple, and in
this case, $\partial\Delta$ is regard as a natural edge without endpoints.
If $\Gamma$ has only one natural vertex $p_{0}\in\partial\Delta,$ then, by the
definition, $q_{0}=f(p_{0})=0,1$ or $\infty,$ and $\Gamma$ must be also
contained in some great circle $C_{2}$ of $S$ so that
$C_{2}\cap\\{0,1,\infty\\}=\\{q_{0}\\}$ and $\Gamma$ must be a simple path
from $q_{0}$ to $q_{0}.$ In this case $\partial\Delta$ will be regarded as a
natural edge with endpoints coinciding at the unique natural vertex $q_{0}$.
###### Definition 2.3.
Let $\Gamma=f(z),z\in\partial\Delta,$ be a _polygonal closed curve_ and assume
that $p_{1}\in\partial\Delta$ is a natural vertex of $\Gamma$. Then there
uniquely exist a finite number of points $p_{j}\in\partial\Delta,j=1,\dots,n,$
with (2.2) such that $p_{1},\dots,p_{n}$ is an enumeration of all natural
vertices of $\Gamma.$ In this case,
(2.3) $\Gamma=\Gamma_{1}+\Gamma_{2}+\dots+\Gamma_{n}$
is called a _natural partition_ of $\Gamma,$ where each $\Gamma_{j}$ is the
restriction of $\Gamma$ to the section $\alpha_{j}$ of $\partial\Delta$ from
$p_{j}$ to $p_{j+1}$ $(p_{n+1}=p_{n}),$ and
(2.4) $\partial\Delta=\alpha_{1}+\alpha_{2}+\dots+\alpha_{n}$
is also called a _natural partition_ of $\partial\Delta$ corresponding to
(2.3).
###### Remark 2.1.
For the sake of simplicity and avoiding confusions, we make the following
conventions.
(1) When we say that $\Gamma^{\prime}$ is a natural edge of a polygonal closed
curve $\Gamma=f(z),z\in\partial\Delta,$ we always mean that $\Gamma$ and
$\partial\Delta$ have natural partitions (2.3) and (2.4), respectively, such
that $\Gamma^{\prime}$ is the restriction $\Gamma_{j}=f(z),z\in\alpha_{j},$
for some $j.$
(2) When we use (2.3) to denote a natural partition of $\Gamma,$ we always
mean that there is a natural partition (2.4) corresponding to (2.3). Then, in
the above definition we also call $q_{j}=f(p_{j}),$ which should be understood
to be the pair $(p_{j},q_{j}),$ a _natural vertex_ of $\Gamma$ for
$j=1,\dots,n$.
###### Definition 2.4.
A partition
$\Gamma=\Gamma_{1}+\Gamma_{2}+\dots+\Gamma_{n}$
of a closed polygonal curve in $S$ is called a _permitted partition_ if each
$\Gamma_{j}$ is contained in some natural edge of $\Gamma.$
A polygonal Jordan curve in $S$ that is composed of exactly three line
segments is called a _triangle_. Note that a vertex of a triangle may not be a
natural vertex. Any great circle may be regarded as a triangle, while it has
no any natural vertex.
###### Definition 2.5.
Let $\Gamma=\Gamma(z),z\in\partial\Delta,$ be a closed polygonal curve in $S$.
(1) For a point $p_{0}\in\partial\Delta,$ $\Gamma$ is called _convex_ at
$p_{0}$, if $p_{0}$ has a neighborhood $I$ in $\partial\Delta$ such that the
following two conditions (a) and (b) hold.
(a) The restriction $\Gamma|_{I}$ of $\Gamma$ to $I$ is a simple path.
(b) Either $\Gamma|_{I}$ is straight or
$\Gamma^{\prime}=\Gamma|_{I}+\overline{p^{\prime\prime}p^{\prime}},$ in which
$p^{\prime}\ $and $p^{\prime\prime}$ are the initial and terminal point of
$\Gamma|_{I},$ respectively, is a triangle which encloses777By definition,
“encloses” means the triangle domain is “on the left hand side of” of the
triangle $\Gamma^{\prime}$. a convex triangle domain in $S.$
(2) $\Gamma$ is called _strictly convex_ at $p_{0}\in\partial\Delta$ if
$\Gamma$ is convex at $p_{0}$ and for any neighborhood $I$ of $p_{0}$ in
$\partial\Delta,$ $\Gamma|_{I}$ is not straight.
(3). For a point $q_{0}\in S,\ \Gamma$ is called _convex_ at $q_{0}\in S$ if
and only if for each $p\in\partial\Delta$ with $\Gamma(p)=q_{0},$ $\Gamma$ is
convex at $p.$
(4). For a set $T\subset S,$ the closed curve $\Gamma$ is called _locally
convex in_ $T$ if and only if $\Gamma$ is convex at each point $q_{0}\in T.$
It is clear that if $\Gamma$ is convex at $q_{0}\in S,$ then for some
neighborhood $T$ of $q_{0}$ in $S,$ $\Gamma$ is locally convex in $T.$
###### Definition 2.6.
A path
$\Gamma=\Gamma(t),t\in[0,1],$
in $S,$ is called a _polygonal path_ if and only if $[0,1]$ has a partition
(2.5) $0=t_{0}<t_{1}<\dots<t_{n}=1,$
such that the section $\Gamma_{j}=\Gamma(t),t\in[t_{j-1},t_{j}]$, is a locally
simple and locally straight path, $j=1,\dots,n,$ and in this case
$\Gamma=\Gamma_{1}+\dots+\Gamma_{n}$
is called a _partition_ of $\Gamma.$
Natural vertices, natural edges, natural partition, permitted partitions and
convex vertices of a polygonal path $\Gamma=\Gamma(t),t\in[0,1],$ in $S,$ can
be defined as that for polygonal closed curves. But convex vertices are only
defined in the open interval $(0,1)$ of $[0,1]$ and we don’t call the
endpoints $0$ and $1$ natural vertices. To avoid confusions we write these
definitions completely.
###### Definition 2.7.
Let $\Gamma=f(t),t\in[0,1],$ be a polygonal path in $S$.
(1) A point $p_{0}\in(0,1)$ is called a _natural vertex_ of $\Gamma$ if and
only if one of the following conditions holds:
(a) $f(p_{0})\in E=\\{0,1,\infty\\}.$
(b) $f(p_{0})\notin E$ and for any neighborhood $I_{p_{0}}$ of $p_{0}$ in
$(0,1),$ the restriction $\Gamma|_{I_{p_{0}}}=f(t),t\in I_{p_{0}},$ can not be
a straight and simple path.
(2) A closed interval $I$ in $[0,1]$ is called a _natural edge_ of $\Gamma$ if
and only if each endpoint of $I$ is either $0,$ or $1,$ or a natural vertex of
$\Gamma,$ and the interior of $I$ does not contain any natural vertex of
$\Gamma.$
(3) If $I$ is a natural edge of $\Gamma,$ the restriction
$\Gamma|_{I}=f(t),t\in I,$ is also called a _natural edge_ of $\Gamma.$
###### Definition 2.8.
For a polygonal path $\Gamma=f(t),t\in[0,1],$ a partition
(2.6) $\Gamma=\Gamma_{1}+\dots+\Gamma_{n}$
is called a _natural partition_ of $\Gamma,$ if and only if $[0,1]$ has a
partition
(2.7) $0=t_{0}<t_{1}<\dots<t_{n}=1,$
such that each $[t_{j-1},t_{j}]$ is a natural edge of $\Gamma,$ and
$\Gamma_{j}$ is the restriction of $\Gamma$ to $[t_{j-1},t_{j}],j=1,\dots,n$,
in this case (2.7) is also called a _natural partition_ of $[0,1]$
corresponding to (2.6).
###### Remark 2.2.
We make similar conventions as in Remark 2.1 for polygonal paths.
(1) When we say that $\Gamma^{\prime}$ is a natural edge of a polygonal path
$\Gamma=f(t),t\in[0,1],$ we always mean that $\Gamma$ and $[0,1]$ have natural
partitions (2.6) and (2.7), respectively, such that $\Gamma^{\prime}$ is the
restriction $\Gamma_{j}=f(t),t\in[t_{j-1},t_{j}],$ for some $j.$
(2) When we use (2.6) to denote a natural partition of $\Gamma,$ we always
mean that there is a natural partition (2.7) corresponding to (2.6). Then, in
the above definition we also call $q_{j}=f(t_{j}),$ which should be understood
to be the pair $(t_{j},q_{j}),$ a natural vertex of $\Gamma$ for
$j=1,\dots,n-1$.
###### Definition 2.9.
Let $\Gamma=\Gamma(t),t\in[0,1],$ be a polygonal path in $S$.
(1) For a point $p_{0}\in(0,1),$ $\Gamma$ is called _convex_ at $p_{0}$, if
there is a closed interval $I\subset(0,1)$ such that (a) and (b) in Definition
2.5 (1) hold.
(2) $\Gamma$ is called _strictly convex_ at $p_{0}\in\partial\Delta$ if
$\Gamma$ is convex at $p_{0}$ and for any neighborhood $I$ of $p_{0}$ in
$(0,1),$ $\Gamma|_{I}$ is not straight.
(3). For a point $q_{0}\in S,\ \Gamma$ is called _convex_ at $q_{0}\in S$ if
and only if for each $p\in(0,1)$ with $\Gamma(p)=q_{0},$ $\Gamma$ is convex at
$p.$
(4). For a set $T\subset S,$ the closed curve $\Gamma$ is called _locally
convex in_ $T$ if and only if $\Gamma$ is convex at each point $q_{0}\in T.$
Geometrically, a locally convex path (or curve) has the property that when we
go ahead along the path (or curve) with our heads pointing to the center of
the sphere $S$, we always go straight or turn left.
###### Remark 2.3.
The term “closed polygonal path” and “closed polygonal curve” have distinct
meaning in some sense. If a polygonal path $\Gamma$ given by its natural
partition (2.6), the natural vertices mean $t_{1},\dots,t_{n-1}$. But when
$f(0)=f(1)$ and $\Gamma$ is regarded as a _closed curve,_
$t_{1},\dots,t_{n-1}$ are still natural vertices of $\Gamma,$ $t_{0}=0$,
identified with $t_{n}=1,$ may or may not be a natural vertex of the _closed
curve_ $\Gamma.$ Closed polygonal paths still emphasize the initial and
terminal points, while for a closed polygonal curve, there is no initial and
terminal points, all points on it have equality.
###### Remark 2.4.
A _locally convex polygonal Jordan path_ that is closed may not be a _locally
convex polygonal Jordan curve_ , by the definition.
###### Definition 2.10.
A polygonal Jordan curve in $S$ that is either a great circle, or is composed
of exactly two straight edges is called a biangle. A biangle divides the
sphere $S$ into two biangle domains.
Note that a biangle may contains more than two natural edges, in the case that
it contains $0$, $1$ or $\infty$ in its straight edges.
###### Definition 2.11.
A triangle in $S$ is called a _generic_ triangle if it encloses a triangle
domain whose three angles are all strictly less than $\pi.$
###### Definition 2.12.
A Jordan curve $\Gamma$ in $S$ is called convex if the domain
$D_{\Gamma}\subset S$ inside $\Gamma$ is a convex domain in the sense that for
any two points $q_{1}$ and $q_{2}$ in $D_{\Gamma},$ there is a line segment
$L\subset S$ with endpoints $q_{1}$ and $q_{2}$ such that $L\subset
D_{\Gamma}.$
###### Remark 2.5.
By the definition, each locally convex polygonal Jordan curve is a convex
curve and is contained in some closed hemisphere, while any locally convex
curve that is not simple may not be contained in any closed hemisphere.
###### Remark 2.6.
Any triangle $\Gamma$ in $S$ all of whose edges have length $\leq\pi$ has a
orientation so that $\Gamma$ is a convex polygonal Jordan curve. But when a
triangle $\Gamma$ in $S$ has an edge with length $>\pi,$ $\Gamma,$ with either
orientation, may not be a locally convex triangle.
###### Remark 2.7.
For any convex triangle $\Gamma$ in $S$, the triangle domain enclosed888By
definition, “enclosed” means “on the left hand side of”. by $\Gamma$ is
contained in some hemisphere of $S.$ Conversely, any triangle domain whose
closure is contained in some open hemisphere of $S$ is enclosed by a generic
convex triangle in the same open hemisphere.
## 3\. Definition and some properties of Normal mappings
The proof of the main theorem is based on the investigation of so called
normal mappings defined in this section, which are the mappings satisfying
condition (a) in Section 1. But we will use another definition.
###### Definition 3.1.
Let $D$ be a Jordan domain in $\mathbb{C}$. A mapping
$f:\overline{D}\rightarrow S$ is called a _normal mapping_ if the following
five conditions are satisfied:
(a) The boundary curve $\Gamma_{f}=f(z),z\in\partial D,$ is a polygonal closed
curve.
(b) For each $p\in D,$ there exist a neighborhood $U\subset D$ of $p$, a disk
$V\ $in $S$ centered at $q=f(p)$ and homeomorphisms $h_{1}:U\rightarrow\Delta$
and $h_{2}:V\rightarrow\Delta,$ such that
$h_{2}\circ f|_{U}\circ h_{1}^{-1}(\zeta)=\zeta^{d},\zeta\in\Delta$
for some positive integer $d.$
(c) For each $p\in\partial D,$ there exists a neighborhood $U$ of $p$ in
$\overline{D}$, a disk $V\ $in $S$ centered at $q=f(p)$ and homeomorphisms
$h_{1}:\overline{U}\rightarrow\overline{\Delta^{+}}$ and
$h_{2}:\overline{V}\rightarrow\overline{\Delta},$ such that
$\displaystyle h_{1}\left(\overline{U}\cap\partial D\right)$ $\displaystyle=$
$\displaystyle[-1,1],$ $\displaystyle h_{2}\circ f|_{\overline{U}}\circ
h_{1}^{-1}(\zeta)$ $\displaystyle=$
$\displaystyle\zeta^{d},\zeta\in\overline{\Delta^{+}},$
for some positive integer $d$, where $\Delta^{+}$ is the upper half disk
$\\{\zeta\in\Delta,\mathrm{Im}\zeta>0\\}.$
(d) $f(D)\cap\\{0,1,\infty\\}=\emptyset.$
(e) $f$ is orientation preserved in the sense that $P^{-1}\circ f$ is
orientation preserved, where $P$ is the stereographic projection.
The reader should be aware of that a normal mapping satisfies condition (a) in
Section 1. Conversely, a mapping that is orientation preserved and satisfies
(a) in Section 1 must be a normal mapping, but this is not important for us.
In the above definition if for some point $p\in D,$ the corresponding $d\geq
2,$ then $p$ is called a _ramification point_ , $f(q)$ is called a _branched
point_ , $v_{f}(p)=d$ is called the _multiplicity_ of $f$ at $p,$ and
$b_{f}(p)=d-1$ is called the _branched number_ of $f$ at $p.$
If for some $p\in\partial D,$ the corresponding $d\geq 3,$ then $p$ is called
a _ramification point_ , $f(q)$ is called a _branched point_ ,
$v_{f}(p)=\left[\frac{d}{2}\right]$ is called the multiplicity of $f$ at $p,$
and $b_{f}(p)=\left[\frac{d+1}{2}\right]-1$ is called the branched number of
$f$ at $p.$
In the definition, “orientation preserved” means that for any regular point
$p\in\overline{\Delta}$ of $f,$ there is a closed Jordan domain $K_{p}$ in
$\overline{\Delta}$ that is a neighborhood999This means that if $p\in\Delta,$
$p$ is contained in the interior of $K_{p}$ in $\mathbb{C},$ and if
$p\in\partial\Delta,$ $p$ is contained in the interior of the arc
$K_{p}\cap\partial\Delta$ in $\partial\Delta.$ of $p$ in $\overline{\Delta}$
such that $\widetilde{f}=P^{-1}\circ f$ or $\frac{1}{\widetilde{f}}$ maps
$K_{p}$ homeomorphically onto a Jordan domain $K^{\prime}$ in $\mathbb{C}$
such that when $z$ goes along $\partial K_{p}$ anticlockwise,
$\widetilde{f}(z)$ goes along $\partial K^{\prime}$ anticlockwise.
For a normal mapping $f:\overline{D}\rightarrow S$, $f$ has only finitely many
ramification points. $p\in\overline{D}$ is called a regular point of $f$ if
$v_{p}(f)=1.$
The reader may be puzzled by the definition of $v_{f}(p)$ and $b_{f}(p)$ when
$p\in\partial D.$ As a matter of fact, the definition in this case follows
from the fact that we can extend the mapping $f$ to be a normal mapping so
that $p$ becomes an interior ramification point with multiplicity
$\left[\frac{d+1}{2}\right].$
For a Jordan domain $D$ and a normal mapping $f:\overline{D}\rightarrow S,$
the boundary curve $\Gamma_{f}=f(z),z\in\partial D,$ is a polygonal closed
curve. Then the term _natural vertex_ , _natural edge,_ _natural partition,
permitted partition,_ etc. introduced in Section 2 are well defined for
$\Gamma_{f}.$
###### Definition 3.2.
For a Jordan domain $D$ and a normal mapping $f:\overline{D}\rightarrow S.$ We
define $V(f)$ to be the number of natural vertices of the boundary curve
$\Gamma_{f}=f(z),z\in\partial D;$ define $V_{E}(f)$ to be the number of
natural vertices of $\Gamma_{f}$ that is contained in $E$ and define
$V_{NE}(f)=V(f)-V_{E}(f),$
which is the number of natural vertices of $\Gamma_{f}$ that is not contained
in $E.$
Recall that $E$ always denotes the set $\\{0,1,\infty\\}$ in $S.$
Let $f:\overline{D}\rightarrow S$ be a normal mapping. Then by the definition,
$\overline{D}$ has a triangulation such that each ramification point of $f$ is
a vertex of the triangulation and $f$ restricted to each triangle of the
triangulation of $\overline{D}$ is a homeomorphism onto a real triangle on
$S,$ i.e., each edge of the triangle is straight. Then $f$ and the
triangulation of $\overline{D}$ induce a triangulation of the Riemann surface
of $f;$ which is consisted of real triangles in $S.$ Therefore, the following
two lemmas are obvious.
###### Lemma 3.1.
Let $D$ be a Jordan domain in $\mathbb{C}$ and let $f:\overline{D}\rightarrow
S$ be a normal __ mapping. Then for any Jordan domain $D_{1}$ contained in
$D,$ the restriction of $f$ to $\overline{D_{1}}$ is a normal mapping,
provided that the curve $f(z),z\in\partial D_{1},$ is a polygonal curve.
###### Lemma 3.2.
Let $D$ be a Jordan domain in $\mathbb{C}$, let $\alpha$ be a Jordan path in
$\overline{D}$ such that the interior101010This means the curve $\alpha$
without endpoints. of $\alpha$ is contained in $D$ and $\alpha$ has two
distinct endpoints lying on $\partial D$, let $D_{1}$ and $D_{2}$ be the two
components of $D\backslash\alpha$, and let $f_{j}:\overline{D_{j}}\rightarrow
S$ be two normal mappings, $j=1,2$. If $f_{1}(z)=f_{2}(z)$ for each
$z\in\alpha,$ then the mapping
$F=\left\\{\begin{array}[]{l}f_{1}(z),z\in\overline{D_{1}},\\\ f_{2}(z),z\in
D\backslash\overline{D_{1}},\end{array}\right.$
is a normal mapping defined on $\overline{D}.$
###### Lemma 3.3.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and let $q\in
f(\overline{\Delta}).$ Then, for sufficiently small disk $D(q)$ in $S$
centered at $q,$ $f^{-1}(\overline{D(q)})$ is a union of disjoint closed
domains $\overline{U_{j}}$ in $\overline{\Delta},j=1,2,\dots,n,$ such that for
each $j,$ $\overline{U_{j}}$ is the closure of a (relatively) open subset
$U_{j}$ of $\overline{\Delta},$ $U_{j}\cap f^{-1}(q)$ contains exactly one
point $x_{j}$ and the followings holds:
(i). If $x_{j}\in\Delta,$ then $f$ restricted to $\overline{U_{j}}$ is a
branched covering mapping onto $\overline{D(q)}$ such that $x_{j}$ is the
unique possible ramification point.
(ii). If $x_{j}\in\partial\Delta,\ $then $f(\overline{U_{j}})=\overline{D(q)}$
or $f(\overline{U_{j}})$ is a closed sector of $\overline{D(q)}$, and there
exist homeomorphisms $\phi_{j}$ from $\overline{U_{j}}$ onto the closed half
disk $\overline{\Delta^{+}}$ and $\psi_{j}$ from $D(q)$ onto
$\overline{\Delta}$ such that
$\phi_{j}(x_{j})=0,\ \phi_{j}(\overline{U_{j}}\cap\partial\Delta)=[-1,1],$
and
$\psi_{j}\circ
f\circ\phi_{j}^{-1}(\xi)=\xi^{d_{j}},\xi\in\overline{\Delta^{+}},$
for some positive integer $d_{j}$.
###### Proof.
The proof is quite simple and standard. Note that in (ii)
$f(\overline{U_{j}}\cap\Delta)$ may be the disk $D(q)$ omitting a radius. ∎
###### Corollary 3.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping that has a
ramification point $p_{0}\in\partial\Delta$. Then $\partial\Delta$ has a
section $\alpha_{1}$ from $p_{0}$ to some point in
$\partial\Delta\backslash\\{p_{0}\\}$ such that $\beta=f(z),z\in\alpha_{1},$
is a simple path in $S$ starting from $f(p_{0})$ and, lifted by $f,$ $\beta$
has $b=b_{f}(p_{0})$ lifts $\alpha_{2},\dots,\alpha_{b+1}$ that start from
$p_{0}$ and satisfy
$\alpha_{j}\backslash\\{p_{0}\\}\subset\Delta,j=2,\dots,b+1.$
###### Lemma 3.4.
Let $D$ be a Jordan domain in $\mathbb{C}$ and let
$\alpha_{j}=\alpha_{j}(t),t\in[0,1],$ be two paths contained in $\partial D$
such that $\alpha_{1}(0)=\alpha_{2}(0)$ and $\alpha_{1}\cap\alpha_{2}$
contains at most two points. Let $f:\overline{D}\rightarrow S$ be a normal
mapping such that
$f(\alpha_{1}(t))=f(\alpha_{2}(t)),t\in[0,1].$
If $\alpha_{1}(1)\neq\alpha_{2}(1),$ then $f$ can be regarded as a normal
mapping $g:\overline{\Delta}\rightarrow S$ such that
$A(g,\Delta)=A(f,D),L(g,\partial\Delta)=L(f,\left(\partial
D\right)\backslash\\{\alpha_{1}\cup\alpha_{2}\\}),$
and $\Gamma_{g}=g(z),z\in\partial\Delta,$ is the same as the closed curve
$\Gamma_{f}=f(z),z\in\left\\{\left(\partial
D\right)\backslash\left[\alpha_{1}\cup\alpha_{2}\right]\right\\}\cup\\{\alpha_{1}(1)\\},$
ignoring a parameter transformation.
If $\alpha_{1}(1)=\alpha_{2}(1),$ then $f$ can be regard as an open continuous
mapping $g$ from the sphere $S$ onto itself. And so, $f$ takes every value in
$S.$
###### Proof.
The proof is the standard gluing argument that glue the domain $D$ by
identifying $\alpha_{1}(t)$ and $\alpha_{2}(t)$ for each $t\in[0,1]$. ∎
###### Lemma 3.5.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and let
$p_{0}\in\overline{\Delta}$ be a ramification point of $f$. Assume that
$\beta=\beta(t),t\in[0,1],$ is a polygonal Jordan path in $S$ such that the
followings hold.
(a) $\beta(0)=f(p_{0}),$ $\beta$ has two distinct lifts
$\alpha_{j}=\alpha_{j}(t),t\in[0,1],$ in $\overline{\Delta}$ by $f,$ with
$\alpha_{j}(0)=p_{0}$ and
(3.1) $f(\alpha_{1}(t))=f(\alpha_{2}(t))=\beta(t),t\in[0,1],j=1,2.$
(b) The interior $\alpha_{j}^{\circ}=\alpha_{j}(t),t\in(0,1),$ of $\alpha_{j}$
is contained in $\Delta,j=1,2,\ $and
$\\{\alpha_{1}(1),\alpha_{2}(1)\\}\subset\partial\Delta.$
(c) $f$ has no ramification point in the interior of $\alpha_{1}$ and
$\alpha_{2}.$
Then $\alpha_{1}(1)\neq\alpha_{2}(1).$
###### Proof.
Since there is no ramification point in the interiors of $\alpha_{1}$ and
$\alpha_{2},$ we have
(3.2) $\alpha_{1}\cap\alpha_{2}\subset\\{\alpha_{1}(0),\alpha_{1}(1)\\}.$
By (3.1) and (b),
(3.3) $\beta(t)\neq 0,1,\infty,t\in(0,1).$
If $\alpha_{1}(1)=\alpha_{2}(1),$ then $\alpha_{1}-\alpha_{2},$ or
$\alpha_{2}-\alpha_{1},$ encloses a Jordan domain $D$ in $\overline{\Delta},$
and then by Lemma 3.4, $f(\overline{D})=S.$ But $f$ is a normal mapping, and
then $f(D)\subset f(\Delta)\subset S\backslash\\{0,1,\infty\\},$ and then by
(3.2) and (3.3) we have
$f^{-1}(\\{0,1,\infty\\})\subset\\{\alpha_{1}(0),\alpha_{2}(1)=\alpha_{2}(1)\\}.$
Therefore we have $f(\overline{D})\neq S$. This is a contradiction. ∎
## 4\. A classical isoperimetric inequality of the unit hemisphere
In this section we use Bernstein’s isoperimetric inequality to prove theorems
4.3 and 4.4, which will be used in Section 14.
The following result is obtained by Bernstein in 1905.
###### Theorem 4.1 (Bernstein inequality [2]).
Let $\Gamma$ be a simple curve in some hemisphere $S^{\ast}$ of $S.$ Then the
length $L=L(\Gamma)$ and the area $A$ of the domain in $S^{\ast}$ enclosed by
$\Gamma$ satisfy
$L^{2}\geq 4\pi A-A^{2},$
equality holds if and only if $\Gamma$ is a circle.
The following inequality is another version of Bernstein inequality.
###### Corollary 4.1.
Under the same hypothesis and additional condition $L(\Gamma)\leq 2\pi,$
$A\leq 2\pi\left(1-\sqrt{1-R^{2}}\right),$
equality holds if and only if $\Gamma$ is a circle, where
$R=\frac{L(\Gamma)}{2\pi}.$
In fact, any circle in $S$ with Euclidian radius $R$ divides the sphere into
two spherical disks with areas $2\pi\left(1\pm\sqrt{1-R^{2}}\right).$ The
following result is obtained by Ladó in 1935.
###### Theorem 4.2 (Ladó [5]).
Any closed curve in $S$ with length less than $2\pi$ is contained in some open
hemisphere.
###### Corollary 4.2.
Let $l$ be a given positive number with
$\pi<l<\sqrt{2}\pi,$
let $l_{1}$ and $l_{2}$ be positive numbers with
$l_{1}+l_{2}=l\ \mathrm{and\ }l_{j}\geq\frac{\pi}{2},j=1,2,$
and, for $j=1,2,$ let $\gamma_{j}$ be a circular path in $S$ such that
$\gamma_{j}$ has endpoints $\\{0,1\\}$, $L(\gamma_{j})=l_{j},$ and
$\gamma=\gamma_{1}+\gamma_{2}$ is a Jordan curve that encloses a domain
$D_{\gamma}$ in some hemisphere of $S.$ Then the area of $D_{\gamma}$ assumes
the maximum if and only if $l_{1}=l_{2}=\frac{1}{2}l$ and $D_{\gamma}$ is
convex.
By this corollary, $D_{\gamma}$ assume the maximum if and only $D_{\gamma}$ is
congruent with the domain $D_{l}$ defined in Section 1.
###### Proof.
This follows from Corollary 4.1 and Theorem 4.2 directly. Let $\Gamma_{1}$ be
a circle passing through $0$ and $1$ in $S$ so that the length of the section
$\alpha_{1}$ of $\Gamma_{1}$ from $0$ to $1$ is $\frac{l}{2}$ and $\Gamma_{1}$
is convex in the sense that the disk inside $\Gamma_{1}$ is contained in some
open hemisphere of $S$. Then by the assumption, we have
$L(\alpha_{1})<L(\Gamma_{1}\backslash\alpha_{1}),$ and then there is a point
$p\in\Gamma_{1}\backslash\alpha_{1}$ so that the section $\alpha_{2}$ of
$\Gamma_{1}$ from $1$ to $p$ has length $\frac{l}{2}$ as well.
We replace $\alpha_{j}$ with $\gamma_{j}^{\prime}$ so that
$\gamma_{j}^{\prime}$ is congruent with $\gamma_{j},j=1,2,$ and that the
circle $\Gamma_{1}$ becomes a Jordan curve $\Gamma_{2}$ that is convex
everywhere, except at $0,1$ and $p,$ in the sense that the triangle
$\overline{0,1,p,0}$ is inside the closure of the domain inside $\Gamma_{2}$.
It is clear that $L(\Gamma_{1})=L(\Gamma_{2})<2\pi,$ and thus by Theorem 4.2,
$\Gamma_{j}$ is contained in some hemisphere $S_{j}$ of $S,j=1,2$. Then by
Theorem 4.1 $A_{\Gamma_{1}}\geq A_{\Gamma_{2}},$ the equality holds if and
only if $\Gamma_{2}$ is a circle, where $A_{\Gamma_{j}}$ is the area enclosed
by $\Gamma_{j}$ in $S_{j},j=1,2.$ From this, the conclusion follows. ∎
###### Lemma 4.1.
Let $l<2\pi$ be a positive number and let $l_{1},l_{2},\dots,l_{n}$ be
nonnegative numbers with
$0\leq l_{1}\leq l_{2}\leq\dots\leq l_{n}\ \mathrm{and\
}l_{1}+l_{2}+\dots+l_{n}=l.$
Then
$\sum_{k=1}^{n}\left(2\pi-\sqrt{(2\pi)^{2}-l_{k}^{2}}\right)\leq
2\pi-\sqrt{(2\pi)^{2}-l^{2}},$
the equality holds if and only if $l_{1}=\dots=l_{n-1}=0$ and $l_{n}=l.$
###### Proof.
There is a standard way in calculus to prove this. In fact, it also follows
from Bernstein’s inequality and Ladó’s theorem directly. ∎
###### Theorem 4.3.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that
$L(f,\partial\Delta)<2\pi.$
Then
(4.1) $A(f,\Delta)\leq
A(D_{f})=\frac{1-\sqrt{1-R_{f}^{2}}}{R_{f}}L(f,\partial\Delta)<L(f,\partial\Delta)$
with $R_{f}=\frac{L(f,\partial\Delta)}{2\pi}$ and $D_{f}$ is a disk in some
open hemisphere of $S$ with $L(\partial D_{f})=L(f,\partial\Delta).$
If, in addition, $L(f,\partial\Delta)\geq\sqrt{2}\pi,$ then
(4.2) $4\pi+A(f,\Delta)<4L(f,\partial\Delta).$
###### Proof.
We first show that
(4.3) $A(f,\Delta)\leq
2\pi-\sqrt{4\pi^{2}-\left(L(f,\partial\Delta)\right)^{2}}.$
We may assume that
(a) The boundary curve $\Gamma_{f}=f(z),z\in\partial\Delta,$ has finitely many
multiple points, i.e. there is a finite set $Q\subset\partial\Delta,$ such
that $f$ restricted to $\left(\partial\Delta\right)\backslash Q$ is injective.
If (a) fails, we may conside the restriction $f_{1}=f|_{\overline{D}}$ of $f$
to some closed Jordan domain $\overline{D}\subset\overline{\Delta},$ such that
the boundary curve
$\Gamma_{f_{1}}=f_{1}(z)=f(z),z\in\partial D,$
of $f_{1}$ satisfies (a), while $|A(f,\Delta)-A(f_{1},D)|$ and
$|L(f,\partial\Delta)-L(f_{1},\partial D)|$ may be made arbitrarily small.
Then we prove (4.3) for $f_{1},$ which implies (4.3) for $f$.
By Theorem 4.2, $f(\partial\Delta)$ is contained in some open hemisphere
$S^{\prime}$ of $S,$ and then $f(\partial\Delta)\cap\left(S\backslash
S^{\prime}\right)=\emptyset.$ If $f(\Delta)\cap\left(S\backslash
S^{\prime}\right)\neq\emptyset,$ then, since $f$ is normal and a normal
mapping is an open mapping, it is clear that $S\backslash S^{\prime}\subset
f(\Delta),$ which implies that $f(\Delta)\cap E\neq\emptyset$ (recall that
$E=\\{0,1,\infty\\}),$ for $S\backslash S^{\prime}$ is a closed hemisphere of
$S$ and a closed hemisphere of $S$ must contain at least one point of $E.$ But
this contradicts that $f$ is a normal mapping. Thus, the followings holds.
(b) $f(\overline{\Delta})$ is contained in $S^{\prime}.$
For each positive integer $j,$ let $\Delta_{j}$ be the set that for each point
$p\in\Delta_{j},$ $f(z)=p$ has at least $j$ solutions in $\Delta$, counted
with multiplicities. Since $f$ is normal, there exists a positive integer $n$
such that $n$ is the largest number with $\Delta_{n}\neq\emptyset.$ Then
(4.4) $A(f,\Delta)=\sum_{j=1}^{n}A(\Delta_{j}),$
and by (a), considering that $f$ is a normal mapping, it is clear that for any
pair $\\{j,k\\}$ with $j\neq k,$
$\left(\partial\Delta_{j}\right)\cap\left(\partial\Delta_{k}\right)$ is a
finite set and
(4.5) $L(f,\partial\Delta)=\sum_{j=1}^{n}L(\partial\Delta_{j}).$
For each $j\leq n,$ $\Delta_{j}$ is a union of finitely many components
$\Delta_{jk},k=1,2,\dots,k_{j},$ each of which is a domain also contained in
$S^{\prime}$ (by (b)) and is enclosed by a finite number of polygonal Jordan
curves. For each $j$ and each $k\leq k_{j},$ Let $\Delta_{jk}^{\ast}$ be the
domain which is the complement of the component of $S\backslash\Delta_{jk}$ in
$S$ that contains $\partial S^{\prime}.$ Then $\Delta_{jk}^{\ast}$ is a
polygonal Jordan domain with
$\partial\Delta_{jk}^{\ast}\subset\partial\Delta_{jk}\mathrm{\ but\
}\Delta_{jk}^{\ast}\supset\Delta_{jk},$
and then by Corollary 4.1 we have
$A(\Delta_{jk})\leq A(\Delta_{jk}^{\ast})\leq
2\pi-\sqrt{4\pi^{2}-L(\partial\Delta_{jk}^{\ast})^{2}}\leq
2\pi-\sqrt{4\pi^{2}-L(\partial\Delta_{jk})^{2}},$
i.e.
$A(\Delta_{jk})\leq 2\pi-\sqrt{4\pi^{2}-L(\partial\Delta_{jk})^{2}},$
for each $j\leq n$ and each $k\leq k_{j}.$ Then, by Lemma 4.1 we have
$\displaystyle\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}A(\Delta_{jk})$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}(2\pi-\sqrt{4\pi^{2}-L(\partial\Delta_{jk})^{2}})$
$\displaystyle\leq$ $\displaystyle
2\pi-\sqrt{4\pi^{2}-\left(\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}L(\partial\Delta_{jk})\right)^{2}},$
the second equality holds if and only if $n=1$ and $\Delta_{1}=f(\Delta).$ By
(4.4) and (4.5), considering that
$\sum_{j=1}^{n}A(\Delta_{j})=\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}A(\Delta_{jk}),$
and
$\sum_{j=1}^{n}L(\partial\Delta_{j})=\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}L(\partial\Delta_{jk}),$
we have (4.3).
Let $R_{f}=\frac{L(f,\partial\Delta)}{2\pi}.$ Then by (4.3), considering that
$R_{f}<1,$ we have
$\displaystyle A(f,\Delta)$ $\displaystyle\leq$ $\displaystyle
2\pi(1-\sqrt{1-R_{f}^{2}})$ $\displaystyle=$
$\displaystyle\frac{1-\sqrt{1-R_{f}^{2}}}{R_{f}}L(f,\partial\Delta)$
$\displaystyle<$ $\displaystyle L(f,\partial\Delta),$
and (4.1) is proved.
On the other hand, under the additional assumption
$L(f,\partial\Delta)\geq\sqrt{2}\pi,$ we have
$\frac{4\pi}{L(f,\partial\Delta)}\leq 2\sqrt{2},$ and then by (4.1), we have
$A(f,\Delta)+4\pi<L(f,\partial\Delta)+4\pi\leq\left(1+2\sqrt{2}\right)L(f,\partial\Delta)<4L(f,\partial\Delta).$
This completes the proof. ∎
###### Corollary 4.3.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that $f$ maps
the diameter $I=[-1,1]$ of $\overline{\Delta}$ homeomorphically onto the line
segment $\gamma=\overline{0,1}$ in $S$ and
(4.6) $L(f,\partial\Delta)<\sqrt{2}\pi.$
Then
$A(f,\Delta)\leq A(D_{l}),$
where $D_{l}$ is the convex Jordan domain in $S$ which is contained in the
spherical disk in $S$ with diameter $\overline{0,1}$ and is enclosed by the
two circular arcs in $S,$ each of which has endpoints $\\{0,1\\}$ and length
$l=\frac{1}{2}L(f,\partial\Delta).$
###### Remark 4.1.
The domain $D_{l}$ defined here is congruent with the domain $D_{l}$ defined
in Section 1 and the convexity of $D_{l}$ is ensured by (4.6).
###### Proof.
This follows from Corollaries 4.2 and Theorem 4.3. Without loss of generality,
we assume that the orientation of $f([-1,1])\subset S$ is from $0$ to $1.$
Let
$\alpha^{+}=\\{z\in\partial\Delta;\mathrm{Im}z\geq 0\\},\
\alpha^{-}=\\{z\in\partial\Delta;\mathrm{Im}z\leq 0\\},$
$\Delta^{+}=\\{z\in\Delta;\mathrm{Im}z>0\\},\
\Delta^{-}=\\{z\in\Delta;\mathrm{Im}z<0\\}.$
Then by (4.6) there uniquely exists a circle $C$ in $S$ passing through $0$
and $1$ such that the interior $\overline{0,1}^{\circ}$ of $\overline{0,1}$ is
contained in the disk $K$ enclosed by $C$ and the section $c_{1}$ of $C$ from
$1$ to $0$ has length $L(f,\alpha^{+}).$ Then, by the assumption, it is clear
that
$L(c_{1})=L(f,\alpha^{+})\geq\frac{\pi}{2}.$
If $f(\alpha^{+})=\frac{\pi}{2},$ then by the assumption we have
$f(\partial\Delta^{+})=\overline{0,1},$ and then $f(\Delta^{+})$ must contains
$\infty,$ for normal mappings are open mappings. But this contradicts the
assumption that $f$ is normal and as a normal mapping $f(z)\neq 0,1,\infty$
for all $z\in\Delta$. Thus we have $L(f,\alpha^{+})>\frac{\pi}{2},$ which
implies
(4.7) $L(\partial K)=L(C)<2\pi.$
Thus $\overline{0,1}+c_{1}$ encloses a Jordan domain $D_{1}$ and
$D_{2}=K\backslash\overline{D_{1}}$ is also a Jordan domain.
We may extend $f|_{\overline{\Delta^{+}}}$ to be a continuous mapping
$F:\overline{\Delta}\rightarrow S$ such that $F$ restricted to
$\overline{\Delta^{-}}$ is a homeomorphism onto $\overline{D_{2}}$ and
restricted to $\alpha^{-}$ is a homeomorphism onto $C\backslash c_{1}.$ Then,
we have
$L(F,\partial\Delta)=L(f,\alpha^{+})+L(F,\alpha^{-})=L(c_{1})+L(C\backslash
c_{1})=L(C),$
which, with (4.7), implies
(4.8) $L(F,\partial\Delta)=L(\partial K)=L(C)<2\pi.$
$F$ is not a normal mapping, and so we can not apply Theorem 4.3 to $F$
directly. But by (4.8) we can apply Theorem 4.3 to a normal mapping $g$ so
that $|L(g,\partial\Delta)-L(F,\partial\Delta)|$ and
$|A(g,\Delta)-A(F,\Delta)|$ can be made arbitrarily small, and finally obtain
(4.9) $A(F,\Delta)\leq A(D_{F}),$
where $D_{F}$ is a disk in some hemisphere of $S$ with
(4.10) $L(\partial D_{F})=L(F,\partial\Delta).$
$F$ is not normal just because the boundary curve
$\Gamma_{F}=F(z),z\in\partial\Delta,$ is not polygonal. Since
$F(\alpha^{+})=f(\alpha^{+})$ is already polygonal, $F(\alpha^{-})=C\backslash
c_{1}$ and $\partial\Delta=\alpha^{+}\cup\alpha^{-},$ the mapping $g$
mentioned above can be obtained by restricting $F$ to a domain
$\Delta_{g}\subset\Delta$ with $\Delta_{g}\supset\Delta^{+}.$
By (4.8) and (4.10) we have $L(\partial D_{F})=L(\partial K),$ which implies
$A(D_{F})=A(K).$ Thus, by (4.9) we have
$A(F,\Delta)\leq A(K).$
Therefore, by the facts $A(K)=A(D_{1})+A(D_{2})$ and
$A(F,\Delta)=A(f,\Delta^{+})+A(D_{2})$ we have $A(f,\Delta^{+})\leq A(D_{1}).$
Similarly, we can show that $A(f,\Delta^{-})\leq A(D_{1}^{\prime}),$ where
$D_{1}^{\prime}$ is the convex domain in some hemisphere of $S$ and is
enclosed by $\overline{1,0}$ and the circular arc $c_{2}$ from $0$ to $1$ with
$L(c_{2})=L(f,\alpha^{-}).$ Then $\gamma=c_{1}+c_{2}$ encloses a Jordan domain
$D_{\gamma}$ with $A(D_{\gamma})=A(D_{1})+A(D_{1}^{\prime})$ and
$A(f,\Delta)\leq A(D_{1})+A(D_{1}^{\prime})=A(D_{\gamma}),$
and by Corollary 4.2, the desired result follows. This completes the proof. ∎
Let $\alpha$ be a circular path in the upper half plane $\mathrm{Im}z\geq 0$
from $1$ to $0$ and let $\mathfrak{A}_{\alpha}$ be the domain in $\mathbb{C}$
enclosed by $\alpha$ and the interval $[0,1]$ and assume
$L(\alpha)\leq\frac{\sqrt{2}}{2}\pi,$ which means that $\alpha$ is contained
in the closed half-disk
$\\{z\in\mathbb{C};\ \mathrm{Im}z\geq 0\ \mathrm{and\
}|z-\frac{1}{2}|<\frac{1}{2}\\}.$
Then
$\frac{\pi}{2}\leq L(\alpha)\leq\frac{\sqrt{2}}{2}\pi.$
We want to find the relation between the spherical length $L(\alpha)$ and the
spherical area $A(\mathfrak{A}_{\alpha}).$ We will show that both $L(\alpha)$
and $A(\mathfrak{A}_{\alpha})$ is a real analytical function of
$\tau=\sin\theta_{a},0\leq\theta_{\alpha}\leq\frac{\pi}{2}.$
where $\theta_{\alpha}$ is the value of the angle between $\alpha$ and the
interval $[0,1]$ at $0.$
###### Lemma 4.2.
In the above setting, we have
(4.11)
$L(\alpha)=\zeta_{0}(\tau):=\frac{2}{\sqrt{1+\tau^{2}}}(\frac{\pi}{2}-\arctan\frac{\sqrt{1-\tau^{2}}}{\sqrt{1+\tau^{2}}}),\tau\in[0,1],$
and
(4.12)
$A(\mathfrak{A}_{\alpha})=\zeta_{1}(\tau):=2\arcsin\tau-\tau\zeta_{0}(\tau),\tau\in[0,1].$
###### Proof.
Let $c_{\alpha}\in\mathbb{C}$ be the center of the circle containing $\alpha.$
Then $\mathrm{Re}c_{\alpha}=\frac{1}{2},$ and since
$L(\alpha)\leq\frac{\sqrt{2}}{2}\pi,$ $\mathrm{Im}c_{\alpha}\leq 0.$ Let
$d_{\alpha}=2c_{\alpha}.$ Then the triangle in $\mathbb{C}$ with vertices
$0,1$ and $d_{\alpha}$ is a right-angled triangle and $\theta_{\alpha}$ is the
value of the angle at $d_{\alpha}.$
It is clear that
$|d_{\alpha}|=\frac{1}{\sin\theta_{\alpha}}.$
On the other hand, for any point $z\in\alpha,$ it is clear that
$|z|=\sin(\theta_{\alpha}-t)|d_{\alpha}|,$
where $t=\arg z.$ Then we obtain a parameter expression of the circular path
$\alpha$:
$\alpha=\alpha(t)=\frac{\sin(\theta_{\alpha}-t)}{\sin\theta_{\alpha}}e^{it},t\in[0,\theta_{\alpha}],$
Then we have
$|d\alpha(t)|=\frac{|-e^{it}\cos(\theta_{\alpha}-t)+ie^{it}\sin(\theta_{\alpha}-t)|}{\sin\theta_{\alpha}}dt=\frac{dt}{\sin\theta_{\alpha}},$
and
$\displaystyle L(\alpha)$ $\displaystyle=$
$\displaystyle\int_{\alpha}\frac{2|dz|}{1+|z|^{2}}=\int_{0}^{\theta_{\alpha}}\frac{2|d\alpha(t)|}{1+|\alpha(t)|^{2}}$
$\displaystyle=$
$\displaystyle\int_{0}^{\theta_{\alpha}}\frac{2\sin\theta_{\alpha}}{\sin^{2}\theta_{\alpha}+\sin^{2}(\theta_{\alpha}-t)}dt$
$\displaystyle=$
$\displaystyle\int_{0}^{\theta_{\alpha}}\frac{2\sin\theta_{\alpha}}{\sin^{2}\theta_{\alpha}+\sin^{2}x}dx$
$\displaystyle=$
$\displaystyle\frac{2}{\sqrt{1+\sin^{2}\theta_{\alpha}}}\left(\frac{\pi}{2}-\arctan\frac{\sqrt{1-\sin^{2}\theta_{\alpha}}}{\sqrt{1+\sin^{2}\theta_{\alpha}}}\right),$
and we have (4.11).
On the other hand, we have
$\displaystyle A(\mathfrak{A}_{\alpha})$ $\displaystyle=$
$\displaystyle\iint\limits_{\mathfrak{A}_{\alpha}}\frac{4dxdy}{\left(1+|z|^{2}\right)^{2}}$
$\displaystyle=$
$\displaystyle\int_{0}^{\theta_{a}}dt\int_{0}^{|\alpha(t)|}\frac{4rdr}{\left(1+r^{2}\right)^{2}}=2\int_{0}^{\theta_{a}}\left(1-\frac{1}{1+|\alpha(t)|^{2}}\right)dt$
$\displaystyle=$ $\displaystyle
2\theta_{\alpha}-2\int_{0}^{\theta_{a}}\frac{dt}{1+|\alpha(t)|^{2}}$
$\displaystyle=$ $\displaystyle
2\theta_{a}-2\int_{0}^{\theta_{a}}\frac{\sin^{2}\theta_{\alpha}dx}{\sin^{2}\theta_{\alpha}+\sin^{2}x}$
$\displaystyle=$ $\displaystyle 2\theta_{a}-\sin\theta_{\alpha}L(\alpha),$
and we have (4.12). ∎
It is clear from the geometrical sense that the function
$\zeta_{0}(\tau)=\frac{2}{\sqrt{1+\tau^{2}}}\left(\frac{\pi}{2}-\arctan\frac{\sqrt{1-\tau^{2}}}{\sqrt{1+\tau^{2}}}\right),\tau\in[0,1]$
is an injective mapping.
###### Corollary 4.4.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that $f$ maps
the diameter $[-1,1]$ of $\overline{\Delta}$ homeomorphically onto the line
segment $\overline{0,1}$ in $S$ and
$L(f,\partial\Delta)<\sqrt{2}\pi.$
Then
$A(f,\Delta)\leq 2\zeta_{1}(\tau)=4\arcsin\tau-2\tau\zeta_{0}(\tau),$
where $\tau=\zeta_{0}^{-1}(\frac{1}{2}L(f,\partial\Delta))$.
###### Proof.
This follows from Lemma 4.2 and Corollary 4.3 directly. ∎
###### Theorem 4.4.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that $f$ maps
the diameter $[-1,1]$ of $\overline{\Delta}$ homeomorphically onto the
interval $[0,1]$ in $S$ and
$L(f,\partial\Delta)<\sqrt{2}\pi.$
Then
$4\pi+A(f,\Delta)\leq h_{0}L(f,\partial\Delta),$
where $h_{0}$ is given by (1.3), i.e.
$h_{0}=\max_{\tau\in[0,1]}\left[\frac{\sqrt{1+\tau^{2}}\left(\pi+\arcsin\tau\right)}{\mathrm{arccot}\frac{\sqrt{1-\tau^{2}}}{\sqrt{1+\tau^{2}}}}-\tau\right].$
###### Proof.
Let $\tau=\zeta_{0}^{-1}(\frac{1}{2}L(f,\partial\Delta)).$ Then
$L(f,\partial\Delta)=2\zeta_{0}(\tau)$
and by Corollary 4.4.
$A(f,\Delta)\leq 2\zeta_{1}(\tau).$
Then
$\displaystyle 4\pi+A(f,\Delta)$ $\displaystyle\leq$ $\displaystyle
4\pi+2\zeta_{1}(\tau)$ $\displaystyle=$
$\displaystyle\frac{4\pi+2\zeta_{1}(\tau)}{L(f,\partial\Delta)}L(f,\partial\Delta)$
$\displaystyle=$
$\displaystyle\frac{4\pi+2\zeta_{1}(\tau)}{2\zeta_{0}(\tau)}L(f,\partial\Delta)$
$\displaystyle=$
$\displaystyle\frac{2\pi+\zeta_{1}(\tau)}{\zeta_{0}(\tau)}L(f,\partial\Delta)\leq
h_{0}L(f,\partial\Delta),$
where
$\displaystyle h_{0}$ $\displaystyle=$
$\displaystyle\max_{\tau\in[0,1]}\frac{2\pi+\zeta_{1}(\tau)}{\zeta_{0}(\tau)}=\max_{\tau\in[0,1]}\frac{2\pi+2\arcsin\tau-\tau\zeta_{0}(\tau)}{\zeta_{0}(\tau)}$
$\displaystyle=$
$\displaystyle\max_{\tau\in[0,1]}\left[\frac{2\pi+2\arcsin\tau}{\frac{2}{\sqrt{1+\tau^{2}}}\left(\frac{\pi}{2}-\arctan\frac{\sqrt{1-\tau^{2}}}{\sqrt{1+\tau^{2}}}\right)}-\tau\right]$
$\displaystyle=$
$\displaystyle\max_{\tau\in[0,1]}\left[\frac{\sqrt{1+\tau^{2}}\left(\pi+\arcsin\tau\right)}{\mathrm{arccot}\frac{\sqrt{1-\tau^{2}}}{\sqrt{1+\tau^{2}}}}-\tau\right].$
∎
## 5\. Locally convex polygonal paths and curves in the Riemann sphere
The goal of Sections 5–7 is to prove Theorem 7.1, which is the second key step
to prove the main theorem.
In this section we prove some results about locally convex polygonal Jordan
paths and curves, which is used in Sections 6 and 7.
It is clear that a generic convex triangle111111See Definition 2.11. in $S$ is
contained in some open hemisphere of $S,$ and then we have the followings.
###### Lemma 5.1.
Let $T\subset S$ be a triangle domain inside121212By definition, “inside”
means “on the left hand side of”. a generic convex triangle
$\overline{q_{1}q_{2}q_{3}q_{1}}$ in $S.$ Then for any $q\in T$, the notation
$\overline{q_{1}qq_{3}q_{1}}$ makes sense and denotes a generic convex
triangle.
The following result is easy to see.
###### Lemma 5.2.
For any polygonal convex Jordan curve $\Gamma$ in $S$ and any natural
edge131313By definition, natural edges are oriented by the polygonal curve.
$l$ of $\Gamma,$ the domain inside $\Gamma$ is contained in the open
hemisphere of $S$ which is inside the great circle determined141414This means
that the great circle contains $l$ and is oriented by $l$. by $l.$
###### Lemma 5.3.
Let $\Gamma=\Gamma(z),z\in\partial\Delta,$ be a polygonal Jordan curve in $S.$
(i) If $\Gamma$ is convex and contains a pair of antipodal points, then
$\Gamma$ is a biangle, and moreover, if in addition $\Gamma$ has a straight
edge with length $>\pi,$ then $\Gamma$ is a great circle of $S.$
(ii) If $\Gamma$ is convex and has at least three vertices in the usual sense,
i.e. $\Gamma$ can be expressed as
$\Gamma=l_{1}+l_{2}+\dots+l_{m},m\geq 3,$
where each $l_{j}$ is a straight edge151515Note that the interior of $l_{j}$
may contain points in $E,$ and so $l_{j}$ may not be a natural edge of
$\Gamma,$ by the definition of natual edges. of $\Gamma$ whose endpoints are
both strictly convex vertices of $\Gamma$, $j=1,2,\dots,m;$ then for each $j,$
$j=1,2,\dots,m,$
(5.1) $L(l_{j})<\pi$
and for the great circle $C_{l_{j}}$ in $S$ determined by $l_{j},$
(5.2) $\Gamma\cap C_{l_{j}}=l_{j},$
and therefore, $\Gamma$ is contained in some open hemisphere of $S.$
###### Proof.
Assume that $\Gamma$ has a pair of antipodal points $q_{1}$ and $q_{2}$. We
show that the section $\Gamma^{\prime}$ of $\Gamma$ from $q_{1}$ to $q_{2}\
$is straight.
For any natural edge $e$ of $\Gamma^{\prime}$, by Lemma 5.2, $q_{1}$ and
$q_{2}$ are both contained in the closed hemisphere inside the great circle
$C_{e}$ determined by $e,$ and thus $q_{1}$ and $q_{2}$ are both contained in
$C_{e}.$ By the arbitrariness of $e,$ $\Gamma^{\prime}$ must be a straight
path from $q_{1}$ to $q_{2}.$ For the same reason, we can show that
$\Gamma\backslash\Gamma^{\prime}$ is also a straight path. Thus, $\Gamma$ is a
biangle, which implies the second conclusion of (i), and (i) is proved.
Now, we prove (ii). It is clear that (i) implies (5.1) directly, for otherwise
$\Gamma$ is a biangle which contains at most two edges in the usual sense. So
we may write
$\Gamma=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m}q_{1}},m\geq
3,$
where $l_{j}=\overline{q_{j}q_{j+1}}$ with $q_{m+1}=q_{1}.$
We denote by $C_{l_{m}}$, $C_{l_{1}}$ and $C_{l_{2}}$ the great circles in $S$
determined by $l_{m}=\overline{q_{m}q_{1}},l_{1}=\overline{q_{1}q_{2}}\ $and
$l_{2}=\overline{q_{2}q_{3}},$ and denote by $D_{l_{m}}$, $D_{l_{1}}$ and
$D_{l_{2}}$ the domains inside $C_{l_{m}}$, $C_{l_{1}}$ and $C_{l_{2}},$
respectively. Then, $q_{m}$ and $q_{3}$ must be both contained in $D_{l_{1}},$
since, by the assumption, $\Gamma$ is strictly convex at $q_{1}$ and $q_{2}$;
and then, it is clear that
$K=\overline{D_{l_{m}}}\cap\overline{D_{l_{1}}}\cap\overline{D_{l_{2}}}$ is a
closed triangle domain whose three angles are all strictly less than $\pi$,
and then $K$ has a vertex in $D_{l_{1}}$ and
$l_{1}=\overline{q_{1}q_{2}}=K\cap C_{l_{1}}.$
On the other hand, it is clear that $\Gamma\cap C_{l_{1}}\supset l_{1}$ and,
by Lemma 5.2, $K\supset\Gamma.$ Therefore, we have (5.2) for $j=1$. This
completes the proof. ∎
###### Lemma 5.4.
Let $\Gamma$ be a locally convex polygonal Jordan path with initial and
terminal point at $q_{1}.$ Assume $\Gamma$ has the following natural
partition161616By definition, here ”locally convex” means that for each
$j=1,2,\dots,m-1,$ $\overline{q_{j}q_{j+1}q_{j+2}}$ is a convex path from
$q_{j}$ to $q_{j+2}.$ So, as a closed curve, $\Gamma$ may not be convex at
$q_{1}$, i.e., $\overline{q_{m}q_{1}q_{2}}$ may not be a convex path.
(5.3)
$\Gamma=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m}q_{1}},m\geq
3,$
such that
(5.4) $\overline{q_{1}q_{2}\dots q_{m}}\cap[0,+\infty]=\\{q_{1}\\}.$
Then the followings hold.
(i) For each $j=1,\dots,m-2,$ $L_{j}=\overline{q_{1}q_{j+1}q_{j+2}q_{1}}$ is a
generic convex triangle.
(ii) The closure $\overline{T_{\Gamma}}$ of the domain $T_{\Gamma}$ inside
$\Gamma$ is contained in some open hemisphere of $S.$
(iii) For each triangle domain $T_{j}$ inside the triangle $L_{j},$
$T_{j}\cap T_{k}=\emptyset,1\leq j<k\leq m-2,$
and
$\overline{T_{\Gamma}}=\cup_{j=1}^{m-2}\overline{T_{j}}.$
###### Remark 5.1.
(1). Condition (5.4) is used just to ensure that each vertex
$q_{j},j=2,3,\dots,m,$ of $\Gamma$ is a _strictly_ convex vertex. Thus, (5.4)
can be replaced by the condition that $\Gamma$ is strictly convex at
$q_{2},\dots,q_{m}.$ By Definition 2.9, (5.4) can also be replaced by
$\\{q_{2},\dots q_{m}\\}\cap E=\emptyset,$
which, with the assumption that $\Gamma$ is locally convex, implies that
$\Gamma$ is strictly convex at $q_{2},\dots,q_{m}.$
(2). The reader should notice that (5.3) makes sense if and only if
$d(q_{j},q_{j+1})<\pi$ for all $j=1,\dots,m-1,$ by the appointment.
(3). By conclusion (ii), in the case that
$\overline{q_{m}q_{1}}+\overline{q_{1}q_{m}}$ is straight, we have
$L(\overline{q_{m-1}q_{m}}+\overline{q_{m}q_{1}})<\pi.$ Thus if we regard
$\Gamma$ as a closed polygonal Jordan curve, each edge, in the usual sense, of
$\Gamma$ has length $<\pi,$ and thus, each natural edge of $\Gamma$ has length
$<\pi.$
###### Proof.
We regard $\Gamma$ as a closed curve. Then $\Gamma$ is locally convex
everywhere, with at most one exceptional point at $q_{1}.$
Let $T_{\Gamma}$ be the polygonal domain inside $\Gamma.$ Then, it is easy to
see that there is a path $l$ in $\overline{T_{\Gamma}}$ from $q_{1}$ to some
point $q^{\prime}\in\Gamma$ such that the followings hold.
(a) $l\cap\overline{q_{s}q_{s+1}}^{\circ}=\\{q^{\prime}\\}$ for some natural
edge $\overline{q_{s}q_{s+1}}$ of $\Gamma,$ where
$\overline{q_{s}q_{s+1}}^{\circ}$ is the interior of $\overline{q_{s}q_{s+1}}$
(if $s=m,$ $q_{s+1}=q_{1})$.
(b) The interior of $l$ is in the domain $T_{\Gamma}.$
(c) $l$ divides the angle $\Theta_{q_{1}}$ of the polygonal domain
$T_{\Gamma}$ at $q_{1}$ into two angles, each of which has value $<\pi.$
By the fact that any two distinct straight lines in the sphere $S$ only
intersect at a pair of antipodal points, and that (5.3) implies
$L(\overline{q_{1}q_{2}})<\pi$ and $L(\overline{q_{m}q_{1}})<\pi,$ we have
that
(5.5) $l\cap\overline{q_{1}q_{2}}=l\cap\overline{q_{m}q_{1}}=\\{q_{1}\\},$
which implies
(5.6) $2\leq s\leq m-1.$
It is easy to see from (a)–(c) that
$\Gamma_{1}=\overline{q_{1}q_{2}\dots q_{s}q^{\prime}}-l$
is strictly convex at $q_{1}$ and $q^{\prime},$ and then by (5.4) and the
assumption that $\Gamma$ is a locally convex path and that $q_{2},\dots,q_{m}$
are the all natural vertices, $\Gamma_{1}$ is a polygonal convex Jordan curve
that is strictly convex at all points $q_{1},\dots,q_{s},q^{\prime}.$ On the
other hand, since $\Gamma$ is simple, by (5.5) and (5.6) we conclude that
$q_{1},q_{2}$ and $q^{\prime}$ are distinct each other. Therefore,
$\Gamma_{1}$ is a convex polygonal Jordan curve that has at least three
strictly convex vertices, and thus, by Lemma 5.3 (ii),
$\Gamma_{1}\backslash\overline{q_{s}q^{\prime}}$ is contained in the open
hemisphere $S^{\prime}$ inside the great circle determined by
$\overline{q_{s}q_{s+1}}\supset\overline{q_{s}q^{\prime}}$, and for the same
reason,
$\Gamma_{2}=\overline{q^{\prime}q_{s+1}\dots q_{m}q_{1}}+l$
is also a convex polygonal Jordan curve that has at least three strictly
convex vertices and $\Gamma_{2}\backslash\overline{q^{\prime}q_{s+1}}$ is also
contained in $S^{\prime}.$ Thus, $\Gamma\backslash\overline{q_{s}q_{s+1}}$ is
contained in $S^{\prime},$ and, considering that
$L(\overline{q_{s}q_{s+1}})<\pi,$ we have proved (ii).
(i) follows from (ii) and the convexity of $\Gamma_{1}$ and $\Gamma_{2}$; and
(iii) follows from (i) and (ii) directly. This completes the proof. ∎
In the rest of this section we assume that $\gamma_{0}$ is a locally convex
polygonal Jordan path that has the natural partition
(5.7)
$\gamma_{0}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m-1}q_{m}},m\geq
3,$
with
(5.8) $\gamma_{0}\cap[0,+\infty]=\\{q_{1},q_{m}\\}.$
Then $q_{2},\dots,q_{m-1}$ are natural vertices of $\gamma_{0},$ at which
$\gamma_{0}$ is convex, and none of $q_{2},\dots,q_{m-1}$ is contained in $E.$
Thus, by Definitions 2.7 and 2.9 we have that
(a) $\gamma_{0}$ is strictly convex at all its natural vertices, the points
$q_{2},\dots,q_{m-1}.$
###### Lemma 5.5.
Assume $q_{1}\neq q_{m}$ and let $I_{q_{1}q_{m}}$ be the section of
$[0,+\infty]$ from $q_{1}$ to $q_{m}.$ Then the followings hold.
(i) $\Gamma=\gamma_{0}-I_{q_{1}q_{m}}$ is a polygonal Jordan curve that is
convex everywhere, with at most one exceptional point at $q_{1}$ or $q_{m}$.
(ii) $L(I_{q_{1}q_{m}})<\pi,$ and $\Gamma$ and the closure
$\overline{T_{\Gamma}}$ of the domain $T_{\Gamma}$ enclosed by
$\Gamma=\gamma_{0}-I_{q_{1}q_{m}}=\gamma_{0}+\overline{q_{m}q_{1}}$
is contained in some open hemisphere of $S$.
(iii) If, in addition, $q_{1}=0,$ then $\Gamma$ is strictly convex at $q_{m}.$
###### Proof.
It is clear that $\Gamma=\gamma_{0}-I_{q_{1}q_{m}}$ is simple, and by (a) we
have
(b) $q_{2},\dots,q_{m-1}$ are strictly convex vertices of $\Gamma.$
Thus the possible nonconvex vertices of $\Gamma$ are $q_{1}$ and $q_{m}.$ We
show that $\Gamma$ is convex at $q_{1}$ or $q_{m}.$ We assume the contrary
that both $q_{1}$ and $q_{m}$ are nonconvex vertices and without loss of
generality, we assume
(5.9) $q_{1}<q_{m}.$
Then we have171717Note that under this contrary assumption, $\Gamma$ does not
go straight at $q_{1},$ nor at $q_{m}$, but turn right at both $q_{1}$ and
$q_{m}$. On the other hand, $\overline{q_{1}q_{2}},\overline{q_{m-1}q_{m}}$
make sense if and only if $d\\{q_{1},q_{2}\\}<\pi$ and
$d\\{q_{m-1},q_{m}\\}<\pi.$ Thus, $\overline{q_{1}q_{2}}\cap
C_{1m}=\\{q_{1}\\}\ $and and $\overline{q_{m-1}q_{m}}\cap C_{1m}=\\{q_{m}\\}$,
which, with the assumption $q_{1}<q_{m},$ implies (c).
(c) Both $q_{2}$ and $q_{m-1}$ are contained in the open hemisphere
$S^{\prime}$ inside the great circle $C_{1m}$ determined by $I_{q_{1}q_{m}}.$
Then by (5.8), $I_{q_{1}q_{m}}$ has a neighborhood $J_{1}$ in the great circle
$C_{1m}$ determined by $I_{q_{1}q_{m}}\subset[0,+\infty]$ such that
$J_{1}^{\circ}\supset[0,+\infty]$ and $J_{1}\backslash I_{q_{1}q_{m}}\subset
T_{\Gamma},$ where $T_{\Gamma}$ is the domain inside $\Gamma$ and
$J_{1}^{\circ}$ is the interior of $J_{1}.$
It is clear that there are only two cases need to discuss:
Case 1. $C_{1m}\cap\Gamma=I_{q_{1}q_{m}}.$
Case 2. $\left(C_{1m}\cap\Gamma\right)\backslash I_{q_{1}q_{m}}\neq\emptyset.$
Assume Case 1 occurs. Then $C_{1m}\cap\gamma_{0}=\\{q_{1},q_{m}\\},$ and for
the section181818Recall that, by the appointment, a section of a curve
inherits the orientation of the curve, and so $I_{q_{m}q_{1}}^{\prime}$ is the
complementary of $I_{q_{1}q_{m}}^{\circ}$ in $C_{1m}.$
$I_{q_{m}q_{1}}^{\prime}$ of $C_{1m}$ from $q_{m}$ to $q_{1},$ by (5.9) and
(c) we conclude that
$\Gamma^{\prime}=\gamma_{0}+I_{q_{m}q_{1}}^{\prime}$
is a Jordan curve that is strictly convex at $q_{1}$ and $q_{m},$ and then by
(a) and Remark 2.5 we can conclude that $\Gamma^{\prime}$ is a convex
polygonal Jordan curve in $S$ and is strictly convex at $q_{1},\dots,q_{m},$
and thus we have by Lemma 5.3 that $L(I_{q_{m}q_{1}}^{\prime})<\pi,$ but on
the other hand
$L(I_{q_{m}q_{1}}^{\prime})=L(C_{1m})-L(I_{q_{1}q_{m}})\geq
2\pi-L([0,+\infty])=\pi,$
which is a contradiction. Thus, Case 1 can not occur, and then, Case 2 must
occur.
Then, we can extend the path $J_{1}$ past both sides to be a longer path $J$
from $q^{\prime}$ to $q^{\prime\prime}$ such that
(d) $\\{q^{\prime},q^{\prime\prime}\\}\subset\Gamma,\ J$ is oriented by
$I_{q_{1}q_{m}},$ the interior of the section of $J$ from $q^{\prime}$ to
$q_{1}$ and the interior of the section of $J$ from $q_{m}$ to
$q^{\prime\prime}$ are both contained in $T_{\Gamma}.$
Then
(5.10) $L(J)>\pi,$
for $J\supset J_{1}^{\circ}\supset[0,+\infty].$
We first show that $q^{\prime}\neq q^{\prime\prime}.$ We assume the contrary
that $q^{\prime}=q^{\prime\prime}.$ Then it is clear that $q^{\prime}$ is in
the interior $\gamma_{0}^{\circ}$ of $\gamma_{0}$ and, by (d), we have
(5.11) $C_{1m}\cap\gamma_{0}^{\circ}=\\{q^{\prime}\\}.$
Then, by (c), (5.11) and the fact that $\gamma_{0}$ is simple and connected,
we have
(5.12) $\gamma_{0}^{\circ}\backslash\\{q^{\prime}\\}\subset S^{\prime}.$
Since $q^{\prime}\neq q_{1},q_{m},$ $\gamma_{0}$ is convex at $q^{\prime}$ by
the assumption. Then by (5.12), $\gamma_{0}$ is strictly convex at
$q^{\prime},$ and thus the domain $T_{\Gamma}$ is a polygonal Jordan domain
with an angle at $q^{\prime}$ strictly less than $\pi.$ But by (d) and the
assumption $q^{\prime}=q^{\prime\prime},$ $J\backslash I_{q_{1}q_{m}}$ is a
neighborhood of $q^{\prime}$ in $C_{1m}$ and $\left(J\backslash
I_{q_{1}q_{m}}\right)\backslash\\{q^{\prime}\\}\subset T_{\Gamma}$. This is a
contradiction. Thus, $q^{\prime}\neq q^{\prime\prime}.$
Let $\gamma_{0}^{\prime}$ be the section of $\gamma_{0}$ from $q^{\prime}$ to
$q^{\prime\prime}.$ Then
$\Gamma^{\prime}=\gamma_{0}^{\prime}-J$
is a polygonal Jordan curve that is strictly convex at $q^{\prime}\ $and
$q^{\prime\prime},$ for $\gamma_{0}$ is a locally convex path,
$\\{q^{\prime},q^{\prime\prime}\\}\subset\gamma_{0}^{\prime}\subset\gamma_{0}^{\circ}$,
$q^{\prime}$ and $q^{\prime\prime}$ have neighborhoods in $J$ contained in
$T_{\Gamma}.$ Thus, $\Gamma^{\prime}$ is convex everywhere by the assumption
on $\gamma_{0}$, and then $\Gamma^{\prime}$ is convex by Remark 2.5, and then
by (5.10) and Lemma 5.3 (i), $\Gamma^{\prime}$ is a great circle, which is a
contradiction since $\Gamma^{\prime}$ strictly convex at $q^{\prime}.$
Summarizing the above argument, we can conclude that $\Gamma$ must be convex
at $q_{1}$ or $q_{m}$, and (i) is proved.
To prove the inequality in (ii), assume the contrary, that is,
$L(I_{q_{1}q_{m}})\geq\pi.$ Then
$q_{1}=0\ \mathrm{and\ }q_{m}=\infty,$
and so $L(I_{q_{1}q_{m}})=\pi.$ Without loss of generality, by (i), we may
assume that
(e) $\Gamma=\gamma_{0}-I_{q_{1}q_{m}}$ is convex at $q_{1}=0.$
If $\Gamma$ is also convex at $q_{m},$ then $\Gamma$ is a convex curve in $S,$
and then by Lemma 5.3 (i), $\Gamma$ is a biangle with vertices $0$ and
$\infty$, and then $[0,+\infty]$ and $\gamma_{0}$ should be the two straight
edges of the biangle $\Gamma;$ but by (b) this is a contradiction.
We first assume that $\Gamma$ is not convex at $q_{m}.$ Then we can extend
$I_{q_{1}q_{m}}$ past $q_{m}$ to obtain a longer line segment $J^{\prime}$
from $q_{1}$ to $q^{\prime}$ so that
(5.13) $q^{\prime}\in\gamma_{0}\ \mathrm{and\ }\left(J^{\prime}\backslash
I_{q_{1}q_{m}}\right)\backslash\\{q^{\prime}\\}\subset T_{\Gamma}.$
If $q^{\prime}=q_{1},$ then we have $J^{\prime}=C_{1m}\ $and then
(5.14) $J^{\prime}\backslash I_{q_{1}q_{m}}=C_{1m}\backslash[0,+\infty]\subset
T_{\Gamma}.$
But on the other hand, by (e), $\overline{q_{1}q_{2}}\backslash\\{q_{1}\\}$ is
either contained in the open hemisphere $S\backslash\overline{S^{\prime}}$
outside $C_{1m},$ or $\overline{q_{1}q_{2}}\subset C_{1m}.$ Then, in the case
$q^{\prime}=q_{1},$ we have $\overline{q_{1}q_{2}}\backslash\\{q_{1}\\}\subset
S\backslash\overline{S^{\prime}}$ by (5.14), and then $q^{\prime}=q_{1}$ has a
neighborhood in $J^{\prime}$ that is outside $T_{\Gamma},$ which contradicts
(5.14). Thus, $q^{\prime}\neq q_{1}.$
Then $-J^{\prime}$ and the segment of $\gamma_{0}$ from $q_{1}$ to
$q^{\prime}$ compose a polygonal Jordan curve $\Gamma^{\prime},\ $and
$\Gamma^{\prime}$ is strictly convex at $q^{\prime},$ since
$J^{\prime}\backslash I_{q_{1}q_{m}}$ intersects $\Gamma$ at $q^{\prime}$ from
the left hand side of $\Gamma,$ by (5.13), and $\Gamma$ is convex at
$q^{\prime}(\neq q_{1},q_{m}).$ Hence, by (b) and (e), $\Gamma^{\prime}$ is
locally convex polygonal Jordan curve with the straight edge $-J^{\prime}$
with $L(J^{\prime})>\pi,$ which implies that $\Gamma^{\prime}$ is a great
circle in $S$ by Lemma 5.3 (i). But this contradicts that $\Gamma^{\prime}$ is
strictly convex at $q^{\prime},$ and we obtain a contradiction again.
Summarizing the above discussion, we have proved
(5.15) $L(I_{q_{1}q_{m}})<\pi,$
the inequality in (ii). Then, we can write
$-I_{q_{1}q_{m}}=-\overline{q_{1}q_{m}}=\overline{q_{m}q_{1}},$ and
$\Gamma=\overline{q_{1}q_{2}}+\overline{q_{2}q_{2}}+\dots\overline{q_{m-1}q_{m}}+\overline{q_{m}q_{1}}.$
Now we prove that $\Gamma$ is contained in some open hemisphere of $S.$
If $\\{q_{1},q_{m}\\}\subset(0,\infty),$ then by (5.8), neither $q_{2},$ nor
$q_{m-1}$ can lie in $C_{1m}$ and thus by (i) $\Gamma$ is strictly convex at
$q_{1}$ or $q_{m}.$
Assume $q_{1}=0$. Then, by (5.15), $q_{m}\in(0,\infty)$ and by (5.8)
(5.16) $q_{m-1}\notin C_{1m}.$
If $\Gamma$ is not convex at $0,$ then by (i) and by (5.8), $\Gamma$ is
strictly convex at $q_{m}.$ If $\Gamma$ is straight near $q_{1},$ then
$\overline{q_{1}q_{2}}\subset C_{1m}$ and by (b),
$\overline{q_{2}q_{3}}\backslash\\{q_{2}\\}\subset
S\backslash\overline{S^{\prime}},$ and then we can extend
$\overline{q_{3}q_{2}}$ past $q_{2}$ to a point $q_{2}^{\prime}$ so that
$\overline{q_{3}q_{2}^{\prime}}$ makes sense and
$\overline{q_{1}q_{2}^{\prime}q_{3}}$ is still strictly convex at
$q_{2}^{\prime}.$ Then the curve
$\gamma_{0}^{\ast}=\overline{q_{1}q_{2}^{\prime}q_{3}\dots q_{m}}$ satisfies
all the assumptions of $\gamma_{0}$ but the curve
$\Gamma^{\ast}=\gamma_{0}^{\ast}-I_{q_{1}q_{m}}$ is not convex at $q_{1},$ and
thus by (i) and (5.16) $\Gamma^{\ast}$ is strictly convex at $q_{m}.$ But
$\Gamma^{\ast}$ and $\Gamma$ coincide near $q_{m},$ and thus $\Gamma$ is
strictly convex at $q_{m}.$ If $q_{m}=\infty,$ the discussion is similar.
Summarizing the above discussion, we can conclude that $\Gamma$ is strictly
convex at $q_{1}$ or $q_{m},$ and thus, by (b), either
$\overline{q_{1}q_{2}\dots q_{m}q_{1}}$, or $\overline{q_{m}q_{1}\dots
q_{m-1}q_{m}},$ is a locally convex path that is strictly convex at each
natural vertices, and then $\Gamma=\overline{q_{1}q_{2}\dots q_{m}q_{1}}$ is
contained in some open hemisphere of $S.$ The second part of (ii) is proved,
and (ii) is proved completely.
Now, assume $q_{1}=0.$ Then $q_{m}\in(0,+\infty)$ by the assumption and (ii).
Thus, by the assumption, $q_{m-1}$ is either contained in the open hemisphere
$S^{\prime}$ inside the great circle determined by $[0,+\infty],$ or
$q_{m-1}\in S\backslash\overline{S^{\prime}}.$ If $q_{m-1}\in S^{\prime},$
then $\Gamma$ is not convex at $q_{m}$ and the open interval of the great
circle $C$ determined by $[0,+\infty]$ from $q_{m}$ to $\infty$ is contained
in $T_{\Gamma},$ and then we can obtain a contradiction as the above argument
involving $J^{\prime}$. Thus $q_{m-1}\in S\backslash\overline{S^{\prime}},$
i.e. $\Gamma$ is strictly convex at $q_{m},$ and (iii) is proved. ∎
###### Lemma 5.6.
If $q_{1}=0$ and $q_{m}\in(0,+\infty),$ then for each $j=1,\dots,m-2,$
$L_{j}=\overline{q_{1}q_{j+1}q_{j+2}q_{1}}$ is a generic convex triangle and
for the triangle domain $T_{j}$ inside $L_{j},$
$T_{j}\cap T_{k}=\emptyset,1\leq j<k\leq m-2,$
$\overline{T_{j}}\cap[0,+\infty]=\\{0\\},\ \mathrm{for\ }j=1,\dots,m-3,$
$\overline{T_{m-2}}\cap[0,+\infty]=\overline{q_{m}q_{1}}.$
###### Proof.
By (b) in the above proof and by Lemma 5.5 (ii) and (iii),
$\Gamma=\gamma_{0}+\overline{q_{m}q_{1}}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m-1}q_{m}}+\overline{q_{m}q_{1}}$
is contained in some open hemisphere of $S$ and is strictly convex at
$q_{2},\dots,q_{m}$. Then by Lemma 5.4 (iii) and Remark 5.1 (1), the
conclusion follows. ∎
###### Lemma 5.7.
If $q_{2}$ is contained in the open hemisphere $S^{\prime}$ inside the great
circle $C$ determined by $[0,+\infty]$, $q_{m-1}$ is contained in
$S\backslash\overline{S^{\prime}}$, and if
(5.17) $\left\\{q_{1},q_{m}\right\\}\subset(0,+\infty),$
then the followings hold.
(i) $\Gamma=\gamma_{0}+\overline{q_{m}q_{1}}=\overline{q_{1}q_{2}\dots
q_{m}q_{1}}$ is a Jordan curve such that $0$ is contained in the domain
$T_{\Gamma}$ inside $\Gamma$.
(ii) For $q_{0}=0$ and $j=1,\dots,m-1,$
$L_{j}=\overline{q_{0}q_{j}q_{j+1}q_{0}}$ is a generic convex triangle; and
for the triangle domain $T_{j}$ inside $L_{j},$
$\overline{T_{j}}\cap[0,+\infty]=\\{0\\},\ \mathrm{for\ }j=2,\dots,m-2,$
$\overline{T_{1}}\cap[0,+\infty]=\overline{q_{0}q_{1}},\
\overline{T_{m-1}}\cap[0,+\infty]=\overline{q_{m}q_{0}}.$
###### Proof.
For any point $q_{1}^{\prime}$ that is in the interior of
$\overline{q_{1}q_{2}}$ and is sufficient close to $q_{1},$ by the assumption
of the lemma, the polygonal curve
$\gamma_{0}^{\prime}=\overline{0q_{1}^{\prime}q_{2}\dots
q_{m}}=\overline{0q_{1}^{\prime}}+\dots+\overline{q_{m-1}q_{m}}$
is a locally convex Jordan path and satisfies the assumption on $\gamma_{0}$
just with more edges, then applying Lemma 5.6 to $\gamma_{0}^{\prime}$ and
taking $q_{1}^{\prime}\rightarrow q_{1},$ we can obtain (i) and (ii). ∎
###### Lemma 5.8.
If
(5.18) $\left\\{q_{1},q_{m}\right\\}\subset(0,+\infty),$
and
(5.19) $\\{q_{2},q_{m-1}\\}\subset S^{\prime},$
where $S^{\prime}$ is the open hemisphere inside the great circle determined
by $[0,+\infty]$, then the curve
$\Gamma=\gamma_{0}+\overline{q_{m}q_{1}}=\overline{q_{1}q_{2}\dots
q_{m}q_{1}}$ is a convex polygonal Jordan curve, $q_{m}\leq q_{1}$ and
$\Gamma$ is strictly convex at $q_{j},j=1,2,\dots,m.$
###### Proof.
We first assume $q_{1}=q_{m}.$ Then
$\Gamma=\gamma_{0}=\overline{q_{1}q_{2}}+\dots+\overline{q_{m-1}q_{1}}$
is a locally convex Jordan path, and then $m\geq 4$ and, by (a), $\Gamma$ is
strictly convex at $q_{2},\dots,q_{m-1}$, and considering that in this case,
(5.8) is reduced to (5.4), we can conclude by Lemma 5.4 that the closure
$\overline{T_{\Gamma}}$ of the domain $T_{\Gamma}$ enclosed $\Gamma$ is
contained in some open hemisphere of $S.$ On the other hand, by (5.8), (5.18)
and (5.19) and the assumption that $q_{1}=q_{m},$ it is easy to see that, if
$\overline{q_{m-1}q_{1}q_{2}}$ is not convex at $q_{1},$ then
$[0,+\infty]\backslash\\{q_{1}\\}$ will be contained in $T_{\Gamma}$, and then
$T_{\Gamma}$ can not be contained in any open hemisphere of $S.$ This is a
contradiction. Thus, by (5.18) and (5.19), $\Gamma$ is strictly convex at
$q_{1},$ and then by (a), $\Gamma$ is strictly convex at
$q_{j},j=1,2,\dots,m.$
Now, we assume $q_{1}\neq q_{m}.$ Then by Lemma 5.5, $\Gamma$ is convex at
$q_{1}$ or $q_{m}.$ If $q_{1}<q_{m},$ then $\Gamma$ is neither convex at
$q_{1},$ nor at $q_{m}.$ Thus, we must have $q_{m}<q_{1}.$ Then, by (5.18) and
(5.19), $\Gamma$ is strictly convex at $q_{1}$ and $q_{m},$ and then by (a),
$\Gamma$ is strictly convex at $q_{j},j=1,2,\dots,m.$ ∎
## 6\. Lifting Lemmas for normal mappings
In this section, we prove Theorem 6.1 that is used to prove Theorem 7.1.
Theorem 7.1 is the second key step to prove the main theorem.
###### Lemma 6.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and let $D$ be a
polygonal Jordan domain in $S$ such that $f^{-1}$ has a univalent
branch191919”univalent branch” always means that the branch is a
homeomorphism. $g$ defined on $D.$ Then $g$ can be extended to be a
homeomorphism $\widetilde{g}$ from $\overline{D}$ onto
$\widetilde{g}(\overline{D})$.
###### Proof.
There is a simple and standard way to prove this by Lemma 3.3. ∎
The following result is obvious but useful.
###### Lemma 6.2.
Let $D_{1}$ and $D_{2}$ be Jordan domains in $\mathbb{C}$ and let
$f:\overline{D_{1}}\rightarrow\overline{D_{2}}$ be a mapping such that
$f:\overline{D_{1}}\rightarrow f(\overline{D_{1}})$ is a homeomorphism. If
$f(\partial D_{1})\subset\partial D_{2},$ Then
$f(\overline{D_{1}})=\overline{D_{2}}$.
###### Lemma 6.3.
Let $p_{1}$ and $p_{2}$ be two distinct points in $\partial\Delta,$ let
$\alpha$ be the section202020Recall that $\partial\Delta$ is always orientated
anticlockwise, and a section of a curve inherits the orientation of the curve.
of $\partial\Delta$ from $p_{1}$ to $p_{2}$ and let $\beta$ be a Jordan path
in $\overline{\Delta}$ from $p_{2}$ to $p_{1}$ such that $\alpha$ and $\beta$
have a common point $p_{0}$ with $p_{0}\neq p_{1},p_{2}.$ Assume that
$f:\overline{\Delta}\rightarrow S$ is a normal mapping such that the
followings hold.
(a) The curve $\Gamma_{\alpha}=f(z),z\in\alpha,$ and
$\Gamma_{\beta}=f(z),z\in\beta,$ are polygonal paths and are both convex at
$p_{0}$.
(b) $f$ is regular212121This means that $f$ is homeomorphic in a neighborhood
of $p_{0}.$ at $p_{0}.$
Then $p_{0}$ has a neighborhood $\beta^{\prime}$ in $\beta$ such that
$\beta^{\prime}\subset\alpha\subset\partial\Delta$ and $f$ restricted to
$\beta^{\prime}$ is a line segment in $S$.
###### Proof.
By the assumption, $p_{0}$ has a neighborhood $\alpha^{\prime\prime}$ in
$\alpha$ and a neighborhood $\beta^{\prime\prime}$ in $\beta,$ such that the
curves $f(\alpha^{\prime\prime})$ and $f(\beta^{\prime\prime})$ intersect
”tangently”.
∎
###### Lemma 6.4.
Let $p_{j}=e^{i\theta_{j}}$ be a number of $m$ distinct points in
$\partial\Delta$ with
$\theta_{1}<\theta_{2}<\dots<\theta_{m}<\theta_{1}+2\pi,$
let $\alpha_{j}$ be the section of $\partial\Delta$ from $p_{j}$ to
$p_{j+1},j=1,\dots,m-1,$ let $f:\overline{\Delta}\rightarrow S$ be a normal
mapping and let
(6.1) $q_{j}=f(p_{j}),j=1,\dots m.$
Assume that the followings hold.
(a) The section
$\Gamma_{0}=f(z),z\in\alpha_{0}=\alpha_{1}+\dots+a_{m}$
of the boundary curve $\Gamma_{f}=f(z),z\in\partial\Delta,$ is a polygonal
Jordan path and each section $\Gamma_{j}=f(\alpha_{j})$ of $\Gamma_{0}$ is a
natural edge of $\Gamma_{0}$ with
(6.2) $L(\Gamma_{j})<\pi,j=1,\dots,m.$
(b) $L_{j}=\overline{q_{1}q_{j+1}q_{j+2}q_{1}},j=1,\dots,m-2,$ are generic
convex triangles in $S$, the triangle domains $T_{j}$ enclosed by $L_{j}$ are
disjoint each other, and
$\Gamma=\Gamma_{0}+\overline{q_{m}q_{1}}$
is a polygonal Jordan curve.
(c) For the domain $T$ enclosed by $\Gamma,$ $f$ has no branched point in
$\overline{T}\backslash\overline{q_{m}q_{1}}.$
(d) The boundary curve $\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex
in $T.$
Then, the followings hold true.
(i) $f^{-1}$ has a univalent branch $g$ defined on $\overline{T}$ such that
$g$ maps $\Gamma_{0}=\overline{q_{1}q_{2}\dots q_{m}}$ onto
$\alpha_{0}=\alpha_{1}+\dots+\alpha_{m-1}$ with
$g(q_{j})=p_{j},j=1,2,\dots,m.$
(ii) If in addition, for some _open_ interval $\gamma$ of
$\overline{q_{m}q_{1}},$ $f$ has no branched point in $\gamma$ and
$\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex in $\gamma,$ then
either $g(\gamma)\subset\partial\Delta$ or $g(\gamma)\subset\Delta.$
We first prove the following lemma under the same assumption as that in Lemma
6.4. Note that by (6.1) and (6.2), we can write
$\Gamma_{j}=\overline{q_{j}q_{j+1}},j=1,2,\dots,m-1.$
###### Lemma 6.5.
(i). $f^{-1}$ has a univalent branch $g_{1}$ defined on $\overline{T_{1}},$
such that $g_{1}$ restricted to $\overline{q_{1}q_{2}q_{3}}$ is a
homeomorphisms onto $\alpha_{1}+\alpha_{2}$ with $g_{1}(q_{j})=p_{j},j=1,2,3.$
(ii). If $m>3,$ then $\beta_{1}=g_{1}(\overline{q_{1}q_{3}})$ is a Jordan path
in $\overline{\Delta}$ from $p_{1}$ to $p_{3}$ and the interior of $\beta_{1}$
is contained in $\Delta.$
(iii). If $m=3$ and for some _open_ interval $\gamma$ contained
$\overline{q_{m}q_{1}},$ $f$ has no branched point in $\gamma$ and
$\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex in $\gamma,$ then
either $g(\gamma)\subset\partial\Delta$ or $g(\gamma)\subset\Delta.$
###### Proof.
Write $c_{0}=\alpha_{1}+\alpha_{2}$,
$\gamma_{0}=\Gamma_{1}+\Gamma_{2}=\overline{q_{1}q_{2}q_{3}}$ and
$\gamma_{1}=\overline{q_{1}q_{3}}$ .
Let $v_{1}$ be an interior point of $\gamma_{1}=\overline{q_{1}q_{3}}$ and let
$v=v_{s}=v(s)$, $s\in[0,1],$ be a Jordan path that represents the straight
path from $q_{2}$ to $v_{1}$ in the closed triangle domain $\overline{T_{1}}$
enclosed by the triangle
$L_{1}=\overline{q_{1}q_{2}q_{3}q_{1}}=\gamma_{0}-\gamma_{1}$ . Then, by (b)
and Lemma 5.1, for each $s\in(0,1),$ the polygonal Jordan path
$\gamma_{s}=\overline{q_{1}v_{s}q_{3}}$ in $\overline{T_{1}}$ is strictly
convex at $v_{s}$, and $\gamma_{s},s\in[0,1],$ is a family of curves
exhausting the closed domain $\overline{T_{1}}$ and satisfying the following
condition (e).
(e) For each $s\in(0,1],$ the domain $T_{s}$ inside $\gamma_{0}-\gamma_{s}$ is
a (spherical) quadrilateral domain contained in $T_{1}$ and for any pair
$s_{1},s_{2}\in(0,1]$ with $s_{1}<s_{2},$
$T_{s_{1}}\cup(\gamma_{s_{1}}\backslash\\{q_{1},q_{3}\\})=\overline{T_{s_{1}}}\backslash\gamma_{0}\subset
T_{s_{2}}.$
Since $f$ is normal, by the definition, there exists a point $q_{1}^{\prime}$
in the interior of $\Gamma_{1}=\overline{q_{1}q_{2}}$ and there exists a point
$q^{\prime}$ in the domain $T_{1}$ such that $f^{-1}$ has a univalent branch
defined on the closure of the triangle domain inside the triangle
$\overline{q_{1}q_{1}^{\prime}q^{\prime}q_{1}}\subset\overline{T_{1}}$ and
this branch restricted to $\overline{q_{1}q_{1}^{\prime}}$ is a homeomorphism
onto a section of $\alpha_{1}\ $from $p_{1}$ to some interior point of
$\alpha_{1}.$ At $q_{3}$ we can do this similarly. On the other hand,
considering that $\overline{q_{1}q_{2}q_{3}}$ is simple and $f$ is a normal
mappings, by (b) and (c), we can conclude that for each $q_{0}$ contained in
the interior222222Note that the interior of $\gamma_{0}$ does not intersects
$\overline{q_{m}q_{1}},$ and thus $f$ has no branched point in the interior of
$\gamma_{0}.$ of
$\gamma_{0}=\Gamma_{1}+\Gamma_{2}=\overline{q_{1}q_{2}q_{3}},$ there exists a
disk $V_{q_{0}}$ in $S$ such that $f^{-1}$ has a univalent branch defined on
$V_{q_{0}}\cap\overline{T_{1}}$ and this branch maps $\gamma_{0}\cap
V_{q_{0}}$ onto a section of $c_{0}.$ Summarizing these discussion, we
conclude that, for sufficiently small $\delta>0,$ $\delta$ satisfies the
following property:
(f) $f^{-1}$ has a univalent branch $g_{\delta}$ defined on
$\overline{T_{\delta}}$ with
$g_{\delta}(q_{j})=p_{j},j=1,2,3,$
$g_{\delta}$ restricted to $\gamma_{0}=\overline{q_{1}q_{2}q_{3}}$ is a
homeomorphism onto $c_{0}=\alpha_{1}+\alpha_{2}$ and
$c_{\delta}=g_{\delta}(\gamma_{\delta})$ is a Jordan path from $p_{1}$ to
$p_{3}$ whose interior is contained in $\Delta.$
If $\delta$ satisfies (f) and $\delta<1$, then $c_{0}-c_{\delta}$ is a Jordan
curve, the domain $\widetilde{\Delta}_{\delta}$ inside $c_{0}-c_{\delta}$ is a
Jordan domain, $\overline{\Delta}\backslash\widetilde{\Delta}_{\delta}$ is a
closed Jordan domain232323By (f), $\alpha_{\delta}$ divides $\Delta$ into two
Jordan domains. and $f$ restricted
$\overline{\Delta}\backslash\widetilde{\Delta}_{\delta}$ is a normal mapping
(note that
$f(\partial(\overline{\Delta}\backslash\widetilde{\Delta}_{\delta}))$ is
polygonal). In this case, replacing $\overline{\Delta}$ by
$\overline{\Delta}\backslash\widetilde{\Delta}_{\delta},$ $c_{0}$ by
$c_{\delta},$ $\gamma_{0}$ by $\gamma_{\delta}$ and applying the above
argument once more, we can also prove the following property for $\delta:$
(g) For each $\delta\in(0,1),$ if $\delta$ satisfies (f), then for
sufficiently small $\varepsilon>0$, $\delta+\varepsilon$ satisfies (f) as
well.
On the other hand, it is clear that, if $\delta$ satisfies (f), then each
positive number $\delta^{\prime}<\delta$ satisfies (f) as well. Thus, for
$\delta_{0}=\sup\\{\delta\in(0,1);\ \delta\ \mathrm{satisfies}\
\mathrm{(f)}\\},$
we have
(h) Each $\delta\in(0,\delta_{0})$ satisfies (f).
To show $\delta_{0}=1,$ we first show that $\delta_{0}$ satisfies (f) if
$\delta_{0}<1$.
By (e), (f) and (h), $f^{-1}$ has a univalent branch
$\widetilde{g}_{\delta_{0}}$ defined on $T_{\delta_{0}}\cup\gamma_{0}$. By
Lemma 6.1, $\widetilde{g}_{\delta_{0}}$ can be extended to be a homeomorphism
$g_{\delta_{0}}$ defined on $\overline{T_{\delta_{0}}}.$ Thus,
$\gamma_{\delta_{0}}$ has a lift
$c_{\delta_{0}}=g_{\delta_{0}}(w),w\in\gamma_{\delta_{0}},$ by $f,$ and
$c_{\delta_{0}}$ is a Jordan path from $p_{1}$ to $p_{3}$ in
$\overline{\Delta}.$ Let
$\overline{\widetilde{\Delta}_{\delta_{0}}}=g_{\delta_{0}}(\overline{T_{\delta_{0}}}),$
then $f$ restricted to $\overline{\widetilde{\Delta}_{\delta_{0}}}$ is a
homeomorphism onto $\overline{T_{\delta_{0}}}$, and maps $c_{\delta_{0}}$ onto
$\gamma_{\delta_{0}}$.
Now, we show that the following hold.
(j) If $\delta_{0}<1,$ then the interior of $c_{\delta_{0}}$ is contained in
$\Delta.$
Assume $\delta_{0}<1$ and let $p_{0}\in c_{\delta_{0}}$ be any interior point
of $c_{\delta_{0}}\ $with $p_{0}\in\partial\Delta.$ Then
$p_{0}\in\alpha:=\left(\partial\Delta\right)\backslash c_{0}$ and $f(p_{0})$
is in the interior of $\gamma_{\delta_{0}},$ and then $f(p_{0})\in T_{1}.$
Thus, by (d), the curves
$\Gamma_{\alpha}=f(z),z\in\alpha=\left(\partial\Delta\right)\backslash c_{0},$
and $\Gamma_{\beta}=f(z),z\in\beta=c_{\delta_{0}},$ are both convex at $p_{0}$
(note that $\Gamma_{\beta}$ is the path $\gamma_{\delta_{0}})$. Therefore, by
(c) and Lemma 6.3, $p_{0}$ has a neighborhood $\beta^{\prime}$ in
$\beta=c_{\delta_{0}}$ such that
$\beta^{\prime}\subset\alpha=\left(\partial\Delta\right)\backslash c_{0}$ and
$f(\beta^{\prime})$ is straight. But then, with a continuation argument, we
can prove that the whole of $f(c_{\delta_{0}})$ is also straight, which
contradicts the fact that $\gamma_{\delta_{0}}=f(c_{\delta_{0}})$ is not
straight if $\delta_{0}<1.$ Thus, the interior of $c_{\delta_{0}}$ must be in
$\Delta$ and (j) is proved.
(j) implies that $\delta_{0}$ satisfies (f) if $\delta_{0}<1.$ This, with (g),
implies that if $\delta_{0}<1,$ then $\delta_{0}+\varepsilon$ satisfies (f)
for sufficiently small $\varepsilon>0$. This contradicts the definition of
$\delta_{0}.$ Thus we have proved $\delta_{0}=1.$
Now that $\delta_{0}=1,$ by (e)–(h), $f^{-1}$ has a univalent branch
$\widetilde{g}_{1}$ defined on $T_{1}\cup\gamma_{0},$ and by Lemma 6.1,
$\widetilde{g}_{1}$ can be extended to be a homeomorphism $g_{1}$ defined on
$\overline{T_{1}}$. Thus, (i) holds. (ii) can be proved as the proof of (j),
by (i) and Lemma 6.3, and (iii) can be proved similarly. ∎
###### Proof of Lemma 6.4.
If $m=3,$ then Lemma 6.4 follows from (i) and (iii) of Lemma 6.5. So we may
assume $m\geq 4.$ But, without loss of generality, we complete the proof only
for the case $m=4.$
We continue the proof of Lemma 6.5. Let
$\beta_{1}=g_{1}(\overline{q_{1}q_{3}}).$ Then by Lemma 6.5 (ii),
$\beta_{1}^{\circ}\subset\Delta,$ where $\beta_{1}^{\circ}$ is the interior of
$\beta_{1}.$ Then $\beta_{1}$ divides $\Delta$ into two Jordan domains. We
denote by $\Delta_{1}$ the component of $\Delta\backslash\beta_{1}$ that is on
the left hand side of $\beta_{1},$ i.e. $\Delta_{1}=\Delta\backslash
g_{1}(\overline{T_{1}}).$
Then, by the assumption $m=4,$ $\Delta_{1}$ is enclosed by
$\beta_{1}+\alpha_{3}+\alpha^{\ast}$
where $\alpha^{\ast}$ is the section of $\partial\Delta$ from $p_{m}=p_{4}$ to
$p_{1}.$
Again by (a)–(d) and Lemma 6.5 (i), $f^{-1}$ has a univalent branch $g_{2}$
defined on $\overline{T_{2}}$ such that $g_{2}:\overline{T_{2}}\rightarrow
g_{2}(\overline{T_{2}})$ is a homeomorphism, restricted to
$\overline{q_{1}q_{3}q_{4}}$ is a homeomorphism onto $\beta_{1}+\alpha_{3}$
and
$g_{2}(q_{j})=p_{j},j=1,3,4.$
Since $f$ has no branched point in
$\overline{T}\backslash\overline{q_{4}q_{1}}$ (note that $m=4),$ $f$ has no
branched point on $\overline{q_{3}q_{1}}\backslash\\{q_{1}\\}.$ Thus, $g_{1}$
and $g_{2}$ must be identical on $\overline{q_{3}q_{1}}.$ Then $g_{1}$ and
$g_{2}$ make up a univalent branch $g$ of $f^{-1},$ such that
$g:\overline{T}=\overline{T_{1}\cup T_{2}}\rightarrow g(\overline{T})$ is a
homeomorphism with
$g(\overline{q_{1}q_{2}q_{3}q_{4}})=\alpha_{1}+\alpha_{2}+\alpha_{3}$. (i) is
proved.
(ii) can be proved as the proof of (j). This completes the proof of Lemma 6.4.
∎
###### Remark 6.1.
Lemma 6.4 implies an interesting proposition: let
$f:\overline{\Delta}\rightarrow\mathbb{C}$ be an open mapping that is
orientation preserved and is locally homeomorphism. Then, $f$ is a
homeomorphism, provided that the boundary curve
$\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex.
Here “locally convex” means that when $z$ goes around $\partial\Delta$
anticlockwise, $f(z)$ always go straight or turn left. For example, if we
assume that the curve $\Gamma_{f}$ is smooth and is locally straight, or
locally on the left hand side of its tangent line, then $\Gamma_{f}$ is
locally convex.
For later use, we only prove this in a special version for normal mappings,
which is the following corollary.
###### Corollary 6.1.
Let $\alpha_{0}\ $be a section of $\partial\Delta$ from $p_{1}$ to $p_{m}$
with
(6.3) $p_{1}\neq p_{m},$
let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that the
section $\gamma_{0}=f(z),z\in\alpha_{0},$ is a closed Jordan path that has the
natural partition
(6.4)
$\gamma_{0}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m-1}q_{1}},$
with $q_{1}=f(p_{1})=f(p_{m})$ and
$\\{q_{2},\dots,q_{m-1}\\}\cap E=\emptyset.$
Assume that for the domain $T_{\gamma_{0}}\subset S$ enclosed by $\gamma_{0},$
the boundary curve $\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex in
$\overline{T_{\gamma_{0}}}\backslash\\{q_{1}\\}.$ Then $f$ has a branched
point in $\overline{T_{\gamma_{0}}}\backslash\\{q_{1}\\}.$
###### Proof.
Since $\gamma_{0}$ is a closed Jordan path, by (6.4) we have242424Note that
(6.11) makes sense iff each term on the right hand side has spherical length
$<\pi.$ $m\geq 4.$ Since $\Gamma_{f}$ is locally convex in
$\overline{T_{\gamma_{0}}}\backslash\\{q_{1}\\},$ $\gamma_{0}$ is a locally
convex path, and then by Lemma 5.4 and Remark 5.1 (1), for each
$j=1,\dots,m-3,$ $L_{j}=\overline{q_{1}q_{j+1}q_{j+2}q_{1}}$ is a generic
convex triangle such that the triangle domains $T_{j}$ enclosed by $L_{j}$ are
disjoint each other and
$\overline{T_{\gamma_{0}}}=\cup_{j=1}^{m-3}\overline{T_{j}}$.
Assume $f$ has no branched point in
$\overline{T_{\gamma_{0}}}\backslash\\{q\\}$. Then, Lemma 6.4 applies, i.e.
$f^{-1}$ has a univalent branch $g$ defined on $\overline{T_{\gamma_{0}}}$
such that $g$ restricted to $\overline{q_{1}q_{2}\dots q_{m-1}}$ is a
homeomorphism onto a section $\alpha_{0}^{\prime}$ of $\alpha_{0}$ from
$p_{1}$ to some point $p_{m-1}^{\prime}\in\alpha_{0}^{\circ},$ here
$\alpha_{0}^{\circ}$ is the interior $\alpha_{0}\backslash\\{p_{1},p_{2}\\}$
of $\alpha_{0}.$
Let $\alpha_{0}^{\prime\prime}$ be the section of $\alpha_{0}$ from
$p_{m-1}^{\prime}$ to $p_{m},$ then, by the assumption, it is clear that $f$
maps $\alpha_{0}^{\prime\prime}$ homeomorphically onto
$\overline{q_{m-1}q_{m}}=\overline{q_{m-1}q_{1}}.$ Since $f$ has no branched
point on $\overline{T_{\gamma_{0}}}\backslash\\{q_{1}\\}$ and
$q_{m-1}\in\overline{T_{\gamma_{0}}}\backslash\\{q_{1}\\},$ after an argument
of uniqueness of the lifting, we have
$g(\overline{q_{m-1}q_{1}})=\alpha_{0}^{\prime\prime}\subset\partial\Delta.$
Then we have $g(\gamma_{0})\subset\partial\Delta,$ and then
$g(\gamma_{0})=\partial\Delta$ by Lemma 6.2. Thus $f$ is a homeomorphism, and
$\gamma_{0}$ is the whole curve $\Gamma_{f},$ which contradicts (6.3). The
proof is completed. ∎
In the rest of this section, let $p_{j}=e^{i\theta_{j}}$ be $m$ distinct
points in $\partial\Delta,j=1,\dots,m,$ with
$m\geq 3\ \mathrm{and\
}\theta_{1}<\theta_{2}<\dots<\theta_{m}\leq\theta_{1}+2\pi,$
let $\alpha_{j}$ be the section of $\partial\Delta$ from $p_{j}$ to
$p_{j+1},j=1,\dots,m-1,$ and let
$\alpha_{0}=\alpha_{1}+\alpha_{1}+\dots+\alpha_{m-1}.$
###### Definition 6.1.
The family $\mathcal{F}_{m}$ is defined to be the family of all normal
mappings $f:\overline{\Delta}\rightarrow S$ that satisfies all the following
conditions (A)–(E).
(A) The section $\gamma_{0}=f(z),z\in\alpha_{0},$ of the boundary curve
$\Gamma_{f}=f(z),z\in\partial\Delta,$ is a Jordan path.
(B) $\gamma_{0}$ has the natural partition
(6.5)
$\gamma_{0}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m-1}q_{m}},$
with
(6.6) $\gamma_{0}\cap[0,+\infty]=\\{q_{1},q_{m}\\},$
where, $q_{j}=f(p_{j}),$ $j=1,\dots,m,$ and $\overline{q_{j}q_{j+1}}$ is the
section
$\Gamma_{j}=f(z),z\in\alpha_{j},j=1,\dots,m-1.$
(C) The boundary curve $\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex
in $S\backslash\\{0,\infty\\}.$
(D) $f$ has no ramification point in $\overline{\Delta}.$
(E) $f(\Delta)\cap[0,+\infty]=\emptyset.$
Each $f\in\mathcal{F}_{m}$ will be endowed with all the notations in the
definition. By (A) and (B) the curve
(6.7) $\Gamma=\gamma_{0}+\overline{q_{m}q_{1}}$
is a polygonal Jordan curve. Here it is permitted that $q_{1}=q_{m},$ and in
this case $\Gamma=\gamma_{0}.$
Note that by (A), (B), (C) and Definition 2.9, we have
(F) $\gamma_{0}$ is a locally convex polygonal Jordan path that is strictly
convex at $q_{2},\dots,q_{m-1}.$
Then by (B) and Lemma 5.5 (ii), $\overline{q_{m}q_{1}}$ in (6.7) makes sense.
On the other hand, if $q_{1}=q_{m},$ then, by (A) and (6.5), $m\geq 4.$
Therefore, by Lemma 5.4 (for the case $q_{1}=q_{m}$ here) and Lemma 5.5 (for
the case $q_{1}\neq q_{m})$ the following holds true.
(G) The closure $\overline{T_{\Gamma}}$ of the domain $T_{\Gamma}$ enclosed by
$\Gamma=\gamma_{0}+\overline{q_{m}q_{1}}$ is contained in some open hemisphere
of $S.$
###### Theorem 6.1.
Let $f\in\mathcal{F}_{m}$ and denote by $T_{\Gamma}$ the domain enclosed by
$\Gamma.$ Then the followings hold.
(i) The restriction $f|_{\Delta}:\Delta\rightarrow
T_{\Gamma}\backslash[0,+\infty]$ is a homeomorphism. (ii)
$f(\overline{\Delta})$ is contained in some open hemisphere of $S$.
(iii) For $\alpha_{0}^{\circ}=\alpha_{0}\backslash\\{p_{1},p_{m}\\},$
(6.8) $f(\alpha_{0}^{\circ})\cap[0,+\infty]=\emptyset,$ (6.9)
$f(\left(\partial\Delta\right)\backslash\alpha_{0}^{\circ})\subset[0,+\infty],$
and
(6.10) $L(f,\alpha_{0})>L(f,\left(\partial\Delta\right)\backslash\alpha_{0}).$
###### Proof.
By (A) and (B), it is clear that (6.8) holds true. To complete the remained
proof, it suffices to consider three cases.
Case 1.
(6.11) $0=q_{1}\leq q_{m}<+\infty.$
If $q_{1}=q_{m}=0,$ then by (G) we have
$\overline{T_{\Gamma}}\cap[0,+\infty]=\\{0\\},$
and then by (C), (D) and Corollary 6.1 we have $p_{1}=p_{m},$ and then, by
(A), $f$ maps $\alpha_{0}=\partial\Delta$ homeomorphically onto the closed
Jordan curve $\Gamma=\gamma_{0},$ and since $f$ is normal we conclude that
$f:\Delta\rightarrow T_{\Gamma}=T_{\Gamma}\backslash[0,+\infty]$ is a
homeomorphism, and other conclusions of Theorem 6.1 is trivially hold with
$\alpha_{0}=\partial\Delta$, by (G).
If $q_{1}\neq q_{m},$ i.e. $q_{1}=0$ and $q_{m}\in(0,+\infty),$ then by Lemma
5.6 the triangles $L_{j}=\overline{q_{1}q_{j+1}q_{j+2}q_{1}}$ are generic
convex for $j=1,2,\dots,m-2,$ the domains $T_{j}$ enclosed by $L_{j}$ are
disjoint each other and for the domain $T_{\Gamma}$ enclosed by
$\Gamma=\gamma_{0}+\overline{q_{m}q_{1}}$ we have
$\overline{T_{\Gamma}}=\cup_{j}^{m-2}\overline{T_{j}},\
0\notin\overline{T_{\Gamma}},$
and
$\overline{T_{\Gamma}}\backslash\overline{q_{m}q_{1}}=\overline{T_{\Gamma}}\backslash\overline{q_{m}0}\subset
S\backslash\\{0,\infty\\}.$
Then, by (C) and (D), Lemma 6.4 applies, and then, $f^{-1}$ has a univalent
branch $g$ defined on $\overline{T_{\Gamma}}$ such that $g$ restricted to
$\gamma_{0}$ is a homeomorphism onto $\alpha_{0}.$ Let
$\alpha^{\ast}=g(\overline{q_{m}q_{1}}).$ Then $\alpha^{\ast}$ is a Jordan
path in $\overline{\Delta}$ from $p_{m}$ to $p_{1}$ and by (E) we have
$\alpha^{\ast}\subset\partial\Delta,$ and then
$\alpha^{\ast}=\left(\partial\Delta\right)\backslash\alpha_{0}^{\circ}$. This
implies that $g(\partial T_{\Gamma})=\partial\Delta,$ and then
$f:\overline{\Delta}\rightarrow\overline{T_{\Gamma}}$ and $f:\Delta\rightarrow
T_{\Gamma}=T_{\Gamma}\backslash[0,+\infty]$ are homeomorphisms, with
$f(\left(\partial\Delta\right)\backslash\alpha_{0}^{\circ})=f(\alpha^{\ast})=\overline{q_{m}q_{1}}\subset[0,+\infty],$
and
$L(f,\alpha_{0})=L(\gamma_{0})>L(\overline{q_{m}q_{1}})=L(f,\left(\partial\Delta\right)\backslash\alpha_{0}).$
Then, by (G), the proof is complete for Case 1.
Case 2.
(6.12) $\\{q_{1},q_{m}\\}\subset(0,+\infty),$
and
(6.13) $\\{q_{2},q_{m-1}\\}\subset S^{\prime},$
where $S^{\prime}$ is the open hemisphere inside the great circle determined
by $[0,+\infty].$
By (A), (B), (C), (6.12), (6.13) and Lemma 5.8, we have
(H) $\Gamma=\gamma_{0}+\overline{q_{m}q_{1}}$ is a convex Jordan curve that is
strictly convex at all vertices $q_{1},q_{2},\dots,q_{m}.$
We first assume $q_{1}=q_{m}.$ Then the closed curve
$\Gamma=\gamma_{0}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m-1}q_{1}}$
is strictly convex at all its vertices $q_{1},\dots,q_{m-1},$ and, by Lemma
5.3 (ii)
$\overline{T_{\gamma_{0}}}\backslash\\{q_{1}\\}\subset S^{\prime},$
where $T_{\gamma_{0}}$ is the domain enclosed by $\gamma_{0}.$ Then by (C),
(D) and Corollary 6.1,
$\alpha_{0}=\alpha_{1}+\alpha_{1}+\dots+\alpha_{m-1}=\partial\Delta,$ i.e.
$p_{1}=p_{m}.$ This implies that $f$ restricted to $\partial\Delta$ is a
homeomorphism onto $\gamma_{0}$ and then $f$ is a homeomorphism, and the other
conclusions are trivial in this setting.
Now, we assume $q_{1}\neq q_{m}.$ Then
$\Gamma=\gamma_{0}+\overline{q_{m}q_{1}}$ has the following natural partition
$\Gamma=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m-1}q_{m}}+\overline{q_{m}q_{1}},$
and by (H) and Lemma 5.3 (ii), $\overline{T_{\Gamma}}\subset
S^{\prime}\cup\overline{q_{m}q_{1}},$ which, with (6.12), implies that
(6.14) $\overline{T_{\Gamma}}\cap\\{0,\infty\\}=\emptyset\ \mathrm{and\
}\overline{T_{\Gamma}}\cap[0,+\infty]=\overline{q_{m}q_{1}}.$
Then again by (H), the triangles $L_{j}=\overline{q_{1}q_{j+1}q_{j+2}q_{1}}$
are all generic convex triangles and the domains $T_{j}$ enclosed by $L_{j}$
are disjoint each other, and
$\overline{T_{\Gamma}}=\cup_{j}^{m-2}\overline{T_{j}}.$ By (C) and (6.14),
$\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex in
$\overline{T_{\Gamma}}$ and by (D), $f$ has no branched point in
$\overline{T_{\Gamma}}.$ Thus, by Lemma 6.4, $f^{-1}$ has a univalent branch
$g$ defined on $\overline{T_{\Gamma}}$ such that $g$ maps
$\gamma_{0}=\overline{q_{1}q_{2}\dots q_{m}}$ onto $\alpha_{0}$.
Let $\alpha^{\ast}=g(\overline{q_{m}q_{1}}).$ Then $\alpha^{\ast}$ is a Jordan
path in $\overline{\Delta}$ from $p_{m}$ to $p_{1}.$ By (E), we have
$\alpha^{\ast}\subset\partial\Delta,$ and then we have
$\alpha^{\ast}=\partial\Delta\backslash\alpha_{0}^{\circ}$ and $g(\partial
T_{\Gamma})=g(\Gamma)\subset\partial\Delta$, which, with Lemma 6.2, implies
that $g(\overline{T_{\Gamma}})=\overline{\Delta},$ and then
$f:\overline{\Delta}\rightarrow\overline{T_{\Gamma}}$ is a homeomorphism.
Thus $f:\Delta\rightarrow T_{\Gamma}=T_{\Gamma}\backslash[0,+\infty]$ is a
homeomorphism
$f(\left(\partial\Delta\right)\backslash\alpha_{0})=\overline{q_{m}q_{1}}\subset[0,+\infty],$
and, by the fact that $L(\gamma_{0})>L(\overline{q_{m}q_{1}}),$ we have
$L(f,\alpha_{0})>L(f,\alpha^{\ast})=L(f,\left(\partial\Delta\right)\backslash\alpha_{0}).$
Then, by (G), The proof is complete for Case 2.
Case 3.
(6.15) $\\{q_{1},q_{m}\\}\subset(0,+\infty),$
and
(6.16) $q_{2}\in S^{\prime},q_{m-1}\in S\backslash\overline{S^{\prime}}.$
By (A), (B), (C), (6.15) and (6.16), Lemma 5.7 apply to $\gamma_{0}$, and then
we have the following.
(I) $0$ is contained in the domain $T_{\Gamma}$,
$L_{j}=\overline{0q_{j}q_{j+1}0}$ is a generic convex triangle for
$j=1,\dots,m-1$; and for the triangle domain $T_{j}$ enclosed by $L_{j},$
$\overline{T_{j}}\cap[0,+\infty]=\\{0\\},\ \mathrm{for\ }j=2,\dots,m-2,$
$\overline{T_{1}}\cap[0,+\infty]=\overline{0q_{1}},\
\overline{T_{m-1}}\cap[0,+\infty]=\overline{q_{m}0}.$
By (I), we can extend $\overline{q_{1}0}$ past $0$ to some point
$q^{\prime}\in\Gamma$ such that the open line segment
$\overline{q^{\prime}0}^{\circ}$ is contained in $T_{\Gamma}$ (note that by
(G) the notations $\overline{q^{\prime}0}$ and
$\overline{q^{\prime}q_{1}}=\overline{q^{\prime}0}+\overline{0q_{1}}$ make
sense, i.e. $d(q^{\prime},q_{1})<\pi).$ By (G) and (I),
$\overline{q^{\prime}q_{1}}$ divides $T_{\Gamma}$ into two polygonal Jordan
domains $T_{1}^{\ast}$ and $T_{2}^{\ast}$ with $q_{2}\in\partial T_{1}^{\ast}$
and $q_{m-1}\in\partial T_{2}^{\ast}$, both $T_{1}^{\ast}$ and $T_{2}^{\ast}$
are strictly convex at $q^{\prime},$ $T_{1}^{\ast}$ is on the left hand side
of $\overline{q^{\prime}q_{1}}$ and $T_{2}^{\ast}$ is on the right hand side
of $\overline{q^{\prime}q_{1}},$ $q_{1}$ is a strictly convex vertex of of
$T_{1}^{\ast}$ and $q_{m}$ is a strictly convex vertex of $T_{2}^{\ast}$.
Thus, by (F), both $T_{1}^{\ast}$ and $T_{2}^{\ast}$ are polygonal convex
Jordan domains.
Considering that $q^{\prime}$, $q_{1}$ and $q_{2}$ are strictly convex
vertices of $T_{1}^{\ast},$ by Lemma 5.3, we have
(6.17) $\overline{T_{1}^{\ast}}\backslash\overline{q^{\prime}q_{1}}\subset
S^{\prime}.$
Let $\gamma_{1}$ be the section of $\gamma_{0}$ from $q_{1}$ to $q^{\prime}$,
$p^{\prime}$ the unique point in $\alpha_{0}$ such that
$f(p^{\prime})=q^{\prime}$, $\alpha_{0}^{1}$ the section of $\alpha_{0}$ from
$p_{1}$ to $p^{\prime}$ and let $\alpha_{0}^{2}$ be the section of
$\alpha_{0}$ from $p^{\prime}$ to $p_{m}.$ We may assume
$q^{\prime}\in\overline{q_{s}q_{s+1}}^{\circ}$ (in the case $q^{\prime}=q_{s}$
or $q_{s+1},$ the proof is the same). Then
$\partial
T_{1}^{\ast}=\gamma_{1}+\overline{q^{\prime}q_{1}}=\overline{q_{1}q_{2}}+\dots+\overline{q_{s}q^{\prime}}+\overline{q^{\prime}q_{1}},$
and $T_{1}^{\ast}$ is strictly convex at $q_{1},q_{2},\dots,q_{s},q^{\prime}$,
and then
$\overline{q_{1}q_{2}q_{3}q_{1}},\dots,\overline{q_{1}q_{s-1}q_{s}q_{1}},$
$\overline{q_{1}q_{s}q^{\prime}q_{1}}$ are generic convex triangles that
triangulate $\overline{T_{1}^{\ast}}$. Hence, by (C), (D), (6.17), Lemma 6.4
applies to $\overline{T_{1}^{\ast}}$, and then $f^{-1}$ has a univalent branch
$g_{1}$ defined on $\overline{T_{1}^{\ast}}$ such that $g_{1}$ restricted to
$\gamma_{1}$ is a homeomorphism onto $\alpha_{0}^{1}$ with
$g_{1}(q_{1})=p_{1}\text{{and\ }}g_{1}(q^{\prime})=p^{\prime}.$
For the same reason, $f^{-1}$ has a univalent branch $g_{2}$ defined on
$\overline{T_{2}^{\ast}}$ such that $g_{2}$ restricted to
$\gamma_{2}=\overline{q^{\prime}q_{s+1}}+\dots+\overline{q_{m-1}q_{m}}$ is a
homeomorphism onto $\alpha_{0}^{2}$ with
$g_{2}(q_{m})=p_{m}\mathrm{\ and\ }g_{2}(q^{\prime})=p^{\prime}.$
Considering that $f$ has no branched point in $S$ and
$g_{1}(q^{\prime})=g_{2}(q^{\prime}).$ We have
(6.18) $g_{1}(w)=q_{2}(w),w\in\overline{q^{\prime}0},$
and we denote by
$\alpha=g_{1}(\overline{q^{\prime}0})=g_{2}(\overline{q^{\prime}0}).$ By (E)
we have
$g_{1}(0)=g_{2}(0)\in\partial\Delta,$
and thus, the initial and terminal points of $\alpha$, the points $p^{\prime}$
and $g_{1}(0)=g_{2}(0),$ are contained in $\partial\Delta.$
Then we can glue $g_{1}$ and $g_{2}$ along $\overline{q^{\prime}0}$ to be a
multivalent function $G$ such that $G$ restricted to
$T_{\Gamma}\backslash\left(\overline{0q_{1}}\cap\overline{0q_{m}}\right)$ is a
homeomorphism and restricted to $\overline{T_{j}^{\ast}}$ is the homeomorphism
$g_{j}$, $j=1,2.$ Then, it is clear that the interior of
$\alpha=g_{1}(\overline{q^{\prime}0})=g_{2}(\overline{q^{\prime}0})$ is
contained in $\Delta,$ and thus $\alpha$ divides $\Delta$ into two Jordan
domains $\Delta_{1}$ and $\Delta_{2},$ and we assume $\Delta_{1}$ is on the
left hand side of $\alpha.$
Let $\alpha^{\prime}=g_{2}(\overline{q_{m}0})$ and
$\alpha^{\prime\prime}=g_{1}(\overline{0q_{1}}).$ Then by (E),
$\alpha^{\prime}$ is a section of $\partial\Delta$ from $p_{m}$ to $g_{2}(0)$
and $\alpha^{\prime\prime}$ is a section of $\partial\Delta$ from
$g_{1}(0)=g_{2}(0)$ to $p_{1}$, since $g_{1}(q_{1})=p_{1}$ and
$g_{2}(q_{m})=p_{m}.$ Thus, we can conclude that $f$ maps
$\alpha_{0},\alpha^{\prime},\alpha^{\prime\prime}$ homeomorphically onto
$\gamma_{0},\overline{q_{m}0},\overline{0q_{1}},$ respectively, and
$\partial\Delta=\alpha_{0}+\alpha^{\prime}+\alpha^{\prime\prime}\text{{with\
}}\alpha_{0}^{\circ}\cap\left(\alpha^{\prime}+\alpha^{\prime\prime}\right)=\emptyset.$
This implies that $f$ maps $\Delta$ homeomorphically onto
$T_{\Gamma}\backslash\overline{0q_{1}}=T_{\Gamma}\backslash\overline{0q_{m}},$
since $f$ is normal.
On the other hand, it is clear that $L(\gamma_{1})>L(\overline{0q_{1}})$ and
$L(\gamma_{2})>L(\overline{q_{m}0}).$ Thus, we have
$\displaystyle L(f,\alpha_{0})$ $\displaystyle=$ $\displaystyle
L(f(\alpha_{0}))=L(\gamma_{0})=L(\gamma_{1})+L(\gamma_{2})>L(\overline{0q_{1}})+L(\overline{q_{m}0})$
$\displaystyle=$ $\displaystyle
L(f,\alpha^{\prime\prime}+\alpha^{\prime})=L(f,(\partial\Delta)\backslash\alpha_{0}).$
By (G), $f(\overline{\Delta})\subset\overline{T_{\Gamma}}$ is contained in
some open hemisphere of $S$ and it is clear that
$f((\partial\Delta)\backslash\alpha_{0}^{\circ})=\overline{q_{m}0}\cup\overline{0q_{1}}\subset[0,+\infty].$
This completes the proof for Case 3, and we have finally proved Theorem 6.1. ∎
## 7\. Cutting Riemann surfaces along $[0,+\infty]$
In this section we prove the following theorem, which is the second key step
to prove the main theorem in Section 14 and is also used to prove Theorem
13.1. Recall that we denote by $[0,+\infty]$ the line segment in $S$ from $0$
to $\infty$ that passes through $1.$
###### Theorem 7.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and assume that the
followings hold.
(a) Each natural edge of the boundary curve
$\Gamma_{f}=f(z),z\in\partial\Delta,$ has length strictly less than $\pi$.
(b) $\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex in252525See
Definition 2.5. $S\backslash E,E=\\{0,1,\infty\\}.$
(c) $f$ has no branched point in $S\backslash E.$
(d) $\Gamma_{f}\cap[0,+\infty]$ contains at most finitely many points.
Then, in the case $\Delta\cap f^{-1}\left([0,+\infty]\right)=\emptyset,$
$f(\overline{\Delta})$ is contained in some open hemisphere of $S,$
$f:\overline{\Delta}\rightarrow f(\overline{\Delta})$ is a homeomorphism and
$\left(\partial\Delta\right)\cap f^{-1}([0,+\infty])$
contains at most one point; and in the case $\Delta\cap
f^{-1}\left([0,+\infty]\right)\neq\emptyset,$ the following (i)–(v) hold:
(i) Each component of $f^{-1}\left([0,+\infty]\right)\cap\Delta$ is a Jordan
path with distinct endpoints contained in $\partial\Delta$ and divides
$\Delta$ into two Jordan domains.
(ii) Any pair of two distinct components of $f^{-1}([0,+\infty])\cap\Delta$
have at most one common endpoint.
(iii) For each component $D$ of $\Delta\backslash
f^{-1}\left([0,+\infty]\right),$ $D$ is a Jordan domain and $f$ restrict to
$D$ is a homeomorphism.
(iv) For each component $D$ of $\Delta\backslash
f^{-1}\left([0,+\infty]\right),$ $\left(\partial D\right)\cap(\partial\Delta)$
is consisted of a connected open subset $\alpha_{0}$ and a number of finite
points such that
$f(\alpha_{0})\cap[0,+\infty]=\emptyset\ \mathrm{and\ }f(\partial
D\backslash\alpha_{0})\subset[0,+\infty],$
and $f$ restricted to $\alpha_{0}$ is a homeomorphism.
(v) For $\alpha_{0}$ in (iv), if $\alpha_{0}\neq\emptyset$, then
$f(\overline{D})$ is contained in some hemisphere of $S$ and
$L(f(\alpha_{0}))>L(f,\partial D\backslash\alpha_{0}),$
that is
$L(f,\partial D\cap(\partial\Delta))>L(f,\left(\partial
D\right)\backslash(\partial\Delta)).$
This theorem has very simple geometrical explanation: when we cut the Riemann
surface of $f$ along $[0,+\infty]$ in the case
$\Delta\cap f^{-1}\left([0,+\infty]\right)\neq\emptyset,$
we obtain a finite number of pieces, each of which is either the whole sphere
$S$ with folded boundary $[0,+\infty],$ or is contained in some open
hemisphere of $S$ such that the length of the boundary located in
$S\backslash[0,+\infty],$ which is a part of the original boundary of the
Riemann surface of $f,$ is larger than the length of the boundary located in
$[0,+\infty],$ which is a part of the the new boundary, the cut edges.
This geometrical understand of the mapping in the theorem plays an important
role in this paper. We first prove the following lemma.
###### Lemma 7.1.
Let $g:\overline{\Delta}\rightarrow S$ be a normal mapping that satisfies
(a)–(c) of the previous theorem and
(7.1) $g(\Delta)\subset S\backslash[0,+\infty].$
Then,
(i) $g$ restricted to $\Delta$ is a homeomorphism onto $g(\Delta).$
(ii) $\partial\Delta$ has an open connected subset $\alpha_{0}$ of
$\partial\Delta,$ such that
$g(\alpha_{0})\cap[0,+\infty]=\emptyset\ \mathrm{and\
}g(\partial\Delta\backslash\alpha_{0})\subset[0,+\infty].$
(iii) If in (ii) $\alpha_{0}\neq\emptyset$, then $g$ restricted to
$\alpha_{0}$ is a homeomorphism onto the curve $g(\alpha_{0})$ in $S$ and
$L(g(\alpha_{0}))>L(g,\partial\Delta\backslash\alpha_{0})$.
(iv) If in (ii), $\alpha_{0}\neq\emptyset,$ then $g(\overline{\Delta})$ is
contained in some open hemisphere of $S$.
###### Proof.
By condition (c) and (7.1) we have
(e) $g$ has no ramification point in $\overline{\Delta}$.
Let $p$ be any point in $\partial\Delta$ such that
$f(p)=1\in(0,+\infty)\subset S.$
If $\Gamma_{g}=g(z),z\in\partial\Delta,$ is not convex at $p,$ then, since $f$
is normal, there is an open interval $I\subset(0,+\infty)$ whose one endpoint
is $1$ such that $I\subset g(\Delta),$ which contradicts (7.1). Thus,
$\Gamma_{g}$ is convex at262626This means that
$\Gamma_{g}=g(z),z\in\partial\Delta,$ is convex at each point
$p\in\left(\partial\Delta\right)\cap g^{-1}(1),$ by Definition 2.5. $1,$ and
then by (b), we have
(f) $\Gamma_{g}$ is locally convex in $S\backslash\\{0,\infty\\}.$
If $g(\partial\Delta)\cap[0,+\infty]=\emptyset,$ then by (f) $\Gamma_{g}$ is
locally convex everywhere, which implies that $\Gamma_{g}$ is locally simple
by the definition, and then by Corollary 6.1 and (e), $\Gamma_{g}$ is a simple
curve and then $g$ is a homeomorphism from $\overline{\Delta}$ onto
$g(\overline{\Delta}).$ On the other hand, in this case, by (a), (f) and
Definition 2.5, $\Gamma_{g}$ is a locally convex curve and has at least three
natural vertices, at each of which $\Gamma_{g}$ is strictly convex. Thus, by
Lemma 5.4 (ii), the closure $\overline{T_{\Gamma_{g}}}$ of the domain
$T_{\Gamma_{g}}$ enclosed by $\Gamma_{g}$ is contained in some open hemisphere
of $S,$ and thus, $g(\overline{\Delta})\subset\overline{T_{\Gamma_{g}}}$ is
contained in some open hemisphere of $S.$ Hence, putting
$\alpha_{0}=\partial\Delta,$ (i)–(iv) hold.
Consider the case $g(\partial\Delta)\subset[0,+\infty].$ Then
$g(\partial\Delta)$ must be a closed interval in $[0,+\infty]$. If
$g(\partial\Delta)\neq[0,+\infty],$ then by the fact that $g$ is normal,
$g(\Delta)$ contains $0$ or $\infty,$ but this contradicts the assumption.
Thus, $g(\partial\Delta)=[0,+\infty],$ and then by (7.1), $g$ restricted to
$\Delta$ is a covering onto $S\backslash[0,+\infty]$, which, together with
(e), implies that $g$ restricted to $\Delta$ is a homeomorphism, and putting
$\alpha_{0}=\emptyset$, we have (ii). Then, in the case
$g(\partial\Delta)\cap[0,+\infty]=\emptyset\ \mathrm{or\
}g(\partial\Delta)\subset[0,+\infty],$
the lemma is proved.
Now, we assume that $g(\partial\Delta)\cap[0,+\infty]\neq\emptyset$ and
$g(\partial\Delta)\backslash[0,+\infty]\neq\emptyset$.
Then by (a), $\Gamma_{g}$ has a section
(7.2)
$\gamma_{0}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{m-1}q_{m}},m\geq
3,$
such that each $q_{2},\dots,q_{m-1}$ are natural vertices of $\Gamma_{g},$ the
edges $\overline{q_{2}q_{3}},\dots,\overline{q_{m-2}q_{m-1}}$ are natural
edges of $\Gamma_{g}$,
(7.3)
$\left[\\{q_{2},\dots,q_{m-1}\\}\cup\cup_{j=2}^{m-2}\overline{q_{j}q_{j+1}}\right]\cap[0,+\infty]=\emptyset$
and
(7.4) $\\{q_{1},q_{m}\\}\subset[0,+\infty]\ \mathrm{but\
}\gamma_{0}\backslash\\{q_{1},q_{m}\\}\subset S\backslash[0,+\infty].$
By (f) and (7.2)–(7.4), $\gamma_{0}$ is locally simple, i.e.
$\overline{q_{j}q_{j+1}}\cap\overline{q_{j+1}q_{j+2}}=\\{q_{j+1}\\}$ for
$j=1,\dots,m-2.$ Then, by (7.2)–(7.4), in the case that $\gamma_{0}$ is not
simple, there exist integers $s$ and $t$ with $1\leq s<s+1<t\leq m-1$ and a
point
$q_{s}^{\prime}\in\left(\overline{q_{s}q_{s+1}}\cap\overline{q_{t}q_{t+1}}\right)\backslash\\{q_{1},q_{m}\\}$
such that
$\Gamma^{\prime}=\overline{q_{s}^{\prime}q_{s+1}}+\overline{q_{s+1}q_{s+2}}+\dots+\overline{q_{t-1}q_{t}}+\overline{q_{t}q_{s}^{\prime}}$
is a section of $\gamma_{0}$ that is a simple path from $q_{s}^{\prime}$ to
$q_{s}^{\prime},$ and
$\Gamma^{\prime}\cap[0,+\infty]=\emptyset.$
Therefore, by (f) and Definition 2.9, $\Gamma^{\prime}$ is a locally convex
Jordan path272727This does not mean that as a closed curve
$\Gamma^{\prime\prime}$ is convex at $q_{s}^{\prime},$ by the definition of
locally convex path and locally convex closed curves. that is strictly convex
at $q_{s+1},q_{s+2},\dots,q_{t}$, and then, by (a), $\Gamma^{\prime}$ has at
least three strictly convex vertices. Therefore, by Lemma 5.4 (ii),
$\Gamma^{\prime}$ is contained in some open hemisphere, which implies that
$[0,+\infty]\cap\overline{T_{\Gamma^{\prime}}}=\emptyset,$ where
$T_{\Gamma^{\prime}}$ is the domain inside $\Gamma^{\prime},$ and then by (f),
$\Gamma_{g}=g(z),z\in\partial\Delta,$ is locally convex in
$\overline{T_{\Gamma^{\prime}}}.$ Then, $g$ and $\Gamma^{\prime}$ satisfies
the assumption of Corollary 6.1, and then $g$ has a branched point in
$\overline{T_{\Gamma^{\prime}}}$, which contradicts (e). Thus we have proved
(g) $\gamma_{0}$ is a Jordan path.
Then $\partial\Delta$ has an open section $\alpha_{0}$ such that $g$
restricted to $\alpha_{0}$ is a homeomorphism onto $\gamma_{0}^{\circ}.$ Then,
by (e), (f), (g), (7.1), (7.2) and (7.4), we have $g\in\mathcal{F}_{m},$ and
Theorem 6.1 applies. Then $g$ restricted to $\Delta$ is a homeomorphism onto
the domain $T_{\Gamma}\backslash[0,+\infty]\subset S,$ where $T_{\Gamma}$ is
the domain enclosed by $\Gamma=\gamma_{0}+\overline{q_{m}q_{1}},$
$g(\left(\partial\Delta\right)\backslash\alpha_{0})\subset[0,+\infty],$
$L(g,\alpha_{0})>L(g,\left(\partial\Delta\right)\backslash\alpha_{0}),$
and $g(\overline{\Delta})$ is contained in some open hemisphere of $S.$ Thus,
(i)–(v) hold, and the proof is complete. ∎
###### Proof of Theorem 7.1.
We first assume
(7.5) $\Delta\cap f^{-1}([0,+\infty])=\emptyset.$
Then Lemma 7.1 applies, and then $f:\Delta\rightarrow f(\Delta)$ is a
homeomorphism, $f(\overline{\Delta})$ is contained in some open hemisphere of
$S$ and there exists a connected open subset $\alpha_{0}\subset\partial\Delta$
such that
(7.6) $f(\alpha_{0})\cap[0,+\infty]=\emptyset,$ (7.7)
$f(\left(\partial\Delta\right)\backslash\alpha_{0})\subset[0,+\infty],$
and $f$ restricted to $\alpha_{0}$ is also a homeomorphism onto some curve in
$S.$ Then, $\left(\partial\Delta\right)\backslash\alpha_{0}$ is also a
connected section of $\partial\Delta$, and $f$ restricted to
$\Delta\cup\alpha_{0}$ is also a homeomorphism.
By (d) and (7.7),
$f(\left(\partial\Delta\right)\backslash\alpha_{0})\subset
f(\partial\Delta)\cap[0,+\infty]$
is a finite set. Then, since $\left(\partial\Delta\right)\backslash\alpha_{0}$
is connected, $f(\left(\partial\Delta\right)\backslash\alpha_{0})$ is a
singleton, or is empty, which implies that
$\left(\partial\Delta\right)\backslash\alpha_{0}$ is a singleton, or is empty,
which, with (7.6) and the above argument, implies that
$\left(\partial\Delta\right)\cap f^{-1}([0,+\infty])$ contains at most one
point and $f:\overline{\Delta}\rightarrow f(\overline{\Delta})$ is a
homeomorphism. The theorem is proved under the assumption (7.5).
Now, we assume
$f(\Delta)\cap[0,+\infty]\neq\emptyset.$
Then by (c) and (d) and the assumption that $f$ is normal, each component of
$f^{-1}([0,+\infty])\cap\Delta$ is a simple path in $\Delta$ whose endpoints
are distinct and contained in $\partial\Delta,$ i.e. (i) holds true, and
$f^{-1}([0,+\infty])\cap\Delta$ has only a finite number of components. This
implies that $\Delta\backslash f^{-1}([0,+\infty])$ has a finite number of
components, each of which is a Jordan domain.
(ii) follows from Lemma 3.5.
Let $D$ be any component of $\Delta\backslash f^{-1}([0,+\infty]).$ Then by
(i), $D$ is a Jordan domain. Let $g$ be the restricted mapping
$g=f|_{\overline{D}}.$ Then $g$ is a normal mapping and each natural edge of
$\Gamma_{g}=g(z),z\in\partial D,$ is either a natural edge of
$\Gamma_{f}=f(z),z\in\partial\Delta,$ or a section of some natural edge of
$\Gamma_{f},$ or an interval contained in $\overline{0,1}$ or
$\overline{1,\infty}.$ Thus (a) is satisfied by $g.$ By (b) and (c), $g$ also
satisfies (b) and (c), and it is clear that
$g(D)\cap[0,+\infty]=\emptyset.$
Thus $g$ satisfies all the assumption of Lemma 7.1, by ignoring a coordinate
transform that maps $\overline{D}$ homeomorphically onto $\overline{\Delta}.$
Thus, Lemma 7.1 applies to $g$ and (iii) follows. By Lemma 7.1, $\partial D$
has a connected open subset $\alpha_{0}$ of $\partial D,$ such that (7.6) and
(7.7) still hold.
It is clear, by (7.6) and (7.7), that
$\alpha_{0}=(\partial D)\backslash f^{-1}([0,+\infty]),$
and, by (i) and (ii), that
$\left(\partial D\right)\cap\Delta\subset f^{-1}([0,+\infty].$
Then, we have
$\alpha_{0}=\left((\partial D)\backslash\Delta\right)\backslash
f^{-1}([0,+\infty]),$
and, considering that $(\partial D)\backslash\Delta=\left(\partial
D\right)\cap(\partial\Delta),$ we have
$\alpha_{0}=\left[\left(\partial D\right)\cap(\partial\Delta)\right]\backslash
f^{-1}([0,+\infty]).$
Then, considering that, by (d), $(\partial\Delta)\cap f^{-1}([0,+\infty])$ is
a finite set, we conclude that $\left(\partial
D\right)\cap(\partial\Delta)\backslash\alpha_{0}$ is a finite set, and thus
$\alpha_{0}$ is the interior of $\left(\partial D\right)\cap(\partial\Delta)$
in $\partial\Delta.$ Therefore, by (7.6) and (7.7), we have (iv).
(v) follows from (iv) and Lemma 7.1. This completes the proof. ∎
In the above two proofs, we have also proved that:
###### Corollary 7.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping that satisfies all
assumptions of Theorem 7.1 and let $\Delta_{1}$ be any component of
$\Delta\backslash f^{-1}([0,+\infty]).$ Then the restriction
$g=f|_{\overline{\Delta_{1}}}$ satisfies all the assumptions of Lemma 7.1,
and, furthermore, if $g(\partial\Delta_{1})\subset[0,+\infty],$ then
$g(\partial\Delta_{1})=[0,+\infty]$ and $g$ restricted to $\Delta_{1}$ is a
homeomorphism onto $S\backslash[0,+\infty].$
The condition (d) in Theorem 7.1 may be removed by the following lemma.
###### Lemma 7.2.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping satisfying the
conditions (a)–(c) of Theorem 7.1.
Then for any $\varepsilon>0,$ there exists a normal mapping
$g:\overline{\Delta}\rightarrow S$ such that
$A(g,\Delta)\geq
A(f,\Delta),L(g,\partial\Delta)<L(f,\partial\Delta)+\varepsilon,$
and $g$ satisfies all conditions (a)–(d) of Theorem 7.1.
###### Proof.
This can be proved by perturb the natural edges of $f$ lying on $[0,+\infty]$
slightly.
Let $\Gamma_{1}$ be any natural edge of $\Gamma_{f}$ such that
$\Gamma_{1}\cap[0,+\infty]$ contains more that one point. Then since
$\Gamma_{1}$ is a natural edge, it is either contained in the interval $[0,1]$
in $S$, or in the interval $[1,\infty].$ Without loss of generality we assume
$\Gamma_{1}\subset[0,1]$ and the orientation of $\Gamma_{1}$ is the same as
$\overline{0,1}.$ Then there are four cases:
(i) $\Gamma_{1}=\overline{0,1}.$
(ii) $\Gamma_{1}=\overline{0,t_{0}}$ for some $t_{0}\in(0,1).$
(iii) $\Gamma_{1}=\overline{t_{0},t_{1}}$ for some $t_{0},t_{1}\in(0,1).$
(iv) $\Gamma_{1}=\overline{t_{0},1}$ for some $t_{0}\in(0,1).$
In these cases, we can extend the Riemann surface of $f$ by patching a closed
triangle domain along $\Gamma_{1}$ so that the vertex $p_{1}^{\prime}$ is very
close the middle point of $\Gamma_{1}$ and is on the right hand side of
$\Gamma_{1}.$ By Lemma 3.2, the new Riemann surface can be realized by a
normal mapping $f_{1}:\overline{\Delta}\rightarrow S.$ It is clear that when
$p_{1}^{\prime}$ is sufficiently close to $\frac{1}{2}$ in case (i), or
$\frac{t_{0}}{2}$ in case (ii), or $\frac{t_{0}+t_{1}}{2}$ in case (iii), or
$\frac{t_{0}+1}{2}$ in case (iv), $f_{1}$ satisfies (a)–(c) of the lemma and
$\left|L(f_{1},\partial\Delta)-L(f,\partial\Delta)\right|<\frac{\varepsilon}{V(f)},A(f_{1},\Delta)\geq
A(f,\Delta),$
while the number of natural edges that lie on $[0,+\infty]$ is dropped by one.
Then repeating the above argument for $f_{1},$ and so on, and finally we can
obtain the desired mapping. ∎
## 8\. Deformation of edges of normal mappings with length larger than $\pi$
In this section we will prove the following theorem, which is prepared for
proving Theorem 10.1 and Theorem 11.1.
###### Theorem 8.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping whose boundary
curve $\Gamma_{f}=f(z),z\in\partial\Delta,$ has the natural partition
$\Gamma_{f}=\Gamma_{1}+\Gamma_{2}+\dots+\Gamma_{n},n=V(f),$
that satisfies one of the following conditions (a)–(d).
(a) $\pi\leq L(\Gamma_{1})<2\pi,$ $L(\Gamma_{j})<\pi$ for all $j\geq 2,$ and
$\Gamma_{1}$ has an endpoint contained in $E=\\{0,1,\infty\\}.$
(b) $\pi\leq L(\Gamma_{1})<2\pi,$ $L(\Gamma_{j})<\pi$ for all $j\geq 2,$ and
$\Gamma_{1}$ has no endpoint contained in $E.$
(c) $\pi\leq L(\Gamma_{1})<2\pi$ and $\pi\leq L(\Gamma_{j_{0}})<2\pi$ for some
$j_{0}\geq 2,$ $L(\Gamma_{j})<\pi$ for each $j\neq 1,j_{0};$ $\Gamma_{1}$ has
an endpoint contained in $E,$ and so does $\Gamma_{j_{0}}$.
(d) $2\pi\leq L(\Gamma_{1})<3\pi,$ while $L(\Gamma_{j})<\pi$ for all $j\geq
2.$
Then, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$ such
that
(i) $L(g,\partial\Delta)\leq L(f,\partial\Delta)$ and $A(g,\Delta)\geq
A(f,\Delta)$
(ii) Each natural edge of $g$ has spherical length strictly less than $\pi,$
(iii) In case (a), $V_{NE}(g)\leq V_{NE}(f),\mathrm{\ }V_{E}(g)\geq
V_{E}(f)+1,$ and $V(g)\leq V(f)+1;$
In case (b), $V_{NE}(g)\leq V_{NE}(f),\mathrm{\ }V_{E}(g)\geq V_{E}(f)+1,$ and
$V(g)\leq V(f)+2;$
In case (c), $V_{NE}(g)\leq V_{NE}(f),\mathrm{\ }V_{E}(g)\geq V_{E}(f)+2,$ and
$V(g)\leq V(f)+2;$
In case (d), $V_{NE}(g)=V_{NE}(f)+2,V_{E}(g)=V_{E}(f)+1,\ $and $V(g)=V(f)+3.$
The proof is divided into four parts: Lemmas 8.3–8.6.
###### Lemma 8.1.
Let $\Gamma$ be a line segment in $S$ with endpoints $q_{1}$ and $q_{2}$ and
$\pi\leq L(\Gamma)<2\pi,$ and let $q_{0}$ be any point in $S\backslash\Gamma.$
Then
(8.1) $d(q_{0},q_{1})<\pi,d(q_{0},q_{2})<\pi,$
and
$L(\overline{q_{1}q_{0}})+L(\overline{q_{0}q_{2}})\leq L(\Gamma).$
By the first two inequalities, $\overline{q_{1}q_{0}}$ and
$\overline{q_{0}q_{2}}$ make sense, which is the shortest paths.
###### Proof.
Since $L(\Gamma)\geq\pi,$ the antipodal points of $q_{1}$ and $q_{2}$ are both
contained in $\Gamma,$ and thus, neither $q_{1}$, nor $q_{2},$ can be an
antipodal point of $q_{0}\in S\backslash\Gamma.$ This implies (8.1).
Let $q_{1}^{\prime}$ be the antipodal point of $q_{1}$ in $S$. Then
$q_{1}^{\prime}\in\Gamma.$ Let $\Gamma_{1}^{\prime}$ be the section of
$\Gamma$ from $q_{1}$ to $q_{1}^{\prime}$ and let $\Gamma_{1}^{\prime\prime}$
be the section of $\Gamma$ from $q_{1}^{\prime}$ to $q_{2}.$ Then it is clear
that $\overline{q_{1}q_{0}}+\overline{q_{0}q_{1}^{\prime}}$ is a straight path
in $S$ from $q_{1}$ to $q_{1}^{\prime}.$ Thus we have
$\displaystyle L(\Gamma)$ $\displaystyle=$ $\displaystyle
L(\Gamma_{1}^{\prime})+L(\Gamma_{1}^{\prime\prime})=\pi+L(\Gamma_{1}^{\prime\prime})$
$\displaystyle=$ $\displaystyle
L(\overline{q_{1}q_{0}}+\overline{q_{0}q_{1}^{\prime}})+L(\Gamma_{1}^{\prime\prime})$
$\displaystyle=$ $\displaystyle
L(\overline{q_{1}q_{0}})+L(\overline{q_{0}q_{1}^{\prime}}+\Gamma_{1}^{\prime\prime})$
$\displaystyle\geq$ $\displaystyle
L(\overline{q_{1}q_{0}})+L(\overline{q_{0}q_{2}}),$
and equality holds if and only if $q_{1}$ and $q_{2}$ are a pair of antipodal
points of $S.$ ∎
###### Lemma 8.2.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and assume that
$\Gamma_{f}$ has a natural partition
(8.2) $\Gamma_{f}=\Gamma_{1}+\Gamma_{2}+\dots+\Gamma_{n},n=V(f),$
such that $L(\Gamma_{1})\geq\pi$ and the initial point of $\Gamma_{1}$ is in
$E.$ Then, there exists a normal mapping $f_{1}:\overline{\Delta}\rightarrow
S$ such that
$L(f_{1},\partial\Delta)\leq L(f,\partial\Delta),A(f_{1},\Delta)\geq
A(f,\Delta),$
and the boundary curve $\Gamma_{f_{1}}$ has a permitted partition
(8.3)
$\Gamma_{f_{1}}=\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n},$
such that the end point of $\Gamma_{1}^{\prime}$, which is also the initial
point of $\Gamma_{1}^{\prime\prime},$ is contained in $E,$ and
$L(\Gamma_{1}^{\prime})<\pi,L(\Gamma_{1}^{\prime\prime})<\pi.$
See definitions in Section 2 for the terms _natural partition_ and _permitted
partition_. Since (8.2) is a natural partition and the initial point of
$\Gamma_{1}$ is in $E,$ it is clear by (8.3) that the initial point of
$\Gamma_{1}^{\prime},$ which is the terminal point of $\Gamma_{n},$ is still
the initial point of $\Gamma_{1},$ and then $\Gamma_{3},\dots,\Gamma_{n}$ are
still natural edges of $\Gamma_{f_{1}}.$ Then the two endpoints of
$\Gamma_{1}^{\prime}$ are both in $E,$ and so $\Gamma_{1}^{\prime}$ is a
natual edge. But $\Gamma_{2}$ may not be a natural edge of $\Gamma_{f_{1}}$
and $\Gamma_{2}$ is a natural edge if and only if $\Gamma_{1}^{\prime}$ is. In
the case $\Gamma_{2}$ is not a natural edge of $\Gamma_{f_{1}},$
$\Gamma_{1}^{\prime\prime}+\Gamma_{2}$ must be a natural edge.
###### Proof.
Let $q_{1}$ and $q_{2}$ be the initial and terminal point of $\Gamma_{1},$
respectively. Then, $\Gamma_{1}$ contains the antipodal point $q_{1}^{\prime}$
of $q_{1}.$ Let $C$ be the great circle determined282828Recall that this means
that $C$ contains $\Gamma_{1}$ and is oriented by $C.$ by $\Gamma_{1}$ and let
$S^{\prime}$ be the open hemisphere outside292929Recall that this means
$S^{\prime}$ is on the right hand side of $C.$ $C$.
There are only two cases (note that we assumed $q_{1}\in E$ in the lemma):
Case 1. $q_{1}\in E,$ $q_{1}^{\prime}\notin E.$
Case 2. $q_{1}\in E,q_{1}^{\prime}\in E.$
Assume the first case occurs. Then we must have $q_{1}=1$ and
$q_{1}^{\prime}=-1.$ Then $C$ must separate $0$ and $\infty.$ Without loss of
generality, we assume $0\in S^{\prime}.$ Let
$\Gamma_{1}^{\prime}=\overline{q_{1}0}=\overline{1,0},$ which is the shortest
path in $S$ from $q_{1}=1$ to $0$ and let
$\Gamma_{1}^{\prime\prime}=\overline{0q_{2}}.$ Then the curve
$\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}$ is a simple path from
$q_{1}=1$ to $q_{2}$ and
$\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}-\Gamma_{1}$ is a Jordan curve
that encloses a domain $T$ in $S^{\prime}$ such that $T$ is on the right hand
side of $\Gamma_{1},$
$T\cap E=\emptyset,$ $L(\Gamma_{1}^{\prime})=\frac{\pi}{2},$
and by Lemma 8.1,
$L(\Gamma_{1}^{\prime\prime})<\pi\ \mathrm{and\
}L(\Gamma_{1}^{\prime})+L(\Gamma_{1}^{\prime\prime})\leq L(\Gamma_{1}).$
Let
$\partial\Delta=\alpha_{1}+\dots+\alpha_{n}$
be a natural partition of $\partial\Delta$ for $\Gamma_{f},$
corresponding303030See Definition 2.3 and Remark 2.1 (2). to (8.2), let $V$ be
a Jordan domain outside $\Delta$ with $\left(\partial
V\right)\cap\partial\Delta=\alpha_{1},$ and let $g$ be a homeomorphism from
$\overline{V}$ onto $\overline{T}$ such that
$f|_{\alpha_{1}}=g|_{\alpha_{1}}.$
Then, by Lemma 3.2,
$g_{1}=\left\\{\begin{array}[]{l}f(z),z\in\overline{\Delta},\\\
g(z),z\in\overline{V}\backslash\overline{\Delta},\end{array}\right.$
is a normal mapping defined on the closure of the Jordan domain
$D=\Delta\cup\alpha_{1}^{\circ}\cup V,$ where $\alpha_{1}^{\circ}$ is the
interior of $\alpha_{1}.$ Then the boundary curve $\Gamma_{g_{1}}$ of $g_{1}$
has the permitted partition
(8.4)
$\Gamma_{g_{1}}=\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n}$
and $A(g_{1},D)=A(f,\Delta)+A(T).$ Then we have
$L(g_{1},\partial D)\leq L(f,\partial\Delta)\ \mathrm{and\
}A(g_{1},D)>A(f,\Delta).$
Let $h$ be any homeomorphism from $\overline{D}$ onto $\overline{\Delta}.$
Then $f_{1}=g_{1}\circ h^{-1}$ satisfies all the desired conditions in (ii).
Assume Case 2 occur. Then $q_{1}$ and $q_{1}^{\prime}$ must be the pair
$\\{0,\infty\\}$, $q_{1}^{\prime}=q_{2},$ and there is no point in $E$ located
in the interior of $\Gamma_{1},$ for $\Gamma_{1}$ is a natural edge of $f$.
Without loss of generality, we assume that $q_{1}=0$ and
$q_{2}=q_{1}^{\prime}=\infty.$ Let $L$ be the straight path from $0$ to
$\infty$ that passes through $1.$ Then, the domain $T$ enclosed by
$\Gamma_{1}$ and $L$ that is on the right hand side of $\Gamma_{1}$ does not
contains point in $E\ $and $\\{\Gamma_{1}\cup T\\}\cap E=\\{0,\infty\\}.$ Let
$\Gamma_{1}^{\prime}$ be the section of $L$ from $0$ to $1$ and
$\Gamma_{1}^{\prime\prime}$ be the section of $L$ from $1$ to $\infty.$ Then
$L(\Gamma^{\prime})=\frac{\pi}{2},L(\Gamma_{1}^{\prime\prime})=\frac{\pi}{2},$
and repeating the process in Case 1, we can obtain a desired $f_{1}$ satisfies
all the conditions. ∎
###### Lemma 8.3.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping, and let
(8.5) $\Gamma_{f}=\Gamma_{1}+\dots+\Gamma_{n},n=V(f),$
be the natural partition of $\Gamma_{f}=f(z),z\in\partial\Delta.$ Assume that
(a) At least one endpoint of $\Gamma_{1}$ is contained in $E.$
(b) $\pi\leq L(\Gamma_{1})<2\pi,$ but $L(\Gamma_{j})<\pi,j=2,\dots,n.$
Then, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$ such
that
(i) $L(g,\partial\Delta)\leq L(f,\partial\Delta)$ and $A(g,\Delta)\geq
A(f,\Delta)$
(ii) Each natural edge of $g$ has spherical length strictly less than $\pi,$
(iii) $V_{NE}(g)\leq V_{NE}(f)$ and $V_{E}(g)\geq V_{E}(f)+1.$
(iv) $V(g)\leq V(f)+1.$
###### Proof.
Without loss of generality, we assume the initial point $q_{1}$ of
$\Gamma_{1}$ is in $E.$ Then by Lemma 8.2, there exists a normal mapping
$f_{1}:\overline{\Delta}\rightarrow S$ such that
(8.6) $L(f_{1},\partial\Delta)\leq L(f,\partial\Delta),A(f_{1},\Delta)\geq
A(f,\Delta),$
and the boundary curve $\Gamma_{f_{1}}$ has a permitted partition
(8.7)
$\Gamma_{f_{1}}=\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n}$
such that
(8.8) $L(\Gamma_{1}^{\prime})=\frac{\pi}{2},L(\Gamma_{1}^{\prime\prime})<\pi,$
the end point of $\Gamma_{1}^{\prime}$, which is also the initial point of
$\Gamma_{1}^{\prime\prime},$ is contained in $E.$ It is clear that the initial
points of $\Gamma_{1}^{\prime}$ and $\Gamma_{1}$ are the same point and so is
in $E,$ for they are both the terminal point of $\Gamma_{n},$ by (8.5) and
(8.7), and thus, $\Gamma_{1}^{\prime}$ is a natural edge of $\Gamma_{f_{1}}$
whose two endpoints are in $E,$ the terminal point of
$\Gamma_{1}^{\prime\prime}$ is a natural vertex of $\Gamma_{f}$, which may not
be a natural vertex of $\Gamma_{f_{1}}$. On the other hand, by (8.5) and
(8.7), the terminal points of $\Gamma_{2},\dots,\Gamma_{n}$ are still natural
vertices of $\Gamma_{f_{1}}.$ Then we have
(8.9) $V_{NE}(f_{1})\leq V_{NE}(f),V_{E}(f_{1})=V_{E}(f)+1,V(f_{1})\leq
V(f)+1.$
Case 1. If the terminal point of $\Gamma_{1}$ is also in $E,$ then both
$\Gamma_{1}^{\prime}$ and $\Gamma_{1}^{\prime\prime}$ have initial and
terminal points in $E,$ and then they are natural edges of $f_{1}$ and then
(8.7) is a natural partition; therefore, by (8.6)–(8.9) and (b), $g=f_{1}$ is
the desired mapping.
Case 2. Now assume that the terminal point of $\Gamma_{1}$ is not in $E.$
If $\Gamma_{1}^{\prime\prime}$ is still a natural edge, then (8.7) is still a
natural partition, and $g=f_{1}$ is the desired mapping by (b) and (8.9).
Assume $\Gamma_{1}^{\prime\prime}$ is not a natural edge. Then
$\Gamma_{1}^{\prime\prime}+\Gamma_{2}$ will be a natural edge, and then
$\Gamma_{f_{1}}$ has the natural partition
(8.10)
$\Gamma_{f_{1}}=(\Gamma_{1}^{\prime\prime}+\Gamma_{2})+\Gamma_{3}+\dots+\Gamma_{n}+\Gamma_{1}^{\prime},$
the initial point of $\Gamma_{2},$ which is also the terminal point of
$\Gamma_{1}^{\prime\prime}$ is now in the interior of the the natural edge
$\left(\Gamma_{1}^{\prime\prime}+\Gamma_{2}\right)$ and then we have by (8.10)
(8.11) $V_{NE}(f_{1})=V_{NE}(f)-1,V_{E}(f_{1})=V_{E}(f)+1,V(f_{1})=V(f).$
On the other hand, by (b) and (8.8) we have
(8.12) $L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<2\pi.$
If $L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<\pi,$ then $g=f_{1}$ is the
desired mapping.
If $L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})\geq\pi,$ then $f_{1}$ satisfies
all the assumption of the lemma with natural partition (8.10), and the above
argument applies to $f_{1}.$ By (8.10)–(8.12), if we repeat the above argument
once, and if we do not arrive at the desired mapping, then $V_{NE}(\cdot)$
drops by one, $V_{E}(\cdot)$ increases by one and $V(\cdot)$ keep invariant.
But $V_{NE}(\cdot)\geq 0$ in any case, and so we can reach the desired mapping
by repeating the above argument finitely many times. This completes the proof.
∎
###### Lemma 8.4.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping, and let
$\Gamma_{f}=\Gamma_{1}+\Gamma_{2}+\dots+\Gamma_{n},n=V(f),$
be a natural partition of $\Gamma_{f}=f(z),z\in\partial\Delta.$ Assume that
for some positive integer $k_{0}$ with $1<k_{0}\leq n$ the followings hold.
(a) $\Gamma_{1}$ has an endpoint contained in $E,$ and so does
$\Gamma_{k_{0}}$.
(b) $\pi\leq L(\Gamma_{1})<2\pi,\pi\leq L(\Gamma_{k_{0}})<2\pi,$ but
$L(\Gamma_{j})<\pi,j\neq 1,k_{0}.$
Then, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$ such
that
(i) $L(g,\partial\Delta)\leq L(f,\partial\Delta)$ and $A(g,\Delta)\geq
A(f,\Delta)$
(ii) Each natural edge of $g$ has spherical length strictly less than $\pi,$
(iii) $V_{NE}(g)\leq V_{NE}(f)$ and $V_{E}(g)\geq V_{E}(f)+2.$
(iv) $V(g)\leq V(f)+2.$
###### Proof.
Without loss of generality, we assume that the initial point $q_{1}$ of
$\Gamma_{1}$ is in $E.$
By Lemma 8.2, there exists a normal mapping
$f_{1}:\overline{\Delta}\rightarrow S$ such that
(8.13) $L(f_{1},\partial\Delta)\leq L(f,\partial\Delta)\ \mathrm{and\
}A(f_{1},\Delta)\geq A(f,\Delta),$
and the boundary curve $\Gamma_{f_{1}}$ has a permitted partition
(8.14)
$\Gamma_{f_{1}}=\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n}$
such that
(8.15)
$L(\Gamma_{1}^{\prime})=\frac{\pi}{2},L(\Gamma_{1}^{\prime\prime})<\pi,$
and the end point of $\Gamma_{1}^{\prime}$, which is also the initial point of
$\Gamma_{1}^{\prime\prime},$ is contained in $E.$ Then, $\Gamma_{1}^{\prime}$
is a natural edge of $\Gamma_{f_{1}},$ because its endpoints are both in $E.$
If $\Gamma_{1}^{\prime\prime}$ is a natural edge of $f_{1},$ then (8.14) is a
natural partition and by (a), (b), (8.14) and (8.15), $f_{1}$ satisfies all
assumptions of Lemma 8.3, with
$V_{NE}(f_{1})=V_{NE}(f),V_{E}(f_{1})=V_{E}(f)+1,V(f_{1})=V(f)+1,$
and then, by (8.13), we can apply Lemma 8.3 to deform $f_{1}$ to be another
normal mapping $g$ satisfying (i)–(iv) with
$\left\\{\begin{array}[]{l}V_{NE}(g)\leq V_{NE}(f_{1})=V_{NE}(f),\\\
V_{E}(g)\geq V_{E}(f_{1})+1=V_{E}(f)+2,\\\ V(g)\leq
V(f_{1})+1=V(f)+2.\end{array}\right.$
Now, assume that
(c) $\Gamma_{1}^{\prime\prime}$ is not a natural edge of $\Gamma_{f_{1}}$.
We complete the proof by induction on $k_{0}\geq 2.$ We first assume
$k_{0}=2.$
Then by the assumption (c), $\Gamma_{1}^{\prime\prime}+\Gamma_{2}$ must be a
natural edge, $\Gamma_{f_{1}}$ has the natural partition
(8.16)
$\Gamma_{f_{1}}=\Gamma_{1}^{\prime}+\left(\Gamma_{1}^{\prime\prime}+\Gamma_{2}\right)+\Gamma_{3}+\dots+\Gamma_{n},$
with
(8.17) $V_{NE}(f_{1})=V_{NE}(f)-1,V_{E}(f_{1})=V_{E}(f)+1,V(f_{1})=V(f),$
and the initial point of $\Gamma_{2}$ is not contained in $E,$ for, otherwise,
$\Gamma_{1}^{\prime\prime}$ has two endpoints in $E$, which implies that
$\Gamma_{1}^{\prime\prime}$ is a natural edge. Therefore, by (a), the initial
and terminal points of the natural edge $\Gamma_{1}^{\prime\prime}+\Gamma_{2}$
of $\Gamma_{f_{1}}$ are both contained in $E.$
By the definition, each natural edge of a closed polygonal curve with initial
and terminal points in $E$ is simple and has length $\frac{\pi}{2},$ $\pi,$ or
$2\pi.$ Then by (b) and the fact that
$L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})>L(\Gamma_{2})\geq\pi$ we have
$L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})=2\pi.$
Note that we are in the situation that $\Gamma_{1}^{\prime\prime}+\Gamma_{2}$
is a natural edge of $\Gamma_{f_{1}}$ and a natural edge never contains any
point of $E$ in its interior, and then we conclude that
$C=\Gamma_{1}^{\prime\prime}+\Gamma_{2}$
is a great circle passing through $1$ with $C\cap E=\\{1\\}$ (so, $1$ is the
initial and terminal point of $C).$
Then $C$ separates $0$ and $\infty,$ without loss of generality we assume $0$
is on the right hand side of $C.$ Let
$\partial\Delta=\alpha_{1}+\alpha_{2}+\dots+\alpha_{n}$
be a natural partition of $\partial\Delta$ corresponding to (8.16) (see
Definition 2.3). Then $\Gamma_{1}^{\prime\prime}+\Gamma_{2}$ is the section of
$\Gamma_{f_{1}}$ restricted to $\alpha_{2}$. Let $V$ be a bounded Jordan
domain in $\mathbb{C}$ that is outside $\Delta$ with
$\left(\partial\Delta\right)\cap\left(\partial V\right)=\alpha_{2}$, let
$p_{1}$ and $p_{2}$ be the initial and terminal points of $\alpha_{2},$
respectively, and let $T$ be the hemisphere on the right hand side of $C$ with
the path $\overline{0,1}$ being removed. Then, there exists a continuous
mapping $\tau$ from $\overline{V}$ onto $\overline{T}$ such that
$\tau|_{\alpha_{2}}=f_{1}|_{\alpha_{2}},$ $\tau$ restricted to $\alpha_{2}\cup
V$ is a homeomorphism onto $(\Gamma_{1}^{\prime\prime}+\Gamma_{2})\cup T$ with
$\tau(\alpha_{2})=\Gamma_{1}^{\prime\prime}+\Gamma_{2},$ and $\tau$ restricted
to $\left(\partial V\right)\backslash\alpha_{2}=\left(\partial
V\right)\backslash\overline{\Delta}$ is a folded $2$ to $1$ mapping onto
$\overline{0,1}$. Then by Lemma 3.2, the mapping
$f^{\ast}=\left\\{\begin{array}[]{l}f_{1}(z),z\in\overline{\Delta},\\\
\tau(z),z\in\overline{V}\backslash\overline{\Delta},\end{array}\right.$
is a normal mapping defined on the closure of the Jordan domain
$\Delta^{\ast}=\Delta\cup V\cup\alpha_{2}\backslash\\{p_{1},p_{2}\\},$
with
$A(f^{\ast},\Delta^{\ast})=A(f_{1},\Delta)+A(T),\ $
and
$\displaystyle L(f^{\ast},\partial\Delta^{\ast})$ $\displaystyle=$
$\displaystyle
L(f_{1},\left(\left(\partial\Delta\right)\backslash\alpha_{2}\right))+L(f,\left(\partial
V\right)\backslash\alpha_{2})$ $\displaystyle=$ $\displaystyle
L(f_{1},\partial\Delta)-L(f_{1},\alpha_{2})+L(f,\left(\partial
V\right)\backslash\alpha_{2})$ $\displaystyle=$ $\displaystyle
L(f_{1},\partial\Delta)-L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})+L(\overline{1,0})+L(\overline{0,1})$
$\displaystyle=$ $\displaystyle
L(f_{1},\partial\Delta)-2\pi+\frac{\pi}{2}+\frac{\pi}{2}$ $\displaystyle<$
$\displaystyle L(f_{1},\partial\Delta),$
and the boundary curve $\Gamma_{f^{\ast}}$ has a natural partition
$\Gamma_{f^{\ast}}=\Gamma_{1}^{\prime}+\Gamma_{2}^{\prime}+\Gamma_{2}^{\prime\prime}+\Gamma_{3}+\dots+\Gamma_{n},$
and we have $V_{E}(f^{\ast})=V_{E}(f_{1})+1$ and $V(f^{\ast})=V(f_{1})+1,$ and
then by (8.17) we have
$V_{E}(f^{\ast})\geq V_{E}(f)+2,V(f^{\ast})\leq V(f)+2.$
Considering that $V(f^{\ast})=V_{E}(f^{\ast})+V_{NE}(f^{\ast})$ and regarding
$\Delta^{\ast}$ as a disk, we obtained the desired mapping $g=f^{\ast}$ that
satisfies (i)–(iv). The proof is complete for the case $k_{0}=2$ under the
assumption (c).
Then we have in fact prove the lemma in the case $k_{0}=2.$
Now, assume that for some positive integer $m$ with $2\leq m<n=V(f),$ Lemma
8.4 holds true for all $k_{0}$ with $2\leq k_{0}\leq m.$ We prove that Lemma
8.4 holds true for $k_{0}=m+1.$
To prove the lemma for $k_{0}=m+1,$ it is suffices to prove the lemma under
the assumption (c).
By the assumption (c), $\Gamma_{1}^{\prime\prime}+\Gamma_{2}$ is still a
natural edge of $\Gamma_{f_{1}}$ and since $k_{0}=m+1\geq 3$ we have, by (b)
and (8.15), that
$L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<2\pi,$
the initial point of $\Gamma_{1}^{\prime\prime}+\Gamma_{2}$ is in $E$, and
(8.17) still holds.
If $L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<\pi,$ then $f_{1}$ also satisfies
all assumptions of Lemma 8.3 and by (8.17) we can again deform $f_{1}$ to be
another normal mapping $g$ such that (i)–(iv) hold.
If
$\pi\leq L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<2\pi,$
then, considering that by (8.14) $\Gamma_{f_{1}}$ also has the following
natural partition
$\Gamma_{f_{1}}=\left(\Gamma_{1}^{\prime\prime}+\Gamma_{2}\right)+\Gamma_{3}+\dots+\Gamma_{n}+\Gamma_{1}^{\prime},$
$f_{1}$ satisfies all the assumption of Lemma 8.4, with $k_{0}=m.$ Then by the
induction hypothesis, the proof is complete. ∎
###### Lemma 8.5.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and let
$\Gamma_{f}=\Gamma_{1}+\dots+\Gamma_{n},n=V(f),$
be a natural partition of $\Gamma_{f}=f(z),z\in\partial\Delta.$ Assume that
the following hold.
(a) $\pi\leq L(\Gamma_{1})<2\pi$ but $L(\Gamma_{j})<\pi$ for all
$j=2,\dots,n.$
(b) The two endpoints of $\Gamma_{1}$ are outside $E.$
Then, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$ such
that
(i) $L(g,\partial\Delta)\leq L(f,\partial\Delta)$ and $A(g,\Delta)\geq
A(f,\Delta)$
(ii) Each natural edge of $g$ has spherical length strictly less than $\pi,$
(iii) $V_{NE}(g)\leq V_{NE}(f)\ \mathrm{and\ }V_{E}(g)\geq V_{E}(f)+1.$
(iv) $V(g)\leq V(f)+2.$
###### Proof.
Let $C$ be the great circle determined by $\Gamma_{1}.$ Then there are two
cases:
Case 1. $C\cap E\neq\emptyset.$
Case 2. $C\cap E=\emptyset.$
Assume Case 1 occurs. Then by (a), $C$ contains only one point $p_{0}$ in $E$
and this point must be $1.$ Otherwise, $C$ contains the antipodal points $0$
and $\infty,$ and either $0$ or $\infty$ is in the interior of $\Gamma_{1}$ by
(a) and (b), which contradicts the assumption that $\Gamma_{1}$ is a natural
edge. Then $C$ must separates $0$ and $\infty,$ without loss of generality,
assume $0$ is on the right hand side of $C.$ Let $q_{j}$ be the initial point
of $\Gamma_{j},j=1,\dots,n.$ Then $q_{j+1}$ is the endpoint of
$\Gamma_{j},j=1,\dots,n,$ where $q_{n+1}=q_{1}.$ Let
(8.18) $\Gamma_{1}^{\prime}=\overline{q_{1}0}\ \mathrm{and\
}\Gamma_{1}^{\prime\prime}=\overline{0q_{2}}.$
Then, by Lemma 8.1, $\Gamma_{1}^{\prime}$ and $\Gamma_{1}^{\prime\prime}$ make
sense and
(8.19) $L(\Gamma_{1}^{\prime})<\pi,L(\Gamma_{1}^{\prime\prime})<\pi\
\mathrm{and\ }L(\Gamma_{1}^{\prime})+L(\Gamma_{1}^{\prime\prime})\leq
L(\Gamma_{1}),$
and $\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}-\Gamma_{1}$ encloses a
domain $T$ that is on the right hand side of $C$ and $(\Gamma_{1}\cup T)\cap
E=\emptyset.$
By Lemma 3.2, ignoring a coordinate transform, there exists a normal mapping
$g_{1}:\overline{\Delta}\rightarrow S$, which will be regarded as an extension
of $f,$ such that $\Gamma_{g_{1}}$ has the permitted partition
$\Gamma_{g_{1}}=\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n}$
and
(8.20) $L(g_{1},\partial\Delta)\leq L(f,\partial\Delta),A(g_{1},\Delta)\geq
A(f,\Delta).$
It is clear that we have
(8.21) $V_{NE}(g_{1})\leq V_{NE}(f),V_{E}(g_{1})=V_{E}(f)+1,V(g_{1})\leq
V(f)+1,$
and we can rewrite the permitted partition of $\Gamma_{g_{1}}$ as
$\Gamma_{g_{1}}=\Gamma_{n}+\Gamma_{1}^{\prime}+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n-1}.$
Then there are three cases:
Case 1.1. Both $\Gamma_{1}^{\prime}$ and $\Gamma_{1}^{\prime\prime}$ are
natural edges of $\Gamma_{g_{1}}.$
Case 1.2. One of $\Gamma_{1}^{\prime}$ and $\Gamma_{1}^{\prime\prime}$ is a
natural edge, while the other is not.
Case 1.3. Neither $\Gamma_{1}^{\prime}$ nor $\Gamma_{1}^{\prime\prime}$ is a
natural edge.
In Case 1.1, it is clear that $g=g_{1}$ satisfies all the desired conclusions
with
(8.22) $V_{NE}(g_{1})=V_{NE}(f),\mathrm{\
}V_{E}(g_{1})=V_{E}(f)+1,V(g_{1})=V(f)+1.$
Assume Case 1.2 occurs. Without loss of generality, assume that
$\Gamma_{1}^{\prime\prime}$ is a natural edge. Then $\Gamma_{g_{1}}$ has the
natural partition
$\Gamma_{g_{1}}=\left(\Gamma_{n}+\Gamma_{1}^{\prime}\right)+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n-1}$
where $\left(\Gamma_{n}+\Gamma_{1}^{\prime}\right)\ $is a natural edge, and
(8.21) becomes
(8.23) $V_{NE}(g_{1})=V_{NE}(f)-1,\mathrm{\ }V_{E}(g_{1})=V_{E}(f)+1\
\mathrm{and\ }V(g_{1})=V(f).$
Then, in the case $L(\Gamma_{n}+\Gamma_{1}^{\prime})<\pi,$ by (a) and (8.19),
$g=g_{1}$ satisfies (i)–(iv) with (8.23); and in the case
$L(\Gamma_{n}+\Gamma_{1}^{\prime})\geq\pi,$ by (a), (8.18) and (8.19),
$\pi\leq L(\Gamma_{n}+\Gamma_{1}^{\prime})<2\pi$ and $g_{1}$ satisfies the
assumption of Lemma 8.3 with (8.23), and then, by (8.20), there exists a
normal mapping $g:\overline{\Delta}\rightarrow S$ that satisfies (i) and (ii),
and
$V_{NE}(g)\leq V_{NE}(g_{1}),\mathrm{\ }V_{E}(g)\geq V_{E}(g_{1})+1\
\mathrm{and\ }V(g)\leq V(g_{1})+1,$
and so by (8.23), (iii) and (iv) are satisfied by $g\ $with
(8.24) $V_{NE}(g)\leq V_{NE}(f)-1,\mathrm{\ }V_{E}(g)\geq V_{E}(f)+2\
\mathrm{and\ }V(g)\leq V(f)+1.$
Assume Case 1.3 occurs. Then both $\Gamma_{n}+\Gamma_{1}^{\prime}$ and
$\Gamma_{1}^{\prime\prime}+\Gamma_{2}$ are natural edges of $g_{1},$
$\Gamma_{g_{1}}$ has the natural partition
$\Gamma_{g_{1}}=\left(\Gamma_{n}+\Gamma_{1}^{\prime}\right)+\left(\Gamma_{1}^{\prime\prime}+\Gamma_{2}\right)+\Gamma_{3}+\dots+\Gamma_{n-1}$
and (8.21) becomes
(8.25) $V_{NE}(g_{1})=V_{NE}(f)-2,V_{E}(g_{1})=V_{E}(f)+1,V(g_{1})=V(f)-1.$
By (a) and (8.19) we have
(8.26) $L(\Gamma_{n}+\Gamma_{1}^{\prime})<2\pi\ \mathrm{and\
}L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<2\pi.$
Then, by (a), in the case
(8.27) $L(\Gamma_{n}+\Gamma_{1}^{\prime})<\pi\ \mathrm{and\
}L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<\pi,$
$g=g_{1}$ is the desired mapping satisfying (i)–(iv) with (8.25); and in the
case that (8.27) fails, by (a), (8.18) and (8.26), Lemma 8.3 or Lemma 8.4
applies, and then there exists a normal mapping $g$ satisfies (i), (ii) and
$V_{NE}(g)\leq V_{NE}(g_{1}),V_{E}(g)\geq V_{E}(g_{1})+1,V(g)\leq V(g_{1})+2,$
which, with (8.25), implies
$V_{NE}(g)\leq V_{NE}(f),V_{E}(g)\geq V_{E}(f)+1,V(g)\leq V(f)+1,$
i.e. (iii) and (iv) hold. This completes the proof in Case 1.3.
Now, assume Case 2 occurs. Then the hemisphere $S^{\prime}$ outside $C$
contains one or two points of $E.$ If $S^{\prime}$ contains only one point of
$E,$ the proof is exactly the same as the above arguments. So, we assume that
$S^{\prime}$ contains two points $q_{0}$ and $q_{0}^{\prime}$ of $E.$ Then
either $\\{q_{0},q_{0}^{\prime}\\}=\\{0,1\\}$ or $\\{1,\infty\\},$ and then
there are two cases:
Case 2.1. The great circle of $S$ containing $q_{0}$ and $q_{0}^{\prime}$
intersects $C\backslash\Gamma_{1}$ .
Case 2.2. The great circle containing $q_{0}$ and $q_{0}^{\prime}$ does not
intersects $C\backslash\Gamma_{1}$.
In Case 2.1, the argument for Case 1 exactly applies.
In Case 2.2, it is easy to show that the exists two points $r_{1}$ and
$r_{1}^{\prime}$ on $\Gamma_{1}$ such that $r_{1}$ close to $q_{1}$ and
$r_{1}^{\prime}$ close to $q_{2}$ (in $\Gamma_{1}),$ and
$r_{1},q_{0},q_{0}^{\prime},r_{1}^{\prime}$ or
$r_{1},q_{0}^{\prime},q_{0},r_{1}^{\prime}$ are in order on the geodesic path
from $r_{1}$ to $r_{1}^{\prime}$ in $S^{\prime},$ (then $r_{1}$ and
$r_{1}^{\prime}$ are antipodal). We assume
$r_{1},q_{0},q_{0}^{\prime},r_{1}^{\prime}$ is ordered in the orientation of
the geodesic path from $r_{1}$ to $r_{1}^{\prime}$ in $S^{\prime}.$ It is
clear that the notations
$\Gamma_{1}^{\prime}=\overline{q_{1}q_{0}},\gamma=\overline{q_{0}q_{0}^{\prime}},\Gamma_{1}^{\prime\prime}=\overline{q_{0}^{\prime}q_{2}}.$
make sense. Then
(8.28)
$L(\Gamma_{1}^{\prime})<\pi,L(\Gamma_{1}^{\prime\prime})<\pi,L(\gamma)=\frac{\pi}{2},$
and it is also clear that
(8.29)
$L(\Gamma_{1}^{\prime})+L(\gamma)+L(\Gamma_{1}^{\prime\prime})=L(\Gamma_{1}^{\prime})+\frac{\pi}{2}+L(\Gamma_{1}^{\prime\prime})<L(\Gamma_{1})<2\pi,$
and $\Gamma_{1}^{\prime}+\gamma+\Gamma_{1}^{\prime\prime}-\Gamma_{1}$ encloses
a Jordan domain $T$ in $S^{\prime}$ with $\left(\Gamma_{1}\cup T\right)\cap
E=\emptyset.$
By Lemma 3.2, there exists a normal mapping $g_{1}$, such that
$\Gamma_{g_{1}}$ has the permitted partition
$\Gamma_{g_{1}}=\Gamma_{1}^{\prime}+\gamma+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n}$
and by (8.29),
$L(g_{1},\partial\Delta)\leq L(f,\partial\Delta),A(f,\Delta)\geq
A(g_{1},\Delta).$
It is clear that we have
(8.30) $V_{NE}(g_{1})\leq V_{NE}(f),V_{E}(g_{1})=V_{E}(f)+2\ \mathrm{and\
}V(g_{1})\leq V(f)+2$
and we rewrite the permitted partition of $\Gamma_{g_{1}}$ as
(8.31)
$\Gamma_{g_{1}}=\Gamma_{n}+\Gamma_{1}^{\prime}+\gamma+\Gamma_{1}^{\prime\prime}+\Gamma_{2}+\dots+\Gamma_{n-1}.$
Note that $\gamma$ is always a natural edge of $\Gamma_{g_{1}},$ because the
endpoints of $\gamma$ are both in $E.$
Then there are three cases:
Case 2.2.1. Both $\Gamma_{1}^{\prime}$ and $\Gamma_{1}^{\prime\prime}$ are
natural edges of $\Gamma_{g_{1}}$.
Case 2.2.2. One of $\Gamma_{1}^{\prime}$ and $\Gamma_{1}^{\prime\prime}$ is a
natural edge, while the other is not.
Case 2.2.3. Neither $\Gamma_{1}^{\prime}$ nor $\Gamma_{1}^{\prime\prime}$ is a
natural edge.
In Case 2.2.1, (8.31) is a natural partition, and by (8.28), $g=g_{1}$
satisfies (i)–(iv) with (8.30).
In Case 2.2.2, we may assume $\Gamma_{1}^{\prime}$ is a natural edge, and then
by (8.31), $\Gamma_{g_{1}}$ has the natural partition
$\Gamma_{g_{1}}=\Gamma_{n}+\Gamma_{1}^{\prime}+\gamma+\left(\Gamma_{1}^{\prime\prime}+\Gamma_{2}\right)+\Gamma_{3}\dots+\Gamma_{n-1}.$
Then, by (8.30) and (8.31),
(8.32) $V_{NE}(g_{1})\leq V_{NE}(f),V_{E}(g_{1})=V_{E}(f)+2\ \mathrm{and\
}V(g_{1})=V(f)+1.$
Then, by (a) and (8.28), in the case
$L\left(\Gamma_{1}^{\prime\prime}+\Gamma_{2}\right)<\pi,$ $g=g_{1}$ satisfies
(i)–(iv) with (8.30), and otherwise, Lemma 8.3 applies to $g_{1},$ and then
there exists a normal mapping $g:\overline{\Delta}\rightarrow S$ satisfying
(i)–(iv), by (8.32), with
(8.33) $\left\\{\begin{array}[]{l}V_{NE}(g)\leq V_{NE}(g_{1})\leq
V_{NE}(f),\\\ V_{E}(g)\geq V_{E}(g_{1})+1=V_{E}(f)+3,\\\ V(g)\leq
V(g_{1})+1=V(f)+2.\end{array}\right.$
In Case 2.2.3, $\Gamma_{n}+\Gamma_{1}^{\prime}$ and
$\Gamma_{1}^{\prime\prime}+\Gamma_{2}$ are two natural edges of
$\Gamma_{g_{1}}$ with
$L(\Gamma_{n}+\Gamma_{1}^{\prime})<2\pi\text{ and
}L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<2\pi;$
$\Gamma_{g_{1}}$ has the natural partition
$\Gamma_{g_{1}}=\left(\Gamma_{n}+\Gamma_{1}^{\prime}\right)+\gamma+\left(\Gamma_{1}^{\prime\prime}+\Gamma_{2}\right)+\Gamma_{3}+\dots+\Gamma_{n-1}.$
and (8.30) becomes
(8.34) $V_{NE}(g_{1})\leq V_{NE}(f),V_{E}(g_{1})=V_{E}(f)+2\ \mathrm{and\
}V(g_{1})=V(f).$
Then, by (a) and (8.28), in the case $L(\Gamma_{n}+\Gamma_{1}^{\prime})<\pi$
and $L(\Gamma_{1}^{\prime\prime}+\Gamma_{2})<\pi,$ $g=g_{1}$ satisfies
(i)–(iv) with (8.34), and in other cases, Lemma 8.3 or Lemma 8.4 applies to
$g_{1},$ and then there exists a normal mapping
$g:\overline{\Delta}\rightarrow S$ satisfying (i)–(iv) with
(8.35) $\left\\{\begin{array}[]{l}V_{NE}(g)\leq V_{NE}(g_{1})\leq
V_{NE}(f),\\\ V_{E}(g)\geq V_{E}(g_{1})+1=V_{E}(f)+3,\\\ V(g)\leq
V(g_{1})+2=V(f)+2.\end{array}\right.$
This completes the proof. ∎
###### Lemma 8.6.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping, and let
(8.36) $\Gamma_{f}=\Gamma_{1}+\Gamma_{2}+\dots+\Gamma_{n},n=V(f),$
be a natural partition of $\Gamma_{f}=f(z),z\in\partial\Delta.$ Assume that
(8.37) $2\pi\leq L(\Gamma_{1})<3\pi,$
and
(8.38) $L(\Gamma_{j})<\pi\ \mathrm{for\ }j=2,\dots,n.$
Then, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$ such
that
(i) $L(g,\partial\Delta)\leq L(f,\partial\Delta)$ and $A(g,\Delta)\geq
A(f,\Delta)$
(ii) Each natural edge of $g$ has spherical length strictly less than $\pi,$
(iii) $V_{NE}(g)=V_{NE}(f)+2$, $V_{E}(g)=V_{E}(f)+1$ and $V(g)=V(f)+3.$
###### Proof.
Let $C$ be the great circle determined by $\Gamma_{1}.$ Then by (8.37) we have
$C\subset\Gamma_{1},$ and by the definition of natural edges, in the case that
$L(\Gamma_{1})=2\pi,$ the only possible point of $E=\\{0,1,\infty\\}$
contained in $C$ is $1,$ and in the case that $2\pi<L(\Gamma_{1})<3\pi,$ $C$
does not intsects $E,$ for otherwise the interior $\alpha_{1}^{\circ}$ of
$\alpha_{1}$ contains at least one point of $f^{-1}(E).$
Let $S^{\prime}$ be the hemisphere outside313131This means that $S^{\prime}$
is on the right hand side of $C.$ Note that $C$ is oriented by $\Gamma_{1}.$
$C$. Then
(8.39) $1\leq\\#\left(S^{\prime}\cap E\right)\leq 2.$
Let
$\partial\Delta=\alpha_{1}+\dots+\alpha_{n}$
be a natural partition of $\partial\Delta$ corresponding the partition (8.36)
and let $p_{1}$ and $p_{4}$ be the initial and terminal point of $\alpha_{1}$,
respectively. Then by (8.37) and (8.39), summarizing what we have, there
exists $p_{2}$ and $p_{3}$ in the interior of $\alpha_{1}$ such that
$p_{1},p_{2},p_{3},p_{4}$ are in order anticlockwise and the followings hold.
(a) $f$ restricted to each section $\alpha_{j}^{\prime}$ of $\alpha_{1}$ from
$p_{j}$ and $p_{j+1}$ is a homeomorphism, $j=1,2,3.$
(b) For the sections $\Gamma_{j}^{\prime}=f(z),z\in\alpha_{j},j=1,2,3$
$L(\Gamma_{2}^{\prime})=\pi,\ L(\Gamma_{1}^{\prime})<\pi,\
L(\Gamma_{3}^{\prime})<\pi.$
(c) Any shortest path from $q_{2}\mathrm{\ }$to $q_{3}\ $contains at most one
point of $E$.
Then by the definition of natural edges,
$\Gamma_{2}^{\prime}\cap E=\emptyset,$
and by (b), $q_{2}$ and $q_{3}\ $are antipodal. Therefore, by (8.39) and (c),
there exists a unique shortest path $L$ from $q_{2}$ to $q_{3}$ such that
$L-\Gamma_{2}^{\prime}$ enclose a domain $T$ such that $\overline{T}\cap E$
contains exactly one point $q\in E$, which lies in $L\cap S^{\prime}.$ We
denote by $\Gamma^{\prime}$ the section of $L$ from $q_{2}=f(p_{2})$ to
$q=f(q)$ and by $\Gamma^{\prime\prime}$ the section of $L$ from $q$ to
$q_{3}=f(p_{3}).$
Then we can extend the Riemann surface of $f$ to be a new Riemann surface so
that in the new Riemann surface, $\overline{T}$ is patched along
$\Gamma_{2}^{\prime}.$ By Lemma 3.2, this can be realized by a normal mapping
$g:\overline{\Delta}\rightarrow S.$ Then the boundary curve
$\Gamma_{g}=g(z),\in\partial\Delta,$ has the following natural partition
(8.40)
$\Gamma_{g}=\Gamma_{1}^{\prime}+\Gamma^{\prime}+\Gamma^{\prime\prime}+\Gamma_{3}^{\prime}+\Gamma_{2}+\dots+\Gamma_{n},$
because $\Gamma_{1}^{\prime},\Gamma_{3}^{\prime}$ is in $\Gamma_{1}$ and
$\Gamma^{\prime}$ and $\Gamma^{\prime\prime}$ are clearly natural edges.
It is clear that $\Gamma_{g}$ satisfies (ii) and
(8.41)
$L\left(\Gamma_{1}^{\prime}\right)+L\left(\Gamma^{\prime}\right)+L\left(\Gamma^{\prime\prime}\right)+L\left(\Gamma_{3}^{\prime}\right)=L\left(\Gamma_{1}^{\prime}\right)+L(\Gamma_{2}^{\prime})+L\left(\Gamma_{3}^{\prime}\right)=L(f,\alpha_{1}),$
and then by (8.40) we have
$L(g,\partial\Delta)=L(f,\partial\Delta).$
On the other hand, it is also clear that
$A(g,\Delta)=A(f,\Delta)+A(T)>A(f,\Delta).$
Thus, $g$ satisfies (i).
On the other hand, by (8.40), considering that all the natural vertices of
$\Gamma_{f}$ are natural vertices of $\Gamma_{g}$ and
$q_{2}^{\prime},q_{3}^{\prime}$ and $q^{\prime}$ are the three new natural
vertices of $g,$ we have
$V_{NE}(g)=V_{NE}(f)+2,V_{E}(g)=V_{E}(f)+1,V(g)=V(f)+3.$
Thus, (iii) is satisfied by $g.$ This completes the proof. ∎
## 9\. Movement of branched points
This section is prepared for prove Theorem 10.1.
###### Lemma 9.1.
Let $f:\overline{\Delta^{+}}\rightarrow\overline{\Delta}$ be an orientation
preserved open mapping that satisfies the following conditions:
(a) f restricted to the upper half circle
$\left(\partial\Delta\right)^{+}=\\{z\in\partial\Delta;\mathrm{Im}z\geq 0\\}$
is given by $f(e^{i\theta})=e^{\phi(\theta)i},$ where $\phi$ is a strictly
increasing function defined on $[0,\pi]$ with $\phi(0)=0$ and
$\phi(\pi)=(2d+1)\pi$, where $d$ is a positive integer.
(b) $f$ maps the interval $[-1,-1]$ homeomorphically onto the interval
$[-1,1].$
(c) $p_{0}\in\Delta^{+}$ is the unique ramification point of $f$ in
$\overline{\Delta^{+}}.$
Then there exists an orientation preserved open mapping
$g:\overline{\Delta^{+}}\rightarrow\overline{\Delta}$ such that the followings
hold.
(I) $z=0$ is the unique ramification point of $g$ in $\overline{\Delta^{+}}$
and $g(0)=0.$
(II)
$g|_{\left(\partial\Delta\right)^{+}}=f|_{\left(\partial\Delta\right)^{+}}$
and $g$ restricted to the interval $[-1,1]$ is a homeomorphism onto the
interval $[-1,1].$
###### Remark 9.1.
$g$ acts as an orientation preserved open mapping that moves the unique
ramification point $p_{0}$ of $f$ into the boundary of $\Delta^{+}$ with the
same branched number, while none other ramification points appear.
###### Proof.
There exists an orientation preserved homeomorphism $f_{1}$ from
$\overline{\Delta^{-}}$ onto $\overline{\Delta^{-}}$ such that
$f_{1}|_{[-1,1]}=f|_{[-1,1]}.$
Then
$f_{2}(z)=\left\\{\begin{array}[]{l}f(z),z\in\overline{\Delta^{+}},\\\
f_{1}(z),z\in\overline{\Delta^{-}}\backslash\overline{\Delta^{+}},\end{array}\right.$
is an orientation preserved $d+1$ to $1$ covering mapping from
$\overline{\Delta}$ onto $\overline{\Delta}$ such that $p_{0}$ is the unique
ramification point of $f_{2},$ and then there exists another covering mapping
$f_{3}:\overline{\Delta}\rightarrow\overline{\Delta}$ such that
$f_{3}|_{\partial\Delta}=f_{2}|_{\partial\Delta}$
and that $0$ is the unique ramification point of $f_{3}$ with $f_{3}(0)=0.$
It is clear that the path $\beta=\beta(t),t\in[-1,1]$ in $\overline{\Delta}$
has a unique lift $\alpha=\alpha(t),t\in[-1,1],$ in $\overline{\Delta}$ such
that
$\alpha(-1)=-1,\alpha(1)=1,$
$f_{3}$ restricted to $\alpha$ is a homeomorphism with
$f_{3}(\alpha(t))=\beta(t)=t,t\in[-1,1],$
and $\alpha$ is a Jordan path and the interior of $\alpha$ is contained in
$\Delta.$ Thus $\alpha$ divides $\Delta$ into two Jordan domains and one of
these domains is enclosed by $\left(\partial\Delta\right)^{+}$ and $\alpha,$
and we denote this domain by $U^{+}.$ Then $f_{3}$ restricted to
$\overline{\Delta^{+}\backslash U^{+}}$ is a homeomorphism.
Now consider the restriction $f_{3}|_{\overline{U^{+}}}.$ Let $h$ be a
homeomorphism from $\overline{\Delta^{+}}$ onto $\overline{U^{+}}$ such that
$h$ restricted to $\left(\partial\Delta\right)^{+}$ is an identity mapping,
restricted to the interval $[-1,1]$ is a homeomorphism onto $\alpha$ with
$h(0)=0$ (note that $0\in\alpha),$ and finally let
$g=f_{3}\circ h(z),z\in\overline{\Delta^{+}}.$
Then $g$ satisfies all the desired conditions. ∎
###### Remark 9.2.
By the proof we may construct that $g$ such that $g(0)=t$ for any fixed
$t\in(-1,1)$, $0$ is the unique ramification point of $g$ and all other
conclusions hold.
###### Lemma 9.2.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping, let
$p_{0}\in\Delta$ be a ramification point of $f$ with $v_{f}(p_{0})=d+1,$ and
assume that $\beta=\beta(t),t\in[0,1],$ is a polygonal Jordan path in $S$ that
satisfies the followings:
(a) $\beta(0)=f(p_{0}),$ $\beta(0)\neq\beta(1)$ and $\beta$ has a number of
$d+1$ lifts $\alpha_{j}=\alpha_{j}(t),t\in[0,1],$ by $f$ in
$\overline{\Delta},$ such that
$\cup_{j=2}^{d+1}\alpha_{j}\subset\Delta,$
$f(\alpha_{j}(t))=\beta(t),t\in[0,1],j=1,\dots,d+1,$
and
$\alpha_{j}(0)=p_{0},\mathrm{\ }j=1,\dots,d+1,$
(b) $\alpha_{1}(t)\in\Delta$ for all $t\in[0,1)\ $but
$p_{1}=\alpha_{1}(1)\in\partial\Delta.$
(c) $f$ has no ramification point on $\cup_{j=1}^{d+1}\alpha_{j}(0,1],$ where
$\alpha_{j}(0,1]$ is the curve $\alpha_{j}(t),t\in(0,1],$ which is the curve
$\alpha_{j}$ without initial point, $j=1,2,\dots d+1.$
(d) $f$ restricted to a neighborhood of $p_{1}=\alpha_{1}(1)$ in
$\overline{\Delta}$ is a homeomorphism.
Then, there exist a normal mapping $g:\overline{\Delta}\rightarrow S$ such
that
$A(g,\Delta)=A(f,\Delta),L(g,\partial\Delta)=L(f,\partial\Delta),$
and the followings hold.
(I) The ramification point $p_{0}$ of $f$ is no longer a ramification point of
$g,$ while the regular point $p_{1}$ of $f$ is a regular point of $g$ with
$g(p_{1})=\beta(1)=f(p_{1})\mathrm{\ and\ }v_{f}(p_{0})=v_{g}(p_{1}).$
(II) The boundary curves $\Gamma_{f}=f(e^{i\theta}),\theta\in[0,2\pi],$ and
$\Gamma_{g}=g(e^{i\theta}),\theta\in[0,2\pi],$ are the same curves after a
parameter transformation.
(III) The ramification point sets of $f$ and $g$ in
$\overline{\Delta}\backslash\\{p_{0},p_{1}\\}$ are the same, and $f$ and $g$
coincide in a neighborhood of this ramification point set.
.
###### Remark 9.3.
$g$ acts as a normal mapping that moves the ramification point $p_{0}$ of $f$
into the boundary $\partial\Delta$ with the same branched number, while all
other ramification points, as well as their branched number, remain unchanged,
and no other new ramification point appear, and the length and the area also
remains unchanged.
###### Proof.
Let $\delta<\frac{1}{2}$ be a positive number, let $D_{\delta}$ be the disk
$|z-p_{1}|<\delta$ in $\mathbb{C},$ let
$c=\left(\partial\Delta\right)\cap\overline{D_{\delta}}$, which will be
regarded as a path from $p_{2}\in\partial\Delta$ to $p_{3}\in\partial\Delta$
(anticlockwise). Then $p_{1}$ is the middle point of $c$. We write
$q_{j}=f(p_{j}),j=1,2,3;$
$D_{\delta}^{+}=D_{\delta}\backslash\overline{\Delta},$
$\Delta^{\ast}=\Delta\cup D_{\delta}\cup c\backslash\\{p_{2},p_{3}\\};$
$\gamma=f(c);$
$e=\partial\Delta^{\ast}\cap\overline{D_{\delta}}=\left(\partial
D_{\delta}\right)\backslash\Delta,$
which is the boundary of $\Delta^{\ast}$ outside $\Delta,$ and is the boundary
of $\partial D_{\delta}$ outside $\Delta$ as well. It is clear that
$\Delta^{\ast}$ is a Jordan domain (note that $\delta<\frac{1}{2})$ and
$\partial\Delta^{\ast}=e\cup\left(\left(\partial\Delta\right)\backslash
c\right),\ e\cap\overline{\left(\partial\Delta\right)\backslash
c}=\\{p_{2},p_{3}\\}.$
Since $f$ is normal and $\alpha_{2}\subset\Delta$ (by (a)), we have that
$f(p_{1})=f(\alpha_{1}(1))=f(\alpha_{2}(1))\notin E.$
On the other hand, by (c) and (d), we may take the number $\delta$
sufficiently small such that the following conditions (e)–(f) are satisfied.
(e) $f$ restricted to $c$ is a homeomorphism onto
$\gamma=f(c)=\overline{q_{2}q_{1}q_{3}},$ i.e., $\gamma$ is a polygonal Jordan
path with only one possible vertex at $q_{1}.$
(f) There exists a point $q_{1}^{\prime}$ in $S\backslash\gamma$ such that
$q_{1}^{\prime}$ is very close to $\gamma$ and is on the right hand side of
$\gamma,$ and the quadrangle $\overline{q_{2}q_{1}^{\prime}q_{3}q_{1}q_{2}}$
encloses a domain $T$ that is on the right hand side of $\gamma,$ with
$\overline{T}\cap E=\emptyset,$ and $f^{-1}(\overline{T})$ has $d$ components
$A_{j}$ with $f(\alpha_{j}(1)\in A_{j}$ and $f$ restricted to each $A_{j}$ is
a homeomorphism onto $\overline{T},$ for $j=2,\dots,d+1.$
(g) $\beta$ intersects $\overline{T}$ only at $q_{1}=f(p_{1}).$
Let $f_{1}$ be an orientation preserved homeomorphism from
$\overline{D_{\delta}^{+}}$ onto $\overline{T}$ such that $f_{1}$ and $f$
restricted to $c$ are equal to each other. Then
$f_{2}(z)=\left\\{\begin{array}[]{l}f(z),z\in\overline{\Delta},\\\
f_{1}(z),z\in\overline{D_{\delta}^{+}}\backslash\overline{\Delta},\end{array}\right.$
is a normal mapping defined on $\overline{\Delta^{\ast}}$.
The above argument show that $T$ can be extended to be a polygonal Jordan
domain $T^{\ast}$ such that the followings hold.
(h) $\beta\subset T^{\ast}$, the path
$\gamma^{\prime}=\overline{q_{2}q_{1}^{\prime}q_{3}}$ is still a section of
$\partial T^{\ast},$ and
$(\gamma\cup T)\backslash\\{q_{2},q_{3}\\}\subset T^{\ast}.$
(i) $f_{2}$ restricted to the component $\overline{U}$ of
$f_{2}^{-1}(\overline{T^{\ast}})$ with $p_{0}\in U$ is a $d+1$ to $1$ covering
with the unique ramification point $p_{0},$ and
$f_{2}(\overline{U})=\overline{T^{\ast}}.$
(j) The boundary of $U$ is composed of $e$ and a Jordan path $\alpha$ in
$\overline{\Delta}$ whose interior is in $U\ $and endpoints are $p_{2}$ and
$p_{3}.$
Then $V=U\cap\Delta$ is also a Jordan domain. Let $h_{1}$ be a homeomorphism
from $\overline{V}$ onto $\overline{\Delta^{+}}$ such that $h_{1}$ maps
$\alpha$ homeomorphically onto $\left(\partial\Delta\right)^{+},$ maps $c$
homeomorphically onto the interval $[-1,1]\ $with
$h_{1}(p_{1})=0;$
let $h_{2}$ be a homeomorphism from $\overline{T^{\ast}}$ onto
$\overline{\Delta}$ such that $h_{2}$ maps $\left(\partial
T^{\ast}\right)\backslash\\{\gamma^{\prime}\backslash\\{q_{2},q_{3}\\}\\}$
homeomorphically onto $\left(\partial\Delta\right)^{+}$, maps
$\gamma^{\prime}$ homeomorphically onto $\left(\partial\Delta\right)^{-},$ and
maps $\gamma$ homeomorphically onto the interval $[-1,1]$ with
$h_{2}(q_{1})=0;$
and finally let
$g_{1}=h_{2}\circ f_{2}|_{\overline{V}}\circ
h_{1}^{-1}(\zeta):\overline{\Delta^{+}}\rightarrow\overline{\Delta}.$
Then $g_{1}$ is an orientation preserved open mapping that satisfies all the
assumptions of Lemma 9.1, and then there exists an orientation preserved open
mapping $g_{2}:\overline{\Delta^{+}}\rightarrow\overline{\Delta}$ such that
the followings hold.
(k) $0$ is the unique ramification point of $g_{2}$ in $\overline{\Delta^{+}}$
and $g_{2}(0)=0.$
(l)
$g_{2}|_{\left(\partial\Delta\right)^{+}}=f|_{\left(\partial\Delta\right)^{+}}$
and both $f$ and $g_{2}$ restricted to the interval $[-1,1]$ are
homeomorphisms onto the interval $[-1,1].$
Let
$g_{3}=h_{2}^{-1}\circ g_{2}\circ h_{1}(z),z\in\overline{V}.$
Then $g_{3}$ restricted to a neighborhood of $\alpha$ in $\overline{V}$ is a
homeomorphism, $g_{3}$ maps $c$ homeomorphically onto $\gamma$ and $g_{3}$
restricted to $\alpha$ equals the restriction of $f$ to $\alpha$ and
$A(g_{3},V)=\left(d+1\right)A(T^{\ast})-A(T)=A(f,U)-A(f,D_{\delta}^{+})=A(f,V).$
Now,
$g(z)=\left\\{\begin{array}[]{l}f(z),z\in\overline{\Delta}\backslash\overline{V},\\\
g_{3}(z),z\in\overline{V},\end{array}\right.$
is the desired mapping. ∎
###### Lemma 9.3.
Let
$\alpha_{1}=\alpha_{1}(\theta)=e^{i\theta},\theta\in[\theta_{1},\theta_{2}]$
with $\theta_{1}<\theta_{2}<\theta_{1}+2\pi,$ be a section of $\partial\Delta$
and let
$p_{j}=\alpha_{1}(e^{i\theta_{j}}),j=1,2;$
let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that $p_{1}$
is a ramification point of $f$ with
$v_{f}(p_{1})=d.$
Assume that the section
$\beta=\beta(\theta)=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{2}],$
of $\Gamma_{f}=f(z),z\in\partial\Delta,\ $is a Jordan path with
$\beta\cap E=\emptyset,$
and $\beta$ has $d=v_{f}(p_{1})$ distinct lifts
$\alpha_{j}=\alpha_{j}(\theta),\theta\in[\theta_{1},\theta_{2}],j=1,\dots,d$
in $\overline{\Delta}$ by $f,$ such that
(a) For each $j=1,\dots,d,$
$f(\alpha_{j}(\theta))=f(\alpha_{1}(\theta))=\beta(\theta)$ for
$\theta\in[\theta_{1},\theta_{2}]\ $and $\alpha_{j}(\theta_{1})=p_{1}.$
(b) For each $j=2,\dots,d,$ $\alpha_{j}(\theta)\in\Delta$ for
$\theta\in(\theta_{1},\theta_{2}].$
(c) There is no ramification point of $f$ in $\cup_{j=1}^{d}\alpha_{j}$ other
than $p_{1}.$
(d) $f$ restricted to a neighborhood of $p_{j}$ in $\partial\Delta$ is a
homeomorphism, for $j=1,2.$
Then, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$ such
that
$A(g,\Delta)=A(f,\Delta),L(g,\partial\Delta)=L(f,\partial\Delta),$
and the followings hold.
(I) The ramification point $p_{1}=e^{i\theta_{1}}$ of $f$ is no longer a
ramification point of $g,$ while the regular point $p_{2}=e^{i\theta_{2}}$ of
$f$ is a ramification point of $g$ with $g(p_{2})=\beta(\theta_{2})=f(p_{2})$
and
$b_{f}(p_{1})=b_{g}(p_{2}).$
(II) The boundary curves $f(e^{i\theta})$ and $g(e^{i\theta})$ are the same
after a parameter transform.
(III) In $\overline{\Delta}\backslash\\{p_{1},p_{2}\\},$ $f$ and $g$ has the
same set of ramification points and $f$ and $g$ coincide in a neighborhood of
this ramification point set.
###### Proof.
There are two ways to prove this lemma. One way is to use Remark 9.2. Here we
use Lemma 9.2 to give another proof.
We will first construct a normal mapping $f_{2}$ that is defined on some
closed Jordan domain $\overline{\Delta^{\prime}}\ni p_{2}$ such that the
length and the area concerned in the lemma unchanged, the boundary curve
$\Gamma_{f_{2}}$ of $f_{2}$ is the same as that of $f,$ $f$ and $f_{2}$ have
the same set $B$ of ramification points in
$\overline{\Delta}\backslash\\{p_{1},p_{2}\\}$, $f_{2}$ and $f$ coincide in a
neighborhood of this ramification point set, and $f_{2}$ has only one more
ramification point $p_{1}^{\prime}$ outside $B,$ while $p_{1}^{\prime}$ is in
the interior of the domain $\Delta^{\prime},$ and there is a path $\beta_{3}$
whose interior and initial point are located in $\Delta^{\prime}$ and the
terminal point is $p_{2}\in\partial\Delta^{\prime},$ and $f_{2}$ and
$\beta_{3}$ satisfies all assumptions of Lemma 9.2 if $\Delta^{\prime}$ is
regarded as a disk. Then by applying Lemma 9.2, we obtain the desired
conclusion.
Let $\delta<\frac{1}{2}$ be a positive number, $D_{\delta}$ the disk
$|z-p_{1}|<\delta,$ $c$ the section of $\partial\Delta$ that is contained in
$\overline{D_{\delta}}$ and regarded as a path from $s_{1}$ to $s_{2}$
anticlockwise, $e$ the section of $\partial D_{\delta}$ that is outside
$\Delta$ $V$ the part of $D_{\delta}$ outside $\Delta$ and write .
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle f(c),$ $\displaystyle
t_{1}$ $\displaystyle=$ $\displaystyle f(s_{1}),t_{2}=f(s_{2}),$
$\displaystyle q_{1}$ $\displaystyle=$ $\displaystyle
f(p_{1}),q_{2}=f(p_{2}),$ $\displaystyle V^{\ast}$ $\displaystyle=$
$\displaystyle\Delta\cup D_{\delta},$ $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle f(c),\gamma^{\prime}=f(e).$
By the assumption, we may assume that $\delta$ is sufficiently small such that
the followings hold.
(e) $f$ can be extended to be a normal mapping $f_{1}$ defined on
$\overline{\Delta^{\ast}}.$
(f) $f_{1}$ restricted to $\overline{V}$ is a homeomorphism onto the closure
of a polygonal Jordan domain $T.$
(g) $q_{1}=f(p_{1})$ has a neighborhood $T^{\ast}$ such that $T^{\ast}\supset
T\cup\gamma\backslash\\{t_{1},t_{2}\\}$, $T^{\ast}$ is a polygonal Jordan
domain and for the component $U$ of $f_{1}^{-1}(\overline{T^{\ast}})$ with
$p_{1}\in U,$ $f_{1}$ restricted to $\overline{U}$ is a $d$ to $1$ covering
mapping onto $\overline{T^{\ast}},$ with the unique ramification point at
$p_{1}.$
(h) $\beta\cap\partial T^{\ast}=\\{t_{2}\\}$.
Then there is another normal mapping
$f_{2}:\overline{\Delta^{\ast}}\rightarrow S$ such that
$f_{2}|_{\overline{\Delta^{\ast}}\backslash
U}=f_{1}|_{\overline{\Delta^{\ast}}\backslash U},$
the restriction $f_{2}|_{\overline{U}}$ is also a $d$ to $1$ covering with a
unique ramification point $p_{1}^{\prime}$ in $U$ such that
$p_{1}^{\prime}\in\Delta$ and $q_{1}^{\prime}=f_{2}(p_{1}^{\prime})\in
T^{\ast}\backslash\overline{T}.$
Consider the lift of the path $\gamma=f(c)=f_{1}(c)$ by $f_{2}$. Since
$f_{2}|_{\overline{U}}$ is a covering with the unique ramification point
$p_{1}^{\prime}$ with $f_{2}(p_{1}^{\prime})=q_{1}^{\prime}\notin\gamma,$
$\gamma=f(c)$ has a unique lift $\alpha$ in $\overline{U}$ by $f_{2}$ such
that the interior of $\alpha$ is in $U$ with endpoints $s_{1}$ and $s_{2}$
(note that $\overline{T}$ can be lifted by $f_{2}|_{\overline{U}},$ because
$\overline{T}$ is simple connected and there is no branched point in $T).$
Then $\alpha$ divides $\Delta^{\ast}$ in to two Jordan domains
$\Delta^{\prime}$ and $\Delta^{\prime\prime}$ such that
$f_{2}(\partial\Delta^{\prime\prime})=\gamma\cup\gamma^{\prime}=\partial T,$
and $f_{2}$ restricted to $\partial\Delta^{\prime\prime}=\alpha\cup e$ is a
homeomorphism onto the boundary $\partial T.$ Then
$f_{2}|_{\overline{\Delta^{\prime\prime}}}$ is a homeomorphism from
$\Delta^{\prime\prime}$ onto $T.$
Then the restriction of $f_{2}$ to $\overline{\Delta^{\prime}}$ is a normal
mapping such that
$A(f_{2},\Delta^{\prime})=A(f_{2},\Delta^{\ast})-A(f_{2},\Delta^{\prime\prime})=A(f_{1},\Delta^{\ast})-A(T)=A(f,\Delta),$
and $\Gamma_{f}$ and $\Gamma_{f_{2}|_{\overline{\Delta^{\prime}}}}$ are the
same, ignoring a parameter transformation.
By (h), there exists a unique $\theta_{1}^{\prime}\in(\theta_{1},\theta_{2})$
such that $t_{2}=\beta(\theta_{1}^{\prime}),$ which is the unique point in
$\beta\cap\partial T^{\ast}.$ Let
$\beta_{1}(\theta),\theta\in[\theta_{1},\theta_{1}^{\prime}],$ be a polygonal
Jordan path in $T^{\ast}\backslash T$ such that
$\beta_{1}(\theta_{1})=q_{1}^{\prime},\beta_{1}(\theta_{1}^{\prime})=t_{2},$
and the interior of $\beta_{1}$ is contained in
$T^{\ast}\backslash\overline{T}.$ Then since $f_{2}|_{\overline{U}}$ is a
covering with the unique ramification point $p_{1}^{\prime}$ and
$f_{2}(p_{1}^{\prime})=q_{1}^{\prime},$ $\beta_{1}$ has $d$ lifts
$\alpha_{j}^{\ast}=\alpha_{j}^{\ast}(\theta),\theta\in[\theta_{1},\theta_{1}^{\prime}],j=1,\dots,d,$
by $f_{2}|_{\overline{U}},$ such that
(i) $\cup_{j=2}^{d}\alpha_{j}^{\ast}\subset\Delta^{\prime},$ the interior of
$\alpha_{1}^{\ast}$ is also contained in $\Delta^{\ast},$ while
$\alpha_{1}^{\ast}(\theta_{1}^{\prime})=s_{2}=\alpha_{1}(\theta_{1}^{\prime})\in\partial\Delta^{\prime}.$
(j)
$\alpha_{j}^{\ast}(\theta_{1}^{\prime})=\alpha_{j}(\theta_{1}^{\prime}),$and
$\alpha_{j}^{\ast}(\theta_{1})=p_{1}^{\prime},j=1,\dots,d.$
Let
$\beta_{2}(\theta)=\left\\{\begin{array}[]{c}\beta_{1}(\theta),\theta\in[\theta_{1},\theta_{1}^{\prime}],\\\
\beta(\theta),\theta\in[\theta_{1}^{\prime},\theta_{2}];\end{array}\right.$
and let
$\alpha_{j}^{\prime\prime}=\left\\{\begin{array}[]{c}\alpha_{j}^{\prime}(\theta),\theta\in[\theta_{1},\theta_{1}^{\prime}],\\\
\alpha_{j}(\theta),\theta\in[\theta_{1}^{\prime},\theta_{2}].\end{array}\right.$
Then, by the assumption of the lemma, we have
$f_{2}(\alpha_{j}^{\prime\prime}(\theta))=\beta_{2}(\theta),\theta\in[\theta_{1},\theta_{2}],j=1,\dots,d$
$\alpha_{j}^{\prime\prime}\subset\Delta^{\prime},j=2,\dots,d,$
and $f_{2}$ has no ramification point in
$\cup_{j=1}^{d}\alpha_{j}^{\prime\prime}\backslash\\{p_{1}^{\prime}\\}.$ Since
$\Gamma_{f_{2}|_{\overline{\Delta^{\prime}}}}=f_{2}(z),z\in\partial\Delta^{\prime}$,
is polygonal, there exists another polygonal Jordan path
$\beta_{3}(\theta),\theta\in[\theta_{1},\theta_{2}],$ such that
$\beta_{3}\cap\beta=\\{q_{2}\\}$, $\beta_{3}$ is so close to $\beta_{2}$ that
$\beta_{3}$ has a number of $d$ lifts
$\gamma_{j}=\gamma_{j}(\theta),\theta\in[\theta_{1},\theta_{2}]$ such that
$f_{2}(\gamma_{j}(\theta))=\beta_{3}(\theta),\theta\in[\theta_{1},\theta_{2}],j=1,\dots,d,$
$\gamma_{j}(\theta_{1})=p_{1}^{\prime},j=1,\dots,d,$
$\cup_{j=2}^{d}\gamma_{j}\subset\Delta^{\prime},\gamma\backslash\\{p_{2}\\}\subset\Delta^{\prime}$
and $f_{2}$ has no ramification point in
$\cup_{j}^{d}\gamma_{j}\backslash\\{p_{1}^{\prime}\\}.$ Then by Lemma 9.2,
there exists a normal mapping $f_{3}$ defined on $\Delta^{\prime}$ such that
the followings hold.
(k) $p_{2}=e^{i\theta_{2}}$ is a ramification point of $f_{3}$ with
$f_{3}(p_{2})=q_{2}^{\prime}=\beta(\theta_{2}),\ $while $p_{1}$ is not a
ramification point of $f_{3}$, and
$b_{f_{2}}(p_{1})=b_{f_{3}}(p_{2}),$
(l) The boundary curves $f_{3}(e^{i\theta})$ and $f_{2}(e^{i\theta})$ are the
same after a parameter transform.
(m) In $\overline{\Delta^{\prime}}\backslash\\{p_{1}^{\prime},p_{2}\\},$
$f_{3}$ and $f_{2}$ has the same set of ramification points and $f$ and $g$
coincide in a neighborhood of this ramification point set.
(n) $A(f_{3},\Delta^{\prime})=A(f_{2},\Delta^{\prime}).$
Let $B$ be the set of all ramification points of $f_{3},$ then it is clear
that $B\backslash p_{2}\subset\Delta\cap\Delta^{\prime}.$ Let $h$ be a
homeomorphism from $\overline{\Delta^{\prime}}$ to $\overline{\Delta},$ such
that $h$ restricted to a neighborhood of $B\backslash p_{2}$ is an identity,
and let $g=f_{2}\circ h^{-1}.$ Then $g$ is the desired mapping. ∎
## 10\. Cutting and Gluing Riemann surfaces of normal mappings
In this section, we will prove the following theorem, which is used in the
proof of Theorem 12.1.
###### Theorem 10.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and assume that
each natural edge of $f$ has spherical length strictly less than $\pi$. If $f$
has a branched point in $S\backslash E$, then there exist two normal mappings
$f_{j}:\overline{\Delta}\rightarrow S,j=1,2,$ such that the followings hold.
(i) Each natural edge of $f_{j}$ has spherical length strictly less than
$\pi,j=1,2.$
(ii) $\sum_{j=1}^{2}L(f_{j},\partial\Delta)\leq
L(f,\partial\Delta),\sum_{j=1}^{2}A(f_{j},\Delta)\geq A(f,\Delta)$.
(iii) $V_{NE}(f_{1})+V_{NE}(f_{2})\leq V_{NE}(f)+2,$
$V_{E}(f_{1})+V_{E}(f_{2})\geq V_{E}(f).$
(iv) $V(f_{1})+V(f_{2})\leq V(f)+2.$
The proof will be put to the end of this section, after we establish some
results for cutting and gluing the Riemann surface of $f.$
###### Lemma 10.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping, let
$p_{0}\in\Delta$ be a ramification point of $f$ and let
$\beta=\beta(t),t\in[0,1],$ be a polygonal Jordan path in $S$ with distinct
endpoints. Assume that the followings hold.
(a) Each natural edge $f$ has spherical length strictly less that $\pi.$
(b) $\beta(0)=f(p_{0}),$ $\beta$ has two lifts
$\alpha_{j}=\alpha_{j}(t),t\in[0,1],$ in $\overline{\Delta}$ by $f,$ with
(10.1) $\alpha_{j}(0)=p_{0}\ \text{{and\
}}f(\alpha_{j}(t))=\beta(t),t\in[0,1],j=1,2.$
(c) $\alpha_{j}(t)\in\Delta$ for all $t\in[0,1),$ $j=1,2,$ but
$\\{\alpha_{1}(1),\alpha_{2}(1)\\}\subset\partial\Delta.$
(d) $f$ has no ramification point in the interior
$\alpha_{j}(0,1)=\\{\alpha_{j}(t),t\in(0,1)\\},j=1,2.$
Then there exist normal mappings $f_{1},f_{2}:\overline{\Delta}\rightarrow S,$
such that the following conditions hold.
(i) Each natural edge $f_{j}$ has spherical length strictly less that $\pi,$
$j=1,2$.
(ii) $L(f,\Delta)\geq L(f_{1},\Delta)+L(f_{2},\Delta)$ and $A(f,\Delta)\leq
A(f_{1},\Delta)+A(f_{2},\Delta).$
(iii) $V_{NE}(f_{1})+V_{NE}(f_{2})\leq V_{NE}(f)+2$ and
$V_{E}(f_{1})+V_{E}(f_{2})\geq V_{E}(f).$
(iv) $V(f_{1})+V(f_{2})\leq V(f)+2.$
###### Proof.
By Lemma 3.5, we have $\alpha_{1}(1)\neq\alpha_{2}(1),$ and by (b) and (d),
$\alpha_{1}$ and $\alpha_{2}$ are Jordan paths that intersect only at $p_{0}.$
Then by the assumption, $\alpha=\alpha_{2}^{-}+\alpha_{1}$ compose a Jordan
path in $\overline{\Delta}$ from $\alpha_{2}(1)$ to $\alpha_{1}(1),$ such that
the interior of $\alpha$ is contained in $\Delta.$ Thus, $\alpha$ divides the
disk $\overline{\Delta}$ into two parts, of which one is on the left hand side
of $\alpha$ and is denoted by $\Delta_{1},$ and the other, denoted by
$\Delta_{2},$ is on the right hand side of $\alpha$.
Now, we consider the restrictions
$f|_{\overline{\Delta_{j}}}:\overline{\Delta}\rightarrow S,j=1,2.$
By Lemma 3.4, these two normal mapping can be regarded as two normal mappings
$g_{1}$ and $g_{2}$ defined on $\overline{\Delta}$ as follows.
Let $\gamma_{j}=\partial\Delta\cap\partial\Delta_{j},$ which is the section of
the boundary of $\Delta_{j}$ that is on the circle $\partial\Delta,$ and let
$h_{j}:\overline{\Delta_{j}}\rightarrow\overline{\Delta}$ be a continuous
mapping such that $h_{j}|_{\Delta}$ is a homeomorphism onto
$\Delta_{[0,1]}=\Delta\backslash[0,1],$ in which $[0,1]$ is the interval of
the real numbers, $h_{j}\ $maps the interior of $\gamma_{j}$ homeomorphically
onto $\left(\partial\Delta\right)\backslash\\{1\\},$ and
$h_{j}(\alpha_{j}(t))=t,t\in[0,1].$
Then define $g_{j}=f_{j}\circ h_{j}^{-1},$ and this is the glued mappings from
$\overline{\Delta}\ $into $S$. Then the followings hold.
(e) The boundary curves of $g_{1}$ and $g_{2}$ compose the boundary curve of
$f,$ i.e. the curve $\Gamma_{g_{1}}=g(z),z\in\partial\Delta,$ and the section
of the curve $\Gamma_{f}=f(z),$ in which $z$ runs on $\partial\Delta$ from
$\alpha_{1}(1)$ to $\alpha_{2}(1)$ are the same and the curve
$\Gamma_{g_{2}}=g_{2}(z),z\in\partial\Delta,$ and the the section of the curve
$\Gamma_{f}=f(z),$ in which $z$ runs on $\partial\Delta$ from $\alpha_{2}(1)$
to $\alpha_{1}(1)\ $are the same; and
(10.2) $A(f,\Delta)=A(g_{1},\Delta)+A(g_{2},\Delta)\ \mathrm{and\
}L(f,\Delta)=L(g_{1},\Delta)+L(g_{2},\Delta).$
Let $p_{j}=e^{i\theta_{j}},j=1,\dots,n,$ be an enumeration of all natural
vertices of $f$ that are in order anticlockwise, let
$c_{j}=e^{i\theta},\theta\in[\theta_{j},\theta_{j+1}]$
($\theta_{n+1}=\theta_{1}+2\pi$) be the section of $\partial\Delta$ from
$p_{j}$ to $p_{j+1}$ and write $q_{j}=f(p_{j}).$ Then
$\displaystyle\Gamma_{f}$ $\displaystyle=$
$\displaystyle\Gamma_{1}+\Gamma_{2}+\Gamma_{3}+\dots+\Gamma_{n}$
$\displaystyle=$
$\displaystyle\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{n-1}q_{n}}+\overline{q_{n}q_{1}}$
is a natural partition of the boundary curve
$\Gamma_{f}=f(e^{i\theta}),\theta\in[0,2\pi],$ with (by (a))
$L(\Gamma_{j})<\pi,j=1,2,\dots,n,$
and
$n=V(f).$
Without loss of generality, assume $\alpha_{1}(1)\in c_{1}$ and
$\alpha_{2}(1)\in c_{j_{0}}$ for some $j_{0}\leq n.$
Let
$q^{\prime}=f(\alpha_{1}(1))=f(\alpha_{2}(1)).$
Then, it is clear that the boundary curves
$\Gamma_{g_{j}}(z),z\in\partial\Delta,j=1,2,$ have the permitted partitions
$\displaystyle\Gamma_{g_{1}}$ $\displaystyle=$
$\displaystyle\overline{q_{j_{0}}q^{\prime}}+\overline{q^{\prime}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{j_{0}-1}q_{j_{0}}}$
$\displaystyle=$
$\displaystyle\Gamma_{11}+\Gamma_{12}+\Gamma_{2}+\dots+\Gamma_{j_{0}-1},$
and
$\displaystyle\Gamma_{g_{2}}$ $\displaystyle=$
$\displaystyle\overline{q_{1}q^{\prime}}+\overline{q^{\prime}q_{j_{0}+1}}+\overline{q_{j_{0}+1}q_{j_{0}+2}}+\dots+\overline{q_{n-1}q_{n}}+\overline{q_{n}q_{1}}$
$\displaystyle=$
$\displaystyle\Gamma_{21}+\Gamma_{22}+\Gamma_{j_{0}+1}+\dots+\Gamma_{n},$
respectively, such that
$L(\Gamma_{ij})<\pi,i,j=1,2,$
where
(10.6)
$\Gamma_{11}=\overline{q_{j_{0}}q^{\prime}},\Gamma_{12}=\overline{q^{\prime}q_{2}},\Gamma_{21}=\overline{q_{1}q^{\prime}},\Gamma_{22}=\overline{q^{\prime}q_{j_{0}+1}}.$
If $\alpha_{1}(1)$ (or $\alpha_{2}(1))$ is one of the endpoint of $c_{1}$ (or
$c_{j_{0}}),$ then the discussion is similar and easier than the followings,
since in this case some of edges in (10.6) reduce to points, and the
discussion is left to the reader. So, we assume $\alpha_{1}(1)$ is in the
interior of $c_{1}$ and $\alpha_{2}(1)$ is in the interior of $c_{2}.$ Then
$q^{\prime}\notin E.$
and it is clear that
(10.7) $\left\\{\begin{array}[]{l}V_{NE}(g_{1})+V_{NE}(g_{2})\leq
V_{NE}(f)+2,\\\ V_{E}(g_{1})+V_{E}(g_{2})=V_{E}(f),\\\ V(f_{1})+V(f_{2})\leq
n+2,\end{array}\right.$
and
$\displaystyle L(\Gamma_{1})+L(\Gamma_{j_{0}})$ $\displaystyle=$
$\displaystyle L(\Gamma_{11}+\Gamma_{12})+L(\Gamma_{21}+\Gamma_{22})$
$\displaystyle=$ $\displaystyle
L(\Gamma_{11}+\Gamma_{22})+L(\Gamma_{21}+\Gamma_{12}).$
Now, there are two cases need to discuss.
Case 1.
$\Gamma_{1}^{\prime}=\Gamma_{11}+\Gamma_{12}=\overline{q_{j_{0}}q^{\prime}}+\overline{q^{\prime}q_{2}}$
is not a natural edge of $\Gamma_{g_{1}}.$
Case 2.
$\Gamma_{1}^{\prime}=\Gamma_{11}+\Gamma_{12}=\overline{q_{j_{0}}q^{\prime}}+\overline{q^{\prime}q_{2}}$
is a natural edge of $\Gamma_{g_{1}}.$
In Case $1,$
$\Gamma_{2}^{\prime}=\Gamma_{21}+\Gamma_{22}=\overline{q_{1}q^{\prime}}+\overline{q^{\prime}q_{j_{0}+1}}$
not a natural as well. Then the partitions (10) and (10) are natural
partitions, since (10) is a natural partition, and then $g_{1},g_{2}$ are the
two desired mappings by (10.2) and (10.7).
In Case 2, $\Gamma_{2}^{\prime}=\Gamma_{21}+\Gamma_{22}$ is a natural edge as
well. Then
$\Gamma_{g_{1}}=\Gamma_{1}^{\prime}+\Gamma_{2}+\dots+\Gamma_{j_{0}-1},$
and
$\Gamma_{g_{2}}=\Gamma_{2}^{\prime}+\Gamma_{j_{0}+1}+\dots+\Gamma_{n},$
are natural partition of $\Gamma_{g_{1}}$ and $\Gamma_{g_{2}},$ respectively,
and then (10.7) changes into
(10.9) $\left\\{\begin{array}[]{l}V_{NE}(g_{1})+V_{NE}(g_{2})\leq
V_{NE}(f),\\\ V_{E}(g_{1})+V_{E}(g_{2})=V_{E}(f),\\\
V(f_{1})+V(f_{2})=n.\end{array}\right.$
If $L(\Gamma_{1}^{\prime})<\pi$ and $L(\Gamma_{2}^{\prime})<\pi,$ then $g_{1}$
and $g_{2}$ are the desired mappings with (10.9).
If $L(\Gamma_{1}^{\prime})\geq\pi,$ then by (10) and the assumption of the
lemma, $L(\Gamma_{2}^{\prime})<\pi$. This is because that by the assumption of
the lemma,
$L(\Gamma_{1}^{\prime})+L(\Gamma_{2}^{\prime})=L(\Gamma_{1})+L(\Gamma_{j_{0}})<2\pi.$
Then applying Theorem 8.1, there exists a normal mapping
$f_{1}:\overline{\Delta}\rightarrow S$ such that
$L(f_{1},\Delta)\leq L(g_{1},\partial\Delta),\mathrm{\
}A(f_{1},\Delta)=A(g_{1},\Delta),$
and
$V_{NE}(f_{1})\leq V_{NE}(g_{1}),\mathrm{\ }V_{E}(f_{1})\geq
V_{E}(g_{1})+1,V(f_{1})\leq V(g_{1})+2,$
and each natural edges of $\Gamma_{f_{1}}$ has spherical length strictly less
than $\pi.$ Then we have by (10.2) and (10.9) that
$L(f_{1},\partial\Delta)+L(g_{2},\partial\Delta)\leq
L(f,\partial\Delta),A(f_{1}.\Delta)+A(g_{2},\Delta)=A(f,\Delta)$
$V_{NE}(f_{1})+V_{NE}(g_{2})\leq V_{NE}(g_{1})+V_{NE}(g_{2})\leq V_{NE}(f),$
$V_{E}(f_{1})+V_{E}(g_{2})\geq V_{E}(g_{1})+1+V_{E}(g_{2})=V_{E}(f)+1,$
and
$V(f_{1})+V(g_{2})\leq V(g_{1})+2+V(g_{2})\leq V(f)+2.$
Thus, $f_{1}$ and $g_{2}$ is the desired mappings. This completes the proof. ∎
###### Corollary 10.1.
Assume that $f,\beta,\alpha_{1}$ and $\alpha_{2}$ satisfy all the assumptions
in Lemma 10.1 and, in addition, $\beta(1)\in E.$ Then there exist normal
mappings $g_{1},g_{2}:\overline{\Delta}\rightarrow S,$ such that the
followings hold.
(i) Each natural edge of $\Gamma g_{j}$ is a natural edge of
$\Gamma_{f},j=1,2,$ and each natural edge of $\Gamma_{f}$ is a natural edge of
either $g_{1}$ or $g_{2}$.
(ii) $L(g_{1},\Delta)+L(g_{2},\Delta)=L(f,\Delta)$ and
$A(g_{1},\Delta)+A(g_{2},\Delta)=A(f,\Delta).$
(iii)
$V_{NE}(g_{1})+V_{NE}(g_{2})=V_{NE}(f),V_{E}(g_{1})+V_{E}(g_{2})=V_{E}(f),$
$V(g_{1})+V(g_{2})=V(f).$
(iv)
(10.10) $\sum_{p\in\overline{\Delta}\backslash
g_{1}^{-1}(E)}b_{g_{1}}(p)+\sum_{p\in\overline{\Delta}\backslash
g_{2}^{-1}(E)}b_{g_{2}}(p)\leq\sum_{p\in\overline{\Delta}\backslash
f^{-1}(E)}b_{f}(p).$
###### Proof.
By repeating the above proof from the beginning to (10.6) and considering
that, in current situation, $q^{\prime}=f(\alpha(1))=f(\alpha_{2}(1))$ must be
a natural vertex of $f,g_{1}$ and $g_{2},$ we can conclude that all the
conclusion follows, except the inequality (iv).
By the assumption and the definition of $g_{j}$s, it is clear that
(10.11) $b_{g_{1}}(0)+b_{g_{2}}(0)=b_{f}(p_{0})-1.$
Next, we show that
(10.12) $b_{g_{1}}(1)+b_{g_{2}}(1)\leq
b_{f}(\alpha_{1}(1))+b_{f}(\alpha_{2}(1))+1.$
Let $l_{j}$ be the circular arc of the circle
$C_{j}:|z-\alpha_{j}(1)|=\varepsilon$ inside $\overline{\Delta}$ and
$\gamma_{j}$ be the section of $\partial\Delta$ inside $C_{j},j=1,2;$ let $l$
be the circular arc of the circle $C:|z-1|$ inside $\overline{\Delta}$ and
$\gamma^{\prime}$ be the section of $\partial\Delta$ inside $C,j=1,2;$ where
$\varepsilon$ is a sufficiently small positive number. Let
$s_{1},s_{2},s_{1}^{\prime},s_{2}^{\prime}$ be smooth and orientation
preserved diffeomorphisms from neighborhoods of $f(a_{1}(1))$,
$f(\alpha_{2}(1)),g_{1}(1)$ and $g_{2}(1)$ onto the disk $\Delta$ with
$s_{j}(f(\alpha_{j}(1))=0,s_{j}^{\prime}(g_{j}(1))=0,$
such that they keep the angles at $f(a_{1}(1))$, $f(\alpha_{2}(1)),g_{1}(1)$
and $g_{2}(1)$ and maps
$f(\gamma_{1}),f(\gamma_{2}),g_{1}(\gamma_{1}^{\prime})$ and
$g_{2}(\gamma_{2}^{\prime})$ onto angles (broken lines) with vertices
$f(a_{1}(1))$, $f(\alpha_{2}(1)),g_{1}(1)$ and $g_{2}(1),$ respectively, in
$\mathbb{C}$. Then we can define the rotation numbers
$\tau_{j}=\frac{1}{2\pi}\int_{l_{j}}\frac{d\left(s_{j}\circ
f(z)\right)}{s_{j}\circ f(z)}\ \mathrm{and\
}\tau_{j}^{\prime}=\frac{1}{2\pi}\int_{l}\frac{d\left(s_{j}^{\prime}\circ
g_{j}(z)\right)}{s_{j}^{\prime}\circ g_{j}(z)},$
which is invariant for sufficiently small $\varepsilon$, independent of
$s_{j}$s and $s_{j}^{\prime}$s by the assumption and all are positive because
$s_{j}$s and $s_{j}^{\prime}$s are orientation preserved and $g_{j}$s and $f$
are normal. It is clear that
(10.13) $\tau_{1}^{\prime}+\tau_{2}^{\prime}=\tau_{1}+\tau_{2}.$
Then there exists $k_{j}$ and $k_{j}^{\prime},j=1,2,$ such that
$k_{j}^{\prime}<\tau_{j}^{\prime}\leq k_{j}^{\prime}+1,k_{j}<\tau_{j}\leq
k_{j}+1,j=1,2,$
and then we have
$\displaystyle b_{g_{1}}(1)+b_{g_{2}}(1)$ $\displaystyle=$ $\displaystyle
k_{1}^{\prime}+k_{2}^{\prime}$ $\displaystyle<$
$\displaystyle\tau_{1}^{\prime}+\tau_{2}^{\prime}=\tau_{1}+\tau_{2}\leq
k_{1}+k_{2}+2$ $\displaystyle=$ $\displaystyle
b_{f}(\alpha(1))+b_{f}(\alpha_{2}(1))+2,$
but branched numbers are integers, we have (10.12).
It is clear that, by the definition of $g_{j}$s
$\sum_{p\in\overline{\Delta}\backslash\\{\alpha_{1}(1),\alpha_{2}(1)\\}}b_{f}(p)=\sum_{j=1}^{2}\sum_{p\in\overline{\Delta}\backslash\\{1\\}}b_{g_{j}}(p),$
which, together (10.11) and (10.12), implies (10.10). ∎
###### Lemma 10.2.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping, let
$\alpha_{1}=\alpha_{1}(\theta)=e^{i\theta},\theta\in[\theta_{1},\theta_{2}],$
be a section of $\partial\Delta$ and denote
$\beta=\beta(\theta)=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{2}].$ Assume
that the followings hold:
(a) Each natural edge $f_{j}$ has spherical length strictly less that $\pi.$
(b) $\beta$ is a Jordan path with distinct endpoints and
$\beta(\theta)\notin E\ \mathrm{for\ each\ }t\in[\theta_{1},\theta_{2}).$
(c) $\beta$ has a lift
$\alpha_{2}=\alpha_{2}(\theta),\theta\in[\theta_{1},\theta_{2}],$ in
$\overline{\Delta},$ with
$\alpha_{2}(\theta_{1})=\alpha_{1}(\theta_{1})=e^{i\theta_{1}},$ and
$f(e^{i\theta})=f(\alpha_{1}(\theta))=\beta(\theta),\theta\in[\theta_{1},\theta_{2}].$
(d) $f$ has no ramification point in the interior of $\alpha_{1}$ and
$\alpha_{2}.$
(e) The interior of $\alpha_{2},$ which means the open curve
$\alpha_{2}(\theta),\theta\in(\theta_{1},\theta_{2}),$ is contained in
$\Delta,\ $but $\\{\alpha_{2}(\theta_{2})\\}\subset\partial\Delta.$
Then there exist two normal mappings $f_{1},f_{2}:\overline{\Delta}\rightarrow
S,$ such that the followings hold.
(i) Each natural edge $f_{j}$ has spherical length strictly less that $\pi,$
$j=1,2$.
(ii) $A(f,\Delta)\leq A(f_{1},\Delta)+A(f_{2},\Delta)\ \mathrm{and\
}L(f,\Delta)\geq L(f_{1},\Delta)+L(f_{2},\Delta).$
(iii) $V_{NE}(f_{1})+V_{NE}(f_{2})\leq V_{NE}(f)+2,$ and
$V_{E}(f_{1})+V_{E}(f_{2})\geq V_{E}(f).$
(iv) $V(f_{1})+V(f_{2})\leq V(f)+2.$
###### Proof.
By Lemma 3.5, we have $\alpha_{1}(\theta_{2})\neq\alpha_{2}(\theta_{2}).$
$\alpha_{2}$ divides the disk $\overline{\Delta}$ into two parts, one of which
denoted by $\Delta_{1},$ is on the left hand side of $\alpha_{2},$ and the
other, denoted by $\Delta_{2},$ is on the right hand side. Then, $\alpha_{1}$
is a section of $\partial\Delta_{2}.$ By ignoring a coordinate transform, we
may regard the restriction $f|_{\overline{\Delta_{1}}}$ as a normal mapping
$g_{1}$ defined on $\overline{\Delta}.$
Consider the Jordan domain $\Delta_{2}.$ By Lemma 3.4, we can glue the
$\alpha_{1}$ and $\alpha_{2}$ so that the restriction
$f|_{\overline{\Delta_{2}}}$ can be regarded as a normal mapping
$g_{2}:\overline{\Delta}\rightarrow S,$ as we did in the proof of Lemma 10.1.
Then the followings hold:
(e) The boundary curves of $g_{1}$ and $g_{2}$ compose the boundary curve of
$f,$ i.e. the curve $\Gamma_{g_{1}}=g(z),z\in\partial\Delta,$ and the section
of the curve $\Gamma_{f}=f(z),$ in which $z$ runs on $\partial\Delta$ from
$\alpha_{1}(1)$ to $\alpha_{2}(1)$ are the same and the curve
$\Gamma_{g_{2}}=g_{2}(z),z\in\partial\Delta,$ and the the section of the curve
$\Gamma_{f}=f(z),$ in which $z$ runs on $\partial\Delta$ from $\alpha_{2}(1)$
to $\alpha_{1}(1)\ $are the same; and
(10.14) $A(f,\Delta)=A(g_{1},\Delta)+A(g_{2},\Delta)\ \mathrm{and\
}L(f,\Delta)=L(g_{1},\Delta)+L(g_{2},\Delta).$
Then, as in the proof of Lemma 10.1, there exist normal mappings
$f_{j}:\overline{\Delta}\rightarrow S,j=1,2,$ satisfied the conclusions. ∎
###### Corollary 10.2.
Assume that $f,\beta,\alpha_{1}$ and $\alpha_{2}$ satisfy all the assumptions
in Lemma 10.2 and, in addition, $\beta(1)\in E.$ Then all the conclusions of
Corollary 10.1 hold.
###### Proof.
Repeat the above proof from the beginning to (10.14) and repeat the proof of
Corollary 10.1. ∎
###### Lemma 10.3.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that each
natural edge of $f$ has spherical length strictly less than $\pi$. If $f$ has
a ramification point in $\Delta,$ then, one of the following conditions (A)
and (B) is satisfied.
(A) There exists a normal mappings $f_{1}:\overline{\Delta}\rightarrow S$ such
that, the followings hold.
(i) The boundary curve $\Gamma_{f_{1}}=f_{1}(z),z\in\partial\Delta$, is the
same as that of $f$.
(ii)
$L(f_{1},\partial\Delta)=L(f,\partial\Delta),A(f_{1},\Delta)=A(f,\Delta).$
(iii) $f_{1}$ has no ramification point in $\Delta.$
(iv) $f_{1}$ has at least one ramification point in
$\left(\partial\Delta\right)\backslash f_{1}^{-1}(E).$
(B) There exist normal mappings $f_{j}:\overline{\Delta}\rightarrow S,j=1,2,$
such that the followings hold.
(i) Each natural edge of $f_{j}$ has spherical length strictly less than
$\pi,j=1,2$.
(ii) $\sum_{j=1}^{2}L(f_{j},\partial\Delta)\leq
L(f,\partial\Delta),\sum_{j=1}^{2}A(f_{j},\Delta)\geq A(f,\Delta).$
(iii) $V_{NE}(f_{1})+V_{NE}(f_{2})\leq E(f)+2,$ $V_{E}(f_{1})+V_{E}(f_{2})\geq
V_{E}(f).$
(iv) $V(f_{1})+V(f_{2})\leq V(f)+2.$
###### Proof.
Let $p_{0}\in\Delta$ be any ramification point of $f$ and let
$\beta=\beta(t),t\in[0,1],$ be a polygonal Jordan path in $S$ from
$q_{0}=f(p_{0})$ to some point $q_{1}\in E=\\{0,1,\infty\\}$ such that the
interior of $\beta$ does not contain any point in $E$. Then
$\beta(0)\neq\beta(1),$ since $f$ is normal.
We may assume that
(a) There is no branched point of $f$ in the interior of $\beta$ (otherwise,
we deform $\beta$ slightly$).$
Let $d=v_{f}(p_{0}).$ Then, by (a) and the fact that $f(\Delta)\cap
E=\emptyset$ (note that $f$ is normal) we conclude that there are only two
cases:
Case 1. There exists a positive number $t_{1}\leq 1,$ such that the followings
hold.
(a1) The section $\beta[0,t_{1}]=\\{\beta(t);t\in[0,t_{1}]\\}$ of $\beta$ has
two lifts $\alpha_{j}=\alpha_{j}(t),t\in[0,t_{1}],$ in $\overline{\Delta}$ by
$f,$ such that
$\alpha_{j}(0)=p_{0}\text{{and\
}}f(\alpha_{j}(t))=\beta(t),t\in[0,t_{1}];j=1,2.$
(b1) For $j=1$ and $2,$ $\alpha_{j}(t)\in\Delta$ for all $t\in[0,t_{1}],$ but
$\\{\alpha_{1}(t_{1}),\alpha_{2}(t_{1})\\}\subset\partial\Delta.$
(c1) $f$ has no ramification point in the interior of $\alpha_{1}$ and
$\alpha_{2},$ i.e. $f$ has no ramification point on
$\alpha_{1}(0,1)\cup\alpha_{2}(0,1),$ where $\alpha_{j}(0,1)$ is the open
curve $\alpha_{j}(t),t\in(0,1),$ which is the curve $\alpha_{j}$ without end
points, $j=1,2.$
Case 2. There exists a positive number $t_{1}\leq 1,$ such that the followings
hold.
(a2) The section $\beta(t),t\in[0,t_{1}],$ of $\beta$ has a number of
$d=v_{f}(p_{0})$ lifts $\alpha_{j}=\alpha_{j}(t),t\in[0,t_{1}],$ in
$\overline{\Delta},$ such that
(10.15) $\cup_{j=2}^{d}\alpha_{j}\subset\Delta,$
$f(\alpha_{j}(t))=\beta(t),t\in[0,t_{1}],\mathrm{\ }j=1,\dots,d,$
and
$\alpha_{j}(0)=p_{0},\mathrm{\ }j=1,\dots,d.$
(b2) $\alpha_{1}(t)\in\Delta$ for all $t\in[0,t_{1})\ $but
$p_{1}^{\prime}=\alpha_{1}(t_{1})\in\partial\Delta.$
In Case 1, by Lemmas 10.1, (B) is satisfied.
Now, assume Case 2 occurs. Then, we must have $t_{1}<1.$ Otherwise, we have
$f(\alpha_{2}(t_{1}))=\beta(1)=q_{1}\in E,$ and then by the fact
$f(\Delta)\cap E=\emptyset,$ we have $\alpha_{2}(t_{1})\in\partial\Delta,$
contradicting (10.15). Thus by (a) we have:
(c2) $f$ has no ramification point on $\cup_{j=1}^{d}\alpha_{j}(0,t_{1}],$
where $\alpha_{j}(0,t_{1}]$ is the curve $\alpha_{j}(t),t\in(0,t_{1}],$ which
is the curve $\alpha_{j}$ without initial point, $j=1,2,\dots d.$
Then, by (a2), (b2) and (c2), Lemma 9.2 applies, and then, there exists a
normal mapping $g_{1}:\overline{\Delta}\rightarrow S$ such that (recall that
$p_{1}^{\prime}=\alpha_{1}(t_{1}))$
(10.16) $\\#\\{p\in\Delta;b_{g_{1}}(p)>1\\}=\\#\\{p\in\Delta;b_{f}(p)>1\\}-1,$
(10.17) $b_{g_{1}}(p_{1}^{\prime})=b_{f}(p_{0})$ (10.18)
$L(g_{1},\partial\Delta)=L(f,\partial\Delta),A(g_{1},\Delta)=A(f,\Delta),$
and
(d) The boundary curve of $g_{1}$ is the same as that of $f.$
Then, $g_{1}$ satisfies (i), (ii) of condition (A).
By (10.17), $p_{1}^{\prime}\in\partial\Delta$ is a ramification point of
$g_{1}.$ On the other hand, by (b2) and (10.15)
$g_{1}(\partial\Delta)\ni
g_{1}(p_{1}^{\prime})=f(p_{1}^{\prime})=f(\alpha_{1}(t_{1}))=f(\alpha_{2}(t_{1}))\in
f(\Delta),$
,i.e. $g_{1}(p_{1}^{\prime})\in g_{1}(\partial\Delta)\cap f(\Delta),$ and then
$g_{1}(p_{1}^{\prime})\notin E$ since $f$ is normal. Thus, (iv) of (A) hold.
If $g_{1}$ does not satisfies (iii) in (A), then $g_{1}$ satisfies all
assumptions of the lemma under proving, but the number of ramification points
of $g_{1}$ located in $\Delta$ is dropped by one (by (10.16)), and then apply
the above argument, we again reach Case 1 or Case 2. Since there are finitely
many ramification point of $f$ in $\Delta,$ after repeating the above
arguments finitely many time, we can show that either (A) or (B) holds. ∎
###### Lemma 10.4.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that
(a) Each natural edge of $f$ has length strictly less than $\pi,$ and
(b) $f$ has no ramification point in $\Delta$.
Assume that there exist $\theta_{1}$ and $\theta_{3}$ with
$\theta_{1}<\theta_{3}<\theta_{1}+2\pi$ such that the followings hold.
(c) $p_{1}=e^{i\theta_{1}}\in\partial\Delta\backslash f^{-1}(E)$ is a
ramification point of $f.\ $Note that $E=\\{0,1,\infty\\}.$
(d) $f(e^{i\theta_{3}})\in E\ $but $f(e^{i\theta})\notin E$ for each
$\theta\in(\theta_{1},\theta_{3}).$
(e) Each point $e^{i\theta}\in\partial\Delta$ with
$\theta\in(\theta_{1},\theta_{3})$ is not a ramification point of $f$.
Then, there exist two normal mappings $f_{j}:\overline{\Delta}\rightarrow
S,j=1,2,$ such that the followings holds.
(i) Each natural edge of $f_{j}$ has length strictly less than $\pi$, $j=1,2.$
(ii) $\sum_{j=1}^{2}L(f_{j},\partial\Delta)\leq
L(f,\partial\Delta),\sum_{j=1}^{2}A(f_{j},\Delta)\geq A(f,\Delta).$
(iii) $V_{NE}(f_{1})+V_{NE}(f_{2})\leq V_{NE}(f)+2,$
$V_{E}(f_{1})+V_{E}(f_{2})\geq V_{E}(f).$
(iv) $V(f_{1})+V(f_{2})\leq V(f)+2.$
###### Proof.
By the assumption, there are only two cases need to discuss.
Case 1. There exist $\theta_{4},\theta_{5}\in(\theta_{1},\theta_{2})\ $with
$\theta_{4}<\theta_{5}$ such that
(f) $e^{i\theta_{4}}$ is a natural vertex of $f.$
(g) Both of the sections
$\Gamma_{14}=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{4}],$ and
$\Gamma_{45}=f(e^{i\theta}),\theta\in[\theta_{4},\theta_{5}]$ are Jordan paths
with $\Gamma_{14}(\theta_{1})\neq\Gamma_{14}(\theta_{4})$ and
$\Gamma_{45}(\theta_{4})\neq\Gamma_{45}(\theta_{5})$, but
$\Gamma_{14}+\Gamma_{45}$ is not a Jordan curve.
Case 2. The section
$\Gamma_{13}=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{3}],$ is a Jordan
path.
Let $d=v_{f}(p_{0}).$ We first assume Case 1 occur.
Then, there exists a positive number $\delta$ and there exist a number of $d$
Jordan paths
$\alpha_{j,\delta}=\alpha_{j,\delta}(\theta),\theta\in[\theta_{1},\theta_{1+\delta}],j=1,\dots,d,$
such that
$\alpha_{1,\delta}(\theta)=e^{i\theta},\theta\in[\theta_{1},\theta_{1+\delta}],$
$\alpha_{j,\delta}(\theta_{1})=p_{1},j=1,\dots,d,$
$\alpha_{j,\delta}(\theta)\in\Delta,\theta\in(\theta_{1},\theta_{1+\delta}),j=2,3,\dots
d,$
and
$f(\alpha_{j,\delta}(\theta))=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{1+\delta}],j=1,\dots,d.$
Since $f$ has no ramification point in $\Delta,$ there are only two further
cases for Case 1.
Case 1.1. Each $\alpha_{j,\delta}$ can be extended to be a Jordan path
$\alpha_{j}=\alpha_{j}(\theta),\theta\in[\theta_{1},\theta_{4}],$ such that
$\alpha_{j}(\theta)\in\Delta,\theta\in(\theta_{1},\theta_{4}],j=2,3,\dots,d,$
and
$f(\alpha_{j}(\theta))=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{4}],j=2,3,\dots,d.$
Case 1.2. For some $j_{0}\in\\{2,3,\dots,d\\},$ there exists
$\theta_{2}\in(\theta_{1},\theta_{4}]$ such that $\alpha_{j_{0},\delta}$ can
be extended to be a Jordan path
$\alpha_{j_{0}}=\alpha_{j_{0}}(\theta),\theta\in[\theta_{1},\theta_{2}],$ such
that
$\alpha_{j_{0}}(\theta_{2})\in\partial\Delta,$
$\alpha_{j_{0}}(\theta)\in\Delta,\theta\in(\theta_{1},\theta_{2}),$
$f(\alpha_{j_{0}}(\theta))=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{2}].$
In Case 1.1, since $f$ has no ramification point in $\Delta$, $f$ has no
ramification point in the interior of each $\alpha_{j},j=2,\dots,d,$ and then
$f,\alpha_{1},\dots,\alpha_{d}$ and
$\beta=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{4}]$ satisfy all
assumptions of Lemma 9.3 by (e). Then Lemma 9.3 apply, and then there exists a
normal mapping $g:\overline{\Delta}\rightarrow S$ such that
$g(e^{i\theta})=f(e^{i\theta}),\theta\in[0,2\pi],$
$b_{g}(p_{1})=0,b_{g}(p_{2})=b_{f}(p_{1}),$ $b_{g}(p)=b_{f}(p)\ \mathrm{for\
a}\text{{l}}\mathrm{l\ }p\in\overline{\Delta}\backslash\\{p_{1},p_{2}\\},$
and
$L(g,\partial\Delta)=L(f,\partial\Delta),A(g,\Delta)=A(f,\Delta).$
Then $g$ satisfies all the assumptions in the lemma under proving by replacing
$\theta_{1}$ with $\theta_{4}$. But now, the number of loops of the section
$g(e^{i\theta}),\theta\in[\theta_{4},\theta_{3}],$ is dropped by one. Then, by
repeating the same argument several times, we can find a number
$\theta_{1}^{\prime}\in[\theta_{1},\theta_{3}),$ and a normal mapping
$f_{1}:\overline{\Delta}\rightarrow S$ such that
$f_{1}(e^{i\theta})=f(e^{i\theta}),\theta\in[0,2\pi],$
$f_{1}$, $\theta_{1}=\theta_{1}^{\prime}$ and $\theta_{3}$ satisfy all
assumptions of the lemma and fit case Case 2.
In Case 1.2, $f$ also has no ramification point in the interior of each
$\alpha_{j},j=1,j_{0}.$ Without loss of generality, we assume $j_{0}=2.$ Then,
$f,\alpha_{1},\alpha_{2}=\alpha_{j_{0}}$ and
$\beta=\beta=f(e^{i\theta}),\theta\in[\theta_{1},\theta_{2}]$, satisfy all
assumptions of Lemma 10.2. Then by Lemmas 10.2, There exist two normal
mappings $f_{1},f_{2}:\overline{\Delta}\rightarrow S,$ satisfying (i)–(ii).
Now assume Case 2 occurs. Then since $f(\Delta)\cap E=\emptyset$ (note that
$f$ is normal), by (a)–(e), there exists
$\theta_{2}\in(\theta_{1},\theta_{3}]$ such that the sections
$\alpha_{1}=e^{i\theta}$ and $\beta=f(e^{i\theta})$ with
$\theta\in[\theta_{1},\theta_{2}]$ satisfy all assumptions of Lemma 10.2, and
the arguments in Case 1.2 apply. This completes the proof. ∎
###### Proof of Theorem10.1.
Assume $f$ has a branched point in $f(\overline{\Delta})\backslash E.$ There
are two cases:
Case 1. $f$ has a ramification point in $\Delta.$
Case 2. $f$ has no ramification point in $\Delta,$ but has a ramification
point in $\partial\Delta\backslash f^{-1}(E).$
In Case 1, Lemma 10.3 applies, and we have the following conclusions (A) or
(B).
(A) There exists a normal mappings $g_{1}:\overline{\Delta}\rightarrow S$ such
that, the followings hold.
(1) The boundary curve $\Gamma_{g_{1}}=g_{1}(z),z\in\partial\Delta$, is the
same as that of $f$.
(2) $L(g_{1},\partial\Delta)=L(f,\partial\Delta),A(g_{1},\Delta)=A(f,\Delta).$
(3) $g_{1}$ has no ramification point in $\Delta.$
(4) $g_{1}$ has at least one branched point in $f(\partial\Delta)\backslash
E.$
(B) The conclusions of Theorem 10.1 hold true.
If (A) occurs, then $g_{1}$ has a ramification point
$p_{1}\in\left(\partial\Delta\right)\backslash f^{-1}(E),$ and then we can
found $\theta_{1}$ and $\theta_{3}$ such that $g_{1}$, $\theta_{1}$ and
$\theta_{3}$ satisfy all assumptions of Lemma 10.4, and then (B) holds.
In Case 2, Lemma 10.4applies, and so, the conclusions of Theorem 10.1 hold
true again. ∎
## 11\. Deformation of normal mappings that have nonconvex vertices
In this section we will prove the following theorem, which is used to prove
Theorem 12.1. Theorem 12.1 is the first key step to prove the main theorem.
###### Theorem 11.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and assume that
each natural edge of $\Gamma_{f}$ has length strictly less than $\pi$. If
$\Gamma_{f}$ is not convex at some natural vertex $q$ and $q\notin E.$ Then
there exists a normal mapping $g:\overline{\Delta}\rightarrow S,$ such that
$L(g,\partial\Delta)\leq L(f,\partial\Delta),A(g,\Delta)\geq A(f,\Delta),$
each natural edge of $g$ has spherical length strictly less than $\pi,$ and
$V_{NE}(g)\leq V_{NE}(f)-1,\mathrm{\ }V_{E}(g)\geq V_{E}(f)\ \mathrm{and}\
V(g)\leq V(f)+1.$
###### Proof.
We divide the proof into four parts, which is the coming Lemmas 11.1–11.4. ∎
Before we introduce these lemmas, we first make some conventions.
We fix the normal mapping $f:\overline{\Delta}\rightarrow S$ and assume
(11.1) $\Gamma_{f}=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}+\dots+\Gamma_{n}$
is a natural partition of $\Gamma_{f},$ with $n=V(f)$,
$\partial\Delta=\gamma_{1}+\dots+\gamma_{n}$
is the corresponding natural partition of $\partial\Delta$ for $f,$ and denote
by $p_{j}=e^{i\theta_{j}},j=1,\dots,n,$ the initial point of
$\gamma_{j},j=1,\dots,n$, with $\theta_{j+1}=\theta_{1}$ and
$\theta_{1}<\theta_{2}<\dots<\theta_{n}<\theta_{1}+2\pi;$
and assume that
(I) All natural edges of $f$ has spherical length strictly less than $\pi.$
Then, $q_{j}=f(p_{j})$ is the initial point of $\Gamma_{j}$ for each
$j=1,2,\dots,n,$ and by (I), the notation $\overline{q_{j}q_{j+1}}$ makes
sense, which is the unique shortest path from $q_{j}$ to $q_{j+1},$ and
$\Gamma_{j}=\overline{q_{j}q_{j+1}},j=1,2,\dots,n.$ Therefore, the natural
partition (11.1) can be written
$\Gamma_{f}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{n-1}q_{n}}.$
We will also assume that
(II) $\Gamma_{1}+\Gamma_{2}$ is not convex at $q_{2}\notin E.$
The assumption (II) means that either $\Gamma_{1}+\Gamma_{2}$ can be regarded
as a perigon angle, or the oriented triangle $\overline{q_{1}q_{3}q_{2}q_{1}}$
is a convex triangle. When $\Gamma_{1}+\Gamma_{2}$ is a perigon angle, there
is only one case need to discuss.
Case A. $q_{3}\in\Gamma_{1}=\overline{q_{1}q_{2}}$ or
$q_{1}\in\Gamma_{2}=\overline{q_{2}q_{3}}$.
When $\overline{q_{1}q_{3}q_{2}q_{1}}$ is a convex triangle, it encloses a
triangle domain $T$ that is on the right hand side of $\Gamma_{1}+\Gamma_{2},$
and there are only three cases need to discuss:
Case B. $\left(\overline{\mathbf{T}}\backslash\\{q_{1},q_{3}\\})\cap
E\right)=\emptyset.$
Case C. There is only one point $q_{1}^{\prime}$ in $E=\\{0,1,\infty\\}$ that
is located in $\overline{T}\backslash\\{\Gamma_{1}+\Gamma_{2}\\}.$
Case D. There exist two points $q_{1}^{\prime}$ and $q_{1}^{\prime\prime}$ in
$E=\\{0,1,\infty\\}$ that is located in the triangle domain $T.$
Under these settings, we can execute deformations of $f\ $which will be stated
in the following Lemmas 11.1–11.4.
###### Lemma 11.1.
In Case A, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$
such that
(11.2) $L(g,\partial\Delta)<L(f,\partial\Delta)\text{{\ and }}A(g,\Delta)\geq
A(f,\Delta),$
each natural edge of $\Gamma_{g}$ has spherical length strictly less than
$\pi,$ and
(11.3) $V_{NE}(g)\leq V_{NE}(f)-1,V_{E}(g)\geq V_{E}(f),V(g)=V(f).$
###### Proof.
Assume Case A occurs. Then, without loss of generality, we may assume
$q_{3}\in\Gamma_{1}.$ Let $p^{\prime}=e^{i\theta_{1}^{\prime}}$,
$\theta_{1}^{\prime}\in(\theta_{1},\theta_{2}),$ such that
$f(p^{\prime})=q_{3}.$ Then we can glue the section of $\partial\Delta$ from
$p^{\prime}$ to $p_{2}$ and the section of $\partial\Delta$ from $p_{2}$ to
$p_{3}$ and regard $f$ as a mapping $g$ of the glued closed set, which can be
regard as a closed disk, such that the boundary curve of $g$ has a permitted
partition
(11.4) $\Gamma_{g}=\Gamma_{1}^{\prime}+\Gamma_{3}+\dots+\Gamma_{n},$
where $\Gamma^{\prime}=\overline{q_{1}q^{\prime}}=\overline{q_{1}q_{3}}$ is
the section of $\Gamma_{1}$ from $q_{1}=f(p_{1})$ to
$q^{\prime}=f(p^{\prime})=q_{3},$ and
(11.5) $L(g,\partial\Delta)<L(f,\partial\Delta),\ A(g,\Delta)=A(f,\Delta).$
By (11.4), we have (11.3).
If (11.4) is a natural partition, then $g$ also satisfies (I) and then $g$ is
the desired mapping.
Assume that (11.4) is not a natural partition, which is only in the case that
$\Gamma_{1}^{\prime\prime}=\Gamma_{1}^{\prime}+\Gamma_{3}=\overline{q_{1}q_{3}}+\overline{q_{3}q_{4}}$
is a natural edge of $\Gamma_{g}$ .
But in this case,
$\Gamma_{g}=\Gamma_{1}^{\prime\prime}+\Gamma_{4}+\dots+\Gamma_{n}$
is a natural partition, and then we have
(11.6) $V_{NE}(g)=V_{NE}(f)-2,V_{E}(g)=V_{E}(f),\ V(g)=V(f)-2.$
By (I) we have
$L(\Gamma_{1}^{\prime\prime})<2\pi.$
If $L(\Gamma_{1}^{\prime\prime})<\pi,$ then $g$ already satisfies all the
conclusions of Lemma 11.1.
If $L(\Gamma_{1}^{\prime\prime})\geq\pi,$ then $g$ satisfies (a) or (b) of
Theorem 8.1, and then there exists a normal mapping
$f_{1}:\overline{\Delta}\rightarrow S$ such that
$L(f_{1},\partial\Delta)\leq L(g,\partial\Delta),A(f_{1},\Delta)\geq
A(g,\Delta),$
each natural edge of $f_{1}$ has spherical length strictly less than $\pi,$
and
$V_{NE}(f_{1})\leq V_{NE}(g),V_{E}(f_{1})\geq V_{E}(g)+1,V(f_{1})\leq V(g)+2.$
Then by (11.5) and (11.6) we have
$L(f_{1},\partial\Delta)<L(f,\partial\Delta),A(f_{1},\Delta)\geq A(f,\Delta),$
and
$V_{NE}(f_{1})\leq V_{NE}(f)-2,V_{E}(f_{1})\geq
V_{E}(f)+1>V_{E}(f),V(f_{1})\leq V(f).$
Thus, $f_{1}$ satisfies all the conclusion of Lemma 11.1. ∎
###### Lemma 11.2.
In Case B, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$
such that
$L(g,\partial\Delta)<L(f,\partial\Delta)\text{{\ and
}}A(g,\Delta)>A(f,\Delta),$
each natural edge of $\Gamma_{g}$ has spherical length strictly less than
$\pi,$ and
$V_{NE}(g)\leq V_{NE}(f)-1,\text{{\ }}V_{E}(g)\geq V_{E}(f),V(g)\leq V(f).$
###### Proof.
Putting
$\Gamma_{1}^{\prime}=\overline{q_{1}q_{3}},$
by (I) and (II), we have
(11.7) $L(\Gamma_{1}^{\prime})=L(\overline{q_{1}q_{3}})<\pi.$
as in the previous proof, by Lemma 3.2, there exists a normal mapping $g$,
which will be regarded as an extension of $f,$ such that $\Gamma_{g}$ has the
permitted partition
(11.8) $\Gamma_{g}=\Gamma_{1}^{\prime}+\Gamma_{3}+\dots+\Gamma_{n},$
and
$L(g,\partial\Delta)<L(f,\partial\Delta),A(g,\Delta)>A(f,\Delta).$
Then
$V_{NE}(g)\leq V_{NE}(f)-1,V_{E}(g)=V_{E}(f),V(g)\leq V(f)-1,$
and there are four cases:
Case 1. Neither $\overline{q_{n}q_{1}q_{3}}$ nor $\overline{q_{1}q_{3}q_{4}}\
$is a natural edge of $\Gamma_{g}.$
Case 2. $\overline{q_{n}q_{1}q_{3}}\ $is a natural edge of $\Gamma_{g},$ while
$\overline{q_{1}q_{3}q_{4}}\ $is not.
Case 3. $\overline{q_{n}q_{1}q_{3}}$ is not a natural edge of $\Gamma_{g}$,
while $\overline{q_{1}q_{3}q_{4}}$ is.
Case 4. Both $\overline{q_{n}q_{1}q_{3}q_{4}}\ $is a natural edge of
$\Gamma_{g}$.
In Case 1, (11.8) is a natural partition, and $g$ is the desired mapping.
In Case 2, $g$ has a natural partition
$\Gamma_{g}=\Gamma_{1}^{\prime\prime}+\Gamma_{3}+\dots+\Gamma_{n-1},$
where
$\Gamma_{1}^{\prime\prime}=\Gamma_{n}+\Gamma_{1}^{\prime}=\overline{q_{n}q_{1}q_{3}},$
and it is clear that
$V_{NE}(g)=V_{NE}(f)-2,V_{E}(g)=V_{E}(f),V(g)=V(f)-2,$
and by (I) and (11.7),
(11.9) $L(\Gamma_{1}^{\prime\prime})<2\pi.$
If $L(\Gamma_{1}^{\prime\prime})<\pi,$ the $g$ satisfies all the conclusions.
If $L(\Gamma_{1}^{\prime\prime})\geq\pi,$ then by (I), (11.9) and Theorem 8.1
for the cases (a) and (b), there exists a normal mapping
$f_{1}:\overline{\Delta}\rightarrow S$ such that
$L(f_{1},\partial\Delta)\leq L(g,\partial\Delta),A(f_{1},\Delta)\geq
A(g,\Delta),$
each natural edge of $f_{1}$ has spherical length strictly less than $\pi,$
and
$V_{NE}(f_{1})\leq V_{NE}(g_{1}),\mathrm{\ }V_{E}(f_{1})\geq
V_{E}(g)+1,V(f_{1})\leq V(g)+2.$
Then $f_{1}$ satisfies all the desired conditions in the lemma with
$V_{NE}(f_{1})\leq V_{NE}(f)-2,V_{E}(f_{1})\geq V_{E}(f)+1\ \mathrm{and\
}V(f_{1})\leq V(f).$
Case $3$ can be treated as Case $2.$
In case $4$ we have
(11.10) $V_{NE}(g)=V_{NE}(f)-3,V_{E}(g)=V_{E}(f),V(g)=V(f)-3,$
and $g$ has a natural partition
(11.11)
$\Gamma_{g}=\Gamma_{1}^{\prime\prime\prime}+\Gamma_{4}+\dots+\Gamma_{n-1},$
where
$\Gamma_{1}^{\prime\prime\prime}=\overline{q_{n}q_{1}}+\overline{q_{1}q_{3}}+\overline{q_{3}q_{4}}=\Gamma_{n}+\Gamma_{1}^{\prime}+\Gamma_{3}.$
Then by (I) and (11.7) we have
(11.12) $\Gamma_{1}^{\prime\prime\prime}<3\pi.$
If $L(\Gamma_{1}^{\prime\prime\prime})<\pi,$ then by (I), (11.10) and (11.11),
$g$ is the desire mapping.
If $L(\Gamma_{1}^{\prime\prime\prime})\geq\pi,$ then by (I), (11.11) and
Theorem 8.1 (a), (b) and (d), there exists a normal mapping
$g_{1}:\overline{\Delta}\rightarrow S$ such that
$L(g_{1},\partial\Delta)\leq L(g,\partial\Delta),A(g_{1},\Delta)\geq
A(g,\Delta),$
each natural edge of $g_{1}$ has spherical length strictly less than $\pi,$
and
$V_{NE}(g_{1})\leq V_{NE}(g_{1})+2,\mathrm{\ }V_{E}(g_{1})\geq
V_{E}(g)+1,V(g_{1})\leq V(g)+3.$
Then $g_{1}$ satisfies all the desired conditions in the lemma with (by
(11.10))
$V_{NE}(g_{1})\leq V_{NE}(f)-1,V_{E}(g_{1})\geq V_{E}(f)+1\ \mathrm{and\
}V(g_{1})\leq V(f).$
This completes the proof. ∎
###### Lemma 11.3.
In Cases C, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$
such that
$L(g,\partial\Delta)<L(f,\partial\Delta)\text{{\ and
}}A(g,\Delta)>A(f,\Delta),$
each natural edge of $\Gamma_{g}$ has spherical length strictly less than
$\pi,$ and
$V_{NE}(g)\leq V_{NE}(f)-1,V_{E}(g)\geq V_{E}(f)+1\ \mathrm{and}\ V(g)\leq
V(f).$
###### Proof.
Assume Case C occurs and let
$\Gamma_{1}^{\prime}=\overline{q_{1}q_{1}^{\prime}}$ and
$\Gamma_{2}^{\prime}=\overline{q_{1}^{\prime}q_{2}}.$ Then, considering that
$q_{1},q_{1}^{\prime},q_{2}$ are contained in the closure of the triangle
domain $T$ which in on the left hand side of the convex triangle
$\overline{q_{1}q_{3}q_{2}q_{1}},$ we have
(11.13) $L(\Gamma_{1}^{\prime})<\pi,L(\Gamma_{2}^{\prime})<\pi,\
L(\Gamma_{1}^{\prime}+\Gamma_{2}^{\prime})<L(\Gamma_{1}+\Gamma_{2}),$
and it is clear that
$\Gamma_{1}^{\prime}+\Gamma_{2}^{\prime}-\Gamma_{2}-\Gamma_{1}$
is a quadrilateral and encloses a domain $T^{\prime}$ in $T$ that is on the
right hand side of $\Gamma_{1}+\Gamma_{2}.$ Then, by (11.13), replacing the
the domain $T$ in the proof of Lemma 11.2 by $T^{\prime}$ and repeating the
extension arguments, we can obtain a normal mapping
$g:\overline{\Delta}\rightarrow S$ such that
(11.14) $L(g,\partial\Delta)<L(f,\partial\Delta)\text{{\ and
}}A(g,\Delta)>A(f,\Delta),$
and the boundary curve $\Gamma_{g}$ of $g$ has the following permitted
partition
$\Gamma_{g}=\Gamma_{1}^{\prime}+\Gamma_{2}^{\prime}+\Gamma_{3}+\dots+\Gamma_{n},$
which implies another permitted partition
$\displaystyle\Gamma_{g}$ $\displaystyle=$
$\displaystyle\Gamma_{n}+\Gamma_{1}^{\prime}+\Gamma_{2}^{\prime}+\Gamma_{3}+\dots+\Gamma_{n-1}.$
$\displaystyle=$
$\displaystyle\overline{q_{n}q_{1}}+\overline{q_{1}q_{1}^{\prime}}+\overline{q_{1}^{\prime}q_{2}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{n-1}q_{n}}.$
But here the terminal point $q_{1}^{\prime}$ of $\Gamma_{1}^{\prime}$, which
is the initial point of $\Gamma_{2}^{\prime},$ is in $E,$ and so we have
$V_{NE}(g)\leq V_{NE}(f)-1,V_{E}(g)=V_{E}(f)+1\ \mathrm{and\ }V(g)\leq V(f).$
Now, there are four cases need to discuss.
Case 1. Neither
$\Gamma_{n}+\Gamma_{1}^{\prime}=\overline{q_{n}q_{1}q_{1}^{\prime}}\ $nor
$\Gamma_{2}^{\prime}+\Gamma_{3}=\overline{q_{1}^{\prime}q_{2}q_{3}}\ $is a
natural edge of $\Gamma_{g}.$
Case 2. $\overline{q_{n}q_{1}q_{1}^{\prime}}$ is a natural edge of
$\Gamma_{g},$ while $\overline{q_{1}^{\prime}q_{2}q_{3}}$ is not.
Case 3. $\overline{q_{n}q_{1}q_{1}^{\prime}}$ is a natural edge of
$\Gamma_{g}$, while $\overline{q_{1}^{\prime}q_{2}q_{3}}\ $is not.
Case 4. Both $\overline{q_{n}q_{1}q_{1}^{\prime}}\ $and
$\overline{q_{1}^{\prime}q_{2}q_{3}}\ $are natural edges of $\Gamma_{g}.$
In Case 1, (11) is a natural partition, and $g$ is the desired mapping.
In Case 2, $g$ has the natural partition
$\Gamma_{g}=\Gamma_{1}^{\prime\prime}+\Gamma_{2}^{\prime}+\Gamma_{3}+\dots+\Gamma_{n-1},$
where
$\Gamma_{1}^{\prime\prime}=\Gamma_{n}+\Gamma_{1}^{\prime}=\overline{q_{n}q_{1}q_{1}^{\prime}},$
and it is clear that
(11.16) $V_{NE}(g)=V_{NE}(f)-2,V_{E}(g)=V_{E}(f)+1\ \mathrm{and\
}V(g)=V(f)-1.$
and by (I) and (11.13)
$L(\Gamma_{1}^{\prime\prime})<2\pi.$
If $L(\Gamma_{1}^{\prime\prime})<\pi,$ then $g$ is the desired mapping with
(11.16).
If $\pi\leq L(\Gamma_{1}^{\prime\prime})<2\pi,$ then by (I) and Theorem 8.1
(a) (note that $q_{1}^{\prime}\in E$ is the terminal point of
$\Gamma_{1}^{\prime\prime})$, there exists a normal mapping
$g_{1}:\overline{\Delta}\rightarrow S$ such that
$L(g_{1},\partial\Delta)\leq L(g,\partial\Delta),A(g_{1},\Delta)\geq
A(g,\Delta),$
each natural edge of $g_{1}$ has spherical length strictly less than $\pi,$
and
$V_{NE}(g_{1})\leq V_{NE}(g),\mathrm{\ }V_{E}(g_{1})\geq V_{E}(g)+1\
\mathrm{and\ }V(f_{1})\leq V(g)+1.$
Then, by (11.14) and (11.16), $g_{1}$ satisfies all the desired conclusions in
Lemma 11.3 with
$V_{NE}(f_{1})\leq V_{NE}(f)-2,\mathrm{\ }V_{E}(f_{1})\geq V_{E}(f)+2\
\mathrm{and\ }V(f_{1})\leq V(f).$
Case $3$ can be treated as Case $2.$
In case $4$ we have
(11.17) $V_{NE}(g)=V_{NE}(f)-3,V_{E}(g)=V_{E}(f)+1\ \mathrm{and\ }V(g)\leq
V(f)-3$
and $g$ has a natural partition
(11.18)
$\Gamma_{g}=\Gamma_{1}^{\prime\prime}+\Gamma_{2}^{\prime\prime}+\Gamma_{4}+\dots+\Gamma_{n-1},$
where
$\Gamma_{1}^{\prime\prime}=\Gamma_{n}+\Gamma_{1}^{\prime}=\overline{q_{n}q_{1}q_{1}^{\prime}}\
$and
$\Gamma_{2}^{\prime\prime}=\Gamma_{2}^{\prime}+\Gamma_{3}=\overline{q_{1}^{\prime}q_{2}q_{3}}.$
By (11.13) and (I), we have
(11.19) $L(\Gamma_{1}^{\prime\prime})<2\pi,L(\Gamma_{2}^{\prime\prime})<2\pi.$
If
(11.20) $L(\Gamma_{1}^{\prime\prime})<\pi,L(\Gamma_{2}^{\prime\prime})<\pi,$
then by (I), (11.14), (11.17) and (11.18), $g$ is the desired mapping.
If (11.20) does not hold, then by (I), (11.18), (11.19) and the fact that both
$\Gamma_{1}^{\prime\prime}$ and $\Gamma_{2}^{\prime\prime}$ have endpoints in
$E,$ Theorem 8.1 (a) or (c) applies to $g$, and then, there exists a normal
mapping $f_{1}:\overline{\Delta}\rightarrow S$ such that
$L(f_{1},\partial\Delta)\leq L(g,\partial\Delta),A(f_{1},\Delta)\geq
A(g,\Delta),$
each natural edge of $f_{1}$ has spherical length strictly less than $\pi,$
and
$V_{NE}(f_{1})\leq V_{NE}(g_{1}),\mathrm{\ }V_{E}(f_{1})\geq V_{E}(g)+1,\
\mathrm{and\ }V(f_{1})\leq V(g)+2.$
Then $f_{1}$ satisfies all the desired conclusions of Lemma 11.3 with (by
(11.17))
$V_{NE}(f_{1})\leq V_{NE}(f)-3,\mathrm{\ }V_{E}(f_{1})\geq V_{E}(f)\
\mathrm{and}\ V(f_{1})\leq V(f)-1.$
This completes the proof. ∎
###### Lemma 11.4.
In Case D, there exists a normal mapping $g:\overline{\Delta}\rightarrow S$
such that
$L(g,\partial\Delta)<L(f,\partial\Delta)\text{{\ and
}}A(g,\Delta)>A(f,\Delta),$
each natural edge of $\Gamma_{g}$ has spherical length strictly less than
$\pi,$ and
$V_{NE}(g)\leq V_{NE}(f)-1,V_{E}(g)\geq V_{E}(f)+1\ \mathrm{and\ }V(g)\leq
V(f)+1.$
###### Proof.
In Case D, $q_{1}^{\prime}\in T$ and $q_{2}^{\prime}\in T$ are the only points
in $\overline{T}\cap E.$ Let $L$ be the line segment in $\overline{T}$ that
passes through $q_{1}^{\prime}$ and $q_{2}^{\prime}$ and has endpoints in
$\partial T.$ Then there are two cases:
Case 1. $L$ intersects $\overline{q_{1}q_{3}}$.
Case 2. $L$ does not intersect $\overline{q_{1}q_{3}}$.
Assume Case 1 occurs and, without loss of generality, assume $q_{2}^{\prime}$
is closer to $\overline{q_{1}q_{3}}$ than $q_{1}^{\prime}.$ Let
$\Gamma_{1}^{\prime}=\overline{q_{1}q_{1}^{\prime}}$ and
$\Gamma_{2}^{\prime}=\overline{q_{1}^{\prime}q_{2}}$ (a)). Then
$\Gamma_{1}^{\prime}$ and $\Gamma_{2}^{\prime}$ satisfy all the conditions in
the proof of Lemma 11.3, and in this case, we can prove Lemma 11.4 by exactly
repeating the proof of Lemma 11.3.
Assume Case 2 occurs. Then one endpoint $q_{1}^{\prime\prime}$ of $L$ is in
the interior of $\Gamma_{1}$ and the other endpoint $q_{2}^{\prime\prime}$ of
$L$ is in the interior of $\Gamma_{2}.$ Without loss of generality, assume
$q_{1}^{\prime\prime},q_{1}^{\prime},q_{2}^{\prime}$ and
$q_{2}^{\prime\prime}$ are arranged in order on $L.$ Let
$\Gamma_{1}^{\prime}=\overline{q_{1}q_{1}^{\prime}},\Gamma^{\prime\prime}=\overline{q_{1}^{\prime}q_{2}^{\prime}}$
and $\Gamma_{2}^{\prime}=\overline{q_{2}^{\prime}q_{3}}$. Then, considering
that $T$ is on the left hand side of the convex triangle
$\overline{q_{1}q_{3}q_{2}q_{1}},$ we have that
$L(\Gamma_{1}^{\prime})<\pi,L(\Gamma^{\prime\prime})=\frac{\pi}{2},L(\Gamma_{2}^{\prime})<\pi,$
$L(\Gamma_{1}^{\prime}+\Gamma^{\prime\prime}+\Gamma_{2}^{\prime})<L(\Gamma_{1}+\Gamma_{2});$
and the domain $T$ enclosed by
$\Gamma_{1}^{\prime}+\Gamma^{\prime\prime}+\Gamma_{2}^{\prime}-\Gamma_{2}-\Gamma_{1}$
is a polygonal Jordan domain on the right hand side of $\Gamma_{1}+\Gamma_{2}$
with
$\overline{T}\cap E=\\{q_{1},q_{2}\\}.$
Then by Lemma 3.2 and the extension arguments, there exists a normal mapping
$g:\overline{\Delta}\rightarrow S$ such that
$L(g,\partial\Delta)<L(f,\partial\Delta)\text{{\ and
}}A(g,\Delta)>A(f,\Delta).$
and $\Gamma_{g}$ has a permitted partition
$\Gamma_{g}=\Gamma_{1}^{\prime}+\Gamma^{\prime\prime}+\Gamma_{2}^{\prime}+\Gamma_{3}+\dots+\Gamma_{n},$
which implies the following permitted partition
$\displaystyle\Gamma_{g}$ $\displaystyle=$
$\displaystyle\Gamma_{n}+\Gamma_{1}^{\prime}+\Gamma^{\prime\prime}+\Gamma_{2}^{\prime}+\Gamma_{3}+\dots+\Gamma_{n-1}$
$\displaystyle=$
$\displaystyle\overline{q_{n}q_{1}}+\overline{q_{1}q_{1}^{\prime}}+\overline{q_{1}^{\prime}q_{2}^{\prime}}+\overline{q_{2}^{\prime}q_{3}}+\overline{q_{2}q_{3}}+\dots+\overline{q_{n-1}q_{n}}.$
Since $q_{1}^{\prime},q_{2}^{\prime}\in E,$ it is clear that
$V_{NE}(g)\leq V_{NE}(f)-1,V_{E}(g)\geq V_{E}(f)+2\ \mathrm{and\ }V(g)\leq
V(f)+1.$
Now, there are four cases:
Case 2.1. None of
$\Gamma_{n}+\Gamma_{1}^{\prime}=\overline{q_{n}q_{1}q_{1}^{\prime}}$ and
$\Gamma^{\prime\prime}+\Gamma_{2}^{\prime}=\overline{q_{1}^{\prime}q_{2}^{\prime}q_{3}}$
is a natural edge of $\Gamma_{g}.$
Case 2.2. $\overline{q_{n}q_{1}q_{1}^{\prime}}\ $is a natural edge of
$\Gamma_{g},$ while $\overline{q_{1}^{\prime}q_{2}^{\prime}q_{3}}\ $is not.
Case 2.3. $\overline{q_{n}q_{1}q_{1}^{\prime}}$ is not a natural edge of
$\Gamma_{g},$ while $\overline{q_{1}^{\prime}q_{2}^{\prime}q_{3}}\ $is.
Case 2.4. Both $\overline{q_{n}q_{1}q_{1}^{\prime}}$ and
$\overline{q_{1}^{\prime}q_{2}^{\prime}q_{3}}$ are natural edges of
$\Gamma_{g}.$
The discussion for these cases is almost the same as that for the four Cases
1–4 in the proof of Lemma 11.3, just with a little difference which leads to
that the desired mapping may has a number of $V(f)+1$ natural edges. ∎
## 12\. Decomposition and deformation of Riemann surfaces of normal mappings
In this section, we prove the following theorem, which is the first key step
to prove the main theorem in Section 14.
###### Theorem 12.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and assume that
each natural edge of $f$ has spherical length strictly less than $\pi$. Then,
there exist a finite number of normal mappings
$f_{j}:\overline{\Delta}\rightarrow S,j=1,\dots m,$ with $m\geq 1,$ such that
$\sum_{j=1}^{m}L(f_{j},\partial\Delta)\leq
L(f,\partial\Delta),\sum_{j=1}^{m}A(f_{j},\Delta)\geq A(f,\Delta),$
and for each $j\leq m$ the followings hold.
(i) Each natural edge of $f_{j}$ has spherical length strictly less than
$\pi$.
(ii) The boundary curve $\Gamma_{f_{j}}=f_{j}(z),z\in\partial\Delta$, is
locally convex in $S\backslash E,$ where $E=\\{0,1,\infty\\}.$
(iii) $f_{j}$ has no branched point in $S\backslash E.$
We first prove several lemmas before we prove this theorem.
###### Lemma 12.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and assume that
each natural edge of $\Gamma_{f}=f(z),z\in\partial\Delta,$ has spherical
length strictly less than $\pi.$ Then $V(f)\geq 3,$ and if in addition
$V_{NE}(f)=0,$ then $V(f)\geq 4.$
###### Proof.
If $V(f)=1$, then $\Gamma_{f}$ itself is a natural edge that is a straight and
closed curve in $S$, and $L(f,\partial\Delta)<\pi.$ This is impossible.
Assume $V(f)=2$ and $\Gamma_{f}=\Gamma_{1}+\Gamma_{2}$ is a natural partition.
Since $\Gamma_{f}$ is a closed curve, $L(\Gamma_{j})<\pi$ and $\Gamma_{j}$ is
straight, $j=1,2$, we have $\Gamma_{1}=-\Gamma_{2}$ (ignoring a transformation
of parameter) with $L(\Gamma_{1})=L(\Gamma_{2})<\pi.$ Then,
$S\backslash\Gamma_{f}$ contains at least one point in $E=\\{0,1,\infty\\}.$
Considering that $f$ is normal, we conclude that $f(\Delta)\supset
S\backslash\Gamma_{f}$ contains at least one point of $E,$ which contradicts
the assumption that $f$ is normal. Thus, $V(f)\geq 3.$
If in addition $V_{NE}(f)=0,$ then by the assumption, each natural edge of
$\Gamma_{f}$ must be $\overline{0,1},\overline{1,0},\overline{1,\infty}$ or
$\overline{\infty,1},$ and then since $\Gamma_{f}$ is a closed curve and
$V(f)\geq 3$, we have $V(f)\geq 4.$ ∎
###### Lemma 12.2.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and assume that the
followings hold.
(a) each natural edge of $\Gamma_{f}=f(z),z\in\partial\Delta,$ has spherical
length strictly less than $\pi.$
(b) $V(f)=3.$
Then $f:\overline{\Delta}\rightarrow f(\overline{\Delta})$ is a homeomorphism
and $\Gamma_{f}$ is a generic convex triangle.
###### Proof.
Let
$\alpha=\alpha_{1}+\alpha_{2}+\alpha_{2}$
be a natural partition of $\partial\Delta$ for $f$ and let
$\Gamma_{f}=\overline{q_{1}q_{2}}+\overline{q_{2}q_{3}}+\overline{q_{3}q_{1}}$
be the corresponding natural partition of
$\Gamma_{f}=f(z),z\in\partial\Delta.$ Then by (a), $f$ restricted to each
$\alpha_{j}$ is a homeomorphism onto $\Gamma_{j}=\overline{q_{j}q_{j+1}},$
where $q_{4}=q_{1}.$
We first show that $\overline{q_{1}q_{2}q_{3}}$ can not be contained in any
great circle of $S.$ Otherwise, by (a) and the definition of natural edges,
either $q_{3}\in\overline{q_{1}q_{2}}^{\circ}$ or
$q_{1}\in\overline{q_{2}q_{3}}^{\circ},\ $where
$\overline{q_{1}q_{2}}^{\circ}$ denotes the interior of
$\overline{q_{1}q_{2}}.$ But in the first case, $q_{3}$ is not a natural
vertex of $\Gamma_{f}$ and in the second case, $q_{1}$ is not a natural vertex
of $\Gamma_{f}.$ Thus $\overline{q_{1}q_{2}q_{3}}$ is not contained in any
great circle of $S.$
Then $\Gamma_{f}$ must be a triangle that is contained in some open hemisphere
$S^{\prime}$ of $S$ and $f$ maps $\partial\Delta$ homeomorphically onto
$\Gamma_{f}\ $and then, since $f$ is normal,
$f:\overline{\Delta}\rightarrow\overline{T}$ is a homeomorphism, where $T$ is
the domain inside $\Gamma_{f}.$ Since $f$ is normal, we also have
$f(\Delta)\cap E=\emptyset.$ Thus, $\overline{T}\subset S^{\prime}$ and then
$\Gamma_{f}$ is a generic convex triangle. ∎
###### Lemma 12.3.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping such that each
natural edge of $f$ has spherical length strictly less than $\pi$. If
$V_{NE}(f)=0,\ $then $L(f,\partial\Delta)\geq 2\pi,$ and if $V_{NE}(f)=1,$
then $L(f,\partial\Delta)\geq\pi.$
###### Proof.
If $V_{NE}(f)=0,$ then by Lemma 12.1, $V(f)\geq 4,$ and in this case each
natural edge of $\Gamma_{f}=f(z),z\in\partial\Delta,$ has spherical length
$\frac{\pi}{2},$ and then $L(f,\partial\Delta)\geq 2\pi$. If $V_{NE}(f)=1,$
then by Lemma 12.1, $V(f)\geq 3,$ and then $f(\partial\Delta)$ contains at
least two point of $E;$ and since $\Gamma_{f}$ is closed, we have
$L(f,\partial\Delta)\geq\pi.$ ∎
###### Lemma 12.4.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping and let $p_{0}$ be
a ramification point of $f$. Assume $\beta=\overline{q_{0}q_{1}}\subset S$
satisfies the followings.
(a) $q_{0}=f(p_{0})$ and $q_{1}\in E=\\{0,1,\infty\\}$.
(b) The interior $\beta^{\circ}$ of $\beta$ has a neighborhood $N$ in $S$ such
that $N$ is a polygonal Jordan domain and $f$ has no branched point in $N$.
(c) The boundary curve $\Gamma_{f}=\Gamma_{f}(z),z\in\partial\Delta,$ has no
natural vertex in $N.$
(d) Either
(12.1) $f(\partial\Delta)\cap N=\emptyset,$
or
(12.2) $p_{0}\in\partial\Delta\ \mathrm{and}\ f(\partial\Delta)\cap
N=\beta^{\circ}.$
Then there exist normal mappings $g_{1},g_{2}:\overline{\Delta}\rightarrow S,$
such that the followings hold.
(i) Each natural edge of $\Gamma g_{j}$ is a natural edge of
$\Gamma_{f},j=1,2,$ and each natural edge of $\Gamma_{f}$ is a natural edge of
either $g_{1}$ or $g_{2}$.
(ii) $L(g_{1},\Delta)+L(g_{2},\Delta)=L(f,\Delta)$ and
$A(g_{1},\Delta)+A(g_{2},\Delta)=A(f,\Delta).$
(iii)
$V_{NE}(g_{1})+V_{NE}(g_{2})=V_{NE}(f),V_{E}(g_{1})+V_{E}(g_{2})=V_{E}(f),$
$V(g_{1})+V(g_{2})=V(f).$
###### Proof.
By Lemma 3.3 or Corollary 3.1, there exist a point $q_{2}$ in $\beta^{\circ}$
such that the section $\overline{q_{0}q_{2}}$ of $\beta=\overline{q_{0}q_{1}}$
has a lift $\gamma\subset\overline{\Delta}$ from $p_{0}$ to some point
$p_{2}\in\Delta$ with $\gamma\backslash\\{p_{0}\\}\subset\Delta.$ Let
$q^{\ast}\in\beta$ be the closest point to $q_{1}$ in $\beta$ such that the
section $\overline{q_{0}q^{\ast}}$ has a lift $\alpha_{2}$ that is an
extension of the lift $\gamma$ and that $\alpha_{2}^{\circ}\subset\Delta.$ We
show that $q^{\ast}=q_{1}.$
Let $p^{\ast}$ be the terminal point of $\alpha_{2}$. Assume $q^{\ast}\neq
q_{1},$ i.e. $q^{\ast}\in\beta^{\circ}.$ If $p^{\ast}\in\Delta,$ then by (b)
and Lemma 3.3, $\alpha_{2}$ can be extended past $p^{\ast}$ to be a longer
lift so that the extended part is still in $\Delta$, which contradicts the
definition of $p^{\ast}$ and $q^{\ast}.$ Thus, we have
$p^{\ast}\in\partial\Delta.$
Then by (c) and the definition of natural vertices there is a neighborhood
$A_{p^{\ast}}$ of $p^{\ast}$ in $\partial\Delta,$ such that $f$ restricted to
$A_{p^{\ast}}\ $is a homeomorphism onto a section of $\beta^{\circ}.$ On the
other hand, by (b), (c) and Lemma 3.3, there is a neighborhood $U_{p^{\ast}}$
of $p^{\ast}$ in $\overline{\Delta}$ such that $f$ restricted to
$U_{p^{\ast}}$ is a homeomorphism onto $f(U_{p^{\ast}})$ with
$f(U_{p^{\ast}})\subset N$ and $f(U_{p^{\ast}})$ is a half-disc whose boundary
diameter is contained in $\beta^{\circ}\cap f(U_{p^{\ast}})$. Thus, by (d),
$U_{p^{\ast}}\cap\partial\Delta=U_{p^{\ast}}\cap f^{-1}([0,+\infty]),$ and
then $\alpha_{2}\cap U_{p^{\ast}}\subset U_{p^{\ast}}\cap\partial\Delta.$ This
is a contradiction, since $\alpha_{2}^{\circ}\subset\Delta.$ Thus we have
proved that $\alpha_{2}$ is a lift of the whole path $\beta$ with
$\alpha_{2}^{\circ}\subset\Delta.$
Since $p_{0}$ is a ramification point, in case $p_{0}\in\Delta$, by Lemma 3.3,
$\beta$ has another lift $\alpha_{1}$ starting from $p_{0}$ such that
$\alpha_{1}^{\circ}\subset\Delta.$ Since $f(\Delta)\cap E=\emptyset\ $and the
terminal point $q_{1}$ of $\beta$ is in $E,$ the terminal points of
$\alpha_{1}$ and $\alpha_{2}$ must land on $\partial\Delta,$ and by Lemma 3.5,
these terminal points are distinct each other. Thus, $f,\alpha_{1},\alpha_{2}$
and $\beta$ satisfy all assumptions of Corollary 10.1, and then the desired
$g_{1}$ and $g_{2}$ follow.
In case (12.2), by (c) there is a section $\alpha_{1}$ of $\partial\Delta$
starting from $p_{0}$ so that $\alpha_{1}$ is a lift of $\beta,$ and by Lemma
3.5, the terminal points of $\alpha_{1}$ and $\alpha_{2}$ are also distinct.
Then, $f,\alpha_{1},\alpha_{2}$ and $\beta$ satisfy all assumptions of
Corollary 10.2, and then the desired $g_{1}$ and $g_{2}$ follow as well. This
completes the proof of the lemma. ∎
Now, we can prove Theorem 12.1 in some special cases.
###### Lemma 12.5.
Theorem 12.1 holds true if $V_{NE}(f)=0$ or $V_{NE}(f)=1.$
###### Proof.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping that satisfies the
assumption of Theorem 12.1 and $V_{NE}(f)=0$ or $V_{NE}(f)=1.$
If $\Gamma_{f}$ is locally convex in $S\backslash E$ and $f$ has no branched
point in $S\backslash E,$ then $f$ itself satisfies the conclusion of Theorem
12.1, and there is nothing to proof.
If $\Gamma_{f}$ is not locally convex in $S\backslash E,$ then $V_{NE}(f)=1$
and by Theorem 11.1, there exists a normal mapping
$g:\overline{\Delta}\rightarrow S,$ such that each natural edge of $g$ has
spherical length strictly less than $\pi,$
$L(g,\partial\Delta)\leq L(f,\partial\Delta),A(g,\Delta)\geq A(f,\Delta),$
and
$V_{NE}(g)\leq V_{NE}(f)-1=0.$
and then $V_{NE}(g)=0,$ and in this case $\Gamma_{g}$ is locally convex in
$S\backslash E.$ If Theorem 12.1 holds for $g,$ then it is clear that Theorem
12.1 holds for $f.$ Thus we may assume that
(a) $\Gamma_{f}$ is locally convex in $S\backslash E.$
If $f$ has no branched point in $S\backslash E,$ then there is nothing to
prove again. Thus, we may complete the proof under the assumption that
(b) $\Gamma_{f}$ is locally convex in $S\backslash E$ and $f$ has a branched
point in $S\backslash E.$
Let $p_{0}\in\overline{\Delta}\backslash f^{-1}(E)$ be a ramification point of
$f$ and let $q_{0}=f(p_{0}).$ Since $V_{NE}(f)=0$ or $1,$ $\Gamma_{f}$ has at
most one natural vertex $q^{\ast}$ outside $E.$ Then by (a), there is a
shortest path $\beta=\overline{q_{0}q_{1}}$ from $q_{0}$ to some point
$q_{1}\in E$ such that either $\beta\cap f(\partial\Delta)=\\{q_{1}\\}$ or
$\beta\cap f(\partial\Delta)=\overline{q_{0}q_{1}}.$ The later case occurs if
and only if $q_{0}\in\Gamma_{f}\backslash E.$
We may assume $f$ has no branched point in $\beta\backslash\\{q_{0},q_{1}\\},$
otherwise we take the branched point in the interior of $\beta$ that is
closest to $q_{1}.$ Then $\beta^{\circ}$ has a neighborhood $N$ satisfying the
hypothesis of Lemma 12.4, and then, by Lemma 12.4, there exist two normal
mappings $f_{1},f_{2}:\overline{\Delta}\rightarrow S,$ satisfying the
following two conditions.
(c) Each natural edge of $f_{j}$ is a natural edge of $f,$ $j=1,2$, and each
natural edge of $\Gamma_{f}$ is a natural edge of $f_{1}$ or $f_{2}.$
(d) $L(f,\partial\Delta)=L(f_{1},\partial\Delta)+L(f_{2},\partial\Delta)$ and
$A(f,\Delta)=A(f_{1},\Delta)+A(f_{2},\Delta).$
By (c), we have
(12.3) $n=V(f)=V(f_{1})+V(f_{2}).$
It is clear by (c) that if $V_{NE}(f)=0,$ then
$V_{NE}(f_{1})=V_{NE}(f_{2})=0,$ and if $V_{NE}(f)=1,$ then the the unique
natural vertex $q^{\ast}$ of $\Gamma_{f}$ outside $E$ can not be contained in
both $\Gamma_{f_{1}}$ and $\Gamma_{f_{2}},$ but $q^{\ast}$ must be a convex
natural vertex of $\Gamma_{f_{1}}$ or $\Gamma_{f_{2}}$ and both $f_{1}$ and
$f_{2}$ satisfy the assumption of Theorem 12.1. Summarizing, we may assume
(e) $V_{NE}(f_{1})=0,$ $V_{NE}(f_{2})=1,$ and $f_{1}$ and $f_{2}$ satisfy (a).
On the other hand, by Lemma 12.1 and (e) we have
(12.4) $V(f_{1})\geq 4\ \mathrm{and\ }V(f_{2})\geq 3.$
Thus, we have
$n=V(f)\geq 7,$
and by (12.3) we have
(12.5) $V(f_{1})\leq V(f)-3\ \mathrm{and\ }V(f_{2})\leq V(f)-3.$
We have in fact proved that under the assumption (b), $n\geq 7.$ Thus, Theorem
12.1 holds true in case (a) with $n\leq 6.$ From this and the above arguments
for the existence of $f_{1}$ and $f_{2}$ satisfying (c), (d), (e) and (12.5)
we can prove the theorem, under the assumption (a), by induction on $n=V(f)$.
This completes the proof. ∎
###### Proof of Theorem 12.1.
We prove Theorem 12.1 by induction on the sum $V_{NE}(f)+V(f).$
By Lemma 12.1 we have $V(f)\geq 3,$ and then $V_{NE}+V(f)=3$ holds only in the
case $V_{NE}=0,$ but by Lemma 12.1, $V_{NE}=0$ implies $V(f)\geq 4.$ Thus
$V_{NE}(f)+V(f)\geq 4,$
and equality holds if and only if $V_{NE}(f)=0$ and $V(f)=4$, or $V_{NE}(f)=1$
and $V(f)=3.$ Thus, by Lemma 12.5, Theorem 12.1 holds true in the case
$V_{NE}(f)+V(f)=4.$
Now, let $k>4$ be a positive integer and assume that we have proved Theorem
12.1 for the case $4\leq V_{NE}(f)+V(f)\leq k.$ Let $f$ be any normal mapping
that satisfies the assumption of Theorem 12.1 with
(12.6) $V_{NE}(f)+V(f)=k+1.$
We call this that $f$ is at the level $k+1,$ and will show that Theorem 12.1
holds true for $f.$
Then, there are only three cases need to be discussed.
Case 1. The boundary curve $\Gamma_{f}=f(z),z\in\partial\Delta$, is locally
convex in $S\backslash E$ and $f$ has no branched point in $S\backslash E.$
Case 2. $\Gamma_{f}$ is not convex at some natural vertex
$p_{1}\in\left(\partial\Delta\right)\backslash f^{-1}(E)$ of $f.$
Case 3. $\Gamma_{f}$ is locally convex in $S\backslash E,$ and $f$ has a
branched point in $S\backslash E.$
If Case 1 occurs, then $f$ itself satisfies the conclusion of Theorem 12.1,
and then there is nothing to proof.
Discussion of Case 2. In this case, by Theorem 11.1 it is clear that there
exists a normal mapping $f_{1}:\overline{\Delta}\rightarrow S,$ such that each
natural edge of $f_{1}$ has spherical length strictly less than $\pi,$
(12.7) $L(f_{1},\partial\Delta)\leq L(f,\partial\Delta),A(f_{1},\Delta)\geq
A(f,\Delta),$
and
(12.8) $V_{NE}(f_{1})\leq V_{NE}(f)-1\text{{\ and\ }}V(f_{1})\leq V(f)+1.$
If one of the equalities of (12.8) fails, then by (12.6), $f_{1}$ is at the
level of $k,$ and by the induction hypothesis, Theorem 12.1 holds for $f_{1}$
and by (12.7), Theorem 12.1 holds for $f.$ If both of the equalities in (12.8)
hold true, then $f_{1}$ is still at the level of $k+1,$ but
(12.9) $V_{NE}(f_{1})=V_{NE}(f)-1.$
Then, $f_{1}$ satisfies the assumption of Theorem 12.1, and then we return to
Cases 1, 2, or 3. If Case 1 occurs for $f_{1},$ then Theorem 12.1 holds for
$f_{1},$ and then Theorem 12.1 holds for $f$ by (12.7). If Case 2 occurs, then
we can replace $f$ by $f_{1}$ and repeat the discussion of Case 2. By (12.9),
we can not always return to Case 2 from Case 2. Thus, Repeating discussion for
Case 2 finitely many times, we return to either Case 1 or Case 3. If we return
to Case 1, the proof is completed, and if we return to Case 3, we continue the
following discussion.
Discussion of Case 3. By Theorem 10.1, there exist two normal mappings
$g_{j}:\overline{\Delta}\rightarrow S,j=1,2,$ such that the followings hold.
(a1) Each natural edge of $g_{j}$ has spherical length strictly less than
$\pi,j=1,2.$
(a2) $\sum_{j=1}^{2}L(g_{j},\partial\Delta)\leq
L(f,\partial\Delta),\sum_{j=1}^{2}A(g_{j},\Delta)\geq A(f,\Delta)$.
(a3) $V_{NE}(g_{1})+V_{NE}(g_{2})\leq V_{NE}(f)+2.$
(a4) $V(g_{1})+V(g_{2})\leq V(f)+2.$
By Lemma 12.1 and (a1), $V(g_{j})\geq 3,j=1,2,$ and so by (a4) we have
(a5) $V(g_{j})\leq V(f)-1,j=1,2.$
Then there are only two cases:
Case 2.1 $V_{NE}(g_{j})\geq 2,j=1,2.$
Case 2.2 $V_{NE}(g_{1})\leq V_{NE}(g_{2})$ and $V_{NE}(g_{1})=0$ or $1.$
Discussion of Case 2.1. In this case, by (a3) we have $V_{NE}(g_{j})\leq
V_{NE}(f),$ and then by (a5) and (12.6) we have
$V_{NE}(g_{j})+V(g_{j})\leq V_{NE}(f)+V(f)-1=k,j=1,2,$
i.e. both $g_{1}$ and $g_{2}$ are at level $\leq k.$ Then by (a1) and the
induction hypothesis, Theorem 12.1 holds for $g_{1}$ and $g_{2},$ and then by
(a2) and the induction hypothesis, Theorem 12.1 holds for $f.$
Discussion of Case 2.2. Assume
(12.10) $V_{NE}(g_{1})=0\ \mathrm{or\ }1.$
Then by Lemma 12.3
(12.11) $L(g_{1},\partial\Delta)\geq\pi.$
We first show that
(12.12) $V_{NE}(g_{2})+V(g_{2})\leq V_{NE}(f)+V(f)\leq k+1.$
If $V_{NE}(g_{1})=0,$ then by Lemma 12.1 we have $V(g_{1})\geq 4,$ and then by
(a3) and (a4) we have (12.12). If $V_{NE}(g_{1})=1,$ then by (a3) and (a5) we
still have (12.12)
If $V_{NE}(g_{2})+V(g_{2})<k+1,$ then Theorem 12.1 holds for $g_{2}$ by the
induction hypothesis; and on the other hand, by (12.10) and (a1), Theorem 12.1
holds for $g_{1}$; and therefore, by (a2) Theorem 12.1 holds for $f$.
Now, we assume
$V_{NE}(g_{2})+V(g_{2})=k+1,$
i.e., $g_{2}$ is in the level $k+1.$ Then we can return to Cases 1–3 and
repeat the same discussion for $g_{2}.$ We can prove Theorem 12.1 by repeating
the above arguments finitely many times, since by (12.11), Case 2.2 can not
occur infinitely times. This completes the proof. ∎
## 13\. Decomposition of fat mappings
In this section we prove Theorem 13.1. This is the third key step to prove the
main theorem in Section 14.
A normal mapping $g:\overline{\Delta}\rightarrow S$ is called _fat_ if and
only if $\Delta\backslash f^{-1}([0,+\infty])$ has a component $D$ such that
$f:D\rightarrow S\backslash[0,+\infty]$ is a homeomorphism.
By Corollary 7.1, if $g$ satisfies all assumptions of Theorem 7.1, then $g$ is
fat if and only if
$f(\partial D)\subset[0,+\infty].$
###### Theorem 13.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping that satisfies
(a)–(d) of Theorem 7.1, that is, the following conditions (a)–(d) hold.
(a) Each natural edge of the boundary curve
$\Gamma_{f}=f(z),z\in\partial\Delta,$ has length strictly less than $\pi$.
(b) $\Gamma_{f}=f(z),z\in\partial\Delta,$ is locally convex in $S\backslash
E,E=\\{0,1,\infty\\}.$
(c) $f$ has no branched point in $S\backslash E.$
(d) $\Gamma_{f}\cap[0,+\infty]$ contains at most finitely many points.
If $f$ is fat, then there exist two normal mappings
$f_{j}:\overline{\Delta}\rightarrow S,j=1,2,$ such that $f_{1}$ and $f_{2}$
satisfy (a)–(d) and
$A(f_{1},\Delta)+A(f_{2},\Delta)=A(f,\Delta)-4\pi,$
$L(f_{1},\partial\Delta)+L(f_{2},\partial\Delta)=L(f,\partial\Delta),$
$f_{1}$ maps $[-1,1]\subset\overline{\Delta}$ homeomorphically onto
$\overline{0,1}\subset S$ and $f_{2}$ maps $[-1,1]$ homeomorphically onto
$\overline{1,\infty}\subset S.$
The geometrical meaning of this theorem is that we can cut off the whole
Riemann sphere $S,$ with $[0,+\infty]$ being removed, from the interior of the
Riemann surface of $f$ and then sew up the cut edges along $[0,+\infty].$ Then
we obtain two Riemann surfaces that are only joint at $1\in S,$ and the
boundary curve of these two surfaces compose the boundary curve $\Gamma_{f}.$
###### Proof.
Let $\Delta_{1}$ be a component of $\Delta\backslash f^{-1}([0,+\infty])$ such
that $f(\partial\Delta_{1})\subset[0,+\infty].$
Then by Corollary 7.1,
$f|_{\overline{\Delta_{1}}}:\overline{\Delta_{1}}\rightarrow S$ is normal and
surjective, $f(\partial\Delta_{1})=[0,+\infty]$ and $f$ restricted to
$\Delta_{1}$ is a homeomorphism onto $S\backslash[0,+\infty]$. Then the
restriction
$f:\partial\Delta_{1}\rightarrow[0,+\infty]$
is a folded two to one mapping and we can express $\partial\Delta_{1}$ to be
$\partial\Delta_{1}=\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}$
such that $f$ maps $\beta_{1},\beta_{2},\beta_{3},\beta_{4}$ homeomorphically
onto $\overline{0,1},\overline{1,\infty},\overline{\infty,1},$
$\overline{1,0},$ respectively and $\beta_{1},\beta_{2},\beta_{3},\beta_{4}$
are arranged anticlockwise in $\partial\Delta_{1}.$ Denote by $p_{j}\ $the
initial points of $\beta_{j},j=1,2,3,4.$ Then
$f(p_{1})=0,f(p_{2})=f(p_{4})=1,f(p_{3})=\infty,$
which implies
$p_{j}\in\partial\Delta,j=1,2,3,4,$
by the definition of normal mappings. We denote by $\alpha_{j}$ the section of
$\partial\Delta$ from $p_{j}$ to $p_{j+1},j=1,2,3,4$ ($p_{5}=p_{1}).$
We first show that the interior of $\beta_{j}$ is contained in $\Delta$ for
$j=1,2,3,4.$ Assume $\beta_{1}$ has an interior point $p_{0}$ with
$p_{0}\in\partial\Delta.$ Then it is clear that $p_{0}$ is in the interior of
$\alpha_{1}$ and $p_{0}\neq 0,1,\infty,$ and then, by (b) the section
$\Gamma_{\alpha_{1}}=f(z),z\in\alpha_{1},$ of $\Gamma_{f}$ is convex at
$p_{0},$ and by (c), $f$ is regular at $p_{0}.$ On the other hand,
$\Gamma_{\beta_{1}}=f(z),z\in-\beta_{1},$ which is the simple path
$\overline{1,0},$ is obviously convex by the definition. Therefore, Lemma 6.3
applies to $p_{0},\alpha_{1},-\beta_{1}$ and $f$, and then $\alpha_{1}$ has a
neighborhood contained in $\beta_{1}.$ This contradicts (d), for
$\alpha_{1}\subset\partial\Delta$ and
$\beta_{1}=\overline{0,1}\subset[0,+\infty].$ Thus the interior of $\beta_{1}$
is contained in $\Delta.$ For the same reason, the interiors of
$\beta_{2},\beta_{3}$ and $\beta_{4}$ are all contained in $\Delta$ .
We have proved that $\Delta\backslash\Delta_{1}$ contains four disjoint Jordan
domains $D_{j}$ enclosed by $\alpha_{j}-\beta_{j}$ with
$\overline{D_{j}}\cap\overline{D_{j+1}}=\\{p_{j+1}\\},j=1,2,3,4$
($D_{5}=D_{1}$ and $p_{5}=p_{1}).$
Now, we glue $\overline{D_{1}}$ and $\overline{D_{4}}$ along $\beta_{1}$ and
$-\beta_{4}$ so that $x\in\beta_{1}$ and $y\in-\beta_{4}$ are identified if
and only if $f(x)=f(y).$ The glued closed domain can be understood to be the
unit disk $\overline{\Delta}$ so that $\beta_{1}$ and $-\beta_{4}$ both become
the diameter $[-1,1]$ of $\overline{\Delta}.$ In this way we have in fact
glued the restrictions $f|_{\overline{D_{1}}}$ and $f|_{\overline{D_{4}}}$ to
be a normal mapping $f_{1}:\overline{\Delta}\rightarrow S$ such that $f_{1}$
maps $[-1,1]\subset\overline{\Delta}$ homeomorphically onto
$\overline{0,1}\subset S.$
Similarly, we can glue the restrictions $f|_{\overline{D_{2}}}$ and
$f|_{\overline{D_{3}}}$ to be a normal mapping
$f_{2}:\overline{\Delta}\rightarrow S$ such that $f_{2}$ maps $[-1,1]$
homeomorphically onto $\overline{1,\infty}.$
It is clear that $f_{1}$ and $f_{2}$ satisfies all the conclusions of the
Theorem. As a matter of fact, the above process just cut off $f(\Delta_{1}),$
the sphere $S$ with $[0,+\infty]$ being removed, from the interior of the
Riemann surface of $f$ and then sew up the cut edges along $[0,+\infty].$ Then
we obtain two Riemann surfaces that are only joint at $1\in S,$ and the
boundary curve of these two surfaces compose the boundary curve $\Gamma_{f}.$
This completes the proof. ∎
## 14\. Proof of the main theorem
We first prove the main theorem under certain conditions.
###### Lemma 14.1.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping that is not fat and
satisfies (a)–(d) of Theorem 7.1. Then
$A(f,\Delta)<2L(f,\partial\Delta),$
and
$A(f,\Delta)<h_{0}L(f,\partial\Delta)-A(f,\Delta),$
where $h_{0}$ is given by (1.3).
###### Proof.
The second inequality follows from the first one directly, for $h_{0}>4.$ So
we only prove the first inequality.
Since $f$ is not fat, for each component $D$ of $\Delta\backslash
f^{-1}([0,+\infty]),$ $f(\partial D)\backslash[0,+\infty]\neq\emptyset,$ and
so by (d) the interior $\alpha_{0}$ of $\left(\partial
D\right)\cap(\partial\Delta)$ is not empty, and then by Theorem 7.1,
$L(f,\left(\partial D\right)\cap(\partial\Delta))>L(f,(\partial
D)\backslash\left(\partial\Delta\right)),$
$f(\overline{D})$ is contained in some hemisphere of $S,$ which implies
$A(f(D))<2\pi,$
and $f$ restricted to $D$ is a homeomorphism.
Then we have
(14.1) $2L(f,\left(\partial D\right)\cap(\partial\Delta))>L(f,\left(\partial
D\right)\cap(\partial\Delta))+L(f,(\partial
D)\backslash\left(\partial\Delta\right))=L(f,\partial D).$
If $L(f,\partial D)\geq 2\pi,$ then we have
$A(f,D)=A(f(D))<2\pi\leq L(f,\partial D),$
and if $L(f,\partial D)<2\pi,$ then by Theorem 4.3 we have
$A(f,D)<L(f,\partial D),$
and then by (14.1), in both cases we have
(14.2) $A(f,D)<2L(f,\left(\partial D\right)\cap(\partial\Delta)).$
By Theorem 7.1 it is clear that for any pair $D_{1}$ and $D_{2}$ of distinct
components of $\Delta\backslash f^{-1}([0,+\infty]),$ $\left(\partial
D_{1}\right)\cap(\partial\Delta)$ and $\left(\partial
D_{2}\right)\cap(\partial\Delta)$ contains at most two common points, and on
the other hand, by the same theorem, $\Delta\backslash f^{-1}([0,+\infty])$
contains only finitely many components. Thus, we have
$L(f,\partial\Delta)=\sum_{D}L(f,\left(\partial
D\right)\cap(\partial\Delta)),$
where the sum runs over all components $D$ of $\Delta\backslash
f^{-1}([0,+\infty])$. Then, summing up (14.2), we have
$A(f,\Delta)=\sum_{D}A(f,D)<2\sum_{D}L(f,(\partial
D)\cap(\partial\Delta))=2L(f,\partial\Delta).$
This completes the proof. ∎
###### Lemma 14.2.
Let $f:\overline{\Delta}\rightarrow S$ be a normal mapping that is fat and
satisfies (a)–(d) of Theorem 7.1. Then
$A(f,\Delta)\leq h_{0}L(f,\partial\Delta)-4\pi.$
where $h_{0}$ is given by (1.3).
###### Proof.
By Theorem 13.1, there exist normal mappings
$g_{j}:\overline{\Delta}\rightarrow S,j=1,2,\dots,n+1,$ such that for each
$j\leq n+1$ the followings hold.
(e) Each $g_{j}$ satisfies all assumptions (a)–(d) of theorem 7.1,
$j=1,2,\dots,n+1$.
(f) Each $g_{j}$ is not fat, $j=1,2,\dots,n+1$.
(g) $g_{j}$ maps the diameter $[-1,1]$ of $\overline{\Delta}$ homeomorphically
onto the real interval $\overline{0,1}$ in $S$ or onto $\overline{1,\infty}$
in $S.$
(h)
$A(f,\Delta)=4n\pi+\sum_{j=1}^{n+1}A(g_{j},\Delta),L(f,\partial\Delta)=\sum_{j=1}^{n+1}L(g_{j},\partial\Delta).$
Let $j$ be any positive integer with $j\leq n+1$ and consider the mapping
$g_{j}.$ By (e) and (f), Lemma 14.1 applies. Then we have
(14.3) $A(g_{j},\Delta)<2L(g_{j},\partial\Delta),$
and then
$\displaystyle 4\pi+A(g_{j},\Delta)$ $\displaystyle<$ $\displaystyle
2L(g_{j},\partial\Delta)+4\pi$ $\displaystyle=$
$\displaystyle(2+\frac{4\pi}{L(g_{j},\partial\Delta)})L(g_{j},\partial\Delta),$
and, considering that $2+\frac{4\pi}{L(g_{j},\partial\Delta)}\leq 4$ in the
case $L(g_{j},\partial\Delta)\geq 2\pi,$ we have
(k) If $L(g_{j},\partial\Delta)\geq 2\pi,$ then
$4\pi+A(g_{j},\Delta)<4L(g_{j},\partial\Delta).$
By Theorem 4.3, we have
(l) If $\sqrt{2}\pi\leq L(g_{j},\partial\Delta)<2\pi,$ then
$4\pi+A(g_{j},\Delta)<4L(g_{j},\partial\Delta).$
By (g) and Theorem 4.4, we have
(m) If $L(g_{j},\partial\Delta)<\sqrt{2}\pi,$ then
$4\pi+A(g_{j},\Delta)\leq h_{0}L(g_{j},\partial\Delta),$
where $h_{0}$ is given by (1.3).
Summarizing (k)–(m) and the fact that $h_{0}>4$, we have in any case
$4\pi+A(g_{j},\Delta)\leq h_{0}L(g_{j},\partial\Delta),j=1,\dots,n+1,$
and then by (h), we have
$\displaystyle A(f,\Delta)$ $\displaystyle=$ $\displaystyle
4n\pi+\sum_{j=1}^{n+1}A(g_{j},\Delta)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n+1}\left(4\pi+A(g_{j},\Delta)\right)-4\pi$
$\displaystyle\leq$ $\displaystyle
h_{0}\sum_{j=1}^{n+1}L(g_{j},\partial\Delta)-4\pi$ $\displaystyle=$
$\displaystyle h_{0}L(f,\partial\Delta)-4\pi.$
This completes the proof. ∎
###### Proof of the Main Theorem.
Let $f:\overline{\Delta}\rightarrow S$ be any nonconstant holomorphic mapping
such that $f(\Delta)\cap E=\emptyset.$ Then for any positive number
(14.4) $\varepsilon<\frac{1}{4h_{0}}\min\\{4\pi,A(f,\Delta)\\},$
there exists a Jordan domain $D\subset\Delta$ such that $f$ restricted to $D$
is a normal mapping,
(14.5) $A(f,D)\geq A(f,\Delta)-\varepsilon\mathrm{\ and\ }L(f,\partial
D)<L(f,\Delta)+\varepsilon,$
and the following condition holds.
(1) Each natural edge of the restricted mapping $f|_{\overline{D}}$ has
spherical length strictly less than $\pi$.
Let $h$ be a homeomorphism from $\overline{\Delta}$ onto $\overline{D}$ and
let $F=f\circ h.$ Then by (1) $F:\overline{\Delta}\rightarrow S$ is a normal
mapping satisfying the assumption of Theorem 12.1 with $A(F,\Delta)=A(f,D)\
$and $L(F,\partial\Delta)=L(f,\partial D).$ Then by (14.5) we have that
(14.6) $A(F,\Delta)\geq A(f,\Delta)-\varepsilon\ \mathrm{and}\
L(F,\partial\Delta)<L(f,\Delta)+\varepsilon;$
and by Theorem 12.1 there exist a number of $m$ normal mappings
$f_{j}:\overline{\Delta}\rightarrow S,j=1,\dots m,$ such that
(14.7) $\sum_{j=1}^{m}A(f_{j},\Delta)\geq A(F,\Delta)\ \mathrm{and\
}\sum_{j=1}^{m}L(f_{j},\partial\Delta)\leq L(F,\partial\Delta),$
and for each $j$ the followings hold.
(A) Each natural edge of the boundary curve
$\Gamma_{f_{j}}=f_{j}(z),z\in\partial\Delta$, has spherical length strictly
less than $\pi$.
(B) $\Gamma_{f_{j}}=f_{j}(z),z\in\partial\Delta$, is locally convex in
$S\backslash E.$
(C) $f_{j}$ has no branched point in $S\backslash E.$
Then, by Lemma 7.2, for the above $\varepsilon$ and each $j\leq m,$ there
exists a normal mapping $g_{j}:\overline{\Delta}\rightarrow S$ such that
(14.8) $A(g_{j},\Delta)\geq
A(f_{j},\Delta),L(g_{j},\partial\Delta)<L(f_{j},\partial\Delta)+\frac{\varepsilon}{m},$
and $g_{j}$ satisfies (d) in Theorem 7.1 and (A)–(C), say, $g_{j}$ satisfies
all hypotheses of Theorem 7.1. Then by Lemmas 14.1 and 14.2 we have
(14.9) $A(g_{j},\Delta)\leq
h_{0}L(g_{j},\partial\Delta)-\min\\{A(g_{j},\Delta),4\pi\\},j=1,\dots,m.$
On the other hand, if for some $j_{0}\leq m,$ $A(g_{j_{0}},\Delta)\geq 4\pi,$
then we have
$\sum_{j=1}^{m}\min\\{A(g_{j},\Delta),4\pi\\}\geq 4\pi,$
and if $A(g_{j},\Delta)<4\pi$ for all $j\leq m,$ then we have, by (14.8),
(14.7), (14.6) and (14.4), that
$\displaystyle\sum_{j=1}^{m}\min\\{A(g_{j},\Delta),4\pi\\}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}A(g_{j},\Delta)\geq\sum_{j=1}^{m}A(f_{j},\Delta)\geq
A(F,\Delta)$ $\displaystyle>$ $\displaystyle
A(f,\Delta)-\varepsilon>\frac{4h_{0}-1}{4h_{0}}A(f,\Delta);$
and thus, in both cases, we have
(14.10)
$\sum_{j=1}^{m}\min\\{A(g_{j},\Delta),4\pi\\}\geq\min\\{\frac{4h_{0}-1}{4h_{0}}A(f,\Delta),4\pi\\}.$
Summing up the inequalities of (14.9), by (14.10) we have
$\displaystyle\sum_{j=1}^{m}A(g_{j},\Delta)$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{m}h_{0}L(g_{j},\partial\Delta)-\sum_{j=1}^{m}\min\\{A(g_{j},\Delta),4\pi\\}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{m}h_{0}L(g_{j},\partial\Delta)-\min\\{\frac{4h_{0}-1}{4h_{0}}A(f,\Delta),4\pi\\}.$
By (14.8), (14.7) and (14.6) we have
$\displaystyle\sum_{j=1}^{m}h_{0}L(g_{j},\partial\Delta)$ $\displaystyle<$
$\displaystyle\sum_{j=1}^{m}h_{0}L(f_{j},\partial\Delta)+\varepsilon h_{0}\leq
h_{0}L(F,\partial\Delta)+\varepsilon h_{0}$ $\displaystyle<$ $\displaystyle
h_{0}L(f,\Delta)+2\varepsilon h_{0},$
i.e.
(14.12)
$\sum_{j=1}^{m}h_{0}L(g_{j},\partial\Delta)<h_{0}L(f,\Delta)+2\varepsilon
h_{0}.$
By (14.6)–(14.8) we have
(14.13) $A(f,\Delta)\leq\sum_{j=1}^{m}A(g_{j},\Delta)+\varepsilon.$
Therefore, we have by (14)–(14.13)
$\displaystyle A(f,\Delta)$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{m}A(g_{j},\Delta)+\varepsilon$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{m}h_{0}L(g_{j},\partial\Delta)-\min\\{\frac{4h_{0}-1}{4h_{0}}A(f,\Delta),4\pi\\}$
$\displaystyle<$ $\displaystyle h_{0}L(f,\Delta)+2\varepsilon
h_{0}+\varepsilon-\min\\{\frac{4h_{0}-1}{4h_{0}}A(f,\Delta),4\pi\\}$
$\displaystyle\leq$ $\displaystyle
h_{0}L(f,\Delta)+\frac{2h_{0}+1}{4h_{0}}\min\\{A(f,\Delta),4\pi\\}-\min\\{\frac{4h_{0}-1}{4h_{0}}A(f,\Delta),4\pi\\},$
and considering that $h_{0}>4,$ we have
$A(f,\Delta)<h_{0}L(f,\Delta).$
It remains to show that the lower is sharp.
We give an example to show that $h_{0}$ given by (1.3) is a sharp lower bound
of the constant $h$ in (1.1).
As in Section 1, we denote by $D$ the spherical disk in $S$ with diameter
$\overline{1,\infty},$ the shortest path in $S$ from $1$ to $\infty,$ and for
$l\in[\pi,\sqrt{2}\pi]$ denote by $D_{l}$ the domain contained in the disk $D$
such that the boundary $\partial D_{l}$ is composed of two congruent circular
arcs, each of which has endpoints $\\{1,\infty\\}$ and spherical length
$\frac{l}{2}$. Then, $l=L(\partial D_{l})$ and by (1.4) and (1.5), the number
$h_{0}$ given by (1.3) is the maximum value of the function
$\frac{4\pi+A(D_{l})}{l}$ and
$h_{0}=\frac{4\pi+A(D_{l_{0}})}{l_{0}}$
for some $l_{0}\in(\pi,\sqrt{2}\pi)$.
It is clear that $D_{l_{0}},$ regarded as a domain in $\mathbb{C},$ is an
angular domain whose vertex is $1$ and bisector is the ray $[1,+\infty)$ in
$\mathbb{C}$. We denote by $2\theta_{0}$ the value of the angle of this
angular domain. Then it is clear that $\theta_{0}<\frac{\pi}{2}.$
Let $M_{1}$ be the angular domain in $\mathbb{C}$ defined by
$M_{1}=\\{re^{i\theta};\ r>0,0<\theta<\frac{\theta_{0}}{m}\\}$
and let $\Sigma_{1}$ be the angular domain in $\mathbb{C}$ defined by
$\Sigma_{1}=\\{1+re^{i\theta};\ 0<r<+\infty,-\theta_{0}<\theta<\pi\\}.$
Then it is easy to construct a homeomorphism $f_{0}$ from the closure
$\overline{M_{1}}$ of $M_{1}$ in $\overline{\mathbb{C}}$ onto the closure
$\overline{\Sigma_{1}}$ of $\Sigma_{1}$ in $\overline{\mathbb{C}},$ such that
$f_{0}$ maps the ray $\arg z=\frac{\theta_{0}}{m}$ onto the ray $\arg z=\pi,$
maps the interval $[0,1]$ onto itself increasingly, maps the ray $[1,+\infty]$
onto the ray
$z=1+re^{-i\theta_{0}},r\in[0,+\infty]$
and $f_{0}$ is holomorphic on $M_{1}.$
Let
$M_{2}=e^{\frac{\theta_{0}}{m}}M_{1}=\\{e^{\frac{\theta_{0}}{m}}z;\ z\in
M_{1}\\}.$
Then, by the Schwarz symmetry principle, we can extend $f_{0}$ to be an open
and continuous mapping $f_{1}$ from the closed angular domain
$A_{1}=\overline{M_{1}\cup M_{2}}=\\{z\in\overline{\mathbb{C}};0\leq\arg
z\leq\frac{2\theta_{0}}{m}\\}$
onto $\overline{\mathbb{C}}$ such that $f_{1}\ $maps the segments
$l_{k}=\left\\{re^{i\frac{k\theta_{0}}{m}},r\in[0,1]\right\\},k=0,1,$
homeomorphically onto the interval $[0,1],$ respectively; $f_{1}$ maps the
segments
$L_{k}=\left\\{re^{i\frac{k\theta_{0}}{m}},r\in[1,+\infty]\right\\},k=0,1,$
homeomorphically onto the segments
$l^{-}=\left\\{1+re^{-i\theta_{0}},r\in[0,+\infty]\right\\}$
and
$l^{+}=\left\\{1+re^{i\theta_{0}},r\in[0,+\infty]\right\\},$
respectively; and $f_{1}$ restricted to the domain $A_{1}$ is a holomorphic
mapping that covers the domain $D_{l_{0}}$ two times and covers the domain
$\mathbb{C}\backslash\overline{D_{l_{0}}}$ one times.
Let
$A_{1}^{\ast}=A_{1}^{\circ}\cup l_{0}^{\circ}\cup l_{1}^{\circ},$
where $A_{1}^{\circ}$ is the interior of $A_{1},$ and let
$A_{k}^{\ast}=e^{i\frac{2\left(k-1\right)\theta_{0}}{m}}A_{1}^{\ast}=\\{e^{i\frac{2\left(k-1\right)\theta_{0}}{m}}z;z\in
A_{1}^{\ast}\\},k=2,\dots,m.$
Then for each $k=1,\dots,m$ we can defined a continuous function $f_{k}$ on
$A_{k}^{\ast}$ inductively: $f_{k+1}$ is obtained from $f_{k}$ by Schwarz
symmetry principle cross the symmetry axis
$l_{k}=\\{re^{i\frac{2k\theta_{0}}{m}},r\in[0,1]\\}.$
Let $H^{+}$ be the upper half plane $\mathrm{Im}z>0$ in $\mathbb{C}$ and let
$K=H^{+}\backslash\cup_{k=1}^{m-1}\\{re^{i\frac{2k\theta_{0}}{m}},r\in[1,+\infty)\\}.$
Then $K=\left(\cup_{k=1}^{m}A_{k}^{\ast}\right)\backslash(l_{0}\cup l_{m})$
and $f_{1},\dots,f_{m}$ can be patched to be a holomorphic function $f$
defined on $K\cup(-\infty,+\infty)$, by the Schwarz symmetric principle. It is
clear that $K$ is a simply connected domain and there is a conformal mapping
$h$ from $\Delta$ onto $K$ such that $h$ can be extended to be a continuous
function $\widetilde{h}$ such that when $z$ goes around $\partial\Delta$ once,
$\widetilde{h}(z)$ describes the boundary section $(-\infty,+\infty)$ of $K$
once and the boundary sections
$\\{re^{i\frac{k\theta_{0}}{m}},r\in[1,+\infty)\\}$ twice for $k=1,\dots,m-1.$
Let $g=f\circ\widetilde{h}.$ Then
$g:\overline{\Delta}\rightarrow\overline{\mathbb{C}}$ is a continuous mapping
that is holomorphic in $\Delta$ and when we regard $g$ as a mapping from
$\overline{\Delta}$ to $S,$ $g$ restricted to $\Delta$ covers $S\backslash
D_{l_{0}}$ by $m$ times and covers $D_{l_{0}}$ by $2m$ times and the boundary
curve $\Gamma_{g}=g(z),z\in\partial\Delta,$ covers $\partial D_{l_{0}}$ by $m$
times and covers the shortest path $\overline{0,1}$ in $S$ from $0$ to $1$ by
$2$ times.
Then we have
$A(g,\Delta)=4m\pi+mA(D_{l_{0}})$
and
$L(g,\partial\Delta)=2L(\overline{0,1})+mL(\partial D_{l_{0}})=\pi+ml_{0},$
and then
$\frac{A(g,\Delta)}{L(g,\partial\Delta)}=\frac{4m\pi+mA(D_{l_{0}})}{\pi+mL(\partial
D_{l_{0}})}<\frac{4\pi+A(D_{l_{0}})}{l_{0}}=h_{0}.$
It is clear that as $m\rightarrow+\infty,$
$\frac{A(g,\Delta)}{L(g,\partial\Delta)}=\frac{4m\pi+mA(D_{l_{0}})}{\pi+mL(\partial
D_{l_{0}})}$ converges to $h_{0}.$ This completes the proof. ∎
## References
* [1] L. Ahlfors, Zur Theorie der Üherlagerung-Sflächen, Acta Math., 65 (1935), 157-194.
* [2] F. Bernstein, Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene, Math. Ann., vol. 60 (1905), pp. 117-136.
* [3] J. Dufresnoy, Sur les. domaines couverts. parles valeurs d’une. fonction. méromorphe. ou. algébroide, Ann. Sci. École. Norm. Sup. 58. (1941),. 179-259.
* [4] W.K. Hayman, Meromorphic functions, Oxford, 1964.
* [5] T. Radó, The Isoperimetric Inequality on the Sphere, Amer. Jour. Math., Vol.57, No.4. (Oct.,1935), pp.765-770.
|
arxiv-papers
| 2009-03-20T07:50:13 |
2024-09-04T02:49:01.275296
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guang Yuan Zhang",
"submitter": "Guang Yuan Zhang",
"url": "https://arxiv.org/abs/0903.3460"
}
|
0903.3517
|
# Soliton stability and collapse in the discrete nonpolynomial Schrödinger
equation with dipole-dipole interactions
Goran Gligorić1, Aleksandra Maluckov2, Ljupčo Hadžievski1, and Boris A.
Malomed3 1 Vinča Institute of Nuclear Sciences, P.O. Box 522,11001 Belgrade,
Serbia
2 Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, 18001
Niš, Serbia
3 Department of Physical Electronics, School of Electrical Engineering,
Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
###### Abstract
The stability and collapse of fundamental unstaggered bright solitons in the
discrete Schrödinger equation with the nonpolynomial on-site nonlinearity,
which models a nearly one-dimensional Bose-Einstein condensate trapped in a
deep optical lattice, are studied in the presence of the long-range dipole-
dipole (DD) interactions. The cases of both attractive and repulsive contact
and DD interaction are considered. The results are summarized in the form of
stability/collapse diagrams in the parametric space of the model, which
demonstrate that the the attractive DD interactions stabilize the solitons and
help to prevent the collapse. Mobility of the discrete solitons is briefly
considered too.
###### pacs:
03.75.Lm; 05.45.Yv
## I Introduction
It is well established that the mean-field description of Bose-Einstein
condensates (BECs) trapped in a deep optical lattice can be reduced, starting
from the one-dimensional (1D) Gross-Pitaevskii equation with the cubic
nonlinearity, to the discrete nonlinear Schrödinger (DNLS) equation DNLS-BEC ;
DNLS-BEC-review . However, while the DNLS equation with the cubic on-site
nonlinearity readily predicts solitons Panos , it cannot describe the
dynamical collapse, which is often observed experimentally in the self-
attractive BEC Strecker02 , Cornish .
In the general case, the reduction of the 3D Gross-Pitaevskii equation for the
BEC trapped in a “cigar-shaped” configuration to the 1D form leads, instead of
the “naive” cubic Schrödinger equation, to ones with a nonpolynomial
nonlinearity sala1 ; Canary . Such models are useful in various settings,
making it possible to obtain results in a relatively simple form, which
compare well to direct simulations of the underlying 3D equations various . In
particular, the version of this model corresponding to the combination of the
tight transverse trap and strong optical-lattice potential acting in the
longitudinal direction takes the form of the discrete nonpolynomial
Schrödinger equation (DNPSE), which was introduced recently luca . An
essential property of both the continual nonpolynomial equation and its
discrete counterpart is that they make it possible to model the collapse.
A new variety of the BEC dynamics, dominated by long-range (nonlocal)
interactions, occurs in dipolar condensates, which can be composed of
magnetically polarized 52Cr atoms, as demonstrated in a series of experimental
works Cr . In particular, the dipole-dipole (DD) attraction in the condensate
may give rise to a specific $d$-wave mode of the collapse d-collapse . On the
other hand, the 52Cr condensate can be efficiently stabilized against the
collapse, adjusting the scattering length of the contact interaction by means
of the Feshbach-resonance (FR) technique experim1 . Another theoretically
analyzed possibility is to create a condensate dominated by DD interactions
between electric dipole moments induced in atoms by a strong external dc
electric field dc . A similar situation may be realized in BEC formed by
dipolar molecules hetmol . In particular, recent experimental work LiCs has
reported the creation of LiCs dipolar molecules in a mixed ultracold gas.
A possibility of making 2D solitons in dipolar condensates was predicted in
several works. In particular, isotropic solitons Pedri05 and vortices Ami2
may exist if the sign of the DD interaction is inverted by means of rapid
rotation of the dipoles reversal . On the other hand, stable anisotropic
solitons can be supported by the ordinary DD interaction, if the dipoles are
polarized in the 2D plane Ami1 . Solitons supported by nonlocal interactions
were also predicted and realized in optics, making use of the thermal
nonlinearity Krolik .
A natural extension of the consideration of the dipolar BEC includes a strong
optical lattice potential, which leads to the discrete model with the long-
range DD interactions between lattice sites gpe ; Santos . In particular, 1D
unstaggered solitons in this model with the cubic on-site nonlinearity were
studied in Ref. gpe . It was shown that the DD interactions might enhance the
solitons’ stability. This conclusion suggests a possibility of suppressing the
collapse by means of the long-range DD forces, but the study of the collapse
is not possible in the discrete system with the cubic nonlinearity. The
objective of the present work is to introduce the DD interaction into DNPSE
model and analyze the effects of the long-range interactions between dipoles
localized at site of the discrete lattice on the soliton’s collapse in the
sufficiently dense self-attractive condensate trapped in the deep optical
lattice potential. We here focus on unstaggered solitons, leaving the
consideration of staggered ones for another work.
The paper is structured as follows. The discrete model including the on-site
nonpolynomial nonlinearity and long-range DD interactions is formulated in
Section II. Results for on-site and inter-site solitons, including the study
of their stability, and the possibility of the collapse onset, are presented
in Section III. The mobility of discrete solitons in the DNPSE model is also
briefly considered in Section III. The paper is concluded by Section IV.
## II The model
Adding the long-range DD interaction to the scaled nonpolynomial Schrödinger
equation (in its continual form sala1 ; various ) leads to the following 1D
equation:
$\displaystyle i\frac{\partial F}{\partial t}$ $\displaystyle=$
$\displaystyle\left[-\frac{1}{2}\frac{\partial^{2}}{\partial
z^{2}}-V_{0}\cos{(2qz)}+\frac{1-(3/2)\aleph|F|^{2}}{\sqrt{1-\aleph|F|^{2}}}\right]F$
(1) $\displaystyle+$ $\displaystyle
GF(z)\int_{-\infty}^{+\infty}\frac{|F(z^{\prime})|^{2}}{|z-z^{\prime}|^{3}}dz^{\prime}.$
Here, $V_{0}$ and $\pi/q$ are the strength and period of the longitudinal
optical lattice potential, $F(z,t)\equiv\sqrt{|\gamma|}f(z,t)$, where $f$ is
the 1D mean-field wave function subject to normalization
$\int_{-\infty}^{+\infty}\left|f(z,t)\right|^{2}dz=1$, and
$\gamma=-2Na_{s}\sqrt{m\omega_{\bot}/\hbar}$ is the effective strength of the
local interaction, with $N$ the total number of atoms in the condensate,
$a_{s}$ the scattering length of atomic collisions ($a_{s}<0$ corresponds to
the attraction), $m$ the atom mass, and $\omega_{\bot}$ the transverse
trapping frequency sala1 . Further,
$\aleph\equiv\mathrm{sgn}\left(a_{s}\right)$ is the sign of the local
interaction, and $G=g\left(1-3\cos^{2}\theta\right)$ is the coefficient which
defines the ratio of the DD and contact interactions, where $g$ is a positive
coefficient and $\theta$ the angle between the $z$ axis and the orientation of
the dipoles. One obvious case of interest in the 1D geometry corresponds to
the dipoles polarized along the $z$ axis, i.e., $\theta=0$ and $G=-2g$, when
the long-range interaction is _attractive_. Another interesting case
corresponds to the orientation of the dipoles perpendicular to the $z$ axis
($\theta=\pi/2$ , i.e. $G=g$), which implies the repulsive DD interaction.
Assuming a sufficiently deep optical lattice potential, and approximating the
wave function by a superposition of localized Wannier modes, similar to how it
was done in Refs. luca and gpe , one can derive the discrete version of Eq.
(1), i.e., the DNPSE with the DD term:
$\displaystyle i\frac{\partial F_{n}}{\partial t}$ $\displaystyle=$
$\displaystyle-C\left(F_{n+1}+F_{n-1}-2F_{n}\right)+\frac{1-\left(3/2\right)\aleph\left|F_{n}\right|^{2}}{\sqrt{1-\aleph\left|F_{n}\right|^{2}}}F_{n}$
(2) $\displaystyle-$ $\displaystyle\Gamma\sum_{n^{\prime}\neq
n}\frac{\left|F_{n^{\prime}}\right|^{2}}{\left|n-n^{\prime}\right|^{3}}F_{n},$
where the linear-coupling $C$ and the ratio between DD and contact interaction
coefficients $\Gamma=G/\left|\gamma\right|$, are defined as in Ref. gpe
($\Gamma>0$ for attractive and $\Gamma<0$ for the repulsive DD interactions).
Two dynamical quantities are conserved by Eq. (2), viz., norm
$P=\sum_{n}\left|F_{n}\right|^{2}$ and Hamiltonian
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n}\left[C\left|F_{n}-F_{n+1}\right|^{2}+\sqrt{1-\aleph\left|F_{n}\right|^{2}}\left|F_{n}\right|^{2}\right.$
(3) $\displaystyle\left.-\Gamma\sum_{n\neq
n^{\prime}}\frac{\left|F_{n^{\prime}}\right|^{2}\left|F_{n}\right|^{2}}{\left|n-n^{\prime}\right|^{3}}\right].$
Note that the staggering transformation,
$F_{n}\equiv(-1)^{n}\exp\left(-4iCt\right)\tilde{F}_{n}$, can be used to
change the sign of $C$ if it is negative, but this transformation cannot be
used to invert the signs of nonlinearity coefficients in Eq. (2).
Experimentally adjustable parameters are the relative strength of the
DD/contact interactions, $\Gamma$, and the norm of stationary wave function,
$P$. The latter may be expressed in terms of the total number of atoms in the
condensate, $N$ luca ; gpe : $P\sim
N\left|a_{s}\right|\sqrt{m\omega_{\bot}/\hbar}$. In particular, $a_{s}\approx
5$ nm for 52Cr atoms (far from the FR); assuming the transverse-confinement
width to be $(\sqrt{m\omega_{\bot}/\hbar})^{-1}\sim 5$ $\mathrm{\mu}$m, we
conclude that $P\sim 1$ may correspond to $\sim 1000$ atoms in the condensate.
A typical value of the relative interaction strength, if estimated as the
ratio of the effective scattering lengths corresponding to the DD and contact
interactions in the 52Cr condensate, without the use of the FR technique, is
$~{}\left|\Gamma\right|\simeq 0.15$ experim1 . Actually, $\Gamma$ can be made
both positive and negative, and its absolute value may be altered within broad
limits by means of the FR Cr . In particular, $\left|\Gamma\right|$ can be
given very large values in the experimentally possible situation when the
strength of the contact interactions is almost nullified with the help of the
FR experim1 .
Stationary solutions to Eq. (2), with chemical potential $\mu$, are sought for
as $F_{n}=u_{n}\exp(-i\mu t)$, with real discrete function $u_{n}$ satisfying
a stationary equation,
$\displaystyle\mu u_{n}$ $\displaystyle=$
$\displaystyle-C\left(u_{n+1}+u_{n-1}-2u_{n}\right)+\frac{1-\left(3/2\right)\aleph
u_{n}^{2}}{\sqrt{1-\aleph u_{n}^{2}}}u_{n}$ (4) $\displaystyle-$
$\displaystyle\Gamma u_{n}\sum_{n^{\prime}\neq
n}\frac{u_{n^{\prime}}^{2}}{\left|n-n^{\prime}\right|^{2}}.$
In the case of the attractive contact interaction ($\aleph=+1$), $u_{n}^{2}$
cannot exceed the maximum value, $\left(u_{n}^{2}\right)_{\max}=1$, as seen
from Eq. (4). In fact, the presence of the singularity in Eq. (2) at
$\left|F_{n}\right|^{2}=1$ makes it possible to study the onset of collapse in
the framework of this equation luca .
Stationary equation (4) was numerically solved by an algorithm based on the
modified Powell minimization method luca ; gpe . Initial ansätze used to
construct on-site and inter-site-centered discrete solitons were taken,
respectively, as $\left\\{u_{n}^{(0)}\right\\}=(...,\,0,\,A,\,0,\,...)$ and
$(...,\,0,\,A,\,A,\,0,\,...)$, where $A$ is a real constant obtained from Eq.
(4) in the corresponding approximation. Results reported below were obtained
in the lattice composed of $101$ or $100$ sites, for the on-site and inter-
site configurations, respectively. It was checked that the results do not
alter if a larger lattice had been used.
## III Results and discussion
### III.1 The case of the attractive contact interaction
Families of fundamental unstaggered solitons of on-site and inter-site types
for the local attraction ($\aleph=+1$) and either sign of the DD interaction
were obtained from the numerical solution of Eq. (4). In contrast to the 1D
discrete model with the cubic on-site nonlinearity, where bright unstaggered
solitons exist only for sufficiently weak repulsive DD interaction gpe , in
the present model solitons can be also found if the repulsive DD interaction
is strong. The solitons were categorized as stable ones if they met two
conditions: the slope (Vakhitov-Kolokolov) criterion, according to which the
slope of the $P(\mu)$ dependence must be negative (or may be very close to
zero, see below), $dP/d\mu\leq 0$, and, simultaneously, the spectral
condition, according to which the corresponding eigenvalues, found from
linearized equations for small perturbations, must not have a positive real
part gpe ; stability .
In order to draw general conclusions about the influence of the DD interaction
on the solitons, the analysis is presented below for two different values of
the lattice coupling constant $C$, viz., $C=0.8$ and $C=0.2$. These cases
correspond, respectively, to the proximity to the continuum limit, and to a
strongly discrete system.
#### III.1.1 The quasi-continual model ($C=0.8$)
A global characteristic of soliton families is the $P(\mu)$ dependence, i.e.,
the scaled norm as a function of the chemical potential. For $C=0.8$ and four
different values of DD parameter $\Gamma$, the $P(\mu)$ curves are displayed
in Fig. 1. In the absence of the DD interaction, $\Gamma=0$, two subfamilies
of on-site solitons are found, one (which occupies a narrow interval of $\mu$)
obeying the slope criterion, and the other one violating it, in a broad
interval. With the increase of the strength of the attractive DD interactions
($\Gamma=5$), the region where the slope condition is met spreads out, and,
when the DD interaction is dominant ($\Gamma=12$), the condition is satisfied
for all on-site solitons. On the other hand, the slope condition is fulfilled
for all inter-site solitons, regardless of the value of $\Gamma$. It is worthy
to note that the difference between the $P(\mu)$ curves for the on-site and
inter-site solitons vanishes as the attractive DD interaction strengthens.
In the case of the repulsive DD interaction, the $P(\mu)$ curves for on-site
and inter-site solitons are completely separated, see Fig. 1(d). Actually, the
slope of $P(\mu)$ curves for both on-site and inter-site solitons tends to be
very small in this case, which makes the application of the slope criterion
doubtful. In these areas, the solitons feature the amplitude close to the
upper limit admitted by the model, $u_{\max}^{2}=1$, and a very small width,
corresponding to a situation when nearly all atoms are collected in a single
well of the underlying potential lattice.
The spectral stability was examined by linearizing Eq. (2) for small
perturbations $\delta F_{n}$ around the soliton, and finding the respective
eigenvalues in a numerical form. The results of the stability analysis for the
on-site solitons, with $C=0.8$, are summarized in Fig. 2, in the form of the
stability diagram in the plane of $\left(\mu,\Gamma\right)$, where contours of
constant norm $P$ for the on-site solitons are included too. The collapse
condition, $u_{\max}^{2}=1$, is attained at the black solid line, which is,
simultaneously, a stability border. In direct simulations, the on-site
solitons which are predicted to be stable survived as long as the simulations
ran [Fig. 3(a)], while the solitons classified as unstable ones suffered the
collapse (destruction of the solution after attaining the level of
$u_{\max}^{2}=1$) in a finite time.
On the other hand, all inter-site solitons are unstable, as the spectrum of
eigenvalues for small perturbations around them always contains real
eigenvalues. However, unstable inter-site solitons which have stable on-site
counterparts with the same norm avoid the collapse, evolving into robust
breathers oscillating around the corresponding stable on-site solitons, as
shown in Fig. 3(b). Unstable inter-site solitons do collapse if no stable on-
site soliton with the same norm can be found, see Fig. 3(c). Therefore, the
border line for the collapse of the inter-site solitons coincides with the
stability border for on-site solitons in Fig. 2.
For the repulsive DD interaction, the spectral stability condition for all
inter-site solitons does not hold either. Because, in this case, curves
$P(\mu)$ for the inter-site and on-site solitons are strongly separated
[Fig.1(d)], unstable inter-site solitons do not find stable on-site
counterparts with the same norm, therefore they suffer the collapse. As for
the on-site solitons, there exists a region where the spectral stability
condition holds for them. This region expands as the repulsive DD interaction
gets stronger, although the respective curve $P(\mu)$ becomes almost
horizontal, and the slope condition cannot be accurately verified. Direct
numerical simulations confirm the predictions of the stability analysis for
the on-site solitons in this case too.
#### III.1.2 The strongly discrete model ($C=0.2$)
In this case, the slope condition is satisfied for all on-site solitons in the
absence of the DD interaction ($\Gamma=0$), see Fig. 4(a). As the strength of
the attractive DD interaction grows, a pair of subfamilies emerge, the slope-
stable and unstable ones, the respective $P(\mu)$ curves being similar to
those observed in the case of the strong coupling, $C=0.8$ [Fig. 4(b)].
Eventually, when the attractive DD interaction becomes dominant, the region
where the slope condition is satisfied spreads over the entire parameter
space, as seen in Fig. 4(c).
In the model with $C=0.2$, all inter-site solitons again satisfy the slope
condition. With the strengthening of the DD interaction, the separation
between the $P(\mu)$ curves for the on-site branch, which satisfies the slope
condition, and its inter-site counterpart vanishes. On the other hand, in the
case of the repulsive DD interaction, the corresponding $P(\mu)$ curves for
on-site and inter-site soliton families are strongly separated, see Fig. 4(d).
Results of the stability analysis for the on-site solitons are summarized in
the stability diagram displayed in Fig. 5(a) in parametric space
$(\Gamma,\mu)$. Unlike the case of $C=0.8$, cf. Fig. 2, in the present case
there appears a region where the spectral stability condition is violated for
on-site solitons as the DD interaction grows stronger, as well as a region
where the stability condition holds for inter-site solitons. These regions
disappear again with the further growth of $\Gamma$. Also in contrast to the
case of $C=0.8$, the line at which the collapse is attained in the family of
on-site solitons _does not_ coincide with the border between stable and
unstable parts of the family. Rather, the collapse line passes trough the
unstable region where the spectral-stability condition does not hold, see Fig.
5(a). Direct simulations demonstrate that unstable on-site solitons which have
stable inter-site counterparts with the equal norm evolve into breathers
oscillating around the stable counterparts. On the other hand, if unstable on-
site solitons cannot find stable inter-site counterparts with the same (or
close) value of the norm, they undergo the collapse. In fact, the existence of
the two different scenarios of the instability development – the formation of
the breather and collapse – explains the above-mentioned fact that the
collapse line does not coincide with the instability border. In the present
case too, direct simulations corroborate the stability of those solitons which
do not have unstable eigenvalues.
Extending the above-mentioned trend, unstable inter-site solitons which have
stable on-site counterparts with the same norm evolve into persistent
breathers, while those unstable solitons that are devoid of stable equal-norm
counterparts suffer the collapse. Accordingly, the line at which the collapse
is attained does not coincide with the instability border, as seen in Fig.
5(b). It is worthy to note the existence of a stability region for inter-site
solitons in Fig. 5(b) which is adjacent to the collapse line. In the latter
case, the stable inter-site solitons do not have on-site counterparts with the
same value of the norm.
For the repulsive DD interactions, the spectral stability condition always
holds for on-site solitons and does not hold for inter-site modes, similar to
the case of $C=0.8$. Again, in some part of the parameter space, the
corresponding $P(\mu)$ curves for the on-site solitons are nearly horizontal
lines with zero slope, being completely separated from the $P(\mu)$ line for
the inter-site modes. Direct simulations demonstrate that the on-site solitons
are indeed stable in this case, while the unstable inter-site solitons undergo
the collapse.
### III.2 The case of repulsive contact interactions
If the local interaction is repulsive, the existence of the unstaggered
solitons may only be supported by the attractive DD interaction. In the model
with the self-repulsive cubic on-site nonlinearity, unstaggered solitons were
found only when the relative strength of the attractive DD interaction was
large enough, $\Gamma\geq 0.4$ gpe . In the present model, unstaggered
solitons (with large amplitudes) are obtained for smaller values of $\Gamma$
as well. Nevertheless, general results obtained in the present model for the
case of the local repulsion are not drastically different from those reported
in the model with the cubic on-site nonlinearity in Ref. gpe . In particular,
differences between $P(\mu)$ curves for on-site and inter-site solitons are
small at all values of $\mu$, the slope condition is always satisfied for both
types of the solitons, and there is an exchange between regions where the
spectral stability condition is fulfilled for on-site and inter-site solitons.
Further, unstable solitons evolve into breathers oscillating around their
stable counterparts. The similarity of these results for discrete solitons in
the present DNPSE model and its cubic counterpart is not surprising, as in the
case of the local repulsion (unlike attraction) there is no dramatic
difference between the nonpolynomial and cubic nonlinearities, therefore the
competition of the local term of either type (nonpolynomial or cubic) with the
DD attraction gives rise to similar solitary modes.
### III.3 Moving discrete solitons
Finally, we briefly summarize results concerning mobility of localized modes,
with respect to the concept of the Peierls-Nabarro barrier mobil . To this
end, we follow the lines of the analysis developed in Refs. luca and gpe .
Examination of the mobility has shown that the Peierls-Nabarro barrier in the
DNPSE model depends on the strength of the DD interaction in the same way as
it was in the discrete model with the cubic on-site nonlinearity. Namely, in
the case of the local attraction, the repulsive or attractive DD interaction
decreases or increases, respectively, a region in plane $(P,\mu)$ where mobile
localized modes can be found. On the other hand, in the case of the contact
repulsion, all localized modes can be set in motion by an initial kick,
regardless of the value of DD coefficient $\Gamma$. In all cases when mobile
discrete solitons exist, the vanishing of the Peierls-Nabarro barrier
coincides with the disappearance of the separation between $P(\mu)$ curves for
the on-site and inter-site soliton families, cf. Refs. luca ; gpe .
## IV Conclusion
The purpose of this work was to achieve a better understanding on the
influence of the long-range DD (dipole-dipole) interactions on the stability
and collapse of localized nonlinear modes in the cigar-shaped Bose-Einstein
condensate trapped in a deep optical-lattice potential. To this end, we have
introduced the model based on the one-dimensional DNPSE (discrete
nonpolynomial Schrödinger equation), which includes the contact (on-site) and
DD nonlinear terms. Both attractive and repulsive signs of the contact and DD
interactions were considered. The main conclusion is that the presence of the
attractive DD interaction enhances the soliton’s stability and helps to
prevent the collapse. Our analysis was limited to unstaggered solitons, the
consideration of staggered localized modes being a subject of a separate work.
G.G., A.M. and Lj.H. acknowledge support from the Ministry of Science, Serbia
(through project 141034).
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## Figures
Figure 1: The $P(\mu)$ (norm versus chemical potential) dependencies for
families of on- and inter-site unstaggered solitons (the solid and dashed
lines, respectively) in the case of the attractive contact interaction, for
$C=0.8$ and relative strength of the DD interaction $\Gamma=0$ (a), $5$ (b),
$12$ (c), and $-5$ (d). Figure 2: The stability/collapse diagram for on-site
solitons, as determined by the stability analysis performed for $C=0.8$. The
gray and white areas are occupied by unstable and stable solitons,
respectively. The soliton norm keeps constant values along thin contour lines,
as indicated in the figure. The chain of bold dots connected by the black line
shows a curve along which the collapse is attained. Figure 3: (a) An example
of a stable on-site soliton. (b,c) Development of the instability of inter-
site solitons: (b) the case when a stable on-site soliton exists whose norm is
equal to that of the unstable inter-site sooliton. In this situation, the
unstable soliton evolves into a breather oscillating around the stable on-site
soliton. (c) The case without the stable on-site counterpart with the equal
norm. In the latter case, the unstable soliton collapses. Figure 4: $P(\mu)$
dependencies for families of on- and inter-site (solid and dashed lines,
respectively) unstaggered solitons in the strongly discrete model ($C=0.2$)
with the attractive contact interaction. The relative strength of the DD
interaction is $\Gamma=0$ (a), $5$ (b), $15$ (c), and $-5$ (d). Figure 5: The
stability/collapse diagram for (a) on-site (a) and inter-site (b) unstaggered
solitons in the strongly discrete version of the model, with $C=0.2$. The
notation has the same meaning as in Fig. 2.
|
arxiv-papers
| 2009-03-20T13:46:07 |
2024-09-04T02:49:01.306798
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Gligoric, A. Maluckov, Lj. Hadzievski, B. A. Malomed",
"submitter": "Aleksandra Maluckov",
"url": "https://arxiv.org/abs/0903.3517"
}
|
0903.3653
|
# Cohomological rigidity and the number of homeomorphism types for small
covers over prisms
Xiangyu Cao and Zhi Lü School of Mathematical Sciences, Fudan University,
Shanghai, China xiangyu.cao08@gmail.com Institute of Mathematics, School of
Mathematical Sciences, Fudan University, Shanghai, 200433, P.R.China.
zlu@fudan.edu.cn
###### Abstract.
In this paper we give a method of constructing homeomorphisms between two
small covers over simple convex polytopes. As a result we classify, up to
homeomorphism, all small covers over a 3-dimensional prism $\mathrm{P}^{3}(m)$
with $m\geq 3$. We introduce two invariants from colored prisms and other two
invariants from ordinary cohomology rings of small covers. These invariants
can form a complete invariant system of homeomophism types of all small covers
over a prism in most cases. Then we show that the cohomological rigidity holds
for all small covers over a prism $\mathrm{P}^{3}(m)$ (i.e., cohomology rings
of all small covers over a $\mathrm{P}^{3}(m)$ determine their homeomorphism
types). In addition, we also calculate the number of homeomorphism types of
all small covers over a $\mathrm{P}^{3}(m)$.
###### Key words and phrases:
Small covers, prism, classification
This work is supported by grants from FDUROP (No. 080705) and NSFC (No.
10671034)
## 1\. Introduction
In 1991, Davis and Januszkiewicz [DJ] introduced and studied a class of
${\mathbb{Z}}_{2}^{n}$-manifolds (called small covers), which belong to the
topological version of toric varieties. An $n$-dimensional small cover $M^{n}$
is a closed $n$-manifold with a locally standard ${\mathbb{Z}}_{2}^{n}$-action
such that its orbit space is a simple convex $n$-polytope $P^{n}$. As shown in
[DJ], $P^{n}$ naturally admits a characteristic function $\lambda$ defined on
the facets of $P^{n}$ (here we also call $\lambda$ a
${\mathbb{Z}}_{2}^{n}$-coloring on $P^{n}$), so that the geometrical topology
and the algebraic topology of the samll cover $M^{n}$ can be completely
determined by the pair $(P^{n},\lambda)$. In other words, the Davis-
Januszkiewicz theory for small covers indicates the following two key points:
1. $\bullet$
Each small cover $\pi:M^{n}\longrightarrow P^{n}$ can be reconstructed from
$(P^{n},\lambda)$, and this reconstruction is denoted by $M(\lambda)$. Thus,
all small covers over a simple convex polytope $P^{n}$ correspond to all
${\mathbb{Z}}_{2}^{n}$-colorings on $P^{n}$, i.e., all small covers over a
simple convex polytope $P^{n}$ are given by
$\Gamma(P^{n})=\\{M(\lambda)|\lambda$ is a ${\mathbb{Z}}_{2}^{n}$-coloring on
$P^{n}$$\\}$.
2. $\bullet$
The algebraic topology of a small cover $\pi:M^{n}\longrightarrow P^{n}$, such
as equivariant cohomology, mod 2 Betti numbers and ordinary cohomology etc.,
can be explicitly expressed in terms of $(P^{n},\lambda)$.
In the recent years, much further research on small covers has been carried on
(see, e.g. [I], [GS], [NN], [CCL], [C], [LM], [LY], [KM], [M1]-[M3]). In some
sense, the classification up to equivariant homeomorphism of small covers over
a simple convex polytope has been understood very well. Actually, this can be
seen from the following two kinds of viewpoints: One is that two small covers
$M(\lambda_{1})$ and $M(\lambda_{2})$ over a simple convex polytope $P^{n}$
are equivariantly homeomorphic iff there is an automorphism
$h\in\text{Aut}(P^{n})$ such that $\lambda_{1}=\bar{h}\circ\lambda_{2}$ where
$\bar{h}$ induced by $h$ is an automorphism on all facets of $P^{n}$ (see
[LM]); the other is that two small covers $M(\lambda_{1})$ and
$M(\lambda_{2})$ over a simple convex polytope $P^{n}$ are equivariantly
homeomorphic iff their equivariant cohomologies are isomorphic as
$H^{*}(B{\mathbb{Z}}_{2}^{n};{\mathbb{Z}}_{2})$-algebras (see [M1]). However,
in non-equivariant case, the classification up to homeomorphism of small
covers over a simple convex polytope is far from understood very well except
for few special polytopes (see, e.g. [GS], [KM], [M2], [M3], [LY]).
In this paper we shall introduce an approach (called the sector method) of
constructing homeomorphisms between two small covers. The basic idea of sector
method is simply stated as follows: we first cut a
$\mathbb{Z}_{2}^{n}$-colored simple convex $n$-polytope $(P^{n},\lambda)$ in
${\mathbb{R}}^{n}$ into two parts $P_{1}$ and $P_{2}$ by using an
$(n-1)$-dimensional hyperplane $H$ such that the section $S$ cut out by $H$ is
an $(n-1)$-dimensional simple convex polytope with certain property. Of
course, $S$ naturally inherits a coloring from $(P^{n},\lambda)$. Then by
using automorphisms of $S$ and automorphisms of ${\mathbb{Z}}_{2}^{n}$ we can
construct new colored polytopes $(P^{\prime n},\lambda^{\prime})$ from
$(P^{n},\lambda)$ (note that generally $P^{\prime n}$ may not be
combinatorially equivalent to $P^{n}$), and further obtain new small covers
$M(\lambda^{\prime})$ from those new colored polytopes $(P^{\prime
n},\lambda^{\prime})$ by the reconstruction of small covers. Moreover, we can
study how to construct the homeomorphisms between $M(\lambda)$ and
$M(\lambda^{\prime})$. In particular, we give the method of constructing the
homeomorphisms between $M(\lambda)$ and $M(\lambda^{\prime})$ (see Theorems
3.2-3.3).
As an application, up to homeomorphism we shall classify all small covers over
prisms. Let $\mathrm{P}^{3}(m)$ denote a 3-dimensional prism that is the
product of $[0,1]$ and an $m$-gon where $m\geq 3$. Let
$\Lambda(\mathrm{P}^{3}(m))=\\{\lambda\big{|}\lambda\text{ is a
${\mathbb{Z}}_{2}^{3}$-coloring on $\mathrm{P}^{3}(m)$}\\}$, and let
$\Gamma(\mathrm{P}^{3}(m))=\\{M(\lambda)\big{|}\lambda\in\Lambda(\mathrm{P}^{3}(m))\\}$.
Using the sector method, we first study rectangular sectors and find seven
rectangular sectors with a good twist $\Psi(\psi,\rho,v_{0})$ in Section 4
(see Definition 3.3 for $\Psi(\psi,\rho,v_{0})$). We then use these seven
rectangular sectors to define some operations on the coloring sequences of
side-faces of colored polytopes $(\mathrm{P}^{3}(m),\lambda)$ in Section 5.
Furthermore, we show that all $(\mathrm{P}^{3}(m),\lambda)$ can be reduced to
some canonical forms without changing the homeomorphism type of the small
covers $M(\lambda)$ (see Propositions 5.2-5.4 and 5.9). In particular, we
introduce two combinatorial invariants $m_{\lambda}$ and $n_{\lambda}$ in
Section 5, and then show in Section 6 that $(m_{\lambda},n_{\lambda})$ is
actually a complete homeomorphism invariant of a class of small covers
$M(\lambda)$ (see Corollary 6.6). In addition, we also introduce two algebraic
invariants $\Delta(\lambda)$ and ${\mathcal{B}}(\lambda)$ in
$H^{*}(M(\lambda);\mathbb{Z}_{2})$, which can become a complete homeomorphism
invariant in most cases. Both $(m_{\lambda},n_{\lambda})$ and
$(\Delta(\lambda),{\mathcal{B}}(\lambda))$ are of interest because of the
nature of colored polytopes. With the help of these invariants, we can obtain
the following cohomological rigidity theorem.
###### Theorem 1.1 (Cohomological rigidity).
Two small covers $M(\lambda_{1})$ and $M(\lambda_{2})$ in
$\Gamma(\mathrm{P}^{3}(m))$ are homeomorphic if and only if their cohomologies
$H^{*}(M(\lambda_{1});{\mathbb{Z}}_{2})$ and
$H^{*}(M(\lambda_{2});{\mathbb{Z}}_{2})$ are isomorphic as rings.
###### Remark 1.1.
Kamishima and Masuda showed in [KM] that the cohomological rigidity holds for
small covers over a cube. In addition, Masuda in [M2] also gave a
cohomological non-rigidity example of dimension 25. A further question is what
extent the cohomological rigidity can extend to small covers.
In addition, we also determine the number of homeomorphism classes of all
small covers in $\Gamma(\mathrm{P}^{3}(m))$.
###### Theorem 1.2 (Number of homeomorphism classes).
Let $N(m)$ denote the number of homeomorphism classes of all small covers in
$\Gamma(\mathrm{P}^{3}(m))$. Then
$N(m)=\begin{cases}2&\text{ if $m=3$}\\\ 4&\text{ if $m=4$}\\\ \sum_{0\leq
k\leq{m\over 2}}([{k\over 2}]+1)+6&\text{ if $m>4$ is even}\\\ \sum_{1\leq
k\leq{m\over 2}}([{k\over 2}]+1)+4&\text{ if $m>4$ is odd.}\\\ \end{cases}$
The arrangement of this paper is as follows. In Section 2 we review the basic
theory about small covers. In Section 3 we introduce the sector method. In
Section 4 we apply the sector method to colored prisms and discuss the
rectangular sectors. In Section 5 we use the rectangular sectors to define
some operations on the coloring sequences of colored prisms and then determine
the canonical forms of all colored prisms. In Section 6 we introduce two
algebraic invariants of cohomology and then calculate such two invariants of
all small covers. Finally, we complete the proofs of Theorems 1.1 and 1.2 in
Section 7.
## 2\. Theory of small covers
The purpose of this section is to briefly review the theory of small covers.
Throughout the following assume that $\pi:M^{n}\longrightarrow P^{n}$ is a
small cover over a simple convex $n$-polytope $P^{n}$. Note that a simple
convex $n$-polytope $P^{n}$ means that exactly $n$ faces of codimension-one
(i.e., facets) meet at each of its vertices. Let
$\mathcal{F}(P^{n})=\\{F_{1},...,F_{\ell}\\}$ denote the set of all facets of
$P^{n}$.
### 2.1. Coloring and Reconstruction
Take a $k$-face $F^{k}$ of $P^{n}$, an easy observation shows (see also [DJ,
Lemma 1.3]) that $\pi^{-1}(F^{k})\longrightarrow F^{k}$ is still a
$k$-dimensional small cover. In particular, for any
$x\in\pi^{-1}(\text{int}F^{k})$, its isotropy subgroup $G_{x}$ is independent
of the choice of $x$, denoted by $G_{F}$. $G_{F}$ is isomorphic to
$\mathbb{Z}_{2}^{n-k}$, and $G_{F}$ fixes $\pi^{-1}(F^{k})$ in $M^{n}$. In the
case $k=n-1$, $F^{n-1}$ is a facet and $G_{F}$ has rank $1$, so that $G_{F}$
uniquely corresponds to a nonzero vector $v_{F}$ in ${\mathbb{Z}}_{2}^{n}$.
Then there is a natural map (called characteristic function)
$\lambda:\mathcal{F}(P)\longrightarrow\mathbb{Z}_{2}^{n}$
by mapping each facet $F$ to its corresponding nonzero vector $v_{F}$ in
${\mathbb{Z}}_{2}^{n}$ with the property $(\star)$: whenever the intersection
of some facets $F_{i_{1}},...,F_{i_{r}}$ in $\mathcal{F}(P^{n})$ is nonempty,
$\lambda(F_{i_{1}}),...,\lambda(F_{i_{r}})$ are linearly independent in
$\mathbb{Z}_{2}^{n}$. Note that if each nonzero vector of $\mathbb{Z}_{2}^{n}$
is regarded as being a color, then the characteristic function $\lambda$ means
that each facet is colored by a color. Thus, we also call $\lambda$ a
$\mathbb{Z}_{2}^{n}$-coloring on $P^{n}$ here. By $\Lambda(P^{n})$ we denote
the set of all $\mathbb{Z}_{2}^{n}$-colorings on $P^{n}$.
###### Remark 2.1.
Since $P^{n}$ is simple, for each $k$-face $F^{k}$, there are $n-k$ facets
$F_{i_{1}},...,F_{i_{n-k}}$ such that $F^{k}=F_{i_{1}}\cap\cdots\cap
F_{i_{n-k}}$ and $\pi^{-1}(F^{k})$ is a transversal intersection of
$\pi^{-1}(F_{i_{1}})$, $...,$ $\pi^{-1}(F_{i_{n-k}})$. Then the group $G_{F}$
determined by $F^{k}$ is actually generated by $\lambda(F_{i_{1}})$, $...,$
$\lambda(F_{i_{n-k}})$.
Davis and Januszkiewicz [DJ] gave a reconstruction of $M^{n}$ by using the
$\mathbb{Z}_{2}^{n}$-coloring $\lambda$ and the product bundle
$P^{n}\times\mathbb{Z}_{2}^{n}$ over $P^{n}$. Geometrically this
reconstruction is exactly done by gluing $2^{n}$ copies of $P^{n}$ along their
boundaries via $\lambda$. Thus this reconstruction can be written as follows:
$M(\lambda):=P^{n}\times\mathbb{Z}_{2}^{n}/(p,v)\sim(p,v+\lambda(F))\text{ for
}p\in F\in\mathcal{F}(P^{n}).$
Then we have
###### Theorem 2.1 (Davis-Januszkiewicz).
All small covers over $P^{n}$ are given by
$\Gamma(P^{n})$$=\\{M(\lambda)|\lambda\in\Lambda(P^{n})\\}$.
There is a natural action of $\text{GL}(n,\mathbb{Z}_{2})$ on $\Lambda(P^{n})$
defined by $\lambda\longmapsto\alpha\circ\lambda$, and it is easy to see that
such an action is free, and also induces an action of
$\text{GL}(n,\mathbb{Z}_{2})$ on $\Gamma(P^{n})$ by $M(\lambda)\longmapsto
M(\alpha\circ\lambda)$. Following [DJ], two small covers $M(\lambda_{1})$ and
$M(\lambda_{2})$ in $\Gamma(P^{n})$ are said to be Davis-Januszkiewicz
equivalent if there is a $\alpha\in\text{GL}(n,\mathbb{Z}_{2})$ such that
$\lambda_{1}=\alpha\circ\lambda_{2}$. Thus, each Davis-Januszkiewicz
equivalence class in $\Gamma(P^{n})$ is actually an orbit of the action of
$\text{GL}(n,\mathbb{Z}_{2})$ on $\Gamma(P^{n})$.
### 2.2. Betti numbers and $h$-vector
The notion of the $h$-vector plays an essential important role in the theory
of polytopes, while the notion of Betti numbers is also so important in the
topology of manifolds. Davis-Januszkiewicz theory indicates that the Dehn-
Sommerville relations for the $h$-vectors and the Poincaré duality for the
Betti numbers are essentially consistent in the setting of small covers.
Let $P^{*}$ be the dual of $P^{n}$ that is a simplicial polytope. Then the
boundary $\partial P^{*}$ denoted by $K_{P}$ is a finite simplicial complex of
dimension $n-1$. For $0\leq i\leq n-1$, by $f_{i}$ one denotes the number of
all $i$-faces in $K_{P}$. Then the vector $(f_{0},f_{1},...,f_{n-1})$ is
called the $f$-vector of $P^{n}$, denoted by ${\bf f}(P^{n})$. Then the
$h$-vector denoted by ${\bf h}(P^{n})$ of $P^{n}$ is an integer vector
$(h_{0},h_{1},...,h_{n})$ defined from the following equation
$h_{0}t^{n}+\cdots+h_{n-1}t+h_{n}=(t-1)^{n}+f_{0}(t-1)^{n-1}+\cdots+f_{n-1}.$
###### Theorem 2.2 (Davis-Januszkiewicz).
Let $\pi:M^{n}\longrightarrow P^{n}$ be a small cover over a simple convex
polytope $P^{n}$. Then
${\bf h}(P^{n})=(h_{0},...,h_{n})=(b_{0},...,b_{n})$
where $b_{i}=\dim H^{i}(M^{n};{\mathbb{Z}}_{2})$.
###### Remark 2.2.
We see from Theorem 2.1 that the Poincaré duality $b_{i}=b_{n-i}$ agrees with
the Dehn-Sommerville relation $h_{i}=h_{n-i}$.
###### Example 2.1.
For the prism $\mathrm{P}^{3}(m)$, ${\bf h}(\mathrm{P}^{3}(m))=(1,m-1,m-1,1)$,
so any small cover over $\mathrm{P}^{3}(m)$ has the mod 2 Betti numbers
$(b_{0},b_{1},b_{2},b_{3})=(1,m-1,m-1,1)$.
### 2.3. Stanley-Reisner face ring and equivariant cohomology
Stanley-Reisner face ring is a basic combinatorial invariant, and equivariant
cohomology is an essential invariant in the theory of transformation groups.
Davis-Januszkiewicz theory indicates that these two kinds of invariants are
also essentially consistent in the setting of small covers.
Let $P^{n}$ be a simple convex polytope with facet set
$\mathcal{F}(P^{n})=\\{F_{1},...,F_{\ell}\\}$. Following [DJ], the Stanley-
Reisner face ring of $P^{n}$ over $\mathbb{Z}_{2}$, denoted by
$\mathbb{Z}_{2}(P^{n})$, is defined as follows:
$\mathbb{Z}_{2}(P^{n})=\mathbb{Z}_{2}[F_{1},...,F_{\ell}]/I$
where the $F_{i}$’s are regarded as indeterminates of degree one, and $I$ is a
homogenous ideal generated by all sequence free monomials of the form
$F_{i_{1}}\cdots F_{i_{s}}$ with $F_{i_{1}}\cap\cdots\cap
F_{i_{s}}=\emptyset$.
###### Example 2.2.
Let $P^{n}$ be an $n$-simplex $\Delta^{n}$ with $n+1$ facets
$F_{1},...,F_{n+1}$. Then
$\mathbb{Z}_{2}(\Delta^{n})=\mathbb{Z}_{2}[F_{1},...,F_{n+1}]/(F_{1}\cdots
F_{n+1}).$
###### Example 2.3.
Let $F_{1},...,F_{2n}$ be $2n$ facets of an $n$-cube $I^{n}$ with $F_{i}\cap
F_{i+n}=\emptyset,i=1,...,n$. Then
$\mathbb{Z}_{2}(I^{n})=\mathbb{Z}_{2}[F_{1},...,F_{2n}]/(F_{i}F_{i+n}|i=1,...,n).$
###### Theorem 2.3 (Davis-Januszkiewicz).
Let $\pi:M^{n}\longrightarrow P^{n}$ be a small cover over a simple convex
polytope $P^{n}$. Then its equivariant cohomology
$H^{*}_{\mathbb{Z}_{2}^{n}}(M^{n};\mathbb{Z}_{2})\cong\mathbb{Z}_{2}(P^{n}).$
### 2.4. Ordinary cohomology
Let $\pi:M^{n}\longrightarrow P^{n}$ be a small cover over a simple convex
polytope $P^{n}$ with $\mathcal{F}(P^{n})=\\{F_{1},...,F_{\ell}\\}$, and
$\lambda:\mathcal{F}(P^{n})\longrightarrow\mathbb{Z}_{2}^{n}$ its
$\mathbb{Z}_{2}^{n}$-coloring. Now let us extend
$\lambda:\mathcal{F}(P^{n})\longrightarrow\mathbb{Z}_{2}^{n}$ to a linear map
$\widetilde{\lambda}:\mathbb{Z}_{2}^{\ell}\longrightarrow\mathbb{Z}_{2}^{n}$
by replacing $\\{F_{1},...,F_{\ell}\\}$ by the basis
$\\{e_{1},...,e_{\ell}\\}$ of $\mathbb{Z}_{2}^{\ell}$. Then
$\widetilde{\lambda}:\mathbb{Z}_{2}^{\ell}\longrightarrow\mathbb{Z}_{2}^{n}$
is surjective, and $\widetilde{\lambda}$ can be regarded as an
$n\times\ell$-matrix $(\lambda_{ij})$, which is written as follows:
$(\lambda(F_{1}),...,\lambda(F_{\ell})).$
It is well-known that
$H_{1}(B\mathbb{Z}_{2}^{\ell};\mathbb{Z}_{2})=H_{1}(E\mathbb{Z}_{2}^{n}\times_{\mathbb{Z}_{2}^{n}}M^{n};\mathbb{Z}_{2})=\mathbb{Z}_{2}^{\ell}$
and $H_{1}(B\mathbb{Z}_{2}^{n};\mathbb{Z}_{2})=\mathbb{Z}_{2}^{n}$. So one has
that
$p_{*}:H_{1}(E\mathbb{Z}_{2}^{n}\times_{\mathbb{Z}_{2}^{n}}M^{n};\mathbb{Z}_{2})\longrightarrow
H_{1}(B\mathbb{Z}_{2}^{n};\mathbb{Z}_{2})$ can be identified with
$\widetilde{\lambda}:\mathbb{Z}_{2}^{\ell}\longrightarrow\mathbb{Z}_{2}^{n}$,
where $p:E\mathbb{Z}_{2}^{n}\times_{\mathbb{Z}_{2}^{n}}M^{n}\longrightarrow
B\mathbb{Z}_{2}^{n}$ is the fibration of the Borel construction associating to
the universal principal $\mathbb{Z}_{2}^{n}$-bundle
$E\mathbb{Z}_{2}^{n}\longrightarrow B\mathbb{Z}_{2}^{n}$. Furthermore,
$p^{*}:H^{1}(B\mathbb{Z}_{2}^{n};\mathbb{Z}_{2})\longrightarrow
H^{1}_{\mathbb{Z}_{2}^{n}}(M^{n};\mathbb{Z}_{2})$ is identified with the dual
map
$\widetilde{\lambda}^{*}:\mathbb{Z}_{2}^{n*}\longrightarrow\mathbb{Z}_{2}^{\ell*}$,
where $\widetilde{\lambda}^{*}=\widetilde{\lambda}^{\top}$ as matrices.
Therefore, column vectors of $\widetilde{\lambda}^{*}$ can be understood as
linear combinations of $F_{1},...,F_{\ell}$ in the face ring
$\mathbb{Z}_{2}(P^{n})=\mathbb{Z}_{2}[F_{1},...,F_{\ell}]/I$. Write
$\lambda_{i}=\lambda_{i1}F_{1}+\cdots+\lambda_{i\ell}F_{\ell}.$
Let $J_{\lambda}$ be the homogeneous ideal $(\lambda_{1},...,\lambda_{n})$ in
$\mathbb{Z}_{2}[F_{1},...,F_{\ell}]$. Davis and Januszkiewicz calculated the
ordinary cohomology of $M^{n}$, which is stated as follows.
###### Theorem 2.4 (Davis-Januszkiewicz).
Let $\pi:M^{n}\longrightarrow P^{n}$ be a small cover over a simple convex
polytope $P^{n}$. Then its ordinary cohomology
$H^{*}(M^{n};\mathbb{Z}_{2})\cong\mathbb{Z}_{2}[F_{1},...,F_{\ell}]/I+J_{\lambda}.$
The following result which will be used later is due to Nakayama and Nishimura
[NN].
###### Proposition 2.5.
Let $\pi:M^{n}\longrightarrow P^{n}$ be a small cover over a simple convex
polytope $P^{n}$, and
$\lambda:\mathcal{F}(P^{n})=\\{F_{1},...,F_{\ell}\\}\longrightarrow\mathbb{Z}_{2}^{n}$
its $\mathbb{Z}_{2}^{n}$-coloring. Then $M^{n}$ is orientable if and only if
there exists an automorphism $\sigma\in\text{\rm GL}(n,\mathbb{Z}_{2})$ such
that $\lambda^{\prime}=\sigma\circ\lambda$ satisfies
$\sum_{j=1}^{n}\lambda^{\prime}_{jl}\equiv 1\mod 2$ for all $1\leq l\leq\ell$,
where
$\widetilde{\lambda^{\prime}}=(\lambda^{\prime}_{ij}):\mathbb{Z}_{2}^{\ell}\longrightarrow\mathbb{Z}_{2}^{n}$
is the linear extension of $\lambda^{\prime}$, as before.
## 3\. Sector Method
Each point of a simple convex polytope $P^{n}$ has a neighborhood which is
affinely isomorphic to an open subset of ${\mathbb{R}}^{n}_{\geq 0}$, so
$P^{n}$ is an $n$-dimensional nice manifold with corners (see [D]). An
automorphism of $P^{n}$ is a self-homeomorphism of $P^{n}$ as a manifold with
corners, and by $\text{Aut}(P^{n})$ we denote the group of automorphisms of
$P^{n}$. All faces of $P^{n}$ forms a poset by inclusion. An automorphism of
$\mathcal{F}(P^{n})$ is a bijection from $\mathcal{F}(P^{n})$ to itself which
preserves the poset structure of all faces of $P^{n}$, and by
$\text{Aut}(\mathcal{F}(P^{n}))$ we denote the group of automorphisms of
$\mathcal{F}(P^{n})$. Each automorphism of $\text{Aut}(P^{n})$ naturally
induces an automorphism of $\mathcal{F}(P^{n})$. It is well-known (see [BP] or
[Z]) that two simple convex polytopes are combinatorially equivalent if and
only if they are homeomorphic as manifolds with corners. Thus, the natural
homomorphism
$\Phi:\text{Aut}(P^{n})\longrightarrow\text{Aut}(\mathcal{F}(P^{n}))$ is
surjective.
###### Definition 3.1.
Let $P$ be a simple $n$-polytope and $S$ be a simple $(n-1)$-polytope. An
embedding $i:S\longrightarrow P$ is called a _sector_ if the following
conditions are satisfied:
1. (a)
$P\backslash i(S)$ have two connected components such that $i(S)$ is the
common facet of $P_{1}$ and $P_{2}$, where $P_{1},P_{2}$ denote the closures
of the two components respectively, called the sub-polytopes of $i$.
2. (b)
For every face $F\subset S$ of dimension $k$, $i(F)$ is the subset of a unique
$(k+1)$-dimensional face of $P^{n}$.
Suppose that $i:S\longrightarrow P$ is a sector with $P_{1}$ and $P_{2}$ as
its two sub-polytopes. Let $i_{r}:S\rightarrow P_{r},r=1,2$, be the embeddings
induced by $p\mapsto i(p)$. Then
$P=P_{1}\coprod P_{2}/i_{1}(s)\sim i_{2}(s).$
Furthermore, we can get an induced map
$i_{*}:\mathcal{F}(S)\rightarrow\mathcal{F}(P)$, which is poset structure-
preserving. Of course, $i_{*}$ is injective. Now set
$\bar{\lambda}:=\lambda\circ i_{*}$, which is called the _derived coloring_ of
$i_{*}$ and $\lambda$. Obviously, the derived coloring $\bar{\lambda}$ assigns
to each facet of $S$ a vector in $\mathbb{Z}_{2}^{n}$ and satisfies the
independence condition: whenever the intersection of some facets
$F_{l_{1}},...,F_{l_{r}}$ in $\mathcal{F}(S)$ is nonempty,
$\bar{\lambda}(F_{l_{1}}),...,\bar{\lambda}(F_{l_{r}})$ are linearly
independent. Let $(\psi,\rho)$ be a pair of $\psi\in\text{Aut}(S)$ and
$\rho\in\text{GL}(n,\mathbb{Z}_{2})$. $(\psi,\rho)$ is called an _auto-
equivalence_ of $S$ if $\bar{\lambda}\circ\psi=\rho\circ\bar{\lambda}$, where
we just abuse $\varphi$ and the automorphism of $\mathcal{F}(S)$ induced by
$\varphi$.
Now given a $\mathbb{Z}_{2}^{n}$-colored simple $n$-polytope $(P,\lambda)$
with a sector $i:S\longrightarrow P$ and its two sub-polytopes $P_{1},P_{2}$,
and fix $(\psi,\rho)$ an auto-equivalence of $S$. Suppose that $P^{\prime}$ is
another simple $n$-polytope with $j:S\rightarrow P^{\prime}$ another sector,
cutting $P^{\prime}$ into ${P^{\prime}}_{1}$ and ${P^{\prime}}_{2}$, such that
there are $f_{r}:P_{r}\rightarrow{P^{\prime}}_{r},r=1,2$, which are
combinatorially equivalent with $f_{1}\circ i_{1}=j_{1}$ and $f_{2}\circ
i_{2}\circ\psi=j_{2}$. Thus
$P^{\prime}=P_{1}^{\prime}\coprod P_{2}^{\prime}/j_{1}(s)\sim
j_{2}(s)=f_{1}(P_{1})\coprod f_{2}(P_{2})/f_{1}\circ i_{1}(s)\sim f_{2}\circ
i_{2}\circ\psi(s).$
###### Remark 3.1.
Generally $P$ is not combinatorially equivalent to $P^{\prime}$ although
$P_{r}$ is combinatorially equivalent to $P^{\prime}_{r}$ $(r=1,2)$.
Then we can define a $\mathbb{Z}_{2}^{n}$-coloring $\lambda^{\prime}$ on
$P^{\prime}$ as follows: for each facet $F\in\mathcal{F}(P^{\prime})$, if
$F\cap{P_{1}}^{\prime}\neq\emptyset$, then there is a unique facet
$F_{1}\in\mathcal{F}(P)$ such that $f_{1}(F_{1}\cap P_{1})\subset F$.
Similarly, if $F\cap{P^{\prime}}_{2}\neq\emptyset$, then there is also a
unique facet $F_{2}\in\mathcal{F}(P^{n})$ such that $f_{2}(F_{2}\cap
P_{2})\subset F$. Furthermore, define
$\lambda^{\prime}:\mathcal{F}(P^{\prime})\rightarrow\mathbb{Z}_{2}^{n}$ in the
following way:
$\lambda^{\prime}(F)=\begin{cases}\lambda(F_{1})&\text{ if
$F\cap{P_{1}}^{\prime}\neq\emptyset$}\\\ \rho^{-1}\circ\lambda(F_{2})&\text{
if $F\cap{P_{2}}^{\prime}\neq\emptyset$.}\end{cases}$
Such $\lambda^{\prime}$ is well-defined. In fact, if $F$ has nonempty
intersection with both ${P^{\prime}}_{1}$ and $P^{\prime}_{2}$, then $F$ must
lie in the image of $j_{*}$, say $F=j_{*}(f)$ where $f\in\mathcal{F}(S)$. Then
from $f_{1}\circ i_{1}=j_{1}$ we see that $F_{1}=i_{*}(f)$, and from
$f_{2}\circ i_{2}\circ\psi=j_{2}$ we see that $F_{2}=i_{*}(\psi(f))$. So
$\lambda(F_{1})=\bar{\lambda}(f)=\rho^{-1}\circ\bar{\lambda}\circ\psi(f)=\rho^{-1}\circ\lambda\circ
i_{*}\circ\psi(f)=\rho^{-1}\circ\lambda(F_{2})$ as desired. In summary, we now
have two colored polytopes $(P,\lambda)$ and
$({P}^{\prime},\lambda^{\prime})$. Then using the reconstruction of small
covers, we obtain two small covers
$M(\lambda)=P\times\mathbb{Z}_{2}^{n}/(p,v)\sim(p,v+\lambda(F))\text{ for
}p\in F\in\mathcal{F}(P)$ and
$M(\lambda^{\prime})=P^{\prime}\times\mathbb{Z}_{2}^{n}/(p,v)\sim(p,v+\lambda^{\prime}(F))\text{
for }p\in F\in\mathcal{F}(P^{\prime})$.
Now let us look at two small covers $\pi:M(\lambda)\longrightarrow P$ and
$\pi^{\prime}:M(\lambda^{\prime})\longrightarrow P^{\prime}$.
Set $M_{r}={\pi}^{-1}(P_{r}),r=1,2$. Let
$\mathcal{S}=S\times\mathbb{Z}_{2}^{n}/\\{(s,v)\sim(s,v+\bar{\lambda}(f))|s\in
f\in\mathcal{F}(S)\\}.$
Then it is easy to see that $\mathcal{S}$ is an $(n-1)$-dimensional closed
manifold (but possibly disconnected), called the sector manifold here. The
sector $i:S\longrightarrow P$ naturally induces an embedding
$\iota:\mathcal{S}\hookrightarrow M(\lambda)$ defined by
$\\{(s,v)\\}\longmapsto\\{(i(s),v)\\}$, and $i_{r}:S\longrightarrow P_{r}$
also induces the embedding $\iota_{r}:\mathcal{S}\hookrightarrow M_{r}$
$(r=1,2)$. Obviously, $\partial
M_{r}=\iota_{r}(\mathcal{S})={\pi}^{-1}(i(S))=\iota(\mathcal{S})$. Using this
terminology one can write
$M(\lambda)=M_{1}\coprod M_{2}/\iota_{1}(x)\sim\iota_{2}(x)=M_{1}\coprod
M_{2}/\\{(i_{1}(s),v)\\}\sim\\{(i_{2}(s),v)\\},$
i.e., $M(\lambda)$ is obtained by gluing $M_{1}$ and $M_{2}$ together along
their common boundary $\iota(\mathcal{S})$ via $\iota_{1}$ and $\iota_{2}$.
Similarly, set $M^{\prime}_{r}={\pi^{\prime}}^{-1}(P^{\prime}_{r}),r=1,2$.
Since ${P^{\prime}}_{r}$ is combinatorially equivalent to $P_{r}$ for $r=1,2$,
$M^{\prime}_{r}$ is homeomorphic to $M_{r}$. More precisely,
$f_{1}:P_{1}\longrightarrow P^{\prime}_{1}$ induces an equivariant
homeomorphism $\widetilde{f}_{1}:M_{1}\longrightarrow M^{\prime}_{1}$ by
mapping $x=\\{(p,v)\\}\longmapsto\\{(f_{1}(p),v)\\}$, while
$f_{2}:P_{2}\longrightarrow P^{\prime}_{2}$ induces a weakly equivariant
homeomorphism $\widetilde{f}_{2}:M_{2}\longrightarrow M^{\prime}_{2}$ by
mapping $x=\\{(p,v)\\}\longmapsto\\{(f_{2}(p),\rho^{-1}(v))\\}$. Let
$\iota^{\prime}_{r}$ $(r=1,2)$ be the embedding $\mathcal{S}\hookrightarrow
M^{\prime}_{r}$ induced by $j_{r}:S\longrightarrow P^{\prime}_{r}$ and
$\iota^{\prime}$ the embedding $\mathcal{S}\hookrightarrow
M(\lambda^{\prime})$ induced by $j:S\longrightarrow P^{\prime}$. Then
$\partial
M^{\prime}_{r}=\iota^{\prime}_{r}(\mathcal{S})={\pi^{\prime}}^{-1}(i(S))=\iota^{\prime}(\mathcal{S})$.
Furthermore, one has that
$M(\lambda^{\prime})=M^{\prime}_{1}\coprod
M^{\prime}_{2}/\iota^{\prime}_{1}(x)\sim\iota^{\prime}_{2}(x)=M^{\prime}_{1}\coprod
M^{\prime}_{2}/\\{(j_{1}(s),v)\\}\sim\\{(j_{2}(s),v)\\}.$
On the other hand, take an arbitrary $v_{0}\in\mathbb{Z}_{2}^{n}$, using the
relation $\bar{\lambda}\circ\psi=\rho\circ\bar{\lambda}$ we see easily that
the auto-equivalence $(\psi,\rho)$ of $S$ also induces a weakly equivariant
homeomorphism $\Psi:\mathcal{S}\rightarrow\mathcal{S}$ defined by
$\\{(s,v)\\}\mapsto\\{(\psi(s),\rho(v)+v_{0})\\}$, so that
$\iota_{2}\circ\Psi$ gives a new embedding of $\mathcal{S}$ in $M_{2}$ and
$\widetilde{f}_{2}\circ\iota_{2}\circ\Psi$ also gives a new embedding of
$\mathcal{S}$ in $M^{\prime}_{2}$. Let $\widetilde{M}(\lambda)=M_{1}\coprod
M_{2}/\iota_{1}(x)\sim\iota_{2}(\Psi(x))$. The one has that
###### Lemma 3.1.
$\widetilde{M}(\lambda)$ is homeomorphic to $M(\lambda^{\prime})$.
###### Proof.
Let $z=\\{(p,v)\\}\in\widetilde{M}(\lambda)$. Define
$\Pi:\widetilde{M}(\lambda)\longrightarrow M(\lambda^{\prime})$ by
$\Pi(z)=\begin{cases}\widetilde{f}_{1}(z)&\text{if $z\in M_{1}$}\\\
\widetilde{f^{\prime}}_{2}(z)&\text{if $z\in M_{2}$}\end{cases}$
where
$\widetilde{f^{\prime}}_{2}(z)=\\{(f_{2}(p),\rho^{-1}(v)+\rho^{-1}(v_{0}))\\}$.
To show that $\Pi$ is a homeomorphism, it suffices to prove that for
$x\in\mathcal{S}$, if $\iota_{1}(x)\sim\iota_{2}(\Psi(x))$ in
$\widetilde{M}(\lambda)$, then
$\Pi\circ\iota_{1}(x)\sim\Pi\circ\iota_{2}(\Psi(x))$ in $M(\lambda^{\prime})$.
Since $\Pi\circ\iota_{1}=\widetilde{f}_{1}\circ\iota_{1}=\iota^{\prime}_{1}$,
it needs only to check that $\Pi\circ\iota_{2}\circ\Psi=\iota^{\prime}_{2}$.
Let $x=\\{(s,v)\\}$. Then
$\Pi\circ\iota_{2}\circ\Psi(x)=\widetilde{f^{\prime}}_{2}\circ\iota_{2}\circ\Psi(x)=\widetilde{f^{\prime}}_{2}\circ\iota_{2}(\\{(\psi(s),\rho(v)+v_{0})\\})=\widetilde{f^{\prime}}_{2}(\\{(i_{2}\circ\psi(s),\rho(v)+v_{0})\\}=\\{(f_{2}\circ
i_{2}\circ\psi(s),\rho^{-1}(\rho(v)+v_{0})+\rho^{-1}(v_{0}))\\}=\\{(f_{2}\circ
i_{2}\circ\psi(s),v)\\}=\iota^{\prime}_{2}(x)$, as desired. ∎
###### Remark 3.2.
It is easy to see that generally $\widetilde{M}(\lambda)$ is not (weakly)
equivariant homeomorphic to $M(\lambda^{\prime})$ except that $\rho$ is the
identity of $\mathbb{Z}_{2}^{n}$. Also, clearly the definition of $\Psi$
depends on the auto-equivalence $(\psi,\rho)$ of $S$ and $v_{0}$, so we may
write $\Psi$ as $\Psi(\psi,\rho,v_{0})$ to indicate this dependence.
Now let us consider when $M(\lambda)$ and $M(\lambda^{\prime})$ are
homeomorphic.
###### Definition 3.2.
We say that $\Psi(\psi,\rho,v_{0})$ is extendable to a self-homeomorphism of
$M_{2}$ if there is a self-homeomorphism $\widetilde{\Psi}:M_{2}\rightarrow
M_{2}$ such that $\widetilde{\Psi}\circ\iota_{2}=\iota_{2}\circ\Psi$.
###### Theorem 3.2.
If $\Psi(\psi,\rho,v_{0})$ is extendable to a self-homeomorphism of $M_{2}$,
then $M(\lambda)$ is homeomorphic to $M(\lambda^{\prime})$.
###### Proof.
Let $\widetilde{\Psi}:M_{2}\rightarrow M_{2}$ be a self-homeomorphism with
$\widetilde{\Psi}\circ\iota_{2}=\iota_{2}\circ\Psi$. By Lemma 3.1, one may
identify $M(\lambda^{\prime})$ with $\widetilde{M}(\lambda)=M_{1}\coprod
M_{2}/\iota_{1}(x)\sim\iota_{2}(\Psi(x))$. Define $H:M(\lambda)\longrightarrow
M(\lambda^{\prime})$ by
$H(x)=\begin{cases}x&\text{if $x\in M_{1}\subset M(\lambda)$}\\\
\widetilde{\Psi}(x)&\text{if $x\in M_{2}\subset M(\lambda)$.}\end{cases}$
An easy observation shows that $H$ is a well-defined homeomorphism. ∎
###### Remark 3.3.
Take an automorphism $\widetilde{\psi}$ of $P_{2}$ such that
$\widetilde{\psi}(i_{2}(S))=i_{2}(S)$. Then the restriction
$\widetilde{\psi}|_{i_{2}(S)}$ gives an automorphism of $i_{2}(S)$. If we can
choose $(\psi,\rho)$ with the property that $\psi\circ
i_{2}=\widetilde{\psi}|_{i_{2}(S)}$ and
$\rho\circ\lambda=\lambda\circ\widetilde{\psi}$, then it is easy to check that
$(\widetilde{\psi},\rho)$ can induce a self-homeomorphism
$\widetilde{\Psi}(\widetilde{\psi},\rho,v_{0})$ of $M_{2}$, which is defined
by $\\{(p,v)\\}\longmapsto\\{(\widetilde{\psi}(p),\rho(v)+v_{0})\\}$, so that
$\widetilde{\Psi}(\widetilde{\psi},\rho,v_{0})$ is an extension of
$\Psi(\psi,\rho,v_{0})$. In this case, we see that $M(\lambda)$ is
homeomorphic to $M(\lambda^{\prime})$. However, the condition
$\rho\circ\lambda=\lambda\circ\widetilde{\psi}$ results in a little bit
difficulty for the choice of $(\widetilde{\psi},\rho)$.
Next let us further analyze $\Psi(\psi,\rho,v_{0})$.
###### Theorem 3.3.
If $\Psi(\psi,\rho,v_{0})$ is isotopic to the identity, then $M(\lambda)$ is
homeomorphic to $M(\lambda^{\prime})$.
###### Proof.
Since the image of the embedding $i:S\rightarrow P$ does not contain any
vertex of $P$, we can extend $i$ to an embedding
$\widetilde{i}:S\times[-1,1]\rightarrow P$ such that
$i=\widetilde{i}(\cdot,0)$ and each $\widetilde{i}(\cdot,t)$ is a sector. We
can further assume that $\widetilde{i}(S\times[-1,0])\subset P_{1}$ and
$\widetilde{i}(S\times[0,1])\subset P_{2}$. Now in the world of topology,
$\widetilde{i}$ corresponds to an embedding
$\widetilde{\iota}:\mathcal{S}\times[-1,1]\rightarrow M(\lambda)$ such that
$\iota=\widetilde{\iota}(\cdot,0)$, and
$\widetilde{\iota}(\mathcal{S}\times[0,1])\subset M_{2}$. Now let
$\bar{\Psi}:\mathcal{S}\times[0,1]\rightarrow\mathcal{S}$ be an isotopy such
that $\bar{\Psi}(\cdot,0)=\Psi$ and $\bar{\Psi}(\cdot,1)=\text{id}$. Then
define $\widetilde{\Psi}:M_{2}\rightarrow M_{2}$ by
$\widetilde{\Psi}(y)=\begin{cases}\widetilde{\iota}(\bar{\Psi}(x,t),t)&\text{if
$y=\widetilde{\iota}(x,t)\in\text{Im}\widetilde{\iota}$}\\\ y&\text{if
$y\not\in\text{Im}\widetilde{\iota}$.}\end{cases}$
One checks easily that $\widetilde{\Psi}$ is a self-homeomorphism of $M_{2}$
such that $\widetilde{\Psi}|_{\iota_{2}(\mathcal{S})}=\iota_{2}\circ\Psi$.
Moreover, Theorem 3.3 follows by applying Theorem 3.2. ∎
###### Definition 3.3.
$\Psi(\psi,\rho,v_{0})$ is called a _good twist_ if $\Psi(\psi,\rho,v_{0})$ is
isotopic to the identity.
We note that the homeomorphism type of $M(\lambda^{\prime})$ doesn’t depend on
the choice of $v_{0}$. So to apply Theorem 3.3 we can choose a suitable
$v_{0}$ such that $\Psi(\psi,\rho,v_{0})$ meets the conditions.
###### Remark 3.4.
The sector method above provides a way of how to construct a homeomorphism
between two small covers $M(\lambda)\longrightarrow P$ and
$M(\lambda^{\prime})\longrightarrow P^{\prime}$ regardless of whether $P$ is
combinatorially equivalent to $P^{\prime}$ or not. In addition, the sector
method also gives an approach of how to construct a new colored polytope
$(P^{\prime},\lambda^{\prime})$ from $(P,\lambda)$ by the auot-equivalence
$(\psi,\rho)$ at the sector $i:S\longrightarrow P$.
## 4\. Application to prisms: Rectangular Sector Method
The objective of this section is to give the application of the sector method
to prisms.
Let $\mathrm{P}^{3}(m)$ denote a 3-dimensional prism that is the product of
$[0,1]$ and an $m$-gon where $m\geq 3$. When $m\neq 4$, let $c,f$ (the
_ceiling_ and the _floor_) be the two 2-faces of $\mathrm{P}^{3}(m)$ that are
$m$-gons. For the 3-cube (i.e., $m=4$), we specify two opposite 2-faces and
distinguish them as ceiling and floor. For convenience, we identify other
2-faces (i.e., side 2-faces) with $s_{1},...,s_{m}$ in the natural way. Let
$\Lambda(\mathrm{P}^{3}(m))=\\{\lambda\big{|}\lambda\text{ is a
${\mathbb{Z}}_{2}^{3}$-coloring on $\mathrm{P}^{3}(m)$}\\}.$
### 4.1. Rectangular sector
Generally, any polygon can become a sector in the setting of all 3-polytopes.
However, here we shall put attention on rectangular sectors because this will
be sufficient enough to the classification of all small covers over prisms.
Throughout the following, choose the rectangle
$S=\\{(x,y)\in\mathbb{R}^{2}\big{|}|x|\leq 1,|y|\leq 1\\}$ in a plane
$\mathbb{R}^{2}$. Clearly $S$ can always be embedded as a sector in any
$\mathbb{Z}_{2}^{3}$-colored simple 3-polytope $(P^{3},\lambda)$. Fix
$\\{e_{1},e_{2},e_{3}\\}$ as a basis of $\mathbb{Z}_{2}^{3}$, then it is easy
to see that up to Davis-Januszkiewicz equivalence, all possible derived
colorings $\bar{\lambda}:\mathcal{F}(S)\longrightarrow\mathbb{Z}_{2}^{3}$ and
corresponding sector manifolds $\mathcal{S}$ can be stated as follows:
| top edge | left edge | bottom edge | right edge | sector manifold $\mathcal{S}$
---|---|---|---|---|---
$\bar{\lambda}_{1}$ | $e_{1}$ | $e_{2}$ | $e_{1}$ | $e_{2}$ | union of 2 tori
$\bar{\lambda}_{2}$ | $e_{1}$ | $e_{2}$ | $e_{1}+e_{2}$ | $e_{2}$ | union of 2 Klein bottles
$\bar{\lambda}_{3}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{2}$ | torus
$\bar{\lambda}_{4}$ | $e_{3}$ | $e_{1}$ | $e_{1}+e_{3}$ | $e_{2}$ | Klein bottle
$\bar{\lambda}_{5}$ | $e_{3}$ | $e_{1}$ | $e_{1}+e_{2}+e_{3}$ | $e_{2}$ | torus
It is well-known that the symmetric group $\mathrm{Aut}(S)$ of $S$ as a 4-gon
is isomorphic to the dihedral group $\mathcal{D}_{4}$ of order 8, which just
contains four reflections. Clearly, each reflection of $S$ may be expressed as
a matrix. For example, the reflection along $y$-axis can be written as
$\text{diag}(-1,1)$, and the reflection along $x$-axis can be written as
$\text{diag}(1,-1)$.
### 4.2. Construction of new colored polytopes from
$(\mathrm{P}^{3}(m),\lambda)$
Given a pair $(\mathrm{P}^{3}(m),\lambda)$ in $\Lambda(\mathrm{P}^{3}(m))$. We
use the convention that all embedded rectangular sectors of
$(\mathrm{P}^{3}(m),\lambda)$ used here are always orthogonal to the ceiling
and floor of $\mathrm{P}^{3}(m)$. Given such a sector
$i:S\longrightarrow\mathrm{P}^{3}(m)$ (note: here we don’t need that $i$ must
map the top edge and the bottom edge of $S$ into the ceiling and the floor of
$\mathrm{P}^{3}(m)$, respectively), it is easy to see that there are two side
faces $s_{k},s_{l}$ $(k\neq l$ and $k<l)$ such that
$S\longrightarrow\mathrm{P}^{3}(m)$ is essentially determined by $s_{k}$ and
$s_{l}$ and it is called the sector _at_ $s_{k},s_{l}$ and is denoted by
$i(k,l)$, where $S=\\{(x,y)\in\mathbb{R}^{2}\big{|}|x|\leq 1,|y|\leq 1\\}$.
Let $P_{1}$ and $P_{2}$ be two prisms cut out by
$i(k,l):S\longrightarrow\mathrm{P}^{3}(m)$ from $\mathrm{P}^{3}(m)$.
Throughout the following, one also uses the convention that $P_{2}$ contains
side faces $s_{k},...,s_{l}$ of $\mathrm{P}^{3}(m)$.
Now, using the sector method we discuss how to construct new colored
3-polytopes $(\mathrm{P}^{\prime},\lambda^{\prime})$ from
$(\mathrm{P}^{3}(m),\lambda)$ by auto-equivalences $(\psi,\rho)$ at the sector
$i(k,l):S\longrightarrow\mathrm{P}^{3}(m)$. For our purpose, we wish that (1)
$\mathrm{P}^{\prime}$ is still combinatorially equivalent to
$\mathrm{P}^{3}(m)$, and (2) $\Psi(\psi,\rho,v_{0})$ is a good twist, so that
each $M(\lambda^{\prime})$ is homeomorphic to $M(\lambda)$ by Theorem 3.3.
This will depend upon the choices of $\psi$ and $\rho$. Actually, the
construction of $\mathrm{P}^{\prime}$ depends upon the choice of $\psi$, and
the definition of $\lambda^{\prime}$ depends upon the choice of $\rho$. In
particular, $v_{0}$ will provide a convenience for the choices of
$(\psi,\rho)$.
Convention: $\psi=\text{\rm diag}(-1,1)$ or $\text{\rm diag}(1,1)$ means that
$i(k,l)$ maps the top edge and the bottom edge of $S$ into the ceiling and the
floor of $\mathrm{P}^{3}(m)$, respectively, and $\psi=\text{\rm diag}(1,-1)$
means that $i(k,l)$ maps the left edge and the right edge of $S$ into the
ceiling and the floor of $\mathrm{P}^{3}(m)$, respectively.
###### Lemma 4.1.
If $\psi=\text{\rm diag}(-1,1)$ or $\text{\rm diag}(1,-1)$ or $\text{\rm
diag}(1,1)$, then we can construct a new polytope $\mathrm{P}^{\prime}$ from
$\mathrm{P}^{3}(m)$ by $\psi$ at the sector
$i(k,l):S\longrightarrow\mathrm{P}^{3}(m)$ such that $\mathrm{P}^{\prime}$ is
combinatorially equivalent to $\mathrm{P}^{3}(m)$.
###### Proof.
The case $\psi=\text{\rm diag}(1,1)$ is trivial. Actually, in this case we can
just choose $P^{\prime}_{r}=P_{r}$ and let $f_{r}:P_{r}\longrightarrow
P^{\prime}_{r}$ be the identity, where $r=1,2$. If $\psi=\text{diag}(-1,1)$,
to construct $\mathrm{P}^{\prime}$, we first choose $P^{\prime}_{1}=P_{1}$ and
$f_{1}$ as the identity from $P_{1}\longrightarrow P^{\prime}_{1}$, and then
choose $P^{\prime}_{2}$ as the image of mirror reflection $R$ of $P_{2}$ along
a 2-plane $H$ orthogonal to the ceiling and floor of $P_{2}$ with $H\cap
P_{2}=\emptyset$ (i.e., intuitively $P^{\prime}_{2}$ is obtained by reserving
the ordering of the side faces $s_{k},...,s_{l}$ of $P_{2}$) and $f_{2}$ as
the homeomorphism induced by the reflection $R$, as shown in the following
figure.
Now, we clearly see that $\mathrm{P}^{\prime}$ can be defined as $P_{1}\coprod
P^{\prime}_{2}/i_{1}(s)\sim f_{2}\circ i_{2}\circ\psi(s)$, which is
combinatorially equivalent to $\mathrm{P}^{3}(m)$. In a similar way, we can
prove the case $\psi=\text{diag}(1,-1)$. ∎
Now suppose that $\mathrm{P}^{\prime}$, which is just constructed from
$\mathrm{P}^{3}(m)$ by a $\psi$ at the sector
$i(k,l):S\longrightarrow\mathrm{P}^{3}(m)$, is combinatorially equivalent to
$\mathrm{P}^{3}(m)$. Then, as stated in Section 3, we can use $\rho$ to give a
coloring $\lambda^{\prime}$ on $\mathrm{P}^{\prime}$ as long as $\rho$
satisfies the equation $\rho\circ\bar{\lambda}=\bar{\lambda}\circ\psi$. To
guarantee that $M(\lambda^{\prime})$ is homeomorphic to $M(\lambda)$, we need
choose $(\psi,\rho)$ carefully such that $\Psi(\psi,\rho,v_{0})$ is a good
twist.
Based upon the possible values of $\bar{\lambda}$ (see the table above), we
find some good twists and list them as follows:
Sector | $(S,\bar{\lambda})$ | $\Psi(\psi,\rho,v_{0})$ | $\psi$ | $\rho(e_{1})$ | $\rho(e_{2})$ | $\rho(e_{3})$ | $v_{0}$
---|---|---|---|---|---|---|---
$S(1)$ | $(S,\bar{\lambda}_{1})$ | $\Psi(\psi_{1},\rho_{1},v_{0}^{(1)})$ | $\text{diag}(1,-1)$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{1}$
$S(2_{1})$ | $(S,\bar{\lambda}_{2})$ | $\Psi(\psi_{21},\rho_{21},v_{0}^{(21)})$ | $\text{diag}(1,1)$ | $e_{1}$ | $e_{2}$ | $e_{3}+e_{2}$ | $0$
$S(2_{2})$ | $(S,\bar{\lambda}_{2})$ | $\Psi(\psi_{22},\rho_{22},v_{0}^{(22)})$ | $\text{diag}(1,-1)$ | $e_{1}+e_{2}$ | $e_{2}$ | $e_{3}$ | $e_{1}$
$S(3_{1})$ | $(S,\bar{\lambda}_{3})$ | $\Psi(\psi_{31},\rho_{31},v_{0}^{(31)})$ | $\text{diag}(-1,1)$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{2}$
$S(3_{2})$ | $(S,\bar{\lambda}_{3})$ | $\Psi(\psi_{32},\rho_{32},v_{0}^{(32)})$ | $\text{diag}(1,-1)$ | $e_{3}$ | $e_{2}$ | $e_{1}$ | $e_{1}$
$S(4)$ | $(S,\bar{\lambda}_{4})$ | $\Psi(\psi_{4},\rho_{4},v_{0}^{(4)})$ | $\text{diag}(1,-1)$ | $e_{1}$ | $e_{2}$ | $e_{3}+e_{1}$ | $e_{3}$
$S(5)$ | $(S,\bar{\lambda}_{5})$ | $\Psi(\psi_{5},\rho_{5},v_{0}^{(5)})$ | $\text{diag}(-1,1)$ | $e_{2}$ | $e_{1}$ | $e_{3}$ | $e_{1}$
###### Remark 4.1.
It should be pointed out that we have not listed all such good twists. At
least we omit compositions of good twists that have already appeared in the
above table. However, as we shall see, those good twists listed above are
sufficient in the further applications.
Here we only give a detailed argument of $S(1)$ because all other cases can be
checked similarly. In $S(1)$, we know that
$\mathcal{S}={[-1,1]}^{2}\times\mathbb{Z}_{2}^{3}/\sim$ is the disjoint union
of two tori. Then we can write
$\mathcal{S}=\\{(z_{1},z_{2},\alpha)\big{|}z_{k}\in
S^{1}\subset{\mathbb{C}},k=1,2,\alpha\in\\{0,1\\}\\}$, such that for
$\mathbf{x}=\\{((x_{1},x_{2}),v=a_{1}e_{2}+a_{2}e_{1}+a_{3}e_{3})\\}\in\mathcal{S}$,
there is the following one-one correspondence
$z_{k}(\mathbf{x})=\begin{cases}\mathrm{exp}(\mathbf{i}x_{k}\pi/2)&\text{ if
}a_{k}=0\\\ \mathrm{exp}(\mathbf{i}(\pi-x_{k}\pi/2))&\text{ if }a_{k}\neq
0\end{cases}$
for $k=1,2$ and $\alpha(\mathbf{x})=a_{3}$. Now we consider the map
$\Psi(\psi,\rho,v_{0})$ where $\psi(x_{1},x_{2})=(x_{1},-x_{2})$,
$\rho=\text{id}$ and $v_{0}=e_{1}$. An easy computation yields that
$\Psi(z_{1},z_{2},\alpha)=(z_{1},-z_{2},\alpha)$, which is clearly isotopic to
the identity via the homotopy $((z_{1}$, $z_{2}$,
$\alpha),t)$$\mapsto$$(z_{1}$, $z_{2}\mathrm{exp}({\mathbf{i}\pi t})$,
$\alpha)$ where $t\in[0,1]$. Thus $\Psi(\psi,\rho,v_{0})$ is a good twist.
## 5\. Operations on coloring sequences and canonical forms
Now we apply the developed rectangular sector method to study small covers
over prisms.
Given a pair $(\mathrm{P}^{3}(m),\lambda)$ in $\Lambda(\mathrm{P}^{3}(m))$ and
a sector $i(k,l):S\longrightarrow\mathrm{P}^{3}(m)$. We have known how to
construct a $(\mathrm{P}^{\prime},\lambda^{\prime})$ from
$(\mathrm{P}^{3}(m),\lambda)$ by an auto-equivalence $(\psi,\rho)$ at the
sector $S\longrightarrow\mathrm{P}^{3}(m)$. Indeed, by Lemma 4.1, if
$\psi=\text{id}$ or $\text{\rm diag}(-1,1)$ or $\text{\rm diag}(1,-1)$, then
$\mathrm{P}^{\prime}$ is also a $\mathrm{P}^{3}(m)$ with the same ceiling and
floor coloring, and $\lambda^{\prime}$ has the same side coloring sequence as
$\lambda$ on sides faces from $s_{l+1}$ to $s_{k-1}$. For $k\leq r\leq l$, if
$\psi=\text{id}$, then $\lambda^{\prime}(s_{r})=\rho^{-1}\lambda(s_{r})$; if
$\psi=\text{\rm diag}(-1,1)$ or $\text{\rm diag}(1,-1)$,
$\lambda^{\prime}(s_{r})=\rho^{-1}\lambda(s_{k+l-r})$, that is, we reflect the
sequence from $s_{k}$ to $s_{l}$ and apply the linear transformation $\rho$.
By Theorem 3.3, when the derived coloring $\bar{\lambda}$ of the sector at
$s_{k},s_{l}$ and $(\psi,\rho)$ match a case in the table of last section, we
can conclude that $M(\lambda^{\prime})$ is homeomorphic to $M(\lambda)$. Thus
we can reduce $(\mathrm{P}^{3}(m),\lambda)$ to
$(\mathrm{P}^{3}(m),\lambda^{\prime})$ without changing the homeomorphism type
of the small cover. In this case, both $(\mathrm{P}^{3}(m),\lambda)$ and
$(\mathrm{P}^{3}(m),\lambda^{\prime})$ are said to be sector-equivalent,
denoted by
$(\mathrm{P}^{3}(m),\lambda)\approx(\mathrm{P}^{3}(m),\lambda^{\prime})$ or
simply $\lambda\approx\lambda^{\prime}$.
Based upon this rectangular sector method, we shall show that
$\Lambda(\mathrm{P}^{3}(m))$ contains some basic colored pairs, called
“canonical forms”, such that any pair in $\Lambda(\mathrm{P}^{3}(m))$ is
sector-equivalent to one of canonical forms. This means that up to
homeomorphism, those canonical forms determine all small covers over
$\mathrm{P}^{3}(m)$.
For a convenience, after fixing the colorings of ceiling and floor, we use the
convention that a coloring on $\mathrm{P}^{3}(m)$ will simply be described as
a sequence by writing its side face colorings in order, keeping in mind that
the first one is next to the last.
###### Definition 5.1.
A coloring $\lambda\in\lambda(\mathrm{P}^{3}(m))$ is said to be
_2-independent_ if all $\lambda(s_{i}),i=1,...,m$, span a 2-dimensional
subspace of $\mathbb{Z}_{2}^{3}$; otherwise it’s said to be _3-independent_.
If $\lambda(c)=\lambda(f)$, then $\lambda$ is said to be _trivial_ ; otherwise
_nontrivial_.
The argument is divided into two cases: (i) $\lambda$ is trivial; (ii)
$\lambda$ is nontrivial.
### 5.1. Trivial colorings
Given a pair $(\mathrm{P}^{3}(m),\lambda)$ in $\Lambda(\mathrm{P}^{3}(m))$,
throughout the following suppose that $\lambda$ is trivial with
$\lambda(c)=\lambda(f)=e_{1}$. Let $\\{\alpha,\beta,e_{1}\\}$ be a basis of
$\mathbb{Z}_{2}^{3}$, and let $\gamma=\alpha+\beta$. Write
$\bar{\alpha}=\alpha+e_{1}$, $\bar{\beta}=\beta+e_{1}$ and
$\bar{\gamma}=\gamma+e_{1}$. We say that $\lambda$ satisfies the property
$(\star)$ if all three letters $\alpha,\beta,\gamma$ (with or without bar we
don’t care) appear in its coloring sequence.
Applying sectors $S(1),S(2_{1}),S(2_{2})$ and $S(3_{2})$ to the trivial
coloring $\lambda$ gives the following four fundamental operations on its
coloring sequence $(\lambda(s_{1}),...,\lambda(s_{m}))$:
1. $\text{O}_{1}$
Take two side faces $s_{k},s_{l}$ $(k<l)$ with the same coloring and then use
$S(1)$ to reflect the coloring sequence of $s_{k},s_{k+1},...,s_{l}$.
2. $\text{O}_{21}$
Take two faces $s_{k},s_{l}$ $(k<l)$ with
$\lambda(s_{k})=\lambda(s_{l})+\lambda(c)$ (without loss of generality, assume
that $\\{\lambda(s_{k}),\lambda(s_{l})\\}=\\{\alpha,\bar{\alpha}\\}$), and
then by using $S(2_{1})$, we can do a linear transform
$(e_{1},\alpha,\beta)\mapsto(e_{1},\alpha,\bar{\beta})$ to change the coloring
sequence of $s_{k},s_{k+1},...,s_{l}$.
3. $\text{O}_{22}$
Take two faces $s_{k},s_{l}$ $(k<l)$ with
$\lambda(s_{k})=\lambda(s_{l})+\lambda(c)$ and
$\\{\lambda(s_{k}),\lambda(s_{l})\\}=\\{\alpha,\bar{\alpha}\\}$ as above, then
by using $S(2_{2})$ we can reflect the coloring sequence of
$s_{k},s_{k+1},...,s_{l}$ and do a linear transform
$(e_{1},\alpha,\beta)\mapsto(e_{1},\bar{\alpha},\beta)$ to change the
reflected coloring sequence.
4. $\text{O}_{32}$
Take $s_{k},s_{l}$ $(k<l)$ with $\lambda(s_{k}),\lambda(s_{l}),e_{1}$
independent, and then by using $S(3_{2})$ we can reflect the coloring sequence
of $s_{k},s_{k+1},..,s_{l}$ and do a linear transform
$(\lambda(s_{k}),\lambda(s_{l}),e_{1})\mapsto(\lambda(s_{l}),\lambda(s_{k}),e_{1})$
to change the reflected coloring sequence.
###### Lemma 5.1.
The trivial coloring $\lambda$ with the property $(\star)$ is always sector-
equivalent to a coloring whose coloring sequence contains only one of both
$\mathrm{\gamma}$ and $\mathrm{\bar{\gamma}}$.
###### Proof.
Let $\widetilde{\gamma}=\gamma$ or $\bar{\gamma}$. With no loss, assume that
the time number $\ell$ of $\widetilde{\gamma}$ appearing in the coloring
sequence of $\lambda$ is greater than one. Up to Davis-Januszkiewicz
equivalence, one also may assume that $\ell<m/2$. By the definition of
$\lambda$, it is easy to see that any two $\widetilde{\gamma}$’s in the
coloring sequence cannot become neighbors. Let
$\widetilde{\gamma},x_{1},...,x_{r},\widetilde{\gamma},y$ with
$x_{i},y\not=\widetilde{\gamma}$ be a subsequence of the coloring sequence. If
$r>1$, we proceed as follows:
1. (1)
When $x_{1}=y$, by doing the operation $\text{O}_{1}$ on
$x_{1},...,x_{r},\widetilde{\gamma},y$, we may only change the subsequence
$\widetilde{\gamma},x_{1},...,x_{r},\widetilde{\gamma},y$ into
$\widetilde{\gamma},y,\widetilde{\gamma},x_{r},...,x_{1}$ in the coloring
sequence, and the value of $\ell$ is unchanged.
2. (2)
When $x_{1}-y=e_{1}$, with no loss one may assume that
$x_{1}={\alpha},y={\bar{\alpha}}$. Then by doing the operation $\text{O}_{22}$
on $x_{1}=\alpha,x_{2},...,x_{r},\widetilde{\gamma},y={\bar{\alpha}}$, we may
only change the subsequence
$\widetilde{\gamma},\alpha,x_{2},...,x_{r},\widetilde{\gamma},{\bar{\alpha}}$
into
$\widetilde{\gamma},\alpha,\widetilde{\gamma},x^{\prime}_{r},...,x^{\prime}_{2},\bar{\alpha}$
with $x^{\prime}_{i}\not=\widetilde{\gamma}$, and the value of $\ell$ is
unchanged.
3. (3)
When $x_{1},y,e_{1}$ are linearly independent, with no loss one may assume
that $x_{1}=\alpha,y=\beta$. Then by doing the operation $\text{O}_{32}$ on
$x_{1}=\alpha,x_{2},...,x_{r}$, $\widetilde{\gamma},y=\beta$, we may only
change the subsequence
$\widetilde{\gamma},\alpha,x_{2},...,x_{r},\widetilde{\gamma},\beta$ into
$\widetilde{\gamma}$, $\alpha$,
$\widetilde{\gamma},x^{\prime}_{r},...,x^{\prime}_{2},\beta$ with
$x^{\prime}_{i}\not=\widetilde{\gamma}$, and the value of $\ell$ is unchanged.
Thus, we may reduce the coloring $\lambda$ to another coloring with the
following coloring sequence
(5.1)
$(\widetilde{\gamma},y_{1},\widetilde{\gamma},y_{2},...,\widetilde{\gamma},y_{\ell-1},\widetilde{\gamma},y_{\ell},z_{1},...,z_{m-2\ell})\text{
with }m-2\ell>0.$
Without loss of generality, one may assume that $y_{\ell-1}=\alpha$. If
$y_{\ell}=\beta$ or $\bar{\beta}$, by doing the operation $\text{O}_{32}$ on
$\widetilde{\gamma},y_{\ell-1},\widetilde{\gamma},y_{\ell}$, one may change
$\widetilde{\gamma},y_{\ell-1},\widetilde{\gamma},y_{\ell}$ into
$\widetilde{\gamma},y_{\ell},y_{\ell-1},y_{\ell}$, so that the coloring
sequence (5.1) is reduced to $(\widetilde{\gamma}$, $y_{1}$,
$\widetilde{\gamma}$, $y_{2},...$, $\widetilde{\gamma},y_{\ell-2}$,
$\widetilde{\gamma}$, $y_{\ell},y_{\ell-1},y_{\ell}$,
$z_{1},...,z_{m-2\ell})$. If $y_{\ell}=\alpha$ or $\bar{\alpha}$, then
$z_{1}=\beta$ or $\bar{\beta}$. By doing the operation $\text{O}_{32}$ on
$\widetilde{\gamma},y_{\ell-1},\widetilde{\gamma},y_{\ell},z_{1}$, one may
change $\widetilde{\gamma},y_{\ell-1},\widetilde{\gamma},y_{\ell},z_{1}$ into
$\widetilde{\gamma},y_{\ell},z_{1},y_{\ell-1},z_{1}$, so that the coloring
sequence (5.1) is reduced to $(\widetilde{\gamma}$, $y_{1}$,
$\widetilde{\gamma}$, $y_{2},...$, $\widetilde{\gamma},y_{\ell-2}$,
$\widetilde{\gamma}$, $y_{\ell},z_{1},y_{\ell-1}$, $z_{1},...,z_{m-2\ell})$.
So we have managed to reduce the number $\ell$ of $\widetilde{\gamma}$’s by 1.
We can continue this process until we reach $\ell=1$, as desired. ∎
Now let us determine the “canonical form” of the trivial coloring $\lambda$ on
$\mathrm{P}^{3}(m)$.
First let us consider the case in which $\lambda$ is 2-independent
###### Proposition 5.2.
Suppose that $\lambda$ is 2-independent. Then
1. (1)
If $\lambda$ doesn’t possess the property $(\star)$, then $m$ is even and
$\lambda$ is sector-equivalent to the canonical form $\lambda_{C_{1}}$ with
the coloring sequence $\mathcal{C}_{1}=(\alpha$, $\beta,$$...,\alpha,\beta)$.
2. (2)
If $\lambda$ possesses the property $(\star)$, then $\lambda$ is sector-
equivalent to one of the following two canonical forms: $\mathrm{(a)}$
$\lambda_{C_{2}}$ with $m$ even and with the coloring sequence
$\mathcal{C}_{2}=(\alpha,\gamma,\alpha,\beta,...,\alpha,\beta)$;
$\mathrm{(b)}$ $\lambda_{C_{3}}$ with $m$ odd and with the coloring sequence
$\mathcal{C}_{3}=(\alpha,\gamma,\beta,\alpha,\beta,...,\alpha,\beta)$.
###### Proof.
If $\lambda$ doesn’t possess the property $(\star)$, then it is easy to see
that $\lambda$ is unique up to Davis-Januszkiewicz equivalence and $m$ is
even. So Proposition 5.2(1) follows from this. By Lemma 5.1, an easy
observation shows that Proposition 5.2(2) holds. ∎
Next let us consider the case in which $\lambda$ is 3-independent.
###### Proposition 5.3.
If $\lambda$ is 3-independent without the property $(\star)$, then $m$ is even
and $\lambda$ is sector-equivalent to the canonical form $\lambda_{C_{4}}$
with the coloring sequence
$\mathcal{C}_{4}=(\bar{\alpha},\beta,\alpha,\beta,...,\alpha,\beta)$.
###### Proof.
Without loss of generality assume that each element in the coloring sequence
$\mathcal{C}$ of $\lambda$ is in the set
$\\{\alpha,\beta,\bar{\alpha},\bar{\beta}\\}$ and that both $\alpha$ and
$\bar{\alpha}$ must appear in $\mathcal{C}$. Since $\alpha$ and $\bar{\alpha}$
(or $\beta$ and $\bar{\beta}$) can become neighbors, one has that $m$ must be
even.
Similarly to the argument of Lemma 5.1, by using the operations $\text{O}_{1}$
and $\text{O}_{22}$, we may reduce $\lambda$ to a coloring with the coloring
sequence
(5.2)
$(\bar{\alpha},x_{1},...,\bar{\alpha},x_{r},\alpha,x_{r+1},...,{\alpha},x_{{m\over
2}})$
where $r\geq 1$ and $x_{i}=\beta$ or $\bar{\beta}$, and this reduction doesn’t
change the number of bars on $\alpha$’s. If $r>1$, by doing the operation
$\text{O}_{22}$ on $\bar{\alpha},x_{r-1},\bar{\alpha},x_{r},\alpha$, we may
reduce the sequence (5.2) to
$(\bar{\alpha},x_{1},...,\bar{\alpha},x_{r-2},\bar{\alpha},x_{r},\alpha,x_{r-1},\alpha,x_{r+1},...,{\alpha},x_{{m\over
2}}),$
reducing the number of bars on $\alpha$’s by one. This process can be carried
out until the sequence (5.2) is reduced to
(5.3)
$(\bar{\alpha},x_{r},\alpha,x_{1},...,{\alpha},x_{r-2},\alpha,x_{r-1},\alpha,x_{r+1},...,{\alpha},x_{{m\over
2}}).$
Next, we claim that by using the operation $\text{O}_{21}$, we may remove all
possible bars on $\beta$’s in the sequence (5.3). In fact, if
$x_{r}=\bar{\beta}$, then applying the operation $\text{O}_{21}$ on
$\bar{\alpha},x_{r},\alpha$, we may remove the bar on $x_{r}$. Generally, with
no loss, one may assume that $x_{j}=\bar{\beta}$ and $x_{r}=x_{l}=\beta$ where
$l\in\\{1,...,j-1\\}$ if $j\leq r+1$ and $j\not=r$, and
$l\in\\{1,...,r-1,r+1,...,j-1\\}$ if $j>r+1$. Applying the operation
$\text{O}_{21}$ on $\bar{\alpha},x_{r},...,\alpha,x_{j}=\bar{\beta},\alpha$,
one may remove the bar on $x_{j}=\bar{\beta}$, but add the bar on $x_{r}$ and
$x_{l}$’s. Again applying the operation $\text{O}_{21}$ on
$\bar{\alpha},\bar{x}_{r},...,\alpha,\bar{x}_{l},...,\alpha,\bar{x}_{j-1},\alpha$,
one may remove all bars on $\bar{x}_{r}$ and $\bar{x}_{l}$’s. Thus, by
carrying on this procedure, the above claim holds. This completes the proof. ∎
###### Proposition 5.4.
If $\lambda$ is 3-independent with the property $(\star)$ and $m>4$, then
$\lambda$ is sector-equivalent to one of the following six canonical forms:
1. (1)
$\lambda_{C_{5}}$ with $m$ odd and with the coloring sequence
$\mathcal{C}_{5}=(\bar{\gamma},\alpha,\beta,...,\alpha,\beta).$
2. (2)
$\lambda_{C_{6}}$ with $m$ odd and with the coloring sequence
$\mathcal{C}_{6}=(\bar{\gamma},\bar{\alpha},\beta,\alpha,\beta,...,\alpha,\beta).$
3. (3)
$\lambda_{C_{7}}$ with $m$ odd and with the coloring sequence
$\mathcal{C}_{7}=({\gamma},\bar{\alpha},\beta,\alpha,\beta,...,\alpha,\beta).$
4. (4)
$\lambda_{C_{8}}$ with $m$ even and with the coloring sequence
$\mathcal{C}_{8}=(\bar{\alpha},\gamma,\bar{\alpha},\beta,\alpha,\beta,...,\alpha,\beta).$
5. (5)
$\lambda_{C_{9}}$ with $m$ even and with the coloring sequence
$\mathcal{C}_{9}=(\alpha,\gamma,\bar{\alpha},\beta,\alpha,\beta,...,\alpha,\beta).$
6. (6)
$\lambda_{C_{10}}$ with $m$ even and with the coloring sequence
$\mathcal{C}_{10}=(\alpha,\bar{\gamma},\alpha,\beta,...,\alpha,\beta).$
###### Proof.
By Lemma 5.1, one may assume that $\widetilde{\gamma}(=\gamma$ or
$\bar{\gamma})$ appears only one time in the coloring sequence $\mathcal{C}$
of $\lambda$.
Case (I): $m$ is odd. If only one of both $\alpha$ and $\bar{\alpha}$ appears
in $\mathcal{C}$ and the same thing also happens for both $\beta$ and
$\bar{\beta}$, then $\widetilde{\gamma}$ must be $\bar{\gamma}$ so
$\mathcal{C}$ is just $\mathcal{C}_{5}$. Otherwise we can carry out our
argument as in Proposition 5.3 on the subsequence in $\mathcal{C}$ of
containing no $\widetilde{\gamma}$, so that $\mathcal{C}$ is sector-equivalent
to $\mathcal{C}_{6}$ or $\mathcal{C}_{7}$.
Case (II): $m$ is even. Consider two neighbors of $\widetilde{\gamma}$, since
$m$ is even, such two neighbors must be the same letter. Up to Davis-
Januszkiewicz equivalence, one may assume that they are
$\\{\alpha,{\alpha}\\}$ or $\\{\alpha,\bar{\alpha}\\}$.
If two neighbors of $\widetilde{\gamma}$ are $\\{\alpha,\bar{\alpha}\\}$, with
no loss assume that
$\mathcal{C}=(\alpha,\widetilde{\gamma},\bar{\alpha},x_{4},...,x_{m})$. Then
we can carry out our argument as in Proposition 5.3 on
$\bar{\alpha},x_{4},...,x_{m}$, so that $\mathcal{C}$ may be reduced to
$\mathcal{C}^{\prime}=(\alpha,\widetilde{\gamma},\bar{\alpha},\beta,\alpha,\beta,...,\alpha,\beta)$.
If $\widetilde{\gamma}=\gamma$, then $\mathcal{C}^{\prime}$ is just
$\mathcal{C}_{8}$. If $\widetilde{\gamma}=\bar{\gamma}$, applying the
operation $\text{O}_{22}$ on $\alpha,\bar{\gamma},\bar{\alpha}$, one may
further reduce $\mathcal{C}^{\prime}$ to $\mathcal{C}_{9}$.
Now suppose that two neighbors of $\widetilde{\gamma}$ are
$\\{\alpha,{\alpha}\\}$. If $\mathcal{C}$ only contains $\alpha$ and $\beta$
except for $\widetilde{\gamma}$, then $\mathcal{C}$ is just
$\mathcal{C}_{10}$. Otherwise, with no loss assume that $\mathcal{C}$ also
contains $\bar{\alpha}$. Up to Davis-Januszkiewicz equivalence, by using the
linear transformation
$(e_{1},\alpha,\beta)\longmapsto(e_{1},\bar{\alpha},\beta)$ one may write
$\mathcal{C}=(\bar{\alpha},\widetilde{\gamma},\bar{\alpha},x_{4},...,x_{m})$.
Furthermore, Then we can carry out our argument as in Proposition 5.3 to
reduce $\mathcal{C}$ to
$(\bar{\alpha},\widetilde{\gamma},\bar{\alpha},\beta,\alpha,\beta,...,\alpha,\beta)$.
If $\widetilde{\gamma}$ is not $\gamma$, applying the operation
$\text{O}_{22}$ on $\alpha,\bar{\gamma},\bar{\alpha}$, one may further reduce
$\mathcal{C}^{\prime}$ to $\mathcal{C}_{8}$. ∎
###### Remark 5.1.
An easy observation shows that for a 3-independent trivial coloring $\lambda$
with the property $(\star)$, if $m=3$, then $\lambda$ is just sector-
equivalent to the following canonical form
$\lambda_{C^{3}}\text{ with the coloring sequence
}\mathcal{C}^{3}=(\bar{\gamma},\alpha,\beta)$
and if $m=4$, then $\lambda$ is just sector-equivalent to one of the following
two canonical forms
1. (1)
$\lambda_{C_{1}^{4}}$ with the coloring sequence
$\mathcal{C}_{1}^{4}=(\alpha,\gamma,\bar{\alpha},\beta).$
2. (2)
$\lambda_{C_{2}^{4}}$ with the coloring sequence
$\mathcal{C}_{2}^{4}=(\alpha,\bar{\gamma},\alpha,\beta).$
Combining Propositions 5.2-5.4 and Remark 5.1 gives the following
###### Corollary 5.5.
The number of homeomorphism classes of small covers over $\mathrm{P}^{3}(m)$
with trivial colorings is at most
$N_{t}(m)=\begin{cases}2&\text{ if }m=3\\\ 4&\text{ if $m>3$ is odd}\\\
6&\text{ if $m$ is even}\end{cases}$
By Proposition 2.5, a direct observation shows that
###### Corollary 5.6.
$M(\lambda_{C_{i}}),i=1,5,10$, are orientable, and
$M(\lambda_{C_{i}}),i=2,3,4,6,7,8,9$, are non-orientable.
### 5.2. Nontrivial Prism Small Covers
Given a pair $(\mathrm{P}^{3}(m),\lambda)$ in $\Lambda(\mathrm{P}^{3}(m))$,
throughout suppose that $\lambda$ is nontrivial, i.e.,
$\lambda(c)\not=\lambda(f)$.
###### Definition 5.2.
Let
$M_{\lambda}=\\{s_{i}\big{|}\text{Span}\\{\lambda(s_{i-1}),\lambda(s_{i}),\lambda(s_{i+1})\\}=\mathbb{Z}_{2}^{3}\\}$
and let $m_{\lambda}:=|M_{\lambda}|$ denote the number of side faces in
$M_{\lambda}$. Set $\lambda_{0}:=\lambda(c)-\lambda(f)$. Let
$N_{\lambda}=\\{s_{i}\big{|}\lambda(s_{i})=\lambda_{0}\\}$, and let
$n_{\lambda}:=|N_{\lambda}|$ denote the number of side faces in $N_{\lambda}$.
###### Lemma 5.7.
Let $\lambda$ be a nontrivial coloring on $\mathrm{P}^{3}(m)$ with $m>3$. Then
1. (1)
$m_{\lambda}\leq n_{\lambda}\leq m/2$. In particular, if $m$ is odd, then
$n_{\lambda}>0$.
2. (2)
$m_{\lambda}$ is even.
###### Proof.
First, $n_{\lambda}\leq m/2$ is obvious since any two faces in $N_{\lambda}$
are not adjacent. To show that $m_{\lambda}\leq n_{\lambda}$, take one
$s_{i}\in N_{3}$. Then $\lambda(s_{i-1}),\lambda(s_{i}),\lambda(s_{i+1})$ are
linearly independent. Furthermore, the linear independence of
$\\{\lambda(s_{i-1}),\lambda(s_{i}),\lambda(c)\\}$ and
$\\{\lambda(s_{i}),\lambda(s_{i+1}),\lambda(c)\\}$ implies that $\lambda(c)$
must be either $\lambda(s_{i-1})+\lambda(s_{i+1})$ or
$\lambda(s_{i-1})+\lambda(s_{i})+\lambda(s_{i+1})$. This is also true for
$\lambda(f)$. Now $\lambda(c)\neq\lambda(f)$ makes sure that
$\lambda(c)-\lambda(f)=\lambda(s_{i})$, so that $s_{i}\in N_{\lambda}$. Thus,
$M_{\lambda}\subseteq N_{\lambda}$, i.e., $m_{\lambda}\leq n_{\lambda}$.
Moreover, if $n_{\lambda}=0$ then $m_{\lambda}=0$, so that the coloring
sequence of $\lambda$ is 2-independent. However, $n_{\lambda}=0$ means that
one has only two choices of colors. This forces $m$ to be even.
With no loss, assume that $m_{\lambda}>0$ (since $0$ is even). For each $i$,
let $V_{i}$ denote the subspace spanned by $\lambda(s_{i})$ and
$\lambda(s_{i+1})$. Obviously, if $s_{i}\in M_{\lambda}$ then
$V_{i}\not=V_{i-1}$, and if $s_{i}\not\in M_{\lambda}$ then $V_{i}=V_{i-1}$.
Thus, $s_{i}\in M_{\lambda}$ if and only if $V_{i}\not=V_{i-1}$. Next we claim
that for any $i$, $\lambda_{0}\in V_{i}$. In fact, if $s_{i}\in M_{\lambda}$,
since $M_{\lambda}\subseteq N_{\lambda}$, we have that $s_{i}\in N_{\lambda}$
so $\lambda_{0}=\lambda(s_{i})\in V_{i}$. If $s_{i}\not\in M_{\lambda}$ then
$V_{i}=V_{i-1}$. Since $V_{i}=V_{i-1}$ contains no $\lambda(c)$ and
$\lambda(f)$, $\lambda_{0}$ must be in $V_{i}=V_{i-1}$. This proves the claim.
Furthermore, for any $i$, $V_{i}$ must be either
$\text{Span}\\{\lambda_{0},\alpha\\}$ or
$\text{Span}\\{\lambda_{0},\lambda(c)+\alpha\\}$, where $\alpha$ is a nonzero
element such that $\alpha$, $\lambda(c)$ and $\lambda(f)$ are linearly
independent. Now we clearly see that there is a switch of choosing either
$\text{Span}\\{\lambda_{0},\alpha\\}$ or
$\text{Span}\\{\lambda_{0},\lambda(c)+\alpha\\}$ exactly when we pass
$s_{i}\in M_{\lambda}$. But the total number of switches must be even. So
$m_{\lambda}$ is even. ∎
###### Remark 5.2.
An easy observation shows that if $m=3$, then there is a possibility that
$m_{\lambda}=3$ but still $n_{\lambda}=1<3/2$. This is exactly an exception
only for $m_{\lambda}$ in the case $m=3$.
Throughout the following, assume that $m>3$.
Applying sectors $S(2_{1}),S(3_{1}),S(4)$ and $S(5)$ to the nontrivial
coloring $\lambda$ gives the following four fundamental operations on its
coloring sequence:
1. $\bar{\text{O}}_{21}$
Take $s_{k},s_{l}\in N_{\lambda}$ with $k<l$, by using $S(2_{1})$, we may do a
linear transformation
$(\lambda(c),\lambda_{0},\lambda(s_{k+1})\mapsto(\lambda(c),\lambda_{0},\lambda(s_{k+1})+\lambda_{0})$
to change the coloring sequence of $s_{k},s_{k+1},...,s_{l}$.
2. $\text{O}_{31}$
Take $s_{k},s_{l}$ $(k<l)$ with
$\lambda(s_{k})=\lambda(s_{l})\neq\lambda_{0}$, and use $S(3_{1})$ to reflect
the coloring sequence of $s_{k},s_{k+1},...,s_{l}$.
3. $\text{O}_{4}$
Take $s_{k},s_{l}$ $(k<l)$ with $\lambda(s_{k})-\lambda(s_{l})=\lambda(c)$ or
$\lambda(f)$, we use $S(4)$ to reflect the coloring sequence of
$s_{k},s_{k+1},...,s_{l}$ and then to do a linear transformation
$(\lambda_{0},\lambda(s_{k}),\lambda(s_{l}))\mapsto(\lambda_{0},\lambda(s_{l}),\lambda(s_{k}))$
to change the reflected coloring sequence.
4. $\text{O}_{5}$
Take $s_{k},s_{l}$ $(k<l)$ with $\lambda(s_{k})-\lambda(s_{l})=\lambda_{0}$,
we use $S(4)$ to reflect the coloring sequence of $s_{k},s_{k+1},...,s_{l}$
and then to do a linear transformation $(\lambda(c),\lambda(s_{k}),$
$\lambda(s_{l}))$ $\mapsto(\lambda(c),\lambda(s_{l}),\lambda(s_{k}))$ to
change the reflected coloring sequence.
It is easy to check the following
###### Lemma 5.8.
The operations $\bar{\text{\rm O}}_{21}$, $\text{\rm O}_{31}$, $\text{\rm
O}_{4}$ and $\text{\rm O}_{5}$ above will not change $m_{\lambda},n_{\lambda}$
of the nontrivial coloring $\lambda$.
Without loss of generality, throughout the following one assumes that
$\lambda(c)=e_{1},\lambda(f)=e_{1}+e_{2}$, so that $\lambda_{0}=e_{2}$, where
$\\{e_{1},e_{2},e_{3}\\}$ is a basis of $\mathbb{Z}_{2}^{3}$.
###### Proposition 5.9.
For $m>3$, each nontrivial coloring $\lambda$ is sector-equivalent to the
following canonical form $\lambda_{C_{*}}$ with the coloring sequence
(5.4)
$\mathcal{C}_{*}=(e_{2},x_{1},e_{2},...,e_{2},x_{m_{\lambda}},e_{2},y_{1},...,e_{2},y_{n_{\lambda}-m_{\lambda}},z_{1},...,z_{m-2n_{\lambda}})$
where $x_{i}=\begin{cases}e_{1}+e_{3}&\text{if $i$ is odd}\\\ e_{3}&\text{if
$i$ is even}\end{cases}$ and for all $1\leq i\leq n_{\lambda}-m_{\lambda}$,
$y_{i}=e_{3}$ and $z_{i}=\begin{cases}e_{2}+e_{3}&\text{if $i$ is odd}\\\
e_{3}&\text{if $i$ is even.}\end{cases}$
###### Proof.
If $n_{\lambda}\leq 1$ then clearly the coloring $\lambda$ can be reduced to a
coloring with the coloring sequence
$\begin{cases}(e_{2},e_{3},e_{2}+e_{3},e_{3},...,e_{2}+e_{3},e_{3})&\text{ if
$n_{\lambda}=1$ and $m$ is even}\\\
(e_{2},e_{3},e_{2}+e_{3},e_{3},...,e_{2}+e_{3},e_{3},e_{2}+e_{3})&\text{ if
$n_{\lambda}=1$ and $m$ is odd}\\\
(e_{2}+e_{3},e_{3},...,e_{2}+e_{3},e_{3}).&\text{ if
$n_{\lambda}=0$}\end{cases}$
If $n_{\lambda}\geq 2$, we may choose two $s_{k}$ and $s_{l}$ in $N_{\lambda}$
with $k<l$. Consider the coloring sub-sequence
(5.5)
$(\lambda(s_{k})=)e_{2},r_{1},...,r_{l-2},e_{2}(=\lambda(s_{l})),r_{l-1}$
of $s_{k},...,s_{l},s_{l+1}$, it is easy to see that
$r_{1}-r_{l-1}\in\text{Span}\\{e_{1},e_{2}\\}$. Then when when
$r_{1}-r_{l-1}=0$ (resp. $e_{1}$ or $e_{1}+e_{2}$, $e_{2}$), we may do the
operation $\text{O}_{31}$ (resp. $\text{O}_{4}$ or $\text{O}_{5}$) on
$r_{1},...,r_{l-2},e_{2},r_{l-1}$ from $s_{k+1}$ to $s_{l+1}$, and change
(5.5) into $e_{2},r_{l-1},e_{2},r^{\prime}_{l-2},...,r^{\prime}_{2},r_{1}$.
With this understood, assume that $N_{\lambda}=\\{s_{1}$, $s_{3},$$...,$
$s_{2n_{\lambda}-1}\\}$, so we may write the coloring sequence of $\lambda$ as
follows:
$\mathcal{C}=(e_{2},\alpha_{1},...,e_{2},\alpha_{n_{\lambda}},\beta_{1},...,\beta_{m-2n_{\lambda}})$
with $\alpha_{n_{\lambda}},\beta_{i}\in\\{e_{2}+e_{3},e_{3}\\}$. By doing the
operation $\bar{\text{O}}_{21}$ on
$\alpha_{n_{\lambda}},\beta_{1},...,\beta_{m-2n_{\lambda}}$, we may reduce
$\mathcal{C}$ to
$\mathcal{C}^{\prime}=(e_{2},\alpha_{1},...,\alpha_{n_{\lambda}-1},e_{2},e_{3},z_{1},...,z_{m-2n_{\lambda}})$
such that $z_{i}$ is $e_{2}+e_{3}$ if $i$ is odd, and $e_{3}$ if $i$ is even.
Then we may further use the operation $\text{O}_{4}$ to reduce
$\mathcal{C}^{\prime}$ to $\mathcal{C}^{\prime\prime}$ with
$M_{\lambda}=\\{s_{1},...,s_{2m_{\lambda}-1}\\}$ and without changing the part
$\mathcal{C}^{\prime}-\\{e_{2},\alpha_{1},...,\alpha_{n_{\lambda}-1},e_{2}\\}$.
Finally, by using the operation $\bar{\text{O}}_{21}$, we may reduce
$\mathcal{C}^{\prime\prime}$ to $\mathcal{C}_{*}$ as desired. ∎
Together with Theorem 3.3, Lemmas 5.7-5.8 and Proposition 5.9, it easily
follows that
###### Corollary 5.10.
Let $\lambda_{1},\lambda_{2}$ be two nontrivial colorings on
$\mathrm{P}^{3}(m)$ with $m>3$. If
$(m_{\lambda_{1}},n_{\lambda_{1}})=(m_{\lambda_{2}},n_{\lambda_{2}})$, then
$M(\lambda_{1})$ and $M(\lambda_{2})$ are homeomorphic.
###### Corollary 5.11.
For $m>3$, let $(k,l)$ be a pair such that $(1)$ $l\leq k\leq m/2$ and if
$2\nmid m$ then $k>0$; and $(2)$ $l$ is even. Then there is a nontrivial
coloring $\lambda$ on $\mathrm{P}^{3}(m)$ with
$(n_{\lambda},m_{\lambda})=(k,l)$.
As a consequence of Proposition 2.5 and Proposition 5.9, one also has
###### Corollary 5.12.
Let $\lambda$ be a nontrivial coloring on $\mathrm{P}^{3}(m)$ with $m>3$. Then
$M(\lambda)$ is orientable if $n_{\lambda}=0$, and non-orientable if
$n_{\lambda}>0$.
## 6\. Mod 2 cohomology rings and two invariants
Given a pair $(\mathrm{P}^{3}(m),\lambda)$ in $\Lambda(\mathrm{P}^{3}(m))$,
one knows that the mod 2 cohomology ring of $M(\lambda)$ is
$H^{*}(M(\lambda);\mathbb{Z}_{2})=\mathbb{Z}_{2}[c,f,s_{1},...,s_{m}]/I+J_{\lambda}$
where $I$ is the ideal generated by $cf$ and $s_{i}s_{j}$ with $s_{i}\cap
s_{j}=\emptyset$, and $J_{\lambda}$ is the ideal generated by three linear
relations (determined by the $3\times(m+2)$ matrix
$(\lambda(c),\lambda(f),\lambda(s_{1}),...,\lambda(s_{m}))$).
### 6.1. Two invariants $\Delta(\lambda)$ and $\mathcal{B}(\lambda)$
Now let us introduce two invariants in $H^{*}(M(\lambda)$; $\mathbb{Z}_{2})$.
Set
$\mathcal{H}_{\lambda}^{1}=\\{x\in
H^{1}(M(\lambda);\mathbb{Z}_{2})\big{|}x^{2}=0\\}$
$\mathcal{H}_{\lambda}^{2}=\\{x^{2}\big{|}x\in
H^{1}(M(\lambda);\mathbb{Z}_{2})\\}$
and
$\mathcal{K}_{\lambda}=\text{Span}\\{xy\big{|}x\in
H^{1}(M(\lambda);\mathbb{Z}_{2}),y\in\mathcal{H}_{\lambda}^{1}\\}.$
Clearly, they are all vector spaces over $\mathbb{Z}_{2}$, and
$\dim\mathcal{H}_{\lambda}^{2}=\dim
H^{1}(M(\lambda);\mathbb{Z}_{2})-\dim\mathcal{H}_{\lambda}^{1}=m-1-\dim\mathcal{H}_{\lambda}^{1}.$
Note that $\dim H^{1}(M(\lambda);\mathbb{Z}_{2})=m-1$ by Example 2.1.
Obviously, $\dim\mathcal{H}_{\lambda}^{1}$ is an invariant of the cohomology
ring $H^{*}(M(\lambda);\mathbb{Z}_{2})$, denoted by $\Delta(\lambda)$.
Define a bilinear map
$\omega:H^{1}(M(\lambda);\mathbb{Z}_{2})\times\mathcal{H}_{\lambda}^{1}\longrightarrow\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$
by $(x,y)\longmapsto[xy]$, which is surjective. Let
$\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$
be the dual space of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$.
Take a
$\theta\in\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$,
one can obtain a bilinear map
$\theta\circ\omega:H^{1}(M(\lambda);\mathbb{Z}_{2})\times\mathcal{H}_{\lambda}^{1}\longrightarrow\mathbb{Z}_{2},$
which corresponds an $(m-1)\times\Delta(\lambda)$-matrix. Let $b_{r}(\lambda)$
denote the number of those
$\theta\in\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$
such that $\text{rank }\theta\circ\omega=r$ where $1\leq
r\leq\Delta(\lambda)$. Then we obtain an integer vector
$\mathcal{B}(\lambda)=(b_{1}(\lambda),...,b_{\Delta(\lambda)}(\lambda)),$
called the bilinear vector. It is not difficult to see that
$\mathcal{B}(\lambda)$ is an invariant of the cohomology ring
$H^{*}(M(\lambda);\mathbb{Z}_{2})$.
It should be pointed out that we shall only calculate $b_{1}(\lambda)$ and
$b_{2}(\lambda)$ in $\mathcal{B}(\lambda)$ because basically this will be
sufficient enough to reach our purpose. By $\bar{\mathcal{B}}(\lambda)$ we
denote $(b_{1}(\lambda),b_{2}(\lambda))$. We also use the convention that
$b_{2}(\lambda)=0$ if $\Delta(\lambda)=1$.
### 6.2. Calculation of $\Delta(\lambda)$
First we shall deal with the case in which $\lambda$ is nontrivial.
###### Lemma 6.1.
If $\lambda$ is nontrivial, then
$\Delta(\lambda)=\begin{cases}n_{\lambda}&\text{if $n_{\lambda}>0$ and
$m_{\lambda}=0$}\\\ n_{\lambda}-1&\text{if $m_{\lambda}>0$}\\\ 1&\text{if
$n_{\lambda}=0$ (so $m$ is even).}\end{cases}$
###### Proof.
By Proposition 5.9, each $\lambda$ is sector-equivalent to the canonical form
$\lambda_{C_{*}}$ with the coloring sequence $\mathcal{C}_{*}$ and without
changing $m_{\lambda}$ and $n_{\lambda}$, so it suffices to consider the
$\lambda_{C_{*}}$.
If $n_{\lambda}>0$ and $m_{\lambda}=0$, we can obtain from (5.4) that
$\lambda_{C_{*}}$ determines the following three linear relations in
$H^{1}(M(\lambda_{C_{*}});\mathbb{Z}_{2})$
(6.1) $c+f=0$ (6.2) $f+\sum_{i\text{ is odd}}s_{i}=0$ (6.3)
$s_{2}+\cdots+s_{2n_{\lambda}}+\sum_{2n_{\lambda}<i\leq m}s_{i}=0.$
So we may choose $B_{1}=\\{f,s_{2},s_{3},...,s_{m-1}\\}$ as a basis of
$H^{1}(M(\lambda_{C_{*}});\mathbb{Z}_{2})$. Since $cf=0$ and $s_{i}s_{j}=0$
with $s_{i}\cap s_{j}=\emptyset$ in
$H^{*}(M(\lambda_{C_{*}});\mathbb{Z}_{2})$, one can easily obtain from (6.1)
and (6.3) that
$B_{2}=\\{f,s_{2},s_{4},...,s_{2n_{\lambda}-2}\\}\subset\mathcal{H}_{\lambda_{C_{*}}}^{1}$,
and $B_{2}\subset B_{1}$. Thus,
$\dim\mathcal{H}_{\lambda}^{1}=\dim\mathcal{H}_{\lambda_{C_{*}}}^{1}\geq
n_{\lambda}$. On the other hand, an easy argument shows that
$B_{3}=\\{s_{3}^{2},...,s_{2n_{\lambda}-1}^{2},s_{2n_{\lambda}}^{2},...,s_{m-1}^{2},fs_{2},s_{3}s_{4},s_{5}s_{6},...,s_{2n_{\lambda}-1}s_{2n_{\lambda}}\\}$
forms a basis of $H^{2}(M(\lambda_{C_{*}});\mathbb{Z}_{2})$. Now observe that
the square of each element of $B_{1}\setminus B_{2}$ is in $B_{3}$, so
$\dim\mathcal{H}_{\lambda}^{2}=\dim\mathcal{H}_{\lambda_{C_{*}}}^{2}\geq
m-1-n_{\lambda}$. Furthermore, $\dim\mathcal{H}_{\lambda}^{1}\leq
n_{\lambda}$. Therefore, $\Delta(\lambda)=n_{\lambda}$.
If $m_{\lambda}>0$, then $\lambda_{C_{*}}$ determines the following three
linear relations
$\begin{cases}c+f+s_{2}+\cdots+s_{2m_{\lambda}-2}=0\\\ f+\sum_{i\text{ is
odd}}s_{i}=0\\\ s_{2}+\cdots+s_{2n_{\lambda}}+\sum_{2n_{\lambda}<i\leq
m}s_{i}=0.\end{cases}$
In this case, we choose $B_{4}=\\{s_{1},s_{2},...,s_{m-1}\\}$ as a basis of
$H^{1}(M(\lambda_{C_{*}});\mathbb{Z}_{2})$. Then one sees that
$B_{5}=\\{s_{2},s_{4},...,s_{2n_{\lambda}-2}\\}\subset\mathcal{H}_{\lambda_{C_{*}}}^{1}$.
Furthermore, we choose
$B_{6}=\\{s_{1}^{2},s_{3}^{2},...,s_{2n_{\lambda}-1}^{2},s_{2n_{\lambda}}^{2},s_{2n_{\lambda}+1}^{2},...,s_{m-1}^{2},s_{2}s_{3},s_{4}s_{5},...,s_{2n_{\lambda}-2}s_{2n_{\lambda}-1}\\}$
as a basis of $H^{2}(M(\lambda_{C_{*}});\mathbb{Z}_{2})$. A similar argument
as above shows that $\Delta(\lambda)=n_{\lambda}-1$.
If $n_{\lambda}=0$, in a similar way as above, it is easy to see that we may
choose $B_{7}=\\{f,s_{3},...,s_{m}\\}$ as a basis of
$H^{1}(M(\lambda_{C_{*}});\mathbb{Z}_{2})$ and $B_{8}=\\{f\\}$ forms a basis
of $\mathcal{H}_{\lambda_{C_{*}}}^{1}$. Thus, $\Delta(\lambda)=1$. ∎
Next we consider the case in which $\lambda$ is trivial.
###### Lemma 6.2.
Let $\lambda$ be trivial. Then
$\Delta(\lambda)=\begin{cases}m-1&\text{if $\lambda\approx\lambda_{C_{1}}$}\\\
m-2&\text{if $\lambda\approx\lambda_{C_{i}},i=2,3,4$}\\\ m-3&\text{if
$\lambda\approx\lambda_{C_{i}}$ with $m>4$, $i=5,6,7,8,9,10$.}\end{cases}$
In particular, if $m=3$ then $\Delta(\lambda_{C^{3}})=0$, and if $m=4$ then
$\Delta(\lambda_{C_{1}^{4}})=\Delta(\lambda_{C_{2}^{4}})=1$.
###### Proof.
The argument is similar to that of Lemma 6.1, and is not quite difficult. Here
we only list the three linear relations and the bases of
$H^{i}(M(\lambda);\mathbb{Z}_{2})(i=1,2)$ and $\mathcal{H}_{\lambda}^{1}$, but
for the detailed proof, we would like to leave it to readers as an exercise.
$\lambda$ | $m$ | Three linear relations by determined by $J_{\lambda}$
---|---|---
$\lambda_{C_{1}}$ | even | $c+f=0,\sum_{i\text{ is odd}}s_{i}=0,\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{2}}$ | even | $c+f=0,s_{2}+\sum_{i\text{ is odd}}s_{i}=0,\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{3}}$ | odd | $c+f=0,s_{2}+\sum_{i\text{ is odd}}s_{i}=0,\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{4}}$ | even | $c+f+s_{1}=0,\sum_{i\text{ is odd}}s_{i}=0,\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{5}}$ | odd | $c+f+s_{1}=0,\sum_{i\text{ is odd}}s_{i}=0,s_{1}+\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{6}}$ | odd | $c+f+s_{1}+s_{2}=0,\sum_{i\text{ is odd}}s_{i}=0,s_{1}+\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{7}}$ | odd | $c+f+s_{2}=0,\sum_{i\text{ is odd}}s_{i}=0,s_{1}+\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{8}}$ | even | $c+f+s_{1}+s_{3}=0,s_{2}+\sum_{i\text{ is odd}}s_{i}=0,\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{9}}$ | even | $c+f+s_{3}=0,s_{2}+\sum_{i\text{ is odd}}s_{i}=0,\sum_{i\text{ is even}}s_{i}=0$
$\lambda_{C_{10}}$ | even | $c+f+s_{2}=0,s_{2}+\sum_{i\text{ is odd}}s_{i}=0,\sum_{i\text{ is even}}s_{i}=0$
$\lambda$ | Basis of $H^{1}(M(\lambda);\mathbb{Z}_{2})$ | Basis of $\mathcal{H}_{\lambda}^{1}$ | Basis of $H^{2}(M(\lambda);\mathbb{Z}_{2})$
---|---|---|---
$\lambda_{C_{1}}$ | $\\{f,s_{3},...,s_{m}\\}$ | $\\{f,s_{3},...,s_{m}\\}$ | $\\{s_{3}s_{4},fs_{3},...,fs_{m}\\}$
$\lambda_{C_{2}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{f,s_{2},s_{4},...,s_{m-1}\\}$ | $\\{s_{1}^{2},fs_{2},...,fs_{m-1}\\}$
$\lambda_{C_{3}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{f,s_{4},...,s_{m}\\}$ | $\\{s_{1}^{2},fs_{1},...,fs_{m-2}\\}$
$\lambda_{C_{4}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{s_{3},...,s_{m}\\}$ | $\\{s_{1}s_{2},fs_{3},...,fs_{m}\\}$
$\lambda_{C_{5}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{s_{3},...,s_{m-1}\\}$ | $\\{s_{1}s_{2},fs_{2},...,fs_{m-1}\\}$
$\lambda_{C_{6}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{s_{3},...,s_{m-1}\\}$ | $\\{f^{2},s_{2}^{2},fs_{2},...,fs_{m-2}\\}$
$\lambda_{C_{7}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{s_{3},...,s_{m-1}\\}$ | $\\{f^{2},s_{2}^{2},fs_{3},...,fs_{m-1}\\}$
$\lambda_{C_{8}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{s_{2},s_{4},...,s_{m-1}\\}$ | $\\{f^{2},s_{3}^{2},fs_{1},fs_{4},...,fs_{m-1}\\}$
$\lambda_{C_{9}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{s_{2},s_{4},...,s_{m-1}\\}$ | $\\{f^{2},s_{3}^{2},fs_{1},fs_{4},...,fs_{m-1}\\}$
$\lambda_{C_{10}}$ | $\\{f,s_{2},...,s_{m-1}\\}$ | $\\{s_{2},s_{4},...,s_{m-1}\\}$ | $\\{f^{2},s_{3}^{2},fs_{3},...,fs_{m-1}\\}$
∎
###### Remark 6.1.
Although it is not mentioned in this paper, the authors have calculated the
first Betti number under $\mathbb{Z}$-coefficients of all small covers over
prisms and discovered that the number is always equal to $\Delta(\lambda)$ in
the $\mathbb{Z}_{2}$-cohomology ring. One can check that this is also true for
all closed surfaces (i.e., 2-dimensional small covers). It should be
reasonable to conjecture that this is true for all small covers.
###### Proposition 6.3.
Let $\lambda_{1}$ and $\lambda_{2}$ be two colorings in
$\Lambda(\mathrm{P}^{3}(m))$ such that $\lambda_{1}$ is trivial but
$\lambda_{2}$ is nontrivial. If $m>6$, then both $M(\lambda_{1})$ and
$M(\lambda_{1})$ cannot be homeomorphic.
###### Proof.
Suppose that $M(\lambda_{1})$ and $M(\lambda_{1})$ are homeomorphic. Then
their cohomologies are isomorphic, so
$\Delta(\lambda_{1})=\Delta(\lambda_{2})$. However, by Lemmas 6.1 and 6.2, one
has that $\Delta(\lambda_{1})\geq m-3$ and $\Delta(\lambda_{2})\leq m/2$.
Furthermore, if $m>6$, then $\Delta(\lambda_{1})\geq
m-3>m/2\geq\Delta(\lambda_{2})$, so
$\Delta(\lambda_{1})\not=\Delta(\lambda_{2})$, a contradiction. ∎
###### Remark 6.2.
We see from the proof of Proposition 6.3 that $\Delta(\lambda_{1})$ and
$\Delta(\lambda_{2})$ can coincide only if $m\leq 6$. For $m=5,6$, all
possible cases that $\Delta(\lambda_{1})=\Delta(\lambda_{2})$ happens are
stated as follows: when $(n_{\lambda_{C_{*}}},m_{\lambda_{C_{*}}})=(m-3,0)$,
one has that $\Delta(\lambda_{C_{*}})=\Delta(\lambda_{C_{i}})$,
$i=5,6,7,8,9,10$. For $m=3,4$, we know from [LY] and [M3] that up to
homeomorphism, there are only two small covers over $\mathrm{P}^{3}(3)$:
${\mathbb{R}}P^{3}$ and $S^{1}\times{\mathbb{R}}P^{2}$, and there are only
four small covers over $\mathrm{P}^{3}(4)$: $(S^{1})^{3}$, $S^{1}\times K$, a
twist $(S^{1})^{2}$-bundle over $S^{1}$ and a twist $K$-bundle over $S^{1}$,
where $K$ is a Klein bottle. In particular, the cohomological rigidity holds
in this case.
### 6.3. Calculation of $\bar{\mathcal{B}}(\lambda)$
Let $\lambda\in\Lambda(\mathrm{P}^{3}(m))$ with $m>4$. Choose an ordered basis
$B^{\prime}$ of $H^{1}(M(\lambda);\mathbb{Z}_{2})$ and an ordered basis
$B^{\prime\prime}$ of $\mathcal{H}_{\lambda}^{1}$, let $A_{0}$ denote an
$(m-1)\times\Delta(\lambda)$ matrix $(a_{ij})$, where
$a_{ij}=[u_{i}v_{j}]\in\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$,
$u_{i}$ is the $i$-th element in $B^{\prime}$ and $v_{j}$ the $j$-th element
in $B^{\prime\prime}$, so each element in $B^{\prime}$ corresponds to a row
and each element in $B^{\prime\prime}$ a column. It follows that for any
$\theta\in\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$,
$\theta(A_{0})=(\theta(a_{ij}))$ is a representation matrix of
$\theta\circ\omega$.
First let us look at the case in which $\lambda$ is nontrivial.
###### Lemma 6.4.
Let $\lambda$ be nontrivial. Then
$\bar{\mathcal{B}}(\lambda)=\begin{cases}(0,0)&\text{if
$(n_{\lambda},m_{\lambda})=(0,0)$}\\\ (1,0)&\text{if
$(n_{\lambda},m_{\lambda})=(1,0)$ or $(2,2)$}\\\ (1,3)&\text{if
$(n_{\lambda},m_{\lambda})=(2,0)$}\\\ (0,n_{\lambda})&\text{if $n_{\lambda}>2$
and $m_{\lambda}=0$}\\\ (n_{\lambda}-m_{\lambda},{{m_{\lambda}-1}\choose
1}+{{m_{\lambda}-1}\choose 2}+{{n_{\lambda}-m_{\lambda}}\choose 2})&\text{if
$n_{\lambda}>2$ and $m_{\lambda}>0$}\end{cases}$
###### Proof.
By Proposition 5.9, one may assume that $\lambda=\lambda_{C_{*}}$. Then our
argument proceeds as follows.
(1) If $n_{\lambda}>0$ and $m_{\lambda}=0$, then Lemma 6.1 we may take
$B^{\prime}=B_{1}$ and $B^{\prime\prime}=B_{2}$. Thus one has that $A_{0}$ is
equal to
$\begin{bmatrix}0&[fs_{2}]&[fs_{4}]&[fs_{6}]&\cdots&[fs_{2n_{\lambda}-6}]&[fs_{2n_{\lambda}-4}]&[fs_{2n_{\lambda}-2}]\\\
[s_{2}f]&0&0&0&{3}&0\\\ [s_{3}f]&[s_{3}s_{2}]&[s_{3}s_{4}]&0&{3}&0\\\
[s_{4}f]&0&0&0&{3}&0\\\ {8}\\\
[s_{2n_{\lambda}-3}f]&0&{3}&0&[s_{2n_{\lambda}-3}s_{2n_{\lambda}-4}]&[s_{2n_{\lambda}-3}s_{2n_{\lambda}-2}]\\\
[s_{2n_{\lambda}-2}f]&0&0&0&{3}&0\\\
[s_{2n_{\lambda}-1}f]&0&0&0&{2}&0&[s_{2n_{\lambda}-1}s_{2n_{\lambda}-2}]\\\
[s_{2n_{\lambda}}f]&0&0&0&{3}&0\\\ {8}\\\ [s_{m-1}f]&0&0&0&{3}&0\end{bmatrix}$
By direct calculations one knows from (6.2) and (6.3) that
$s_{2n_{\lambda}}s_{2n_{\lambda}+1}+s_{2n_{\lambda}+1}^{2}+\cdots+s_{m}^{2}=0$
so $[s_{2n_{\lambda}}s_{2n_{\lambda}+1}]=0$ and
$\begin{cases}s_{2i}s_{2i+1}=s_{2i+1}s_{2i+2}&\text{when $1\leq i\leq
n_{\lambda}-1$}\\\ fs_{i}=s_{i}^{2}\text{ so $[fs_{i}]=0$}&\text{when either
$i$ is odd or $i>2n_{\lambda}$ is even}\\\
fs_{i}=s_{i-1}s_{i}+s_{i}s_{i+1}&\text{when $2\leq i\leq 2n_{\lambda}$ is
even.}\end{cases}$
Set $x_{1}=[fs_{2}]$ and $x_{i}=[s_{2i-1}s_{2i}]$ for $2\leq i\leq
n_{\lambda}$. Then $[fs_{2i}]=x_{i}+x_{i+1}$ for $2\leq i\leq n_{\lambda}-1$
and $[fs_{2n_{\lambda}}]=x_{n_{\lambda}}$. Thus, we see that
$\\{x_{1},...,x_{n_{\lambda}}\\}$ forms a basis of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$,
and the corresponding rows of $f,s_{2},...,s_{2n_{\lambda}}$ in $A_{0}$ are
nonzero. Now we may reduce $A_{0}$ to $A$ by deleting those zero rows of
$A_{0}$, so that for each
$\theta\in\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$,
$\text{rank}(\theta(A_{0}))=\text{rank}(\theta(A))$ still holds. Write $A$ as
follows:
$\begin{bmatrix}0&x_{1}&x_{2}+x_{3}&x_{3}+x_{4}&\cdots&x_{n_{\lambda}-3}+x_{n_{\lambda}-2}&x_{n_{\lambda}-2}+x_{n_{\lambda}-1}&x_{n_{\lambda}-1}+x_{n_{\lambda}}\\\
x_{1}&0&0&0&{3}&0\\\ 0&x_{2}&x_{2}&0&{3}&0\\\ x_{2}+x_{3}&0&0&0&{3}&0\\\
{8}\\\ 0&0&{3}&0&x_{n_{\lambda}-1}&x_{n_{\lambda}-1}\\\
x_{n_{\lambda}-1}+x_{n_{\lambda}}&0&0&0&{3}&0\\\
0&0&0&0&{2}&0&x_{n_{\lambda}}\\\ x_{n_{\lambda}}&0&0&0&{3}&0\\\ \end{bmatrix}$
Let $\\{\theta_{i}\big{|}i=1,...,n_{\lambda}\\}$ be the dual basis of
$\\{x_{1},...,x_{n_{\lambda}}\\}$ in
$\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$.
Take any
$\theta\in\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$,
one may write $\theta=\sum_{i\in S}\theta_{i}$ where
$S\subset\\{1,...,n_{\lambda}\\}$. Obviously, if $n_{\lambda}=1$ then
$b_{1}(\lambda)=1$ and $b_{2}(\lambda)=0$. If $n_{\lambda}\geq 2$, it is easy
to see that $b_{1}(\lambda)=0$ since $\text{rank}\theta(A)$ cannot be 1
whenever $S$ is empty or non-empty. If $n_{\lambda}=2$, then
$\text{rank}\theta(A)=2$ only when $S=\\{1\\},\\{2\\},\\{1,2\\}$, so
$b_{2}(\lambda)=3$. If $n_{\lambda}>2$, by direct calculations, one has that
only when $S=\\{i\\}(i\not=2)$ or $\\{1,2\\}$, $\text{rank}\theta(A)=2$, so
$b_{2}(\lambda)=n_{\lambda}$.
(2) If $n_{\lambda}>0$, then Lemma 6.1 we may take $B^{\prime}=B_{4}$ and
$B^{\prime\prime}=B_{5}$. Moreover, we see that in $A_{0}$, only corresponding
rows of $s_{1},s_{3},...,s_{2n_{\lambda}-1}$ are nonzero, so we may delete the
other rows from $A_{0}$ to obtain $A$ so that for each
$\theta\in\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$,
$\text{rank}(\theta(A_{0}))=\text{rank}(\theta(A))$. Now we can write down $A$
after simple calculations:
$A=\begin{bmatrix}[s_{1}s_{2}]&0&{2}&0\\\ [s_{2}s_{3}]&[s_{2}s_{3}]&0&{1}&0\\\
0&[s_{4}s_{5}]&[s_{4}s_{5}]&0&{1}\\\ {5}\\\
{2}&0&[s_{2n_{\lambda}-4}s_{2n_{\lambda}-3}]&[s_{2n_{\lambda}-4}s_{2n_{\lambda}-3}]\\\
{3}&0&[s_{2n_{\lambda}-2}s_{2n_{\lambda}-1}]\end{bmatrix}$
Set
$x_{i}=[s_{2i}s_{2i+1}]\in\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$,
for $i=1,...,n_{\lambda}-1$. A direct calculation shows that
$[s_{1}s_{2}]=x_{1}+x_{2}+\cdots+x_{m_{\lambda}-1}$. So we see that
$\\{x_{i}\big{|}i=1,...,n_{\lambda}-1\\}$ forms a basis of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$.
Let $\\{\theta_{i}\big{|}i=1,...,n_{\lambda}-1\\}$ be its dual basis in
$\text{Hom}(\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}),\mathbb{Z}_{2})$.
Then one may write $\theta=\sum_{i\in S}\theta_{i}$ where
$S\subset\\{1,...,n_{\lambda}-1\\}$. Now $i\in S$ implies that the $(i+1)$-th
row of $\theta(A)$ is nonzero. In order that $\text{rank}(\theta(A))=1$, one
must have $\sharp(S)=1$ since $S$ cannot be empty. If $n_{\lambda}=2$ then
$m_{\lambda}=2$, $\mathcal{H}^{1}_{\lambda}$ is 1-dimensional and
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$
has only a nonzero element, so $b_{1}(\lambda)=1$ and $b_{2}(\lambda)=0$. If
$n_{\lambda}>2$, $\text{rank}(\theta_{i}(A))=1$ if and only if
$\theta_{i}([s_{1}s_{2}])=0$, which is equivalent to that
$\theta_{i}(x_{1}+x_{2}+...+x_{m_{\lambda}-1})=0\Leftrightarrow
i>m_{\lambda}-1$. Therefore, $b_{1}(\lambda)=n_{\lambda}-m_{\lambda}$. In this
case, an easy argument shows that $b_{2}(\lambda)={{m_{\lambda}-1}\choose
1}+{{m_{\lambda}-1}\choose 2}+{{n_{\lambda}-m_{\lambda}}\choose 2}$.
(3) If $n_{\lambda}=0$, then Lemma 6.1 we may take $B^{\prime}=B_{4}$ and
$B^{\prime\prime}=B_{5}$, so
$A_{0}=(0,[s_{3}f],...,[s_{m}f]).$
However, a direct calculation shows that for each $i$, $s_{i}f=s_{i}^{2}$, so
$[s_{i}f]=0$ in
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$.
Thus,
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=0$,
and so $\bar{\mathcal{B}}(\lambda)=(0,0)$. ∎
###### Theorem 6.5.
Let $\lambda_{1},\lambda_{2}$ be two nontrivial colorings on
$\mathrm{P}^{3}(m)$ with $m>4$. Then $M(\lambda_{1})$ and $M(\lambda_{2})$ are
homeomorphic if and only if their cohomologies
$H^{*}(M(\lambda_{1});\mathbb{Z}_{2})$ and
$H^{*}(M(\lambda_{2});\mathbb{Z}_{2})$ are isomorphic as rings.
###### Proof.
It suffices to show that if $H^{*}(M(\lambda_{1});\mathbb{Z}_{2})$ and
$H^{*}(M(\lambda_{2});\mathbb{Z}_{2})$ are isomorphic, then $M(\lambda_{1})$
and $M(\lambda_{2})$ are homeomorphic. Now suppose that
$H^{*}(M(\lambda_{1});\mathbb{Z}_{2})\cong
H^{*}(M(\lambda_{2});\mathbb{Z}_{2})$. Then one has that
$\bar{\mathcal{B}}(\lambda_{1})=\bar{\mathcal{B}}(\lambda_{2})$. We claim that
$(m_{\lambda_{1}},n_{\lambda_{1}})=(m_{\lambda_{2}},n_{\lambda_{2}})$. If not,
then by Lemma 6.4, the possible case in which this happens is
$\bar{\mathcal{B}}(\lambda_{1})=\bar{\mathcal{B}}(\lambda_{2})=(1,0)$. Without
loss of generality, assume that $(m_{\lambda_{1}},n_{\lambda_{1}})=(1,0)$ and
$(m_{\lambda_{2}},n_{\lambda_{2}})=(2,2)$. Then by Lemma 6.1, one has
$\Delta(\lambda_{1})=\Delta(\lambda_{2})=1$, so
$\mathcal{H}^{1}_{\lambda_{1}}$ and $\mathcal{H}^{1}_{\lambda_{2}}$ contains
only a nonzero element. Let $z_{0}^{(i)}$ be the unique nonzero element of
$\mathcal{H}^{1}_{\lambda_{i}},i=1,2$. For each $i$, define a linear map
$\Phi_{i}:H^{1}(M(\lambda_{i});\mathbb{Z}_{2})\longrightarrow
H^{2}(M(\lambda_{i});\mathbb{Z}_{2})$ by $x\longmapsto z_{0}^{(i)}x$.
When $i=1$, by Lemma 6.1 one may choose
$B_{1}=\\{f,s_{2},s_{3},...,s_{m-1}\\}$ as a basis of
$H^{1}(M(\lambda_{1});\mathbb{Z}_{2})$ and $B_{2}=\\{f\\}$ as a basis of
$\mathcal{H}^{1}_{\lambda_{1}}$, so $z_{0}^{(1)}=f$. By direct calculations,
one has that for $3\leq j\leq m-1$, $fs_{j}=s_{j}^{2}$. Since
$fs_{2},s_{3}^{2},...,s_{m-1}^{2}$ are linearly independent, one knows that
$\Phi_{1}$ has rank $m-2$.
When $i=2$, by Lemma 6.1 one may choose $B_{4}=\\{s_{1},s_{2},...,s_{m-1}\\}$
as a basis of $H^{1}(M(\lambda_{2});\mathbb{Z}_{2})$ and $B_{5}=\\{s_{2}\\}$
as a basis of $\mathcal{H}^{1}_{\lambda_{2}}$, so $z_{0}^{(1)}=s_{2}$. Since
$s_{2}^{2}=s_{2}s_{j}=0,j\geq 4$, one sees that $\Phi_{2}$ has rank at most
$2$.
Now since $m>4$, one has that
$\text{rank}\Phi_{1}=m-2>2\geq\text{rank}\Phi_{2}$, but this is impossible.
Thus, one must have
$(m_{\lambda_{1}},n_{\lambda_{1}})=(m_{\lambda_{2}},n_{\lambda_{2}})$.
Moreover, the theorem follows from Corollary 5.10. ∎
###### Corollary 6.6.
Let $\lambda_{1},\lambda_{2}$ be two nontrivial colorings on
$\mathrm{P}^{3}(m)$ with $m>4$. Then $M(\lambda_{1})$ and $M(\lambda_{2})$ are
homeomorphic if and only if
$(m_{\lambda_{1}},n_{\lambda_{1}})=(m_{\lambda_{2}},n_{\lambda_{2}})$.
Furthermore, by Corollary 5.11 one has
###### Corollary 6.7.
The number of homeomorphism classes of small covers over $\mathrm{P}^{3}(m)$
$(m>4)$ with nontrivial colorings is exactly
$N_{nt}(m)=\begin{cases}\sum_{0\leq k\leq{m\over 2}}([{k\over 2}]+1)&\text{ if
$m$ is even }\\\ \sum_{1\leq k\leq{m\over 2}}([{k\over 2}]+1)&\text{ if $m$ is
odd.}\\\ \end{cases}$
Next let us look at the case in which $\lambda$ is trivial. By Lemma 6.2 we
divide our argument into two cases: (I) $\Delta(\lambda)$ is odd; (II)
$\Delta(\lambda)$ is even.
Case (I): $\Delta(\lambda)$ is odd.
###### Lemma 6.8.
Let $\lambda$ be trivial such that $\Delta(\lambda)$ is odd. Then
$\bar{\mathcal{B}}(\lambda)=\begin{cases}(0,2^{m-2}-1)&\text{if
$\lambda\approx\lambda_{C_{1}}$}\\\ (0,2^{m-3}-1)&\text{if
$\lambda\approx\lambda_{C_{3}}$}\\\ (2^{m-4}-1,0)&\text{if
$\lambda\approx\lambda_{C_{8}}$}\\\ (2^{m-3}-1,0)&\text{if
$\lambda\approx\lambda_{C_{9}}$}\\\ (2^{m-4}-1,0)&\text{if
$\lambda\approx\lambda_{C_{10}}$}\end{cases}$
###### Proof.
If $\lambda\approx\lambda_{C_{1}}$, using Lemma 6.2 and by direct
calculations, one has that
$s_{1}s_{2}=s_{2}s_{3}=\cdots=s_{m-1}s_{m}=s_{m}s_{1}$, so $A_{0}$ may be
written as follows:
$\begin{bmatrix}0&x_{2}&x_{3}&x_{4}&\cdots&x_{m-2}&x_{m-1}\\\
x_{2}&0&x_{1}&0&\cdots&0&0\\\ x_{3}&x_{1}&0&x_{1}&\cdots&0&0\\\
x_{4}&0&x_{1}&0&\cdots&0&0\\\ {7}\\\ x_{m-2}&0&0&0&\cdots&0&x_{1}\\\
x_{m-1}&0&0&0&\cdots&x_{1}&0\end{bmatrix}$
where $x_{1}=[s_{3}s_{4}]$ and $x_{i}=[fs_{i+1}],i=2,...,m-1$. We see easily
that $\\{x_{1},...,x_{m-1}\\}$ forms a basis of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$
so
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-1$.
Then one may conclude that $\bar{\mathcal{B}}(\lambda)=(0,2^{m-2}-1)$. Also,
it is easy to see that in this case $b_{\Delta(\lambda)}$ is nonzero.
In a similar way as above, if $\lambda\approx\lambda_{C_{3}}$, one has that
$[s_{2}s_{3}]=[s_{3}s_{4}]=[s_{4}s_{5}]=\cdots=[s_{m-1}s_{m}]=0$, so $A_{0}$
may be written as follows:
$\begin{bmatrix}0&x_{3}&x_{4}&x_{5}&\cdots&x_{m-3}&\sum_{j\text{ is
odd}}x_{j}&x_{1}+\sum_{j\text{ is even}}x_{j}\\\ x_{1}&0&0&0&\cdots&0&0&0\\\
x_{2}&0&0&0&\cdots&0&0&0\\\ x_{3}&0&0&0&\cdots&0&0&0\\\ {8}\\\
x_{m-3}&0&0&0&\cdots&0&0&0\\\ \sum_{j\text{ is
odd}}x_{j}&0&0&0&\cdots&0&0&0\end{bmatrix}$
where $x_{i}=[fs_{i+1}],i=1,...,m-3$. And $\\{x_{1},...,x_{m-3}\\}$ forms a
basis of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$
so
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-3$.
A direct observation shows that $\bar{\mathcal{B}}(\lambda)=(0,2^{m-3}-1)$.
If $\lambda\approx\lambda_{C_{8}}$ or $\lambda_{C_{9}}$, then
$[s_{1}s_{2}]=[s_{2}s_{3}]=\cdots=[s_{m-1}s_{m}]=[s_{m}s_{1}]=0$, so $A_{0}$
can be reduced to a $1\times(m-3)$ matrix
$([fs_{2}],[fs_{4}],...,[fs_{m-1}]).$
Also, we easily see that $\\{s_{3}^{2},fs_{2},fs_{4},...,fs_{m-1}\\}$ can be
used as a basis of $\mathcal{K}_{\lambda}$ and $\\{f^{2},s_{3}^{2}\\}$ forms a
basis of $\mathcal{H}_{\lambda}^{2}$ (note that
$\dim\mathcal{H}_{\lambda}^{2}=m-1-\Delta(\lambda)$=2). However, when
$\lambda\approx\lambda_{C_{8}}$, by direct calculations one has that
$f^{2}=fs_{2}+\sum_{j>4\text{ is odd}}fs_{j}$, so
$f^{2}\in\mathcal{K}_{\lambda}$ and
$\mathcal{H}_{\lambda}^{2}\subset\mathcal{K}_{\lambda}$. Thus,
$\dim\mathcal{H}_{\lambda}^{2}\cap\mathcal{K}_{\lambda}=2$,
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-4$
and $\\{[fs_{4}],...,[fs_{m-1}]\\}$ forms a basis of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$.
Moreover, one has that $\bar{\mathcal{B}}(\lambda_{C_{8}})=(2^{m-4}-1,0)$.
When $\lambda\approx\lambda_{C_{9}}$, it is not difficult to check that
$\dim\mathcal{H}_{\lambda}^{2}\cap\mathcal{K}_{\lambda}=1$ and
$[fs_{2}],[fs_{4}],...,[fs_{m-1}]$ are linearly independent, so
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-3$.
Thus, $\bar{\mathcal{B}}(\lambda_{C_{9}})=(2^{m-3}-1,0)$.
If $\lambda\approx\lambda_{C_{10}}$, then
$[s_{1}s_{2}]=[s_{2}s_{3}]=\cdots=[s_{m-1}s_{m}]=[s_{m}s_{1}]=[s_{3}^{2}]=0$
and $fs_{2}=f^{2}$, so $A_{0}$ can be reduced to a $1\times(m-3)$ matrix
$(0,[fs_{4}],...,[fs_{m-1}])$. It is easy to see that
$\dim\mathcal{H}_{\lambda}^{2}\cap\mathcal{K}_{\lambda}=1$ and
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-4$,
so $\bar{\mathcal{B}}(\lambda_{C_{10}})=(2^{m-4}-1,0)$. ∎
Case (II): $\Delta(\lambda)$ is even.
###### Lemma 6.9.
Let $\lambda$ be trivial such that $\Delta(\lambda)$ is even. Then
$\bar{\mathcal{B}}(\lambda)=\begin{cases}(1,2^{m-2}-2)&\text{if
$\lambda\approx\lambda_{C_{2}}$}\\\ (2^{m-2}-1,0)&\text{if
$\lambda\approx\lambda_{C_{4}}$}\\\ (2^{m-3}-1,0)&\text{if
$\lambda\approx\lambda_{C_{5}}$}\\\ (2^{m-4}-1,0)&\text{if
$\lambda\approx\lambda_{C_{6}}$}\\\ (2^{m-3}-1,0)&\text{if
$\lambda\approx\lambda_{C_{7}}$}\end{cases}$
###### Proof.
If $\lambda\approx\lambda_{C_{2}}$, then one can obtain by Lemma 6.2 that
$[s_{1}s_{2}]=[s_{2}s_{3}]=\cdots=[s_{m-1}s_{m}]=[s_{m}s_{1}]=[s_{1}^{2}]=0$
and so $A_{0}$ can be reduced to the following matrix
$\begin{bmatrix}0&[fs_{2}]&[fs_{4}]&\cdots&[fs_{m-1}]\\\
[fs_{2}]&0&0&\cdots&0\\\ [fs_{3}]&0&0&\cdots&0\\\ [fs_{4}]&0&0&\cdots&0\\\
{5}\\\ [fs_{m-1}]&0&0&\cdots&0\end{bmatrix}$
One may easily show that $\\{[fs_{2}],...,[fs_{m-1}]\\}$ is a basis of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$.
Then a direct observation can obtain that
$\bar{\mathcal{B}}(\lambda_{C_{2}})=(1,2^{m-2}-2)$.
If $\lambda\approx\lambda_{C_{4}}$, then one has that
$[s_{1}s_{2}]=[s_{2}s_{3}]=\cdots=[s_{m-1}s_{m}]$, so $A_{0}$ can be reduced
to the following matrix
$\begin{bmatrix}[fs_{3}]&[fs_{4}]&[fs_{5}]&\cdots&[fs_{m-3}]&[fs_{m-2}]&[fs_{m-1}]&[fs_{m}]\\\
[s_{1}s_{2}]&0&0&\cdots&0&0&0&0\\\ 0&[s_{1}s_{2}]&0&\cdots&0&0&0&0\\\
[s_{1}s_{2}]&0&[s_{1}s_{2}]&\cdots&0&0&0&0\\\ {8}\\\
0&0&0&\cdots&[s_{1}s_{2}]&0&[s_{1}s_{2}]&0\\\
0&0&0&\cdots&0&[s_{1}s_{2}]&0&[s_{1}s_{2}]\end{bmatrix}$
and $\\{[s_{1}s_{2}],[fs_{3}],...,[fs_{m}]\\}$ is a basis of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$
so
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-1$.
Furthermore, one knows that
$\bar{\mathcal{B}}(\lambda_{C_{4}})=(2^{m-2}-1,0)$. Note that in this case
$b_{\Delta(\lambda)}$ is nonzero.
If $\lambda\approx\lambda_{C_{5}}$, then one has that
$[s_{1}^{2}]=[s_{1}s_{2}]=[s_{2}s_{3}]=\cdots=[s_{m-1}s_{m}]=0$, so $A_{0}$
can be reduced to a $1\times(m-3)$ matrix
$([fs_{3}],[fs_{4}],...,[fs_{m-1}])$, and
$\\{[fs_{3}],[fs_{4}],...,[fs_{m-1}]\\}$ is a basis of
$\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})$
so
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-3$.
Note that in this case
$\dim\mathcal{H}_{\lambda}^{2}\cap\mathcal{K}_{\lambda}=1$. Thus,
$\bar{\mathcal{B}}(\lambda_{C_{5}})=(2^{m-3}-1,0)$, but
$b_{\Delta(\lambda)}=0$.
If $\lambda\approx\lambda_{C_{6}}$ or $\lambda_{C_{7}}$, similarly to the case
$\lambda\approx\lambda_{C_{5}}$, then one has that
$[s_{1}^{2}]=[s_{1}s_{2}]=[s_{2}s_{3}]=\cdots=[s_{m-1}s_{m}]=0$, so $A_{0}$
can be reduced to a $1\times(m-3)$ matrix
$([fs_{3}],[fs_{4}],...,[fs_{m-1}]).$
As in the proof of cases $\lambda\approx\lambda_{C_{8}}$ or $\lambda_{C_{9}}$,
we see that $\\{s_{2}^{2},fs_{3},fs_{4},...,fs_{m-1}\\}$ can be used as a
basis of $\mathcal{K}_{\lambda}$ and $\\{f^{2},s_{2}^{2}\\}$ forms a basis of
$\mathcal{H}_{\lambda}^{2}$. However, when $\lambda\approx\lambda_{C_{6}}$, it
is easy to check that $\mathcal{H}_{\lambda}^{2}\subset\mathcal{K}_{\lambda}$,
so $\dim\mathcal{H}_{\lambda}^{2}\cap\mathcal{K}_{\lambda}=2$ and
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-4$.
Moreover, $\bar{\mathcal{B}}(\lambda_{C_{6}})=(2^{m-4}-1,0)$ and
$b_{\Delta(\lambda)}=0$. When $\lambda\approx\lambda_{C_{7}}$, one may check
that $\dim\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}=1$ and then
$\dim\mathcal{K}_{\lambda}/(\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2})=m-3$.
Thus $\bar{\mathcal{B}}(\lambda_{C_{7}})=(2^{m-3}-1,0)$ and
$b_{\Delta(\lambda)}=0$. ∎
###### Remark 6.3.
We see that for $\lambda_{C_{5}}$ and $\lambda_{C_{7}}$,
$\Delta(\lambda_{C_{5}})=\Delta(\lambda_{C_{7}})$ and
$\bar{\mathcal{B}}(\lambda_{C_{5}})=\bar{\mathcal{B}}(\lambda_{C_{7}})$.
However, we can still distinguish them by using the first Stiefel-Whitney
class. Let $w_{1}(\lambda)\in H^{1}(M(\lambda);\mathbb{Z}_{2})$ denote the
first Stiefel-Whitney class. It is well-known that $w_{1}(\lambda)=0$ if and
only if $M(\lambda)$ is orientable. Then, by Corollary 5.6 one knows that if
$\lambda\approx\lambda_{C_{5}}$, then $w_{1}(\lambda_{C_{5}})=0$; but if
$\lambda\approx\lambda_{C_{7}}$, $w_{1}(\lambda_{C_{7}})\not=0$. This also
happens for $\lambda_{C_{8}}$ and $\lambda_{C_{10}}$. But we can use the
number $\dim\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}$ to distinguish
them. Actually, by Lemma 6.8, if $\lambda\approx\lambda_{C_{8}}$,
$\dim\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}=2$; but if
$\lambda\approx\lambda_{C_{10}}$,
$\dim\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}=1$.
###### Theorem 6.10.
Let $\lambda_{1},\lambda_{2}$ be two trivial colorings on $\mathrm{P}^{3}(m)$
with $m>4$. Then $M(\lambda_{1})$ and $M(\lambda_{2})$ are homeomorphic if and
only if their cohomologies $H^{*}(M(\lambda_{1});\mathbb{Z}_{2})$ and
$H^{*}(M(\lambda_{2});\mathbb{Z}_{2})$ are isomorphic as rings.
###### Proof.
This follows immediately from Lemmas 6.2, 6.8-6.9 and Remark 6.3. ∎
As a consequence of Collorary 5.5 and Theorem 6.10, one has
###### Corollary 6.11.
The number of homeomorphism classes of small covers over
$\mathrm{P}^{3}(m)(m>4)$ with trivial colorings is exactly
$N_{t}(m)=\begin{cases}4&\text{ if $m$ is odd}\\\ 6&\text{ if $m$ is
even}\end{cases}$
## 7\. Proofs of Theorems 1.1 and 1.2
Now let us finish the proofs of Theorems 1.1 and 1.2.
Proof of Theorem 1.1. It suffices to show that if their cohomologies
$H^{*}(M(\lambda_{1});{\mathbb{Z}}_{2})$ and
$H^{*}(M(\lambda_{2});{\mathbb{Z}}_{2})$ are isomorphic as rings, then
$M(\lambda_{1})$ and $M(\lambda_{2})$ are homeomorphic. By Propositions 6.3,
6.5 and 6.10, this is true when $m>6$. It remains to consider the case $m\leq
6$. As stated in Remark 6.2, the cohomological rigidity holds when $m\leq 4$
(see also [LY] and [M3]). Next, we only need put our attention on the case
$5\leq m\leq 6$. By Lemmas 6.1, 6.2, 6.4, 6.8, 6.9 and Remark 6.3, we may list
all possible $\lambda$ with mentioned invariants in the case $5\leq m\leq 6$
whichever $\lambda$ is trivial or nontrivial.
(A) Case $m=5$:
$\lambda$ | Trivialization | $\Delta(\lambda)$ | $\bar{\mathcal{B}}(\lambda)$ | $(n_{\lambda},m_{\lambda})$ | $w_{1}(\lambda)$
---|---|---|---|---|---
$\lambda_{C_{*}}$ | nontrivial | 1 | $(1,0)$ | $(1,0)$ |
$\lambda_{C_{*}}$ | nontrivial | 2 | $(1,3)$ | $(2,0)$ |
$\lambda_{C_{*}}$ | nontrivial | 1 | $(1,0)$ | $(2,2)$ |
$\lambda_{C_{3}}$ | trivial | 3 | $(0,3)$ | |
$\lambda_{C_{5}}$ | trivial | 2 | $(3,0)$ | | 0
$\lambda_{C_{6}}$ | trivial | 2 | $(1,0)$ | |
$\lambda_{C_{7}}$ | trivial | 2 | $(3,0)$ | | nonzero
(B) Case $m=6$:
$\lambda$ | Trivialization | $\Delta(\lambda)$ | $\bar{\mathcal{B}}(\lambda)$ | $(n_{\lambda},m_{\lambda})$ | $\dim\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}$
---|---|---|---|---|---
$\lambda_{C_{*}}$ | nontrivial | 1 | $(1,0)$ | $(0,0)$ |
$\lambda_{C_{*}}$ | nontrivial | 1 | $(1,0)$ | $(1,0)$ |
$\lambda_{C_{*}}$ | nontrivial | 2 | $(1,3)$ | $(2,0)$ |
$\lambda_{C_{*}}$ | nontrivial | 3 | $(0,3)$ | $(3,0)$ |
$\lambda_{C_{*}}$ | nontrivial | 1 | $(1,0)$ | $(2,2)$ |
$\lambda_{C_{*}}$ | nontrivial | 2 | $(1,1)$ | $(3,2)$ |
$\lambda_{C_{1}}$ | trivial | 5 | $(0,15)$ | |
$\lambda_{C_{2}}$ | trivial | 4 | $(1,14)$ | |
$\lambda_{C_{4}}$ | trivial | 4 | $(15,0)$ | |
$\lambda_{C_{8}}$ | trivial | 3 | $(3,0)$ | | 2
$\lambda_{C_{9}}$ | trivial | 3 | $(7,0)$ | |
$\lambda_{C_{10}}$ | trivial | 3 | $(3,0)$ | | 1
We clearly see from two tables above that by using invariants
$\Delta(\lambda)$, $\bar{\mathcal{B}}(\lambda)$, $(n_{\lambda},m_{\lambda})$,
$w_{1}(\lambda)$ and $\dim\mathcal{K}_{\lambda}\cap\mathcal{H}_{\lambda}^{2}$,
we can distinguish all $M(\lambda)$ up to homeomorphism when $m=5,6$. This
completes the proof. $\Box$
Furthermore, Theorem 1.2 follows immediately from Theorem 1.1, Corollaries
6.7, 6.11 and Remark 6.2.
Finally, let us return to the invariants $\Delta(\lambda)$ and
$\mathcal{B}(\lambda)$ again. We see that generally these invariants can
always be defined for any small cover over a simple convex polytop $P^{n}$. We
would like to pose the following problems:
1. $\bullet$
Under what condition can $\Delta(\lambda)$ and $\mathcal{B}(\lambda)$ become
the combinatorial invariants?
2. $\bullet$
If $\Delta(\lambda)$ and $\mathcal{B}(\lambda)$ are the combinatorial
invariants, then how can one calculate them in terms of polytopes $P^{n}$?
## References
* [BP] V.M. Buchstaber and T.E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, vol. 24, Amer. Math. Soc., Providence, RI, 2002.
* [CCL] M. Z. Cai, X. Chen, and Z. Lü, Small covers over prisms, Topology Appl. 154 (2007) 2228-2234.
* [C] S. Choi, The number of small covers over cubes, Algebr. Geom. Topol. 8 (2008), 2391-2399.
* [D] M. W. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Annals of Mathematics, 117 (1983), 293-324.
* [DJ] M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus Actions, Duke Mahtematicial Journal, 62 (1991), 417-451.
* [GS] A. Garrison, R. Scott, Small covers of the dodecahedron and the 120-cell, Proc. Amer. Math. Soc.131 (2002) 963-971.
* [I] I. V. Izmestiev, Three-dimensional manifolds defined by a coloring of the faces of a simple polytope, Math. Notes 69 (2001), no. 3-4, 340–346.
* [KM] Y. Kamishima and M. Masuda, Cohomological rigidity of real Bott manifolds, arXiv:0807.4263.
* [LM] Z. Lü and M. Masuda, Equivariant classification of $2$-torus manifolds, arXiv:0802.2313.
* [LY] Z. Lü and L. Yu, Topological types of 3-dimensianl small covers (accepted), to appear in Forum Math., arXiv:0710.4496.
* [M1] M. Masuda, Equivariant cohomology distinguishes toric manifolds, Adv. Math. 218 (2008), 2005-2012.
* [M2] M. Masuda, Cohomological non-rigidity of generalized real Bott manifolds of height 2, preprint, arXiv:0809.2215.
* [M3] M. Masuda, Classification of real Bott manifolds, preprint, arXiv:0809.2178.
* [NN] H. Nakayama and Y. Nishimura, The orientability of small covers and coloring simple polytopes, Osaka J. Math. 42 (2005), 243-256.
* [Z] G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Math., Springer-Verlag, Berlin, 1994.
|
arxiv-papers
| 2009-03-23T18:50:36 |
2024-09-04T02:49:01.316653
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiangyu Cao and Zhi L\\\"u",
"submitter": "Zhi L\\\"u",
"url": "https://arxiv.org/abs/0903.3653"
}
|
0903.3669
|
# Comment on ”Language Trees and Zipping”
Xiu-Li Wang wangxiuli@ahu.edu.cn Departmant of Chinese Literature and
Language Anhui University Hefei Anhui 230039 China
###### Abstract
every encoding has priori information if the encoding represents any semantic
information of the un- verse or object.Encoding means mapping from the un-
verse to the string or strings of digits. The semantic here is used in the
model-theoretic sense or denotation of the object.if encoding or strings of
symbols is the adequate and true mapping of model or object,and the mapping is
recursive or computable ,the distance between two strings(text)is mapping the
distance between models.We then are able to measure the distance by computing
the distance be- tween the two strings.Oherwise,we may take a misleading
course.”language tree” may not be a family tree in the sense of historical
linguistics.Rather it just means the similarity
###### pacs:
showpacs
## I Comment on ”Language Trees and Zipping”
Several statements that Benedetto et al.make in their Letter Benedetto et al.
(2002, 2003)are not certainly true.First,We claim a statement that Benedetto
et al.. make in their Letter and their reply Benedetto et al. (2002, 2003)has
mixed strings of symbols with the objects or models the strings denote.In
another word ,strings of symbols are different from the object or model the
strings denote except when the strings only denote themselves.Moreover,a
statement of the comment on the Letter by Dmitry V. Khmelev et al.is
inaccurate Khmelev and Teahan (2003).That is ,”Notice that the language tree
(LT) diagram [1] does not include the Russian language (Slavic family of Indo-
European family of languages: $288\times 10^{6}$speakers). Our computations
show that once Russian is included, it does not cluster with the other members
of the Slavic group. Obviously, certain Cyrillic alphabet based languages were
left out of the study , which improves results significantly and shows that a
priori information about the alphabet is being taken advantage of to achieve
the results outlined in their Letter .”.
String of symbols and symbol may self-refer or refer to other object.When It
refer to or denote another object ,we say the object is model of the string of
symbols or meaning (semantics) of the string of symbols Simpson (1998); Otto
(2002).The string of symbols represents the object or the model.Obviously when
It refer to or denote Itself,the meaning or model and the symbol or string of
symbols are the same.The alphabet or text(string of symbols) are not
language.They are symbols or strings of symbols that just record the language
Clearly ,every encoding has priori information if the encoding represents any
semantic information of the unverse or object.Encoding means mapping from the
unverse to the string or strings of digits. The semantic here is used in the
model-theoretic sense or denotation of the object .By choosing a string or
code that maps the entities,relation and function in the unverse to symbols
and the relation,function of the symbols ,We encode our knowledge about the
model or object too.If we encode the object by randomly assigning the object
to a string everyone or machine can not recognize or get any information about
the unverse or the object without the assignment.For instance,by isomorphism
,a group is mapped to a group which maintain any information of the former one
such as relations function etc.If the group is mapped to an other structure
randomly ,we can not get any information about the former one from the latter
one without the mapping,even when we know there exist a mapping from the group
to the structure. We may consider the a logical sentence as the code of its
model.A more concrete example is the binary code of integer.If the mapping
from integer to binary code is random,we can not recover the integer from its
binary code without the mapping.Even the mapping is not random ,that is, the
mapping is recursive or computable ,we have to make effort to get the
information if we know there exists a mapping that is recursive,or we are
unable to get any information about the integer.Afterall ,the mapping and the
model a string correspond to are priori information that human being provide.
Therefore,it is true that every encoding has priori information which is
symbolization(mapping to symbol) of part or all of the human being’s knowledge
about the model.Even when ”As for the objection concerning the coding chosen
for our texts, one has to remember that a zipper reads the sequences of
characters which one inputs to it, nothing more than this. The idea of
comparing languages written with different alphabets cannot forget this simple
statement. In order to compare languages written with different alphabets one
should, for instance, consider texts written with the phonetic alphabet. This
is the reason for not having included in our preliminary analysis of the
language tree languages such as Chinese, Greek, Russian, etc.”, the phonetic
alphabet with which the texts are written encodes the knowledge of human about
the language.
Hence,if the distance that Benedetto et al.define is capable of the measure of
similarity of the compressed text,It at most measures the similarity between
the two text compared .If the alphabet computationally represent some
information of language ,the distance resulted from the comparison is the
measure of the similarity of information of the language.Otherwise It is just
the measure of the similarity of the text.
When the compression technique is applied to DNA sequence to cluster DNA,the
distance is just the measure of the similarity.Only under the presupposition
that DNA is mapping of features of creature can we get some information of
creature such as evolution relation or family tree.
Secondly ,the language tree may not be a family tree .Indo-European family of
languages is not a concept that describe the family composed of descendants
and their ancestor H.Robins (1973).
Many Languages are descendants of a same archaic one.They are very similar in
spelling,syntax even meaning or semantics when they inherit or use the same
alphabet.Historical linguist compare language in spelling (phonetics),syntax
and meaning to reconstruct their ancestor.But unfortunately these effort and
results are proved not to be solid or reliable in many cases without data such
as historical text record .Rather,We know that similarity may be because of
type of languages that happen to be similar in some aspect ,interaction
between languages which is called linguistic union or being descendant of a
same ancient father.There is no genetic relationship between languages, but
they still share features, and they are spoken in the same region .Balkan
linguistic union or sprachbunds, such as Albanian, Greek, Bulgarian and
Romanian are all IE languages .However, they are not closely related.
Classification of languages may be genetic typological or areal(linguistic
union) H.Robins (1973).So,what does the term ”language tree” mean?It may not
be a family tree in the sense of historical linguistics.Rather it just means
the similarity H.Robins (1973).By the technique,Benedetto et al.just show the
similarity between the texts ,or the similarity between the languages that may
not be similarity among members of family only if the similarity between the
text (strings or symbols) is the mapping of the similarity between the
languages adequately and truly.The language tree is not able to be considered
as a family tree in the sense of historical linguistics.
Thirdly,the distance Benedetto et al.define in their Letter is similar to the
NID definition by Li Ming Li and Vitanyi (1997).As we discuss relation between
the encoding and model above,if encoding or strings of symbols is the adequate
and true mapping of model or object,and the mapping is recursive or computable
,the distance between two strings(text)is mapping the distance between
models.We then are able to measure the distance by computing the distance
between the two strings.Oherwise,we may take a misleading course.
There is intention (presupposition) in pure mathematic research that the
mapping from model to string is not considered as a key question.But
application to practical problem may cause trouble or error.In fact,it has to
be solved firstly to decide wether mapping from model to string or strings
contains the information of the model,although we often do the mapping that is
heuristic and valid. As everyone knows,theory of physics is the ”strings”,and
experiments of physics is to test or check wether the mapping is valid.The
empirical science may be consider as searching for and testing mapping.
###### Acknowledgements.
Thank Ming-Hui Zhang who works as a faculty in Physics Department of Anhui
University for helpful discussion.
## References
* Benedetto et al. (2002) D. Benedetto, E. Caglioti, and V. Loreto, Phys. Rev. Lett. 88, 048702 (2002).
* Benedetto et al. (2003) D. Benedetto, E. Caglioti, and V. Loreto, Phys. Rev. Lett. 90, 089804 (2003).
* Khmelev and Teahan (2003) D. V. Khmelev and W. J. Teahan, Phys. Rev. Lett. 90, 089803 (2003).
* Simpson (1998) S. G. Simpson, _Model Theory_ (1998), URL http://www.math.psu.edu/simpson/courses/math563.
* Otto (2002) M. Otto, _Algorithmic Model Theory for Specific Semantic Domains_ (2002), URL http://www-compsci.swan.ac.uk/~csmartin/amt.html.
* H.Robins (1973) R. H.Robins, Current Trends in Linguistics 11, 3 (1973).
* Li and Vitanyi (1997) M. Li and P. M. B. Vitanyi, _An Introduction to Kolmogorov Complexity and Its Applications_ (Springer-Verlag, Berlin, 1997), second edition ed.
|
arxiv-papers
| 2009-03-21T14:29:11 |
2024-09-04T02:49:01.331321
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiuli Wang",
"submitter": "Xiu-Li Wang",
"url": "https://arxiv.org/abs/0903.3669"
}
|
0903.3786
|
# Multiple-Input Multiple-Output Gaussian Broadcast Channels with Confidential
Messages
Ruoheng Liu, Tie Liu, H. Vincent Poor, and Shlomo Shamai (Shitz) This research
was supported by the United States National Science Foundation under Grants
CNS-06-25637 and CCF-07-28208, the European Commission in the framework of the
FP7 Network of Excellence in Wireless Communications NEWCOM++, and the Israel
Science Foundation.Ruoheng Liu and H. Vincent Poor are with the Department of
Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
(e-mail: {rliu,poor}@princeton.edu).Tie Liu is with the Department of
Electrical and Computer Engineering, Texas A&M University, College Station, TX
77843, USA (e-mail: tieliu@tamu.edu).Shlomo Shamai (Shitz) is with the
Department of Electrical Engineering, Technion-Israel Institute of Technology,
Technion City, Haifa 32000, Israel (e-mail: sshlomo@ee.technion.ac.il).
###### Abstract
This paper considers the problem of secret communication over a two-receiver
multiple-input multiple-output (MIMO) Gaussian broadcast channel. The
transmitter has two independent messages, each of which is intended for one of
the receivers but needs to be kept asymptotically perfectly secret from the
other. It is shown that, surprisingly, under a matrix power constraint both
messages can be simultaneously transmitted at their respective maximal secrecy
rates. To prove this result, the MIMO Gaussian wiretap channel is revisited
and a new characterization of its secrecy capacity is provided via a new
coding scheme that uses artificial noise and random binning.
###### Index Terms:
Artificial noise, broadcast channel, channel enhancement, information-
theoretic security, multiple-input multiple-output (MIMO) communications,
wiretap channel
## I Introduction
Rapid advances in wireless technology are quickly moving us toward a
pervasively connected world in which a vast array of wireless devices, from
iPhones to biosensors, seamlessly communicate with one another. The openness
of the wireless medium makes wireless transmission especially susceptible to
eavesdropping. Hence, security and privacy issues have become increasingly
critical for wireless networks. Although wireless technologies are becoming
more and more secure, eavesdroppers are also becoming smarter. Sole reliance
on cryptographic keys in large distributed networks where terminals can be
compromised is no longer sustainable from the security perspective.
Furthermore, in wireless networks, secure initial key distribution is
difficult and, in fact, can be performed in perfect secrecy only via physical
layer techniques. Therefore, tackling security at the very basic physical
layer is of critical importance.
In this paper, we study the problem of secret communication over the multiple-
input multiple-output (MIMO) Gaussian broadcast channel with two receivers.
The transmitter is equipped with $t$ transmit antennas, and receiver $k$,
$k=1,2$, is equipped with $r_{k}$ receive antennas. A discrete-time sample of
the channel can be written as
$\mathbf{Y}_{k}[m]=\mathbf{H}_{k}\mathbf{X}[m]+\mathbf{Z}_{k}[m],\quad k=1,2$
(1)
where $\mathbf{H}_{k}$ is the (real) channel matrix of size $r_{k}\times t$,
and $\\{\mathbf{Z}_{k}[m]\\}_{m}$ is an independent and identically
distributed (i.i.d.) additive vector Gaussian noise process with zero mean and
identity covariance matrix. The channel input $\\{\mathbf{X}[m]\\}_{m}$ is
subject to the matrix power constraint:
$\frac{1}{n}\sum_{m=1}^{n}\left(\mathbf{X}[m]\mathbf{X}^{\intercal}[m]\right)\preceq\mathbf{S}$
(2)
where $\mathbf{S}$ is a positive semidefinite matrix, and “$\preceq$” denotes
“less than or equal to” in the positive semidefinite ordering between real
symmetric matrices. Note that (2) is a rather general power constraint that
subsumes many other important power constraints including the average total
and per-antenna power constraints as special cases.
Figure 1: MIMO Gaussian broadcast channel with confidential messages.
Consider the communication scenario in which there are two independent
messages $W_{1}$ and $W_{2}$ at the transmitter. Message $W_{1}$ is intended
for receiver 1 but needs to be kept secret from receiver 2, and message
$W_{2}$ is intended for receiver 1 but needs to be kept secret from receiver
2. (See Fig. 1 for an illustration of this communication scenario.) The
confidentiality of the messages at the unintended receivers is measured using
the normalized information-theoretic quantities [1, 2]:
$\frac{1}{n}I(W_{1};\mathbf{Y}_{2}^{n})\rightarrow
0\quad\mbox{and}\quad\frac{1}{n}I(W_{2};\mathbf{Y}_{1}^{n})\rightarrow 0$
where $\mathbf{Y}_{k}^{n}:=(\mathbf{Y}_{k}[1],\ldots,\mathbf{Y}_{k}[n])$, and
the limits are taken as the block length $n\rightarrow\infty$. The goal is to
characterize the entire secrecy rate region
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})=\\{(R_{1},R_{2})\\}$
that can be achieved by any coding scheme.
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ is usually known
as the _secrecy capacity region_ of the channel.
In recent years, information-theoretic study of secret MIMO communication has
been an active area of research. (See [3] for a recent survey of progress in
this area.) Most noticeably, the secrecy capacity of the MIMO Gaussian wiretap
channel was characterized in [4, 5, 6] for the multiple-input single-output
(MISO) case and [7, 8, 9, 10] for the general MIMO case. The secrecy capacity
region of the MIMO Gaussian broadcast channel with a common and a confidential
messages was characterized in [11]. The problem of communicating two
confidential messages over the two-receiver MIMO Gaussian broadcast channel
was first considered in [12], where it was shown that under the average total
power constraint, secret dirty-paper coding (S-DPC) based on double binning
[13] achieves the secrecy capacity region for the MISO case. For the general
MIMO case, however, characterizing the secrecy capacity region remained as an
open problem.
The main result of this paper is a precise characterization of the secrecy
capacity region of the (general) MIMO Gaussian broadcast channel, summarized
in the following theorem.
###### Theorem 1
The secrecy capacity region
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ of the MIMO
Gaussian broadcast channel (1) with confidential messages $W_{1}$ (intended
for receiver 1 but needing to be kept secret from receiver 2) and $W_{2}$
(intended for receiver 2 but needing to be kept secret from receiver 1) under
the matrix power constraint (2) is given by the set of nonnegative rate pairs
$(R_{1},R_{2})$ such that
$\displaystyle R_{1}$
$\displaystyle\leq\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|\right)$
$\displaystyle\text{and}\qquad R_{2}$
$\displaystyle\leq\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}}\right|\right)$
(3)
where $\mathbf{I}_{r_{k}}$ denotes the identity matrix of size $r_{k}\times
r_{k}$.
###### Remark 1
Note that the rate region (3) is _rectangular_. This implies that under the
matrix power constraint, both confidential messages $W_{1}$ and $W_{2}$ can be
_simultaneously_ transmitted at their respective maximal secrecy rates (as if
over two separate MIMO Gaussian wiretap channels). The secrecy capacity of the
MIMO Gaussian wiretap channel under the matrix power constraint was
characterized in [9], by which the rate region (3) can be rewritten as the set
of nonnegative rate pairs $(R_{1},R_{2})$ such that
$\displaystyle R_{1}$
$\displaystyle\leq\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|\right)$
$\displaystyle\text{and}\qquad R_{2}$
$\displaystyle\leq\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|\right).$
(4)
###### Remark 2
Also note that if $\mathbf{B}^{\star}$ is an optimal solution to the
optimization program:
$\displaystyle\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|-\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|\right),$
(5)
then $\mathbf{B}^{\star}$ _simultaneously_ maximizes both objective functions
on the right-hand side (RHS) of (3). On the other hand, the optimization
programs on the RHS of (4) do not, in general, admit the same optimal
solution. As we will see, this makes (3) a better choice when it comes to
proving the achievability part of the theorem.
It is rather surprising to see that under the matrix power constraint, both
confidential messages $W_{1}$ and $W_{2}$ can be simultaneously transmitted at
their respective maximal secrecy rates over the MIMO Gaussian broadcast
channel (1). As we will see, this is due to the fact that there are in fact
two different coding schemes: one uses only random binning, and the other uses
both random binning and _artificial noise_. Both of them can achieve the
secrecy capacity of the MIMO Gaussian wiretap channel. Through S-DPC (double
binning) [13], both schemes can be _simultaneously_ implemented in
communicating confidential messages $W_{1}$ and $W_{2}$ over the MIMO Gaussian
broadcast channel (1).
As a corollary, we have the following characterization of the secrecy capacity
region under the average total power constraint. The result is a simple
consequence of [14, Lemma 1].
###### Corollary 1
The secrecy capacity region
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},P)$ of the MIMO Gaussian
broadcast channel (1) with confidential messages $W_{1}$ (intended for
receiver 1 but needing to be kept secret from receiver 2) and $W_{2}$
(intended for receiver 2 but needing to be kept secret from receiver 1) under
the average total power constraint:
$\frac{1}{n}\sum_{m=1}^{n}\|\mathbf{X}[m]\|^{2}\leq P$ (6)
is given by the set of nonnegative rate pairs $(R_{1},R_{2})$ such that
$\displaystyle R_{1}$
$\displaystyle\leq\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}_{1}\mathbf{H}_{1}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}_{1}\mathbf{H}_{2}^{\intercal}\right|$
$\displaystyle\mbox{and}\quad\quad R_{2}$
$\displaystyle\leq\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}(\mathbf{B}_{1}+\mathbf{B}_{2})\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}_{1}\mathbf{H}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}(\mathbf{B}_{1}+\mathbf{B}_{2})\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}_{1}\mathbf{H}_{1}^{\intercal}}\right|$
(7)
for some positive semidefinite matrices $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$
such that ${\sf Tr}(\mathbf{B}_{1}+\mathbf{B}_{2})\leq{P}$.
###### Remark 3
Unlike Theorem 1, under the average total power constraint, the secrecy
capacity region of the MIMO Gaussian broadcast channel is, in general, _not_
rectangular.
The rest of the paper is devoted to the proof of Theorem 1. As mentioned
previously, the rectangular nature of the rate region (3) suggests that the
result is intimately connected to the secrecy capacity of the MIMO Gaussian
wiretap channel. The secrecy capacity of the MIMO Gaussian wiretap channel
under the matrix power constraint was previously characterized in [9], where
it was shown that Gaussian random binning _without_ prefix coding is optimal.
In Section II, we revisit the MIMO Gaussian wiretap channel problem and show
that Gaussian random binning _with_ prefix coding can also achieve the secrecy
capacity, provided that the prefix channel is appropriately chosen. In Section
III, we prove Theorem 1 using two different characterizations of the secrecy
capacity of the MIMO Gaussian wiretap channel and S-DPC (double binning) [13].
Numerical examples are provided in Section IV to illustrate the theoretical
results. Finally, in Section V, we conclude the paper with some remarks.
## II MIMO Gaussian Wiretap Channel Revisited
In this section, we revisit the problem of the MIMO Gaussian wiretap channel
under a matrix power constraint. The problem was first considered in [9],
where a precise characterization of the secrecy capacity was provided. The
goal of this section is to provide an alternative characterization of the
secrecy capacity which will facilitate proving Theorem 1. More specifically,
we wish to provide a MIMO wiretap channel bound on the secrecy rate $R_{2}$
which will match the RHS of (3).
For that purpose, consider again the MIMO Gaussian broadcast channel (1) but
this time with only one confidential message $W$ at the transmitter. Message
$W$ is intended for receiver 2 (the legitimate receiver) but needs to be kept
secret from receiver 1 (the eavesdropper). The confidentiality of $W$ at
receiver 1 is measured using the normalized information-theoretic quantity [1,
2]:
$\frac{1}{n}I(W;\mathbf{Y}_{1}^{n})\rightarrow 0.$
The channel input $\\{\mathbf{X}[m]\\}_{m}$ is subject to the matrix power
constraint (2). The goal is to characterize the secrecy capacity
$C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})$111In our notation, the first
argument in $C_{s}(\cdot)$ represents the channel matrix for the legitimate
receiver, and the second argument represents the channel matrix for the
eavesdropper., which is the maximum achievable secrecy rate for message $W$.
This communication scenario, as illustrated in Fig. 2, is widely known as the
MIMO Gaussian wiretap channel [4, 6, 5, 7, 8, 9].
Figure 2: MIMO Gaussian wiretap channel.
In their seminal work [2], Csiszár and Körner provided a single-letter
characterization of the secrecy capacity:
$\displaystyle
C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})=\max_{(U,\mathbf{X})}\left[I(U;\mathbf{Y}_{2})-I(U;\mathbf{Y}_{1})\right]$
(8)
where $U$ is an auxiliary variable, and the maximization is over all jointly
distributed $(U,\mathbf{X})$ such that
$U\rightarrow\mathbf{X}\rightarrow(\mathbf{Y}_{1},\mathbf{Y}_{2})$ forms a
Markov chain and ${\sf E}[\mathbf{X}\mathbf{X}^{\intercal}]\preceq\mathbf{S}$.
Here, $I(U,\mathbf{Y}_{k})$ denotes the mutual information between $U$ and
$\mathbf{Y}_{k}$. As shown in [2], the secrecy rate on the RHS of (8) can be
achieved by a coding scheme that combines random binning and prefix coding
[2]. More specifically, the auxiliary variable $U$ represents a precoding
signal, and the conditional distribution of $\mathbf{X}$ given $U$ represents
the prefix channel. In [9], Liu and Shamai further studied the optimization
problem on the RHS of (8) and showed that a Gaussian $U=\mathbf{X}$ is an
optimal solution. Hence, a matrix characterization of the secrecy capacity is
given by [9]
$\displaystyle
C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|\right).$
(9)
We may conclude that Gaussian random binning _without_ prefix coding is an
optimal coding strategy for the MIMO Gaussian wiretap channel.
Next, we show that a different coding scheme that combines Gaussian random
binning _and_ prefix coding can also achieve the secrecy capacity of the MIMO
Gaussian wiretap channel. This leads to a new characterization of the secrecy
capacity, summarized in the following theorem.
###### Theorem 2
The secrecy capacity $C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})$ of the
MIMO Gaussian broadcast channel (1) with a confidential message $W$ (intended
for receiver 2 but needing to be kept secret from receiver 1) under the matrix
power constraint (2) is given by:
$\displaystyle
C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}}\right|\right).$
(10)
###### Remark 4
The achievability of the secrecy rate on the RHS of (10) can be obtained from
the Csiszár-Körner expression (8) by choosing $\mathbf{X}=U+V$, where $U$ and
$V$ are two independent Gaussian vectors with zero means and covariance
matrices $\mathbf{S}-\mathbf{B}$ and $\mathbf{B}$, respectively. This choice
of $(U,\mathbf{X})$ differs from that for (9) in two important ways:
1. 1.
In (10), the input vector $\mathbf{X}$ always has a full covariance matrix
$\mathbf{S}$. For (9), the covariance matrix of $\mathbf{X}$ needs to be
chosen to solve an optimization program; the full covariance matrix
$\mathbf{S}$ is _not_ always an optimal solution.
2. 2.
In (10), the conditional distribution of $\mathbf{X}$ given $U$ may form a
_nontrivial_ prefix channel. For (9), $U\equiv\mathbf{X}$ so prefix coding is
never applied.
###### Remark 5
Note that the prefix channel in (10) is an additive vector Gaussian noise
channel, so the auxiliary variable $V$ represents an _artificial_ noise [15]
sent (on purpose) by the transmitter to confuse the eavesdropper. Since the
artificial noise has no structure to it, it will add to the noise floor at
both legitimate receiver and the eavesdropper.
The converse part of the theorem can be proved using a _channel-enhancement_
argument, similar to that in [9]. The details of the proof are provided in
Appendix A.
## III MIMO Gaussian Broadcast Channel with Confidential Messages
In this section, we prove Theorem 1. To prove the converse part of the
theorem, we will consider a single-message, wiretap channel bound on the
secrecy rates $R_{1}$ and $R_{2}$. More specifically, note that both messages
$W_{1}$ and $W_{2}$ can be transmitted at the maximum secrecy rate when the
other message is absent from the transmission. Therefore, to bound from above
the secrecy rate $R_{1}$, we assume that only $W_{1}$ needs to be communicated
over the channel. This is precisely a MIMO Gaussian wiretap channel problem
with receiver 1 as legitimate receiver and receiver 2 as eavesdropper.
Reversing the roles of receiver 1 and 2, we have from (9) that
$\displaystyle R_{1}$ $\displaystyle\leq
C_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|\right).$
(11)
Similarly, to bound from above the secrecy rate $R_{2}$, let us assume that
only $W_{2}$ needs to be communicated over the channel. This is, again, a MIMO
Gaussian wiretap channel problem with receiver 2 playing the role of
legitimate receiver and receiver 1 playing the role of eavesdropper. By
Theorem 2,
$\displaystyle R_{2}$ $\displaystyle\leq
C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})$
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}}\right|\right).$
(12)
Putting together (11) and (12), we have proved the converse part of the
theorem.
Next, we show that every rate pair $(R_{1},R_{2})$ within the secrecy rate
region (3) is achievable. Note that (3) is rectangular, so we only need to
show that the corner point $(R_{1},R_{2})$ given by
$\displaystyle R_{1}$
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|\right)$
$\displaystyle\text{and}\qquad R_{2}$
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}}\right|\right)$
(13)
is achievable.
Recall from [13] that for any jointly distributed $(V_{1},V_{2},\mathbf{X})$
such that
$(V_{1},V_{2})\rightarrow\mathbf{X}\rightarrow(\mathbf{Y}_{1},\mathbf{Y}_{2})$
forms a Markov chain and ${\sf
E}[\mathbf{X}\mathbf{X}^{\intercal}]\preceq\mathbf{S}$, the secrecy rate pair
$(R_{1},R_{2})$ given by
$\displaystyle R_{1}$
$\displaystyle=I(V_{1};\mathbf{Y}_{1})-I(V_{1};V_{2},\mathbf{Y}_{2})$
$\displaystyle\text{and}\qquad R_{2}$
$\displaystyle=I(V_{2};\mathbf{Y}_{2})-I(V_{2};V_{1},\mathbf{Y}_{1})$ (14)
is achievable for the MIMO Gaussian broadcast channel (1) under the matrix
power constraint (2). In [13], the achievability of the rate pair (14) was
proved using a _double-binning_ scheme. Specifically, the auxiliary variables
$V_{1}$ and $V_{2}$ represent the precoding signals for the confidential
messages $W_{1}$ and $W_{2}$, respectively.
Now let $\mathbf{B}$ be a positive semidefinite matrix such that
$\mathbf{B}\preceq\mathbf{S}$, and let
$\displaystyle V_{1}$ $\displaystyle=\mathbf{U}_{1}+\mathbf{F}\mathbf{U}_{2}$
$\displaystyle V_{2}$ $\displaystyle=\mathbf{U}_{2}$
$\displaystyle\mbox{and}\quad\quad\mathbf{X}$
$\displaystyle=\mathbf{U}_{1}+\mathbf{U}_{2}$ (15)
where $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$ are two independent Gaussian
vectors with zero means and covariance matrices $\mathbf{B}$ and
$\mathbf{S}-\mathbf{B}$, respectively, and
$\displaystyle\mathbf{F}$
$\displaystyle:=\mathbf{B}\mathbf{H}_{1}^{\intercal}(\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal})^{-1}\mathbf{H}_{1}.$
(16)
By (15),
$\mathbf{Y}_{k}=\mathbf{H}_{k}(\mathbf{U}_{1}+\mathbf{U}_{2})+\mathbf{Z}_{k}$
for $k=1,2$. Note that the matrix $\mathbf{F}$ defined in (16) is precisely
the _precoding_ matrix for suppressing $\mathbf{U}_{2}$ from $\mathbf{Y}_{1}$
[16, Theorem 1]. Hence,
$\displaystyle I(V_{1};\mathbf{Y}_{1})-I(V_{1};V_{2})$
$\displaystyle=I(V_{1};\mathbf{Y}_{1})-I(V_{1};\mathbf{U}_{2})$
$\displaystyle=\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|.$
(17)
Moreover,
$\displaystyle I(V_{1};\mathbf{Y}_{2}|V_{2})$
$\displaystyle=I(\mathbf{U}_{1}+\mathbf{F}\mathbf{U}_{2};\mathbf{H}_{2}(\mathbf{U}_{1}+\mathbf{U}_{2})+\mathbf{Z}_{2}|\mathbf{U}_{2})$
$\displaystyle=I(\mathbf{U}_{1};\mathbf{H}_{2}\mathbf{U}_{1}+\mathbf{Z}_{2}|\mathbf{U}_{2})$
$\displaystyle=I(\mathbf{U}_{1};\mathbf{H}_{2}\mathbf{U}_{1}+\mathbf{Z}_{2})$
$\displaystyle=\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|$
(18)
where the third equality follows from the fact that $\mathbf{U}_{1}$ and
$\mathbf{U}_{2}$ are independent. Putting together (17) and (18), we have
$\displaystyle I(V_{1};\mathbf{Y}_{1})-I(V_{1};V_{2},\mathbf{Y}_{2})$
$\displaystyle=[I(V_{1};\mathbf{Y}_{1})-I(V_{1};V_{2})]-I(V_{1};\mathbf{Y}_{2}|V_{2})$
$\displaystyle=\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|.$
(19)
Similarly,
$\displaystyle I(V_{1},V_{2};\mathbf{Y}_{1})$
$\displaystyle=I(\mathbf{U}_{1}+\mathbf{F}\mathbf{U}_{2},\mathbf{U}_{2};\mathbf{H}_{1}(\mathbf{U}_{1}+\mathbf{U}_{2})+\mathbf{Z}_{2})$
$\displaystyle=I(\mathbf{U}_{1},\mathbf{U}_{2};\mathbf{H}_{1}(\mathbf{U}_{1}+\mathbf{U}_{2})+\mathbf{Z}_{2})$
$\displaystyle=\frac{1}{2}\log\left|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}\right|.$
(20)
Thus,
$\displaystyle I(V_{2};V_{1},\mathbf{Y}_{1})$
$\displaystyle=I(V_{2};\mathbf{Y}_{1}|V_{1})+I(V_{2};V_{1})$
$\displaystyle=I(V_{1},V_{2};\mathbf{Y}_{1})-[I(V_{1};\mathbf{Y}_{1})-I(V_{1};V_{2})]$
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}}\right|$
(21)
where the last equality follows from (17) and (20). Moreover,
$\displaystyle I(V_{2};\mathbf{Y}_{2})$
$\displaystyle=I(\mathbf{U}_{2};\mathbf{H}_{2}(\mathbf{U}_{1}+\mathbf{U}_{2})+\mathbf{Z}_{2})$
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}}\right|.$
(22)
Putting together (21) and (22), we have
$\displaystyle I(V_{2};\mathbf{Y}_{2})-I(V_{2};V_{1},\mathbf{Y}_{1})$
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}}\right|.$
(23)
Finally, let $\mathbf{B}$ be an optimal solution to the optimization program
(5). As mentioned previously in Remark 2, such a choice will _simultaneously_
maximize the RHS of (19) and (23). Thus, the corner point (13) is indeed
achievable. This completes the proof of the theorem.
###### Remark 6
Note that in standard dirty-paper coding (DPC), the precoding matrix
$\mathbf{F}$ is chosen to cancel the known interference. In our scheme, such a
choice plays two important roles. First, it helps to cancel the precoding
signal representing message $W_{2}$, so message $W_{1}$ sees an interference-
free legitimate receiver channel. Second, it helps to boost the security for
message $W_{2}$ by causing interference to its eavesdropper. For this reason,
we call our scheme S-DPC, to differentiate from the standard DPC.
###### Remark 7
In S-DPC, both the legitimate receiver and the eavesdropper for message
$W_{1}$ are interference free. On the other hand, for message $W_{2}$, both
the legitimate receiver and the eavesdropper are subject to interference from
the precoding signal representing message $W_{1}$. As we have seen in Section
II, the secrecy capacity of the MIMO Gaussian wiretap channel can be achieved
with or without interference in place. Therefore, both secrecy capacity
achieving schemes can be simultaneously implemented via S-DPC to
simultaneously communicate both confidential messages at their respective
maximal secrecy rates.
## IV Numerical Examples
In this section, we provide numerical examples to illustrate the secrecy
capacity region of the MIMO Gaussian wiretap channel with confidential
messages. As shown in (3) and (7), under both matrix and average total power
constraints, the secrecy capacity regions
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ and
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},P)$ are expressed in terms of
matrix optimization programs (though implicit in (7)). In general, these
optimization programs are not convex, and hence, finding the boundary of the
secrecy capacity regions is nontrivial.
In [12], a precise characterization of the secrecy capacity region
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},P)$ was obtained for the
_MISO_ Gaussian broadcast channel using the generalized eigenvalue
decomposition [17, Ch. 6.3]. For the _aligned_ MIMO Gaussian wiretap channel,
[10] provided an explicit, closed-form expression for the secrecy capacity. In
the following, we generalize the results of [10] and [12] to the general MIMO
Gaussian broadcast channel under the matrix power constraint.
Let $\phi_{j}$, $j=1,\ldots,t$, be the generalized eigenvalues of the pencil
$\displaystyle\left(\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{H}_{1}^{\intercal}\mathbf{H}_{1}\mathbf{S}^{\frac{1}{2}},\;\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{H}_{2}^{\intercal}\mathbf{H}_{2}\mathbf{S}^{\frac{1}{2}}\right).$
(24)
Since both
$\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{H}_{1}^{\intercal}\mathbf{H}_{1}\mathbf{S}^{\frac{1}{2}}$
and
$\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{H}_{2}^{\intercal}\mathbf{H}_{2}\mathbf{S}^{\frac{1}{2}}$
are strictly positive definite, we have $\phi_{j}>0$ for $j=1,\dots,t$.
Without loss of generality, we may assume that these generalized eigenvalues
are ordered as
$\phi_{1}\geq\dots\geq\phi_{\rho}>1\geq\phi_{\rho+1}\geq\dots\geq\phi_{t}>0,$
i.e., a total of $\rho$ of them are assumed to be greater than $1$. We have
the following characterization of the secrecy capacity of the MIMO Gaussian
wiretap channel under the matrix power constraint, which is a natural
extension of [10].
###### Theorem 3
The secrecy capacity $C_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ of the
MIMO Gaussian broadcast channel (1) with confidential message $W$ (intended
for receiver 1 but needing to be kept secret from receiver 2) under the matrix
power constraint (2) is given by
$\displaystyle
C_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})=\frac{1}{2}\sum_{j=1}^{\rho}\log\phi_{j}$
(25)
where $\phi_{j}$, $j=1,\ldots,\rho$, are the generalized eigenvalues of the
pencil (24) that are greater than 1.
###### Remark 8
Note that
$\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{H}_{2}^{\intercal}\mathbf{H}_{2}\mathbf{S}^{\frac{1}{2}}$
is invertible, so computing the generalized eigenvalues of the pencil (24) can
be reduced to the problem of finding standard eigenvalues of a related
semidefinite matrix [17, Ch. 6.3]. Hence, the secrecy capacity expression (25)
is computable.
A proof of the theorem following the approach of [10] is provided in Appendix
B. As a corollary, we have the following characterization of the secrecy
capacity region of the MIMO Gaussian broadcast channel with confidential
messages under the matrix power constraint.
###### Corollary 2
The secrecy capacity region
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ of the MIMO
Gaussian broadcast channel (1) with confidential messages $W_{1}$ (intended
for receiver 1 but needing to be kept secret from receiver 2) and $W_{2}$
(intended for receiver 2 but needing to be kept secret from receiver 1) under
the matrix constraint (2) is given by the set of nonnegative rate pairs
$(R_{1},R_{2})$ such that
$\displaystyle R_{1}$
$\displaystyle\leq\frac{1}{2}\sum_{j=1}^{\rho}\log\phi_{j}$
$\displaystyle\text{and}\qquad R_{2}$
$\displaystyle\leq\frac{1}{2}\sum_{j=\rho+1}^{t}\log\frac{1}{\phi_{j}}$ (26)
where $\phi_{j}$, $j=1,\ldots,\rho$, are the generalized eigenvalues of the
pencil (24) that are greater than 1, and $\phi_{j}$, $j=\rho+1,\ldots,t$, are
the generalized eigenvalues of the pencil (24) that are less than or equal to
1.
###### Proof:
By Theorem 1, we only need to show that the secrecy capacity
$\displaystyle
C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})=\frac{1}{2}\sum_{j=\rho+1}^{t}\log\frac{1}{\phi_{j}}.$
Consider the pencil
$\displaystyle\left(\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{H}_{2}^{\intercal}\mathbf{H}_{2}\mathbf{S}^{\frac{1}{2}},\;\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{H}_{1}^{\intercal}\mathbf{H}_{1}\mathbf{S}^{\frac{1}{2}}\right).$
(27)
Note that the pencils (24) and (27) are generated by the same pair of
semidefinite matrices but with different order. Therefore, the generalized
eigenvalues of the pencil (27) are given by
$0<\frac{1}{\phi_{1}}\leq\dots\leq\frac{1}{\phi_{\rho}}<1\leq\frac{1}{\phi_{\rho+1}}\leq\dots\leq\frac{1}{\phi_{t}}.$
Applying Theorem 3 for $C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})$
completes the proof of the corollary. ∎
Under the average total power constraint, we have not been able to find a
computable secrecy capacity expression for the general MIMO case. We can,
however, write [14, Lemma 1]
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},P)=\bigcup_{\mathbf{S}\succeq
0,\;{\sf Tr}(\mathbf{S})\leq
P}{\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S}).$
For any given semidefinite $\mathbf{S}$,
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ can be computed
as given by (26). Then, the secrecy capacity region
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},P)$ can be found through an
exhaustive search over the set $\\{\mathbf{S}:\;\mathbf{S}\succeq
0\;\mbox{and}\;{\sf Tr}(\mathbf{S})\leq P\\}$.
(a) $r_{1}=r_{2}=1$
(b) $r_{1}=2$, $r_{2}=1$
(c) $r_{1}=1$, $r_{2}=2$
(d) $r_{1}=r_{2}=2$
Figure 3: Secrecy rate regions of the MIMO Gaussian broadcast channel under
the average total power constraint.
Let $\mathbf{h}_{11}=(0.3\;2.5)$, $\mathbf{h}_{12}=(2.2\;1.8)$,
$\mathbf{h}_{21}=(1.3\;1.2)$, $\mathbf{h}_{22}=(1.5\;3.9)$ and $P=12$, and let
$\displaystyle\mathbf{H}_{k}$
$\displaystyle=\left(\begin{array}[]{c}\mathbf{h}_{k1}\\\ \mathbf{h}_{k2}\\\
\end{array}\right),\quad k=1,2.$
The secrecy capacity regions
${\mathcal{C}}_{s}(\mathbf{h}_{11},\mathbf{h}_{22},P)$,
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{h}_{22},P)$,
${\mathcal{C}}_{s}(\mathbf{h}_{11},\mathbf{H}_{2},P)$ and
${\mathcal{C}}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},P)$ are illustrated in Fig.
3. For comparison, we have also plotted the secrecy rate regions achieved by
the simple zero-forcing (ZF) strategy. In ZF, each of the confidential
messages is encoded using a vector Gaussian signal. To guarantee
confidentiality, the covariance matrices of the transmit signals are chosen in
the _null_ space of the channel matrix at the unintended receiver. Hence, the
achievable secrecy rate region is given by
$\displaystyle{\mathcal{R}}_{S}^{\rm
ZF}(\mathbf{H}_{1},\mathbf{H}_{2},P)=\bigcup_{\begin{subarray}{c}\mathbf{B}_{1}\succeq
0,\;\mathbf{B}_{2}\succeq 0,\;{\sf Tr}(\mathbf{B}_{1}+\mathbf{B}_{2})\leq P\\\
\mathbf{H}_{2}\mathbf{B}_{1}=0,\;\mathbf{H}_{1}\mathbf{B}_{2}=0\end{subarray}}\left\\{(R_{1},R_{2})\left|\;\begin{array}[]{l}R_{1}\leq\frac{1}{2}\log|\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}_{1}\mathbf{H}_{1}^{\intercal}|\\\
R_{2}\leq\frac{1}{2}\log|\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}_{2}\mathbf{H}_{2}^{\intercal}|\end{array}\right.\right\\}.$
(30)
Note that unlike the secrecy capacity region expression (7), computing the
rate region (30) only involves solving convex optimization programs. As shown
in Fig. 3, in all four scenarios, ZF is strictly suboptimal as compared with
S-DPC. In particular, if the channel matrix of the unintended receiver has
full row rank, ZF cannot achieve any positive secrecy rate for the
corresponding confidential message. On the other hand, S-DPC can always
achieve positive secrecy rates for both confidential messages unless the MIMO
Gaussian broadcast channel is degraded.
Figure 4: Rate regions of the MIMO Gaussian broadcast channel under the power
matrix constraint.
Finally, let
$\displaystyle\mathbf{H}_{1}$
$\displaystyle=\left(\begin{matrix}1.8&-2.0&2.0\\\
1.0&-6.0&3.0\end{matrix}\right)$ $\displaystyle\mathbf{H}_{2}$
$\displaystyle=\left(\begin{matrix}2.3&2.0&-3\\\
2.0&1.2&-1.5\end{matrix}\right)$
and
$\displaystyle\mathbf{S}=\left(\begin{matrix}5.0&-0.7&-2.0\\\ -0.7&3.8&-2.5\\\
-2.0&-2.5&5.0\end{matrix}\right).$
Fig. 4 illustrates the secrecy capacity region
$\mathcal{C}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ of the MIMO
Gaussian broadcast channel (1) under the matrix power constraint (2). Here,
the secrecy capacity region
$\mathcal{C}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ is plotted based
on the computable expression (26). Also in the figure are the secrecy rate
region $\mathcal{R}_{s}^{\rm ZF}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$
achieved by ZF strategy and the _nonsecrecy_ capacity region $\mathcal{R}^{\rm
DPC}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$ achieved by standard DPC [14].
As expected, we have $\mathcal{R}_{s}^{\rm
ZF}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})\subset\mathcal{C}_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})\subset\mathcal{R}^{\rm
DPC}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$.
## V Concluding Remarks
In this paper, we have considered the problem of communicating two
confidential messages over the two-receiver MIMO Gaussian broadcast channel.
Each of the confidential messages is intended for one of the receivers but
needs to be kept asymptotically perfectly secret from the other. Precise
characterizations of the secrecy capacity region have been provided under both
matrix and average total power constraints. Surprisingly, under the matrix
power constraint, both confidential messages can be transmitted simultaneously
at their respective maximal secrecy rates.
To prove this result, we have revisited the problem of the MIMO Gaussian
wiretap channel and proposed a new coding scheme that achieves the secrecy
capacity of the channel. Unlike the previous scheme considered in [4, 6, 5, 7,
8, 9] where prefix coding is not applied, the new coding scheme uses
artificial vector Gaussian noise as a way of prefix coding. Moreover, the
optimal covariance matrix of the artificial noise coincides with that of the
transmit signal in the previous scheme. This allows both schemes to be
overlayed via S-DPC without sacrificing the secrecy rate performance for
either of them. We believe that the new understanding of the MIMO Gaussian
wiretap channel problem gained in this work will help to solve some other
multiuser secret communication problems.
## Appendix A Proof of Theorem 2
In this appendix, we prove Theorem 2. As mentioned previously in Remark 4, the
secrecy rate on the RHS of (10) can be achieved by a coding scheme that
combines Gaussian random binning and prefix coding. We therefore concentrate
on the converse part of the theorem.
Following [9], we will first prove the converse result for the special case
where the channel matrices $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ are square
and invertible. Next, we will broaden the result to the general case by
approximating arbitrary channel matrices $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$
by square and invertible ones. For brevity, we will term the special case as
the aligned MIMO Gaussian wiretap channel and the general case as the general
MIMO Gaussian wiretap channel.
### A-A Aligned MIMO Gaussian Wiretap Channel
Consider the special case of the MIMO Gaussian broadcast channel (1) where the
channel matrices $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ are square and
invertible. Multiplying both sides of (1) by $\mathbf{H}_{k}^{-1}$, the
channel model can be equivalently written as
$\mathbf{Y}_{k}[m]=\mathbf{X}[m]+\mathbf{Z}_{k}[m],\quad k=1,2$ (31)
where $\\{\mathbf{Z}_{k}[m]\\}_{m}$ is an i.i.d. additive vector Gaussian
noise process with zero mean and covariance matrix
$\displaystyle\mathbf{N}_{k}$
$\displaystyle=\mathbf{H}_{k}^{-1}\mathbf{H}_{k}^{-\intercal}.$ (32)
Denote by $C_{s}(\mathbf{N}_{2},\mathbf{N}_{1},\mathbf{S})$ the secrecy
capacity of (31) (viewed as a MIMO Gaussian wiretap channel with receiver 2 as
legitimate receiver and receiver 1 as eavesdropper) under the matrix power
constraint (2). We have the following characterization of
$C_{s}(\mathbf{N}_{2},\mathbf{N}_{1},\mathbf{S})$.
###### Lemma 1
The secrecy capacity
$\displaystyle C_{s}(\mathbf{N}_{2},\mathbf{N}_{1},\mathbf{S})$
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{\mathbf{B}+\mathbf{N}_{1}}\right|\right).$
(33)
###### Proof:
The achievability of the secrecy rate on the RHS of (33) follows from the
achievability of the secrecy rate on the RHS of (10) for the general case and
the definition of $\mathbf{N}_{k}$ in (32). To prove the converse result, we
will follow [9] and consider a channel-enhancement argument as follows.
Let us first assume that $\mathbf{S}\succ 0$. In this case, let
$\mathbf{B}^{\star}$ be an optimal solution to the optimization program on the
RHS of (33). Then, $\mathbf{B}^{\star}$ must satisfy the following Karush-
Kuhn-Tucker conditions [9]:
$\displaystyle(\mathbf{B}^{\star}+\mathbf{N}_{1})^{-1}+\mathbf{M}_{1}$
$\displaystyle=(\mathbf{B}^{\star}+\mathbf{N}_{2})^{-1}+\mathbf{M}_{2}$ (34a)
$\displaystyle\mathbf{B}^{\star}\mathbf{M}_{1}$ $\displaystyle=0$ (34b)
$\displaystyle\text{and}\quad\quad(\mathbf{S}-\mathbf{B}^{\star})\mathbf{M}_{2}$
$\displaystyle=0$ (34c)
where $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ are positive semidefinite
matrices. Let $\widetilde{\mathbf{N}}_{1}$ be a real symmetric matrix such
that
$\displaystyle(\mathbf{B}^{\star}+\widetilde{\mathbf{N}}_{1})^{-1}$
$\displaystyle=(\mathbf{B}^{\star}+\mathbf{N}_{1})^{-1}+\mathbf{M}_{1}.$ (35)
From Eqns. (23), (25), (31) and (34) of [9], we have
$\displaystyle
0\prec\widetilde{\mathbf{N}}_{1}\preceq\\{\mathbf{N}_{1},\mathbf{N}_{2}\\},$
(36)
$\displaystyle\left|\frac{\mathbf{B}^{\star}+\widetilde{\mathbf{N}}_{1}}{\widetilde{\mathbf{N}}_{1}}\right|=\left|\frac{\mathbf{B}^{\star}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|$
(37)
and
$\displaystyle\left|\frac{\mathbf{S}+\widetilde{\mathbf{N}}_{1}}{\mathbf{B}^{\star}+\widetilde{\mathbf{N}}_{1}}\right|$
$\displaystyle=\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{\star}+\mathbf{N}_{2}}\right|.$
(38)
Now consider an enhanced MIMO Gaussian broadcast channel:
$\displaystyle\mathbf{Y}_{1}[m]$
$\displaystyle=\mathbf{X}[m]+\mathbf{Z}_{1}[m]$
$\displaystyle\text{and}\qquad\mathbf{Y}_{2}[m]$
$\displaystyle=\mathbf{X}[m]+\tilde{\mathbf{Z}}_{1}[m]$ (39)
where $\\{\mathbf{Z}_{1}[m]\\}_{m}$ and $\\{\tilde{\mathbf{Z}}_{1}[m]\\}_{m}$
are i.i.d. additive vector Gaussian noise processes with zero means and
covariance matrices $\mathbf{N}_{1}$ and $\widetilde{\mathbf{N}}_{1}$,
respectively. Denote by
$C_{s}(\widetilde{\mathbf{N}}_{1},\mathbf{N}_{1},\mathbf{S})$ the secrecy
capacity of (39) (viewed as a MIMO Gaussian wiretap channel with receiver 2 as
legitimate receiver and receiver 1 as eavesdropper) under the matrix
constraint (2). Note from (36) that
$\widetilde{\mathbf{N}}_{1}\preceq\mathbf{N}_{1}$, so the enhanced MIMO
Gaussian wiretap channel (39) is _degraded_. Hence,
$\displaystyle C_{s}(\widetilde{\mathbf{N}}_{1},\mathbf{N}_{1},\mathbf{S})$
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{S}+\widetilde{\mathbf{N}}_{1}}{\widetilde{\mathbf{N}}_{1}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{\mathbf{N}_{1}}\right|$
$\displaystyle=\frac{1}{2}\log\left(\left|\frac{\mathbf{S}+\widetilde{\mathbf{N}}_{1}}{\mathbf{S}+\mathbf{N}_{1}}\right|\left|\frac{\mathbf{N}_{1}}{\widetilde{\mathbf{N}}_{1}}\right|\right)$
$\displaystyle=\frac{1}{2}\log\left(\left|\frac{\mathbf{S}+\widetilde{\mathbf{N}}_{1}}{\mathbf{S}+\mathbf{N}_{1}}\right|\left|\frac{\mathbf{B}^{\star}+\mathbf{N}_{1}}{\mathbf{B}^{\star}+\widetilde{\mathbf{N}}_{1}}\right|\right)$
$\displaystyle=\frac{1}{2}\log\left(\left|\frac{\mathbf{S}+\widetilde{\mathbf{N}}_{1}}{\mathbf{B}^{\star}+\widetilde{\mathbf{N}}_{1}}\right|\left|\frac{\mathbf{B}^{\star}+\mathbf{N}_{1}}{\mathbf{S}+\mathbf{N}_{1}}\right|\right)$
$\displaystyle=\frac{1}{2}\log\left(\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{\star}+\mathbf{N}_{2}}\right|\left|\frac{\mathbf{B}^{\star}+\mathbf{N}_{1}}{\mathbf{S}+\mathbf{N}_{1}}\right|\right)$
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{\star}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{\mathbf{B}^{\star}+\mathbf{N}_{1}}\right|$
(40)
where the first equality follows from [9, Theorem 1]; the third equality
follows from (37); and the fifth equality follows from (38).
Finally, note from (36) that
$\widetilde{\mathbf{N}}_{1}\preceq\mathbf{N}_{2}$, i.e., the legitimate
receiver in the enhanced wiretap channel (39) receives a better signal that
the legitimate receiver in the original wiretap channel (31). Therefore,
$\displaystyle C_{s}(\mathbf{N}_{2},\mathbf{N}_{1},\mathbf{S})$
$\displaystyle\leq
C_{s}(\widetilde{\mathbf{N}}_{1},\mathbf{N}_{1},\mathbf{S})$
$\displaystyle=\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{2}}{\mathbf{B}^{\star}+\mathbf{N}_{2}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{S}+\mathbf{N}_{1}}{\mathbf{B}^{\star}+\mathbf{N}_{1}}\right|$
where the last equality follows from (40). This proved the desired converse
result for $\mathbf{S}\succ 0$.
For the case when $\mathbf{S}\succeq 0$, $|\mathbf{S}|=0$, let
$\theta={\sf Rank}(\mathbf{S})<t.$
Following the same footsteps as in the proof of [14, Lemma 2], we can define
an _equivalent_ aligned MIMO Gaussian wiretap channel with $\theta$ transmit
and receive antennas and a new covariance matrix power constraint that is
strictly positive definite. Hence, we can convert the case when
$\mathbf{S}\succeq 0$, $|\mathbf{S}|=0$ to the case when $\mathbf{S}\succ 0$
with the same secrecy capacity. This argument can be formally described as
follows.
Since $\mathbf{S}$ is positive semidefinite, we can write
$\displaystyle\mathbf{S}=\mathbf{Q}_{\mathbf{S}}{\bf\Lambda_{\mathbf{S}}}\mathbf{Q}_{\mathbf{S}}^{\intercal}$
where $\mathbf{Q}_{\mathbf{S}}$ is an orthogonal matrix and
$\displaystyle{\bf\Lambda_{\mathbf{S}}}={\sf
Diag}(\underbrace{0,\dots,0}_{t-\theta},s_{1},\dots,s_{\theta})$
is diagonal with $s_{j}>0$, $j=1,\dots,\theta$. For $k=1,2$, write
$\displaystyle\mathbf{Q}_{\mathbf{S}}^{\intercal}\mathbf{N}_{k}\mathbf{Q}_{\mathbf{S}}=\left(\begin{matrix}\mathbf{C}_{k}&\mathbf{D}_{k}\\\
\mathbf{D}_{k}^{\intercal}&\mathbf{E}_{k}\end{matrix}\right)$
where $\mathbf{C}_{k}$, $\mathbf{D}_{k}$ and $\mathbf{E}_{k}$ are
(sub)matrices of size $(t-\theta)\times(t-\theta)$, $(t-\theta)\times\theta$
and $\theta\times\theta$, respectively. Let
$\displaystyle\mathbf{A}_{k}:=\left(\begin{matrix}\mathbf{I}_{t-\theta}&0_{(t-\theta)\times\theta}\\\
-\mathbf{D}_{k}^{\intercal}\mathbf{C}_{k}^{-1}&\mathbf{I}_{\theta}\end{matrix}\right),\quad
k=1,2.$
We now define an intermediate and equivalent channel by multiplying both sides
of (31) by an _invertible_ matrix
$\mathbf{A}_{k}\mathbf{Q}_{\mathbf{S}}^{\intercal}$:
$\displaystyle\mathbf{Y}_{k}^{\prime}[m]$
$\displaystyle=\mathbf{X}^{\prime}[m]+\mathbf{Z}_{k}^{\prime}[m],\quad k=1,2$
(41)
where
$\displaystyle\mathbf{Y}_{k}^{\prime}[m]$
$\displaystyle=\mathbf{A}_{k}\mathbf{Q}_{\mathbf{S}}^{\intercal}\mathbf{Y}_{k}[m]$
$\displaystyle\mathbf{X}^{\prime}[m]$
$\displaystyle=\mathbf{A}_{k}\mathbf{Q}_{\mathbf{S}}^{\intercal}\mathbf{X}[m]$
$\displaystyle\mbox{and}\quad\quad\mathbf{Z}_{k}^{\prime}[m]$
$\displaystyle=\mathbf{A}_{k}\mathbf{Q}_{\mathbf{S}}^{\intercal}\mathbf{Z}_{k}[m].$
Then, the covariance matrix $\mathbf{N}_{k}^{\prime}$ of the additive Gaussian
noise vector $\mathbf{Z}_{k}^{\prime}[m]$ is given by
$\displaystyle\mathbf{N}_{k}^{\prime}$
$\displaystyle=\left(\begin{matrix}\mathbf{C}_{k}&0\\\
0&\mathbf{E}_{k}-\mathbf{D}_{k}^{\intercal}\mathbf{C}_{k}^{-1}\mathbf{D}_{k}\end{matrix}\right).$
(42)
and the matrix power constraint (2) becomes
$\displaystyle\frac{1}{n}\sum_{m=1}^{n}\mathbf{X}^{\prime}[m]{\mathbf{X}^{\prime}}^{\intercal}[m]\preceq\mathbf{S}^{\prime}$
(43)
where
$\displaystyle\mathbf{S}^{\prime}$
$\displaystyle=\mathbf{A}_{k}\mathbf{Q}_{\mathbf{S}}^{\intercal}\mathbf{S}\mathbf{Q}_{\mathbf{S}}\mathbf{A}_{k}^{\intercal}$
$\displaystyle=\mathbf{A}_{k}{\bf\Lambda_{\mathbf{S}}}\mathbf{A}_{k}^{\intercal}$
$\displaystyle={\bf\Lambda_{\mathbf{S}}}.$ (44)
Note from (44) that $\mathbf{S}^{\prime}$ is diagonal with first $t-\theta$
diagonal elements equal to zero. Thus, the matrix constraint (43) requires
that the first $t-\theta$ elements of $\mathbf{X}^{\prime}[m]$ be zero.
Moreover, from (42), the first $t-\theta$ and the rest of $\theta$ elements of
$\mathbf{Z}^{\prime}_{k}[m]$ are uncorrelated and hence must be independent as
$\mathbf{Z}^{\prime}_{k}[m]$ is Gaussian. Therefore, only the latter $\theta$
antennas transmit/receive information regarding message $W$. This allows us to
define another _equivalent_ aligned MIMO Gaussian broadcast channel with
$\theta$ antennas at the transmitter and each of the receivers:
$\displaystyle\overline{\mathbf{Y}}_{k}[m]$
$\displaystyle=\overline{\mathbf{X}}[m]+\overline{\mathbf{Z}}_{k}[m],\quad
k=1,2$ (45)
where
$\displaystyle\overline{\mathbf{Y}}_{k}[m]$
$\displaystyle=\overline{\mathbf{A}}\mathbf{Y}^{\prime}_{k}[m]$
$\displaystyle\overline{\mathbf{X}}[m]$
$\displaystyle=\overline{\mathbf{A}}\mathbf{X}^{\prime}[m]$
$\displaystyle\overline{\mathbf{Z}}_{k}[m]$
$\displaystyle=\overline{\mathbf{A}}\mathbf{Z}^{\prime}_{k}[m]$
and
$\overline{\mathbf{A}}=\left[0_{\theta\times(t-\theta)}\;\mathbf{I}_{\theta}\right]$.
Now, the matrix power constraint (43) becomes
$\displaystyle\frac{1}{n}\sum_{m=1}^{n}\overline{\mathbf{X}}[m]\overline{\mathbf{X}}^{\intercal}[m]\preceq\overline{\mathbf{S}}$
(46)
where
$\displaystyle\overline{\mathbf{S}}$
$\displaystyle=\overline{\mathbf{A}}\mathbf{S}^{\prime}\overline{\mathbf{A}}^{\intercal}$
$\displaystyle={\sf Diag}(s_{1},\dots,s_{\theta}).$ (47)
Note that the matrix power constraint $\overline{\mathbf{S}}$ is _strictly_
positive definite, so we can apply the previous result to the new wiretap
channel (45). This completes the proof of the lemma. ∎
### A-B General MIMO Gaussian Wiretap Channel
For the general case, we may assume that the channel matrices $\mathbf{H}_{1}$
and $\mathbf{H}_{2}$ are square but not necessarily invertible. If that is not
the case, we can use singular value decomposition (SVD) to show that there is
an equivalent channel which does have $t\times t$ square channel matrices.
That is, we can find a new channel with square channel matrices which are
derived from the original ones via matrix multiplications. The new channel is
equivalent to the original one in preserving the secrecy capacity under the
same power constraint.
Consider using SVD to write the channel matrices as follows:
$\mathbf{H}_{k}=\mathbf{U}_{k}\boldsymbol{\Lambda}_{k}\mathbf{V}_{k}^{\intercal},\quad
k=1,2$
where $\mathbf{U}_{k}$ and $\mathbf{V}_{k}$ are $t\times t$ orthogonal
matrices, and $\boldsymbol{\Lambda}_{k}$ is diagonal. We now define a new MIMO
Gaussian broadcast channel which has invertible channel matrices:
$\displaystyle\mathbf{Y}_{k}[m]$
$\displaystyle=\overline{\mathbf{H}}_{k}\mathbf{X}[m]+\mathbf{Z}_{k}[m],\quad
k=1,2$ (48)
where
$\overline{\mathbf{H}}_{k}=\mathbf{U}_{k}(\boldsymbol{\Lambda}_{k}+\alpha\mathbf{I}_{t})\mathbf{V}_{k}^{t}$
for some $\alpha>0$, and $\\{\mathbf{Z}_{k}[m]\\}_{m}$ is an i.i.d. additive
vector Gaussian noise process with zero mean and identity covariance matrix.
Note that the channel matrices $\overline{\mathbf{H}}_{k}$, $k=1,2$, are
invertible. By Lemma 1, the secrecy capacity
$C_{s}(\overline{\mathbf{H}}_{2},\overline{\mathbf{H}}_{1},\mathbf{S})$ of
(31) (viewed as a MIMO Gaussian wiretap channel with receiver 2 as legitimate
receiver and receiver 1 as eavesdropper) under the matrix power constraint (2)
is given by
$C_{s}(\overline{\mathbf{H}}_{2},\overline{\mathbf{H}}_{1},\mathbf{S})=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\frac{\mathbf{I}_{t}+\overline{\mathbf{H}}_{2}\mathbf{S}\overline{\mathbf{H}}_{2}^{\intercal}}{\mathbf{I}_{t}+\overline{\mathbf{H}}_{2}\mathbf{B}\overline{\mathbf{H}}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{t}+\overline{\mathbf{H}}_{1}\mathbf{S}\overline{\mathbf{H}}_{1}^{\intercal}}{\mathbf{I}_{t}+\overline{\mathbf{H}}_{1}\mathbf{B}\overline{\mathbf{H}}_{1}^{\intercal}}\right|\right).$
Finally, let $\alpha\downarrow 0$. We have
$\overline{\mathbf{H}}_{k}\rightarrow\mathbf{H}_{k}$, $k=1,2$ and hence
$\displaystyle
C_{s}(\overline{\mathbf{H}}_{2},\overline{\mathbf{H}}_{1},\mathbf{S})\rightarrow\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}}\right|\right).$
Moreover, by Eqns. (45) and (46) of [9],
$\displaystyle C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})$
$\displaystyle\leq
C_{s}(\overline{\mathbf{H}}_{2},\overline{\mathbf{H}}_{1},\mathbf{S})+\mathcal{O}(\alpha)$
(49)
where $\mathcal{O}(\alpha)\rightarrow 0$ in the limit as $\alpha\downarrow 0$.
Thus, we have the desired converse result
$\displaystyle C_{s}(\mathbf{H}_{2},\mathbf{H}_{1},\mathbf{S})$
$\displaystyle\leq\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{S}\mathbf{H}_{2}^{\intercal}}{\mathbf{I}_{r_{2}}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}}\right|-\frac{1}{2}\log\left|\frac{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{S}\mathbf{H}_{1}^{\intercal}}{\mathbf{I}_{r_{1}}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}}\right|\right)$
by letting $\alpha\downarrow 0$ on the RHS of (49). This completes the proof
of the theorem.
## Appendix B Proof of Theorem 3
In this appendix, we prove Theorem 3. Without loss of generality, we may
assume that the matrix power constraint $\mathbf{S}$ is strictly positive
definite and the channel matrices $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ are
square but not necessarily invertible. We start with the following simple
lemma.
###### Lemma 2
For any $t\times t$ matrices $\mathbf{B}$ and $\mathbf{H}$ such that
$\mathbf{B}\succeq 0$, we have
$\displaystyle\left|\mathbf{I}_{t}+\mathbf{H}\mathbf{B}\mathbf{H}^{\intercal}\right|$
$\displaystyle=\left|\mathbf{I}_{t}+\mathbf{H}^{\intercal}\mathbf{H}\mathbf{B}\right|.$
(50)
In particular, if $\mathbf{B}=\mathbf{I}_{t}$, we have
$\displaystyle\left|\mathbf{I}_{t}+\mathbf{H}\mathbf{H}^{\intercal}\right|$
$\displaystyle=\left|\mathbf{I}_{t}+\mathbf{H}^{\intercal}\mathbf{H}\right|.$
(51)
###### Proof:
Note that if $\mathbf{H}$ is invertible, the equalities in (50) and (51) are
trivial. Otherwise, consider using SVD to rewrite $\mathbf{H}$ as
$\displaystyle\mathbf{H}=\mathbf{U}\boldsymbol{\Lambda}\mathbf{V}^{\intercal}$
where $\mathbf{U}$ and $\mathbf{V}$ are $t\times t$ orthogonal matrices, and
$\displaystyle{\bf\Lambda}={\sf
Diag}(\underbrace{0,\dots,0}_{t-b},\lambda_{1},\dots,\lambda_{b})$
is diagonal with $\lambda_{j}>0$, $j=1,\dots,b$. Write
$\displaystyle\mathbf{V}^{\intercal}\mathbf{B}\mathbf{V}=\left(\begin{matrix}\mathbf{C}_{\mathbf{B}}&\mathbf{D}_{\mathbf{B}}\\\
\mathbf{D}_{\mathbf{B}}^{\intercal}&\mathbf{E}_{\mathbf{B}}\end{matrix}\right)$
where $\mathbf{C}_{\mathbf{B}}$, $\mathbf{D}_{\mathbf{B}}$ and
$\mathbf{E}_{\mathbf{B}}$ are (sub)matrices of size $(t-b)\times(t-b)$,
$(t-b)\times b$ and $b\times b$, respectively. Then,
$\displaystyle\left|\mathbf{I}_{t}+\mathbf{H}\mathbf{B}\mathbf{H}^{\intercal}\right|$
$\displaystyle=\left|\mathbf{I}_{t}+\mathbf{U}\boldsymbol{\Lambda}\mathbf{V}^{\intercal}\mathbf{B}\mathbf{V}\boldsymbol{\Lambda}\mathbf{U}^{\intercal}\right|$
$\displaystyle=\left|\mathbf{I}_{t}+\boldsymbol{\Lambda}\mathbf{V}^{\intercal}\mathbf{B}\mathbf{V}\boldsymbol{\Lambda}\right|$
$\displaystyle=\left|\mathbf{I}_{b}+\overline{\boldsymbol{\Lambda}}\mathbf{E}_{\mathbf{B}}\overline{\boldsymbol{\Lambda}}\right|$
(52)
where $\overline{\boldsymbol{\Lambda}}={\sf
Diag}(\lambda_{1},\dots,\lambda_{b})$. On the other hand,
$\displaystyle\left|\mathbf{I}_{t}+\mathbf{H}^{\intercal}\mathbf{H}\mathbf{B}\right|$
$\displaystyle=\left|\mathbf{I}_{t}+\mathbf{V}\boldsymbol{\Lambda}^{2}\mathbf{V}^{\intercal}\mathbf{B}\right|$
$\displaystyle=\left|\mathbf{I}_{t}+\boldsymbol{\Lambda}^{2}\mathbf{V}^{\intercal}\mathbf{B}\mathbf{V}\right|$
$\displaystyle=\left|\mathbf{I}_{b}+\overline{\boldsymbol{\Lambda}}^{2}\mathbf{E}_{\mathbf{B}}\right|$
$\displaystyle=\left|\mathbf{I}_{b}+\overline{\boldsymbol{\Lambda}}\mathbf{E}_{\mathbf{B}}\overline{\boldsymbol{\Lambda}}\right|$
(53)
where the last equality follows from the fact that
$\overline{\boldsymbol{\Lambda}}$ is invertible. Putting together (52) and
(53) proves the equality in (50). This completes the proof of the lemma. ∎
We are now ready to prove Theorem 3, following the approach of [10]. Let
$\displaystyle\mathbf{O}_{k}:=\mathbf{H}_{k}^{\intercal}\mathbf{H}_{k}\quad
k=1,2,$ (54)
and let $\boldsymbol{\Phi}$ denote the generalized eigenvalue matrix of the
pencil
$\left(\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{O}_{1}\mathbf{S}^{\frac{1}{2}},\;\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{O}_{2}\mathbf{S}^{\frac{1}{2}}\right)$
such that
$\displaystyle\boldsymbol{\Phi}=\left(\begin{matrix}\overline{\boldsymbol{\Phi}}_{1}&0\\\
0&\overline{\boldsymbol{\Phi}}_{2}\end{matrix}\right)$
where $\overline{\boldsymbol{\Phi}}_{1}={\rm
Diag}\\{\phi_{1},\dots,\phi_{\rho}\\}$ and
$\overline{\boldsymbol{\Phi}}_{2}={\rm
Diag}\\{\phi_{\rho+1},\dots,\phi_{t}\\}$. Let $\mathbf{G}$ be the
corresponding generalized eigenvector matrix such that
$\displaystyle\mathbf{G}^{\intercal}\left(\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{O}_{1}\mathbf{S}^{\frac{1}{2}}\right)\mathbf{G}$
$\displaystyle=\boldsymbol{\Phi}$
$\displaystyle\text{and}\qquad\mathbf{G}^{\intercal}\left(\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{O}_{2}\mathbf{S}^{\frac{1}{2}}\right)\mathbf{G}$
$\displaystyle=\mathbf{I}_{t}.$ (55)
Now define
$\displaystyle\widetilde{\mathbf{O}}:=\mathbf{S}^{-\frac{1}{2}}\left[\mathbf{G}^{-\intercal}\left(\begin{matrix}\overline{{\boldsymbol{\Phi}}}_{1}&0\\\
0&\mathbf{I}_{t-\rho}\end{matrix}\right)\mathbf{G}^{-1}-\mathbf{I}_{t}\right]\mathbf{S}^{-\frac{1}{2}}.$
(56)
Since the generalized eigenvalues are ordered as
$\phi_{1}\geq\dots\geq\phi_{\rho}>1\geq\phi_{\rho+1}\geq\dots\geq\phi_{t}>0,$
we have
$\displaystyle\left(\begin{matrix}\overline{{\boldsymbol{\Phi}}}_{1}&0\\\
0&\mathbf{I}_{t-\rho}\end{matrix}\right)\succeq\boldsymbol{\Phi}$
$\displaystyle\mbox{and}\quad\quad\left(\begin{matrix}\overline{{\boldsymbol{\Phi}}}_{1}&0\\\
0&\mathbf{I}_{t-\rho}\end{matrix}\right)\succeq\mathbf{I}_{t}.$
Hence by (55) and (56),
$\displaystyle\widetilde{\mathbf{O}}\succeq\\{\mathbf{O}_{1},\mathbf{O}_{2}\\}.$
(57)
It follows that
$\displaystyle C_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{H}_{1}\mathbf{B}\mathbf{H}_{1}^{\intercal}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{H}_{2}\mathbf{B}\mathbf{H}_{2}^{\intercal}\right|\right)$
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{B}^{\frac{1}{2}}\mathbf{H}_{1}^{\intercal}\mathbf{H}_{1}\mathbf{B}^{\frac{1}{2}}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{B}^{\frac{1}{2}}\mathbf{H}_{2}^{\intercal}\mathbf{H}_{2}\mathbf{B}^{\frac{1}{2}}\right|\right)$
(58)
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{B}^{\frac{1}{2}}\mathbf{O}_{1}\mathbf{B}^{\frac{1}{2}}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{B}^{\frac{1}{2}}\mathbf{O}_{2}\mathbf{B}^{\frac{1}{2}}\right|\right)$
(59)
$\displaystyle\leq\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{B}^{\frac{1}{2}}\widetilde{\mathbf{O}}\mathbf{B}^{\frac{1}{2}}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{B}^{\frac{1}{2}}\mathbf{O}_{2}\mathbf{B}^{\frac{1}{2}}\right|\right)$
(60)
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{t}+\widetilde{\mathbf{O}}^{\frac{1}{2}}\mathbf{B}\widetilde{\mathbf{O}}^{\frac{1}{2}}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{O}_{2}^{\frac{1}{2}}\mathbf{B}\mathbf{O}_{2}^{\frac{1}{2}}\right|\right)$
(61)
$\displaystyle=\frac{1}{2}\log\left|\mathbf{I}_{t}+\widetilde{\mathbf{O}}^{\frac{1}{2}}\mathbf{S}\widetilde{\mathbf{O}}^{\frac{1}{2}}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{O}_{2}^{\frac{1}{2}}\mathbf{S}\mathbf{O}_{2}^{\frac{1}{2}}\right|$
(62)
$\displaystyle=\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\widetilde{\mathbf{O}}\mathbf{S}^{\frac{1}{2}}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{S}^{\frac{1}{2}}\mathbf{O}_{2}\mathbf{S}^{\frac{1}{2}}\right|$
(63)
$\displaystyle=\frac{1}{2}\log\left|\overline{{\boldsymbol{\Phi}}}_{1}\right|$
(64) $\displaystyle=\frac{1}{2}\sum_{j=1}^{\rho}\log\phi_{j}$ (65)
where (58), (61) and (63) follow from (51); (59) follows from the definition
of $\mathbf{O}_{1}$ in (54); (60) follows from the fact that
$\mathbf{O}_{1}\preceq\widetilde{\mathbf{O}}$ (see (57)); (62) follows from
the fact that $\mathbf{O}_{2}\preceq\widetilde{\mathbf{O}}$ (see (57)); and
(64) follows (55) and the definition of $\widetilde{\mathbf{O}}$ in (56).
To prove the reverse inequality, let
$\mathbf{G}=[\mathbf{G}_{1}\,\mathbf{G}_{2}]$ where $\mathbf{G}_{1}$ and
$\mathbf{G}_{2}$ are (sub)matrices of size $t\times\rho$ and $t\times\rho$,
respectively, and let
$\displaystyle\mathbf{B}^{\star}:=\mathbf{S}^{\frac{1}{2}}\mathbf{G}\left(\begin{matrix}\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}&0\\\
0&0\end{matrix}\right)\mathbf{G}^{\intercal}\mathbf{S}^{\frac{1}{2}}.$ (66)
Then, $\mathbf{B}^{\star}$ is positive semidefinite. Moreover, we may verify
that $\mathbf{B}^{\star}\preceq\mathbf{S}$ as follows. Note that $\mathbf{G}$
is invertible, so it is enough to show that
$\displaystyle\left(\begin{matrix}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}&0\\\
0&0\end{matrix}\right)\preceq\left(\mathbf{G}^{\intercal}\mathbf{G}\right)^{-1}.$
Note that
$\displaystyle\mathbf{G}^{\intercal}\mathbf{G}$
$\displaystyle=\left(\begin{matrix}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}&\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\\\
\mathbf{G}_{2}^{\intercal}\mathbf{G}_{1}&\mathbf{G}_{2}^{\intercal}\mathbf{G}_{2}\end{matrix}\right).$
Using block inversion, we may obtain
$\displaystyle\left(\mathbf{G}^{\intercal}\mathbf{G}\right)^{-1}$
$\displaystyle=\left(\begin{matrix}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}+(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\mathbf{E}_{\mathbf{G}}^{-1}\mathbf{G}_{2}^{\intercal}\mathbf{G}_{1}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}&(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\mathbf{E}_{\mathbf{G}}^{-1}\\\
\mathbf{E}_{\mathbf{G}}^{-1}\mathbf{G}_{2}^{\intercal}\mathbf{G}_{1}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}&\mathbf{E}_{\mathbf{G}}^{-1}\end{matrix}\right)$
where
$\displaystyle\mathbf{E}_{\mathbf{G}}=\mathbf{G}_{2}^{\intercal}\mathbf{G}_{2}-\mathbf{G}_{2}^{\intercal}\mathbf{G}_{1}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}.$
Since $\mathbf{G}^{\intercal}\mathbf{G}$ is positive definite, we have
$\mathbf{E}_{\mathbf{G}}\succ 0$
and hence
$\displaystyle\left(\mathbf{G}^{\intercal}\mathbf{G}\right)^{-1}-\left(\begin{matrix}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}&0\\\
0&0\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\mathbf{E}_{\mathbf{G}}^{-1}\mathbf{G}_{2}^{\intercal}\mathbf{G}_{1}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}&(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\mathbf{E}_{\mathbf{G}}^{-1}\\\
\mathbf{E}_{\mathbf{G}}^{-1}\mathbf{G}_{2}^{\intercal}\mathbf{G}_{1}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}&\mathbf{E}_{\mathbf{G}}^{-1}\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\mathbf{I}_{\rho}&(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\\\
0&\mathbf{I}_{t-\rho}\end{matrix}\right)\left(\begin{matrix}0&0\\\
0&\mathbf{E}_{\mathbf{G}}^{-1}\end{matrix}\right)\left(\begin{matrix}\mathbf{I}_{\rho}&0\\\
\mathbf{G}_{2}^{\intercal}\mathbf{G}_{1}(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1})^{-1}&\mathbf{I}_{t-\rho}\end{matrix}\right)$
$\displaystyle\succeq 0.$
By (59),
$\displaystyle C_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$
$\displaystyle=\max_{0\preceq\mathbf{B}\preceq\mathbf{S}}\left(\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{B}^{\frac{1}{2}}\mathbf{O}_{1}\mathbf{B}^{\frac{1}{2}}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+\mathbf{B}^{\frac{1}{2}}\mathbf{O}_{2}\mathbf{B}^{\frac{1}{2}}\right|\right)$
$\displaystyle\geq\frac{1}{2}\log\left|\mathbf{I}_{t}+{\mathbf{B}^{\star}}^{\frac{1}{2}}\mathbf{O}_{1}{\mathbf{B}^{\star}}^{\frac{1}{2}}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+{\mathbf{B}^{\star}}^{\frac{1}{2}}\mathbf{O}_{2}{\mathbf{B}^{\star}}^{\frac{1}{2}}\right|$
$\displaystyle=\frac{1}{2}\log\left|\mathbf{I}_{t}+{\mathbf{B}^{\star}}\mathbf{O}_{1}\right|-\frac{1}{2}\log\left|\mathbf{I}_{t}+{\mathbf{B}^{\star}}\mathbf{O}_{2}\right|$
(67)
where the last equality follows from (50). From (55), we have
$\displaystyle\mathbf{O}_{1}$
$\displaystyle=\mathbf{S}^{-\frac{1}{2}}\left(\mathbf{G}^{-\intercal}\boldsymbol{\Phi}\mathbf{G}^{-1}-\mathbf{I}_{t}\right)\mathbf{S}^{-\frac{1}{2}}$
$\displaystyle\text{and}\qquad\mathbf{O}_{2}$
$\displaystyle=\mathbf{S}^{-\frac{1}{2}}\left(\mathbf{G}^{-\intercal}\mathbf{G}^{-1}-\mathbf{I}_{t}\right)\mathbf{S}^{-\frac{1}{2}}.$
(68)
Hence,
$\displaystyle\mathbf{B}^{\star}\mathbf{O}_{1}$
$\displaystyle=\mathbf{S}^{\frac{1}{2}}\mathbf{G}\left(\begin{matrix}\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}&0\\\
0&0\end{matrix}\right)\mathbf{G}^{\intercal}\left(\mathbf{G}^{-\intercal}\boldsymbol{\Phi}\mathbf{G}^{-1}-\mathbf{I}_{t}\right)\mathbf{S}^{-\frac{1}{2}}$
$\displaystyle=\mathbf{S}^{\frac{1}{2}}\mathbf{G}\left[\left(\begin{matrix}\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}&0\\\
0&0\end{matrix}\right)\boldsymbol{\Phi}-\left(\begin{matrix}\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}&0\\\
0&0\end{matrix}\right)\mathbf{G}^{\intercal}\mathbf{G}\right]\mathbf{G}^{-1}\mathbf{S}^{-\frac{1}{2}}$
$\displaystyle=\mathbf{S}^{\frac{1}{2}}\mathbf{G}\left[\left(\begin{matrix}\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}&0\\\
0&0\end{matrix}\right)\left(\begin{matrix}\overline{\boldsymbol{\Phi}}_{1}&0\\\
0&\overline{\boldsymbol{\Phi}}_{2}\end{matrix}\right)-\left(\begin{matrix}\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}&0\\\
0&0\end{matrix}\right)\left(\begin{matrix}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}&\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\\\
\mathbf{G}_{2}^{\intercal}\mathbf{G}_{1}&\mathbf{G}_{2}^{\intercal}\mathbf{G}_{2}\end{matrix}\right)\right]\mathbf{G}^{-1}\mathbf{S}^{-\frac{1}{2}}$
$\displaystyle=\mathbf{S}^{\frac{1}{2}}\mathbf{G}\left(\begin{matrix}\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}\overline{\boldsymbol{\Phi}}_{1}-\mathbf{I}_{\rho}&-\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\\\
0&0\end{matrix}\right)\mathbf{G}^{-1}\mathbf{S}^{-\frac{1}{2}}$
giving
$\displaystyle\left|\mathbf{I}_{t}+{\mathbf{B}^{\star}}\mathbf{O}_{1}\right|=\left|\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right|^{-1}\left|\overline{\boldsymbol{\Phi}}_{1}\right|.$
(69)
Similarly, we may obtain
$\displaystyle\mathbf{B}^{\star}\mathbf{O}_{2}$
$\displaystyle=\mathbf{S}^{\frac{1}{2}}\mathbf{G}\left(\begin{matrix}\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}-\mathbf{I}_{\rho}&-\left(\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right)^{-1}\mathbf{G}_{1}^{\intercal}\mathbf{G}_{2}\\\
0&0\end{matrix}\right)\mathbf{G}^{-1}\mathbf{S}^{-\frac{1}{2}}$
and
$\displaystyle\left|\mathbf{I}_{t}+\mathbf{B}^{\star}\mathbf{O}_{2}\right|$
$\displaystyle=\left|\mathbf{G}_{1}^{\intercal}\mathbf{G}_{1}\right|^{-1}.$
(70)
Substituting (69) and (70) into (67), we may obtain
$\displaystyle C_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$
$\displaystyle\geq\frac{1}{2}\log\left|\overline{\boldsymbol{\Phi}}_{1}\right|$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{\rho}\log\phi_{j}.$ (71)
Putting together (65) and (71) establishes the desired equality
$\displaystyle C_{s}(\mathbf{H}_{1},\mathbf{H}_{2},\mathbf{S})$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{\rho}\log\phi_{j}.$
This completes the proof of the theorem.
## References
* [1] A. D. Wyner, “The wire-tap channel,” _Bell Syst. Tech. J._ , vol. 54, no. 8, pp. 1355–1387, Oct. 1975.
* [2] I. Csiszár and J. Körner, “Broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 24, no. 3, pp. 339–348, May 1978\.
* [3] Y. Liang, H. V. Poor, and S. Shamai (Shitz), _Information Theoretic Security_. Dordrecht, The Netherlands: Now Publishers, 2009.
* [4] Z. Li, W. Trappe, and R. D. Yates, “Secret communication via multi-antenna transmission,” in _Proc. Forty-First Annual Conference on Information Sciences and Systems_ , Baltimore, MD, Mar. 2007.
* [5] A. Khisti and G. Wornell, “Secure transmission with multiple antennas: The MISOME wiretap channel,” _IEEE Trans. Inf. Theory_ , submitted for publication.
* [6] S. Shafiee, N. Liu, and S. Ulukus, “Towards the secrecy capacity of the Gaussian MIMO wire-tap channel: The 2-2-1 channel,” _IEEE Trans. Inf. Theory_ , to appear.
* [7] A. Khisti and G. W. Wornell, “The secrecy capacity of the MIMO wiretap channel,” in _Proc. 45th Annual Allerton Conf. Comm., Contr., Computing_ , Monticello, IL, Sep. 2007.
* [8] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” in _Proc. IEEE Int. Symp. Information Theory_ , Toronto, Canada, July 2008, pp. 524–528.
* [9] T. Liu and S. Shamai (Shitz), “A note on the secrecy capacity of the multiantenna wiretap channel,” _IEEE Trans. Inf. Theory_ , to appear.
* [10] R. Bustin, R. Liu, H. V. Poor, and S. Shamai (Shitz), “A MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel,” _EURASIP Journal on Wireless Communications and Networking (Special Isssue on Wireless Physical Layer Security)_ , submitted November 2008.
* [11] H. D. Ly, T. Liu, and Y. Liang, “MIMO broadcasting with common, private and confidential messages,” in _Proc. Int. Symp. Inform. Theory Applications_ , Auckland, New Zealand, Dec. 2008.
* [12] R. Liu and H. V. Poor, “Secrecy capacity region of a multi-antenna Gaussian broadcast channel with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 3, pp. 1235–1249, Mar. 2009.
* [13] R. Liu, I. Maric, P. Spasojevic, and R. D. Yates, “Discrete memoryless interference and broadcast channels with confidential messages: Secrecy rate regions,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 6, pp. 2493–2507, Jun. 2008.
* [14] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 52, pp. 3936–3964, Sep. 2006.
* [15] S. Goel and R. Negi, “Guaranteeing secrecy using artificial noise,” _IEEE Trans. Wireless Comm._ , vol. 7, pp. 2180–2189, Jun. 2008.
* [16] W. Yu and J. M. Cioffi, “Sum capacity of Gaussian vector broadcast channels,” _IEEE Trans. Inf. Theory_ , vol. 50, pp. 1875–1892, Sep. 2004\.
* [17] G. Strang, _Linear Algebra and Its Applications_. Wellesley, MA: Wellesley-Cambridge Press, 1998.
|
arxiv-papers
| 2009-03-23T04:26:25 |
2024-09-04T02:49:01.337650
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruoheng Liu, Tie Liu, H. Vincent Poor, Shlomo Shamai (Shitz)",
"submitter": "Ruoheng Liu",
"url": "https://arxiv.org/abs/0903.3786"
}
|
0903.3810
|
# Simulations for Terrestrial Planets Formation
Niu ZHANG11affiliation: Graduate School of Chinese Academy of Sciences,
Beijing 100049, China 22affiliation: Purple Mountain Observatory, Chinese
Academy of Sciences, Nanjing 210008, China , Jianghui JI22affiliation: Purple
Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China
33affiliation: National Astronomical Observatory, Chinese Academy of
Sciences,Beijing 100012, China jijh@pmo.ac.cn
###### Abstract
In this paper, we investigate the formation of terrestrial planets in the late
stage of planetary formation using two-planet model. At that time, the
protostar has formed for about 3 Myr and the gas disk has dissipated. In the
model, the perturbations from Jupiter and Saturn are considered. We also
consider variations of the mass of outer planet, and the initial
eccentricities and inclinations of embryos and planetesimals. Our results show
that, terrestrial planets are formed in 50 Myr, and the accretion rate is
about $60\%-80\%$. In each simulation, 3 - 4 terrestrial planets are formed
inside ”Jupiter” with masses of $0.15-3.6M_{\oplus}$. In the 0.5 \- 4 AU, when
the eccentricities of planetesimals are excited, planetesimals are able to
accrete material from wide radial direction. The plenty of water material of
the terrestrial planet in the Habitable Zone may be transferred from the
farther places by this mechanism. Accretion could also happen a few times
between two major planets only if the outer planet has a moderate mass and the
small terrestrial planet could survive at some resonances over time scale of
$10^{8}$ yr. In one of our simulations, com-mensurability of the orbital
periods of planets is very common. Moreover, a librating-circulating 3:2
configuration of mean motion resonance is found.
###### Subject headings:
astrophysics-exoplanet-planetary formation-n-body simulation
## 1\. Introduction
Since the discovery of the first extrasolar planet around Solar-Type star, the
detection of the extrasolar planets develops rapidly. To date, more than $300$
planets are found orbiting their center stars beyond our solar system,
including 35 multiple planetary systems. Recently, scientists have found
evidences of methane and carbon dioxide in the atmosphere of a Hot-Jupiter (HD
189733 b) (http://planetquest.jpl.nasa.gov). Of $\sim 300$ known extrasolar
planets, the minimum mass is generally several Jupiter masses. There are also
several terrestrial planets (Super-Earth), but their orbital characteristic is
unsuitable for the formation or development of life. Along with the
development of survey techniques and incoming high definition space missions,
people will definitely discover more and more Earth-like planets in the
extrasolar planetary systems. The research of formation and evolution of the
terrestrial planet now becomes important topics in astrophysics, astrobiology,
astrochemistry and so on.
Planet formation has certain order (Zhou et al., 2005), and Jupiter-like
planets at greater distance are formed faster than those near the Sun. It is
generally believed that the planet formation may experience the following
stages: The grains condensed in the initial stage grow to km-sized
planetesimals in the early stage, and then, in the middle stage, Moon-to-Mars
sized embryos are formed by accretion of the planetesimals. The size of
embryos correlates with the feeding zone of the planetesimals. According to
the formula of Hill radius: $R_{H}=r(m/3M_{\odot})^{1/3}$ (where $r$,$m$ are
the heliocentric distance and the mass of planetesimal), the distant
planetesimals have wider feeding zones, so the formed embryos are larger. When
the embryos grow to the core of mass ($\sim 10M_{\oplus}$ ), runaway accretion
may take place accordingly. With more atmosphere accreted, the embryos
contract, growing ever denser and more massive, eventually collapse to form
giant Jovian planets (Hu & Xu, 2008; Ida & Lin, 2004). However, at the same
period, the inner planetesimals accrete in respective accretion scope, and
then the embryos of terrestrial planet (namely kind of the terrestrial planet
core matter) are formed. At the end of the third stage, it is around that the
protostar has formed for about $3$ Myr, the gas disk has dissipated. A few
larger bodies with low $e$ and $i$ are in crowds of planetesimals with certain
eccentricities $e$, and inclinations $i$. In the late stage, the terrestrial
planetary embryos are excited to high eccentricity orbit by gravitational
perturbation. Then, the orbital crossing makes the planets accreting material
in the broader radial area. Solid residue is either scattered out of the
planetary system or accreted by the massive planet. However, it also has the
possibility of being captured at the resonance position of the major planets
(Nagasawa & Ida, 2000; Hu & Xu, 2008).
Taking our Solar System as the background, Chambers (2001) made a study of
terrestrial planet formation in the late stage by numerical simulations. He
set $150-160$ Moon-to-Mars size planetary embryos in the area of $0.3-2.0$ AU,
include gravitational perturbations from Jupiter and Saturn. He also examined
two initial mass distributions: approximately uniform masses, and a bimodal
mass distribution. The results show that $2-4$ planets are formed in $50$ Myr,
and finally survive over $200$ Myr timescale. The space distribution and
concentration (see Section 4 in Chambers (2001)) of planets formed in the
simulations are similar to our solar system. However, the planets produced by
the simulations usually have eccentric orbits with higher eccentricities $e$,
and inclinations $i$ than Venus. Raymond, Quinn & Lunine (2004, 2006) also
studied the formation of terrestrial planets. In the simulations, they took
into account Jupiter’s gravitational perturbation, wider distribution of
material ($0.5-4.5$ AU) and higher resolution. The results confirm a leading
hypothesis for the origin of Earth’s water: they may come from the material in
the outer area by impacts in the late stage of planet formation. Raymond,
Mandell, & Sigurdsson (2006) explored the planet formation under planetary
migration of the giant. In the simulations, super Hot Earth form interior to
the migrating giant planet, and water-rich, Earth-size terrestrial planet are
present in the Habitable Zone ($0.8-1.5$ AU) and can survive over $10^{8}$ yr
timescale.
In our model, Solar System is taken as the background. But several changes are
worth noting : 1) we use two-planet (Jupiter and Saturn, see Fig.1) model. 2)
in the model, Jupiter and Saturn are supposed to be formed at the beginning of
the simulation, with two swarms of planetesimals distributed among $0.5-4.2$
AU and $6.2-9.6$ AU respectively. 3) The initial eccentricities and
inclinations of planetesimals are considered. 4) The variations of the mass of
Saturn are examined. 5) The exchange of material in the radial direction is
also studied by the parameter of water mass fraction. 6) We perform the
simulations over longer timescale ($400$ Myr) in order to check the stability
and the dynamical structure evolution of the system. Our results show that the
terrestrial planets produced interior to Jupiter have higher mass accretion
rate, and share the similar architecture as the Solar System. However, the
structure beyond Jupiter correlates with the initial mass of Saturn. Almost
each simulation has a water-rich terrestrial planet in the Habitable Zone
($0.8-1.5$ AU).
In Section 2, the initial conditions, algorithm and integration procedure are
described in detail. Section 3 presents the main results. We conclude the
outcomes in Section 4.
## 2\. Model
### 2.1. Initial conditions
Generally, the time scale for formation of Jupiter-like planet is less than
$10$ Myr (Briceño, 2001). Nevertheless, as we know, the formation scenario of
planet embryos is related to their heliocentric distances and the initial mass
of the star nebular. If we consider the model of $1.5$ MMSN (minimum mass
solar nebular), the upper bound of the time scale for Jupiter-like planet
formation corresponds to the time scale for the embryo formation at $2.5$ AU
(Kokubo & Ida, 2002), which is just at $3:1$ resonance location of Jupiter. In
the region $2.5-4.2$ AU, embryos will be cleared off by strong gravitational
perturbation arising from Jupiter. There should be some much smaller solid
residue among Jupiter and Saturn, even though the ’clearing effect’ may throw
out most of the material in this area. That’s why we set embryos only in the
region $0.5-2.5$ AU and planetesimals in the $0.5-4.2$ AU and $6.2-9.6$ AU.
We adopt the surface density profile as follows (Raymond, Quinn & Lunine,
2004):
$\Sigma(r)=\left\\{\begin{array}[]{ll}\Sigma_{1}r^{-3/2},&r<snow~{}line,\\\
\Sigma_{snow}(\frac{r}{5AU})^{-3/2},&r>snow~{}line.\end{array}\right.$ (1)
In (1), $\Sigma_{snow}=10~{}g/cm^{2}$ is the surface density at snowline, the
snowline is at $2.5$ AU with $\Sigma_{1}=4~{}g/cm^{2}$.As mentioned earlier,
the mass of planetary embryos is proportional to the width of the feeding
zone, which is associated with Hill Radius, $R_{H}$, so the mass of an embryo
increases as
$M_{embryo}\propto r\Sigma(r)R_{H}$ (2)
The embryos among $0.5-2.5$ AU are spaced by $\Lambda$ ($\Lambda$ varying
randomly between 2 and 5) mutual Hill Radii, $R_{H,m}$ , which is defined as
$R_{H,m}=(\frac{a_{1}+a_{2}}{2})(\frac{m_{1}+m_{2}}{3M_{\odot}})^{1/3}$ (3)
where $a_{1,2}$ and $m_{1,2}$ are the semi-major axes and masses of the
embryos respectively. Replacing $R_{H}$ in (2) with $R_{H,m}$ , and
substituting (1) in (2), then, we get relations between the mass of embryos
and the parameter $\Lambda$ as
$M_{embryo}\propto r\Sigma(r)R_{H,m}\propto r^{3/4}\Lambda^{3/2}\Sigma^{3/2}$
(4)
As shown in Fig. 1, the initial planetesimals are spread over $0.5-9.6$ AU
(excluding $3$ Hill Radii around Jupiter); the distribution of them should
meet (1). Here, we equally set the masses of planetesimals inside and outside
Jupiter respectively as shown in (5). Consequently, the number distribution of
the planetesimals is only needed to satisfy $N\propto r^{-1/2}$. Additionally,
we keep the total number of planetesimals and embryos inside Jupiter, and the
number of planetesimals outside Jupiter both equal to $200$.
$\left\\{\begin{array}[]{ll}\sum N_{embryo}+\sum
N_{planetesimal,~{}~{}r<r_{Jupiter}}\\\ ~{}~{}~{}~{}=\sum
N_{planetesimal,~{}r>r_{Jupiter}}=200,\\\ \sum M_{embryo}+\sum
M_{planetesimal,~{}r<r_{Jupiter}}\\\ ~{}~{}~{}~{}=\sum
M_{planetesimal,~{}r>r_{Jupiter}}=7.5M_{\oplus},\\\
M_{planetesimal,~{}r<r_{Jupiter}}=\\\ ~{}~{}~{}~{}(7.5M_{\oplus}-\sum
M_{embryo})/(200-\sum N_{embryo}),\\\
M_{planetesimal,~{}r>r_{Jupiter}}=7.5M_{\oplus}/200.\end{array}\right.$ (5)
The water mass fraction of the bodies is same as Raymond, Quinn & Lunine
(2004), i.e., the planetesimals beyond 2.5 AU have $5\%$ water material by
mass, those between $2-2.5$ AU have $0.1\%$ water material by mass, and the
others have $0.001\%$ water material by mass. The eccentricities and
inclinations vary in ($0-0.02$) and ($0-0.05^{\circ}$), respectively. The mass
of Saturn in simulations 1a/1b,2a/2b and 3a/3b are $0.5M_{\oplus}$,
$5M_{\oplus}$, $50M_{\oplus}$ respectively. Each simulation is carried out
twice with a) considering, b) not considering self-gravitation of
planetesimals among Jupiter and Saturn.
### 2.2. Algorithm
In regular coordinate system, the motion equations of an n-body system are
(Murray & Dermott, 1999)
$\left\\{\begin{array}[]{ll}\frac{dx_{i}}{dt}&=\frac{\partial H}{\partial
p_{i}},\\\ \frac{dp_{i}}{dt}&=-\frac{\partial H}{\partial
x_{i}}.\end{array}\right.$ (6)
where the index $i=1,2,\cdots,n$ denotes the body $i$ , and $x_{i}$ , $p_{i}$
are the generalized coordinate and momentum of the body $i$, respectively. The
Hamiltonian,
$H=\sum\limits_{i=1}^{n}\frac{p_{i}^{2}}{2m_{i}}-G\sum\limits_{i=1}^{n}m_{i}\sum\limits_{j=i+1}^{n}\frac{m_{j}}{r_{ij}}$,
is the sum of the kinetic and potential energy for the system. From (6), we
know that the rate of any quantity, $q$ , can be conveniently expressed in the
following form,
$\begin{array}[]{ll}\frac{dq}{dt}&=\sum\limits_{i=1}^{n}(\frac{\partial
q}{\partial x_{i}}\frac{dx_{i}}{dt}+\frac{\partial q}{\partial
p_{i}}\frac{dp_{i}}{dt})\\\ &=\sum\limits_{i=1}^{n}(\frac{\partial q}{\partial
x_{i}}\frac{\partial H}{\partial p_{i}}-\frac{\partial q}{\partial
p_{i}}\frac{\partial H}{\partial x_{i}}).\end{array}$ (7)
If we define an operator $F=\sum\limits_{i=1}^{n}(\frac{\partial~{}}{\partial
x_{i}}\frac{\partial H}{\partial p_{i}}-\frac{\partial~{}}{\partial
p_{i}}\frac{\partial H}{\partial x_{i}})$ (Chambers, 1999), then we can
rewrite (7) as $\frac{dq}{dt}=Fq$. Integral the differential equation over
time $t_{1}-t_{2}$ ($t_{2}>t_{1}$), and so we get
$q_{2}=e^{(t_{2}-t_{1})F}q_{1},$ (8)
where $q_{1}$ and $q_{2}$ are the values of $q$ corresponding to the time
$t_{1}$ and $t_{2}$ respectively. If we define $h=t_{2}-t_{1}$ ($h$ actually
is the internal time step), then expand (8) at zero, we have
$q_{2}=(1+hF+\frac{h^{2}F^{2}}{2}+\cdots)q_{1},$ (9)
The symplectic integrator is to divide $H$ into pieces, each piece could be
individually solved, and then they approximate the solution of the problem via
applying the solutions once a time. For example, we split the Hamiltonian
$H=H_{1}+H_{2}$ , and hence have the operators $F_{1},F_{2}$ . It is easy to
obtain $q_{2}=e^{h(F_{1}+F_{2})}q_{1}$ from (8). By expanding the exponential
(attn. $F_{1}F_{2}\neq F_{2}F_{1}$), ignoring the second- and higher-order
small quantities of $h$ , then we have
$\begin{array}[]{ll}e^{h(F_{1}+F_{2})}&=e^{hF_{1}}e^{hF_{2}}+\frac{h^{2}(F_{2}F_{1}-F_{1}F_{2})}{2}\\\
&=1+h(F_{1}+F_{2})+\frac{h^{2}(F_{1}^{2}+2F_{1}F_{2}+F_{2}^{2})}{2}+\cdots\end{array}$
(10)
We can get a second-order integrator
$q_{2}=e^{hF_{2}/2}e^{hF_{1}}e^{hF_{2}/2}q_{1}$ by applying a small
equivalence transformation.
The key point for symplectic algorithm is how to split Hamiltonian $H$ into
pieces. Considering a dynamical system composed of $N$ bodies orbiting a
massive central body, we can split the Hamiltonian $H$ into the primary and
the secondary parts. Chambers (1999) proposed a hybrid symplectic algorithm,
in which the Hamiltonian is divided into the following parts:
$\left\\{\begin{array}[]{ll}H_{1}&=\sum\limits_{i=1}^{N}(\frac{p_{i}^{2}}{2m_{i}}-\frac{Gm_{\odot}m_{i}}{r_{i\odot}}),\\\
H_{2}&=-G\sum\limits_{i=1}^{N}\sum\limits_{j=i+1}^{N}\frac{m_{i}m_{j}}{r_{ij}},\\\
H_{3}&=\frac{1}{2m_{\odot}}(\sum\limits_{i=1}^{N}{\bf\it
p_{i}})^{2},\end{array}\right.$ (11)
where $H_{1}$ is the unperturbed Keplerian motion of the $N$ smaller bodies,
$H_{2}$ is the total interaction potential energy of the $N$ smaller bodies,
and $H_{3}$ is the kinetic energy of the center body (Note That: $N$ has
different meaning from $n$ above, $N$ refers to the numbers of bodies
excluding the central body). The term of ’hybrid’ means that, for the
convenience of calculation, heliocentric coordinates and barycentric
velocities are used while solving (11) (Chambers, 1999; Duncan, Levison & Lee,
1998). From (10), we can infer that all of the higher terms depend on both
$F_{1}$ and $F_{1}$. If $F_{2}\sim\epsilon F_{1}$ ($\epsilon=\sum
m_{i}/m_{\odot}$), and therefore, the second-order integrator
$q_{2}=e^{hF_{2}/2}e^{hF_{3}/2}e^{hF_{1}}e^{hF_{3}/2}e^{hF_{2}/2}q_{1}$ is
correct to $O(\epsilon h^{3})$ only when the different parts of Hamiltonian
meet the conditions $H_{1}\gg H_{2},H_{1}\gg H_{3}$. However, when close
encounter occurs between two bodies, the distance between them $r_{ij}$
approaches zero, hence the $H_{1}\gg H_{2}$ can not be satisfied. Chambers
(1999) introduced a changeover function $K(r_{ij})$ to translate part of
$H_{2}$ associated with the close encounter to $H_{1}$, and then integrate it
using Bulirsch-Stoer method (Stoer & Bulirsch, 1980). The modified
$H_{1},H_{2}$ are present as
$\left\\{\begin{array}[]{ll}H_{1}&=\sum\limits_{i=1}^{N}(\frac{p_{i}^{2}}{2m_{i}}-\frac{Gm_{\odot}m_{i}}{r_{i\odot}})\\\
&~{}~{}~{}~{}-G\sum\limits_{i=1}^{N}\sum\limits_{j=i+1}^{N}\frac{m_{i}m_{j}}{r_{ij}}[1-K(r_{ij})],\\\
H_{2}&=-G\sum\limits_{i=1}^{N}\sum\limits_{j=i+1}^{N}\frac{m_{i}m_{j}}{r_{ij}}K(r_{ij}).\end{array}\right.$
(12)
$K(r_{ij})$ tends to zero when $r_{ij}$ is small, while tending to one when
$r_{ij}$ is large (Chambers, 1999).
We use the hybrid symplectic integrator (Chambers, 1999) in MERCURY package to
integrate all the simulations. We take into account that collision and
coalescence will occur, when the minimum distance between any of the two
objects is equal to or less than the sum of their physical radii. While they
were separated by not more than 3 Hill radii, we consider close encounters
will take place. When the distance from the central star is more than $100$
AU, these bodies are removed, because they are so far from the central star
that they play an insignificant role of the interaction. In addition, we adopt
$6$ days as the length of time step, which is a twentieth period of the
innermost body at $0.5$ AU. The $6$ simulations are carried out over $400$ Myr
time scale. At the end of the intergration, the changes of energy and angular
momenta are $10^{-3}$ and $10^{-11}$ respectively. The $6$ simulations are
performed on a workstation composed of $12$ CPUs with $1.2$ GHz, and each
costs roughly $45$ days.
## 3\. Results
All of $6$ simulations exhibit some classical processes on planet formation.
Firstly, we will analyse simulation 2a/2b to discuss the physical processes
which can apply to every simulation. Next, we will make a statistical analysis
in order to find out that how the planet formation may rely on different
physical factors.
### 3.1. Simulation 2a/2b
At the end of the calculation, $3-4$ terrestrial planets are formed in 2a/2b.
Table 1 shows the properties of the terrestrial planets from simulations 2a
and 2b. We label the planets as b, c, d, and so on, according to the
heliocentric distance (hereinafter). The masses of the terrestrial planets
range from several Mars masses to several Earth masses. All of them are water-
rich, except the planet e in simulation 2b. Some parameters of a certain
planet are comparable with the terrestrial planets in solar system. For
example, the orbital eccentricity of planet b in simulation 2b is $0.0309$,
which is very close to that of Earth.
Fig. 2 is a snapshot of simulation 2a. At $0.1$ Myr, it is clear that the
planetesimals are excited at the $3:2$ ($3.97$ AU),$2:1$ ($3.28$ AU) and $3:1$
($2.5$ AU) resonance locations with Jupiter, and this is similar to the
Kirkwood gaps of the asteroidal belt in solar system. For about $1$ Myr,
planetesimals and embryos are deeply intermixed, most of the bodies have large
eccentricities. Collisions and accretions emerge among planetesimals and
embryos. This process continues until about $50$ Myr, the planetary embryos
are mostly formed, and then dynamical evolution is start. The formation time
scale in our work is in accordance with that of (Ida & Lin, 2004). Finally,
inside Jupiter, $3$ terrestrial planets are formed with masses of
$0.15-3.6M_{\oplus}$. However, at the outer region, planetesimals are
continuously scattered out of the system at $0.1$ Myr. For about $10$ Myr,
there are no survivals except at some resonances with the giant planet. As
shown in Fig. 2, there is a small body at the $1:2$ resonance with Jupiter.
Due to the planetesimals’ scattering, Jupiter (Saturn) migrates inward
(outward) $0.13$ AU ($1.19$ AU) toward the sun respectively. Such kind of
migration agrees with the work of Fernandez et al. (Fernandez & Ip, 1984)
Hence, the $2:5$ mean motion resonance is destroyed, then the ratios of
periods between Jupiter and Saturn degenerate to $1:3$. Therefore, the ratio
of periods for Jupiter, small body and Saturn is approximate to $1:2:3$.
Fig. 3 is a snapshot of simulation 2b. In comparison with Fig. 2, it is
apparent that planetesimals are excited more quickly at the $3:2$ ($3.97$ AU),
$2:1$ ($3.28$ AU) and $3:1$ ($2.5$ AU) resonance location with Jupiter. The
several characteristic time scales are the same as simulation 2a for the
bodies within Jupiter. 4 planets are formed in simulation 2b, the changes of
position of Jupiter and Sat-urn are about the same as simulation 2a. We note
simulations 2a and 2b have the same initial conditions, the only difference
between them is whether we consider the self-gravitation among the outer
planetesimals. The results of simulation 2b are shown to be a consequence of
being expected but not surprising. There is a little stack planetesimals
survival over $400$ Myr among $7-8$ AU, located in the area of $2:3$ ($6.63$
AU) and $1:2$ ($8.03$ AU) resonances with Jupiter.
Planets in 55 cnc planetary system have similar spatial distribution to the
solar system (Fischer et al., 2008), From Table 1 and Fig. 3, it is not
difficult to see that 4 terrestrial planets formed in simulation 2b move on
the nearly-circular orbit. Mars is ever regarded as a survivor of an original
planetary embryo, according to its unique chemical and isotopic
characteristics. As a matter of fact, the planet e in simulation 2b is a
survivor of the initial planetary embryos. Planet e in simulation 2b does not
accrete anything over $400$ Myr integration time. Comparing the semi-major
axis of Mars with that of planet e, we notice that the planetesimals of
simulation 2b are located in the asteroidal belt. Furthermore, the ratio of
periods of the planet e and Jupiter is nearly $1:2$. The total mass of the
main-belt in solar system is about $5\times 10^{-5}M_{\oplus}$, being
$0.1\%-0.12\%$ (Hu & Xu, 2008) of the initial solid material. If the assumed
planet e in simulation 2b would break into thousands of fragments, they may
undergo re-accretion or ejection by the perturbation of Jupiter over secular
evolution. If it is similar to the same ratio of mass of the belt in solar
system, then the leftovers of the solid materials almost bear a total mass of
$6.3\times 10^{-5}-7.56\times 10^{-5}M_{\oplus}$ . In this sense, an
asteroidal belt is very likely to form in the system quite similar to that of
our solar system.
Fig. 4a is the mass curve of terrestrial planets for simulation 2b. We can
find that the accrete velocity is not uniform. At $10$ Myr, planets reach half
mass of their final mass, and then, the accretion velocity slows down, because
the planetesimals are only a quarter left. Until about $50$ Myr, the
terrestrial planets are formed. The accretion rate and mass concentration (the
mass rate of the largest terrestrial planet and the total formed objects) are
$73\%$ and $43\%$, respectively. The corresponding parameters in simulation 2a
are $60\%$ and $81\%$ respectively. However, in the area outside Jupiter,
$81\%$ initial material is scattered out of the system.
Planet embryos are formed from feeding zones where the planetesimals are
located. A feeding zone has unique chemical and isotopic characteristics. It
is helpful to study the trace of the planetesimals to understand the
composition of terrestrial planet, vice versa, for example, if we can
investigate the origin and formation process by revealing the chemical or
isotopic characteristics of the moon. In Fig. 4b is shown the trace of
survivals. We note that all the materials accreted by terrestrial planets come
from the inner swarm of planetesimals or embryos. Here Jupiter is like a wall,
which separates the inner and outer planetesimals from exchanging materials.
Once again, Fig. 4b verifies that Jupiter may protect the inner terrestrial
planets from colliding with the outer bodies (Wetherill, 1990). We set a water
mass fraction on each body, it is easy to work out how much water-material of
the finally terrestrial planet bears. Take planet c in simulation 2b for
example, the water material is approximately $1.1\times 10^{22}kg$ , about 8
times than Earth. Fig. 4b can also verify that terrestrial planet accrete
material in broad radial direction.
### 3.2. Statistical analysis
The production efficiency of the terrestrial planet in our model is high, and
the accretion rate inside Jupiter is $60\%-80\%$ in the simulations. $3-4$
terrestrial planets formed in $50$ Myr. 5 of 6 simulations have a terrestrial
planet in the Habitable Zone ($0.8-1.5$ AU) (see Fig. 5). The planetary
systems are formed to have nearly circular orbit and coplanarity, similar to
the solar system (see Table 2). We suppose that the above characteristics are
correlated with the initial small eccentricities and inclinations. Such
adoption could generate more close encounters or collisions in the early
several Myr, which may increase viscosity of the system and then make the
orbits more nested on circular orbit on the orbital plane. The concentration
in Table 2 means the ratio of maximum terrestrial planet formed in the
simulation and the total terrestrial planets mass. It represents different
capability on accretion, and is not associated with self-gravitation. The
average value of this parameter is similar to the solar system. Considering
the self-gravitation of planetesimals among Jupiter and Saturn, the system has
a better viscosity, so that the planetesimals will be excited slower. The
consideration of self-gravitation may not change the formation time scale of
terrestrial planets, but will affect the initial accretion speed and the
eventual accretion rate. In Table 2, the simulations 1b, 2b, 3b have a bit
higher accretion rate. When the self-gravitation is not considered, the
planetesimals may be excited quickly. The accretion has a faster speed at the
early several Myr, so this can promote the accretion rate.
From Fig. 4b, we have to be aware of that Saturn accretes a few planetesimals,
this is uncommon in simulations 1a, 1b, 3a, 3b. And Fig. 5 shows the finally
structure of the simulations, it is clear that different Saturn mass will
affect the outer structure of the system beyond Jupiter. In simulation 3a
(3b), the Saturn mass is $50M_{\oplus}$ . Now it is large enough to clear the
area among Jupiter and Saturn. In simulation 1a (1b), Saturn’s mass is
$0.5M_{\oplus}$ , more or less equal to the embryos’ mass. In this case, it is
too small to clear off any planetesimal amongst the region of Jupiter and
Saturn. Therefore, we draw the conclusion that only the Saturn’s mass is close
to be $5-10M_{\oplus}$ , then accretion may happen. Saturn and Jupiter in our
solar system may form in same stage. If there exist embryos of $10M_{\oplus}$
outside Saturn, the giant Jupiter-mass planets may form.
The scattering of planetesimals could cause the migration of planets, for
example, Jupiter migrated from $5.2$ AU to $5.06$ AU while Saturn traveled
from $9.6$ AU to $10.71$ AU. There are plenty of ratios of semi-major axis of
the survival planets nearly $2:1$. In this case, the planet is easily to be
captured on $2:1$ mean motion resonance (Lee & Peale, 2002; Lee, 2004; Zhou et
al., 2005; Fischer et al., 2008). There are still some ratios of periods
between the survival bodies close to a simple ratio of integers (see Fig. 5).
In the very long process of dynamical evolution after planetary formation, the
planets also have the possibility of been captured onto resonant orbit. For
example, the orbits show the $2:3$ mean motion resonance be-tween Jupiter and
outer small body in simulation 1b, and the $3:1$ resonance between Jupiter and
the planet d in simulation 3a and so on. Many researchers have studied the
resonance and stability of the planetary systems (Ji et al., 2003; Zhou & Sun,
2003; Zhou et al., 2004). As shown in Fig. 6, in simulation 2a, two planets
are on crossing orbits. When a close encounter occurs, $3:2$ mean motion
resonance is formed, with resonance angle
$\theta_{1}=2\lambda_{1}-3\lambda_{2}+\varpi_{2}$ (where $\lambda_{1,2}$ are
the mean longitude and the longitudes of periapse, the footnotes 1, 2 means
the inner and outer planets respectively.) librating around $180^{\circ}$ ,
while $\theta_{2}=2\lambda_{1}-3\lambda_{2}+\varpi_{1}$ circulating, and
$e_{1}$ shows large oscillations. Such ’librating-circulating’ configuration
is similar to the configuration of $2:1$ resonance in HD 73526 planetary
system. There have been several hypotheses about its origin (Tinney et al.,
2006; Sándor & Kley, 2006; Sándor, Kley & Klagyivik, 2007). However, it still
needs further study in the future.
## 4\. Conclusions
We simulate the terrestrial planets formation by using two-planet model. In
the simulation, the variations of the mass of outer planet, the initial
eccentricities and inclinations of embryos and planetesimals are also
considered. The results show that, during the terrestrial planets formation,
planets can accrete material from different regions inside Jupiter. Among
$0.5-4.2$ AU, the accretion rate of terrestrial planet is $60\%-80\%$, i.e.,
about $20\%-40\%$ initial mass is removed during the progress. The
planetesimals will improve the efficiency of accretion rate for certain
initial eccentricities and inclinations, and this also makes the newly-born
terrestrial planets have lower orbital eccentricities. It is maybe a common
phenomenon in the planet formation that the water-rich terrestrial planet is
formed in the Habitable Zone. The structure, which is similar to that of solar
system, may explain the results of disintegration of a terrestrial planet.
Most of the planetesimals among Jupiter and Saturn are scattered out of the
planetary systems, and this migration caused by scattering (Fernandez & Ip,
1984) or long-term orbital evolution can make planets capture at some mean
motion resonance location. Accretion could also happen a few times between two
planets if the outer planet has a moderate mass, and the small terrestrial
planet could survive at some resonances over $10^{8}$ yr time scale.
Structurally, Saturn has little effect on the architecture inside Jupiter,
owing to its protection. However, obviously, a different Saturn mass could
play a vital role of the structure outer Jupiter. Jupiter and Saturn in the
solar system may form over the same period.
In our simulations, neither terrestrial planets are formed within $0.1$ AU,
nor planetesimals or embryos are left. However, a lot of exoplanets with
orbital semi-major less than $0.1$ AU are observed, and several Super-Earths
are discovered. It is usually believed that they were formed far from the
center star and then migrated into current location (Raymond, Mandell, &
Sigurdsson, 2006). We do not consider the migration in the simulations, which
is caused by the interaction between the giant planets or planetesimals in the
gaseous disk (Ou et al., 2007). So the simulation in this work can be applied
to the case of the dissipation of gas disk, in the late stage of planet
formation. In the future study, we will consider the giant planets under
inward migration, and in such circumstances short-period terrestrial planet
could be produced. To date, terrestrial planets are not detected in the
observations, due to the reasons of the selection effect of detection methods
and the low resolution precision. Both Doppler velocities and transit method
are sensitive to the objects moving in smaller orbits. The current research of
extrasolar terrestrial planets has greatly contributed to the origin and
evolution of our own solar system. Kepler has been launched successfully on
March 6, 2009, whose main scientific objective is detecting the Earth-like
terrestrial planets. Along with high accuracy incoming space projects, it is
predictable that more and more extrasolar planetary systems with similar
structure to the solar system will be discovered.
We are very grateful to Prof. Jilin Zhou of Nanjing University for reading
carefully and giving valuable suggestion to improve the manuscript. We thank
Prof. Qinglin Zhou and Dr. Xiaosheng Wan of Ministry of Education Key Modern
Astronomy and Astrophysics Laboratory of Nanjing University for their kind
help. This work is financially supported by the National Natural Science
Foundation of China (Grants 10573040, 10673006, 10833001, 10233020) and the
Foundation of Minor Planets of Purple Mountain Observatory.
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Table 1Properties of terrestrial planets from simulations 2a and 2b. Planet | $a$ (AU) | $e$ | $i(deg)$ | $m(m_{\oplus})$ | water mass
---|---|---|---|---|---
2a b | 0.6174 | 0.1098 | 7.452 | 3.6421 | 0.0316%
c | 1.7796 | 0.1935 | 33.890 | 0.1528 | 0.1040%
d | 2.3304 | 0.3291 | 9.393 | 0.6925 | 0.5218%
2b b | 0.5274 | 0.0309 | 4.041 | 2.3527 | 0.1539%
c | 1.0451 | 0.1080 | 4.915 | 1.3880 | 0.1316%
d | 1.4783 | 0.0520 | 5.189 | 1.6696 | 0.7795%
e | 3.1162 | 0.2086 | 6.421 | 0.0630 | 0.0010%
Table 2Properties of terrestrial planets from different systems System | accretion rate | $n$ | $\bar{m}(m_{\oplus})$ | concentration | $\bar{e}$ | $\bar{i}(^{\circ})$
---|---|---|---|---|---|---
1a | 73.2518% | 3 | 1.8313 | 0.4606 | 0.1381 | 7.6963
1b | 80.3853% | 3 | 2.0096 | 0.4262 | 0.0937 | 1.7790
2a | 59.8322% | 3 | 1.4958 | 0.8116 | 0.2108 | 16.9117
2b | 72.9779% | 4 | 1.3683 | 0.4299 | 0.0999 | 5.1415
3a | 65.1098% | 3 | 1.6277 | 0.5337 | 0.2063 | 5.9153
3b | 66.9694% | 3 | 1.6742 | 0.5040 | 0.1839 | 5.2447
1a-3b | 69.7544% | 3.2 | 1.6678 | 0.5276 | 0.1554 | 7.1148
solar | - | 4 | 0.4943 | 0.5058 | 0.0764 | 3.0624
Figure 1.— Two-planet model and initial conditions. Figure 2.— Snapshot of
simulation 2a with $M_{Saturn}=5M_{\oplus}$. The total mass of embryos is
$2.4M_{\oplus}$, the masses of planetesimals inside Jupiter are
$0.0317M_{\oplus}$, and those outside Jupiter are $0.0375M_{\oplus}$.
Planetesimals among Jupiter and Saturn were nonself-gravitational (see Section
2.1). Note the size of each object is relative, and the value bar is log of
water mass fraction, e.g. the wettest body has water mass fraction
$\log_{10}(5\%)=-1.3$. Figure 3.— Snapshot of simulation 2b. Figure 4.— (a)
Mass curve in simulation 2b. Figure 4.— (b) Trace of the objects’ in
simulation 2b. Figure 5.— Results of six simulations. The $\bullet$ and
$\times$ denote formed terrestrial planets and survival planetesimals. Jupiter
and Saturn locate at about 5 AU and 10 AU, respectively. $H$ zone marked with
dotted lines is so called Habitable Zone in 0.8 - 1.5 AU. Some mean motion
resonance locations with Jupiter are also labeled in the figure. Figure 6.—
Resonant variable $\theta_{1}$ of the 3:2 mean motion resonance of two
terrestrial planets and the curve of eccentricities and semi-major axes in
simulation 2a. ($\theta_{1}=2\lambda_{1}-3\lambda_{2}+\varpi_{2}$, where
$\lambda_{1,2}$ are the mean longitudes and the longitudes of periapse, the
footnotes 1, 2 mean the inner and outer planets respectively.)
|
arxiv-papers
| 2009-03-23T09:02:05 |
2024-09-04T02:49:01.348962
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhang Niu (1,2), Ji Jianghui (2,3) ((1)Graduate School of Chinese\n Academy of Sciences, (2) Purple Mountain Observatory, (3) NAOC)",
"submitter": "Jianghui Ji",
"url": "https://arxiv.org/abs/0903.3810"
}
|
0903.3836
|
11institutetext: † Centre de Physique Théorique de Luminy, Université de la
Méditerranée, F-13288 Marseille FR, and “Sapienza” Università di Roma,
Dipartimento di Fisica and ICRA,
P.le A. Moro 5, 00185 Rome IT, battisti@icra.it
‡ Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette FR, and
“Sapienza” Università di Roma, Dipartimento di Fisica and ICRA, P.le A. Moro
5, 00185 Rome IT, lecian@ihes.fr
§ ICRA, ICRANet, ENEA and “Sapienza” Università di Roma, Dipartimento di
Fisica, P.le A. Moro 5, 00185 Rome IT, montani@icra.it
# GUP vs polymer quantum cosmology: the Taub model
Marco Valerio Battisti† Orchidea Maria Lecian‡ Giovanni Montani§
###### Abstract
The fate of the cosmological singularity in the Taub model is discussed within
the two frameworks. An internal time variable is ruled out and the only
remaining degree of freedom (the anisotropy) of the Universe is quantized
according to such schemes. The resulting GUP Taub Universe is singularity-
free, differently from the second case, where the classical singularity is not
tamed by the polymer-loop quantum effects.
## 1 Introduction
Two different quantum cosmology approaches are applied to the Taub model. The
study is performed at the classical and at the quantum level in both schemes.
In particular, the generalized uncertainty principle (GUP) and the polymer
(loop) frameworks are implemented to this system. In the first case [1], the
cosmological singularity appears to be probabilistically suppressed, while, in
the second one [2], the Universe is still singular. Such a feature then allows
us to better understand the avoidance of the cosmological singularity in other
different quantum gravity toy models.
The Taub Universe arises as a particular case of the Bianchi IX model, i.e.
the most general scheme allowed by the homogeneity constraint (for reviews see
[3]). It is obtained by restricting the dynamics to that of a one-dimensional
particle bouncing against a wall, when only one degree of freedom (the
Universe anisotropy) is taken into account. The relevance of the Taub Universe
in quantum cosmology is then due to the fact that it is a necessary step
towards the Bianchi IX model, being a generalization of other isotropic
models. In particular, it has been used to test the validity of the
minisuperspace scheme [4] and to explore the application of the extrinsic
cosmological time [5].
The paper is organized as follows. In Section 2 the Taub model and the
formalisms are reviewed. Section 3 and 4 are devoted to the classical and
quantum analysis respectively. Comparisons with other approaches follow in
Section 5. We adopt $\hbar=c=16\pi G=1$ units.
## 2 Taub Universe and formalisms
The Taub cosmological model is a particular case of the Bianchi IX Universe,
of which only one anisotropy is taken into account. The Bianchi IX model,
together with Bianchi VIII, is the most general spatially-homogeneous model
and is described by the line element
$ds^{2}=N^{2}dt^{2}-e^{2\alpha}\left(e^{2\gamma}\right)_{ij}\omega^{i}\otimes\omega^{j}$
[3]. Here $N=N(t)$ is the lapse function, $\omega^{i}=\omega^{i}_{a}dx^{a}$
are the $SO(3)$ left-invariant 1-forms, $\alpha=\alpha(t)$ describes the
isotropic expansion of the Universe and $\gamma_{ij}=\gamma_{ij}(t)$ is a
traceless symmetric matrix, which determines the anisotropies via
$\gamma_{\pm}$. The dynamics of this Universe towards the singularity is
described by the motion of a two-dimensional particle (the two physical degree
of freedom of the gravitational field) in a dynamically-closed domain [3]. In
the Misner picture, such a domain depends on the time variable $\alpha$,
while, in the Misner-Chitré one, it becomes stationary in time. Performing the
ADM reduction of the dynamics (according to which the classical constraints
are solved with respect to the given momenta before implementing any
quantization algorithm), an effective Hamiltonian is obtained, which depends
only on the physical degrees of freedom of the system. In particular, the
scalar constraint is solved with respect to the momentum conjugated to the
time variable $p_{\tau}$ (we adopt the time gauge $\dot{\tau}=1$) and,
performing another change of variables, we obtain
$-p_{\tau}\equiv H^{IX}_{ADM}=v\sqrt{p_{u}^{2}+p_{v}^{2}}.$ (1)
The dynamics of such a system is equivalent to a billiard ball on a
Lobatchevsky plane and the three corners of the Misner scheme are replaced by
the points $(0,0)$, $(-1,0)$ and $v\rightarrow\infty$ in the $(u,v)$-plane, as
in Fig. 1.
Figure 1: The dynamical-allowed domain in the $(u,v)$-plane where the dynamics
is restricted.
The Taub cosmological model is described by $\gamma_{-}=0$. The dynamics of
this Universe is equivalent to the motion of a particle in a one-dimensional
closed domain. Such a domain corresponds the choice of only one of the three
equivalent potential walls of the Bianchi IX model. The ADM Hamiltonian (1)
rewrites $H_{ADM}^{T}=vp_{v}$, where $v\in[1/2,\infty)$. This Hamiltonian can
be further simplified defining the new variable $x=\ln v$, and becomes
$H_{ADM}^{T}=p_{x}\equiv p$ (2)
where the configuration variable $x$ is related to the Universe anisotropy
$\gamma_{+}$ by the equation $\gamma_{+}=e^{\tau-x}(e^{2x}-3/4)/\sqrt{3}$. The
Hamiltonian (2) will be the starting point of our analysis. It is worth noting
that the classical singularity now appears for $\tau\rightarrow\infty$.
### 2.1 GUP quantum mechanics
Some issues and results of a non-relativistic quantum mechanics with non-zero
minimal uncertainties in position are briefly reviewed [6]. In one dimension,
we consider the Heisenberg algebra generated by $q$ and $p$ obeying the
commutation relation
$[q,p]=i(1+\beta p^{2}),$ (3)
where $\beta>0$ is a deformation parameter. This commutation relation leads to
the generalized uncertainty relation $\Delta q\Delta
p\geq\frac{1}{2}\left(1+\beta(\Delta p)^{2}+\beta\langle p\rangle^{2}\right)$,
which appears in string theory [7]. The canonical Heisenberg algebra can be
recovered in the limit $\beta=0$, and the generalization to more dimension is
straightforward, leading naturally to a “noncommutative geometry” for the
space coordinates.
It is immediate to verify that a finite minimal uncertainty in position
$\Delta q_{min}=\sqrt{\beta}$ is predicted. The existence of a non-zero
uncertainty in position is relevant since it implies that there cannot be any
physical state that is a position eigenstate. In fact, an eigenstate of an
observable necessarily has a vanishing uncertainty on it. Although it is
possible to construct position eigenvectors, they are only formal eigenvectors
and not physical states. The deformed Heisenberg algebra (3) can be
represented in the momentum space as
$p\psi(p)=p\psi(p),\qquad q\psi(p)=i(1+\beta p^{2})\partial_{p}\psi(p),$ (4)
on a dense domain $S$ of smooth functions. To recover information on position
we have to study those states, which realize the maximally-allowed
localization. Such states $|\psi^{ml}_{\zeta}\rangle$, which are proper
physical states around a position $\zeta$, have the proprieties
$\langle\psi^{ml}_{\zeta}|q|\psi^{ml}_{\zeta}\rangle=\zeta$ and $(\Delta
q)_{|\psi^{ml}_{\zeta}\rangle}=\Delta q_{min}$. We can project an arbitrary
state $|\psi\rangle$ on the maximally-localized states
$|\psi^{ml}_{\zeta}\rangle$ to obtain the probability amplitude for a particle
being maximally localized around the position $\zeta$ (i.e. with standard
deviation $\Delta q_{min}$). We call these projections the quasiposition wave
function $\psi(\zeta)=\langle\psi^{ml}_{\zeta}|\psi\rangle$, and explicitly we
have the generalized Fouries transformation
$\psi(\zeta)\sim\int^{+\infty}_{-\infty}\frac{dp}{(1+\beta
p^{2})^{3/2}}\exp\left(i\frac{\zeta}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}p)\right)\psi(p).$
(5)
As $\beta\rightarrow 0$, the ordinary position wave function
$\psi(\zeta)=\langle\zeta|\psi\rangle$ is recovered.
### 2.2 Polymer quantum mechanics
The polymer representation of quantum mechanics consists in defining abstract
kets, labeled by a real number and assumed to form an orthonormal basis, and
then considering a suitable finite subset of them, whose Hilbert space is
defined by the corresponding inner product [8]. This procedure helps one gain
insight onto some particular features of quantum mechanics, when an underlying
discrete structure is somehow hypothesized. The request that the Hamiltonian
associated to the system be of direct physical interpretation defines the
polymer phase space, and the continuum limit can be recovered by the
introduction of the concept of the scale [9].
In the particular case of a discrete position variable in the momentum
polarization, the Hamiltonian variable $p$ cannot be implemented as an
operator, so that some restrictions on the model have to be required. If the
set of kets is restricted by the introduction of a regular graph
$\gamma_{\mu_{0}}$, the kynetic term of the Hamiltonian is approximated by the
polymer substitution
$p\rightarrow\frac{1}{\mu_{0}}\sin(\mu_{0}p),$ (6)
where the incremental ratio is evluated for an exponentiated operator. The
Hamiltonian operator $H_{\mu_{0}}$, which lives in
$\mathcal{H}_{\gamma_{\mu_{0}}}$, reads
$H_{\mu_{0}}=\frac{\hat{p}_{\mu_{0}}^{2}}{2m}+V(\hat{q}).$ (7)
The definition of a scale, $C_{n}$, eables one to approximate continuous
functions with functions that are constant on the intervals. As a result, at
any given scale $C_{n}$, the kinetic term of the Hamiltonian operator can be
approximated, and effective theories at given scales are related by coarse-
graining maps.
## 3 Deformed classical dynamics
The ordinary Taub model can be interpreted as a massless scalar relativisitic
particle moving in the Lorentzian minisuperspace ($\tau,x$)-plane, whose the
classical trajectory is its light-cone. More precisely, the incoming particle
($\tau<0$) bounces on the wall ($x=x_{0}=\ln(1/2)$) and falls into the
classical cosmological singularity ($\tau\rightarrow\infty$). Investigating
the modification of the dynamics within the two frameworks will show that the
two behaviors can be interpreted as complementary.
### 3.1 GUP framework
The GUP-classical dynamics is contained in the modified symplectic geometry
arising from the classical limit of (3), as soon as the parameter $\beta$ is
regarded as an independent constant with respect $\hbar$. It is then possible
to replace the quantum-mechanical commutator (3) via its Poisson brackets,
i.e. $-i[q,p]\Longrightarrow\\{q,p\\}=(1+\beta p^{2})$. The Poisson brackets
for any two-dimensional phase space function are
$\\{F,G\\}=\left(\frac{\partial F}{\partial q}\frac{\partial G}{\partial
p}-\frac{\partial F}{\partial p}\frac{\partial G}{\partial q}\right)(1+\beta
p^{2}).$ (8)
Applying this scheme to the Hamiltonian (2), we immediately obtain the
equations of motion for the model [1],
$x(\tau)=(1+\beta A^{2})\tau+cost,\qquad p(\tau)=cost=A,$ (9)
where $x\in[x_{0},\infty)$. Therefore, at the classical level, the effects of
the deformed Heisenberg algebra (3) on the Taub Universe are as follows. The
angular coefficient is $(1+\beta A^{2})>1$ for $\beta\neq 0$, and thus the
angle between the two straight lines $x(\tau)$, for $\tau<0$ and $\tau>0$,
becomes smaller as the values of $\beta$ grows. The trajectories of the
particle (Universe), before and after the bounce on the potential wall at
$x=x_{0}\equiv\ln(1/2)$, are closer to each other then in the canonical case
($\beta=0$).
### 3.2 Polymer framework
The polymer-classical dynamics relies on the substitution (6) in the
Hamiltonian of the model (2). This way, the equations of motion rewrite [2]
$\dot{x}=\left\\{x,H\right\\}=\cos(ap),\qquad\dot{p}=\left\\{p,H\right\\}=0,$
(10)
where a dot denotes differentiation with respect to the time variable $\tau$.
The equations of motion are immediately solved as
$x(\tau)=\cos(ap)\tau,\qquad p(\tau)=A,$ (11)
where $A$ is a constant. In the discretized (polymer) case, i.e. for $a\neq
0$, the one-parameter family of trajectories flattens. In fact, the angle
between the incoming trajectory and the outgoing one is greater than $\pi/2$,
since $p\in\left(-\pi/a,\pi/a\right)$. As these trajectories diverge rather
than converge, we expect the polymer quantum effects to be reduced with
respect to the classical case, as we will verify below.
## 4 Deformed quantum dynamics
The quantum dynamics of the Taub Universe is here investigated according to
the two different approaches. Particular attention is paid to the wave-packet
evolution and the consequential fate of the classical cosmological
singularity. In both frameworks, the variable $\tau$ is regarded as a time
coordinate and therefore ($\tau,p_{\tau}$) are treated in the canonical way.
The deformed quantization (GUP or polymer) is then implemented only to the
submanifold describing the only degree of freedom of the Universe, i.e. the
phase space spanned by ($x,p$). We then deal with a Schrödinger-like equation
$i\partial_{\tau}\Psi(\tau,p)=\hat{H}_{ADM}^{T}\Psi(\tau,p),$ (12)
where the operator $\hat{H}_{ADM}^{T}$ accounts for the modifications due to
the two frameworks. We have to square the eigenvalue problem in order to
correctly impose the boundary condition: in agreement with the analysis
developed in [10], we make the well-grounded hypothesis that the
eigenfunctions form be independent of the presence of the square root, since
its removal implies the square of the eigenvalues only. The wave packets,
which are superposition of the eigenfunctions
$\Psi(\tau,x)=\int_{0}^{\infty}dkA(k)\psi_{k}(x)e^{-ik\tau}$, are then
constructed for both models, taking $A(k)$ as a Gaussian-like weighting
function. The differences between the two approaches are due to the features
of the eigenfunctions $\psi_{k}(x)$. Analyzing such an evolution, we show that
the GUP Taub Universe appears to be probabilistically singularity-free,
differently from the polymer case, where the singularity is not tamed by the
cut-off-scale effects.
### 4.1 GUP framework
We now analyze the model in the GUP approach [1]. As explained before, we lost
all informations on the position itself, so that the boundary conditions have
to be imposed on the quasiposition wave function (5), i.e. $\psi(\zeta_{0})=0$
(where $\zeta_{0}=\langle\psi^{ml}_{\zeta}|x_{0}|\psi^{ml}_{\zeta}\rangle$, in
agreement with the previous discussion). The solution of the eigenvalue
problem is the Dirac $\delta$-distribution $\psi_{k}(p)=\delta(p^{2}-k^{2})$,
and therefore the quasiposition wave function (5) reads
$\psi_{k}(\zeta)=\frac{A}{k(1+\beta
k^{2})^{3/2}}\left[\exp\left(i\frac{\zeta}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}k)\right)-\exp\left(i\frac{(2\zeta_{0}-\zeta)}{\sqrt{\beta}}\tan^{-1}(\sqrt{\beta}k)\right)\right],$
(13)
where $A$ is a constant and the boundary condition $\psi(\zeta_{0})=0$ has
been imposed. The deformation parameter $\beta$, i.e. the presence of a non-
zero minimal uncertainty for the configuration variable, is responsible for
the GUP effects on the dynamics. The physical interpretation of $\beta$ is
then a non-zero minimal uncertainty in the anisotropy of the Universe. To
better understand the modifications induced by the deformed Heisenberg algebra
on the canonical Universe dynamics, we have to analyze different
$\beta$-regions. In fact, when $\beta$ becomes more and more important, i.e.
when we are at some scale that allows us to appreciate the GUP effects, the
evolution of the wave packets is different from the canonical case. More
precisely, these effects are present when the product $k_{0}\sqrt{\beta}$
becomes remarkable, i.e. when $k_{0}\sqrt{\beta}\sim\mathcal{O}(1)$, and
therefore when $\beta$ is comparable to $1/k_{0}^{2}$. In fact, the correct
semiclassical behaviors of the model far away from the singularity is
described by wave packets peaked at energies much smaller then
$1/\sqrt{\beta}$ [1]. In particular, for $k_{0}=1$, we can distinguish between
three different $\beta$-regimes:
* •
$\beta\sim\mathcal{O}(10^{-2})$ regime. The wave packets begin to spread and a
constructive and destructive interference between the incoming and outgoing
wave appears. The probability amplitude to find the Universe is still peaked
around the classical trajectory.
* •
$\beta\sim\mathcal{O}(10^{-1})$ regime. It is no more possible to distinguish
an incoming or outgoing wave packet and, at this level, the notion of a wave
packet following a classical trajectory becomes meaningless.
* •
$\beta\sim\mathcal{O}(1)$ regime. A dominant probability peak “near” the
potential wall appears. There are also other small peaks for growing values of
$\zeta$, but they are widely suppressed for bigger $\beta$. The motion of wave
packets shows a stationary behavior, i.e. these are independent of $\tau$. See
Fig. 2.
Following this picture we are able to learn the GUP modifications to the WDW
wave packets evolution. In fact, from small to big values of $\beta$, we can
see how the wave packets escape from the classical trajectories and approach a
stationary state close to the potential wall.
Figure 2: Wave packets $|\Psi(\tau,\zeta)|$ in the GUP framework as $\beta
k_{0}^{2}=1$ ($k_{0}=1$ and $\sigma=4$).
Such a behavior is, in some sense, expected from a classical point of view. In
fact, at classical level the ingoing and the outgoing trajectories shrink each
other. So a quantum probability interference is a fortiori predicted. On the
other hand, the stationarity feature exhibited by the Universe in the
($\beta\sim\mathcal{O}(1)$)-region is a purely quantum GUP effect. Such a
behavior cannot be inferred from a deformed classical analysis. From this
point of view, the classical singularity ($\tau\rightarrow\infty$) is widely
probabilistically suppressed, because the probability to find the Universe is
peaked just around the potential wall. This way we claim that the GUP-Taub
Universe is singularity-free.
### 4.2 Polymer framework
We now analyze the model in the polymer approach [2]. For the quantum analysis
of the model, we choose a discretized $x$ space, and solve the corresponding
eigenvalue problem in the $p$ polarization. Considering the time evolution for
the wave function $\Psi$ as given by $\Psi_{k}(p,\tau)=e^{-ik\tau}\psi_{k}(p)$
and the results of [10], we obtain the following eigenvalue problem
$(p^{2}-k^{2})\psi_{k}(p)=\left[\frac{2}{a^{2}}\left(1-\cos(ap)\right)-k^{2}\right]\psi_{k}(p),$
(14)
solved by
$\displaystyle k^{2}=k^{2}(a)=\frac{2}{a^{2}}\left(1-\cos(ap)\right)\leq
k^{2}_{max}=\frac{4}{a^{2}}$ (15a)
$\displaystyle\psi_{k,a}(p)=A\delta(p-p_{k,a})+B\delta(p+p_{k,a})$ (15b)
$\displaystyle\psi_{k,a}(x)=A\left[\exp(ip_{k,a}x)-\exp(ip_{k,a}(2x_{0}-x))\right]:$
(15c)
(15b) is the momentum wave function, with $A$ and $B$ two arbitrary
integration constant, and (15c) is the coordinate wave function, where an
integration constant has been eliminated by imposing suitable boundary
conditions. Moreover, we have defined the modified dispersion relation
$p_{k,a}\equiv\frac{1}{a}\arccos\left(1-\frac{k^{2}a^{2}}{2}\right)$ (16)
from (15a). Furthermore, we stress that $k^{2}$ is bounded from above, as
illustrated in (15a), but it is its square root, considered for its positive
determination, which accounts for the time evolution of the wave function.
We now construct suitable wave packets $\Psi(x,\tau)$ taking into account the
previous discussion (note that a maximum energy $k_{max}$ is now predicted).
Three relevant cases can be distinguished:
Figure 3: The spread polymer wave packet $|\Psi(x,\tau)|$ as $k_{0}a=1/2$
($a=50$, $k_{0}=0.01$, $\sigma=0.125$).
* •
$k_{0}a\sim\mathcal{O}(1)$ and peaked weighting function. The resulting wave
packet is well approximated by a purely monochromatic wave. A small
interference phenomenon between the wave and the wall is then predicted.
* •
$k_{0}a\sim\mathcal{O}(1)$ and spread weighting function. A strong
interference phenomenon between the wave and the wall now appears.
Nevertheless, this interference phenomenon is not able to probabilistically
tame the singularity, as it takes place in the ’outer’ region, in a way
complementary to that of the GUP approach (see Fig. 3). The polymer-Taub
Universe is then still a singular cosmological model.
* •
$k_{0}a\ll\mathcal{O}(1)$ regime. This can be considered as the semi-classical
limit of the model. In fact, differently from the other cases, the value of
$k_{0}$ around which the wave packet is peaked is not arbitrary, but
constrained by the characteristic scale $a$ under investigation. The ordinary
WDW behavior is therefore recasted.
## 5 Comparison with other approaches
The Taub cosmological model offers a suitable scenario, where different
quantization techniques can be applied. In fact, it is possible to single out
a time variable, so that the anisotropy describes the real degree of freedom
of the Universe. It is therefore reasonable to investigate the fate of the
cosmological singularity without modifying the time variable. The comparison
with analysis of the cosmological singularity in other cosmological models
outlines how the features of the Taub model allow one to pick the cut off
effects out of those due to the choice of the Hamiltonian variables.
In the cosmological isotropic sector of GR, i.e. the FRW models, the
singularity is removed by loop quantum effects. The wave function of the
universe exhibits a non-singular behavior at the classical singularity, and
the big-bang is replaced by a big-bounce, when a free scalar field is taken as
the relational time [11]. The scale factor of the universe is directly
quantized by the use of the polymer (loop) techniques, so that the evolution
itself of the wave packet of the universe is deeply modified by such an
approach. The Hamiltonian constraint does not allow for a constant solution of
the variable conjugated to the scale factor, so that it is not possible to
choose a scale, such that the polymer modifications are negligible throughout
the whole evolution, so that the comparison with the ordinary representation
is not always possible. Anyhow, we stress that, for the Taub model, the
cosmological singularity is probabilistically suppressed, regardless to the
fact whether the system can appreciated or not the cut off during the whole
evolution.
In [12], all the degrees of freedom of the Bianchi cosmological models in the
ADM reduction of the dynamics are quantized by loop techniques. In particular,
also the time variable, i.e. the Universe volume, is treated at the same level
as the others. In most cases, the time variable is defined by a phase space
variable, i.e. it is an internal one. As a result, also the Bianchi Universes
are singularity-free [13]. In this respect, our analysis is based on
considering the time variable as an ordinary Heisenberg variable, while the
cut off is imposed on the anisotropy only.
The GUP dynamics of other cosmological models has been investigated in
different approaches. In particular, the big-bang singularity appears to be
tamed by GUP effects showing a stationary behavior of the wave packets [14].
Such a prediction is in agreement with those achieved in a noncommutative
quantum cosmology [15]. However, in order to predict a big bounce à la LQC, a
Snyder-deformed quantum cosmology has to be addressed [16]. As the last point,
it is interesting to notice that the GUP-Mixmaster Universe is still a chaotic
model [17], as opposite to the LQC one [18], the difference being essentially
based on the application of the deformed scheme to the time variable too.
Acknowledgments. The “Angelo Della Riccia” Fellowship and the “Sapienza CUN 2”
Fellowship are gratefully acknowledged.
## References
* [1] M.V.Battisti and G.Montani, Phys.Rev.D 77 (2008) 023518.
* [2] M.V.Battisti, O.M.Lecian and G.Montani, Phys.Rev.D 78 (2008) 103514.
* [3] G.Montani, M.V.Battisti, R.Benini and G.Imponente, Int.J.Mod.Phys.A 23 (2008) 2353; J.M.Heinzle and C.Uggla, arXiv:0901.0776.
* [4] K.Kuchar and M.P.Ryan, Phys.Rev.D 40 (1989) 3982.
* [5] G.Catren and R.Ferraro, Phys.Rev.D 63 (2001) 023502.
* [6] A.Kempf, G.Mangano and R.B.Mann, Phys.Rev.D 52 (1995) 1108; A.Kempf, J.Math.Phys. 38 (1997) 1347.
* [7] D.J.Gross and P.F.Mendle, Nucl.Phys.B 303 (1988) 407; D.Amati, M.Ciafaloni and G.Veneziano, Phys.Lett.B 216 (1989) 41.
* [8] A.Ashtekar, S.Fairhurst and J.L.Willis, Class.Quant.Grav. 20 (2003) 1031; A.Corichi, T.Vukasinac and J.A.Zapata, Phys.Rev.D 76 (2007) 0440163.
* [9] A.Corichi, T.Vukasinac and J.A.Zapata, Class.Quant.Grav. 24 (2007) 1495.
* [10] R.Puzio, Class.Quant.Grav. 11 (1994) 609.
* [11] A.Ashtekar, T.Pawlowski and P.Singh, Phys.Rev.Lett. 96 (2006) 141301; Phys.Rev.D 73 (2006) 124038.
* [12] M.Bojowald, G.Date and K.Vandersloot, Class.Quant.Grav. 21 (2004) 1253; M.Bojowald, Class.Quant.Grav. 20 (2003) 2595.
* [13] D.W.Chiou, Phys.Rev.D 76 (2007) 124037; G.Date, Phys.Rev.D 71 (2005) 127502.
* [14] M.V.Battisti and G.Montani, Phys.Lett.B 656 (2007) 96.
* [15] B.Vakili and H.R.Sepangi, Phys.Lett.B 651 (2007) 79; H.R.Sepangi, B.Shakerin and B.Vakili, Class.Quant.Grav. 26 (2009) 065003.
* [16] M.V.Battisti, arXiv:0805.1178; J.Phys.Conf.Ser. (at press), arXiv:0810.5039.
* [17] M.V.Battisti and G.Montani, arXiv:0808.0831.
* [18] M.Bojowald and G.Date, Phys.Rev.Lett. 92 (2004) 071302.
|
arxiv-papers
| 2009-03-23T11:25:05 |
2024-09-04T02:49:01.357246
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Valerio Battisti, Orchidea Maria Lecian and Giovanni Montani",
"submitter": "Marco Valerio Battisti",
"url": "https://arxiv.org/abs/0903.3836"
}
|
0903.3837
|
11institutetext: † Centre de Physique Théorique de Luminy, Université de la
Méditerranée, F-13288 Marseille FR, and “Sapienza” Università di Roma,
Dipartimento di Fisica and ICRA,
P.le A. Moro 5, 00185 Rome IT, battisti@icra.it
‡ “Sapienza” Università di Roma, Dipartimento di Fisica and ICRA, P.le A. Moro
5, 00185 Rome IT, riccardo.belvedere@icra.it
§ ICRA, ICRANet, ENEA and “Sapienza” Università di Roma, Dipartimento di
Fisica, P.le A. Moro 5, 00185 Rome IT, montani@icra.it
# Semi-classical isotropization of the Mixmaster Universe
Marco Valerio Battisti† Riccardo Belvedere‡ Giovanni Montani§
###### Abstract
A semi-classical mechanism which leads to the isotropization of the Mixmaster
Universe is developed. A wave function of this Universe, which has a
meaningful probabilistic interpretation, is constructed and it describes the
evolution of the anisotropies of the Universe with respect to the isotropic
scale factor, which plays the external observer-like role. We show that, once
large volume regions are investigated, the closed Friedmann-Robertson-Walker
configuration is deeply privileged.
Quantum cosmology denotes the application of the quantum theory to the entire
Universe [1]. It can be then viewed as a natural arena to investigate as part
of a more general drive to understood quantum gravity. In canonical quantum
gravity, the quantum state $\Psi$ of the system is generally represented by a
wave functional describing the dynamics of the three metric $h_{ij}$ as well
as matter fields $\phi$, i.e. $\Psi=\Psi[h_{ij}(x),\phi(x)]$. This state is
defined on the the Wheeler superspace and, since of the diffeomorphisms
invariance of GR, it has no explicit dependence on time. In particular, it
(formally) satisfies the Dirac quantum implementation of the first-class
constraints of GR.
A complete quantum theory of gravity is not yet available and therefore this
problem is not properly defined. To overcame such a feature the fields are
usually restricted (by hand) to a finite dimensional subspace of the
superspace, i.e. we deal with the minisuperspace representation. Quantum
cosmology is explicitly defined as the minisuperspace quantization of
homogeneous (finite degrees of freedom) cosmological models. From the
peculiarity of the system-Universe, fundamental interpreting difficulties of
the wave function of the Universe $\Psi$ arise. The question about the
interpretation (i.e. extracting physical statements) of quantum cosmology
clearly appears as soon as the differences with respect to ordinary quantum
mechanics are addressed [2, 3]. The standard interpretation of quantum
mechanics (the Copenhagen one) involves the following basic assumptions. (i)
There exist an external observer to the quantum system, i.e. the model under
investigation is not genuinely closed. (ii) Predictions are probabilistic in
nature and performed by a measurement of an external agency. (iii) Time plays
a central and peculiar role. By contrast quantum cosmology is defined by the
following features. (i) The analyzed model is the Universe as whole, i.e. it
is closed without external observers. (ii) No external measurement crutch is
available and an internal one can not plays the observer-like role since of
the extreme conditions a very early Universe is subjected on. (iii) The time
coordinate is not an observable in GR and at quantum level it is known as the
problem of time.
The most developed idea to solve these features relies in accepting that a
meaningfully interpretation of the wave function of the Universe can be only
formulated at semi-classical level. More precisely, it is only possible to
quantum-mechanically interpret a small subsystem of the entire Universe, i.e.
in the domain where at least some of the minisuperspace variables are semi-
classical in the sense of the Wentzel-Kramers-Brillouin (WKB) approximation.
In this work such a scheme is implemented to the most general homogeneous
cosmological model, i.e. the Mixmaster Universe. The relevance of this model
relies on the fact that a generic solution of the Einstein equations toward
the cosmological singularity is formulated by a collection of causal
independent Bianchi IX horizons [4]. In particular, a wave function of the
Universe which has a clear probabilistic interpretation when the isotropic
scale factor $a$ of the Universe is regarded as semi-classical is obtained. It
describes the quantum evolution of the Mixmaster anisotropies and its dynamics
is traced with respect to $a$, which can be regarded as a semi-classical
variable as soon as the Universe expands enough. The main result is that the
wave function of the Universe is spread over all values of anisotropy near the
cosmological singularity but, when the radius of the Universe grows, it is
asymptotically peaked around the isotropic configuration. The closed
Friedmann-Robertson-Walker (FRW) cosmological model is thus the naturally
privileged state far enough from the classical singularity. A semi-classical
isotropization mechanism for the Mixmaster Universe is then predicted.
The dynamics of the homogeneous cosmological models (the Bianchi Universes) is
summarized, in the canonical formalism, by the scalar constraint (for reviews
see [5])
$\mathcal{H}=\kappa\left[-\frac{p_{a}^{2}}{a}+\frac{1}{a^{3}}\left(p_{+}^{2}+p_{-}^{2}\right)\right]+\frac{a}{4\kappa}V(\beta_{\pm})+U(a)=0,$
(1)
where the potential term $V(\beta_{\pm})$ accounts for the spatial curvature
of the model. Here $\kappa=8\pi G$, the variable $a=a(t)$ describes the
isotropic expansion of the Universe and its shape changes (anisotropies) are
associated to $\beta_{\pm}=\beta_{\pm}(t)$. The phase space of this model is
thus six dimensional and the cosmological singularity appears for
$a\rightarrow 0$. In the Universe dynamics we have assumed the matter terms to
be negligible with respect to the cosmological constant $\Lambda$, i.e. the
isotropic potential $U(a)$ reads $U(a)=-a/4\kappa+\Lambda a^{3}/\kappa$. As
matter of fact, far enough from the singularity, the cosmological constant
term dominates on the other ordinary matter fields and such a contribution is
necessary in order to the inflationary scenario takes place [6, 7].
As we said, a correct definition of probability in quantum cosmology can be
formulated by distinguishing between semi-classical and quantum variables [2].
More precisely, the variables which satisfy the Hamilton-Jacobi equation are
regarded as semi-classical and is assumed that the quantum variables do not
affect the dynamics generated by the semi-classical ones. In this respect we
claim that the quantum variables describe a small subsystem of the Universe
and is then natural to regard the isotropic expansion variable $a$ as the
semi-classical one while considering the anisotropy coordinates $\beta_{\pm}$
(the two physical degrees of freedom of the Universe) as the purely quantum
variables. We are thus requiring ab initio that the radius of the Universe is
of different nature with respect to the anisotropies. To implement such a
picture, the wave function of the Universe $\Psi=\Psi(a,\beta_{\pm})$ is
assumed to be [2]
$\Psi=\Psi_{0}\chi=A(a)e^{iS(a)}\chi(a,\beta_{\pm}).$ (2)
This wave function is WKB-like in the $a$ coordinate and the additional
function $\chi$ depends on the quantum variables $\beta_{\pm}$ and only
parametrically, in the sense of the Born-Oppenheimer approximation, on $a$.
The canonical quantization of this model is achieved by the use of the Dirac
prescription for quantizing constrained systems [8], i.e. imposing that the
physical states are those annihilated by the self-adjoint operator
$\hat{\mathcal{H}}$ corresponding to the classical counterpart (1).
Considering (2), we obtain from the quantum operator version of (1) the
Hamilton-Jacobi equation for $S$ and the continuity equation for the amplitude
$A$
$-\kappa
A\left(S^{\prime}\right)^{2}+aUA+\mathcal{V}_{q}=0,\qquad\frac{1}{A}\left(A^{2}S^{\prime}\right)^{\prime}=0,$
(3)
respectively. Here the prime denotes differentiation with respect to the scale
factor $a$ and $\mathcal{V}_{q}=\kappa A^{\prime\prime}$ is the so-called
quantum potential, which in this model is negligible far from the classical
singularity even if the $\hbar\rightarrow 0$ limit is not taken into account
(see below). As usual $S(a)$ defines a congruence of classical trajectories.
The new equation we find is a Schrödinger-like one describing the evolution of
the proper quantum state $\chi$. Neglecting higher order correction terms in
$\hbar$, it reads
$-2ia^{2}S^{\prime}\partial_{a}\chi=\hat{H}_{q}\chi,\qquad
H_{q}=p_{+}^{2}+p_{-}^{2}+\frac{a^{4}}{4\kappa^{2}}V(\beta_{\pm}).$ (4)
Such an equation is in agreement with the assumption that the anisotropies
describe a quantum subsystem of the whole Universe, i.e. that the wave
function $\chi$ depends only on $\beta_{\pm}$ (in the Born-Oppenheimer sense).
As matter of fact, the smallness of such a quantum subsystem can be formulated
requiring that its Hamiltonian $H_{q}$ is of order
$\mathcal{O}(\epsilon^{-1})$, where $\epsilon$ is a small parameter
proportional to $\hbar$. Since the action of the semi-classical Hamiltonian
operator $\hat{H}_{0}=a^{2}\partial_{a}^{2}+a^{3}U/\kappa$ on the wave
function $\Psi$ is of order $\mathcal{O}(\epsilon^{-2})$, the idea that the
anisotropies do not influence the isotropic expansion of the Universe can be
formulated as $\hat{H}_{q}\Psi/\hat{H}_{0}\Psi=\mathcal{O}(\epsilon)$. Such a
requirement is physically reasonable since, the semi-classical proprieties of
the Universe as well as the smallness of the quantum subsystem, are both
related to the fact that the Universe is large enough [2].
A purely Schrödinger equation for the wave function $\chi$ is obtained taking
into account the tangent vector to the classical path. Using
$p_{a}=S^{\prime}$, the equations of motion (3) and considering the time gauge
$da/dt=1$, is possible to define the new time variable $\tau$ such that
$d\tau=(N\kappa/a^{3})da$. In the asymptotic interesting region ($a\gg
l_{\Lambda}\equiv 1/\sqrt{\Lambda}$) the evolution equation (4) rewrites as
$i\partial_{\tau}\chi=\left(-\Delta_{\beta}+\frac{a^{4}}{4\kappa^{2}}V(\beta_{\pm})\right)\chi,$
(5)
where $\tau=(\kappa/12\sqrt{\Lambda})a^{-3}+\mathcal{O}(a^{-5})$. This is the
Schrödinger equation for the wave function $\chi$ describing the quantum
variables $\beta_{\pm}$. The wave function (2) defines a probability
distribution $\rho(a,\beta_{\pm})$ which appears to be
$\rho(a,\beta_{\pm})=\rho_{0}(a)\rho_{\chi}(a,\beta_{\pm})$, where
$\rho_{0}(a)$ is the classical probability distribution for the semi-classical
variable $a$ and $\rho_{\chi}=|\chi|^{2}$ denotes the probability distribution
for the quantum variables $\beta_{\pm}$ on the classical trajectories (3)
where the wave function $\chi$ can be normalized.
In order to enforce the idea that the anisotropies can be considered as the
only quantum degrees of freedom of the Universe, we consider the quasi-
isotropic regime $|\beta_{\pm}|\ll 1$. Moreover, since we are interested at
the link between the isotropic and anisotropic dynamics, the Universe has to
be get through to such a quasi-isotropic era. In this regime, the potential
term reads
$V(\beta_{\pm})=8(\beta_{+}^{2}+\beta_{-}^{2})+\mathcal{O}(\beta^{3})$ and the
Schrödinger equation (5) can be then written as
$i\partial_{\tau}\chi=\frac{1}{2}\left(-\Delta_{\beta}+\omega^{2}(\tau)(\beta_{+}^{2}+\beta_{-}^{2})\right)\chi,$
(6)
where $\omega^{2}(\tau)=C/\tau^{4/3}$ and $C$ being a constant given by
$2C=1/6^{4/3}(\kappa\Lambda)^{2/3}$. In other words, we are dealing with a
time-dependent bi-dimensional harmonic oscillator with frequency
$\omega(\tau)$. The quantum theory of an harmonic oscillator with time-
dependent frequency is well known [9] and the solution of the Schrödinger
equation (6) can be analytically obtained. Through the introduction of the
generalized invariant state, whose eigenstates are connected with those of a
time-independent harmonic oscillator, and via an unitary transformation, the
wave function $\chi_{n}=\chi_{+}\chi_{-}$ reads
$\chi_{\pm}=\chi_{n}(\beta_{\pm},\tau)=A\frac{e^{i\alpha_{n}(\tau)}}{\sqrt{\rho}}h_{n}(\beta_{\pm}/\rho)\exp\left[\frac{i}{2}\left(\dot{\rho}\rho^{-1}+i\rho^{-2}\right)\beta_{\pm}^{2}\right].$
(7)
In this formula $A$ denotes the normalization constant, $h_{n}$ are the usual
Hermite polynomial of order $n$ and $\rho(\tau)$ and the phase $\alpha(\tau)$
are respectively given by
$\rho=\sqrt{\frac{\tau}{\sqrt{C}}\left(1+\frac{\tau^{-2/3}}{9C}\right)},\qquad\alpha_{n}=-\left(n+\frac{1}{2}\right)\int\frac{d\tau}{\rho^{2}(\tau)}.$
(8)
It is immediate to verify that, when $\omega(\tau)\rightarrow\omega_{0}$ and
$\rho(\tau)\rightarrow\rho_{0}=1/\sqrt{\omega_{0}}$ (namely
$\alpha(\tau)\rightarrow-\omega_{0}(n+1/2)\tau$), the solution of a time-
independent harmonic oscillator is recovered.
Let us now investigate the probability density to find the quantum subsystem
of the Universe at a given state. The anisotropies appear to be
probabilistically suppressed as soon as the Universe expands enough far from
the cosmological singularity (which we remember appears for $a\rightarrow 0$
or $\tau\rightarrow\infty$). Such a feature can be immediately observed from
the behavior of the squared modulus of the wave function (7) which is given by
$|\chi_{n}|^{2}\sim\frac{1}{\rho^{2}}|h_{n_{+}}(\beta_{+}/\rho)|^{2}|h_{n_{-}}(\beta_{-}/\rho)|^{2}e^{-\beta^{2}/\rho^{2}},$
(9)
where $\beta^{2}=\beta^{2}_{+}+\beta^{2}_{-}$ and with $\sim$ we omit the
normalization constant. This probability density is still time-dependent
through $\rho=\rho(\tau)$ since the evolution of the wave function $\chi$ is
not traced by an unitary time operator. As we can see from (9), when a large
enough isotropic cosmological region is considered (namely when the limit
$a\rightarrow\infty$ or $\tau\rightarrow 0$ is taken into account), the
probability density to find the Universe is sharply peaked at the isotropic
configuration, i.e. for $|\beta_{\pm}|\simeq 0$. In this limit (which
corresponds to $\rho\rightarrow 0$) the probability density $|\chi_{n=0}|^{2}$
of the ground state ($n=n_{+}+n_{-}=0$) is given by
$|\chi_{n=0}|^{2}\stackrel{{\scriptstyle\tau\rightarrow
0}}{{\longrightarrow}}\delta(\beta,0)$, thus is proportional to the Dirac
$\delta$-distribution centered on $\beta=0$ (see Fig. 1).
Figure 1: The ground state of the wave function $\chi(\beta_{\pm},\tau)$ far
from the cosmological singularity, i.e. in the $\tau\rightarrow 0$ limit. In
the plot we take $C=1$.
Summarizing, when the Universe moves away from the cosmological singularity,
the probability density to find it is asymptotically peaked around the closed
FRW configuration. Near the initial singularity all values of anisotropy
$\beta$ are almost equally favored from a probabilistic point of view. On the
other hand, as the radius of the Universe grows, the isotropic state become
the most probable one. (For other similar approaches see [10].)
It is worth noting that the key feature of such a result relies on the fact
that the isotropic scalar factor $a$ was considered as an intrensically
different variable with respect to the anisotropies. It was treated as a semi-
classical variable while only the two physical degrees of freedom of the
Universe ($\beta_{\pm}$) were described as real quantum coordinates. This way,
a positive semidefinite probability density for the wave function of the
quantum subsystem of the Universe can be constructed and a clear
interpretation of it considered. The validity of such an assumption can be
verified from the analysis of the Hamilton-Jacobi equations (3). In
particular, the WKB function $\Psi_{0}=\exp(iS+\ln A)$ approaches the quasi-
classical limit $e^{iS}$ as soon as $a\gg l_{\Lambda}$ ($l_{\Lambda}$ being
the inflation characteristic length [7]). To corroborate the model, we have
studied the classical limit too. Splitting the $S$-function in two terms,
respectively for the time variable $a$ and for the anisotropies $\beta_{\pm}$,
we achieve, if $a\gg l_{\Lambda}$, an analogous behavior of the anisotropies,
i.e. them go to reduce themselves once one moves away from the cosmological
singularity.
Acknowledgments. M. V. B. thanks ”Fondazione Angelo Della Riccia” for
financial support.
## References
* [1] D.L.Wiltshire, gr-qc/0101003; C.Kiefer and B.Sandhoefer, arXiv:0804.0672.
* [2] A.Vilenkin, Phys.Rev.D 39 (1989) 1116; Phys.Rev.D 33 (1986) 3560.
* [3] J.Halliwell and S.Hawking, Phys.Rev.D 31 (1985) 1777; J.J.Halliwell, gr-qc/9208001; F.Embacher, gr-qc/9605019.
* [4] V.A.Belinski, I.M.Khalatnikov and E.M.Lifshitz, Adv.Phys. 31 (1982) 639.
* [5] G. Montani, M. V. Battisti, R. Benini and G. Imponente, Int.J.Mod.Phys.A 23 (2008) 2353; J. M. Heinzle and C. Uggla, arXiv:0901.0776.
* [6] S.Coleman and E.Weinberg, Phys.Rev.D 7 (1973) 1888; A.A.Kirillov and G.Montani, Phys.Rev.D 66 (2002) 064010.
* [7] E.W.Kolb and M.S.Turner, The Early Universe, (Adison-Wesley Reading, 1990).
* [8] M.Henneaux and C.Teitelboim, Quantization of Gauge Systems (PUP, Princeton, 1992).
* [9] H.R.Lewis, J.Math.Phys. 9 (1968) 1976; H.R.Lewis and W.B.Riesenfeld, it J.Math.Phys. 10 (1969) 1458.
* [10] V.Moncrief and M.P.Ryan, Phys.Rev.D 44 (1991) 2375; N.Pinto-Neto, A.F.Velasco and R.Colistete, Phys.Lett.A 277 (2000) 194; W.A.Wright and I.G.Moss, Phys.Lett.B 154 (1985) 115.
|
arxiv-papers
| 2009-03-23T11:31:48 |
2024-09-04T02:49:01.363890
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Valerio Battisti, Riccardo Belvedere and Giovanni Montani",
"submitter": "Marco Valerio Battisti",
"url": "https://arxiv.org/abs/0903.3837"
}
|
0903.3852
|
# Accelerating numerical solution of Stochastic Differential Equations with
CUDA
M. Januszewski, M. Kostur Institute of Physics, University of Silesia, 40-007
Katowice, Poland
###### Abstract
Numerical integration of stochastic differential equations is commonly used in
many branches of science. In this paper we present how to accelerate this kind
of numerical calculations with popular NVIDIA Graphics Processing Units using
the CUDA programming environment. We address general aspects of numerical
programming on stream processors and illustrate them by two examples: the
noisy phase dynamics in a Josephson junction and the noisy Kuramoto model. In
presented cases the measured speedup can be as high as $675\times$ compared to
a typical CPU, which corresponds to several billion integration steps per
second. This means that calculations which took weeks can now be completed in
less than one hour. This brings stochastic simulation to a completely new
level, opening for research a whole new range of problems which can now be
solved interactively.
###### keywords:
Josephson junction, Kuramoto, graphics processing unit, advanced computer
architecture, numerical integration, diffusion, stochastic differential
equation, CUDA, Tesla, NVIDIA
## 1 Introduction
The numerical integration of stochastic differential equations (SDEs) is a
valuable tool for analysis of a vast diversity of problems in physics, ranging
from equilibrium transport in molecular motors [1], phase dynamics in
Josephson junctions [2, 3], stochastic resonance [4] to dissipative particle
dynamics [5] to finance [6]. Stochastic simulation, as it is referred to as,
is specially interesting when the dimensionality of the problem is larger than
three, and in that case it is often the only effective numerical method. A
prominent example of this is the stochastic variation of molecular dynamics:
Brownian dynamics.
Direct stochastic simulations require a significant computational effort, and
therefore merely a decade ago have been used mostly as validation tools. The
precise numerical results in theory of low-dimensional stochastic problems
were coming from solutions of the corresponding Fokker-Planck equations. Many
different sophisticated, but often complicated, tools have been applied:
spectral methods [7, 8, 9], finite element methods [10] and numerical path
integrals [11, 12].
Stochastic simulation gained acceptance due to its straightforward
implementation and generic robustness with respect to different sorts of
problems. The continuous increase of the efficiency of available computer
hardware has been acting in favour of stochastic simulation, making it
increasingly more popular. The recent evolution of computer architectures
towards multiprocessor and multicore platforms also resulted in improved
performance of stochastic simulation. Let us note that in the case of a low-
dimensional system, stochastic simulation often uses ensemble averaging to
obtain the values of observables, which in turn is an example of a so-called
,,embarrassingly parallel problem” and it can, though with embarrassment,
directly benefit from a parallel architecture. In other cases, mostly where a
large number of interacting subsystems are investigated, the implementation of
the problem on a parallel architecture is less trivial, but still possible.
The recent emergence of techniques collectively known as general-purpose
computing on graphics processing units (GPUs) has caused a breakthrough in
computational science. The current state of the art GPUs are now capable of
performing computations at a rate of about 1 TFLOPS per single silicon chip.
It must be stressed that 1 TFLOPS is a performance level which only in 1996
was achievable exclusively by huge and expensive supercomputers such as the
ASCI Red Supercomputer (which had a peak performance of $1.8$ TFLOPS [13]) The
numerical simulations of SDEs can easily benefit from the parallel GPU
architecture. This however requires careful redesign of the employed
algorithms and in general cannot be done automatically. In this paper we
present a practical introduction to solving SDEs on NVIDIA GPUs using Compute
Unified Device Architecture (CUDA) [14] based on two examples: the model of
phase diffusion in a Josephson junction and the Kuramoto model of coupled
phase oscillators.
The paper is organized as follows: first, we briefly introduce the features
and capabilities of the NVIDIA CUDA environment and describe the two physical
models, then we present the implementation of stochastic algorithms and
compare their efficiency with a corresponding pure-CPU implementation executed
on an Intel Core2 Duo E6750 processor. We also provide the source code [15] of
three small example programs: PROG1, PROG2, and PROG3, which demonstrate the
techniques described in the paper. They can easily be extended to a broad
range of problems involving stochastic differential equations.
## 2 The CUDA environment
Figure 1: A schematic view of a CUDA streaming multiprocessor with 8 scalar
processor cores.
CUDA (Compute Unified Device Architecture) is the name of a general purpose
parallel computing architecture of modern NVIDIA GPUs. The name _CUDA_ is
commonly used in a wider context to refer to not only the hardware
architecture of the GPU, but also to the software components used to program
that hardware. In this sense, the CUDA environment also includes the NVIDIA
CUDA compiler and the system drivers and libraries for the graphics adapter.
From the hardware standpoint, CUDA is implemented by organizing the GPU around
the concept of a streaming multiprocessor (SM). A modern NVIDIA GPU contains
tens of multiprocessors. A multiprocessor consists of 8 scalar processors
(SPs), each capable of executing an independent thread (see Fig. 1). The
multiprocessors have four types of on-chip memory:
* 1.
a set of 32-bit registers (local, one set per scalar processor)
* 2.
a limited amount of shared memory (16 kB for devices having Compute Capability
1.3 or lower, shared between all SPs in a MP)
* 3.
a constant cache (shared between SPs, read-only)
* 4.
a texture cache (shared between SPs, read-only)
The amount of on-chip memory is very limited in comparison to the total global
memory available on a graphics device (a few kilobytes vs hundreds of
megabytes). Its advantage lies in the access time, which is two orders of
magnitude lower than the global memory access time.
The CUDA programming model is based upon the concept of a _kernel_. A kernel
is a function that is executed multiple times in parallel, each instance
running in a separate thread. The threads are organized into one-, two- or
three-dimensional blocks, which in turn are organized into one- or two-
dimensional grids. The blocks are completely independent of each other and can
be executed in any order. Threads within a block however are guaranteed to be
run on a single multiprocessor. This makes it possible for them to synchronize
and share information efficiently using the on-chip memory of the SM.
In a device having Compute Capability 1.2 or higher, each multiprocessor is
capable of concurrently executing 1024 active threads [16]. In practice, the
number of concurrent threads per SM is also limited by the amount of shared
memory and it thus often does not reach the maximum allowed value.
The CUDA environment also includes a software stack. For CUDA v2.1, it
consists of a hardware driver, system libraries implementing the CUDA API, a
CUDA C compiler and two higher level mathematical libraries (CUBLAS and
CUFFT). CUDA C is a simple extension of the C programming language, which
includes several new keywords and expressions that make it possible to
distinguish between host (i.e. CPU) and GPU functions and data.
## 3 Specific models
In this work, we study the numerical solution of stochastic differential
equations modeling the dynamics of Brownian particles. The two models we
concentrate upon are of particular interest in many disciplines and illustrate
the flexibility of the employed methods of solution.
The first model describes a single Brownian particle moving in a symmetric
periodic potential $V(x)=\sin(2\pi x)$ under the influence of a constant bias
force $f$ and a periodic unbiased driving with amplitude $a$ and frequency
$\omega$:
$\ddot{x}+\gamma\dot{x}=-V^{\prime}(x)+a\cos(\omega t)+f+\sqrt{2\gamma
k_{B}T}\xi(t)$ (1)
where $\gamma$ is the friction coefficient and $\xi(t)$ is a zero-mean
Gaussian white noise with the auto-correlation function
$\langle\xi(t)\xi(s)\rangle=\delta(t-s)$ and noise intensity $k_{B}T$.
Equation 1 is known as the Stewart-McCumber model [3] describing phase
differences across a Josephson junction. It can also model a rotating dipole
in an external field, a superionic conductor or a charge density wave. It is
particularly interesting since it exhibits a wide range of behaviors,
including chaotic, periodic and quasi-periodic motion, as well as the recently
detected phenomenon of absolute negative mobility [17, 18].
The second model we analyze is that of $N$ globally interacting overdamped
Brownian particles, with the dynamics of the $i$-th particle described by:
$\displaystyle\gamma\dot{x_{i}}=\omega_{i}+\sum_{j=1}^{N}K_{ij}\sin(x_{j}-x_{i})+$
$\displaystyle\sqrt{2\gamma k_{B}T}\xi_{i}(t),i=1,\ldots,N$ (2)
This model is known as the Kuramoto model [19] and is used as a simple
paradigm for synchronization phenomena. It has found applications in many
areas of science, including neural networks, Josephson junction and laser
arrays, charge density waves and chemical oscillators.
## 4 Numerical solution of SDEs
Most stochastic differential equations of practical interest cannot be solved
analytically, and thus direct numerical methods have to be used to obtain the
solutions. Similarly as in the case of ordinary differential equations, there
is an abundance of methods and algorithms for solving stochastic differential
equations. Their detailed description can be found in references: [20, 21, 22,
23, 24, 25].
Here, we present the implementation of a standard stochastic algorithm on the
CUDA architecture in three distinctive cases:
1. 1.
Multiple realizations of a system are simulated, and an ensemble average is
performed to calculate quantities of interest. The large degree of parallelism
inherent in the problem makes it possible to fully exploit the computational
power of CUDA devices with tens of multiprocessors capable of executing
hundreds of threads simultaneously. The example system models the stochastic
phase dynamics in a Josephson junction and is implemented in program PROG1
(the source code is available in [15]).
2. 2.
The system consists of $N$ globally interacting particles. In each time step
$N^{2}$ interaction terms are calculated. The example algorithm is named PROG2
and solves the Kuramoto model (Eq. 3.)
3. 3.
The system consists of $N$ globally interacting particles as in the previous
case but the interaction can be expressed in terms of a parallel reduction
operation, which is much more efficient than PROG2. The example algorithm in
PROG3 also solves the Kuramoto model (Eq. 3.)
We will now outline the general patterns used in the solutions of all models.
We start with the model of a single Brownian particle, which will form a basis
upon which the solution of the more general model of $N$ globally interacting
particles will be based.
### 4.1 Ensemble of non-interacting stochastic systems
Algorithm 1 A CUDA kernel to advance a Brownian particle by $m\cdot\Delta t$
in time.
1: local $i\leftarrow blockIdx.x\cdot blockDim.x+threadIdx.x$
2: load $x_{i}$, $v_{i}$ and system parameters $\\{par_{ji}\\}$ from global
memory and store them in local variables
3: load the RNG seed $seed_{i}$ and store it in a local variable
4: for $s=1$ to $m$ do
5: generate two uniform variates $n_{1}$ and $n_{2}$
6: transform $n_{1}$ and $n_{2}$ into two Gaussian variates
7: advance $x_{i}$ and $v_{i}$ by $\Delta t$ using the SRK2 algorithm
8: local $t\leftarrow t_{0}+s\cdot\Delta t$
9: end for
10: save $x_{i}$, $v_{i}$ and $seed_{i}$ back to global memory
Algorithm 2 The Stochastic Runge-Kutta algorithm of the 2nd order (SRK2) to
integrate $\dot{x}=f(x)+\xi(t)$, $\langle\xi(t)\rangle=0$,
$\langle\xi(t)\xi(s)\rangle=2D\delta(t-s)$.
1: $F_{1}\leftarrow f(x_{0})$
2: $F_{2}\leftarrow f(x_{0}+\Delta tF_{1}+\sqrt{2D\Delta t}\psi$) {with
$\langle\psi\rangle=0$, $\langle\psi^{2}\rangle=1$}
3: $x(\Delta t)\leftarrow x_{0}+\frac{1}{2}\Delta t(F_{1}+F_{2})\sqrt{2D\Delta
t}\psi$
For the Josephson junction model described by Eq. 1 we use a single CUDA
kernel, which is responsible for advancing the system by a predefined number
of timesteps of size $\Delta t$.
We employ fine-grained parallelism – each path is calculated in a separate
thread. For CUDA devices, it makes sense to keep the number of threads as
large as possible. This enables the CUDA scheduler to better utilize the
available computational power by executing threads when other ones are waiting
for global memory transfers to be completed [16]. It also ensures that the
code will execute efficiently on new GPUs, which, by the Moore’s law, are
expected to be capable of simultaneously executing exponentially larger
numbers of threads. We have found that calculating $10^{5}$ independent
realizations is enough to obtain a satisfactory level of convergence and that
further increases of the number of paths do not yield better results (see Fig.
5).
In order to increase the number of threads, we structured our code so that Eq.
1 is solved for multiple values of the system parameters in a single run. The
default setup calculates trajectories for $100$ values of the amplitude
parameter $a$. This makes it possible to use our code to efficiently analyze
the behavior of the system for whole regions of the parameter space
$\\{a,\omega,\gamma\\}$.
Multiple timesteps are calculated in a single kernel invocation to increase
the efficiency of the code. We observe that usually only samples taken every
$M$ steps are interesting to the researcher running the simulation, the
sampling frequency $M$ being chosen so that the relevant information about the
analyzed system is retained. In all following examples $M=100$ is used. It
should be noted that the results of the intermediate steps do not need to be
copied to the host (CPU) memory. This makes it possible to limit the number of
global memory accesses in the CUDA threads. When the kernel is launched, path
parameters $x$, $v=\dot{x}$ and $a$ are loaded from the global memory and are
cached in local variables. All calculations are then performed using these
variables and at the end of the kernel execution, their values are written
back to the global memory.
Each path is associated with its own state of the random number generator
(RNG), which guarantees independence of the noise terms between different
threads. The initial RNG seeds for each thread are chosen randomly using a
standard integer random generator available on the host system. Since CUDA
does not provide any random number generation routines by default, we
implemented a simple xor-shift RNG as a CUDA device function. In our kernel,
two uniform variates are generated per time step and then transformed into
Gaussian variates using the Box-Muller transform. The integration is performed
using a Stochastic Runge-Kutta scheme of the 2nd order, which uses both
Gaussian variates for a single time step.
Figure 2: The ensemble of $524288$ Brownian particles, modeling the noisy
dynamics of phase in a Josephson junction described by Eq. 1 is simulated for
time $t\in(0,2000\frac{2\pi}{\omega})$ with time step $\Delta
t=0.01\frac{2\pi}{\omega}$. On the left panel sample trajectories are drawn
with black lines and the background colors represent the coarse-grained
(averaged over a potential period) density of particles in the whole ensemble.
The right panel shows the coarse-grained probability distribution of finding a
particle at time $t=2000\frac{2\pi}{\omega}$ obtained by means of a histogram
with $200$ bins. The histogram is calculated with both single and double
precision on a GPU with Compute Capability v1.3. The same calculation has also
been performed on the CPU but their identical results are not presented for
clarity purposes. The total simulation times were: 20 seconds and 13 minutes
on NVIDIA Tesla 1060C when using single and double precision floating-point
arithmetics, respectively. The CPU-based version of the same algorithm needed
over three hours. Used parameters: $a=4.2$, $\gamma=0.9$, $\omega=4.9$,
$D_{0}=0.001$, $f=0.1$ correspond to the anomalous response regime (cf. [17]).
In the example in Fig. 2 we present the results coming from the simultaneous
solution of $N=2^{19}=524288$ independent Eqs. 1 for the same set of
parameters. The total simulation time was less than $20$ seconds. In this case
the CUDA platform turns out to be extremely effective, outperforming the CPU
by a factor of $675$. In order to highlight the amount of computation, let us
note that the size of the intermediate file with all particle positions used
for generation of the background plot was about $30$ GB.
### 4.2 $N$ globally interacting stochastic systems
Algorithm 3 The AdvanceSystem CUDA kernel.
1: local $i\leftarrow blockIdx.x\cdot blockDim.x+threadIdx.x$
2: local $mv\leftarrow 0$
3: local $mx\leftarrow x_{i}$
4: for all tiles do
5: local $tix\leftarrow threadIdx.x$
6: $j\leftarrow tile\cdot blockDim.x+threadIdx.x$
7: shared $sx_{tix}\leftarrow x_{j}$
8: synchronize with other threads in the block
9: for $k=1$ to $blockDim.x$ do
10: $mv\leftarrow mv+\sin(mx-sx_{k})$
11: end for
12: synchronize with other threads in the block
13: end for
14: $v_{i}\leftarrow mv$
Figure 3: All-pairs interaction of 12 particles calculated using the tile-
based approach with 9 tiles of size 4x4. The chosen number of particles and
the size of the tiles are made artificially low for illustration purposes
only. A small square represents the computation of a single particle-particle
interaction term. The highlighted part of the schematic depicts a single tile.
The bold lines represent synchronization points where data is loaded into the
shared memory of the block. The filled squares with circles represent the
start of computation for a new tile. Threads in the red box are executed
within a single block.
For the general Kuramoto model described by Eqs. 3 or other stochastic systems
of $N$ interacting particles, the calculation of $\mathcal{O}(N^{2})$
interaction terms for all pairs $(x_{j},x_{i})$ is necessary in each
integration step. In this case the program PROG2 is split into two parts,
implemented as two CUDA kernels launched sequentially. The first kernel,
called UpdateRHS calculates the right hand side of Eq. 3 for every $i$. The
second kernel AdvanceSystem actually advances the system by a single step
$\Delta t$ and updates the positions of all particles. In our implementation
the second kernel uses a simple first-order Euler scheme. It is
straightforward to modify the program to implement higher-order schemes by
interleaving calls to the UpdateRHS kernel with calls to kernels implementing
the sub-steps of the scheme.
The UpdateRHS kernel is organized around the concept of _tiles_ , introduced
in [26]. A tile is a group of $T$ particles interacting with another group of
$T$ particles. Threads are executed in blocks of size $T$ and each block is
always processing a single tile. There is a total of $N/T$ blocks in the grid.
The $i$-th thread computes the interaction of the $i$-th particle with all
other particles.
The execution proceeds as follows. The $i$-th thread loads the position of the
$i$-th particle and caches it as a local variable. It then loads the position
of another particle from the current tile, stores it in shared memory and
synchronizes with other threads in the block. When this part is completed, the
positions of all particles from the current tile are cached in the shared
memory. The computation of the interaction is then commenced, with the $i$-th
thread computing the interaction of the $i$-th particle with all particles
from the current tile. Afterwards, the kernel advances to the following tile,
the positions stored in shared memory are replaced with new ones, and the
whole process repeats.
This approach might seem wasteful since it computes exactly $N^{2}$
interaction terms, while only $(N-1)N/2$ are really necessary for a symmetric
interaction. It is however very efficient, as it minimizes global memory
transfers at the cost of an increased number of interaction term computations.
This turns out to be a good trade-off in the CUDA environment, as global
memory accesses are by far the most costly operations, taking several hundred
clock cycles to complete. Numerical computations are comparatively cheap,
usually amounting to just a few clock cycles.
Figure 4: An example result of the integration of the Kuramoto system (Eq. 3).
The time evolution of the probability density $P(x;t)$ is shown for
$\omega_{i}=0$, $K_{ij}=4$, $T=1$. The density is a position histogram of
$2^{24}$ particles. The total time of simulation was approximately $20$
seconds using the single precision capabilities of NVIDIA Tesla C1060.
The special form of the interaction term in the Kuramoto model when
$K_{ij}=K=\mathrm{const}$, allows us to significantly simplify the
calculations. Using the identity:
$\displaystyle\sum_{j=1}^{N}\sin(x_{j}-x_{i})=$
$\displaystyle\cos(x_{i})\sum_{j=1}^{N}\sin(x_{j})-\sin(x_{i})\sum_{j=1}^{N}\cos(x_{j})$
(3)
we can compute two sums: $\sum_{j=1}^{N}\sin(x_{j})$ and
$\sum_{j=1}^{N}\cos(x_{j})$ only once per integration step, which has a
computational cost of $\mathcal{O}(N)$. The calculation of the sum of a vector
of elements is an example of the vector reduction operation, which can be
performed very efficiently on the CUDA architecture. Various methods of
implementation of such an operation are presented in the sample code included
in the CUDA SDK 2.1 [27]. The integration of the Kuramoto system taking
advantage of Eq. 4.2 and using a simple form of a parallel reduction is
implemented in PROG3.
In Fig. 4 we present a solution of the classical Kuramoto system described by
Eqs. 3 for parameters as in Fig. 10 of the review paper [19]. In this case we
apply the program PROG3 which makes use of the relation from Eq. 4.2. The
number of particles $N=2^{24}\approx 16.8\cdot 10^{6}$ and the short
simulation time clearly demonstrate the power of the GPU for this kind of
problems.
## 5 Note on single precision arithmetics
The fact that the current generation of CUDA devices only implements single
precision operations in an efficient way is often considered a significant
limitation for numerical calculations. We have found out that for the
considered models this does not pose a problem. Figure 2 presents sample paths
and position distribution functions of a Brownian particle whose dynamics is
determined by Eq. 1 (colored background on the left panel and right panel).
Let us note that we present coarse-grained distribution functions where the
position is averaged over a potential period by taking a histogram with bin
size being exactly equal to the potential period. We observe that the use of
single precision floating-point numbers does not significantly impact the
obtained results. Results obtained by single precision calculations even after
a relatively long time $t=2000\frac{2\pi}{\omega}$ differ from their double
precision counterparts only up to the statistical error, which in this case
can be estimated by the fluctuations of the relative particle number in a
single histogram bin. Since in the right panel of Fig. 2 we have approximately
$10^{4}$ particles in one bin, the error is of the order of $1\%$. If time-
averaged quantities such as the asymptotic velocity $\langle\langle
v\rangle\rangle=\lim_{t\to\infty}\langle v(t)\rangle$ are calculated, the
differences are even less pronounced. However, the single and double precision
programs produce different individual trajectories as a direct consequence of
the chaotic nature of the system given by Eq. 1. Moreover, we have noticed
that even when changing between GPU and CPU versions of the same program, the
individual trajectories diverged after some time. The difference between paths
calculated on the CPU and the GPU, using the same precision level, can be
explained by differences in the floating-point implementation, both in the
hardware and in the compilers.
When doing single precision calculations special care must be taken to ensure
that numerical errors are not needlessly introduced into the calculations. If
one is used to having all variables defined as double precision floating-point
numbers, as is very often the case on a CPU, it is easy to forget that
operations which work just fine on double precision numbers might fail when
single precision numbers are used instead. For instance, consider the case of
keeping track of time in a simulation by naively increasing the value of a
variable $t$ by a constant $\Delta t$ after every step. By doing so, one is
bound to hit a problem when $t$ becomes large enough, in which case $t$ will
not change its value after the addition of a small value $\Delta t$, and the
simulation will be stuck at a single point in time. With double precision
numbers this issue becomes evident when there is a difference of 17 orders of
magnitude between $t$ and $\Delta t$. With single precision numbers, a
8-orders-of-magnitude difference is enough to trigger the problem. It means
that if, for instance, $t$ is $10^{5}$ and $\Delta t$ is $10^{-4}$, the
addition will no longer work as expected. $10^{5}$ and $10^{-4}$ are values
not uncommon in simulations of the type we describe here, hence the need for
extra care and reformulation of some of the calculations so that very large
and very small quantities are not used at the same time. In our
implementations, we avoided the problem of spurious addition invariants by
keeping track of simulation time modulo the system period $2\pi/\omega$. This
way, the difference between $t$ and $\Delta t$ was never large enough to cause
any issues.
## 6 Performance evaluation
In order to evaluate the performance of our numerical solution of Eqs. 1 and
3, we first implemented Algs. 3 and 1 using the CUDA Toolkit v2.1. We then
translated the CUDA code into C++ code by replacing all kernel invocations
with loops and removing unnecessary elements (such as references to shared
memory, which does not exist on a CPU).
We used the NVIDIA CUDA Compiler (NVCC) and GCC 4.3.2 to compile the CUDA code
and the Intel C++ Compiler (ICC) v11.0 for Linux to compile the C++ version.
We have determined through numerical experiments that enabling floating-point
optimizations significantly improves the performance of our programs (by a
factor of $7$ on CUDA) and does not affect the results in a quantitative or
qualitative way. We have therefore used the -fast -fp-model fast=2 ICC options
and \--use_fast_math in the case of NVCC.
Figure 5: (Left panel) Performance estimate for the programs _PROG1_ -_PROG3_
as a function of the number of particles $N$. (Right panel) Performance
estimate for the programs _PROG1_ -_PROG3_ on an Intel Core2 Duo E6750 CPU and
NVIDIA Tesla C1060 GPU. We have counted $79$, $44+6N$ and $66$ operations per
one integration step of programs _PROG1_ , _PROG2_ and _PROG3_ , respectively.
All tests were conducted on recent GNU/Linux systems using the following
hardware:
* 1.
for the CPU version: Intel Core2 Duo E6750 @ 2.66GHz and 2 GB RAM (only a
single core was used for the calculations)
* 2.
for the GPU version: NVIDIA Tesla C1060 installed in a system with Intel Core2
Duo CPU E2160 @ 1.80GHz and 2 GB RAM
Our tests indicate that speedups of the order of 600 and 100 are possible for
the models described by Eqs. 1 and 3, respectively. The performance gain is
dependent on the number of paths used in the simulation. Figure 5 shows that
it increases monotonically with the number of paths, and then saturates at a
number dependent on the used model: $450$ and $106$ GFLOPS for the Eqs. 1 and
3, respectively (which corresponds to speedups: $675$ and $106$). The
saturation point indicates that for the corresponding number of particles the
full computational resources of the GPU are being exploited.
The problem of lower performance gain for small numbers of particles could be
rectified by dividing the computational work between threads in a different
way, i.e. by decreasing the amount of calculations done in a single thread,
while increasing the total number of threads. This is a relatively
straightforward thing to do, but it increases the complexity of the code. We
decided not to do it since for models like 1 and 3 one is usually interested
in calculating observables for whole ranges of system parameters. Instead of
modifying the code to run faster for lower number of paths, one can keep the
number of paths low but run the simulation for multiple system parameters
simultaneously, which results in a higher number of threads.
## 7 Conclusions
In this paper we have demonstrated the suitability of a parallel CUDA-based
hardware platform for solving stochastic differential equations. The observed
speedups, compared to CPU versions, reached an astonishing value $670$ for
non-interacting particles and $120$ for a globally coupled system. We have
also shown that for this kind of calculations single precision arithmetics
poses no problems with respect to accuracy of the results, provided that some
kind of operations, such as adding small and large numbers, are avoided.
The availability of cheap computer hardware which is over two orders of
magnitude faster clearly announces a new chapter in high performance
computing. Let us note that the development of stream processing technology
for general-purpose computing has just started and its potential is surely not
yet fully revealed. In order to take advantage of the new hardware
architecture, the software and its algorithms must be substantially
redesigned.
## 8 Appendix: Estimation of FLOPS
We counted the floating-point operations performed by the kernels in our code,
and the results in the form of the collective numbers of elementary operations
are presented in Table 1. The number of MAD (Multiply and Add) operations can
vary, depending on how the compiler processes the source code. For the
purposes of our performance estimation, we assumed the most optimistic
version. A more conservative approach would result in a lower number of MADs,
and correspondingly a higher total number of GFLOPS.
Table 1: Number of elementary floating-point operations performed per one time step in the AdvanceSystem kernel for Eq. 1. count | type | FLOPs | total
FLOPs
---|---|---|---
22 | multiply, add | 1 | 22
11 | MAD | 1 | 11
2 | division | 4 | 8
3 | sqrt | 4 | 12
1 | $\sin$ | 4 | 4
5 | $\cos$ | 4 | 20
1 | $\log$ | 2 | 2
TOTAL: | 79
The amount of FLOPs for functions such as sin, log, etc. is based on [16],
assuming $1$ FLOP for elementary arithmetical operations like addition and
multiplication and scaling the FLOP estimate for complex functions
proportionately to the number of processor cycles cited in the manual. The
numbers of floating-point operations are summarized in Table 1.
On a Tesla C1060 device our code PROG1 evaluates $6.178\cdot 10^{9}$ time
steps per second. The cost of each time step is $79$ FLOPs, which implies that
the overall performance estimate accounts for $490$ GFLOPS.
In the case of PROG2 the number of operations per one integration step depends
on the number of particles $N$. A similar operation count as the one presented
in Table 1 resulted in the formula $44+6N$ FLOPs per integration step.
## References
* [1] Reimann, P., Physics Reports 361 (2002) 57 .
* [2] Kostur, M., Machura, L., Talkner, P., Hänggi, P., and Łuczka, J., Physical Review B (Condensed Matter and Materials Physics) 77 (2008) 104509.
* [3] Kautz, R. L., Reports on Progress in Physics 59 (1996) 935.
* [4] Gammaitoni, L., Hänggi, P., Jung, P., and Marchesoni, F., Rev. Mod. Phys. 70 (1998) 223.
* [5] Groot, R. and Warren, P., J. Chem. Phys. 107 (1997) 4423.
* [6] McLeish, D. L., Monte Carlo Simulation and Finance, John Wiley and Sons, 2005.
* [7] Bartussek, R., Reimann, P., and Hänggi, P., Phys. Rev. Lett. 76 (1996) 1166.
* [8] Lindner, B., Schimansky-Geier, L., Reimann, P., Hänggi, P., and Nagaoka, M., Phys. Rev. E 59 (1999) 1417.
* [9] Kalmykov, Y. P., Phys. Rev. E 61 (2000) 6320.
* [10] Kostur, M., Internat. J. Modern Phys. C 13 (2002) 1157.
* [11] Yu, J. and Lin, Y., Internat. J. Non-Linear Mech. 39 (2004) 1493.
* [12] Naess, A., Dimentberg, M. F., and Gaidai, O., Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 78 (2008) 021126.
* [13] http://www.computermuseum.li/testpage/asci-red-supercomputer.htm.
* [14] Nvidia cuda webpage, http://www.nvidia.com/object/cuda_home.html.
* [15] Source code of all examples can be found on http://fizyka.us.edu.pl/cuda.
* [16] Corporation, N., Nvidia cuda programming guide v2.1, available from nvidia cuda webpage, http://www.nvidia.com/object/cuda_home.html, 2008.
* [17] Machura, L., Kostur, M., Talkner, P., Łuczka, J., and Hänggi, P., Physical Review Letters 98 (2007) 040601.
* [18] Speer, D., Eichhorn, R., and Reimann, P., EPL (Europhysics Letters) 79 (2007) 10005 (5pp).
* [19] Acebron, J. A., Bonilla, L. L., Vicente, C. J. P., Ritort, F., and Spigler, R., Reviews of Modern Physics 77 (2005) 137.
* [20] Mannella, R. and Palleschi, V., Phys. Rev. A 40 (1989) 3381.
* [21] Mannella, R., Internat. J. Modern Phys. C 13 (2002) 1177.
* [22] Sancho, J. M., Miguel, M. S., Katz, S. L., and Gunton, J. D., Phys. Rev. A 26 (1982) 1589.
* [23] Fox, R. F., Gatland, I. R., Roy, R., and Vemuri, G., Phys. Rev. A 38 (1988) 5938.
* [24] Honeycutt, R. L., Phys. Rev. A 45 (1992) 600.
* [25] Kloeden, P. E. and Platen, E., Numerical Solution of Stochastic Differential Equations (Stochastic Modelling and Applied Probability), Springer, 2000.
* [26] L. Nyland, M. Harris, J. P., GPU Gems 3 - Fast N-body simulation with CUDA, chapter 31, pages 677–695, Addison-Wesley Professional, 2007.
* [27] Nvidia cuda software development kit, available from nvidia cuda webpage, http://www.nvidia.com/object/cuda_home.html.
|
arxiv-papers
| 2009-03-23T14:13:12 |
2024-09-04T02:49:01.369693
|
{
"license": "Public Domain",
"authors": "M. Januszewski, M. Kostur (University of Silesia, Katowice, Poland)",
"submitter": "Marcin Kostur dr",
"url": "https://arxiv.org/abs/0903.3852"
}
|
0903.4080
|
# A Review of Nucleon Spin Calculations
in Lattice QCD
Huey-Wen Lin
###### Abstract
We review recent progress on lattice calculations of nucleon spin structure,
including the parton distribution functions, form factors, generalization
parton distributions, and recent developments in lattice techniques.
###### Keywords:
Lattice QCD, nucleon spin structures, form factors, GPDs
###### :
12.38.-t 12.38.Gc 13.40.Gp 14.20.Dh 13.60.Fz
## 1 Introduction
Quantum chromodynamics (QCD) has been successful in describing many properties
of the strong interaction. In the weak-coupling regime, we can rely on
perturbation theory to work out the path integral which describes physical
observables of interest. However, for long distances perturbative QCD no
longer converges. Instead, we use a discretization of space and time in a
finite volume to calculate these quantities from first principles numerically;
such research forms the regime of lattice QCDDeGrand and Detar (2006).
To keep the systematic error due to discretization under control, one follows
Symanzik improvement order by order in terms of the ultraviolet cutoff ($a$)
for both the action and operators. However, the breaking of continuous
(Euclidean) SO(4) symmetry allows many new degrees of freedom, leading to
various lattice actions that return to the same continuum action once the
symmetry is restored. Thus, there exist many gauge and fermion actions for us
to choose from. Today, most gauge actions used are $O(a^{2})$-improved and
leave small discretization effects ($O(a^{3}\Lambda_{\rm QCD}^{3})$) due to
gauge choices. On the other hand, most fermion actions are only
$O(a)$-improved and have systematic errors of $O(a^{2}\Lambda_{\rm QCD}^{2})$
that become dominant. For this reason, lattice calculations are generally
distinguished according to the fermion action used. Differences among the
actions are benign once all systematics are included, and the choice of
fermion action is constrained by limits of computational and human power and
by the main physics focus. The commonly used actions are: domain-wall fermions
(DWF), overlap fermions, Wilson/clover fermions, twisted-Wilson fermions and
staggered fermions.
Since the real world is effectively continuous and infinitely large, we will
have to take limits of $a\rightarrow 0$ and $V\rightarrow\infty$ to eliminate
the artifacts introduced in a discretized finite box. With the most state-of-
the-art supercomputer, we are close but yet to simulate at the physical pion
mass. Using calculations at multiple heavier pion masses, which are affordable
for available computational resources, we can apply chiral perturbation theory
to extrapolate quantities of interest to the physical limit. A recent work by
the BMW collaborationDurr et al. (2008) calculating multiple lattice spacings,
volumes and pion masses as light as 180 MeV provided an excellent
demonstration of how ground-state hadron masses with fully understood and
controlled systematics are consistent with experiment. Such calculations with
multiple pion masses also help to determine the low-energy constants of chiral
effective theory.
A typical nucleon interpolating field used in the lattice calculation is
$\chi_{N}=\sum_{\vec{x},a,b,c}e^{i\vec{p}\cdot\vec{x}}\epsilon^{abc}\left[u_{a}^{T}C\gamma_{5}d_{b}\right]u_{c},$
and the nucleon two- and three-point Green functions are obtained from
$\displaystyle\Gamma^{(2)}(t_{\rm src},t)$ $\displaystyle=$
$\displaystyle\langle\chi_{N}(t)\chi_{N}^{\dagger}(t_{\rm src})\rangle$ (1)
$\displaystyle\Gamma^{(3)}(t_{\rm src},t,t_{\rm snk})$ $\displaystyle=$
$\displaystyle\langle\chi_{N}(t_{\rm snk},\vec{p}_{\rm snk})\,{\cal
O}(t,\vec{q})\,\chi^{\dagger}_{N}(t_{\rm src},\vec{p}_{\rm src})\rangle,$ (2)
where ${\cal O}$ is the operator of interest. For the vector (axial) current,
the operator is ${\cal O}=\overline{\psi}\gamma_{\mu}(\gamma_{5})\psi$. For
the structure functions, the operators are
$\langle x^{n}\rangle_{q}$: | ${\cal O}_{\mu_{1}...\mu_{n}}^{q}$ | = | $i^{n-1}\overline{\psi}\gamma_{\\{\mu_{1}}\overleftrightarrow{D}_{\mu_{2}}\cdots\overleftrightarrow{D}_{\mu_{n}\\}}\psi$
---|---|---|---
$\langle x^{n}\rangle_{\Delta q}$: | ${\cal O}_{\mu_{1}...\mu_{n}}^{5q}$ | = | $i^{n-1}\overline{\psi}\gamma_{\\{\mu_{1}}\gamma_{5}\overleftrightarrow{D}_{\mu_{2}}\cdots\overleftrightarrow{D}_{\mu_{n}\\}}\psi$
$\langle x^{n}\rangle_{\delta q}$: | ${\cal O}_{\mu\mu_{1}...\mu_{n}}^{\sigma q}$ | = | $i^{n-1}\overline{\psi}\gamma_{5}\sigma_{\mu\\{\mu_{1}}\overleftrightarrow{D}_{\mu_{2}}\cdots\overleftrightarrow{D}_{\mu_{n}\\}}\psi$,
where
$\overleftrightarrow{D}=\frac{1}{2}(\overrightarrow{D}-\overleftarrow{D})$ is
the difference between forward and backward covariant derivatives. We
calculate only the “connected” diagrams, which means the inserted quark
current is contracted with the valence quarks in the baryon interpolating
fields, as in the majority of lattice three-point calculations. However, due
to isospin symmetry, isovector quantities have a cancellation that removes the
unknown disconnected piece. (Disconnected diagrams are notoriously difficult
to calculate directly on the lattice. They require that expensive fermion-
matrix inversion be applied to numerous source vectors. Much effort has been
devoted to solving this difficulty in the near future with new techniques.)
For more details, please refer to a selection of plenary talks: Refs. plenary
(2008) and references within.
## 2 Lattice QCD Reveals the Structure of the Nucleon
The nucleon axial charge is well measured in neutron $\beta$-decay
experiments, so it is a natural candidate for demonstrating how well lattice
QCD can be extrapolated to the physical pion point. The isovector axial
charges $g_{A}$ are defined as the zero-momentum-transfer limits of the
isovector axial form factors. Results from various collaborations are shown on
the left-hand side of Figure 1. We show a small-scale–expansion fit (gray
band) to the 2+1-flavor (DWF sea quarks) RBC dataRBC (2008) (blue filled
circles). The results are consistent with the LHPC dataLHPC (2006) (DWF
valence with staggered sea); RBC’s 2f and 0f results are consistent with
QCDSF’s 2f Clover and 0f overlap fermion numbersQCDSF (2006) respectively. The
lowest–pion-mass values among RBC’s 2+1f and 2f results may suffer from
sizable finite-volume systematic errors; larger-volume calculations should be
carried out to confirm these suspicions.
Figure 1: (left) Renormalized axial charge versus pseudoscalar mass squared
from various lattice groups. (right) Zeroth moment of transversity from
different lattice groups. The bands in both plots are chiral extrapolations
fit to the RBC 2+1f data.
Results for the zeroth moment of transversity $\langle 1\rangle_{\delta q}$
are given on the right-hand side of Figure 1. We observe rather weak
dependence (roughly linear) on the quark mass, which remains consistent for
calculations with the same number of sea-quark flavors. The chiral
extrapolation formxChPT (2002) is applied to the RBC 2+1f dataRBC (2008),
yielding 0.56(4) at the physical pion mass. However, this is close to what has
been found by LHPC with mixed actionLHPC (2006): about 0.7. These extrapolated
values are significantly smaller than those found at the simulated pion
masses, which are near 1. We urgently need data at the lightest pion mass to
confirm the rapidly decreasing behavior predicted by the chiral effective
theory.
Figure 2 shows the latest calculations of the first moments of the quark
momentum fraction (left) and helicity (right) distributions. Here again the
0f, 2f and 2+1f results are compared between different collaborations with
different choices of fermion actions and are seen to be consistent among
calculations with the same number of sea-quark flavors. The chiral
extrapolationxChPT (2002) is performed using RBC’s 2+1f data, which gives
0.133(13) and 0.203(23) for the first moments of the quark momentum fraction
and helicity distributions respectively, consistent with experiment. We see
strong curvature due to the chiral form; more light-pion points should be
taken to reduce extrapolation uncertainties. These extrapolation numbers are
also consistent with the LHPC’s mixed-action calculationLHPC (2006).
Higher moments of the isovector distributions are calculated by LHPC (SCRI,
SESAM) with 0f and 2f Wilson and clover fermionsLHPC (2006) and by QCDSF with
0f clover fermions at multiple lattice spacingsQCDSF (2006). Consistent
results are seen among different groups: The second and third moments are
about 25% and 10% of the first moments respectively. However, for moments with
$n\geq 4$, divergences occur involving lower-dimension operators at finite
$a$, which limits the number of moments accessible to lattice QCD.
Figure 2: Global comparison of the first moments of the quark momentum
fraction (left) and helicity (right) distributions in terms of $M_{\pi}^{2}$
and their chiral extrapolations. The bands in both plots are chiral
extrapolations fit to the RBC 2+1f data.
The twist-3 first moment of the polarized structure function $d_{n}$ is
another interesting feature to consider. It is related to the polarized
structure functions $g_{1}$ and $g_{2}$ and the Wandzura-Wilczek
relationWandzura and Wilczek (1977). The lowest moment $d_{1}$ from RBC’s 2+1f
data extrapolated to the physical pion mass is consistent with zero
$d_{1}^{\rm bare}=-0.002(2)$. Combined with the small value of $d_{2}$ found
by QCDSF’s 2f calculationQCDSF (2006), we conclude that the Wandzura-Wilczek
relation between the moments of $g_{1}$ and $g_{2}$, which asserts vanishing
$d_{n}$, is at least approximately true.
Studying the momentum-transfer ($Q^{2}$) dependence of the elastic
electromagnetic form factors is important in understanding the structure of
hadrons at different scales. There have been many experimental studies of
these form factors on the nucleon. A recent such experiment, the Jefferson Lab
double-polarization experiment (with both a polarized target and
longitudinally polarized beam) revealed a non-trivial momentum dependence for
the ratio $G_{E}^{p}/G_{M}^{p}$. This contradicts results from the Rosenbluth
separation method, which suggested $\mu_{p}G_{E}^{p}/G_{M}^{p}\approx 1$. The
contradiction has been attributed to systematic errors due to two-photon
exchange that contaminate the Rosenbluth separation method more than the
double-polarization. (For details and further references, see the recent
review articles: Refs. FFs (2008).) Lattice calculations can make valuable
contributions to the study of nucleon form factors, since they allow access to
both the pion-mass and momentum dependence of such form factors. Recently, the
limitations of the largest-available $Q^{2}$ (in terms of the quality of the
signal-to-noise ratios) has been overcomeLins (2008). An exploratory study
using clover fermions extends the range of momentum transfer to 6 GeV2, as
shown in Figure 3. The $Q^{2}$ dependence of the neutron has exceeded the
range of the current existing data. Such calculations will provide interesting
comparisons for data collected after the future 12-GeV upgrade at Jefferson
Lab.
Figure 3: Nucleon form factors with pion masses of 480 (triangles), 720
(circles) and 1080 (diamonds) MeV. The dashed lines are plotted using
experimental form-factor fit parametersFFs (2008).
The generalized parton distributions (GPDs) have been calculated by LHPC (2+1f
mixed action, $M_{\pi}\sim 350$–760 MeV)LHPC (2006) and QCDSF (2f clover
action, $M_{\pi}\sim 340$–950 MeV)QCDSF (2006). One of the main topics of
physics interest derived from GPDs is to study the origins of the nucleon’s
spin. The quark spin ($\Delta\Sigma^{q}$) and total quark contribution
($J^{q}$) to the angular momentum can be connected to GPDs viaJi (1997)
$\frac{1}{2}\Delta\Sigma^{q}=\frac{1}{2}\tilde{A}^{q}_{10}(0);\;\;J^{q}=\frac{1}{2}[{A}^{q}_{20}(0)+{B}^{q}_{20}(0)],$
(3)
where $\tilde{A}^{q}_{n0}$ and $\\{A,B\\}^{q}_{n0}$ are the polarized and
unpolarized generalized form factors. The left-hand side of Figure 4 shows the
quark spin and orbital angular momentum ($L^{q}$) calculated by LHPC and
QCDSF; they are consistent. One found the total $d$ quark angular momentum and
the angular momentum from the sum of $u$ and $d$ quarks to be consistent with
zero. This is because $L^{d}$ and $L^{u}$ are of the same magnitude but have
opposite signs. A similar relation holds for $\Delta\Sigma^{d}$ and $L^{d}$.
Note that in both of the calculations only connected diagrams are included.
For the transverse structure of the proton, QCDSF calculates the lowest two
moments of the transverse spin densities of quarks in the nucleonQCDSF (2006).
An exploratory attempt has been made to calculate transverse-momentum
distributions (TMDs) using lattice link products to approximate the Wilson
line in light-cone frame in spatial coordinates and Fourier transforming into
momentum-space to map the density distributionsLHPC (2006); this calculation
used 2+1f mixed action (DWF on staggered, $M_{\pi}\sim 500$ MeV).
Another new development during the last year is progress on disconnected
diagrams. An indirect way of calculating the strangeness form factors via
charge symmetry in 2+1f lattices is shown on the right-hand side of Figure 4.
The direct approach has also made great progress. For example, the strange-
quark distribution $\langle x\rangle_{s}$ has been calculated by the $\chi$QCD
collaboration on 2+1f latticeschiQCD (2008). The same group has also
calculated the gluon momentum fraction $\langle x\rangle_{g}$ with improved
signal.
Figure 4: (left) The quark-component contributions to quark spin and orbital
angular momentum in the spin of the nucleon from LHPC (squares, diamonds and
upward triangles) and QCDSF (tilted squares and rightward triangles). The
filled/open symbols represent dynamical/quenched data. The bands are chiral
extrapolations based on LPHC’s dynamical data. (right) Experimental results
(ellipses) and various theoretical calculations (points) of
$G_{M}^{s}$-$G_{E}^{s}$
.
This is an exciting era for the use of lattice QCD in nuclear physics: there
have been huge leaps due to increasing computational resources worldwide and
improved algorithms, allowing continual improvement in lighter pion masses,
larger volumes and finer lattice spacings. Various groups have demonstrated
universality with the consistent results coming from independent calculations
using different lattice actions. By reproducing well measured experimental
values, we solidify confidence in lattice predictions of quantities that have
not or cannot be measured by experiment. There are many different aspects of
hadron structure that one can do with lattice QCD; only a few examples have
been presented here.
In the near future, pion masses around 200 MeV or lighter with multiple
volumes and lattice spacings will become commonplace. There will be less need
to depend on chiral perturbation theory for extrapolations, and once the
physical pion mass becomes accessible, we can check its correctness. Full-
contraction calculations (including disconnected diagrams) in all matrix
elements, form factors and GPDs will lead to precision calculations for
individual quark components or individual proton and neutron quantities.
Further improvements in methodology and expanding computational resources will
allow direct calculations of the gluon content of the nucleon. The future is
unlimited with lattice QCD.
Acknowledgements: Authored by Jefferson Science Associates, LLC under U.S. DOE
Contract No. DE-AC05-06OR23177. The U.S. Government retains a non-exclusive,
paid-up, irrevocable, world-wide license to publish or reproduce this
manuscript for U.S. Government purposes.
## References
* DeGrand and Detar (2006) T. DeGrand, and C. E. Detar, new Jersey, USA: World Scientific (2006) 345 p.
* Durr et al. (2008) S. Durr, et al., _Science_ 322, 1224–1227 (2008).
* plenary (2008) J. M. Zanotti, _PoS_ LAT2008, 007 (2008); P. Hagler, _PoS_ LAT2007, 013 (2007); K. Orginos, _PoS_ LAT2006, 018 (2006); H.-W. Lin (2008), arXiv:0812.0411[hep-lat].
* xChPT (2002) W. Detmold et al., _Phys. Rev. Lett._ 87, 172001 (2001); D. Arndt et al., _Nucl. Phys._ A697, 429–439 (2002); J.-W. Chen et al., _Nucl. Phys._ A707, 452–468 (2002); W. Detmold et al., _Phys. Rev._ D66, 054501 (2002).
* RBC (2008) H.-W. Lin (2007), 0707.3844; H.-W. Lin et al., _Phys. Rev._ D78, 014505 (2008); T. Yamazaki et al., _Phys. Rev. Lett._ 100, 171602 (2008); K. Orginos et al., _Phys. Rev._ D73, 094503 (2006).
* LHPC (2006) R. G. Edwards et al., (2006), hep-lat/0610007; D. Dolgov et al., _Phys. Rev._ D66, 034506 (2002); P. Hagler, et al., _Phys. Rev._ D77, 094502 (2008); B. U. Musch, et al. (2008), 0811.1536.
* QCDSF (2006) M. Ohtani, _PoS_ LAT2007, 158 (2007); M. Gockeler et al., _Phys. Rev._ D71, 114511 (2005); M. Gockeler et al., _Phys. Rev._ D63, 074506 (2001); M. Gockeler et al., _Phys. Rev. Lett._ 98, 222001 (2007).
* Wandzura and Wilczek (1977) S. Wandzura, and F. Wilczek, _Phys. Lett._ B72, 195 (1977).
* FFs (2008) J. Arrington et al., _Phys. Rev._ C76, 035205 (2007a); C. F. Perdrisat et al., _Prog. Part. Nucl. Phys._ 59, 694–764 (2007); J. Arrington et al., _J. Phys._ G34, S23–S52 (2007b); J. J. Kelly, _Phys. Rev._ C70, 068202 (2004).
* Lins (2008) H.-W. Lin et al., _Phys. Rev._ D78, 114508 (2008a); H.-W. Lin et al., (2008b), 0810.5141.
* Ji (1997) X.-D. Ji, _Phys. Rev. Lett._ 78, 610–613 (1997), hep-ph/9603249.
* chiQCD (2008) M. Deka et al. (2008), 0811.1779; T. Doi et al. (2008), 0810.2482.
|
arxiv-papers
| 2009-03-24T13:51:50 |
2024-09-04T02:49:01.383159
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huey-Wen Lin",
"submitter": "Huey-Wen Lin",
"url": "https://arxiv.org/abs/0903.4080"
}
|
0903.4197
|
# Oxygen Metallicity Determinations from Optical Emission Lines in Early-type
Galaxies
Alex E. Athey The University of Texas at Austin Applied Research
Laboratories 10000 Burnet Rd Austin, TX 78758 athey@arlut.utexas.edu Joel
N. Bregman University of Michigan Department of Astronomy 500 Church St.
Ann Arbor, MI 48109-1090 jbregman@umich.edu
###### Abstract
We measured the oxygen abundances of the warm (T$\sim 10^{4}K$) phase of gas
in seven early-type galaxies through long-slit observations. A template
spectra was constructed from galaxies void of warm gas and subtracted from the
emission-line galaxies, allowing for a clean measurement of the nebular lines.
The ratios of the emission lines are consistent with photoionization, which
likely originates from the UV flux of post-asymototic giant branch (PAGB)
stars. We employ H II region photoionization models to determine a mean oxygen
metallicity of $1.01\pm 0.50$ solar for the warm interstellar medium (ISM) in
this sample. This warm ISM $0.5$ to $1.5$ solar metallicity is consistent with
modern determinations of the metallicity in the hot (T$\sim 10^{6}-10^{7}K$)
ISM and the upper range of this warm ISM metallicity is consistent with
stellar population metallicity determinations. A solar metallicity of the warm
ISM favors an internal origin for the warm ISM such as AGB mass loss within
the galaxy.
Galaxies:Elliptical and Lenticular, Galaxies:ISM, Galaxies:Abundances
## 1 Introduction
### 1.1 Warm ISM Discovery
Early-type galaxies were once thought to contain very little gas in their
interstellar medium (ISM) (e.g. Mathews & Baker (1971); Bregman (1978); White
& Chevalier (1983)), but are now known to host massive amounts of extremely
hot (T$\sim 10^{6}-10^{7}K$) gas that emits in the X-rays (Forman et al.,
1979). The reason early-type galaxies were originally thought to be void of
gas is that photographic-plate surveys did not reveal the warm (T$\sim
10^{4}K$) ISM that is abundant in spiral galaxies (e.g. Mayall (1936); Sandage
(1957); Mayall (1958)). Only with the advent of large telescopes and efficient
detectors was the warm ISM detected, although the masses are an order of
magnitude lower than that of spirals. Spectroscopic surveys in the mid-1980’s
determined warm ISM detection frequencies in the cores of early-type galaxies
of 30%-50% (Caldwell, 1984; Phillips et al., 1986; Veron-Cetty & Veron, 1986).
Subsequent narrow-band $H\alpha$ imaging studies revealed extended emission in
general agreement with the the stellar morphology and detection rates
consistent with the 50% rate determined from the early spectroscopic surveys
(Demoulin-Ulrich et al., 1984; Kim, 1989).
### 1.2 Key Warm ISM Studies
There are three large sample, multi-wavelength studies of early-type galaxies
that have provided a foundation for our understanding of the cold and warm
ISM. Trinchieri & di Serego Alighieri (1991, hereafter TdSA) conducted one of
the first quantitative early-type galaxy emission-line studies with $H\alpha$
imaging and long-slit optical spectroscopy of 13 early-type galaxies known to
have hot X-ray halos. (Goudfrooij et al., 1994a, b, c; Goudfrooij & de Jong,
1995, hereafter G94-5) investigated the origin and evolution of the ISM in an
extensive study of a complete sample of 56 elliptical galaxies using broad-
band imaging, narrow-band $H\alpha$ imaging, long slit spectroscopy, and
_Infrared Astronomical Satellite_ (IRAS) far-IR imaging. A study of the warm
and cold phases of the ISM in early-type galaxies was carried as part of the
ESO Key-Program 1-004-43K (Macchetto et al., 1996; Ferrari et al., 1999; Caon
et al., 2000; Ferrari et al., 2002, hereafter MFCF). Similar to G94-5, the
MFCF study probed the origin and evolution of the warm and cold ISM phases
through a combination of broad-band, narrow-band and far-IR imaging and long-
slit spectroscopy of 73 early-type galaxies. The two later studies had the
benefit of FIR imaging capabilities and added the cold ISM (i.e. dust) to the
discussion of the ISM in early-type galaxies. These studies sought to
determine the frequency of detection of warm and cold ISM in early-type
galaxies and then measure fundamental properties, such as mass, density, and
luminosity. By correlating these properties with stellar and hot ISM
properties, inferences are made about the origin and evolution of the ISM and
how it relates to the origin and evolution of the galaxy itself. Below we
briefly summarize the findings of these key studies and note the areas where
the studies agree and where questions remain.
The three key studies find similar fractions of their samples that host a warm
ISM, ranging from 60%-80%, despite the differences in sample selection
criteria and sizes of the samples. TdSA found 75% of the 13 early-type
galaxies surveyed had detectable $H\alpha$ emission which was extended over a
range of 5-10 $kpc$. The extent and elliptical shape of the emission confirmed
the galaxy-wide nature of this phenomena and excluded a nuclear confined
process, such as AGN activity. The TdSA survey was conducted on galaxies known
to have significant masses of hot ISM, and the authors concede that the high
detection rate is likely connected to sample selection. The G94-5 optically
complete sample (chosen without regard for X-ray luminosity) offers a less
biased measure of the frequency of the warm ISM; 60% of the early-type
galaxies surveyed were detected to have a warm ISM, while a cold (T$\sim 10K$)
dust component is detected in 40% of these galaxies. Through the FIR imaging,
the G94-5 study was able to show that unlike spiral galaxies, early-type
galaxies display a smooth dust component distributed throughout the galaxy.
The MFCF authors report an ionized gas detection frequency at 78% for the 73
galaxies in the optically complete sample. The cold dust component was
detected at a 75% detection rate with smooth, extended morphologies similar to
that of the warm ISM. Two key observations link the warm and the cold ISM
together; first, the similar spatial extent and morphology of the cold ISM
(dust) and warm ISM and second, the fact that the presence of dust is a strong
predictor of the presence of emission lines. It is concluded that these ISM
phases are closely related and likely of the same origin.
The mass of the warm ISM is orders of magnitude below the X-ray emitting hot
ISM and the stellar mass. TdSA calculated a warm ISM gas mass by assuming
case-B recombination, a filling factor of $10^{-5}-10^{-6}$, and densities
corresponding to the pressure implied by the X-ray observations. The resultant
masses were calculated to be $\sim 10^{5}-10^{6}M_{\sun}$ for a typical galaxy
in their sample. The $H\alpha$ imaging in G94-5 revealed warm ISM masses
similar to the FIR determined dust masses of $\sim 10^{4}-10^{5}M_{\sun}$. The
MFCF study determined masses for the warm ISM in the range of
$10^{3}-10^{5}M_{\sun}$. Masses of dust at $10^{5}M_{\sun}$ have direct
implications on the origin of the dust as the dust destruction rate is high in
the presence of the hot ISM and is further discussed below.
The correlations between the luminosities of $L_{FIR}$, $L_{H\alpha}$,
$L_{X}$, $L_{B}$ and $L_{X}/L_{B}$ explore the links between the difference
phases of the ISM and the stars. For numerous of reasons, the luminosity
correlations have been difficult to accurately measure and there are
contradictions between the key studies. TdSA observed that galaxies with more
massive X-ray halos had stronger line emission, indicating that the warm gas
is somehow linked with the hot gas. No other luminosity correlations were
observed. In a related work to G94-5, the authors find a strong anti-
correlation between the masses of dust and X-ray gas (Goudfrooij, 1994, 1995).
However in a later study with a larger sample (based in part on MFCF data),
Goudfrooij (1997) reveals a positive correlation between the warm ISM with
both the hot ISM and the stellar blue luminosity. MFCF determined the
luminosity of the ionized gas to be correlated with both blue and X-ray
luminosity. The scatter in the $L_{H\alpha}$ and $L_{B}$ relation is greatly
reduced when the blue luminosity is confined to the same region as the
emission lines. The scatter in all the $L_{H\alpha}$ to $L_{X}/L_{B}$
relations is large and plagued by upper limits on non-detections.
In general, the three studies agree that the ionization source for the warm
ISM is consistent with PAGB stars, but all comment that the data are unable to
exclude other excitation sources such as internal shocks, AGN heating or
electron conduction from the hot gas. TdSA show that the observed line
intensities are a factor of $\sim 10^{2}-10^{3}$ too large to result from
cooling flow luminosities (e.g. Fabian (1991)) and conclude that UV flux from
post asymptotic giant branch (PAGB) stars are the most likely ionizing
mechanism. The MFCF observed correlation between $H\alpha$ and blue luminosity
is an indication of photoionization of the warm ISM from PAGB stars. However,
the observed correlation with X-ray luminosity is consistent with warm ISM
excitation via electron conduction from the hot ISM (Sparks et al., 1989);
Neither ionization mechanism is preferred in MFCF’s large sample of 73
galaxies. See Section 4.1.1 for a continued discussion on the warm ISM
ionization source.
The key studies narrow down the origin of this dust and warm gas to either be
mass shed from AGB stars or the result of small galaxy accretion and mergers.
G94-5 reasons that cold dust originated in the circumstellar envelopes of AGB
stars are quickly sputtered away by the hot ISM with a typical dust grain
lifetime ($\sim 10^{7}yrs$) much shorter than the age of the galaxy (Draine &
Salpeter, 1979). And further, the balance between the dust injection and
destruction rate does not lead to the dust masses inferred by the far-IR
imaging (Faber & Gallagher, 1976; Knapp et al., 1992). The surprising result
from the confluence of the G94-5 studies is that the majority of the dust in
elliptical galaxies is external in origin. Since the distribution of the
ionized gas (i.e. the warm ISM) is observed to be similar to the dust
morphology, it is presumed that the warm ISM is likewise external to the
galaxy, and will only be seen in galaxies that either have recent merger
activity or significant inflows. The high detection frequency of the warm ISM,
implies mergers and smaller galaxy acquisitions are common in early-type
galaxies. Adding to the puzzle, in a subsample of the gas-rich galaxies, the
warm ISM was frequently observed to be kinematically distinct from the main
stellar population, also strongly suggesting an external origin for this gas
(Caon et al., 2000).
### 1.3 Warm ISM Open Issues
The warm ISM is important to study in early-type galaxies and despite the
ambitious efforts of the studies described above and subsequent works, several
fundamental questions remain. It is critical to resolve these open issues
since the intertwined relationships between the cold, warm, and hot gas phases
and the stellar population ultimately defines galaxy evolution. Even though
the warm ISM mass is orders of magnitude less than the stellar mass,
discriminating between internal versus external origin of this material
provides pivotal data in formation models. Also, even though minute in mass
compared to the cold and hot phases of the ISM, the optical line emissions of
the warm ISM can locally dominate the energy output for a region. Finally, the
optical emission lines of the warm ISM provides information on the heating and
cooling in these systems which demands further exploration since the observed
hot ISM properties along with the non-detection of cooling flows creates a
“cooling crisis.”
The metallicity of the warm ISM is unknown. Because the metallicities of the
hot ISM and the stellar metallicities (cf. Trager et al. (2000b, a)) are
known, determining the metallicity of the warm ISM can create a link between
the gas phases and the stellar population. Also, the metallicity of the warm
ISM contains important discriminating information concerning the outstanding
issues that the key warm ISM studies were not able to resolve. Specifically,
the line ratios of the ionization species provides information on the
excitation mechanism of the gas; currently, both electron conduction and
photoionization remain viable excitation mechanism.
Also the metallicity of the warm ISM contains clues to the internal versus
external origin debate. The gas injected into the ISM from stellar mass loss
will generally be more metal rich than gas accreted from dwarf galaxies (cf.
Lequeux et al. (1979) and Aaronson (1986)).
Because of the important information contained in the metallicity of the warm
ISM and the unresolved issues concerning its nature and origin, we have
conducted a program to determine the oxygen metallicity of the $T\sim 10^{4}K$
gas in a small sample of early-type galaxies. Section 2 describes the sample
and observations. The data processing techniques and spectral analysis is
discussed in Section 3. Results from the emission line analysis is presented
in Section 4. In Section 5, we conclude with a discussion of the implications
of this work on both the origin of the warm ISM and the relationships between
the stellar population and the other phases of the ISM.
## 2 Observations
The MDM 2.4 meter Hiltner Telescope coupled with a Boller and Chivens
spectrograph was used to observe 21 nearby, early-type galaxies in four
observation runs over a two year period from 2000 to 2002. The sample was
selected to span a range of hot ISM properties as determined from the Brown &
Bregman (1998) X-ray study as well as a range of warm ISM properties
determined in the narrow-band $H\alpha$ imaging surveys by TdSA, G94-5 and
MFCF. The galaxies with their relevant fundamental properties are listed in
Table 1. Our warm ISM isolation and detection method depends on observing
galaxies that contain no emission line gas, therefore, several galaxies were
chosen to have very weak detections of the hot and warm phases.
Table 1: Early-type Galaxy Properties: Warm Phase Sample Galaxy | RAaaValues taken from NED (NASA/IPAC Extragalactic Database). | DecaaValues taken from NED (NASA/IPAC Extragalactic Database). | $B_{T}^{0}$bbRadial velocity distance corrected for local flows from Faber et al. (1989) unless otherwise noted. | DbbRadial velocity distance corrected for local flows from Faber et al. (1989) unless otherwise noted. | $log\,L_{B}/L_{\odot}$bbRadial velocity distance corrected for local flows from Faber et al. (1989) unless otherwise noted. | $log\,L_{x}$ccX-ray luminosity from Brown & Bregman (1998) unless otherwise noted. | $log\,L_{H_{\alpha}}$ddH$\alpha$ narrow-band imaging luminosity. | ${H_{\alpha}}$ Reference eeReference for H$\alpha$ narrow-band imaging luminosity with the following abbreviations: M96 $=$ Macchetto et al. (1996), G94 $=$ Goudfrooij et al. (1994b), and T91 $=$ Trinchieri & di Serego Alighieri (1991).
---|---|---|---|---|---|---|---|---
| (J2000.0) | (J2000.0) | | $(km/s)$ | $log~{}(erg~{}s^{-1})$ |
NGC 1407 | 03 40 11.8 | -18 34 48 | 10.57 | $1990\pm 187$ | 11.16 | 41.34 | 39.32 | M96
NGC 1600 | 04 31 39.8 | -05 05 10 | 11.79 | $4019\pm 489$ | 10.67 | 40.8411O’Sullivan et al. (2001). | 40.15 | M96
NGC 2768 | 09 11 37.5 | +60 02 15 | 10.93 | $1532\pm 325$ | 10.79 | 40.41 | $\cdots$ | $\cdots$
NGC 3115 | 10 05 13.9 | -07 43 07 | 9.95 | $1021\pm 215$ | 10.83 | 39.74 | $<37.6$ | M96
NGC 3377 | 10 47 42.3 | +13 59 08 | 11.13 | $857\pm 126$ | 10.21 | 39.42 | 38.95 | G94
NGC 3379 | 10 47 49.6 | +12 34 55 | 10.43 | $857\pm 126$ | 10.49 | 39.78 | 39.04 | M96
NGC 3489 | 11 00 18.3 | +13 54 05 | 11.1222RC3, de Vaucouleurs et al. (1991). | $1039\pm 101$22RC3, de Vaucouleurs et al. (1991). | 10.3522RC3, de Vaucouleurs et al. (1991). | $\cdots$ | 39.34 | M96
NGC 3607 | 11 16 54.3 | +18 03 10 | 10.53 | $1991\pm 242$ | 11.18 | 40.82 | 39.92 | M96
NGC 4125 | 12 08 05.8 | +65 10 27 | 10.58 | $1986\pm 295$ | 11.16 | 41.01 | 40.30 | T91
NGC 4261 | 12 42 02.4 | +11 38 48 | 11.32 | $2783\pm 590$ | 10.35 | 41.1822RC3, de Vaucouleurs et al. (1991). | 39.38 | G94
NGC 4374 | 12 25 03.7 | +12 53 13 | 10.13 | $1333\pm 71$ | 10.99 | 41.09 | 39.56 | G94
NGC 4406 | 12 26 11.7 | +12 56 46 | 9.87 | $1333\pm 71$ | 11.10 | 41.80 | 40.50 | T91
NGC 4472 | 12 29 46.8 | +08 00 02 | 9.32 | $1333\pm 71$ | 11.32 | 41.77 | 39.60 | T91
NGC 4494 | 12 31 24.1 | +25 46 28 | 10.69 | $695\pm 147$ | 10.20 | 39.28 | $\cdots$ | $\cdots$
NGC 4552 | 12 35 39.8 | +12 33 23 | 10.84 | $1333\pm 71$ | 10.71 | 40.92 | 39.26 | M96
NGC 4636 | 12 42 50.0 | +02 41 17 | 10.20 | $1333\pm 71$ | 10.96 | 41.81 | 39.69 | M96
NGC 4649 | 12 43 39.6 | +11 33 09 | 9.77 | $1333\pm 71$ | 10.96 | 41.48 | 39.83 | T91
NGC 4697 | 12 48 35.9 | -05 48 02 | 10.03 | $794\pm 168$ | 10.58 | 40.13 | 39.63 | G94
NGC 5044 | 13 15 23.9 | -16 23 08 | 11.25 | $2982\pm 314$ | 10.34 | 42.39 | 40.73 | M96
NGC 5322 | 13 49 15.2 | +60 11 26 | 11.09 | $1661\pm 352$ | 10.80 | 40.11 | 39.74 | G94
NGC 5846 | 15 06 29.2 | +01 36 21 | 10.67 | $2336\pm 284$ | 11.26 | 42.01 | 40.25 | M96
The configuration of the spectrograph was chosen to maximize galaxy light
input with a $2.1\arcsec$ wide by $5\arcmin$ long slit, while retaining the
ability to discriminate between the $H\alpha$ and [N II]$\lambda 6583$
emission lines. The available and appropriate grating for these requirements
was a 350 lines/mm grating blazed at $4026$Å, resulting in $7.1$Å/pixel and a
spectral range of $\sim 1600$Å over the 1200x800 pixel Loral CCD. The CCD was
characterized by relatively low read noise at seven electrons with the nominal
gain set at 2.1 electrons per ADU. The one drawback to the chosen spectrograph
configuration is that it was necessary to obtain separate blue and red spectra
for each galaxy in order to obtain all of the important emission lines from [O
II]$\lambda 3727$ to [Si II]$\lambda\lambda 6717,6731$. The grating is servo
controlled from the control computer and our tests indicated a $5$Å accuracy
in repositioning. Therefore, internal HgNe and Ne lamps were observed before
and after each grating reposition for wavelength calibration. Additional
calibrations included bias frames, evening and morning twilight flats when the
skies were clear, internal flats illuminated from an incandescent source, and
spectrophotometric standards.
The four observation runs were in March 2000, May 2000, February 2001, and
February 2002. Galaxy exposure times and program-type (E $=$ emission-line, T
$=$ Template, or L $=$ LINER) are listed in Table 2. The meaning of the
program types is discussed below. During the first three runs, the nights were
cloudy and over half of the full run was lost to weather and instrument
problems. Some of these observations are of marginal value and only reported
here for completeness. The final run was entirely photometric.
Table 2: MDM 2.4 Meter Observations Galaxy | Exp Blue (s) | Exp Red (s) | Emission Type aaEmission Type. Column display how the observations were ultimately used: T= template galaxy, L=LINER galaxy, E=emission galaxy, W=weak emission.
---|---|---|---
NGC 1407 | $8400$ | $6600$ | T
NGC 1600 | $8400$ | $5400$ | W
NGC 2768 | $12000$ | $11400$ | L
NGC 3115 | $6000$ | $4500$ | T
NGC 3377 | $7200$ | $6000$ | W
NGC 3379 | $6000$ | $6000$ | T
NGC 3489 | $6000$ | $4500$ | E
NGC 3607 | $9600$ | $7200$ | E
NGC 4125 | $12000$ | $10800$ | L
NGC 4261 | $6000$ | $4500$ | E
NGC 4374 | $6000$ | $4500$ | E
NGC 4406 | $9600$ | $6000$ | W
NGC 4472 | $9600$ | $6000$ | W
NGC 4494 | $1200$ | $0$ | W
NGC 4552 | $6000$ | $6600$ | W
NGC 4636 | $6000$ | $9900$ | E
NGC 4649 | $6000$ | $4500$ | W
NGC 4697 | $6000$ | $1800$ | W
NGC 5044 | $6000$ | $3600$ | E
NGC 5322 | $9600$ | $9900$ | W
NGC 5846 | $7200$ | $6600$ | E
The observing strategy was chosen to suppress spurious emission line
detections and obtain uniform observations of program and template galaxies.
For each galaxy, we attempted to obtain five integrations, ensuring that
reasonable statistics could be employed to eliminate cosmic rays from the
combined data. Because the emission lines are faint and spread over only a few
pixels in both spectral and spatial directions, a fortuitous single cosmic ray
can result in a spurious warm ISM detection. The total integration time was
chosen to result in a S/N of 5-10 in the template subtracted spectra (See
Section 3). We did not constrain the angle of the slit due to complications
with the instrument rotator during the first two runs; this has little
programatic impact because of the choice of a wide slit, and the non-
preferential orientation of the warm ISM observed in the narrow-band imaging
surveys (TdSA, G94-5, MFCF).
## 3 Reductions and Spectral Analysis
The data were reduced in the standard manner using tasks within IRAF (overscan
fitting and subtraction, bias frame construction and subtraction). Each of the
runs was calibrated separately but in a similar manner. Internal lamps were
used to create a response image and divided through the data, eliminating
differences in pixel-to-pixel sensitivity on the CCD. Twilight flats were used
to construct an illumination image, correcting for large scale structure and
slit illumination. For the first two runs, the incandescent internal lamp
produced too few photons in the far blue and only added noise to the data. For
these runs we used wavelength calibrated twilight flats to produce response
images. A HgNe lamp was used for wavelength calibration of blue-tuned
observations and a Ne lamp was used for red-tuned observations. Residuals in
the linear, spectral solution were much better (typically $<1$Å) than the
resolution element ($\sim 8$Å) over the entire chip.
The galaxies were summed over the spatial dimension based on the visual
inspection of the 2-D spectra, with background regions selected from the outer
regions of the chip. Typically the summed regions for the galaxies were about
$1.2\arcmin$ of the $5.2\arcmin$ unvignetted slit length. The spectra were
flux calibrated from observations of spectrophotometric standards (Massey et
al., 1988) and corrected for extinction with the standard correction for Kitt
Peak distributed with IRAF 2.11.
Figure 1: Composite Blue and Red Spectra of NGC3489. Important emission lines
that are visible in the un-subtracted data are marked aobve the continuum
level. In addition, calcium H & K lines are marked for reference.
Custom code was developed to implement the combining of the individual 1-D
spectra, template matching and subtraction, as well as the extraction of line
fluxes. The 1-D spectra combination procedure employed an iterative
$1.5\sigma$ rejection algorithm which eliminated cosmic rays that were
overlooked in the aperture summing rejection methods. The wavelength
calibration was tuned to night sky lines and the measured redshift was checked
and tuned to strong stellar population lines (e.g. Ca H&K in the blue and [N
II]$\lambda 6583$ in galaxies showing emission lines in the red). An example
summed spectra is shown in Figure 1.
Although we have upper-limits from the $H\alpha$ imaging surveys, it is
unknown which galaxies are void of warm ISM emission. Therefore, the
determination of template galaxies, or galaxies without any detected warm ISM,
is an involved and iterative process. First the $H\alpha$ and [N II]$\lambda
6583$ region of the spectra is examined for obvious emission. Using the local
continuum, upper limits on the line fluxes are determined for galaxies with no
obvious $H\alpha$ and [N II]$\lambda 6583$ emission. Ten of the original 21
galaxies match this criteria and act as potential template galaxies. We take
these ten galaxies and combine them into a master template galaxy. When
combined into the template galaxy, the strongest of these weak emitters will
not be included in the template creation because of the aggressive clipping
algorithm in the combination routine. To determine the the weak emitters the
template is subtracted from the individual galaxy observations and examined
for emission lines. Three of the ten template galaxies revealed oxygen lines
and were removed from the master template. This process was repeated with the
emitters removed and reveal four more weak emitters, leaving three galaxies
without emission line activity: NGC 1407, NGC 3115, NGC 3379. As an
independent check of our template definition methodology, all seven of the
potential template galaxies were subtracted from the galaxy with the strongest
emission lines (NGC 2768) and we determined that the three template galaxies
selected resulted in the high reported fluxes (i.e. lowest background).
The three template galaxies, NGC 1407, NGC 3115 and NGC 3379, lack of warm
emission is consistent with results from the narrow-band imaging detections
for these galaxies. NGC 3379 is reported to have a weak $H\alpha$ in MFCF but
an upper limit below this reported detection is determined in G94-5. NGC 3115
is detected only as an upper limit by MFCF and has been previously used as a
stellar template galaxy by the Palomar spectral survey of the nuclear regions
of 500 nearby galaxies (Ho et al., 1993). NGC 1407 is marginally detected in
both the G94-5 and MFCF studies, but the fluxes reported differ by a factor of
five. The inconsistencies of weak detections in the narrow-band surveys reveal
intrinsic limitations of the narrow-band technique. Broad-band red imaging is
used to subtract an underlying stellar population from the narrow-band data; a
process which is susceptible to scaling errors. The stellar template
constructed from our sample contains $<10^{-16}~{}erg\,s^{-1}\,cm^{-2}$
combined $H\alpha$ and [N II]$\lambda 6583$ flux; this is an order of
magnitude more sensitive as a non-detection than the narrow-band imaging
surveys.
Once the template galaxies are defined, the stellar population was subtracted
from each of the other galaxies, revealing emission lines and any differences
in stellar populations. An example subtraction for NGC 4374 is shown in Figure
2. In all galaxies, the residuals in the subtractions are flat with large-
scale variation at a 8-10% level in the blue and better than 5% in the red.
Once the subtractions are made, the relative line fluxes are measured by
defining a low order polynomial to describe the local background and using a
gaussian profile to extract the line flux. A gaussian decomposition is only
necessary in the crowded [N II]$\lambda 6548$, $H\alpha$, [N II]$\lambda 6583$
region and the fitting method gives equivalent results to raw counts above a
background for all other lines. Note that because of the coarse spectral
sampling and the relatively low velocity dispersions, the width of the
gaussian profiles contain no useful kinematic information and is thus not
reported. In eight of the non-template galaxies, the signal-to-noise in the
data are insufficient for emission line work, or only H$\alpha$ is observed.
For the rest of the sample, extracted relative line fluxes normalized to
$H\beta$ are reported in Table 3. The reported flux ratios have been
dereddened based on an average value for the reddening from Burstein & Heiles
(1984) and Schlegel et al. (1998). The observed relative fluxes reported here
do not correct for internal galaxy reddening, displaying the Balmer decrements
as observed; Below we apply the correction (See Section 4.2). The errors
reported are determined by calculating the range of acceptable background
levels within the noise of the local continuum for isolated lines and
combining this background uncertainty in quadrature with the range of
acceptable gaussian widths that reproduce the total observed line flux for the
crowded [N II]-$H\alpha$ complex.
Table 3: Observed Nebular Emission Line Ratios Relative to H$\beta$ Galaxy | [O II] | [Ne III] | $H\beta$ | [O III] | [O III] | [O I] | [N II] | $H\alpha$ | [N II] | [S II] |
---|---|---|---|---|---|---|---|---|---|---|---
| $\lambda 3727$ | $\lambda 3869$ | $\lambda 4862$ | $\lambda 4959$ | $\lambda 5007$ | $\lambda 6300$ | $\lambda 6548$ | $\lambda 6563$ | $\lambda 6583$ | $\lambda\lambda 6717,6731$ |
NGC 2768 | $70\pm 7$ | $14\pm 6$ | $10\pm 3$ | $1\pm 4$ | $24\pm 4$ | $8\pm 2$ | $20\pm 4$ | $29\pm 5$ | $45\pm 7$ | $50\pm 8$ |
NGC 3489 | $19\pm 5$ | $2\pm 3$ | $10\pm 2$ | $2\pm 5$ | $13\pm 5$ | $3\pm 2$ | $30\pm 8$ | $34\pm 5$ | $57\pm 8$ | $37\pm 7$ |
NGC 3607 | $32\pm 8$ | $6\pm 3$ | $10\pm 3$ | $7\pm 3$ | $20\pm 4$ | $2\pm 4$ | $25\pm 5$ | $48\pm 6$ | $59\pm 8$ | $22\pm 5$ |
NGC 4125 | $74\pm 8$ | $14\pm 4$ | $10\pm 2$ | $23\pm 5$ | $71\pm 10$ | $3\pm 3$ | $19\pm 4$ | $34\pm 8$ | $40\pm 5$ | $23\pm 7$ |
NGC 4261 | $16\pm 4$ | $6\pm 5$ | $10\pm 3$ | $6\pm 2$ | $12\pm 3$ | $3\pm 2$ | $15\pm 5$ | $22\pm 8$ | $39\pm 5$ | $15\pm 6$ |
NGC 4374 | $16\pm 4$ | $<1$ | $10\pm 2$ | $5\pm 2$ | $12\pm 4$ | $2\pm 4$ | $21\pm 9$ | $35\pm 6$ | $49\pm 8$ | $29\pm 5$ |
NGC 4636 | $24\pm 5$ | $2\pm 4$ | $10\pm 3$ | $4\pm 4$ | $13\pm 3$ | $2\pm 5$ | $10\pm 3$ | $33\pm 5$ | $49\pm 7$ | $21\pm 5$ |
NGC 5044 | $24\pm 3$ | $2\pm 3$ | $10\pm 2$ | $6\pm 2$ | $20\pm 3$ | $3\pm 2$ | $33\pm 8$ | $32\pm 6$ | $53\pm 8$ | $33\pm 5$ |
NGC 5846 | $26\pm 5$ | $2\pm 3$ | $10\pm 4$ | $6\pm 3$ | $14\pm 4$ | $<1$ | $8\pm 5$ | $29\pm 4$ | $36\pm 8$ | $13\pm 10$ |
Figure 2: Blue (left) and Red (right) spectra of NGC4374 with template
galaxies subtracted, removing the stellar continuum. Large residuals at 5500Å
and 6800Å are due different night sky lines positions in the rest wavelength
of NGC4374 and template galaxies. The large residual at 3900Å is due to the
imperfect matching between the Ca H&K lines of the template galaxies and NGC
4374.
## 4 Warm ISM Emission Line Results
### 4.1 Excitation Mechanism
#### 4.1.1 Normal Galaxies
There are many possible ionization mechanisms for the warm ISM that have been
previously explored and ruled out by the key warm ISM studies (Section 1.2)
and other works. Photoionization by hot young stars (Kim, 1989; Shields, 1991)
is inconsistent with both the optical colors (G94-5; MFCF) and integrated
optical spectra (Sadler, 1987; Heckman et al., 1988). Ionization from
radiation associated with an AGN (Fosbury et al., 1982) is ruled out by the
large radial extent of the warm ISM observed in the narrow-band imaging (TdSA;
G94-5; MFCF). Shock excitation (Ford & Butcher, 1979; Heckman et al., 1989)
would produce filamentary ionization regions, where smooth emission is
observed (TdSA; G94-5; MFCF). Further, the the input energy from shocks is too
low by two orders of magnitude for typical densities and velocities as noted
by Sparks et al. (1989). Condensation out of the hot phase and into warm phase
filaments has been investigated for a number of X-ray clusters (Cowie et al.,
1980; Hu et al., 1985) but the large number of recombinations required per
hydrogen atom (e.g. Johnstone et al. (1987); TdSA and references therein) is
inconsistent with the observed recombination rates for early-type galaxies
(Sparks et al., 1989).
The three remaining viable excitation mechanisms for the warm ISM are
photoionization from PAGB stars, electron conduction from the hot phase (e.g.
Sparks et al. (1989)), and photoionization by extreme UV (EUV) photons from
the hot phase (Voit & Donahue, 1990; Donahue & Voit, 1991). The data in the
key warm ISM studies are equally consistent with all of these ionization
mechanisms. However, photoionization by PAGB stars is the preferred mechanism
since these old stars are known to populate the galaxies and given the
significant correlation of $L_{H\alpha}$ with $L_{B}$ within a given region
(MFCF) strongly indicates a stellar ionizing source. Further, the presence of
the far ultraviolet (UV) flux has been confirmed in a small sample of 30
early-type galaxies with _International Ultraviolet Explorer_ (IUE) (Burstein
et al. (1988); For a review of Far UV flux in early-type galaxies see
O’Connell (1999)).
However, to adopt PAGB stars as the ionizing source, excludes the observed
correlations between $H\alpha$ and X-ray luminosity (TdSA; (Goudfrooij, 1997),
MFCF). The correlations between the X-ray and warm ISM morphologies (Singh et
al., 1995; Trinchieri et al., 1997; Trinchieri & Goudfrooij, 2002) provide
additional support for photoionization of the warm ISM by the EUV flux from
the hot ISM, but it is possible to construct physical conditions that
reproduce these morphological links (e.g. cooling of the hot ISM). Although
there have been detected correlations between the $L_{H\alpha}$ and
$L_{X}/L_{B}$, the correlations contain much more scatter and are less
significant than the correlation between $L_{H\alpha}$ with $L_{B}$ within a
given region. The large amount of scatter in the X-ray relationship and the
known high $L_{X}/L_{B}$ galaxies which contain weak or no $H\alpha$ emission
and strong $H\alpha$ emitters with low $L_{X}/L_{B}$ provide insight that the
hot and warm ISM link is more tenuous than the link between the stars and the
warm ISM. Finally, the key studies in in Section 1.2 and subsequent work
(Sarzi et al., 2006), indicate a strong connection between the dust (cold ISM)
and the emission line regions, but there are not strong correlations between
$L_{FIR}$ and $L_{X}/L_{B}$ in any of the key studies or in recent Spitzer
Observations (Temi et al., 2007).
Emission line ratios of the different ionization species have the ability to
discriminate between collisionally excitation and photoionization. Comparing
our sample’s line ratios to line ratios predicted for photoionization models
of Allen et al. (1998), Allen et al. (1999), and Binette et al. (1996) and
shock ionization of Allen et al. (2008), we conclude the data are consistent
with photoionization. Unfortunately, these data are not discriminating due to
the large errors in the data and the region populated by these galaxies
happens to land in an overlap region for collisional and photoionization.
Additional support for the photoionization of this gas is seen with the Far
Ultraviolet Space Explorer (FUSE) (Bregman et al., 2005). The observed UV
radiation field produces sufficient photons shortward of 912Å to ionize the
warm ISM and is consistent with the number of predicted extreme horizontal-
branch stars, or equivalently PAGB stars.
When the data presented here is combined with the literature, it presents a
reasonable case for photoionization of the warm ISM. This is the preferred
ionization mechanism in the literature and in discriminating between the two
sources of photoionization, PAGB stars or the hot ISM, it would seem more
probable that PAGB stars provide the necessary ionizing photons.
#### 4.1.2 LINER Galaxies
Four of the galaxies listed in Table 3 are reported as low ionization nuclear
emission-line regions (LINERs) galaxies in the literature: NGC 2768 (Heckman,
1980), NGC 3115 (Ho et al., 1993), and NGC 4125 and NGC 4261 (Ho et al.,
1995). The question of identical treatment of these LINER galaxies and the
normal galaxies arises. LINERs are strictly defined to be galaxies with line
ratios of [O II]$\lambda 3727$ greater than [O III]$\lambda 5007$ and [O
I]$\lambda 6100$ greater than one-third [O III]$\lambda 5007$ (Heckman, 1980).
While useful for separating classes of objects, the LINER definition as noted
by its creator, Heckman, has no physical basis and in reality only provides a
crude boundary between photoionization and a low-level AGN activity.
For our purposes, the LINER identification acts as a flag for further
investigation, and not as an automatic exclusion filter. The line ratios for
the four LINER galaxies are consistent within the photoionization, but the
excitation mechanism and thus the valid regions in diagnostic diagrams are
still uncertain and being debated for LINERs (for a review of the physical
mechanisms behind LINERs see Filippenko (1996) and Barth (2002)). Although it
is not entirely clear what to expect for LINER galaxies, the presence of
strong collisionally excited lines casts a cloud of doubt of pure
photoionization. The [Ne III]$\lambda 3869$ line is an indicator of the
importance of collisional excitation because the collisional cross-section is
over an order of magnitude larger than the photoionization transition
probability for this line. NGC 2768 and NGC 4125 have significant [Ne
III]$\lambda 3869$ fluxes and thus we conclude that these systems have a more
complex physical structure and excitation mechanism than photoionization.
These two galaxies are excluded in the subsequent sections where pure
photoionization is assumed. Note that NGC 4125 and NGC 4261 show no
distinguishing line ratios from the normal galaxies and, the data presented
here do not support the LINER classification. These two galaxies were
classified as LINERs by (Ho et al., 1995) which had a similar observational
aperture as this study, but only had wavelength coverage down to 4200Å,
excluding the useful [O II]$\lambda 3727$ information.
### 4.2 Oxygen Metallicity
For those galaxies where photoionization is the dominant ionization mechanism,
the line ratios can be used to determine the metallicity of the warm phase
gas. There are two critical assumptions to be addressed before using these
line ratios and photoionization models to determine global metallicities. The
first assumption is that UV flux from PAGB stars ionizing a diffuse ISM on a
galactic scale is well represented by photoionization models that were
developed for Str$\ddot{{o}}$mgren-sphere-type H II regions. First, we note
that the models are determining the balance between photoionization by UV
photons and recombinations to the ions; this is fundamentally the same
equation in a galaxy-wide ISM as an HII region, given a source of ionizing
photons internal to a volume of gas. The UV flux and electron densities of the
two environments are similar, although, PAGB stars have different spectral
shapes than high mass main sequence stars, but McGaugh (1991) shows that the
photoionization models are largely insensitive to the spectral shape of the
ionizing star. The models assume the gas is ejected from the ionizing star
(and thus of the same metallicity), this is a feature used to calibrate the
observed line ratios, but a direct link between the ionizing source and the
ejecta is not a necessary condition to employ the models. We argue that the
photoionization models will be applicable to the warm ISM, but we readily
acknowledge that there are differences in these two environments which will
lead to will lead to greater uncertainties in its application here.
The second key assumption is that the analysis of the linear combination of
many individual diffuse ionized regions observed in an integrated galaxy
spectra produces a result that is indicative of the mean properties of the
observed regions. Kobulnicky et al. (1999) conducted a study addressing these
issues in nearby galaxies, observing H II regions individually through small
apertures and globally with drift scanning with a long slit. The authors
conclude that global metallicity determinations, accurate to $0.2$ dex, are
possible with long-slit, integrated spectra.
Table 4: R23 parameter and Oxygen Metallicity Galaxy | log R23 | [O/H] | $Z_{Oxygen}/Z_{\sun}$
---|---|---|---
NGC 3489 | $0.63^{+0.11}_{-0.14}$ | $-3.26\pm 0.2$ | $1.13^{+0.49}_{-0.30}$
NGC 3607 | $0.93^{+0.08}_{-0.11}$ | $-3.66\pm 0.2$ | $0.43^{+0.29}_{-0.18}$
NGC 4261 | $0.51^{+0.06}_{-0.07}$ | $-3.14\pm 0.2$ | $1.44^{+0.47}_{-0.29}$
NGC 4374 | $0.58^{+0.09}_{-0.10}$ | $-3.20\pm 0.2$ | $1.26^{+0.50}_{-0.31}$
NGC 4636 | $0.68^{+0.09}_{-0.11}$ | $-3.31\pm 0.2$ | $0.98^{+0.41}_{-0.26}$
NGC 5044 | $0.75^{+0.05}_{-0.06}$ | $-3.37\pm 0.2$ | $0.86^{+0.36}_{-0.24}$
NGC 5846 | $0.68^{+0.08}_{-0.09}$ | $-3.30\pm 0.2$ | $1.00^{+0.37}_{-0.26}$
Metallicity determinations of ionized gas in individual H II regions are built
upon photoionization models, determining ratios of ionized species relative to
hydrogen. The most common diagnostic is the $R23$ parameterization,
$R23\equiv(([O\,II]\,\lambda 3727+[O\,III]\,\lambda\lambda
4959,\,5007)/H\beta)$, defined by Pagel et al. (1979) and refined by McGaugh
(1991). This parameterization is convenient since it employs the readily
available optical oxygen lines, however, it requires the [O II]$\lambda
3727$/[N II]$\lambda 6583$ information to resolve a degeneracy present in the
$R23$ models.
We take the line ratios reported in Table 3 and correct for reddening internal
to the galaxy using the interstellar extinction curve of Savage & Mathis
(1979) and assuming case B ratio for the $H\alpha$ to $H\beta$ at $T=10^{4}$
(Osterbrock, 1989). For NGC 4261 no reddening correction is applied, as the
Balmer decrement indicates a non-physical negative reddening.
These dereddened lines are used to generate $R23$ values for the galaxies.
Because of the faint nature of the $[O\,III]\,\lambda 4959$ line, we used the
conical, equilibrium value of 1/3 the line flux of $[O\,III]\,\lambda 5007$
when this line was buried in the noise of the subtracted spectra. All of the
galaxies end up on the upper branch of the $R23$ curve as indicated by the [O
II]$\lambda 3727$/[NII]$\lambda 6583$ diagnostic. To determine metallicities,
we employ McGaugh (1991) calibration as parameterized by Kobulnicky et al.
(1999) and reported the metallicity results in Table 4. For the zero point of
the solar metallicity we employ a solar photospheric oxygen abundance of
[O/H]$=-3.3$ (Asplund et al., 2005; Scott et al., 2008), but note the
uncertainty and current debate about the solar determinations. The errors
reported on the oxygen metallicities in these galaxies are at least $0.2$ dex
as these are the dominating and intrinsic uncertainties in the models used to
create the $R23$ grid and its application to global galaxy spectra (cf.
Kobulnicky et al. (1999)). The median metallicity for the seven early-type
galaxies is O/O☉=1.01. Within these $0.2$ dex absolute errors, the none of the
galaxies are significantly different from one another.
## 5 Implications and Discussion
The determined mean oxygen metallicity of solar for the warm phase of the ISM
has implications for the origin of this gas and its relationship to other
phases of the ISM and the stellar population. The limitations of this study
are apparent with a small sample of seven galaxies, large $0.2$ dex errors,
and a less than ideal application of H II region models to the warm ISM.
However, this work does present new information as it appears that there are
no determination of the metallicity of the warm ISM in early-type galaxies in
the literature. Sarzi et al. (2006) conducted the most recent large survey of
ionized gas in early-type galaxies and conclude at the end of the study that
the origin of the ionized gas is still unsolved and state that the metallicity
provides an insightful clue. Below we consider the implications of a near
solar metallicity for the warm ISM.
### 5.1 Origin of the Warm ISM
A warm ISM with a solar metallicity argues against an external origin model
for the emission line gas. The accretion galaxies involved in minor mergers
are presumably dwarf irregular (dIrr) galaxies, since dwarf spheroidal (dSph)
galaxies are most frequently devoid of gas. Gaseous and dusty galaxies with
near solar metallicity are luminous ($>10^{10}L_{\sun}$) and massive(Garnett,
2002). Mergers of these types of objects produce profound disturbances, which
are simply not observed in these early-type galaxies with $H\alpha$ emission.
It is possible that an intense gas enrichment by supernovae type-II occurs
once low metallicity gas is accreted, however, the broad-band colors and
global galaxy spectra (G94-5, MFCF) show little evidence for major star
formation. Further, the observed supernovae rates are too low to account for
such a drastic enrichment (Cappellaro et al., 1999).
We can compare a solar metallicity for the warm ISM with metallicities of the
hot ISM and the stars. The stellar population in early-type galaxies is
generally observed to have a solar or super-solar metallicity (Trager et al.,
2000b, a). A number of the seven galaxies observed in this work have stellar
Fe and $\alpha$ metallicity measurements in the literature, although, there
are some inconsistencies between authors (NGC 4261 is measured to have
$[Fe/H]$ of -0.03, 0.275, 0.29, and 0.188 by Trager et al. (2000b), Thomas et
al. (2005), Howell (2005), and Sánchez-Blázquez et al. (2006), respectively.)
Not all the galaxies have measured stellar metallicities but a five galaxy
average from all of the determinations in the 2005 and 2006 papers listed
above is $[Fe/H]=0.22$, ranging from $[Fe/H]=0.099$ to $[Fe/H]=0.44$
($[\alpha/H]$ is similar). It is important to remember that Trager et al.
(2000a) estimates the _absolute_ error on stellar metallicity determinations
close to a factor of two, while the relative errors are much smaller. The
_Chandra_ X-ray Observatory has the spatial resolution to resolve and remove
the stellar component from the gas which is leading to better data and a
revision of our (mis)understanding of the metallicity in the hot ISM.
Recently, Humphrey & Buote (2006) analyzed a sample of 28 early-type galaxies
observed with _Chandra_ and determine that the hot ISM abundances are
consistent and perhaps higher than the stellar abundances. In summary, a
$0.50$ to $1.50$ solar metallicity in the warm ISM is consistent with both hot
ISM and stellar metallicity determinations.
It has been thought that early-type galaxies would have a problem generating
and retaining dust and warm gas, due to the hostile conditions provided by the
hot ISM and the low injection rate of new material into the ISM from stars.
Parriott & Bregman (2008) have shown that is it possible for dust to condense
in the circumstellar envelopes of AGB stars and then subsequently be injected
into the hot ISM. Although this dust has a short lifetime in the hot ISM, the
injection rate has been observed to be greater than the destruction rate and
could account for the total observed mass given a simple evolution of this
injection rate given that more massive stars evolved off the main sequence in
the past (Athey et al., 2002). An internal origin for the dust is supported by
the G94-5 and MFCF studies indicating that the majority ($\sim 90\%$) of the
dust is distributed smoothly throughout the galaxy. The internal origin for
the cold ISM is further confirmed by a 2.29$\mu$m CO absorption feature study
that finds a lower velocity dispersion for the dust than that of the stars in
a sample of 25 nearby early-type galaxies (Silge & Gebhardt, 2003). Finally
from the analysis of _Spitzer_ observations of 46 galaxies in the FIR
($24\mu$m, $70\mu$m, and $160\mu$m), Temi et al. (2007) propose an AGB mass
loss, internal origin for the observed dust, which sinks to the core of the
galaxy and is periodically driven out by AGN activity. This outward transport
of dust may cool the hot gas down to the warm phase.
The major stumbling block to a galaxy internal origin for dust and gas is the
observations of gas kinematics which are distinct with respect to the stars.
Recent observational evidence has suggested that there may be different
evolutionary tracks for early-type galaxies. The classification of early-type
galaxies into fast and slow rotation categories by Emsellem et al. (2007) and
the key ingredient of gas for the formation and evolution of fast rotators
provides an important dividing line for future discussions on the origin of
the ISM in early-type galaxies.
The authors would like to acknowledge useful discussions with P. Goudfrooij
and E. D. Miller. We would also like to thank an anonymous referee for
suggestions that significantly improved this paper.
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|
arxiv-papers
| 2009-03-24T21:29:45 |
2024-09-04T02:49:01.393550
|
{
"license": "Public Domain",
"authors": "Alex E. Athey, Joel N. Bregman",
"submitter": "Alex Athey",
"url": "https://arxiv.org/abs/0903.4197"
}
|
0903.4344
|
# First Determination of the True Mass of Coronal Mass Ejections: A Novel
Approach to Using the Two STEREO Viewpoints
Robin C. Colaninno George Mason University, Fairfax, VA 22030, USA
robin.colaninno@nrl.navy.mil Angelos Vourlidas Code 7663, Naval Research
Laboratory, Washington, DC 20375, USA vourlidas@nrl.navy.mil
(Received –; Revised –; Accepted –)
###### Abstract
The twin Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI)
COR2 coronagraphs of the Solar Terrestrial Relations Observatory (STEREO)
provide images of the solar corona from two view points in the solar system.
Since their launch in late 2006, the STEREO Ahead (A) and Behind (B)
spacecrafts have been slowly separating from Earth at a rate of 22.5 degrees
per year. By the end of 2007, the two spacecraft were separated by more than
40 degrees from each other. At this time, we began to see large-scale
differences in the morphology and total intensity between coronal mass
ejections (CMEs) observed with SECCHI-COR2 on STEREO-A and B. Due to the
effects of the Thomson scattering geometry, the intensity of an observed CME
is dependent on the angle it makes with the observed plane of the sky. From
the intensity images, we can calculate the integrated line of sight electron
density and mass. We demonstrate that is is possible to simultaneously derive
the direction and true total mass of the CME if we make the simple assumption
that the same mass should be observed in COR2-A and B.
Sun: coronal mass ejections, methods: data analysis, techniques: image
processing
††copyright: ©2008: ††slugcomment: To be submitted to ApJ
## 1 INTRODUCTION
Coronal mass ejections (CMEs) have been extensively studied and their general
properties are well known after a complete solar cycle of observations with
the Large Angle and Spectrometric Coronagraphs (LASCO; Brueckner et al, 1995).
There exists a large body of literature detailing the speed, width, and
position angle of individual events as well as statistics for larger samples
(St. Cyr et al., 2000; Yashiro et al., 2004). There are fewer studies on CME
mass and consequently on their kinetic energy. Vourlidas et al. (2000, 2002)
and Subramanian and Vourlidas (2007) published statistics on the mass and
energies of LASCO CMEs and described their analysis methods.
Because the CME observations are the projection of the three dimensional
erupting structure on the plane of the sky, the measured (width, height,
brightness) and derived (speed, mass, energy) quantities are also projected on
the plane and represent lower limits of the true, un-projected CME properties.
The projection effects on these quantities can be estimated by making
assumptions about the CME propagation direction and shape, but the true three
dimensional properties of the CME remains difficult to estimate reliably
(Vršnak et al., 2007).
In the case of CME total mass, Vourlidas et al. (2000) showed that CME masses
are underestimated by about a factor of two, for most cases. This estimation
was supported by three dimensional magnetohydrodynamics model calculations by
Lugaz et al. (2005). But the models are idealized representations of the CME
structure and are subject to many assumptions, leaving some doubts about the
fidelity of mass and kinetic energy measurements. Multiple viewpoint
observations of CMEs offer the best way so far to derive their true properties
and quantify the validity of the projected CME measurements.
The twin Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI)
COR2 coronagraphs (Howard et al., 2008) of the Solar Terrestrial Relations
Observatory (STEREO; Kaiser et al., 2008) provide such observations. Since
their launch in late 2006, the STEREO Ahead (A) and Behind (B) spacecraft have
separated from Earth at a rate of $22.5^{\circ}$ per year. By the end of 2007,
the two spacecraft were separated from each other by more than $40^{\circ}$.
At that time, we began to see large-scale differences in both the morphology
and total intensity between the same CMEs observed with SECCHI-COR2 on
STEREO-A and B.
The differences in the CME morphology seen by SECCHI are the result of
projecting the complex optically thin structure of the CME through the
different lines of sight of the COR2-A and B coronagraphs. However, the
differences in the total intensity of the CME are mostly due to the different
Thomson scattering geometry through the CME plasma. It has long been
established that the visible emission of the K-corona originates from the
scattering of photospheric light by the coronal electrons (Minnaert, 1930; van
de Hulst, 1950; Billings, 1966) via the Thomson scattering mechanism (Jackson,
1997). The scattering strength for a given electron depends on the angle
$\chi$, between the vector from the electron to the observer and the radius
from the electron to the center of the Sun and the distance from the electron
to the Sun. Along any line of sight (LOS), the maximum emission at a fixed
radial distance occurs at the point $\chi=90^{\circ}$. Within the field of
view of a coronagraph, the maximum emission is approximately along a plane.
This plane is referred to as the plane of sky (POS) of the observer. Away from
the POS, the scattering efficiency decreases. The angle along the LOS away
from the POS is $\theta$. Thus the observed intensity of a CME is dependent on
the angle, $\theta$, its electrons make with the POS. From the intensity
images, we can calculate the electron density and mass for various values of
$\theta$. Historically, mass estimates have been calculated for the $\theta=0$
condition, which is the minimum value of the mass. Corrections for this
conditions where $\theta>0$, increases the true mass.
A goal of the STEREO mission are to determine true properties of CMEs,
including their propagation direction. Ultimately, these goals can be achieved
by full three dimensional reconstruction of the CMEs. In this paper, we
present a novel way to use the two viewpoints of STEREO to locate the CME in
longitude. We simply require that the total mass of a CME be the same when the
mass calculation is corrected for the two viewpoints. We further demonstrate
that in doing so it is possible to simultaneously derive the direction and
true total mass of the CME. In § 2, we begin by calibrating our mass
calculations by comparing the total mass measurements from SOHO-LASCO and
SECCHI. In § 3, we describe in detail our method for estimating the direction
and total mass of the CME using two viewpoints. In this section, we also
present the results for eight CMEs observed in COR2. In § 3.2, we give an
expression for the dependence of the de-projected CME mass with height.
Finally, in § 4, we present a discussion of our method and conclusions of our
work in this paper.
## 2 Mass Analysis Procedure for SECCHI-COR2 Images
To calculate the total mass of a CME, we first calibrate the images to the
customary units of mean solar brightness. We then subtract from the event
sequence an image just prior to the appearance of the CME. This subtraction
removes the background F-corona, static K-corona and any residual stray light
that has not been removed during the calibration. Thus, we are left with the
brightness changes caused by the CME.
Because we do not know their distribution along the LOS a priori, we must make
the usual assumption that all the electrons are located on the POS. We can
then estimate the number of electrons by taking the ratio of the observed
brightness ($B_{obs}$) to the brightness of a single electron at a given
angular distance, $B_{e}(\theta)$. The brightness, $B_{e}(\theta)$, is
calculated analytically from the scattering geometry using the equations in
Billings (1966). To convert the electron density to mass, we assume that the
ejected material comprises a mix of completely ionized hydrogen and $10\%$
helium. The mass at each pixel in the image is then calculated using the
equation :
$m={{B_{obs}}\over{B_{e}(\theta)}}\times 1.97\times 10^{-24}g.$ (1)
We note that there are two significant advantages of these mass (or electron
density) images. First, instrumental effects such as vignetting are removed
and secondly the effect of Thomson scattering is removed. Consequently, the
image brightness is directly related to the number of electrons along the LOS,
regardless of where it is located in the field of view.
Once the brightness value of each pixel in the image is converted to grams, we
calculate the total mass by summing the values in the region of the image
containing the CME. We perform this procedure for all the images of a time
sequence until the leading edge of the CME leaves the COR2 field of view. As
an example, Figure 1 shows the calibrated mass images for the eight CMEs that
we studied in COR2-A and B. The dependence of the CME appearance on the
viewing angle is evident in most events, especially on the 2008 April 26
event.
### 2.1 Cross-Calibration with LASCO Mass Calculations
To verify our mass analysis procedure for SECCHI-COR2 images, we compare COR2
mass measurements to LASCO C2 and C3 measurements. Validation of the COR2 data
is a necessary step since this is the first time that mass measurements from
the COR2 instruments have been presented. The availability of concurrent LASCO
observations is fortunate for the analysis of SECCHI data since the
calibration of the LASCO coronagraphs is very well known (Morrill et al.,
2006) and CME masses have been studied with LASCO data (Vourlidas et al.,
2000, 2002). For the cross-calibration, we chose events that occurred early in
2007 when the STEREO spacecraft were closest to the Sun-Earth line and the
SOHO spacecraft. The POS is essentially the same for all instruments since the
STEREO spacecraft were $\leq 2^{\circ}$ from Earth. In the next section, we
will explore the differences in the observed intensities caused by the
separation of STEREO from Earth.
We calculated the total mass using the procedure describe in the previous
section. The results for the four coronagraphs (LASCO C2 and C3, COR2-A and B)
are shown in Figure 2. The LACSO C2 has field of view (FOV) from $\sim 2.5$ to
$7R_{\odot}$ and the LASCO C3 has a FOV from $\sim 4.0$ to $30R_{\odot}$. For
the cross comparison, we choose to compare the COR2-A data to the LASCO C2 and
the COR2-B data to the LASCO C3. We choose to do the comparison in this way
because, unfortunately, stray-light in the COR2-B data limits the usable inner
FOV and dynamic range early in the mission. Thus, we began the COR2-B
measurements at $4.0R_{\odot}$ for easy of comparison with LASCO C3. In the
COR2-A data, we can observe the CME at $2.5R_{\odot}$ which is comparable to
the LASCO C2. Thus we are observing the same area of the CME in both LASCO C2
and C3 with the COR2-A and B, respectively.
For all three events, the data from the LASCO C2 coronagraph matches well with
the data from the COR2-A and the LASCO C3 data matches the data from COR2-B.
As the CMEs expands, the difference in the inner FOV has less of an effect on
the total mass and the data points converge for the LASCO C3 and COR2-A and B
coronagraphs. The good agreement with LASCO C2 and C3 data demonstrates that
the COR2 images can be used with confidence for analysis of CME masses.
The COR2 mass profiles in Figure 2 provide another important result by
verifying that the CME mass increases with height reaching a constant value in
the middle corona, above $10R_{\odot}$ as was suggested by Vourlidas et al.
(2000). We will further analyze and discuss this behavior in § 3.2.
## 3 A Novel Approach Using the Two STEREO Viewpoints
When simultaneous observations from different viewpoints are available, we can
exploit the resulting differences in the mass estimates to obtain not only the
true mass but also the direction of the CME. Figure 4 shows the calculated
mass versus time for the CME on 2008 March 25 as observed in COR2-A and B for
$\theta=0$. The relationship between the calculated total mass in Figure 4 and
the observed total brightness is :
$M_{A}={{B_{A}}\over{B_{e}(\theta=0)}}m_{ej}$ (2)
$M_{B}={{B_{B}}\over{B_{e}(\theta=0)}}m_{ej}$ (3)
where again, $B_{e}(\theta)$ is the brightness of a single electron at a given
angular distance from the POS and $m_{ej}$ is the mass of the ejected
material. For the 2008 March 25 CME, the calculated total mass in COR2-B
remains less than the COR2-A mass as the CME expands into the field of view of
the coronagraphs. As we have seen previously in Figure 2, both mass curves
converge towards a more or less constant value. Since we are using constant
base difference images, we should only be measuring the mass increase caused
by the CME. We then conclude that the full extent of the CME is visible in
both coronagraphs above $10R_{\odot}$. Thus we can assume that we are
observing the same volume of diffuse material from different angles and we
should calculate the same total mass from both COR2-A and B. If this
assumption is true, the difference in the calculated masses is a result of
using an incorrect angle in our Thomson scattering calculation. The masses
calculated in equations 2 and 3 can be expressed as fractions of the true
total mass of the CME :
$M_{T}f_{m}(\theta_{A})=M_{A}$ (4) $M_{T}f_{m}(\theta_{B})=M_{B}$ (5)
where $M_{T}$ is the true total mass of the CME and $\theta_{A}$ and
$\theta_{B}$ are the angular distances of the CME from the POS of COR2-A and
B, respectively. The function, $f_{m}$, is the ratio of the brightness of an
electron at angle $\theta$ relative to its brightness on the POS. We will
refer to this function as the normalized mass :
$f_{m}(\theta)=\frac{B_{e}(\theta)}{B_{e}(\theta=0)}.$ (6)
The function, $f_{m}$, is plotted in figure 3. If the CME were in the POS of
one of the coronagraphs, we would obtain the true total mass by setting
$B_{e}(\theta=0)$. For CMEs away from the instrument POS, the calculated mass
is some fraction of the true total mass, expressed by $f_{m}$. It is of
interest to note that if the CME were directed towards one of the coronagraphs
then, theoretically, we should not observe any mass.
The angle between the COR2-A and B POS is equal to the STEREO spacecraft
separation. Thus we can define $\theta_{A}$ and $\theta_{B}$, with respect to
a common coordinate system. For this coordinate system, the angle, $\theta$,
is measured $90^{\circ}$ from the Sun-Earth line in a right hand coordinate
system. Thus equations 4 and 5 can be written as :
$M_{T}f_{m}(\theta+\frac{1}{2}\Delta_{sc})=M_{A}$ (7)
$M_{T}f_{m}(\theta-\frac{1}{2}\Delta_{sc})=M_{B}$ (8)
where $\Delta_{sc}$ is the angular separation of the two spacecraft, and
$\theta$ is the angle of propagation of the event. Thus the axis of the
coordinate system is equal distance from the COR2-A and B POS. We can now
equate the difference in the calculated mass in COR2-A and COR2-B to the true
total mass. If we combine equations 7 and 8, we have :
$Mf_{m}(\theta+\frac{1}{2}\Delta_{sc})-Mf_{m}(\theta-\frac{1}{2}\Delta_{sc})=M_{A}-M_{B}.$
(9)
We can calculate the true total mass by inverting this function to find the
longitudinal direction that satisfies equation 9. The mass difference,
$M_{A}-M_{B}$, is plotted as a function of longitudinal direction in Figure 6
for separation angles of $10^{\circ}$ to $90^{\circ}$. The mass difference is
the superposition of two of the functions shown in Figure 3 offset by the
spacecraft separation. The inversion of the function can lead to more than one
solution for a given mass difference. However, some of the solutions can be
eliminated. In Figure 6, the gray part of the curve shows where the CME would
appear on opposite limbs of the Sun in the two coronagraphs. The dotted part
of the curve is where the CME would appear as a halo in one of the
coronagraphs. A simple inspection of the images would immediately reveal which
of the solutions should be chosen and which eliminated.
As the separation of the spacecraft increases, the range of observable mass
differences also increases. The extrema of the mass difference are related to
the normalized mass function by :
$(M_{A}-M_{B})^{\ast}=f_{m}(90^{\circ}-\Delta_{sc}).$ (10)
Thus when the separation is $0^{\circ}$ the extrema is zero and when the
separation is $90^{\circ}$ the extrema are equal to the true total mass. If
the difference in our calculated total mass is outside the range of solution
for equation 9, then we are observing intensity that is not from the CME. An
example of this would be instrumental effects or another solar structure, such
as a streamer, that was not removed adequately by the base difference.
We applied our method to the eight CMEs shown in Figure 1. For the purposes of
comparing the total mass across the COR2-A and B instruments, we use the same
IFOV at $4.0R_{\odot}$. We selected the largest events observed by COR2 for
spacecraft separation greater than $40^{\circ}$. In table 1, we list the total
mass of the CME in COR2 A and B using the POS assumption ($\theta=0^{\circ}$).
We then list the true total mass calculated using the CME direction. The
longitudinal direction derived for each CME with respect to the Sun-Earth line
is listed in the next column. For the majority of the CMEs, the true mass is
not significantly different from the larger of the two masses using the POS
assumption. Figure 3 shows that the CME mass does not vary significantly
between $\pm 20^{\circ}$ from the POS. The studied CMEs are mainly within $\pm
20^{\circ}$ of one of the instrument’s POS. The spacecraft separation is given
in table 1.
### 3.1 Comparison to Forward Modeling Results
As a means of further validating our analysis, we compare our direction
estimates with a completely different approach to estimating the three
dimensional position of the CMEs, namely forward modeling. A complete
description of the forward modeling method can be found in Thernisien, Howard,
and Vourlidas (2006). Briefly, a three dimensional geometric representation of
a flux-rope is fitted to the two spacecraft views of a CME at a single time.
The direction of the CME is taken as the apex of the flux-rope. The directions
from forward modeling for the CMEs in our sample are given in table 1
(Thernisien, Vourlidas, and Howard, 2009). Figure 5 provides a visual
comparison between the direction results from our mass method and the forward
modeling method. We plot the direction of the CME as calculated using the mass
difference (solid line) and the direction found from the forward model (dashed
line). We have good agreement between all of the studied CMEs with the
exception of the 2008 April 26 event. For this event, the CME appears as a
partial halo in COR2-B which results in a limitation to the accuracy of our
method. A portion of the CME is behind the occulter and our assumption that we
are observing the same mass in both views is not valid.
### 3.2 CME Mass Variation with Height and Time
As can be seen in Figures 2 and 4, the total CME mass measurements show a very
specific variation with height and time. Namely, the mass increases rapidly
when the CME front is within $8R_{\odot}$, and reaches a constant value beyond
about $10R_{\odot}$. The same behavior was originally seen in LASCO when only
a specific, large scale feature of the CME is measured (e.g., the core
Vourlidas et al., 2000). The result for LASCO was treated with caution because
the sharpest mass increase occurred in the $5-8R_{\odot}$ range which is the
overlapping region between C2 and C3.
Vourlidas et al. (2000) suggests that the mass increase could have been due to
instrumental differences between C2 and C3 such as calibration, dynamic range,
and resolution. However, the COR2 measurements are taken over an uninterrupted
field of view with the same telescope and clearly show that the CME mass
increases with time and height then reaches a constant value above about
$10R_{\odot}$.
Therefore, this mass variation with height appears to be a fundamental
property of the ejection process. To further quantify this behavior, we have
fitted the observed mass-height profiles with the analytical function :
$M(h)=M_{c}(1-e^{-h/h_{c}}).$ (11)
where $M_{c}$ is the final total mass of the event and $h_{c}$ is the height
where the mass reaches $~{}63\%$ of its final mass. The choice of the function
was dictated by the shape of the mass curves. Also this function has the
desired behavior of approaching a constant value as the height increases. We
have not explored other functions and we are wary of employing a more complex
expression because we do not yet have any theoretical or physical foundation
for the variation of CME mass with height. This is an area where CME modelers
and theoreticians could provide some useful insight.
Before fitting the data, we de-projected both the heights and the CME mass
values using the results in Table 1. We obtained good fits to all eight of the
events in our paper (Figure 7). For all events, the scale height ($h_{c}$) is
relatively low in the corona at approximately $2R_{\odot}$ and $~{}99\%$ of
the final CME mass is reached by $10R_{\odot}$. For all of the events in our
sample, the final mass is of the order of $10^{15}g$. The variations in the
profiles do not seem to correlate with the speed or the width of the events.
It is difficult to reach strong conclusions from our small sample of CMEs
taken during a very low period of solar activity. In the future, we plan to
investigate the behavior of the CME mass with a larger number of events.
However, we are confident that the small variation in the parameters of the
fit suggests that we can adopt the average profile of the eight events :
$M(h)=15.6(1-e^{-h/2.1})$ (12)
as representative of the mass variation with height for a typical CME. Of
course, the mass increase is due to material coming up from below the
occulting disk.
## 4 DISCUSSION & CONCLUSIONS
An implicit assumption in all CME mass calculation methods up to now has been
that the mass of the CME is concentrated into a single plane on the POS.
However, CMEs are three dimensional structures with a considerable depth along
the line of sight. While our two viewpoint method is an improvement on the POS
assumption, $\theta=0$, and results in an estimate of the CME direction, it
still assumes that the CME mass lies in a plane along that direction. The true
width along the LOS remains unknown. But we can easily estimate the error from
this assumption by calculating the mass ratio between the CME of zero width
and CMEs of various widths. Vourlidas et al. (2000) showed that this
simplification could cause the total mass to be underestimated by up to
$15\%$.
To overcome this limitation, we could use observations at larger heights by
combining measurements in SECCHI HI-1 A and B instruments, for example. Or
instead of measuring the total mass, we could try to measure the mass of the
same feature as long as it can be reliably identified in both COR2
instruments.
There are other factors that could affect the accuracy of the mass
calculation. An obvious one is the noise in the mass images. We estimate the
noise from histograms of empty sky regions. As expected, the empty sky values
are a Gaussian distribution around zero. We define the error level as one
standard deviation of this distribution. The noise levels in the COR2
telescopes are similar and the error is $\sim 9\times 10^{9}$ g/pixel. This
error is comparable to the error in the LASCO mass images (Vourlidas, 2005).
The average per pixel signal in the measured CMEs is approximately 5 times the
noise level and therefore the noise is insignificant. While the calculation of
the CME mass has a low noise level, the selection of the CME region for the
mass calculation can effect the total mass significantly. In the quiet corona
there are large dense streamers that obscure or interact with the CME. Since
we are using two viewpoints, it is often the case where the streamer can be
isolated from the CME in one view but cannot in another. An example is the
2008 January 02 event where a streamer is below the CME in the COR2-A image
but bisects it in the COR2-B image. This event also has the second largest
discrepancy with the forward fitting model ($\sim 8^{\circ}$), so the addition
of the streamer may be effecting the direction finding to some extent. In
general, however, it is difficult to quantify this type of error since it is
not always obvious from the images when a streamer is part of the measured
mass. That situation can be best addressed by simultaneous observations from
viewpoints inside and outside the ecliptic plane.
The error in the CME direction arises from the shape of the function of mass
with POS angle (Figure 6). Small changes in the difference between the two
masses can cause large differences in the direction, for small spacecraft
separations. Assuming a typical mass error estimate of $\sim 15\%$, the
direction ambiguity becomes reasonably small ($\lesssim 20^{\circ}$) for
separations larger than about $50^{\circ}$.
Another point of discussion is the implication of SECCHI results on the single
viewpoint mass measurements of past missions. As we have mentioned already,
all previous work assumed a POS angle of zero for the CME mass. In table 1, we
show the total mass in each instrument for $\theta=0$ and the true CME mass.
For the majority of the CMEs, the true mass of the CME is not significantly
different from the larger of the two masses using the POS assumption. In other
words, most of the mass tends to lie near one of the two POS for the events of
our sample. Therefore one has a better chance of observing the true mass of
the CME with two viewpoints for spacecraft separations of
$40^{\circ}-50^{\circ}$. The lower mass is within a factor of 2 of the true
mass for most cases with the exception of the April 26 event which is a factor
of 3 lower. However, this is a halo event and such discrepancies are expected.
Our results validate the assumptions in Vourlidas et al. (2000) and the
modeling results of Lugaz et al. (2005) and suggest that past CME mass
measurements are within a factor of two of the true CME mass, except for halo
events.
We thank A. F. Thernisien for providing the data from his geometric CME model.
The SECCHI data is produced by an international consortium of the NRL, LMSAL
and NASA GSFC (USA), RAL and U. Bham (UK), MPS (Germany), CSL (Belgium), IOTA
and IAS (France).
## References
* Billings (1966) Billings, D. E. 1966, A Guide to the Solar Corona (New York: Academic Press)
* Brueckner et al (1995) Brueckner, G.E. et al. 1995, Sol. Phys., 162, 291
* Howard et al. (1985) Howard, R. A. et al. 1985, J. Geophys. Res., 90, 8173
* Howard et al. (2008) Howard, R. et al. 2008, Space Sci. Rev., 136, 67
* Jackson (1997) Jackson, J. D. 1997, Classical Electrodynamics (3rd ed.; New York: Wiley)
* Kaiser et al. (2008) Kaiser, M. L. et al. 2008, Space Sci. Rev., 136, 5
* Lugaz et al. (2005) Lugaz, N. 2005, ApJ, 627, 1019
* Minnaert (1930) Minnaert, M. 1930, Z. Astrophys., 1, 209
* Morrill et al. (2006) Morrill, J. S. et al. 2006, Sol. Phys., 233, 331
* St. Cyr et al. (2000) St. Cyr, O. C. et al. 2000, J. Geophys. Res., 105, 18169
* Subramanian and Vourlidas (2007) Subramanian, P., & Vourlidas, A. 2007, A&A, 467, 685
* Thernisien, Howard, and Vourlidas (2006) Thernisien, A. F., Howard, R., & Vourlidas, A. 2006, ApJ, 652, 763
* Thernisien, Vourlidas, and Howard (2009) Thernisien, A. F., Vourlidas, A., & Howard, R. 2009, Sol. Phys., in press
* van de Hulst (1950) van de Hulst, H. C. 1950, Bull. Astron. Inst. Netherlands, 11, 135
* Vourlidas et al. (2000) Vourlidas, A. et al. 2000, ApJ, 543, 456
* Vourlidas et al. (2002) Vourlidas, A. et al 2002, in Proc. of the 10th Europ. Sol. Phys. Meet. ’Solar Variability: From Core to Outer Frontiers’, Prague, Czech Rep., Wilson, A. (ed), ESA SP-506, Dec. 2002, p. 91
* Vourlidas (2005) Vourlidas, A. in Coronal and Stellar Mass Ejections, IAU Symp. Proc. of the IAU 226, K. Dere, J. Wang, and Y. Yan (eds). Cambridge: Cambridge University Press, 2005., pp.76-76
* Vršnak et al. (2007) Vršnak, B. et al. 2007, A&A, 469, 339
* Yashiro et al. (2004) Yashiro, S. et al. 2004, J. Geophys. Res., 109, A07105, 10.1029/2003JA010282
Table 1: CME Direction and Mass CME | Mass $10^{15}$g | Direction | Separation | HEE Lon
---|---|---|---|---
| B | A | true | mass | model | | B | A
2007 Dec 04 | 2.57 | 2.23 | 2.57 | 68 | 71 | 42.16 | -21.43 | 20.73
2007 Dec 31 | 7.68 | 7.10 | 7.70 | -100 | -91 | 43.97 | -22.79 | 21.17
2008 Jan 02 | 3.59 | 5.29 | 5.29 | -64 | -51 | 44.07 | -22.88 | 21.20
2008 Feb 12 | 3.05 | 4.49 | 4.49 | 110 | 93 | 45.56 | -23.67 | 21.89
2008 Feb 15 | 2.12 | 3.18 | 3.18 | -72 | -60 | 45.64 | -23.68 | 21.97
2008 Mar 25 | 1.27 | 2.86 | 2.87 | -78 | -84 | 47.17 | -23.69 | 23.48
2008 Apr 05 | 1.89 | 2.84 | 2.84 | 117 | 126 | 47.83 | -23.72 | 24.11
2008 Apr 26 | 0.94 | 2.78 | 2.80 | -48 | -21 | 49.51 | -23.95 | 25.56
Figure 1: Mass images of studied events calculated with $\theta=0$. The images
are shown with the same scaling. The left image of each pair is from COR2-B
while the right is from COR2-A. The dependence of the CME morphology and total
mass on the viewing angle is evident in most events. Figure 2: Cross-
calibration of total mass measurements from LASCO-C2 (plus), LASCO-C3 (star),
SECCHI COR2-A (square) and SECCHI COR2-B (diamond) for CMEs observed on 2007
February 9 (top), 2007 March 21 (middle), and 2007 March 31 (bottom). The good
agreement with LASCO C2 and C3 data demonstrates that the COR2 images can be
used with confidence for analysis of CME masses. Figure 3: The normalized mass
function gives the angular dependence of the total brightness of a single
scattering electron normalized to the brightness at $\theta=0$. We can use
this function to relate the mass calculated using the POS assumption to the
true total mass of the CME. Figure 4: March 25, 2008 calculated total mass
($\theta=0$) as a function of time in COR2-A (square) and COR2-B (diamond).
The difference in the calculated mass is the result of using an incorrect
angle in our Thomson scattering calculation. We will exploit this difference
to derive the direction and true total mass of the CME. Figure 5: Graphical
representation of the CME estimated direction for the events in our sample.
The dashed lines are the results of our mass method, while the solid lines are
obtained from forward modeling (Thernisien, Vourlidas, and Howard, 2009).
Figure 6: Mass difference as a function of direction for spacecraft
separations of $10^{\circ}$ to $90^{\circ}$ in steps of $10^{\circ}$.
Directions where CMEs would appear as halos (dotted) or on opposite sides of
the Sun (gray) can be eliminated as possible solutions. We can calculate the
true total mass by inverting this function to find the longitudinal direction
for a given mass difference and spacecraft separation. Figure 7: The
dependence of CME mass on the height of the CME front for the eight events in
our sample. The fitted final CME mass and the scale height for each event are
also shown.
|
arxiv-papers
| 2009-03-25T14:13:06 |
2024-09-04T02:49:01.405862
|
{
"license": "Public Domain",
"authors": "Robin C. Colaninno, Angelos Vourlidas",
"submitter": "Robin Colaninno",
"url": "https://arxiv.org/abs/0903.4344"
}
|
0903.4375
|
# Renormalized Polyakov Loop in the Deconfined Phase of SU(N) Gauge Theory and
Gauge/String Duality
Oleg Andreev Technische Universität München, Excellence Cluster,
Boltzmannstrasse 2, 85748 Garching, Germany L.D. Landau Institute for
Theoretical Physics, Kosygina 2, 119334 Moscow, Russia
###### Abstract
We use gauge/string duality to analytically evaluate the renormalized Polyakov
loop in pure Yang-Mills theories. For $SU(3)$, the result is in a quite good
agreement with lattice simulations for a broad temperature range.
###### pacs:
12.38.Lg, 12.90.+b
††preprint: SPAG-A1/09
## I Introduction
It is well known that a pure $SU(N)$ gauge theory at high temperature
undergoes a phase transition. This phase transition is of special interest
because of many its aspects can be characterized precisely pol . In
particular, the order parameter is given by the Polyakov loop
$L(T)=\frac{1}{N}\text{tr Pexp}\Bigl{[}ig\int_{0}^{1/T}dt\,A_{0}\Bigr{]}\,,$
(1)
where the trace is over the fundamental representation, $t$ is a periodic
variable of period $1/T$, with $T$ the temperature, $g$ is a gauge coupling
constant, and $A_{0}$ is a vector potential in the time direction. The usual
interpretation of (1) is as a phase factor associated to the propagation of an
infinitely heavy test quark in the fundamental representation of the gauge
group.
Until recently, the lattice formulation, still struggling with limitations and
system errors, and effective field theories were the main computational tools
to deal with non-weakly coupled gauge theories. The Polyakov loop was also
intensively studied (see, for example, pis-rev and references therein). The
situation changed drastically with the invention of the AdS/CFT correspondence
malda1 that resumed interest in another tool, string theory.
In this note we continue a series of recent studies az1 ; az2 ; a-pis devoted
to a search for an effective string description of pure gauge theories. In az1
, the model was presented for computing the heavy quark and multi-quark
potentials at zero temperature. Subsequent comparison white with the available
lattice data has made it clear that the model should be taken seriously.
Later, this model was extended to finite temperature. The results obtained for
the spatial string tension az2 and the thermodynamics a-pis are remarkably
consistent with the lattice, too. As is known, QCD is a very rich theory
supposed to describe the whole spectrum of strong interaction phenomena. The
question naturally arises: How well does the model describe other aspects of
quenched QCD? Here, we attempt to analytically evaluate the Polyakov loop as
an important step toward answering this question az3 . In addition, a good
motivation for this test is lattice data revealed recently by gupta .
Before proceeding to the detailed analysis, let us set the basic framework. As
in az1 ; az2 ; a-pis , we take the following ansatz for the five-dimensional
background geometry
$ds^{2}=G_{nm}dX^{n}dX^{m}=R^{2}w\left(fdt^{2}+d\vec{x}^{2}+\frac{1}{f}dz^{2}\right)\,,\\\
w(z)=\frac{\text{e}^{\mathfrak{s}z^{2}}}{z^{2}}\,,\quad
f(z)=1-\bigl{(}\tfrac{z}{z_{\text{\tiny
T}}}\bigr{)}^{4}\,,\phantom{=\frac{\text{e}^{\mathfrak{s}z^{2}}}{z^{2}}}$ (2)
where $z_{\text{\tiny T}}=1/\pi T$. $\mathfrak{s}$ is a deformation parameter
whose value can be fixed from the critical temperature s . We take a constant
dilaton and discard other background fields.
In discussing the Wilson and Polyakov loops within the gauge/string duality
lit , one first chooses a contour ${\cal C}$ on a four-manifold which is the
boundary of a five-dimensional manifold. Next, one has to study fundamental
strings on this manifold such that the string world-sheet has ${\cal C}$ as
its boundary. In the case of interest, ${\cal C}$ is an interval between $0$
and $1/T$ on the $t$-axis. The expectation value of the Polyakov loop is
schematically given by the world-sheet path integral
$\langle\,L(T)\,\rangle=\int DX\,\text{e}^{-S_{w}}\,,$ (3)
where $X$ denotes a set of world-sheet fields. $S_{w}$ is a world-sheet
action. In principle, the integral (3) can be evaluated approximately in terms
of minimal surfaces that obey the boundary conditions. The result is written
as $\langle\,L(T)\,\rangle=\sum_{n}w_{n}\exp[-S_{n}]$, where $S_{n}$ means a
renormalized minimal area whose weight is $w_{n}$.
## II Calculating the Polyakov Loop
Given the background metric, we can attempt to calculate the expectation value
of the Polyakov loop by using the Nambu-Goto action for $S_{w}$ in (3)
$S=\frac{1}{2\pi\alpha^{\prime}}\int
d^{2}\xi\,\sqrt{\det\,G_{nm}\partial_{\alpha}X^{n}\partial_{\beta}X^{m}\vphantom{\bigl{(}\bigr{)}}}\,.$
(4)
Here $G_{nm}$ is the background metric (2). In the case of interest, this
action describes a fundamental string stretched between the test quark on
${\cal C}$ (at $z=0$) and the horizon at $z=z_{\text{\tiny T}}$. Since we are
interested in static configurations, we choose $\xi_{1}=t$, $\xi_{2}=z$. This
yields
$S=\frac{\mathfrak{g}}{\pi T}\int^{z_{\text{\tiny
T}}}_{0}dz\,w\sqrt{1+f(\vec{x}\,^{\prime})^{2}}\,,$ (5)
where $\mathfrak{g}=\frac{R^{2}}{2\alpha^{\prime}}$. A prime stands for a
derivative with respect to $z$.
Now it is easy to find the equation of motion for $\vec{x}$
$\biggl{[}wf\vec{x}\,^{\prime}/\sqrt{1+f(\vec{x}\,^{\prime})^{2}}\biggr{]}^{\prime}=0\,.$
(6)
It is obvious that Eq.(6) has a special solution $\vec{x}=const$ that
represents a straight string stretched between the boundary and the horizon.
Since this solution makes the dominant contribution, as seen from the
integrand in (5), we won’t dwell on other solutions here.
Having found the solution, we can now compute the corresponding minimal area.
Since the integral (5) is divergent at $z=0$ due to the factor $z^{-2}$ in the
metric, we regularize it by imposing a cutoff $\epsilon$
$S_{\text{\tiny R}}=\frac{\mathfrak{g}}{\pi T}\int^{z_{\text{\tiny
T}}}_{\epsilon}dz\,w\,.$ (7)
Subtracting the $\frac{1}{\epsilon}$ term (quark mass) and letting
$\epsilon=0$, we get a renormalized area
$S_{0}=\frac{\mathfrak{g}}{\pi T}\int^{z_{\text{\tiny
T}}}_{0}dz\,\Bigl{(}w-\frac{1}{z^{2}}\Bigr{)}+c\,,$ (8)
where $c$ is a normalization constant which is scheme-dependent.
Next, we can perform the integral over $z$. The result is
$S_{0}=\mathfrak{g}\biggl{(}\sqrt{\pi}\frac{T_{c}}{T}\text{Erfi}\Bigl{(}\frac{T_{c}}{T}\Bigr{)}+1-\text{e}^{(T_{c}/T)^{2}}\biggr{)}+c\,.$
(9)
In this formula $T_{c}$ is given by $T_{c}=\sqrt{\mathfrak{s}}/\pi$ az2 .
Combining the weight factor with the normalization constant as
$\mathfrak{c}=\ln w_{0}-c$, we find
$L(T)=\exp\biggl{[}\mathfrak{c}-\mathfrak{g}\biggl{(}\sqrt{\pi}\frac{T_{c}}{T}\text{Erfi}\Bigl{(}\frac{T_{c}}{T}\Bigr{)}+1-\text{e}^{(T_{c}/T)^{2}}\biggr{)}\biggr{]}\,,$
(10)
with $\text{Erfi}(z)$ the imaginary error function. This is our main result.
## III Numerical Results and Phenomenological Prospects
It is of great interest to compare the temperature dependence of (10) with
other results for the high temperature phase of $SU(N)$ gauge theory. In doing
so, we start with lattice QCD. Clearly, $N=3$ is of primary importance. In
Fig.1 a comparison is shown with the recent data of gupta . We see that our
model is in a quite good agreement with the lattice for a broad temperature
range $1.05\,T_{c}\lesssim T\lesssim 20\,T_{c}$. The maximum discrepancy
occurred at $T=1.05\,T_{c}$ is of order 15%. It rapidly decreases with
temperature reaching 2% at $T=2.2\,T_{c}$ and becoming almost negligible up to
$20\,T_{c}$. Then, it starts to grow back again.
For completeness, we can fit the value of $\mathfrak{g}$ to be $0.72$ that
significantly improves
Figure 1: The renormalized Polyakov loop in $SU(3)$ gauge theory. The solid
blue curve corresponds to (10) with $\mathfrak{g}=0.62$ as fixed from the
heavy quark potential at zero T in white . The dashed green curve represents
the ”best fit” with $\mathfrak{g}=0.72$. In both cases, the value of
$\mathfrak{c}$ is set to $0.10$. The dots are from lattice simulations of
gupta . The red dots are for $N_{\tau}=4$, while the black dots are for
$N_{\tau}=8$. We do not display any error bars because they are quite small,
comparable to the size of the symbols.
accuracy. For example, at $T=1.05\,T_{c}$ it becomes of order 6%. One possible
explanation for the better fit is that we have evaluated (3) classically (in
terms of strings). If we take into account semi-classical corrections, then
the value of $\mathfrak{g}$ gets renormalized.
For practical purposes, the expression (10) looks somewhat awkward. Following
a-pis , we expand $S_{0}$ and $L$ in powers of $(T_{c}/T)^{2}$. If we ignore
all higher terms, then a final result can be written in two simple forms:
$L(T)\approxeq\exp\biggl{[}\mathfrak{c}-\mathfrak{g}\Bigl{(}\frac{T_{c}}{T}\Bigr{)}^{2}\biggr{]}\,,$
(11)
or
$L(T)\approxeq\text{e}^{\mathfrak{c}}\biggl{(}1-\mathfrak{g}\Bigl{(}\frac{T_{c}}{T}\Bigr{)}^{2}\biggr{)}\,.$
(12)
In Fig.2 we have plotted the results. As can be seen, above $2\,T_{c}$ the
discrepancy between the expression (10) and approximations
Figure 2: A comparison of different $L(T)$ curves for $SU(3)$ gauge theory. As
in Fig.1, the solid blue curve corresponds to (10) and the dots are from
lattice simulations of gupta . The blue dashed curve corresponds to the
exponential law (11). The black dot-dashed curve corresponds to the power law
(12). In all the cases, $\mathfrak{g}=0.62$ and $\mathfrak{c}=0.10$. We
display error bars only if they are comparable to the size of the symbols.
(11)-(12) is negligible. At lower T the approximation (11) (exponential law)
is poor. It shows a significant deviation from the lattice. In particular, the
discrepancy occurred at $T=1.05\,T_{c}$ is of order 27%. On the other hand,
the agreement between the approximation (12) (power law) and the lattice is
spectacular. For the temperature range $1.05\,T_{c}\lesssim T\lesssim
20\,T_{c}$ the power law provides a reliable approximation to lattice QCD with
accuracy better than 5%! Moreover, one can use it to describe all available
lattice data of gupta at lower $T$. Then, the maximum discrepancy occurred at
the lowest available value $T=1.012\,T_{c}$ is of order 7%.
It is worth noting that the exponential law has been suggested in arriola
based on a dimension-two condensate $\langle A^{2}\rangle$ 2con . Such a
condensate as well as its possible links to the UV renormalon and $1/Q^{2}$
corrections got intensively discussed in the QCD literature viz . As was first
shown in 1/q2 , the deformation parameter $\mathfrak{s}$ of the background
geometry (2) is tied into the appearance of the quadratic corrections. It is
not, therefore, surprising that we have recovered (11) in our calculations.
Interestingly, the power law (12) is very similar to that observed for the
pressure in pisa . Indeed, for $T\gtrsim 1.2\,T_{c}$ the pressure is simply
$p/T^{4}\approx f_{\text{\tiny pert}}(1-(T_{c}/T)^{2})$.
## IV Conclusions
In this note we have evaluated the Polyakov loop using the now standard ideas
motivated by gauge/string duality. A key point is the use of the background
metric (2) which is singled out by the earlier works az1 ; az2 ; a-pis . (Note
that there is no need for any free parameters except a scheme-dependent
normalization constant $\mathfrak{c}$.) The overall conclusion is that the
same background metric results in a very satisfactory description of the
Polyakov loop as well. Of course, we still have a lot more to learn before
answering the question posed at the beginning of this note.
Acknowledgments
We would like to thank R.D. Pisarski and P. Weisz for useful discussions, and
S. Hofmann for reading the manuscript. This work is supported in part by DFG
”Excellence Cluster” and the Alexander von Humboldt Foundation under Grant No.
PHYS0167.
## References
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* (2) R.D. Pisarski, ”QCD Phase Diagram”, lectures presented at the ”47 Internationale Universitätswochen für Theoretische Physik”, Schladming, Austria, March 2009;
see also http://physik.uni-graz.at/itp/iutp/iutp-09/
LectureNotes/Pisarski/pisarski-2.pdf.
* (3) J.M. Maldacena, Adv.Theor.Math.Phys. 2, 231 (1998);
S.S. Gubser, I.R. Klebanov, and A.M. Polyakov, Phys.
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* (7) C.D. White, Phys.Lett.B 652, 79 (2007).
* (8) See also the earlier work which deals with numerical estimates, O. Andreev and V.I. Zakharov, JHEP 04, (2007) 100.
* (9) S. Gupta, K. Huebner, and O. Kaczmarek, Phys.Rev. D77, 034503 (2008).
* (10) Alternatively, it may be fixed from the heavy quark potentials az1 ; white .
* (11) While a significant literature on the Wilson loops has grown, there has been relatively little investigation of the Polyakov loop. For some developments, see however E. Witten, Adv.Theor.Math.Phys. 2, 505 (1998); A. Hartnoll and S. Prem Kumar, Phys.Rev.D74, 026001 (2006); M. Headrick, Phys.Rev.D77, 105017 (2008).
* (12) E. Megias, E. Ruiz Arriola, and L.L. Salcedo, JHEP 0601, 073 (2006).
* (13) L.S. Celenza and C.M. Shakin, Phys.Rev.D 34, 1591 (1986).
* (14) For a review, see V.I. Zakharov, Nucl.Phys.Proc.Suppl. 74, 392 (1999) and references therein.
* (15) O. Andreev, Phys.Rev.D 73, 107901 (2006).
* (16) R.D. Pisarski, Prog.Theor.Phys.Suppl.168, 276 (2007).
|
arxiv-papers
| 2009-03-25T15:36:43 |
2024-09-04T02:49:01.412870
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Oleg Andreev",
"submitter": "Oleg Andreev",
"url": "https://arxiv.org/abs/0903.4375"
}
|
0903.4440
|
# To the question of the integration of Plebansky Equation
A. N. Leznov111e-mail: andrey@buzon.uaem.mx Universidad Autonoma del Estado de
Morelos, CCICAp,Cuernavaca, Mexico
###### Abstract
It is shown that corresponding to Plebansky equation of symmetry posses the
infinite set of solutions, which we present in explicit form. This fact leads
to conclusion about possibility to find series solutions of the Plebansky
equation in analytic form. Some classes of explicit solution are presented.
## 1 Introduction
The goal of the present paper is to investigate Plebansky equation on the
subject of its integrability. The Plebansky equation is non homogeneous
complex Monge-Ampher equation
$Det_{n}(\phi_{x_{i},\bar{x}_{j}})=1$ (1)
in the case $n=2$. $\phi$ is unknown function of $2n$ independent variables
$x_{i},\bar{x}_{j}$.
Equation of Plebansky arise in self-dual gravity [1] and we will use it in the
form
$v_{y,\bar{y}}v_{z,\bar{z}}-v_{y,\bar{z}}v_{z,\bar{y}}=1$ (2)
which will call as a basic form of this equation.
At this place we would like to notice that the general solution of homogeneous
Monge-Ampher equation (with zero instead of the unity in the right side of
(1)), was recently found in implicit form in [3].
We remind to the reader some known facts from the theory of differential
equations [2]. With each system of differential equations it is connected
symmetry system of equations, which arise after differentiation of the initial
system by some parameter and denotation such obtained derivative $\dot{a}=A$
as a new unknown function. If linear system of such equations have exact
solution on the class of solutions of the initial system the last one is
exactly integrable. This means that it is possible to present its solution (in
explicit or implicit form) depending on necessary number of arbitrary
functions sufficient for the statement the problem of Cauchy-Kovalevskai. Such
solution is called as the general one. If there exist series solutions of the
symmetry equation then initial system poses the same number of exact
solutions, but each solution is not the general one.
In the present paper we would like to show that in the case of Plebansky
equation second possibility take place. It may be found infinite series
solutions of its symmetry equation but not the general one. Thus it is not
possible to find general solution of this equation in analytic form. But
infinite number of its exact solutions exist and the goal of the present paper
consists in description the ways for obtaining them.
## 2 Preliminary manipulations
Now we would like to rewrite (2) in equivalent form
$(v_{\bar{y}}v_{z,\bar{z}}-v_{\bar{z}}v_{z,\bar{y}})_{y}+(v_{y,\bar{y}}v_{\bar{z}}-v_{y,\bar{z}}v_{\bar{y}})_{z}=2$
(3)
$(v_{y}v_{z,\bar{z}}-v_{z}v_{\bar{z},y})_{\bar{y}}+(v_{y,\bar{y}}v_{z}-v_{\bar{y},z}v_{y})_{\bar{z}}=2$
(4)
The last equations (3) and (4) can be partially resolved as
$v_{\bar{y}}v_{z,\bar{z}}-v_{\bar{z}}v_{z,\bar{y}}=\theta_{z}+y,\quad
v_{y,\bar{y}}v_{\bar{z}}-v_{y,\bar{z}}v_{\bar{y}}=-\theta_{y}+z$
$v_{y}v_{z,\bar{z}}-v_{z}v_{\bar{z},y}=\theta_{\bar{z}}+\bar{y},,\quad
v_{y,\bar{y}}v_{z}-v_{\bar{y},z}v_{y}=-\theta_{\bar{y}}+\bar{z}$
It is important to notice that function $\theta$ satisfy the the symmetry
equation corresponding to Plebansky one [2].
Resolving the last equalities with respect to $v_{\bar{y}},v_{\bar{z}}$ (or to
$v_{y},v_{z}$), taking into account the Plebansky equation (2) for $v$ we
obtain
$v_{\bar{y}}=v_{y,\bar{y}}\theta_{z}-v_{z,\bar{y}}\theta_{y}+zv_{z\bar{y}}+yv_{y,\bar{y}},\quad
v_{\bar{z}}=v_{y,\bar{z}}\theta_{z}-v_{z,\bar{z}}\theta_{y}+zv_{z\bar{z}}+yv_{y,\bar{z}}$
(5)
$v_{y}=v_{y,\bar{y}}\theta_{\bar{z}}-v_{y,\bar{z}}\theta_{\bar{y}}+\bar{z}v_{y,\bar{z}}+\bar{y}v_{y,\bar{y}},\quad
v_{z}=v_{z,\bar{y}}\theta_{\bar{z}}-v_{z,\bar{z}}\theta_{\bar{y}}+\bar{z}v_{z\bar{z}}+\bar{y}v_{z,\bar{y}}$
(6)
Equating second mixed derivatives of $v$ with respect to bar variables (or not
to the bar ones), we conclude that function $\theta$ in both cases is the
solution of the symmetry equation corresponding to (2) namely
$v_{\bar{y},y}\theta_{z,\bar{z}}+\theta_{\bar{y},y}v_{z,\bar{z}}-v_{y,\bar{z}}\theta_{\bar{y},z}-\theta_{y,\bar{z}}v_{\bar{y},z}=0$
(7)
## 3 Recurrent formula for solution of the symmetry equation
Rewriting (7) in two equivalent forms
$v_{\bar{y},y}\theta_{z}-v_{\bar{y},z}\theta_{y}=\tilde{\theta}_{\bar{y}},\quad
v_{\bar{z},y}\theta_{z}-v_{\bar{z},z}\theta_{y}=\tilde{\theta}_{\bar{z}}$ (8)
$v_{\bar{y},y}\theta_{\bar{z}}-v_{\bar{z},y}\theta_{\bar{y}}=\tilde{\theta}_{y},\quad
v_{\bar{y},z}\theta_{\bar{z}}-v_{\bar{z},z}\theta_{\bar{y}}=\tilde{\theta}_{z}$
(9)
By the same way as it was done in the previous section it is possible to check
that $\tilde{\theta}$ functions from (8),(9) satisfy the symmetry equation
(7). Obvious solution of symmetry equation is derivatives of $v$ with respect
to 4 independent arguments of the problem $y,z,\bar{y},\bar{z}$. Thus with
help of (8),(9) it is possible to construct infinite serie solutions of
symmetry equation (compare with [LYM]).
### 3.1 Static case
In this subsection we would like explain the case when symmetry equation has a
exact solution and how this connected with the integrable property of the
initial system.
Let us consider ”time independent” configurations, when
$v=v(z+\bar{z},y,\bar{y})$. In this case the Plebanski equation (2) and main
symmetry equation (5) are reduced correspondingly
$v_{y,\bar{y}}v_{z,z}-v_{y,z}v_{z,\bar{y}}=1$
$v_{\bar{y},y}\theta_{z}-v_{\bar{y},z}\theta_{y}=\tilde{\theta}_{\bar{y}},\quad
v_{z,y}\theta_{z}-v_{z,z}\theta_{y}=\tilde{\theta}_{z}$
Let us sick solution of the symmetry equation in terms of three independent
coordinates $(v_{z},y,\bar{y})$. The last symmetry system in these coordinates
looks as
$\theta_{v_{z}}v_{z,\bar{y}}+\theta_{\bar{y}}=\tilde{\theta}_{v_{z}}-v_{z,\bar{y}}\tilde{\theta},\quad\theta_{v_{z}}v_{z,z}=-v_{z,z}\tilde{\theta}_{y}$
From which follows that functions $\tilde{\theta},\theta$ are connected by the
linear system of equations
$\tilde{\theta}_{\bar{y}}=\theta_{v_{z}},\quad\tilde{\theta}_{v_{z}}=-\theta_{y}$
which is equivalent to three dimensional Laplace equation. Thus in the static
case the symmetry equation possesses the general solution and thus in this
case Plebansky equation is exactly integrable.
Below we present its general solution in implicit form
$v_{\bar{y}}=L_{v_{z}}(v_{z},y,\bar{y})\quad z=L_{y}(v_{z},y,\bar{y})$ (10)
where function $L$ satisfy Laplace equation in three dimension
$L_{v_{z},v_{z}}+L_{y,\bar{y}}=0$. General solution of Laplace equation in
three dimension depend on two arbitrary functions of two arguments. And thus
constructed above is a general solution of Plebansky equation in static case.
This solution may be obtained directly. Indeed Plebansky equation in this case
may be rewritten as
$(v_{\bar{y}})_{y}(v_{z})_{z}-(v_{z})_{y}(v_{\bar{y}})_{z}=1$
and functions $(v_{\bar{y}},v_{z})$ may be considered as transformed impulse
and coordinate with unity Poisson bracket between them. This system is
resolved by canonical transformation with generating function
$G(v_{z},y,\bar{y})$ by usual formulae
$v_{\bar{y}}=G_{v_{z}}(v_{z},y,\bar{y}),\quad z=G_{y}(v_{z},y,\bar{y})$
From condition of equality of the second mixed derivatives of $v$ function we
conclude that $G$ satisfy Laplace equation in three dimension. This is exactly
(10) above.
## 4 General strategy and example explaining it
General (5) is the system of two equations on two unknown functions
$v,\theta$. As a consequence we know that $v$ function satisfy Plebansky
equation and $\theta$ function the symmetry one. But all solutions of symmetry
equation are enumerated in the previous section. And thus if we change
$\theta$ in (5) on one of solution of the previous section we will obtain the
self consistent system of two equations only on one unknown function $v$.
Solution of this system (if it will be possible to find it) will be solution
of Plebansky equation corresponding to such chose solution of symmetry
equation.
To show that such idea is not mean less we at first consider simple example of
(5) under the chose $\theta=v_{\bar{y}}$. But result of solution of Plebansky
equation will be absolutely non trivial one.
We rewrite (5)
$v_{\bar{y}}=(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial
y})v_{\bar{y}},\quad v_{\bar{z}}=1+(z\frac{\partial}{\partial
z}+y\frac{\partial}{\partial y})v_{\bar{z}}$
Two independent ordinary differential equations have obvious solution
$v_{\bar{y}}=(yz)^{1\over 2}X_{\bar{y}}(d,\bar{z},\bar{y}),\quad
v_{\bar{z}}=1+(yz)^{1\over 2}X_{\bar{z}}(d,\bar{z},\bar{y}),\quad
d\equiv{z\over y}$
Further
$v_{\bar{y},y}={1\over 2}d^{1\over 2}X_{\bar{y}}-d^{3\over
2}X_{\bar{y},d},\quad v_{\bar{y},z}={1\over 2}d^{-1\over
2}X_{\bar{y}}+d^{1\over 2}X_{\bar{y},d}$ $v_{\bar{z},y}={1\over 2}d^{1\over
2}X_{\bar{z}}-d^{3\over 2}X_{\bar{z},d},\quad v_{\bar{z},z}={1\over
2}d^{-1\over 2}X_{\bar{z}}+d^{1\over 2}X_{\bar{z},d}$
And equation of Plebansky (2) takes the form
$X_{\bar{y}}X_{\bar{z},D}-X_{\bar{z}}X_{\bar{y},D}=1,\quad D\equiv\ln d$ (11)
Resolving of the last equation is connected with second order ordinary
differential equations of the form $X_{D,D}=F(X,D)$ ($F$ arbitrary functions
of its arguments). Indeed solution of this equation depends on two arbitrary
parameters $c_{1},c_{2}$ and may be represented as some function depending on
3 arguments $X=X(D,c_{1},c_{2})$. Let us differential equation for $X$
function argumentson parameters $c_{i}$. We have
$X_{D,D,c_{i}}=F_{X}(X,D)X_{c_{i}}$. From the last equality we conclude
$(X_{D,c_{1}}X_{c_{2}}-X_{D,c_{2}}X_{c_{1}})_{D}=0$. We will assume that this
value is equal to unity (this is always possible to do by corresponding
canonical transformation). Now we will consider $c_{1},c_{2}$ as arbitrary
functions of the arguments $\bar{y},\bar{z}$. Then equation (11) Looks as
$(X_{c_{1}}X_{c_{2},D}-X_{c_{2}}X_{c_{1},D})((c_{1})_{\bar{y}}(c_{2})_{\bar{z}}-(c_{1})_{\bar{z}}(c_{2})_{\bar{y}})=((c_{1})_{\bar{y}}(c_{2})_{\bar{z}}-(c_{1})_{\bar{z}}(c_{2})_{\bar{y}})=1$
Thus obtained solution of initial Plebansky equation depend on two arbitrary
function $F(X,D)$ defined arbitrary equation of the second order and
generating function of canonical transformation resolving the last equation
for $c_{1},c_{2}$ functions.
We present the second way for solution equation (11). Let us consider
$X_{D},X$ as canonical transformed coordinate and impulse variable and
$\bar{z},\bar{y}$ as the same initial ones. Then generating function of
canonical transformation $W=W(X,\bar{y},D)$ satisfy the equations
$X_{D}=W_{X}(X,\bar{y},D),\quad\bar{z}=W_{\bar{y}}(X,\bar{y},D)$
From the second equation we have $X_{D}=-{W_{\bar{y},D}\over W_{\bar{y},X}}$
and after substitution into the first equation we obtain
$(W_{D}+{W_{X}^{2}\over 2})_{\bar{y}}=0,\quad W_{D}+{W_{X}^{2}\over 2}=F(X,D)$
The second one is Hamilton-Jacobi equation of the particle motion in potential
field $V=F(X,D)$, It leads to second order differential equation considered
above. Hamilton-Jacobi equation reduce the number of independent variables in
generating function $W$ from 3 up to 2. And thus solution of Plebansky
equation is determined by two functions $W,F$ each one of two independent
variables.
## 5 Equation of Plebansky in non usual variables
Let us introduce notations $R=\ln v_{\bar{y}}+\ln v_{\bar{z}},\Delta=\ln
v_{\bar{y}}-\ln v_{\bar{z}}$. Or in other words
$v_{\bar{y}}=\exp({R+\Delta\over 2}),v_{\bar{z}}=\exp({R-\Delta\over 2})$. In
these notations the pair of equations (LABEL:LS) take the form
$R_{y},\theta_{z}-R_{z}\theta_{y}+zR_{z}+yR_{y}=2,\quad\Delta_{y}\theta_{z}-\Delta_{z}\theta_{y}+z\Delta_{z}+y\Delta_{y}=0$
(12)
In the last equations let us pass from independent variables $y,z$ to
independent variables $\theta,d={z\over y}$. Corresponding necessary formulae
are presented below
$y=Y(\theta,d,\bar{y},\bar{z}),\quad 1=Y_{\theta}\theta_{y}-Y_{d}{z\over
y^{2}},\quad 0=Y_{\theta}\theta_{z}+Y_{d}{1\over y}$ (13)
$R_{y}==R_{\theta}\theta_{y}-R_{d}{z\over y^{2}},\quad
R_{z}=R_{\theta}\theta_{z}+R_{d}{1\over y}$
and the same formulae for derivatives of the $\Delta$ function. In all
relations above its necessary to keep in mind that all function under
consideration depend also on bar arguments $\bar{y},\bar{z}$. Such dependence
will be taken into account on some forward steps of calculations.
In variables $\theta,d$ the system (12) looks as
$(Y^{2})_{\theta}-Y^{2}R_{\theta}=-R_{d},\quad(Y^{2}e^{-R})_{\theta}=(e^{-R})_{d},\quad
Y^{2}={\Delta_{d}\over\Delta_{\theta}}$ (14)
To do the last equality more symmetrical to $y,z$ variables let us multiply
the last one on $d={z\over y}$. We have in a consequence
$dY^{2}=yz={d\Delta_{d}\over\Delta_{\theta}}={\Delta_{D}\over\Delta_{\theta}},\quad
D=\ln d$ (15)
Now we would like to satisfy equation of Plebansky
$v_{y,\bar{y}}=({R_{y}+\Delta_{y}\over 2})\exp({R+\Delta\over 2}),\quad
v_{z,\bar{y}}=({R_{z}+\Delta_{z}\over 2})\exp({R+\Delta\over 2})$
$v_{y,\bar{z}}=({R_{y}-\Delta_{y}\over 2})\exp({R-\Delta\over 2}),\quad
v_{z,\bar{z}}=({R_{z}-\Delta_{z}\over 2})\exp({R-\Delta\over 2})$
And thus the equation of Plebansky looks as
$e^{R}(R_{z}\Delta_{y}-R_{y}\Delta_{z})=2$
In the last equation let us pass to variables $\theta,d$ with the help of
above formulae. We obtain
$R_{d}\Delta_{\theta}-R_{\theta}\Delta_{d}=e^{-R}(Y^{2})_{\theta}$
Let us compare the last equation with obtained above (14) ones. As a direct
corollary we have
$e^{-R}=-\Delta_{\theta}$ (16)
The last relations have as a direct consequence two first relations (14).
In all calculations above no information about dependence of all functions
involved with respect to bar arguments was not used. Now let use condition of
equality of the second mixed derivatives of $v$ function with respect to
$\bar{y},\bar{z}$ arguments in notations introduced above.
$(\exp({R+\Delta\over 2}))_{\bar{z}}=(\exp({R-\Delta\over
2}))_{\bar{y}},\quad\exp\Delta(R_{\bar{z}}+\Delta_{\bar{z}})=(R_{\bar{y}}-\Delta_{\bar{y}})$
But functions $R,\Delta$ are connected by (16) and function $\Delta$ depends
in its turn on $\theta$ and thus terms with derivatives on bar arguments look
as (we present left hand side term)
$-(\ln\Delta_{\theta})_{\bar{z}}+\Delta_{\bar{z}}+(-(\ln\Delta_{\theta})_{\theta}+\Delta_{\theta})\theta_{\bar{z}}$
(17)
After some computations using (15) we come to equation,which function $\Delta$
satisfy
$e^{5\Delta}((e^{-\Delta})_{\theta,\bar{z}}(e^{-\Delta})_{\theta,d}-(e^{-\Delta})_{\theta,\theta}(e^{-\Delta})_{\bar{z},d})=(e^{\Delta})_{\theta,\bar{y}}(e^{\Delta})_{\theta,d}-(e^{\Delta})_{\theta,\theta}(e^{\Delta})_{\bar{y},d}$
(18)
In the case when symmetry function depends only from non bar variables
equation (18) looks much more simple (in (17)
$\theta_{\bar{z}}=\theta_{\bar{y}}=0$)
$e^{3\Delta}(e^{-\Delta})_{\theta,\bar{z}}=(e^{\Delta})_{\theta,\bar{y}}$
This equation is equivalent to equation considered in section 4. It will be
explained in one of sections below.
After introduction new function $\theta=\Theta(\Delta,d,\bar{y},\bar{z})$ it
looks as
$e^{\Delta}(\Theta_{\Delta,\Delta}\Theta_{d,\bar{z}}-\Theta_{\Delta,d}\Theta_{\Delta,\bar{z}}+\Theta_{\Delta}\Theta_{d,\bar{z}})=\Theta_{\Delta,\Delta}\Theta_{d,\bar{y}}-\Theta_{\Delta,d}\Theta_{\Delta,\bar{y}})-\Theta_{\Delta}\Theta_{d,\bar{y}}$
(19)
or in variables $d=D(\Delta,\theta,\bar{y},\bar{z})$ it looks as
$e^{\Delta}Det_{3}\pmatrix{D_{\bar{z}}&D_{\Delta}&D_{\theta}\cr
D_{\Delta,\bar{z}}&D_{\Delta,\Delta}+D_{\Delta}&D_{\Delta\theta}\cr
D_{\theta,\bar{z}}&D_{\theta,\Delta}&D_{\theta,\theta}\cr}=Det_{3}\pmatrix{D_{\bar{y}}&D_{\Delta}&D_{\theta}\cr
D_{\Delta,\bar{y}}&D_{\Delta,\Delta}-D_{\Delta}&D_{\Delta\theta}\cr
D_{\theta,\bar{y}}&D_{\theta,\Delta}&D_{\theta,\theta}\cr}$ (20)
Each of three equations above are equivalent to the initial Plebansky equation
(2).
### 5.1 Equation (19) in the integral motion form
Let us introduce the following notation the operators of the differentiations
$L^{\pm}=e^{{\Delta\over 2}}{\frac{\partial}{\partial\bar{z}}}\pm
e^{-{\Delta\over 2}}{\frac{\partial}{\partial\bar{y}}},\quad
L^{0}={\frac{\partial}{\partial\Delta}}$
with the obvious commutation relations
$[L^{0},L^{\pm}]={1\over 2}L^{\mp},\quad[L^{+},L^{-}]=0$
(these are commutation relation of algebra of two dimension plane). In these
notations equation (19) may be rewritten as
$\Theta_{\Delta,\Delta}(L^{-}\Theta)_{d}+2\Theta_{\Delta}(L^{-}\Theta_{d})_{\Delta}-2\Theta_{\Delta}(L^{-}\Theta_{\Delta})_{d}-\Theta_{\Delta,d}(L^{-}\Theta_{\Delta})=0$
Or after dividing on $(\Theta_{\Delta})^{{1\over 2}}$ and trivial regrouping
of the terms we obtain
$((\Theta_{\Delta})^{{1\over
2}}(L^{-}\Theta_{d}))_{\Delta}=((\Theta_{\Delta})^{{1\over
2}}(L^{-}\Theta_{\Delta}))_{d}$ (21)
Or (19) may be rewritten in integral of motion form. We remind the reader that
all equations (18),(19), (20) where obtained from the equality of the
derivatives on the bar variables.
## 6 Some examples of particular solutions
In this section we present some particular solutions of equations (19) or
equivalent to it (18), (20). From this consideration it will be clear that
these equation leads indeed to solution of initial Plebansky equation.
### 6.1 The case when symmetry function do not depend on bar variables
From explicit form of symmetry equation of the second section it is clear that
simplest obvious its solution is $\theta=\theta(y,z)$. Equation describing
this situation are the following ones
$e^{3\Delta}(e^{-\Delta})_{\theta,\bar{z}}=(e^{\Delta})_{\theta,\bar{y}},\quad{\Delta_{d}\over\Delta_{\theta}}=Y^{2}(\theta,D)=yz$
From the second equation that function $e^{\Delta}$ really is the function of
only three variables $\bar{y},\bar{z},g(\theta,D)$. Let us seek solution of
the first equation in a form
$e^{\Delta}=-{X_{\bar{y}}\over X_{\bar{z}}}$
where $X=X(\bar{y},\bar{z},g(\theta,D))$. After such substitution first
equation takes the form
$({X_{\bar{y}}\over
X_{\bar{z}}})^{3}({X_{\bar{y},g}X_{\bar{z}}-X_{\bar{z},g}X_{\bar{y}}\over
X^{2}_{\bar{y}}})_{\bar{z}}=({X_{\bar{y},g}X_{\bar{z}}-X_{\bar{z},g}X_{\bar{y}}\over
X^{2}_{\bar{z}}})_{\bar{y}}$
From the last equality it follows immediately
$X_{\bar{y},g}X_{\bar{z}}-X_{\bar{z},g}X_{\bar{y}}=F(X,g)\to 1$
where $F$ arbitrary function of its two arguments. At last by substitution
$X\to Y(X,g)$ it is possible to equate $F$ to unity and we come back to
equation considered in 4 section. Solution of Plebansky equation id the
following one
$v_{\bar{y}}=\exp({R+\Delta\over
2}=({\exp\Delta\over-\Delta_{\theta}})^{{1\over 2}}={g_{\theta}}^{-{1\over
2}}X_{\bar{y}},\quad v_{\bar{z}}={g_{\theta}}^{-{1\over 2}}X_{\bar{z}},\quad
v={g_{\theta}}^{-{1\over 2}}X$
### 6.2 Solution do not depend on one bar coordinate
Let us assume that solution of equation (19) does not depend on $\bar{z}$
coordinate. Then equation for $\Theta$ function takes the form
$e^{-\Delta}(e^{-\Delta}\Theta_{\Delta})_{\Delta}\Theta_{D,\bar{y}}-e^{-\Delta}\Theta_{\Delta,D}e^{-\Delta}\Theta_{\Delta,\bar{y}})=0$
This is exactly three dimensional subclass of complex Monge-Ampher (four
dimensional equation), solution of which in implicit form is known [3].
It may be expressed via (in terms of) $\psi(\Delta,d,\bar{y})$ function which
is solution of the equation
$e^{\Delta}+F_{\psi}(D,\psi)+\bar{F}_{\psi}(\bar{y},\psi)=0$ (22)
In the last equation for $\psi(\Delta,D,\bar{y})$ two arbitrary functions
$F(D,\psi),\bar{F}(\bar{y},\psi)$ assumed to be known. Of course solution of
this equation (in general case) may be obtained only in implicit form.
After this solution of the equation for $\Theta(D,\bar{y},\Delta)$ function is
resolves as follows
$\Theta_{\Delta}=e^{\Delta}\psi,\quad\Delta_{\Theta}=e^{-\Delta}\psi^{-1},\quad\Theta_{\bar{y}}=-\bar{F}_{\bar{y}},\quad\Theta_{D}=F_{D}$
And this is a general solution of this equation.
The solution of the initial equation of Plebansky in coordinates
$z,y,\bar{z},\bar{y}$ is given by equations
$v_{\bar{y}}=e^{\Delta}(\psi)^{{1\over 2}},\quad v_{\bar{z}}=(\psi)^{{1\over
2}},\quad yz+F_{D}(D,\psi)=0$ (23)
We present below direct checking of the formulae above. Equation of Plebansky
in its basic form after substitution (23) looks as
${1\over 2}(e^{\Delta}_{y}\psi_{z}-e^{\Delta}_{z}\psi_{y})=-{1\over
2}F_{\psi,D}{y\psi_{y}+z\psi_{z}\over yz}=1$
In the last step of calculation necessary use last formulae of (23)
$y+F_{DD}{1\over z}+F_{D,\psi}\psi_{z}=0,\quad z-F_{DD}{1\over
y}+F_{D,\psi}\psi_{y}=0,\quad D=\ln z-\ln y$
#### 6.2.1 Solution of the example from a section 4
Let us sick solution of (19), (21) in a form (of course such form is a direct
consequence of calculations of the 4 section)
$\Theta=(e^{-\Delta}-r(D,\bar{y},\bar{z}))^{-1}$
All necessary derivatives are the following ones
$\Theta_{\Delta}=e^{-{1\over
2}\Delta}\Theta^{2},\quad\Theta_{D}=r_{D}\Theta^{2},\quad\Theta_{\bar{y}}=r_{\bar{y}}\Theta^{2},\quad\Theta_{\bar{z}}=r_{\bar{z}}\Theta^{2}$
After calculation of second order derivatives and substitution into (21) we
obtain equation for $r$ function
$r^{3}({1\over r})_{D,\bar{y}}=r_{D,\bar{z}}$
This equation is exactly integrable (author have not met it in literature
before) with solution
$r=-{X_{\bar{z}}\over X_{\bar{y}}}$
where $X(D,\bar{y},\bar{z})$ is exactly the function from the section 4.
Indeed $r_{D}=-{X_{D,\bar{z}}X_{\bar{y}}-X_{D,\bar{y}}X_{\bar{z}}\over
X^{2}_{\bar{y}}}={1\over X^{2}_{\bar{y}}}$.
#### 6.2.2 Once more possible particular solution
Let us sick solution of (19), (21) in a form
$\Theta=P(\bar{y},\bar{z},\Delta)+Q(d,\Delta)$
After substitution into corresponding equations we come to the linear equation
for determining of the $P_{\Delta}$ function
$(e^{-{\Delta\over 2}}{\frac{\partial}{\partial\bar{y}}}-e^{{\Delta\over
2}}{\frac{\partial}{\partial\bar{z}}})P_{\Delta}=0$
with the obvious solution
$P=\int^{\Delta}d\delta p(e^{{-\delta\over 2}}\bar{z}+e^{{\delta\over
2}}\bar{y},\delta)$
But really for solution of Plebansky equation it is necessary only derivative
of $P$ function with respect to the $\Delta$ argument. Indeed
$v_{\bar{y}}=({e^{\Delta}\over\Delta_{\theta}})^{{1\over 2}},\quad
v_{\bar{z}}=({e^{-\Delta}\over\Delta_{\theta}})^{{1\over
2}},\quad\Delta_{\theta}={1\over\Theta_{\Delta}}={1\over
P_{\Delta}+Q_{\Delta}}$
and at last connection between $\theta$ and $\Delta$ is given by relation
$yz={\Delta_{D}\over\Delta_{\theta}}=-\Theta_{D}=-Q_{D}$.
## 7 Discrete transformation
Let us rewrite main equations of preliminary section once more
$v_{\bar{y}}=v_{y,\bar{y}}\theta_{z}-v_{z,\bar{y}}\theta_{y}+zv_{z\bar{y}}+yv_{y,\bar{y}},\quad
v_{\bar{z}}=v_{y,\bar{z}}\theta_{z}-v_{z,\bar{z}}\theta_{y}+zv_{z\bar{z}}+yv_{y,\bar{z}}$
(24)
and pay attention that in connection with (8) these equations may be rewritten
as
$v_{\bar{y}}=\tilde{\theta}_{\bar{y}}+zv_{z,\bar{y}}+yv_{y,\bar{y}},\quad
v_{\bar{z}}=\tilde{\theta}_{\bar{z}}+zv_{z,\bar{z}}+yv_{y,\bar{z}}$
The last equation may resolved with respect $\tilde{\theta}$ function
$\tilde{\theta}=v-zv_{z}-yv_{y}$. And thus after substitution this expression
for $\tilde{v}$ function we obtain linear system of equations for $\tilde{v}$
in a form
$\tilde{v}_{\bar{y}}=(-(Ov_{z})\frac{\partial}{\partial
y}+(Ov_{y})\frac{\partial}{\partial
z}+O)\tilde{v}_{\bar{y}},\quad\tilde{v}_{\bar{y}}=(-(Ov_{z})\frac{\partial}{\partial
y}+(Ov_{y})\frac{\partial}{\partial z}+O)\tilde{v}_{\bar{z}}$ (25)
where operator $O\equiv(y\frac{\partial}{\partial y}+z\frac{\partial}{\partial
z})$. Equations (25) is linear system of equations for determining
$\tilde{v}_{\bar{y}},\tilde{v}_{\bar{z}}$ functions by known $v$ solution of
Plebansky equation.
At this moment it is unknown to the author the systematic way for resolving
(25).
## 8 Outlook
The main result of the present paper is a new approach to the problem of
Plebansky equation. This approach allows rewrite Plebansky equation in
coordinates involving symmetry function and find series solutions of this
equation. Solutions obtained in such way depends on two arbitrary functions
each of two variables. Thus this is not a general solution of Plebansky
equation which must depend on two arbitrary function each of three variables.
But investigation of symmetry equation show that general solution of Plebansky
equation it is not possible present in analytic form. Situation exactly the
same as for instance in the case of famous nonlinear one dimensional
Schredinger equation, where it is possible to find infinite series solution of
soliton like type but not general one.
Constructed in the present paper solutions are connected with ordinary
differential equation of the second order $X_{z,z}=F(X,z)$ and it arise very
interesting problem to understand what relation has this equation to group of
symmetry of Plebansky equation responsible for its integrable properties.
## 9 Appendix
In this Appendix we consider simplest example from section $(6.0.1)$ chose
arbitrary functions in such simple form that all calculation are possible to
do in explicit form. Let us chose $F={(d+\psi)^{2}\over
2},\bar{F}={(\bar{y}+\psi)^{2}\over 2}$. The main equation allow determine
$\psi$ and all other necessary variables in explicit form in usual for
Plebansky equation variables
$\psi=-{e^{\Delta}+\bar{y}+d\over 2},\quad
yz={\Delta_{d}\over\Delta_{\theta}}=-\Theta_{d}=-(d+\psi)={e^{\Delta}+\bar{y}-d\over
2},$ $e^{\Delta}=2yz+D-\bar{y},\quad\psi=-(yz+d),\quad D=\ln{z\over y}$
In connection with results of section $(6.0.1)$ solution of basic Plebansky
equation looks as
$v=[(2yz+\ln{z\over y})\bar{y}-{(\bar{y})^{2}\over 2}+\bar{z}](yz+\ln{z\over
y})^{1\over 2}$
## References
* [1] J.F.Plebanski Nucl. Phys. B373 (1992) 214-232.,of J.Math.Phys 16,(1975), 2395.
* [2] L.V.Ovsjanikov Group Analysis of differential equations., Acad.Press New-York (1992).
* [3] Fairlie D.B. and A. N. Leznov J. Phys. A Volume 33, Number 25, 30 June 2000 (4657-4661
|
arxiv-papers
| 2009-03-25T19:31:42 |
2024-09-04T02:49:01.419188
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.N.Leznov",
"submitter": "Andrey Leznov",
"url": "https://arxiv.org/abs/0903.4440"
}
|
0903.4559
|
# Effects of Dark Matter Substructures on Gravitational Lensing: Results from
the Aquarius Simulations
D. D. Xu1, Shude Mao1, Jie Wang2,3, V. Springel2, Liang Gao3,4, S.D.M. White2
E-mail: Dandan.Xu@postgrad.manchester.ac.uk Carlos S. Frenk3, Adrian Jenkins3,
Guoliang Li5, Julio F. Navarro6
1 Jodrell Bank Centre for Astrophysics, the University of Manchester, Alan
Turing Building, Manchester M13 9PL, United Kingdom
2 Max-Planck Institut Für Astrophysik, Karl-Schwarzshild-Straße 1, 85740
Garching, Germany
3 Institute of Computational Cosmology, Dept. of Physics, University of
Durham, South Road, Durham DH1 3LE, United Kingdom
4 National Astronomical Observatories, Chinese Academy of Sciences, Beijing,
100012, China
5 Argelander-Institut für Astronomie, University of Bonn, Auf dem Hügel 71,
D-53121 Bonn, Germany
6 Department of Physics and Astronomy, University of Victoria, Victoria, BC,
V8P, 5C2, Canada
(Accepted …… Received …… ; in original form…… )
###### Abstract
We use the high-resolution Aquarius simulations of the formation of Milky Way-
sized haloes in the $\Lambda$CDM cosmology to study the effects of dark matter
substructures on gravitational lensing. Each halo is resolved with $\sim
10^{8}$ particles (at a mass resolution $m_{\rm p}\sim 10^{3}$ to
$10^{4}h^{-1}M_{\odot}$) within its virial radius. Subhaloes with masses
$m_{\rm sub}\ga 10^{5}h^{-1}M_{\odot}$ are well resolved, an improvement of at
least two orders of magnitude over previous lensing studies. We incorporate a
baryonic component modelled as a Hernquist profile and account for the
response of the dark matter via adiabatic contraction. We focus on the
“anomalous” flux ratio problem, in particular on the violation of the cusp-
caustic relation due to substructures. We find that subhaloes with masses less
than $\sim 10^{8}h^{-1}M_{\odot}$ play an important role in causing flux
anomalies; such low mass subhaloes have been unresolved in previous studies.
There is large scatter in the predicted flux ratios between different haloes
and between different projections of the same halo. In some cases, the
frequency of predicted anomalous flux ratios is comparable to that observed
for the radio lenses, although in most cases it is not. The probability for
the simulations to reproduce the observed violations of the cusp lenses is
$\approx 10^{-3}$. We therefore conclude that the amount of substructure in
the central regions of the Aquarius haloes is insufficient to explain the
observed frequency of violations of the cusp-caustic relation. These
conclusions are based purely on our dark matter simulations which ignore the
effect of baryons on subhalo survivability.
###### keywords:
Gravitational lensing - dark matter - galaxies: ellipticals - galaxies:
formation
††pagerange: Effects of Dark Matter Substructures on Gravitational Lensing:
Results from the Aquarius Simulations–References††pubyear: 2002
## 1 INTRODUCTION
Currently there are $\sim 200$ known galaxy-scale lenses, divided roughly
equally in number into lensed active galactic
nuclei111http://www.cfa.harvard.edu/castles/ and lensed background galaxies
(Bolton et al. 2008). These galaxy-scale lenses allow diverse applications
(see the review “Strong Gravitational Lensing” by Kochanek in Schneider et al.
2006) such as a determination of the Hubble constant, a characterisation of
galaxy evolution, and measurements of the mass distribution in galaxies. The
last application will likely be the most important one in the next decade,
since there are few other probes at intermediate redshifts ($z\sim 0.5-1$).
It was noticed quite early on that the flux ratios of the multi-lensed images
are more difficult to reproduce with simple parametric mass models than the
image positions (Kochanek 1991). This has been termed the “anomalous flux
ratio” problem. Image positions and magnifications (flux ratios) are
determined by the first-order and second-order derivatives of the lensing
potential, respectively. Therefore flux ratios, as a high-order derivative,
are expected to be more sensitive to small changes in the lensing potential
than image positions.
In this regard, gravitational lenses with two or three close images deserve
special attention because, in these cases, the sources must be close to either
a fold or a cusp of the caustic. It is well known for any smooth lensing
potential that the close images follow asymptotic flux ratio relations: for a
close pair, their flux ratio approaches unity when their separation goes to
zero, while for a close triple, the ratio of the flux of the middle image to
the sum of the fluxes of the two outer images asymptotically goes to unity
(Mao 1992; Schneider & Weiss 1992; Keeton et al. 2003; Congdon et al. 2008).
However, the observed lensing systems often violate these asymptotic
relations. This was taken to be evidence for substructure in lensing galaxies
(Mao & Schneider 1998; Metcalf & Madau 2001; Metcalf & Zhao 2002; Dalal &
Kochanek 2002; Chiba 2002; Kochanek & Dalal 2004) on the physical scale of the
separation between close images (typically of the order of $\sim$ 1 kpc).
Spectroscopic observations go beyond simple broad-band flux ratios and provide
a promising way to probe substructure in lenses (Metcalf et al. 2004; Chiba et
al. 2005; Sugai et al. 2007). Other suggestive evidence for substructures
comes from astrometry (see Chen et al. 2007 for a general discussion), such as
bent jets (Metcalf 2002), and detailed image structures for B2016+112
(Schneider et al. 2006; Koopmans et al. 2002; More et al. 2009) and B0128+437
(Biggs et al. 2004; Zhang 2008). Substructures may also have detectable
effects on the time delays in gravitational lenses (Keeton & Moustakas 2008).
Evans & Witt (2003) argued that some of these lensing “anomalies” may be
accommodated by changes in the potentials of the main lensing galaxies in
parametric models. However, significant changes are needed in order to explain
the anomalies (Kochanek & Dalal 2004; Congdon & Keeton 2005). The angular
structures of the lenses whenever the measurements are available suggest
ellipsoidal central potentials, where the high amplitude, higher order
multipoles that are required to explain the flux ratio anomalies are not seen
(Kochanek & Dalal 2004; Yoo et al. 2005, 2006). There are strong hints that
substructures may indeed be real in lensing galaxies. First, observationally,
saddle (negative-parity) images are often fainter than the predictions of
simple smooth models. This is expected from lensing by substructure (such as
stars, or subhaloes; Schechter & Wambsganss 2002; Kochanek & Dalal 2004), but
impossible to explain by propagation effects, such as galactic scintillation
and scatter broadening, as earlier postulated (Koopmans et al. 2003). This
arguably constitutes the most convincing evidence for substructure lensing.
Second, in many gravitational lenses, the substructure is directly seen as
luminous satellites. For example, nearly half of the CLASS lenses (Browne et
al. 2003; Myers et al. 2003; Jackson et al. 2009) show luminous satellite
galaxies within a few kpc of the primary lensing galaxies222This fraction is a
factor of $\sim 2$ higher than that claimed in Bryan et al. (2008) as revealed
by a more careful analysis of HST images of the CLASS lenses (Jackson et al.
2009).. Inclusion of satellites in the modelling dramatically improves the fit
to the image positions. In the case of B2045+265, the inclusion of a companion
galaxy helps to explain the flux ratio anomaly (McKean et al. 2007). The
additional dark subhaloes within the main lensing galaxies, as well as the
intergalactic perturbers along the line-of-sight (Chen et al. 2003; Wambsganss
et al. 2005; Metcalf 2005a,b; Miranda & Macciò 2007) may also help to explain
the observed lensing anomalies.
Much of the interest in (milli-)lensing flux anomalies arises because they may
be caused by the elusive substructure generically predicted by the
hierarchical structure formation in the cold dark matter (CDM) cosmology (e.g.
Kauffmann et al. 1993; Klypin et al. 1999; Moore et al. 1999; Ghigna et al.
2000; Gao et al. 2004a,b; Diemand et al. 2007). In this model, large
structures form via merging and accretion of smaller structures. The cores of
these small structures often survive tidal destruction and manifest themselves
as subhaloes (substructure). Recent high-resolution simulations predict many
thousands of subhaloes (down to $m_{\rm sub}\sim 10^{6}M_{\odot}$, or to
circular velocity of $V_{\rm c}\sim 4{\rm\,km\,s^{-1}}$; e.g. Madau et al.
2008; Springel et al. 2008), at least two orders of magnitude more than the
number of observed satellite galaxies in the Milky Way, even after accounting
for the newly discovered faint satellite galaxies from the Sloan Digital Sky
Survey (Belokurov et al. 2007). A possible solution is that star formation may
be strongly suppressed in the vast majority of the low-mass subhaloes (e.g.
Efstathiou 1992; Kauffmann et al. 1993; Thoul & Weinberg 1996; Bullock et al.
2000; Gnedin 2000; Benson et al. 2002), and thus they remain dark and
difficult to detect through light-based methods. If this is the case, then
gravitational lensing can potentially probe this population since it depends
only on the mass but not on whether the lenses are luminous or dark.
Numerical simulations indicate that subhaloes typically account for 5-10 per
cent of the total mass in a galaxy-type halo (e.g. Klypin et al. 1999; Moore
et al. 1999; Ghigna et al. 2000). The study by Dalal & Kochanek (2002)
requires $f_{\rm sub}=$ 0.6% to 7% (with a median of 2%) of the mass to be in
substructures (90% confidence limit) in order to explain the observed flux
anomaly problem. At first sight, the fraction of substructure from simulations
seems to be more than sufficient to explain the flux anomaly. Upon closer
examination, however, a problem emerges: lensing probes the central few kpc
around the line-of-sight through the galaxy, while most substructures are in
the outer regions of its dark matter halo, since those that come close to the
centre are tidally destroyed. Thus it remains unclear whether the predicted
substructure in the inner regions is sufficient or not to explain the observed
flux anomalies (e.g. Bradač et al. 2004; Mao et al. 2004; Macciò & Miranda
2006; Amara et al. 2006). In contrast, on cluster scales, the amount of
predicted substructure seems to be consistent with weak and strong lensing
data (Natarajan et al. 2007).
Previous lensing studies simulated galaxy-sized haloes with $\sim 10^{6}$
particles so that subhaloes were resolved down to $\sim 10^{7}$ to
$10^{8}h^{-1}M_{\odot}$. State-of-the-art simulations can now resolve haloes
with two or even three orders of magnitude more particles, thus reaching
substantially lower mass subhaloes. In this work, we revisit the issue of
substructure lensing using the Aquarius simulations of six galaxy-sized
haloes. These collisionless $N$-body simulations were performed by the Virgo
Consortium in a concordance $\Lambda$CDM universe. The subhaloes in each halo
are resolved down to masses of $m_{\rm sub}\sim 10^{5}h^{-1}M_{\odot}$
(Springel et al. 2008), at least two orders of magnitude better than that in
previous substructure lensing studies.
Our paper is organised as follows. In Section 2, we describe the realisation
and the properties of the simulated lensing galaxies. Our methods and
techniques for the lensing simulations together with our test results are
presented in Section 3. In Section 4, we apply our lensing simulation to the
six simulated galaxy haloes from the Aquarius simulation to derive their
lensing properties, including the cusp relations, and we compare the numerical
results with observations. A summary of the paper and a discussion are given
in Section 5. The cosmology we adopt for the lensing simulation is the same as
that used for the Aquarius simulations (Springel et al. 2005), with a matter
density $\Omega_{\rm m}$ = 0.25, cosmological constant $\Omega_{\Lambda}$ =
0.75, Hubble constant $h=H_{0}/(100{\rm\,km\,s^{-1}}\,{\rm Mpc}^{-1})=0.73$
and linear fluctuation amplitude $\sigma_{8}=0.9$.
## 2 From Dark Matter haloes to Early-Type Lensing Galaxies
In this section, we summarise the properties of dark matter haloes from the
Aquarius simulations relevant to our study, in particular the subhalo
properties. Readers are referred to Springel et al. (2008) for more details.
We will show that dark matter alone is, as expected, insufficient to cause
multiple image splittings, and therefore we must incorporate a stellar
component; we detail such a procedure in §2.2.
### 2.1 The Aquarius simulations
The Aquarius project (Springel et al. 2008) is a suite of simulations of six
galaxy-sized dark matter haloes with five levels of numerical resolution. The
haloes were selected from a 100$h^{-1}$ Mpc simulation box within the
concordance cosmology (for parameters see above). The simulations were run
with GADGET-3, an improved version of the GADGET-2 code (Springel et al. 2001;
Springel 2005). The highest resolution level (level 1) was achieved for only
one halo (“Aq-A-1”) with $\sim 1.5$ billion halo particles. Level-2
simulations were performed for a sample of six dark matter haloes, with about
200 million particles per halo. The softening length is $\sim$0.05$h^{-1}$
kpc, and the mass resolution ranges from $10^{3}$ to $10^{4}h^{-1}M_{\odot}$.
All haloes are Milky-Way type systems in terms of their mass and rotation
curve. We will use the six level-2 haloes (Aq-A-2, Aq-B-2, Aq-C-2, Aq-D-2,
Aq-E-2 and Aq-F-2) at redshift zero for our analysis of substructure lensing.
As we will show later on, the scatter in lensing properties among different
haloes (and for different projections) is large, and so it is important to
examine more than one halo for statistical purposes.
The basic properties of the six haloes at $z=0$ are listed in Table 1. In
particular, all the density profiles are reasonably fit by Navarro, Frenk and
White (NFW) profiles (Navarro et al. 1996, 1997)333An even better fit is found
using the Einasto (1966) profile (Navarro et al. 2008), but here we adopt the
simpler NFW profile which we use later to take into account the adiabatic
contraction of dark matter haloes.:
$\begin{array}[]{c}\displaystyle\rho(r)=\frac{M_{200}}{4\pi
r(r+r_{200}/c)^{2}f(c)},\\\ \displaystyle
M(<r)=\frac{M_{200}f(r\,c/r_{200})}{f(c)},\\\ \displaystyle
f(c)=\ln(1+c)-c/(1+c),\end{array}$ (1)
where $r_{200}$ is the radius within which the mean dark halo mass density is
200 times the critical density, $M_{200}$ is the mass enclosed within
$r_{200}$, and $c\equiv r_{200}/r_{\rm s}$ is the concentration parameter with
$r_{\rm s}$ being the scale radius.
Table 1: Dark matter halo properties in the Aquarius simulations:
Halo Name | $r_{200}$ | $M_{tot}$ | $c$ | Mass Resolution | $N_{200}$ | $N_{\rm sub}$ | $f_{\rm sub}$
---|---|---|---|---|---|---|---
| ($h^{-1}$ kpc) | ($10^{10}h^{-1}M_{\odot}$) | | ($h^{-1}M_{\odot}$) | | | (per cent)
Aq-A-2 | 179.5 | 132.8 | 16.2 | $1.0\times 10^{4}$ | $1.3\times 10^{8}$ | $2.1\times 10^{4}$ | 7.14
Aq-B-2 | 137.1 | 59.5 | 9.7 | $4.7\times 10^{3}$ | $1.3\times 10^{8}$ | $2.5\times 10^{4}$ | 6.98
Aq-C-2 | 177.3 | 127.7 | 15.2 | $1.0\times 10^{4}$ | $1.2\times 10^{8}$ | $1.7\times 10^{4}$ | 4.12
Aq-D-2 | 177.3 | 128.5 | 9.4 | $1.0\times 10^{4}$ | $1.3\times 10^{8}$ | $2.2\times 10^{4}$ | 6.56
Aq-E-2 | 155.0 | 85.7 | 8.3 | $7.0\times 10^{3}$ | $1.2\times 10^{8}$ | $2.3\times 10^{4}$ | 7.28
Aq-F-2 | 153.0 | 80.5 | 9.8 | $4.9\times 10^{3}$ | $1.6\times 10^{8}$ | $2.6\times 10^{4}$ | 11.20
Aq-A-2 ($Z$ = 0.6) | 134.4 | 92.2 | 10.4 | $1.0\times 10^{4}$ | $9.3\times 10^{7}$ | $1.7\times 10^{4}$ | 6.50
Note: Col (1): halo name, Cols (2)-(4): $r_{200}$, $c$ and $M_{200}$ are
defined in eq. (1) for the main halo. $M_{\rm tot}=M_{200}+M_{\rm sub}$, where
$M_{\rm tot}$ and $M_{\rm sub}$ are the total masses of all dark matter and of
all the subhaloes within $r_{200}$. Col (5): Mass resolution
($h^{-1}M_{\odot}$). Col (6): $N_{200}$ is the total number of particles
within $r_{200}$. Col (7): $N_{\rm sub}$ is the number of subhaloes within
$r_{200}$. Col (8): $f_{\rm sub}$ is the mass fraction of subhaloes within
$r_{200}$, defined by $M_{\rm sub}/M_{\rm tot}$.
We artificially put all these haloes (snapshot $z=0.0$) at redshift $z=0.6$
(corresponding roughly to the most likely lens redshift, e.g. Turner et al.
1984), keeping their physical sizes unchanged. However, we also take a
snapshot of the halo Aq-A-2 at redshift $z=0.6$ as a lens, and compare its
lensing properties with those artificially shifted to $z=0.6$. As will be
shown in §4, the scatter among the six haloes is much larger than the
differences between haloes at redshifts $z=0$ and $z=0.6$, and so adopting the
$z=0$ haloes will not significantly change the properties of substructure
lensing. This is also seen in the evolution of density profiles of these
haloes. Fig. 1 shows the density profiles for the halo Aq-D-2 at redshifts 0,
0.50 and 0.99. The changes in the profiles since redshift 1 are relatively
small since the Aquarius haloes form earlier than that.
Figure 1: Density profiles (solid curves), multiplied by $r^{2}$, for the halo
Aq-D-2 at redshifts $z=0$, 0.5 and 0.99. All haloes are reasonably well fitted
by the NFW profile (dotted curves, see Eq. 1), which follows $\rho(r)\propto
r^{-1}$ on small scales and $\rho(r)\propto r^{-3}$ on large scales. The
vertical dashed lines indicate the softening length and $r_{200}$.
As we are primarily interested in the substructure lensing, an important step
is the identification of the subhaloes. We use the SUBFIND routine (Springel
2005) to identify subhaloes exceeding 20 particles, which corresponds to a
minimum subhalo mass of $\sim 10^{5}h^{-1}M_{\odot}$. The number of subhaloes
in each halo ranges from about $1.7\times 10^{4}$ to $2.6\times 10^{4}$ within
$r_{200}$, with 4.1-11.2 per cent of the total halo mass locked up in bound
subhaloes (see Col (8): $f_{\rm sub}$ in Table 1).
The subhalo mass function follows a power-law: ${\rm d}N(m_{\rm sub})/{\rm
d}m_{\rm sub}\propto m_{\rm sub}^{-1.9}$ (Springel et al. 2008). The average
mass of subhaloes (within $r_{200}$) is $\sim 10^{6}$ to
$10^{7}h^{-1}M_{\odot}$ and their average half-mass radius is $\leq 0.2h^{-1}$
kpc, with large scatter. The most massive subhalo has a mass of $\sim 10^{9}$
to $10^{10}h^{-1}M_{\odot}$ and a half-mass radius $\sim 5-10h^{-1}$ kpc.
Figure 2: The left panel shows a contour map of the subhalo surface mass
density fraction, which is the ratio of the surface mass in subhaloes to that
in the total halo, for Aq-D-2 projected along the $Y$-axis. The right panel
shows the mean distribution of subhalo surface mass fraction as a function of
$R/r_{200}$, averaged over the three independent projections of each of the
six Aquarius haloes at redshift $z=0$. The error bars indicate the 68% scatter
among different projections and haloes. The red lines show the fit from Mao et
al. (2004). The blue point indicates the median and 90% confidence level of
the required fraction found by Dalal & Kochanek (2002) (assuming the Einstein
radius to be 0.02 $r_{200}$).
As an example, we again consider halo Aq-D-2 and show in the left panel of
Fig. 2 the $Y$-projection of the surface mass fraction in subhaloes within
$r_{200}$. The right panel in the same figure shows the surface mass fraction
of subhaloes averaged within azimuthal annuli as a function of the normalised
radius $R/r_{200}$. It is clear that the scatter in the projected mass
fraction of subhaloes among different haloes is large. Within 0.1 $r_{200}$,
the mean fraction is $\sim$ 0.005, with a scatter of a factor of 10. The red
line in the same panel shows the results from Mao et al. (2004), which were
obtained from 12 haloes (of galactic, group and cluster masses) and 30 random
projections. Their result lies somewhat higher than found here although still
within the scatter. This is probably due to the inclusion of group- and
cluster-sized haloes in the averaging, which tend to have a higher
substructure fraction due to their later formation times. The blue point
indicates the required substructure mass fraction found by Dalal & Kochanek
(2002) to be 0.02 (median, ranging from 0.006 to 0.07 at 90% confidence).
### 2.2 Adding “light” to dark matter haloes
We put the source redshift $z_{\rm s}$ at 3.0. This is reasonable since many
lensed quasars are at similar redshift. The lensing critical surface density
is given by
$\Sigma_{\rm cr}=\frac{c^{2}}{4\pi G}\frac{D_{\rm s}}{D_{\rm d}D_{\rm ds}},$
(2)
where $D_{\rm s}$, $D_{\rm d}$, and $D_{\rm ds}$ are the angular diameter
distances between the source and the observer, the lens and the observer, and
the source and the lens, respectively. For our adopted source and lens
redshifts, $\Sigma_{\rm cr}=1.82\times 10^{9}$ $M_{\odot}$
kpc${}^{-2}=7.95\times 10^{10}$ $M_{\odot}$ arcsec-2.
To produce multiple images, the maximum surface density of a halo usually has
to be super-critical. The left panel of Fig. 3 shows the surface density
distribution for the halo Aq-D-2 projected along the $Y$-axis. Clearly, the
central surface density of the (initial) NFW dark matter halo is below the
critical value, and thus generally no multiple images can be produced (e.g.
Williams et al. 1999; Rusin & Ma 2001) . This is hardly surprising, since for
galaxy-scale strong lensing, the images form only a few kpc (projected) from
the centre where baryons play a crucial role. Thus, one must incorporate a
baryonic component in order to model the lensing galaxies more realistically,
a topic we turn to next.
Most gravitational lenses are early-type (elliptical) galaxies rather than
late-type (disk) galaxies, as the former are more massive and dominate the
lensing cross-sections (Turner et al. 1984). There have been many hybrid
models used for the lensing galaxies (e.g. Keeton 2001; Kochanek & White 2001;
Oguri 2002; Jiang & Kochanek 2007). We use the spherical Hernquist profile to
model the light distribution, since it approximates the de Vaucouleur’s
profile that has been observed for elliptical galaxies and bulges, and it has
many known, convenient analytical properties.
The three-dimensional density and mass profiles $\rho_{H}(r)$, $M_{H}(r)$ for
the Hernquist distribution are given by (Hernquist 1990):
$\begin{array}[]{c}\displaystyle\rho_{H}(r)=\frac{aM_{\star}}{2\pi
r}\frac{1}{(r+a)^{3}},\\\ \displaystyle
M_{H}(<r)=M_{\star}\frac{r^{2}}{(r+a)^{2}}.\end{array}$ (3)
where $M_{\star}$ is the total baryonic mass, and $a$ is a scale length
related to the effective spherical radius $r_{\rm e}$ (within which half of
the mass is contained) by $a=r_{\rm e}/(\sqrt{2}+1)$.
The profile is specified by two parameters $a$ (or $r_{\rm e}$) and
$M_{\star}$, which are linked with the dark matter halo parameters $r_{200}$
and $M_{200}$ by
$\begin{array}[]{c}\displaystyle f_{\rm re}=\frac{r_{\rm
e}}{r_{200}},~{}~{}\displaystyle
f_{\star}=\frac{M_{\star}}{M_{200}},~{}~{}\displaystyle M_{200}=M_{\rm
DM}+M_{\star}.\end{array}$ (4)
Notice that the mass of the main halo dark matter $M_{\rm DM}$ is reduced by a
factor of (1-$f_{\star}$) to conserve the total mass and the mass of
substructures.
The inclusion of the baryonic component affects the distribution of the dark
matter halo. Many studies have shown that the adjustment of the dark matter
halo can be approximated by an adiabatic contraction (Barnes & White 1984;
Blumenthal et al. 1986). Gnedin et al. (2004) have proposed a modification to
this simple model in order to take into account the fact that particle orbits
in realistic halos are not circular, but it is not clear whether this
modification is able to reproduce accurately the results of numerical
simulations (see, e.g. Abadi et al. 2009). In view of this, we have decided to
follow, for simplicity, the procedure outlined by Mo et al. (1998). Assuming
that both the baryon and dark matter components follow an NFW distribution
initially, baryons ($f_{\star}$ percent of the total matter) then cool to form
the galaxy at the centre, which causes the dark matter halo to contract
adiabatically. After the adiabatic contraction, the dark halo follows a new
profile and hosts a Hernquist galaxy at its centre. Note that we contract all
the particles in different components (i.e., diffuse dark matter and
subhaloes) in the same way.
The two parameters ($f_{\rm re}$ and $f_{\star}$) are chosen according to two
criteria (after adding the baryonic galaxy and accounting for the adiabatic
contraction): (1) the projected dark-matter mass fractions inside the Einstein
radii of the host galaxy haloes should range from 0.4 – 0.7 (Treu & Koopmans
2004); (2) the projected slopes are close to isothermal at a few kpc from the
galactic centre (e.g. Rusin et al. 2003; Rusin & Kochanek 2005; Koopmans et
al. 2006; Gavazzi et al. 2007), or equivalently, the final rotation curves are
roughly flat from a few kpc out to a few tens of kpc (see Fig. 3).
Furthermore, $f_{\star}$ should be smaller than the universal baryonic
fraction of $\sim$ 17.5 per cent (from WMAP-3, Spergel et al. 2007).
We find that $f_{\rm re}=0.05$ and $f_{\star}=0.1$ satisfy these criteria
well. From Fig. 3 (the left panel), it is clearly seen that after inserting
the baryonic galaxy and taking the adiabatic contraction into account, the
total surface density is now super-critical and the corresponding Einstein
radius is of the order of a few kpc, similar to that in many gravitational
lenses. Notice however that our procedure is not self-consistent dynamically,
since the inclusion of a baryonic component will affect the evolution and
survival of subhaloes. We shall return to this point briefly in the
discussion.
Figure 3: The halo Aq-D-2: the left panel shows the surface density profiles
$\Sigma(R)$ projected in the $Y$-direction, and normalised to the critical
surface density. Profiles are for cases before and after adding a Hernquist
galaxy and the dark matter halo’s adiabatic contraction, assuming the added
baryonic component has 10% of the total mass and an effective radius of 5% of
the halo virial radius ($f_{\star}=0.1$, $f_{\rm re}=0.05$). Line symbols are
labelled inside the figure. The isothermal slope ($\Sigma(R)\propto R^{-1}$)
is indicated by the red line at the top right (see §2.2 for details). The
middle panel shows the mass distributions $M(\leq r)$. The right panel shows
the rotation curves $V_{\rm c}(r)$. The final total rotation curve is flat
from $\sim 5h^{-1}\,{\rm kpc}$ out to a few tens of kpc.
## 3 LENSING METHODOLOGY
$N$-body simulations provide us with the positions (and velocities) of
particles. For lensing calculations, we first project the particles onto a
mesh in the lens plane (and tabulate the stellar surface density, and then
smooth the surface density field appropriately. Using the smoothed surface
density map, we can numerically calculate the lensing potential, deflection
angles and magnifications. The details of the numerical procedure are given in
§3.1.
We test the accuracy of our numerical procedure by comparing with known
analytical results, using a singular isothermal sphere realised through Monte
Carlo simulations in §3.2. We then relax the spherical assumption, and further
test our procedure with an isothermal ellipsoid generated with a similar
number of particles as those in the Aquarius simulations; the comparison
results are presented in §3.3.
### 3.1 From particles to lensing images
#### 3.1.1 Coarse and fine particle meshes
We use a Particle-Mesh (PM) code for the lensing potential calculation. The
application of Fast Fourier Transforms ($FFT$) in the PM algorithm makes it
computationally efficient. However it is limited in resolution by the finite
mesh size and so cannot accurately represent regions with rapid density
variations on the scale of the grid size. To increase the accuracy in the
regions of interest (within a few kpc from the centres of galaxies), we
establish two two-dimensional [2D] meshes: a coarse grid used for the
potential field generated by the mass projected outside the central
($20h^{-1}$ kpc)2 region, and a fine grid for the mass within. Both grids have
$1024\times 1024$ pixels, covering $(4\,r_{200})^{2}$ and ($40h^{-1}$ kpc)2
(see §3.1.3) with resolutions $\sim$ 0.6$h^{-1}$ kpc and 0.04$h^{-1}$ kpc for
the coarse and fine grids, respectively (the factor of two increase in the box
size is due to the isolated boundary condition, see §3.1.3). This resolution
ensures that the tangential critical curves are resolved with sufficient
accuracy. In contrast, the inner radial critical curves may not be well
reproduced, due to the finite resolution of the mesh. However, this is not a
major concern since all the bright images that we are interested in form close
to the outer (tangential) critical curves. Furthermore, the resolution of the
fine mesh is similar to the softening length of the simulations, and the
density distributions in the very central regions are not accurately modelled
in the simulations on smaller scales than the gravitational softening in the
first place.
#### 3.1.2 Particle assignment with smoothed particle hydrodynamics kernel
The surface density maps of the Aquarius haloes are obtained by assigning
particles to the potential meshes using the Smooth Particle Hydrodynamics
(SPH) kernel (Monaghan 1992). Although, in the end, we will approximate the
underlying mass distributions of the Aquarius haloes by isothermal ellipsoids
in order to circumvent problems caused by discreteness noise (see §3.3), SPH-
smoothed density fields are used as an intermediate step to generate basic
lensing properties (e.g. critical curves and caustics) to constrain the best-
fit isothermal ellipsoids. For more detail, see §4.
The advantage of the SPH assignment is that it adjusts the smoothing scale
according to the local density environment: particles in a high density region
are mildly smoothed while those in a low density region are smoothed more. For
each particle, a smoothing length ${\cal{H}}$ is calculated according to the
local number density in its 3D neighbourhood. The particle mass is then
assigned to all the mesh cells that are within a circle of $2{\cal{H}}$ in
radius in its neighbourhood. The 3D density kernel can be integrated along the
line of sight analytically to obtain the surface density distribution:
$\Sigma(u)=\frac{1}{\pi{\cal{H}}^{2}}\left\\{{\begin{array}[]{*{20}l}\frac{1}{16}[-(8+52u^{2})\sqrt{1-u^{2}}+(16+26u^{2})\sqrt{4-u^{2}}\\\
-9u^{4}\ln u+3u^{2}(16+4u^{2})\ln(1+\sqrt{1-u^{2}})\\\
-3u^{2}(16+u^{2})\ln(2+\sqrt{4-u^{2}})],~{}~{}{\mathrm{if}~{}1>u\geq 0}\\\
\frac{1}{16}[2\sqrt{4-u^{2}}(8+13u^{2})+3u^{2}(16+u^{2})\ln u\\\
-3u^{2}(16+u^{2})\ln(2+\sqrt{4-u^{2}})],~{}~{}{\mathrm{if}~{}2>u\geq 1}\\\
0,~{}~{}~{}{\mathrm{if}~{}u>2}\end{array}}\right.$ (5)
where $u\equiv r/{\cal{H}}$ is the distance from the cell centre to the
particle normalised to ${\cal{H}}$, and the total mass within $u\leq 2$ is
unity.
The smoothing length ${\cal{H}}$ for each particle depends on its local
density and is controlled by the parameter $N_{\rm ngb}$, the number of
particles that are contained within radius ${\cal{H}}$. A good smoothing
procedure should reduce the numerical noise without smoothing excessively out
the real density fluctuations (e.g. substructures).
The total mass that has been assigned to the neighbouring cells should be
equal to the mass of the particle. However this is only approximately true due
to the discreteness of cells. In particular, mass conservation is quite poorly
observed when the smoothing length ${\cal{H}}$ is only a few mesh cells, which
may happen in a dense environment. For particles with $2{\cal{H}}\leq 15$ cell
sizes (30 cells in diameter), we therefore renormalise each individual kernel
so that the total mass is conserved during the assignment.
We find in practice that SPH assignment is superior to Cloud-In-Cell (CIC)
assignment in terms of reducing discreteness noise. For a singular isothermal
sphere realised with $10^{6}$ particles, the SPH-smoothed ($N_{\rm ngb}=32$)
and CIC-smoothed surface density fields show fluctuations of 2% and 30%
relative to the analytical results, respectively. For a realisation with
$10^{7}$ particles, the fluctuations decrease to 1% for the SPH assignment
($N_{\rm ngb}=320$, with the same smoothing length, ${\cal{H}}\propto N_{\rm
p}^{-1/3}N_{\rm ngb}^{1/3}$ (Li et al. 2006)) and to 10% for the CIC
assignment.
#### 3.1.3 Isolated boundary conditions
Periodic boundary conditions are most natural for Fourier Transforms, but are
not appropriate for lensing galaxies. We follow Hockney & Eastwood (1981) to
eliminate the (aliasing) effects due to “mirror” particles by using a mesh
twice as big as the simulated lens system, padding the region outside the
simulation volume with zeros. A truncated 2D gravitational force kernel is
tabulated onto the same simulation mesh, and then convolved with the assigned
surface density field. The gravitational effect is accurately reproduced
within the region where the mass has been distributed (See Hockney & Eastwood
1981, for more technical details). We adopt this procedure throughout this
work.
#### 3.1.4 Lensing potential, deflection angle and magnification
After the discretisation of the surface density field through SPH assignment
and the tabulation of the truncated 2D gravitational kernels on the meshes,
the potentials and their derivatives are easily calculated by convolutions
which can be efficiently implemented in Fourier space.
In particular, the effective lensing potential $\psi(\vec{\theta})$ is the
convolution of the surface density $\Sigma(\vec{\theta})$ and the 2D kernel
$\ln|\vec{\theta}|$:
$\psi(\vec{\theta})=\frac{1}{\pi}\int\Sigma(\vec{\theta^{\prime}})\ln|\vec{\theta}-\vec{\theta^{\prime}}|\,d^{2}\theta^{\prime}.$
(6)
The deflection angle $\vec{\alpha}(\vec{\theta})$ is the first derivative of
the lensing potential, $\psi(\vec{\theta})$, and is thus the convolution of
the surface density $\Sigma(\vec{\theta})$ and the 2D force kernel
$\vec{\theta}/|\vec{\theta}|^{2}$:
$\vec{\alpha}(\vec{\theta})\equiv\nabla\psi(\vec{\theta})=\frac{1}{\pi}\int\Sigma(\vec{\theta^{\prime}})\frac{\vec{\theta}-\vec{\theta^{\prime}}}{|\vec{\theta}-\vec{\theta^{\prime}}|^{2}}\,d^{2}\theta^{\prime}.$
(7)
The convergence $\kappa(\vec{\theta})$ (the surface density normalised to
$\Sigma_{\rm cr}$) and the shear $\gamma(\vec{\theta})$ are second-order
derivatives of the lensing potential $\psi(\vec{\theta})$:
$\begin{array}[]{c}\displaystyle\kappa=(\psi_{11}+\psi_{22})/2,~{}~{}\displaystyle\gamma_{1}=(\psi_{11}-\psi_{22})/2,\\\
\displaystyle\gamma_{2}=\psi_{12}=\psi_{21},~{}~{}\displaystyle\gamma^{2}=\gamma_{1}^{2}+\gamma_{2}^{2},~{}~{}\displaystyle~{}\psi_{ij}\equiv\frac{\partial^{2}\psi}{\partial\theta_{i}\partial\theta_{j}},\end{array}$
(8)
where the derivatives are taken with respective to the index 1 ($x$) and 2
($y$). Numerically, the convergence and shear can be calculated through 4th-
order finite differencing from the deflection angle
$\vec{\alpha}(\vec{\theta})$. The magnification $\mu(\vec{\theta})$ is related
to the convergence and shear by
$\mu=\frac{1}{(1-\kappa)^{2}-\gamma^{2}}.$ (9)
#### 3.1.5 Image finding and cusp relation
Since all the lensing quantities are now known, it is straightforward to find
the images for any given source position. To this end, we construct a separate
mesh in the image plane, with a resolution ($0.02h^{-1}$ kpc) higher than the
fine potential mesh discussed in §3.1.1; the lensing properties (deflection
angle, magnification etc.) on this ultra-fine mesh are found through bi-linear
interpolation. We then search image positions (and magnifications) using the
Newton-Raphson and triangulation methods (Schneider et al. 1992).
Of particular interest to gravitational lensing are the critical curves and
caustics. Critical curves in the image plane are a set of points where the
magnification is formally infinite for a point source,
$\mu(\vec{\theta})\longrightarrow\infty$. In practice, they are identified
according to the fact that the magnifications have different signs (i.e.,
different parities) for images on different sides of a critical curve.
Critical curves are mapped into caustics in the source plane, which can be
easily obtained through the lens equation.
Most strong lenses occur in elliptical galaxies since they have larger lensing
cross-sections than spiral galaxies (Turner et al. 1984). They typically form
two distinct sets of critical curves and corresponding caustics: the
tangential (“outer”) and radial (“inner”) critical curves, which are mapped
into tangential (“inner”) and radial (“outer”) caustics (see Fig. 5 for an
example). A source inside the central caustic usually produces five images:
four close to the tangential critical curve and one central image which is
usually too faint to be observable (and is of no interest to us for the
present work).
We are particularly interested in sources that are close to the cusps of the
tangential caustic (“cusp sources”). For cusp sources, three close images form
around the tangential critical line, with alternate parities. There are two
different kinds of cusp sources and corresponding image configurations. As
illustrated in Fig. 5, a “major cusp” source forms three images around the
tangential critical curve on the same side of the source (with respect to the
centre of the lens) while a “minor cusp” source forms three close images on
the opposite side of the source.
In any smooth lensing potential, for a source very close to a cusp, the three
close images satisfy an asymptotic magnification relation (the “cusp-caustic
relation”; Blandford & Narayan 1986; Schneider & Weiss 1992; Keeton et al.
2003):
$R_{\rm
cusp}\equiv\frac{|\mu_{A}+\mu_{B}+\mu_{C}|}{|\mu_{A}|+|\mu_{B}|+|\mu_{C}|}\rightarrow
0,$ (10)
with the total absolute magnification
$|\mu_{A}|+|\mu_{B}|+|\mu_{C}|\rightarrow\infty$.
For each of the cusp sources, we define an image opening angle $\Delta\theta$,
ranging from 0 to $\pi$, which measures the angle (from the lens centre) of
the outer images of the close triple. Notice that both $\Delta\theta$ and
$R_{\rm cusp}$ are observable. In a smooth lens potential, as a source moves
to a cusp caustic, both $\Delta\theta$ and $R_{\rm cusp}$ decrease
asymptotically to zero. As can be seen from Fig. 6, there are two leading
patterns on the $R_{\rm cusp}-\Delta\theta$ diagram due to “major” and “minor”
cusp sources. Generally speaking, the major cusp sources have larger $R_{\rm
cusp}$ than the minor cusp sources for the same image opening angle.
The cusp-caustic relation predicts that in smooth lens models $R_{\rm cusp}$
would asymptotically approach zero when a source moves towards the caustic.
However, the presence of (clumpy) substructures will break down the smooth
potential assumption in the asymptotic cusp-caustic relation, resulting in
substantial deviations in $R_{\rm cusp}$ values and other quantities (such as
image positions and time delays) from simple predictions. Therefore the
examination of the cusp-caustic relation is a way to test for the presence of
substructures that are projected near the (tangential) critical curves.
However, caution must be exercised because, even for smooth lens models, a
high $R_{\rm cusp}$ is possible. There are many factors that affect the
$R_{\rm cusp}$ distribution apart from the presence of substructures, e.g. the
mass distribution of the lens (radial profile and the ellipticity), external
shear from the environment, and the selection criteria of the cusp sources
(for more discussions see Keeton et al. 2003).
### 3.2 Singular isothermal sphere
Figure 4: The numerical accuracy of the deflection angle, the convergence, and
the magnification for Monte-Carlo realisations of singular isothermal spheres.
The top panels show the ratios of the numerical to the analytical results as a
function of radius. The deviation in the numerical magnification (on the
right) towards the centre is due to the finite mesh resolution of the
Particle-Mesh code, and that seen near the Einstein radius (at about 0.02
$r_{200}$) is due to the divergent behaviour of the magnification close to the
critical curve. The corresponding probability distributions for images with a
total magnification above 20 (around $r\sim 0.02$ $r_{200}$) are presented in
the bottom panels. The cyan and blue curves are for two $10^{6}$-particle
realisations (with $N_{\rm ngb}=32$) while the red curve is for a
$10^{8}$-particle realisation (with $N_{\rm ngb}=640$).
We test our lensing simulation code with Monte-Carlo realisations of singular
isothermal spheres (SIS), for which analytical lensing properties are known.
Each of our SIS contains a mass of $10^{12}h^{-1}M_{\odot}$ within a virial
radius of 100$h^{-1}$ kpc, realised with $10^{6}$ and $10^{8}$ particles; the
SPH assignment parameter is chosen to be $N_{\rm ngb}=32$ and $N_{\rm
ngb}=640$ for the two cases respectively. Fig. 4 shows the numerical accuracy
of the deflection angle, convergence (surface density) and magnification in
our numerical procedures. For the $10^{6}$ particle case, two Monte-Carlo
realisations (cyan and blue curves) are shown. For the $10^{8}$ particle
realisation, the uncertainties around the Einstein radius (at about 0.02
$r_{200}$, defined by a total magnification $\mu(\vec{\theta})\geq 20$) are
0.2% for the deflection angle, 1% for the convergence, and $<$ 10% for the
lensing magnification (estimated by the half width half maximum of the
probability distributions). The deviation towards the centre is due to the
fact that the finite mesh resolution of the Particle-Mesh code fails to
represent the singular behaviour at the centre of the SIS. The significant
deviation of the magnification seen near the Einstein radius is due to the
divergent behaviour of the magnification close to the critical curve, where
$\mu=1/((1-\kappa)^{2}-\gamma^{2})\longrightarrow\infty$, when
$\kappa=\gamma=0.5$ at the Einstein radius for the singular isothermal sphere.
### 3.3 High-resolution isothermal ellipsoid
We simulate an isothermal ellipsoid (IE) with $10^{6}$ and $10^{8}$ particles
(as in the Aquarius haloes). Such an isothermal ellipsoidal distribution is
modelled as an oblate spheroid with axis ratio $q_{3}$ and with a density
distribution:
$\rho\propto(S_{0}^{2}+R^{2}+z^{2}/q_{3}^{2})^{-1},$ (11)
where $S_{0}$ is a core radius, and $(R,z)$ are the cylindrical coordinates.
It is specified by three parameters (see Keeton & Kochanek 1998 for details):
the effective critical radius $b_{\rm I}$, the eccentricity of the mass
distribution $e=(1-q_{3}^{2})^{1/2}$, and a core radius $S_{0}$. The
parameters for the isothermal ellipsoidal halo are adjusted so that its
critical curves and caustics match those for the halo Aq-F-2 in the
$z$-projection. The parameters are $b_{\rm I}=0.4\arcsec$, $q_{3}=0.8$,
$S_{0}=0.1\arcsec$ and the major-axis of the surface density ellipse is
rotated by $\sim\pi/8$ with respect to the $X$-axis.
Figure 5: Critical curves, caustics, and cusp-caustic relation $R_{\rm cusp}$
maps for a Monte-Carlo realisations of isothermal ellipsoid with $N_{\rm
p}=1.6\times 10^{8}$ and $N_{\rm ngb}=640$. The top panels show the critical
curves and the caustics. The position and corresponding images are shown for a
“major cusp” (solid squares) and a “minor cusp” source (open squares). The
bottom panels show the $R_{\rm cusp}$ maps from the analytical solution (left)
and from the numerical result (right), with contour levels (0.0, 0.05, 0.1,
0.15, 0.2). The numerical tangential caustic from a $N_{\rm p}=10^{6}$ Monte-
Carlo realisation is also presented (blue curve). The swallow-tails due to
numerical noise are more apparent in this case. Figure 6: The cusp-caustic
relations (of the same Monte-Carlo realised isothermal ellipsoidal (IE) halo
as in Fig.5) for caustic sources with image opening angles $\Delta\theta\leq
90^{\circ}$. The left panel shows the source positions with respect to the
tangential caustic. The middle panel shows both the numerical
($10^{8}$-particle realisation) and analytical results of $R_{\rm cusp}$ vs.
$\Delta\theta$. The right panel shows the probability density distributions of
$R_{\rm cusp}$ for the analytical IE (red), and the two Monte-Carlo realised
haloes with $10^{6}$ (cyan) and $10^{8}$ particles (blue), respectively.
Fig. 5 shows analytical and numerical critical curves, caustics and $R_{\rm
cusp}$ maps for sources located inside the diamond caustics. The two critical
curves nearly overlap each other, with barely noticeable wiggles in the
numerical result. The caustics also agree reasonably well for the
$10^{8}$-particle case, but for the $10^{6}$-particle realisation (blue
lines), higher-order singularities such as swallowtails are clearly seen close
to the cusps and along the fold caustics. These arise due to numerical noise.
The numerical $R_{\rm cusp}$ map shows visible distortions compared with the
smooth contours in the analytical results, even in the central region.
Fig. 6 presents the $R_{\rm cusp}$ distributions for caustic sources
(indicated in the left panel) with image opening angle $\Delta\theta\leq
90^{\circ}$. The analytical results show two distinct peaks due to major-cusp
and minor-cusp sources with a sharp dropoff around $R_{\rm cusp}\sim 0.12$. In
contrast, for the numerical distributions, even the $10^{8}$-particle
realisation shows a much broader profile than the analytical one, with an
extended tail out to $R_{\rm cusp}\sim 0.4$ due to numerical noise. Notice
that the numerical noise behaves in a similar way as real substructures in the
$R_{\rm cusp}$ distribution.
In the current numerical set-up with $N_{\rm ngb}=640$ for a $10^{8}$-particle
simulation, the smoothing length ${\cal{H}}$ in the central region (of the
halo) approximately reaches the softening length of the Aquarius simulation,
also roughly the cell resolution of the fine mesh. Increasing ${\cal{H}}$ will
indeed further suppress the noise, but it may also over-smooth the underlying
density field. Below we outline an alternative way to approximate the smooth
underlying density fields for the host galaxy haloes.
## 4 Results
### 4.1 Lensing Predictions for Aquarius Haloes
In this section, we will apply our lensing methodology to the Aquarius haloes
(and a baryonic component modelled as a Hernquist profile), and study the
violation of the cusp-caustic relation due to the dark matter substructures
therein. In principle, we should compare the lensing properties of the
simulated haloes with and without substructures in order to assess the effects
of substructure. However, Fig. 6 shows a substantial broadening of the $R_{\rm
cusp}$ distribution due to numerical noise, which will significantly confuse
signals from real substructures.
As mentioned above, the total (dark matter plus baryons) mass profile of each
halo is adjusted to resemble an isothermal distribution in the central region.
To avoid excessive discreteness noise, we go one step further and adopt a
fitted isothermal ellipsoid (with a density distribution given by eq. [11])
rather than the original particle distribution for subsequent lensing
calculations. The parameters of the isothermal ellipsoid model are adjusted to
match the original critical curves and the caustics of each Aquarius halo
(together with the central galaxy) in a particular projection. We add the
particle distributions of substructure in the Aquarius simulations (assigned
to meshes according to the CIC algorithm) to the isothermal ellipsoid that
fitted to the main galaxy halo, and compare its lensing properties with those
of the smooth underlying isothermal ellipsoid. In this approach, the
analytical solutions of the fitted isothermal ellipsoidal potential and its
derivatives are tabulated on the image grid, including the cusp-caustic
relation. By doing so, no Poisson noise of the underlying main halo is
introduced, and so any confusion to the results from substructures is avoided.
We fit an isothermal ellipsoidal model to each of the three independent
projections of each galaxy halo. The six fitting parameters are: (1) the
effective critical radius $b_{\rm I}$, (2) the axis-ratio of the surface
density ellipse $q_{3}$, (3) the core radius $S_{0}$, (4)-(5) the $X$\- and
$Y$-offsets of the projected centre $X_{\rm c}$, $Y_{\rm c}$, and (6) the
rotation angle of the major axis of the surface density ellipse $RA$. The
uncertainties of the fitted parameters are $\Delta b_{\rm I}=0.003\arcsec$,
$\Delta q_{3}=0.01$, $\Delta S_{0}=0.001$, $\Delta X_{\rm c}=\Delta Y_{\rm
c}=0.002\arcsec$, $\Delta RA=0.004\pi$. The relative errors in the fitted
critical curves and caustics are $\lesssim$ 10 % for different projections of
all the simulated galaxy haloes.
In Table 3, we list the isothermal ellipsoid parameters of the main haloes
($b_{\rm I}$, $q_{3}$ and $S_{0}$). The critical radius $b_{\rm I}$ is of the
order of $0.3\arcsec$ to $0.9\arcsec$. The separation between images ($\sim
2b_{\rm I}$) is in the range of the observed gravitational lenses (which peaks
around $1\arcsec$, see e.g. Browne et al. 2003). The axial ratios also match
the observed lenses quite well. There is one exception, however. The core
radius $S_{0}$ is quite large, of the order of ($0.05\arcsec-0.1\arcsec$,
corresponding to a few hundreds of pc). Such a core size is larger than those
inferred from gravitational lenses which are in general consistent with zero
core radius (e.g. Wallington & Narayan 1993; Rusin & Ma 2001; Oguri et al.
2001; Li & Ostriker 2003). This is a direct result of the implementation of
the Hernquist profiles, which follow a logarithmic density slope of $-1$ in
the central regions (see Fig. 1). However, this artifact should have little
effects on the images we are interested in, which are close to the outer
critical curve.
To examine the violation of the cusp-caustic relation, we generate about 10000
cusp sources in each case, and calculate the resulting $R_{\rm cusp}$
distributions. All these cusp sources are inside the central caustic and close
to the cusps, where the corresponding triple images have opening angles
$\Delta\theta\leq 90^{\circ}$. The results of all 7 studied Aquarius haloes
(in 21 projections) are given in Fig. 7 to Fig. 13. As mentioned above, the
probability density distribution of $R_{\rm cusp}$ often shows two peaks for
the smooth haloes which are produced by the major and minor cusps,
respectively. For the “naked” cusp cases (where the central diamond caustic
protrudes the outer elliptical caustic), the distributions of $R_{\rm cusp}$
vs. the opening angle $\Delta\theta$ are somewhat truncated below certain
opening angles (see Fig. 9 for the halo Aq-C-2’s $Y$-projection for an
example). Empirically, it is rare for massive lensing galaxies to produce
naked cusps. There is only one candidate APM08279 (Lewis et al. 2002), and
that is likely due to lensing by an edge-on spiral rather than an elliptical.
The four naked cusp cases from our simulations are caused by the large cores
in the central density profiles of the lensing galaxies. We exclude these four
naked cusp cases in the final statistic calculations (their inclusion does not
significantly alter our results). Strong violations of the cusp-caustic
relation due to substructures are seen in some cases, e.g. for the
$Y$-projection of the Aq-B-2 halo (see Fig. 8). However, most of these cases
have small cusp-lensing cross-sections (listed in Table 3, Column 8), defined
as the areas covered by cusp sources whose images satisfy $\Delta\theta\leq
90^{\circ}$. The mean probability of cusp violations calculated below are
weighted by the cross-sections (see eq. 12). As can be seen from Fig. 7 to
Fig. 12, the scatter in the cusp violation is large between different
projections of different haloes. Also notice that the halo Aq-A-2 at $z=0.6$
(Fig. 13) does not show a significant difference from the redshift-zero haloes
in the violation of the cusp-caustic relation.
To see which massive substructures cause the cusp-caustic violation, we
calculate the $R_{\rm cusp}$ distribution due to subhaloes more massive than
$10^{5}h^{-1}M_{\odot}$, $10^{6}h^{-1}M_{\odot}$, $10^{7}h^{-1}M_{\odot}$, and
$10^{8}h^{-1}M_{\odot}$, respectively. Fig. 14 shows one typical example, for
the halo Aq-D-2 along the $Z$-projection. We find that in most cases
substructures with masses $m_{\rm sub}\leq 10^{7}$ to $10^{8}h^{-1}M_{\odot}$
dominate the contribution to the violations of the cusp-caustic relation (see
the Col (9): $M_{\rm sub,cr}$ in Table 3). Notice that previous studies on
cusp violations typically resolve haloes larger than $\sim
10^{8}h^{-1}M_{\odot}$, and thus would not have been able to evaluate the
effects of substructure accurately. However, the addition of subhaloes with
$m_{\rm sub}\la 10^{6}h^{-1}M_{\odot}$ does not appear to increase the
violation frequency significantly (compare the three right panels). We return
to the convergence issue as a function of subhalo mass in §5.
Notice that most subhaloes that are projected close to the critical curves are
due to chance alignment. Fig. 15 shows the spherical halocentric distance
distribution for the subhaloes that are within a projected distance of 0.05
$r_{200}$ ($\sim 2.5$ Einstein radii). The fractions of subhaloes that are
physically located within a spherical radius of 0.05 $r_{200}$ are 15%, 18%,
15% and 0% for subhaloes more massive than $10^{5}h^{-1}M_{\odot}$,
$10^{6}h^{-1}M_{\odot}$, $10^{7}h^{-1}M_{\odot}$ and $10^{8}h^{-1}M_{\odot}$,
respectively. The large median halocentric distances, $\sim 0.2$ $r_{200}$ in
all cases, also show that projection effects are substantial.
### 4.2 Comparison with observations
Keeton et al. (2003) summarised 19 published quadruply imaged systems. Seven
of them are detected at radio wavelengths444B0128+437 (Phillips et al. 2000),
B0712+472 (Jackson et al. 1998; Jackson et al. 2000), B1422+231 (Impey et al.
1996; Patnaik & Narasimha 2001), B1555+375 (Marlow et al. 1999), B1608+656
(Koopmans & Fassnacht 1999), B1933+503 (Cohn et al. 2001) and B2045+265
(Fassnacht et al. 1999).. Radio lenses are free from dust extinction. Due to
their large emission regions, they are less likely to be affected by
microlensing. In contrast, microlensing is likely to affect optical/IR flux
ratios and so we treat them differently below.
Dalal & Kochanek (2002) studied seven four-image radio-lensing systems:
MG0414+0534 (Hewitt et al. 1992), B0712+472 (Jackson et al. 1998), PG1115+080
(Weymann et al. 1980), B1422+231 (Patnaik & Narasimha 2001), B1608+656
(Fassnacht et al. 1996), B1933+503 (Sykes et al. 1998) and B2045+265
(Fassnacht et al. 1999) and found that six show anomalous flux ratios, which
might be due to the effects of substructure lensing. Among all the detected
radio lenses, three (B0712+472, B1422+231 and B2045+265) show a typical cusp-
caustic geometry (with $\Delta\theta\leq 90^{\circ}$) and violations of the
cusp-caustic relation. Another two lensing systems observed in the optical/IR
band are also cusp-caustic lenses with $\Delta\theta\leq 90^{\circ}$:
RXJ1131-1231 (Sluse et al. 2003) and RXJ0911+0551 (Bade et al. 1997; Burud et
al. 1998). Both have unexpected large values of $R_{\rm cusp}$, which were
shown to have been affected by microlensing (Morgan et al. 2006; Anguita et
al. 2008). Table 2 lists the $R_{\rm cusp}$ and $\Delta\theta$ values for the
five observed cusp-caustic lenses. Three out of the five cusp lenses are
detected at radio wavelengths, thus their large $R_{\rm cusp}$ values are
unlikely due to microlensing. We treat these three radio lenses as cusp-
caustic violations due to substructure lensing. Below we will calculate the
probability for the simulations to reproduce such an observed violation rate.
For each galaxy and each projection, we calculate the violation probability
that the predicted $R_{\rm cusp}$ is larger than the observed $R_{\rm cusp}$
value 0.187 for B1422+231, which shows the smallest violation (smallest
$R_{\rm cusp}$ value) among the five cusp lenses with $\Delta\theta\leq
90^{\circ}$. The cross-section weighted violation probability is given by
$p_{\sigma}=\sum_{i}f_{\sigma,i}\,p_{i}(R_{\rm cusp}\geq
0.187|\Delta\theta\leq
90^{\circ}),f_{\sigma,i}=\frac{\sigma_{i}}{\sum_{i}\sigma_{i}},$ (12)
where the summation $i=1,\cdot\cdot\cdot,(21-4)$ is for the seven haloes along
the three independent projections of each, excluding the four naked cusp
cases, and $\sigma_{i}$ is the cross-section in the source plane for producing
three close images with opening angle $\Delta\theta\leq 90^{\circ}$. Using the
above formula, we find the mean probability $p_{\sigma}\approx 6.4\%$ for
$R_{\rm cusp}\geq 0.187$. Notice that this probability estimate is only
approximate, since we have not considered the magnification bias (e.g. Turner
et al. 1984).
Table 2: The image opening angle and $R_{\rm cusp}$ for the observed cusp-caustic lenses, taken from Amara et al. (2006). Lens | $\Delta\theta$ | $R_{\rm cusp}$ | Band
---|---|---|---
B0712+472 | 79.8∘ | 0.26 $\pm$ 0.02 | radio
B2045+265 | 35.3∘ | 0.501 $\pm$ 0.035 | radio
B1422+231 | 74.9∘ | 0.187 $\pm$ 0.006 | radio
RXJ1131 - 1231 | 69.0∘ | 0.355 $\pm$ 0.015 | optical/IR
RXJ0911 + 0551 | 69.6∘ | 0.192 $\pm$ 0.011 | optical/IR
To have three (radio) lensing cases with $R_{\rm cusp}\geq 0.187$ (due to
substructure lensing rather than microlensing) out of the five cusp lenses
($\Delta\theta\leq 90^{\circ}$) observed so far, the probability is
$C_{5}^{3}p_{\sigma}^{3}(1-p_{\sigma})^{2}\approx 2.3\times 10^{-3}$. The low
probability suggests that the subhalo populations in the inner regions of the
Aquarius haloes with Hernquist galaxies are insufficient to explain the
observed frequency of flux anomalies in the cusp lenses.
## 5 DISCUSSION AND CONCLUSIONS
In this paper, we have used the ultra-high resolution Aquarius simulations to
study the effects of substructure lensing. We incorporate the effects of
baryons in the main halo by adding a stellar component (modelled as a
Hernquist profile), and then take into account its effects on the dark matter
halo through adiabatic contraction. The density profiles and lensing
properties except the flux ratios are broadly consistent with the observed
gravitational lenses. Using Monte Carlo simulations, we find large numerical
noise for an isothermal halo populated with $10^{8}$ particles, which shows
considerable scatter in the $R_{\rm cusp}$ distribution for cusp lenses. In
the end, we therefore study the substructure lensing by modelling the smooth
underlying galaxy halo as an isothermal ellipsoid and superimposing the
subhalo population from the Aquarius simulations. In this way, we focus on the
lensing effects of subhaloes and avoid any confusion from numerical noise in
the $N$-body realisation of the simulated main haloes.
Our study finds that even with the much better resolved subhalo population of
the Aquarius simulations, the observed cusp lenses still violate the cusp-
caustic relation more frequently than predicted by $N$-body simulations.
The Aquarius haloes are Milky Way type haloes in terms of their masses, while
many lenses are ellipticals, which are more massive. Among the five cusp
lenses we compare our results with, three of them (B2045+265, RXJ1131-1231,
RXJ0911) are more massive than our simulated haloes and have Einstein radii
twice as large as those of our haloes. The other two lenses (B0712+472,
B1422+231) have Einstein radii and velocities (circular velocity or velocity
dispersion) roughly comparable to the relatively massive haloes in the
Aquarius simulations. As shown in Fig. 2 (the right panel) the projected
subhalo mass fraction increases with the projected radius $R$. If we have
under-estimated the Einstein radii $b_{I}$ (e.g. because of uncertainties in
the addition of the central galaxies), we could have potentially under-
estimated the violation rates due to the lack of enough substructures at
smaller radii. We artificially increase the Einstein radii of the simulated
haloes by a factor of two to study the violation probabilities due to a higher
fraction of substructures at larger radii. The mean subhalo mass fraction
within a $0.1\arcsec$-annulus around the new Einstein radius would increase
from $f_{\rm sub,annu}$ $\approx$ 0.19% to 0.24%, and the mean violation
probability would increase from $p_{\sigma}\approx$ 6.4% to 14.0%. The
probability of reproducing the observed violation rate would increase from
0.2% to 2%.
Another concern is that due to the finite particle mass in $N$-body
simulations, the central cusps of the subhaloes may not be resolved, which may
potentially result in an under-estimation of the ability of the subhaloes to
induce perturbations to the lensing potential. We consider an extreme case
assuming all subhaloes are point-like sources with their masses and locations
from the simulations. In this scenario, $f_{\rm sub,annu}$ roughly remains at
0.18%, however, $p_{\sigma}$ increases to 15.1%. The probability to reproduce
the observed violation rate increases to 2.5%.
These low probabilities suggest that the subhalo populations in the central
regions of the Aquarius haloes are not sufficient to explain the observed
frequency of violations of the cusp-caustic relations. It is important to ask
whether our results will change significantly if even lower-mass subhaloes are
resolved. We argue that this is unlikely to be the case. The total subhalo
lensing cross-section is an integral of the cross-section of subhaloes of each
mass weighted by their abundance. As shown in §4, most of the perturbing
subhaloes have relatively low mass ($m_{\rm sub}\leq 10^{7}$ to
$10^{8}h^{-1}M_{\odot}$). Their abundance scales as ${\rm d}N(m_{\rm
sub})/{\rm d}m_{\rm sub}\propto m_{\rm sub}^{-1.9}$. For a galaxy (subhalo)
approximated by a SIS, the lensing cross-section roughly scales as
$\sigma^{4}$ (e.g. Turner et al. 1984) where $\sigma$ is the one-dimensional
velocity dispersion. For Aquarius subhaloes, $m_{\rm sub}\propto V_{\rm
max}^{3}$ (Springel et al. 2008), where $V_{\rm max}$ is the maximum circular
velocity. If $\sigma\propto V_{\rm max}$, then the integrated lensing cross-
section will be $\propto m_{\rm sub}^{0.43}$. On the other hand, for a point
lens or an elliptical galaxy, the lensing cross-section is proportional to the
lens mass, and the integrated lensing cross-section would be $\propto m_{\rm
sub}^{0.1}$. In all these cases, the subhalo lensing cross-sections are biased
towards relatively massive subhaloes in the projected central region, and the
incorporation of even lower mass subhaloes should not change our results
significantly.
We mention in passing that a warm dark matter scenario would suppress the
formation of small subhaloes, making it even more difficult to explain the
observed cusp violations (see e.g. Miranda & Macciò 2007). Below, we compare
our study with previous work, before discussing its limitations and outlining
possible future work.
### 5.1 Comparison with previous studies
There have been a number of studies of substructure lensing using numerically
simulated haloes, including those from hydrodynamical simulations. Below we
compare a few of these studies with our own.
Dalal & Kochanek (2002) concluded that at the 90% confidence level, a
substructure fraction of 0.6% to 7% can explain the observed anomalous flux
ratio. For the Aquarius subhalo population, Table 3 Col (5): $f_{\rm
sub,annu}$ shows such fraction averaged over a thin annulus around the outer
tangential curve, which is always below 1%, sometimes much smaller (not to be
confused with $f_{\rm sub}$ in Table 1 and Fig. 2, which refers to the subhalo
mass fraction within $r_{200}$). This is the primary reason why our predicted
cusp violations are smaller than the observed violation frequencies.
Bradač et al. (2004) used hydrodynamical simulations of Steinmetz & Navarro
(2002) and concluded that the predicted cusp violations due to substructure
are comparable to those observed. Their simulated halo has $\sim 10^{5}$
particles, resolving subhaloes down to $5\times 10^{8}M_{\odot}$. As the
authors pointed out, the numerical noise may be as high as 5%. The observed
high-order singularities in their simulations are much higher than ours
(comparable to the caustic structure shown in Fig. 5 for $10^{6}$ particles).
It is possible that their high numerical noise may have produced too many
artificial violations, although we note that they used Voroni density
estimation to reduce the discreteness noise.
Our conclusion that the dark matter subhalo population may be insufficient to
explain the observed cusp violations is consistent with Mao et al. (2004),
Amara et al. (2006), Macciò et al. (2006) and Macciò & Miranda (2006). The
number of particles used in those studies is roughly two orders of magnitude
smaller than here. In particular, the study by Mao et al. (2004) found large
scatter among different haloes, a conclusion confirmed by our results.
### 5.2 Limitations of the present study and future work
The most severe limitation of our study is that the high-resolution
simulations used here include only dark matter. Without baryons, these haloes
are sub-critical (see §2) and incapable of producing multiple images. We are
therefore forced to incorporate a model for the baryonic galaxy at the centre
of each halo. The galaxy changes not only the overall dark matter profiles
(taken into account by adiabatic contraction), but also the dynamical
evolution of subhaloes, an effect which is not considered here. On the one
hand, the increased baryonic density at the centre of the halo will make the
subhaloes feel stronger tidal forces, particularly those that come close to
the centre. On the other hand, the baryons within subhaloes will make them
more resilient to tidal disruption. It is not clear which effect will
dominate. We comment, however, that the subhaloes that come very close to the
centre may have already been tidally stripped or disrupted, and thus most of
the surviving subhaloes that can be identified by SUBFIND may have quite large
peri-centre passages. As a result, the effects of baryons in the host halo may
not change the results very significantly. However, we caution that, SUBFIND,
like most substructure finders, has difficulties in identifying subhaloes in
the densest regions of the halo and assigning them correct masses. Empirically
the Milky Way does not seem to host many luminous satellites close to the
centre. Hydrodynamical simulations can in principle address this issue
directly (subject to the uncertainties in the treatment of gas processes).
Macciò et al. (2006) found a factor of two increase in the number of surviving
satellite galaxies (with masses above $10^{7}M_{\odot}$) in the centres of
galaxies when including baryons in the CDM simulations, but concluded that
even this was not sufficient to explain the flux anomaly problem.
Observationally, it is interesting that more than one half of the CLASS lenses
appear to show luminous companion galaxies in projection (Bryan et al. 2008;
Jackson et al. 2009), and their inclusion in the models appears to alleviate
the anomalous flux ratio problem (see below). This may be just a statistical
fluke due to the small sample size (22 lenses in total) or some of these may
be due to chance alignment along the line-of-sight (Chen et al. 2003;
Wambsganss et al. 2005; Metcalf 2005a,b; Miranda & Macciò 2007). Nevertheless,
for the three radio lenses that show apparent cusp violations (see Table 2),
the most serious case is B2045+265 with $R_{\rm cusp}\approx 0.5$. Recently,
McKean et al. (2007) found a galaxy, G2, which is about 0.66 $\arcsec$ away
from the main lensing galaxy G1 (at redshift 0.867), and about 3.6 to 4.5
magnitudes fainter than G1 depending on the wavelength. The photometric
redshift of G2 is consistent with that of G1 (although also consistent with a
redshift $\sim 4-5$). The inclusion of this faint satellite galaxy in the
model can explain the flux anomaly reasonably well, although the satellite is
required to be very flattened with an axis ratio of 8:1, which may not be
realistic. This case highlights the potential roles that the luminous
satellites may play in the anomalous flux ratio problem. We note, however,
that numerical simulations by Dolag et al. (2008) showed that star-dominated
galaxies (not traced by dark matter only simulations) appear to contribute
only $\sim 10\%$ of the subhalo population in clusters of galaxies. It is
unclear however whether this cluster-based result can be extrapolated to
galaxy scales where cooling is more efficient. We plan to use semi-analytical
galaxy catalogues in the Aquarius simulations to address this issue more
quantitatively in subsequent work.
Substructures not only perturb the flux ratios, but also affect the image
positions. In Table 3, we show the maximum perturbation of the deflection
angle, $\alpha_{\rm sub,max}$, within the central $2\arcsec\times 2\arcsec$
region, produced by all the subhaloes within $r_{200}$. The maximum deviations
range from a few milli-arcseconds to $<0.1$ arcseconds. They may leave
observable signatures on close pair images such as that observed in MG2016+112
(Koopmans et al. 2002; More et al. 2009). We find that most of these
astrometric deviations are dominated by one large, nearby subhalo. This
clearly warrants further work in the near future.
## Acknowledgements
We thank Ian Browne, Neal Jackson, and Peter Schneider for useful discussions.
We also acknowledge an anonymous referee for constructive comments that
improved the paper. DDX has been supported by a Dorothy Hodgkin fellowship for
her postgraduate studies. LG acknowledges support from a STFC advanced
fellowship, one-hundred-talents program of the Chinese Academy of Sciences
(CAS) and the National basic research program of China (973 program under
grant No. 2009CB24901). SM acknowledges travel support from the Humboldt
Foundation and European Community’s Sixth Framework Marie Curie Research
Training Network Programme, contract number MRTN-CT-2004-505183 “ANGLES”. GL
is supported by the Humboldt Foundation. The simulations for the Aquarius
Project were carried out at the Leibniz Computing Centre, Garching, Germany,
at the Computing Centre of the Max-Planck-Society in Garching, at the
Institute for Computational Cosmology in Durham, and on the ‘STELLA’
supercomputer of the LOFAR experiment at the University of Groningen.
Figure 7: Lensing properties for the halo Aq-A-2, in three independent
projections. The top panels show the critical curves (red) and caustics (blue)
superimposed on top of the subhalo population. The middle panels show $R_{\rm
cusp}$ vs. the image opening angle $\Delta\theta$. Large $R_{\rm cusp}$ values
(red) are due to substructure. The triangle pattern (green) gives predictions
for the smooth counterparts. The bottom panels show the corresponding
probability distribution functions (PDFs) of $R_{\rm cusp}$ for cusp sources
with $\Delta\theta\leq 90^{\circ}$. The violation of the cusp-caustic relation
can be seen from the excess of $R_{\rm cusp}$ at large values (red) over the
smooth counterpart curve (green). The $R_{\rm cusp}$ values for the three
radio and two optical/IR cusp lenses are indicated by vertical solid and
dashed bars (see Table 2).
Figure 8: For the halo Aq-B-2, the symbols are the same as in Fig. 7. The
truncated triangle pattern in the $Z$-projection is due to naked cusps of the
central caustic. The strong violation of the cusp-caustic relation seen in the
$Y$-projection is caused by subhaloes with $m_{\rm sub}\leq
10^{8}h^{-1}M_{\odot}$ with a violation rate $P(R_{\rm cusp}\geq 0.187)=64\%$.
Figure 9: For the halo Aq-C-2, the symbols are the same as in Fig. 7. The
truncated triangle pattern in the $Y$-projection is due to naked cusps of the
central caustic. The halo in this projection has large ellipticity, which
results in large $R_{\rm cusp}$ values. The strong violation in the
$X$-projection is caused by subhaloes with $m_{\rm sub}\leq
10^{8}h^{-1}M_{\odot}$ with a violation rate $P(R_{\rm cusp}\geq 0.187)=19\%$.
Figure 10: For the halo Aq-D-2, the symbols are the same as in Fig. 7. The
strong violations in the $Y$\- and $Z$-projection are caused by subhaloes with
$m_{\rm sub}\leq 10^{7}h^{-1}M_{\odot}$ and $m_{\rm sub}\leq
10^{8}h^{-1}M_{\odot}$, respectively, with violation rates $P(R_{\rm cusp}\geq
0.187)=9.7\%$ ($Y$-projection) and $31\%$ ($Z$-projection).
Figure 11: For the halo Aq-E-2, the symbols are the same as in Fig. 7.
Figure 12: For the halo Aq-F-2, the symbols are the same as in Fig. 7. The
truncated triangle pattern in the $Y$-projection is due to naked cusps of the
central caustic. The strong violation in the $Z$-projection is mainly caused
by subhaloes with $m_{\rm sub}\leq 10^{7}h^{-1}M_{\odot}$ with a violation
rate $P(R_{\rm cusp}\geq 0.187)=6.7\%$.
Figure 13: For the halo Aq-A-2 at $z$=0.6, the symbols are the same as in Fig.
7. The strong violation in the $Y$-projection is caused by subhaloes with
$m_{\rm sub}\leq 10^{8}h^{-1}M_{\odot}$; a cusp violation rate is $P(R_{\rm
cusp}\geq 0.187)=56\%$.
Figure 14: Effects of substructure lensing as a function of the lower cutoff
subhalo mass for the halo Aq-D-2 along the $Z$-projection. The upper panels
show the projected substructures with masses above a threshold. The projected
centres of the subhaloes and the corresponding critical curves are plotted at
the top. From the left to the right, the lower cutoff subhalo mass changes
from $10^{8}h^{-1}M_{\odot}$, $10^{7}h^{-1}M_{\odot}$, $10^{6}h^{-1}M_{\odot}$
to $10^{5}h^{-1}M_{\odot}$. The bottom panels show the corresponding
probability distribution functions of $R_{\rm cusp}$. Most substructures that
survive and are projected within the central few kpc are low-mass subhaloes
($\leq 10^{8}$ $h^{-1}M_{\odot}$), which dominate the violation of the cusp-
caustic relation. Figure 15: Distribution of halocentric distances of the
subhaloes projected within 0.05 $r_{200}$ ($\sim$ 2.5 times the Einstein
radius, indicated by the dotted line in each panel). The solid lines give the
median spherical halocentric distances of the subhaloes; all are around 0.2
$r_{200}$. The average number of subhaloes $\bar{N}$ is indicated inside each
panel.
Table 3: Substructure-lensing parameters of Aquarius haloes:
Halo Name | $b_{\rm I}$ | $q_{3}$ | $S_{0}$ | $f_{\rm sub,annu}$ | $\alpha_{\rm sub,max}$ | $P(R_{\rm cusp}\geq 0.187)$ | $f_{\sigma}(\Delta\theta\leq 90^{\circ})$ | $M_{\rm sub,cr}$
---|---|---|---|---|---|---|---|---
Projection | ($\arcsec$) | | ($\arcsec$) | (per cent) | ($\arcsec$) | (per cent) | (per cent) | ($h^{-1}M_{\odot}$)
Aq-A-2 | | | | | | | |
X-projection | 0.602 | 0.77 | 0.103 | 0.11 | 0.065 | 5.90 | 4.85 | $10^{7}\Downarrow$
Y-projection | 0.657 | 0.78 | 0.092 | 0.69 | 0.059 | 1.26 | 5.76 | $10^{8}\Uparrow$
Z-projection | 0.647 | 0.76 | 0.091 | 0.01 | 0.055 | 0.08 | 6.59 | $10^{7}\Downarrow$
Aq-B-2 | | | | | | | |
X-projection | 0.306 | 0.73 | 0.076 | 0.42 | 0.008 | 5.91 | 2.16 | —
Y-projection | 0.408 | 0.91 | 0.071 | 0.48 | 0.007 | 64.13 | 0.41 | $10^{8}\Downarrow$
Z-projection | 0.286 | 0.64 | 0.080 | 0.09 | 0.012 | 0.09 | 0.77 | —
Aq-C-2 | | | | | | | |
X-projection | 0.846 | 0.91 | 0.085 | 0.13 | 0.015 | 19.09 | 1.07 | $10^{8}\Downarrow$
Y-projection | 0.571 | 0.60 | 0.107 | 0.06 | 0.002 | 13.98 | 21.41 | —
Z-projection | 0.589 | 0.70 | 0.104 | 0.03 | 0.007 | 3.71 | 10.58 | $10^{7}\Downarrow$
Aq-D-2 | | | | | | | |
X-projection | 0.576 | 0.83 | 0.101 | 0.13 | 0.006 | 3.79 | 2.13 | $10^{7}\Downarrow$
Y-projection | 0.655 | 0.91 | 0.088 | 0.06 | 0.005 | 9.72 | 0.57 | $10^{7}\Downarrow$
Z-projection | 0.583 | 0.79 | 0.101 | 0.40 | 0.011 | 30.58 | 4.48 | $10^{8}\Downarrow$
Aq-E-2 | | | | | | | |
X-projection | 0.473 | 0.69 | 0.076 | 0.18 | 0.020 | 3.04 | 7.62 | $10^{8}\Downarrow$
Y-projection | 0.548 | 0.79 | 0.056 | 0.13 | 0.007 | 5.40 | 3.29 | $10^{7}\Downarrow$
Z-projection | 0.474 | 0.82 | 0.069 | 0.05 | 0.006 | 0.65 | 1.59 | $10^{7}\Downarrow$
Aq-F-2 | | | | | | | |
X-projection | 0.416 | 0.86 | 0.080 | 0.19 | 0.021 | 5.83 | 0.67 | $10^{8}\Downarrow$
Y-projection | 0.370 | 0.67 | 0.091 | 0.09 | 0.009 | 1.38 | 3.78 | —
Z-projection | 0.435 | 0.85 | 0.088 | 0.22 | 0.012 | 6.74 | 0.78 | $10^{8}\Downarrow$
Aq-A-2 ($Z$ = 0.6) | | | | | | | |
X-projection | 0.568 | 0.71 | 0.079 | 0.11 | 0.008 | 0.60 | 7.75 | $10^{8}\Downarrow$
Y-projection | 0.731 | 0.89 | 0.054 | 0.70 | 0.022 | 55.81 | 1.74 | $10^{8}\Downarrow$
Z-projection | 0.592 | 0.69 | 0.082 | 0.33 | 0.083 | 7.56 | 12.00 | $10^{8}\Downarrow$
Note: Cols (2-4): $b_{\rm I}$, $q_{3}$ and $S_{0}$, the Einstein radius, axis
ratio and core radius of the fitted isothermal ellipsoid (see eq. [11]); Col
(5): $f_{\rm sub,annu}$ is the subhalo mass fraction within a
$0.1\arcsec$-annulus around the outer critical curve; Col (6): $\alpha_{\rm
sub,max}$ is the maximum magnitude in the projected central $2\arcsec\times
2\arcsec$ region of the deflection angle due to all substructures within
$r_{200}$, usually found close to an individual subhalo; Col (7): $P(R_{\rm
cusp}\geq 0.187)$ is the probability (in per cent) for sources with
$\Delta\theta\leq 90^{\circ}$ (defined as “cusp sources”) to have $R_{\rm
cusp}\geq 0.187$, referred to as the “cusp-caustic violation probability”; Col
(8): $f_{\sigma}(\Delta\theta\leq 90^{\circ})$ is the cross-section fraction
(as defined in eq. [12]) in the source plane for producing three close images
with opening angle $\Delta\theta\leq 90^{\circ}$; Col (9): $M_{\rm sub,cr}$ is
the critical subhalo mass that causes the strong violation of the cusp-caustic
relation. Arrows indicate “above” or “below”. Lenses with naked cusps of the
caustic always have low cusp-caustic violation probability and are labelled as
“—”.
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|
arxiv-papers
| 2009-03-26T12:19:00 |
2024-09-04T02:49:01.428853
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. D. Xu (1), S. Mao (1), J. Wang (2,3), V. Springel (2), L. Gao (3),\n S. D. M. White (2), C. S. Frenk (3), A. Jenkins (3), G. Li (4) and J. F.\n Navarro (5) ((1) Jodrell Bank Centre for Astrophysics, (2) Max-Planck\n Institute for Astrophysics, (3) University of Durham, (4) University of Bonn,\n (5) University of Victoria)",
"submitter": "Dandan Xu",
"url": "https://arxiv.org/abs/0903.4559"
}
|
0903.4619
|
11institutetext: Theoretical Physics, University of Oxford, 1 Keble Road,
Oxford, OX1 3NP, United Kingdom
Institut de Physique Théorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France and
CNRS, URA 2306
Science and Finance, Capital Fund Management, 6 Bd Haussmann, 75009 Paris,
France
Theory and modeling of the glass transition Dynamic critical phenomena
Equilibrium properties near critical points, critical exponents
# Mode-coupling as a Landau theory of the glass transition
A. Andreanov 11 G. Biroli and J.-P. Bouchaud 2233112233
###### Abstract
We derive the Mode Coupling Theory (MCT) of the glass transition as a Landau
theory, formulated as an expansion of the exact dynamical equations in the
difference between the correlation function and its plateau value. This sheds
light on the universality of MCT predictions. While our expansion generates
higher order non-local corrections that modify the standard MCT equations, we
find that the square root singularity of the order parameter, the scaling
function in the $\beta$ regime and the functional relation between the
exponents defining the $\alpha$ and $\beta$ timescales are universal and left
intact by these corrections.
###### pacs:
64.70.Q-
###### pacs:
64.60.Ht
###### pacs:
64.60.F-
The Mode-Coupling Theory of glasses (MCT), developed since the mid-eighties
following the seminal work of Götze [1] and Leutheusser [2], has significantly
contributed to our understanding of the slowing down of supercooled liquids.
One of its cardinal predictions is the appearance of a non trivial
$\beta$-relaxation regime where dynamical correlation functions pause around a
plateau value before finally relaxing to zero. In the vicinity of this plateau
value, the theory predicts two power-law regimes in time (or in frequency),
and the divergence of two distinct relaxation times, $\tau_{\alpha}$ and
$\tau_{\beta}$, at the MCT critical temperature $T_{d}$. Although this
divergence is smeared out by activated events in real liquids, the two-step
relaxation picture suggested by MCT seems to account quite well for
experimental and numerical observations [3], at least in weakly supercooled
liquids and for hard sphere colloidal systems.
Originally, MCT was obtained as an uncontrolled self-consistent approximation
within the Mori-Zwanzig projection operator formalism for Newtonian particles.
This scheme yields an integro-differential equation for the dynamic structure
factor $C({\bf k},t)$ that captures mathematically the slowing down of the
dynamics and the appearance of a two-step relaxation at equilibrium. It
provides very detailed predictions for the scaling properties of $C({\bf
k},t)$ in the vicinity of the plateau value $f_{{\bf k}}S_{{\bf k}}$, where
$S_{{\bf k}}$ is the static structure factor and $f_{{\bf k}}$ is called the
non-ergodicity parameter (akin to the Edwards-Anderson parameter in spin
glasses). Alternative derivations of MCT based on field theory have been
sought for [4] and research on this topic has continued until now [5, 6, 7,
8]. It was also realized that the same integro-differential equations describe
the exact evolution of the correlation function of mean-field p-spin glasses
[9]. This is important for at least two reasons:
* •
Technically, it shows that the MCT approximation is realizable: there is a
well defined system for which it is exact; hence MCT does not violate basic
physical constraints.
* •
Physically, it brings in a very useful interpretation of the MCT freezing
transition in the “energy landscape” parlance. Above the transition $T_{d}$,
the dynamics is dominated by unstable saddles that become progressively less
and less unstable as one approaches the transition; below the transition,
there are only local minima that are separated by infinite barriers in mean-
field, so that the system is forever trapped in one of them [9]. For non mean-
field systems, these barriers are finite and the transition is smoothed. MCT
can be naturally embedded within the broader Random First Order Theory [10] of
the glass transition. In this context it describes the high temperature region
where metastable states are still in embryo.
However, this analogy shows that MCT is (at best) an incomplete theory of real
supercooled liquids, and needs to be corrected and enhanced. An important
question is to know to what extent the quantitative predictions of MCT are
stable with respect to the ignored contributions. In fact, it is genrally
accepted that the quantitative value of the critical temperature (or the
critical density) obtained solving MCT equations is incorrect, as is the
interaction parameter $\lambda$, which is a functional of the static structure
factor and fixes the value of critical exponents. However, when MCT is used to
fit empirical data, it is assumed (without much justification) that the
predictions about the critical behavior remain correct if $T_{d}$ and
$\lambda$ are treated as adjustable parameters. This implicitly assumes that
some MCT predictions are universal, e.g. the square root singularity of
$f_{{\bf k}}$ and the relation between the exponents describing the divergence
of $\tau_{\alpha}$ and $\tau_{\beta}$, and others which are not, e.g. $T_{d}$
and the actual value of the exponents!
Clearly, the present theoretical understanding of MCT needs to be improved.
Assessing the degree of structural stability of the theory and its
universality properties is a crucial issue to resolve both for theoretical and
practical purposes. One step in this direction has been performed by Szamel
[11] and then later generalized by Mayer et al., where a schematic version of
MCT including higher order correlations was proposed and analyzed [12]. The
result is that whenever the theory is truncated at any finite order in the
n-boby correlations, the phenomenology of MCT is exactly recovered with a
finite $T_{d}$, whereas $T_{d}=0$ when the theory is treated exactly to all
orders. Another approach was followed in [6], based on a field-theoretical
formulation of MCT consistent with the Fluctuation-Dissipation Theorem, which
suggests closure schemes different from standard MCT.
The aim of this work is to argue that MCT can be rephrased as a Landau theory
of the glass transition, based on general assumptions about the nature of the
dynamical arrest but without relying on any particular model. Therefore, some
predictions of MCT are indeed generic and should be useful in a certain regime
of time and temperature. Note that a Landau theory for the glass transition
has been developed also in [13], but it has a very different starting point
and it does not focus on MCT.
The Landau theory is a general phenomenological approach to equilibrium phase
transitions [14, 15]. It relies on a number of natural hypotheses, such as
symmetry, genericity and regularity. In the classic example of the Ising model
for ferromagnets, the expansion of the free energy as a function of the
magnetisation $m$ reads, in the homogeneous case and subject to external field
$h$: $F[m]=F_{0}-mh+\frac{b}{2}(T-T_{c})m^{2}+\frac{g}{4!}m^{4}+...$, from
which a certain number of well known mean-field properties can be derived.
Including the first gradient corrections in the inhomogeneous case also allows
one to show that close to $T_{c}$, the divergence of the uniform
susceptibility $\chi({\bf q}=0)$ is accompanied by the divergence of the
correlation length $\xi$, over which magnetisation fluctuations are
correlated. The Landau construction can falter in three distinct ways:
1. 1.
Higher order terms, neglected in the expansion of $F[m]$ as a series of $m$,
could qualitatively change the above predictions (structural instability).
This happens, for example, close to a multicritical point where $g(T_{c})=0$.
But if the transition remains second order, higher order terms are truly
negligible when $\epsilon=|T-T_{c}|\to 0$ and the predictions are universal.
2. 2.
The non-linear feedback of spatial fluctuations on the divergence of the
susceptibility can change all the critical exponents when the dimension of
space is smaller than $d_{u}=4$ in the case of the Ising model. For $d>d_{u}$,
on the other hand, one can prove that the low-$q$ behaviour of $\chi({\bf q})$
is (close to the critical point) identical to that predicted by Landau’s
theory.
3. 3.
Non-perturbative effects can wipe out the transition. This is the case for
example of the spinodal transition: the system is not able to reach the
critical point because of nucleation, which is an activated process.
Even the analogue of point (1) is difficult in the case of MCT; the basic
reason being that the order parameter is a not a scalar, but it is a time
dependent function $C({\bf k},t)$. The proof that MCT is structurally stable
with respect to the addition of higher order terms is already quite complex
and this will be the scope of the present paper. Once this is achieved, one
should still worry about points (2) and (3) above. As already mentioned, it
was recently realized that diverging fluctuations and an upper critical
dimension $d_{u}$ also exist for MCT [16, 17, 18] (see also [19, 20] for
earlier insights). In order to complete our proof that MCT is stable, one
should prove that spatial fluctuations can be safely neglected in $d>d_{u}$
and understand how close one can get to the critical point before non-
perturbative (activated) effects impair the transition. We will completely
disregard these issues in the present paper, and focus only on point (1).
The case of the glass transition is quite different from standard critical
phenomena. Several physical and formal problems prevent a direct analogy. The
glass transition seems to be a purely dynamical phenomenon: simple static,
thermodynamical properties do not present any peculiarities as the liquid
freezes into a glass 111It has however been argued that highly non-trivial
static correlations, called point to set correlations, increase approaching
the glass transition [21].. The above Landau construction simply does not make
sense in the absence of the clear analogy of the free energy. This means that
the order parameter in glasses cannot be a one point function (such as the
magnetisation) but, instead, it is likely to be a two point dynamic
correlation. The slowing down of the dynamics in glasses is found to be
accompanied by the appearance of a plateau value $f_{\bf k}$ in the relaxation
pattern of the dynamical structure factor $C({\bf k},\tau)$. Since the
appearance of a plateau coincides with increasing time scales, one expects
that within a very long time interval (to be specified), the correlation
function can be approximately written as:
$C({\bf k},\tau)\approx f_{\bf k}S_{\bf k}+\delta C({\bf k},\tau),\qquad\delta
C({\bf k},\tau)\ll 1.$ (1)
The idea underlying our construction of a Landau theory for glasses is to
consider $\delta C({\bf k},\tau)$ as the analogue of the order parameter $m$
and construct a general, structurally stable, dynamical equation for $\delta
C({\bf k},\tau)$. A way to construct such an equation is to start from the
exact dynamical evolution for $C$ and the response function $R$ that can be
derived in the framework of various dynamical field theories, for example
based on Dean’s equation [22] for Brownian dynamics, on Fluctuating
Hydrodynamics for Newtonian dynamics [4] (see also [5, 6, 7, 8]), or on
Langevin equations for $p$-spin models [9]. Although these theories give very
different sets of equations, they can all be reduced to the following single
equation in the ergodic region:
$\partial_{\tau}C({\bf k},\tau)+TC({\bf
k},\tau)+\int\limits_{0}^{\tau}du\Sigma({\bf k},\tau-u)\partial_{u}C({\bf
k},u)=0$ (2)
with initial condition $C({\bf k},0)=S_{\bf k}$. The self-energy $\Sigma({\bf
k},\tau)$ (or memory kernel in MCT terminology) is given by the sum of
$2$-particle irreducible ($2$-PI) diagrams built with $C$ and $R$ lines, see
e.g. [6]. We do not specify the details of the field theory underlying this
equation, nor the Feynman rules for the diagrams contributing to $\Sigma$: we
just need that such a theory exists. We also stay in the high-$T$ region, so
that the system is at equilibrium: both $C$ and $R$ are then time translation
invariant and the Fluctuation-Dissipation theorem holds at a diagrammatic
level: $T\,R({\bf k},\tau)=-\partial_{\tau}C({\bf k},\tau)$. In this case the
self-energy is a functional of the correlation function only. Eq. (2) has
exactly the structure of the standard MCT equation for liquids, although there
is no well defined prescription to build a consistent approximation for
$\Sigma$; in this sense MCT is rather arbitrary and difficult to improve upon
in a systematic way. The standard MCT results correponds to a self-consistent
$1$-loop approximation for $\Sigma$, $\Sigma({\bf k},t-s)={\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}}V({\bf k},{\bf p})C({\bf k}-{\bf p},t-s)C({\bf p},t-s)$, where
$V({\bf k},{\bf p})$ is an effective vertex. 222Note however that there are
complications related to fluctuation dissipation relations [5, 6, 7, 8]. When
generalized to higher order diagrams, an important difficulty emerges: the
non-locality in time of the corrections. Our goal is to prove that the main
results of the standard MCT (or $1$-loop) approximation still hold. The proof
is done as for usual theories. We start from some conjectures about the
critical properties, such as the nature of the order parameter and its
critical properties, that are motivated by experimental and numerical
findings. Then we show that they result from a Landau-like expansion, which
allows one to assess their universal character and to fix the value of the
critical exponents. We therefore assume that the order parameter is the
dynamical correlation function and that displays the following features:
* •
There is structural arrest: below some temperature $T_{d}$,
$\lim\limits_{\tau\to\infty}C({\bf k},\tau)=f_{\bf k}S_{\bf k}$ with $f_{\bf
k}>0$.
* •
When $\epsilon=(T-T_{d})/T_{d}\ll 1$, the correlation function exhibits a two-
step pattern with three well separated characteristic time scales, see Fig 1.
We assume that there exists a diverging time scale $\tau_{\beta}(\epsilon)$
where the difference $\delta C({\bf k},\tau)$ between $C$ and the plateau
value is small, of order $r(\epsilon)\ll 1$. More precisely, the correlation
function decay is decomposed into: (i) a short time regime,
$\tau\sim\tau_{0}$, where $C({\bf k},\tau)=C_{0}({\bf k},\tau)$, with
$C_{0}({\bf k},\tau\gg 1)\to f_{\bf k}S_{\bf k}$: (ii) a $\beta$-regime,
$\tau=s\tau_{\beta}(\epsilon)$ with $s=O(1)$: $\delta C({\bf
k},s\tau_{\beta})=r(\epsilon)S_{\bf k}(1-f_{\bf k})^{2}G({\bf k},s)$; (iii) an
$\alpha$-regime, $\tau=s^{\prime}\tau_{\alpha}(\epsilon)$ with
$s^{\prime}=O(1)$: $C({\bf k},s^{\prime}\tau_{\alpha})=C_{\alpha}({\bf
k},s^{\prime})$ describes the final fall off of the relaxation.
Figure 1: A set of relaxation curves $C(\tau)$ close to the MCT transition for
the so called schematic model of MCT [3] and several values of $\lambda$,
which quantifies the deviation from the transition. As $\lambda\to 1$ a
shoulder develops in the relaxation pattern corresponding to the emerging
$\beta$-relaxation regime.
We assume further (and justify later) that the function $G({\bf k},s)$ can
itself be expanded in powers of $r(\epsilon)$: $G({\bf
k},s)=\sum_{n=1}^{\infty}r^{n-1}G_{n}({\bf k},s)$. 333One could have also some
regular (in $\epsilon$) contributions. However, as we shall show the only two
possible values of $r(\epsilon)$ are $\sqrt{\epsilon}$ or $\epsilon$. As a a
consequence, regular contributions will be automatically contained in the
expansion. All functions $G_{n}$ are a priori singular at $s=0$ and
$s=\infty$, reflecting the fact that the behaviour of $C$ must match the short
time regime and $\alpha$ regime, where the deviation from the plateau ceases
to be small. A crucial remark for the following is that any function $G_{n}$
will appear with a prefactor $r^{n}(\epsilon)$. These hypotheses turn out to
be sufficient to generalize the MCT results in the $\beta$-regime. First, it
is clear that the above expansion of $G({\bf k},s)$ generates a similar
expansion of the self-energy $\Sigma[C]$ in the $\beta$-regime: $\Sigma({\bf
k},s)=\sum_{n=1}^{\infty}r^{n-1}\Sigma_{n}({\bf k},s)$, where the $\Sigma_{n}$
do not depend on $\epsilon$, but are some functionals of $C({\bf p},\tau)$.
The most generic functional form for $\Sigma$ a priori includes contributions
from all three regimes:
$\Sigma({\bf k},s)=\Sigma[\\{C_{0}({\bf
p},s^{\prime}\tau_{\beta}),r(\epsilon)G({\bf p},s^{\prime}),C_{\alpha}({\bf
p},s^{\prime}\tau_{\beta}/\tau_{\alpha})\\}],$ (3)
but since the $\Sigma_{n}$ should not depend on $\epsilon$, general arguments
can be used to restrict the actual functional form of $\Sigma_{n}$. Note that
we have used the notation $s^{\prime}$ to stress that this equation is a
functional relation which is non-local in time. Also, even if we had assumed
that $\delta C({\bf k},s\tau_{\beta})$ only contains a single term of the
order of $r(\epsilon)$ then we would have generated corrections of all orders
in $r(\epsilon)$ anyway. The reason is that the self-energy will contain all
orders in $r(\epsilon)$ as it can be found by expanding the above equation to
all order in $r(\epsilon)G({\bf p},s)$; this will feed back, via the
Schwinger-Dyson equations, on $G({\bf p},s)$ itself.
We now illustrate how this works for the lowest order terms $\Sigma_{0}$,
$\Sigma_{1}$ and $\Sigma_{2}$. As we shall see higher orders are in fact
irrelevant for our purpose. Clearly, the zeroth order term $\Sigma_{0}({\bf
k},s)$ can only be a function of the wavevector ${\bf k}$ since in the limit
$\epsilon\to 0$ time scales separate: $\tau_{\beta}\to\infty$ and
$\tau_{\beta}/\tau_{\alpha}\to 0$ and therefore the previous equation implies
that in $\Sigma_{0}({\bf k},s)$ all dependence on $s$ drops out. The first
order contribution $\Sigma_{1}({\bf k},s)$ must read:
$\Sigma_{1}({\bf k},s)=\int\limits_{0}^{\infty}du\int_{{\bf p}}K_{1}({\bf
k},{\bf p};s,u)G_{1}({\bf p},u)\quad$ (4)
where, henceforth, we shall use the notation $\int_{{\bf
p}}={\int\frac{d^{3}{\bf p}}{(2\pi)^{3}}}$. Any other combination containing
some $G_{n}$ gives an extra factor $r^{n}(\epsilon)$ and thus it corresponds
to a higher order contribution. In the original time variables, the kernel
$K_{1}$ must have some regular shape with a span fixed by the microscopic time
scale. Therefore, in the rescaled variables $u,s$, $K_{1}$ must be local in
$u-s$ and, to lowest order, is a $\delta$ function; higher derivatives of the
$\delta$ function correspond to corrections smaller by at least a factor
$\tau_{0}/\tau_{\beta}$ which, as we shall see, turn out to be negligible even
at order $r^{2}$. Therefore, $\Sigma_{1}({\bf k},s)={\int\frac{d^{3}{\bf
p}}{(2\pi)^{3}}}K_{1}({\bf k},{\bf p})G_{1}({\bf p},s)$.
The second order term has a richer structure. First, there is a term similar
to the first order one with $G_{2}$ instead of $G_{1}$: ${\int\frac{d^{3}{\bf
p}}{(2\pi)^{3}}}K_{2}({\bf k},{\bf p})G_{2}({\bf p},s)$. But since the kernel
$K_{2}$ is obtained, as $K_{1}$, from the first derivative of the self energy
with respect to $r(\epsilon)G({\bf p},s)$, one finds $K_{2}=K_{1}$. Second,
there are terms quadratic in $G_{1}$:
$\int_{{\bf k}_{1},{\bf
k}_{2}}\int\limits_{0}^{\infty}du\int\limits_{0}^{\infty}dvK_{11}({\bf q},{\bf
k}_{1},{\bf k}_{2};s,u,v)G_{1}({\bf k}_{1},u)G_{1}({\bf k}_{2},v)$
For the same reasons outlined above, time dependence of $K_{11}({\bf q},{\bf
k}_{1},{\bf k}_{2};s,u,v)$ is composed of $\delta$-functions and their
derivatives. Some thinking about the underlying diagrammatic structure of the
theory allows one to be convinced that the general structure of $K_{11}$ is,
to leading order:
$\displaystyle K_{11}(s,u,v)=K_{11,\ell}\delta(s-u)\delta(s-v)+\hskip
82.51282pt$ $\displaystyle
K_{11,n\ell}\delta(u+v-s)(\partial_{u}+\partial_{v})+\tilde{K}_{11,n\ell}(\partial_{u}+\partial_{v})\delta(u+v-s)$
where to simplify the notation we have dropped all wave-vector dependence in
the above equation. The fact that only the combination $u+v-s$ enters comes
from causality and the separation of time scales. The full justification of
the above form and other technical details444In particular one could think
that the separation of timescales not only leads to delta function terms but
also to constants. The latter are absent. This can be shown using the general
field theoretical expression of the self energy, see [23]. are presented in
[23]. The first local term (in $s$), $K_{11,\ell}$, is like the usual MCT
contribution, but the other two terms do not appear within standard MCT. The
third term actually reduces to the second one plus local terms via integration
by parts.
This expansion is the main result in the construction of the Landau theory.
The zeroth order equation in $r(\epsilon)$ fixes the non-ergodic parameter
such that $\frac{T_{d}f_{\bf k}}{1-f_{\bf k}}=\Sigma_{0}({\bf k})$, as in
standard MCT. Substituting the expansion of $\Sigma$ up to second order in $r$
into (2) and using the expansion of $C$ in the $\beta$ timescale, one finally
obtains, for $T=T_{d}(1+\epsilon)$ and in Laplace space within the
$\beta$-regime (we have dropped zeroth order, as discussed above):
$\displaystyle r\left[T_{d}(1+\epsilon)\hat{G}_{1}({\bf k},z)-\int_{\bf
p}K_{1}({\bf k},{\bf p})\hat{G}_{1}({\bf p},z)\right]+$ $\displaystyle
r^{2}\left[T_{d}(1+\epsilon)\hat{G}_{2}({\bf k},z)-\int_{\bf p}K_{1}({\bf
k},{\bf p})\hat{G}_{2}({\bf p},z)\right]+$ $\displaystyle\frac{T_{d}f_{\bf
k}\epsilon}{z(1-f_{\bf k})}+r^{2}T_{d}(1+\epsilon)(1-f_{\bf
k})z\hat{G}_{1}^{2}({\bf k},z)=$ $\displaystyle r^{2}\int_{\bf p}K_{2}({\bf
k},{\bf p})\hat{G}_{2}({\bf p},z)+$ $\displaystyle+$ $\displaystyle
r^{2}\int_{{\bf k}_{1}}\int_{{\bf k}_{2}}K_{11,\ell}({\bf k},{\bf k}_{1},{\bf
k}_{2}){\cal L}[G_{1}({\bf k}_{1},\tau)G_{1}({\bf k}_{2},\tau)](z)$
$\displaystyle+$ $\displaystyle r^{2}\int_{{\bf k}_{1}}\int_{{\bf
k}_{2}}K_{11,n\ell}({\bf k},{\bf k}_{1},{\bf k}_{2})z\hat{G}_{1}({\bf
k}_{1},z)\hat{G}_{1}({\bf k}_{2},z),$
Identifying the coefficients order by order produces a series of equations.
The first order fixes the yet unknown function $r(\epsilon)$: the expansion
(Mode-coupling as a Landau theory of the glass transition) only contains terms
with integer powers of $\epsilon$. They should be matched with powers of
$r(\epsilon)$. Inspection of (Mode-coupling as a Landau theory of the glass
transition) shows that there are two possibilities555Actually, there are other
possibilities that correspond to higher order MCT singularities, which have
been called $A_{n}$ [27]. In a usual Landau theory these correspond to
tricritical, or even higher order, critical points. We will neglect them here
since they require some fine tuning of the coupling constants.: either
$r=\epsilon$, or $r=\sqrt{\epsilon}$. The first choice yields a time
independent solution for $G_{1}$ which is in contradiction with our hypothesis
of a two-step relaxation with diverging time scales. Hence
$r(\epsilon)=\sqrt{\epsilon}$, precisely as for usual MCT, or at $1$-loop
order. This follows from the presence of a regular in $T$ term in (2).
The equation to order $r=\sqrt{\epsilon}$ now reads:
$T_{d}\hat{G}_{1}({\bf k},z)={\int\frac{d^{3}{\bf p}}{(2\pi)^{3}}}K_{1}({\bf
k},{\bf p})\hat{G}_{1}({\bf p},z)$ (6)
This is the standard eigenvalue problem found within MCT that fixes the value
of the critical temperature $T_{d}$. It constrains $G_{1}$ to be a product of
wave-vector dependent and time dependent amplitudes, thus reproducing the
well-know MCT “factorization property”: $\hat{G}_{1}({\bf
k},z)=\hat{g}(z)H_{1}({\bf k})$, where $H_{1}$ is the right eigenvector of
$K_{1}$ with largest eigenvalue $\Lambda=T_{d}$. At this order however the
scaling function $\hat{g}(z)$ remains unfixed. The second order equation is
trickier:
$\displaystyle T_{d}\hat{G}_{2}({\bf k},z)-\int_{\bf k}K_{1}({\bf k},{\bf
p})\hat{G}_{2}({\bf p},z)=$ $\displaystyle-\frac{T_{d}f_{\bf k}}{z(1-f_{\bf
k})}-T_{d}(1-f_{\bf k})z\hat{g}^{2}(z)H_{1}^{2}({\bf k})+$ (7)
$\displaystyle+\int_{{\bf k}_{1}}\int_{{\bf k}_{2}}K_{11,\ell}({\bf k},{\bf
k}_{1},{\bf k}_{2}){\cal L}[g^{2}](z)H_{1}({\bf k}_{1})H_{1}({\bf k}_{2})+$
$\displaystyle+z\hat{g}^{2}(z)\int_{{\bf k}_{1}}\int_{{\bf
k}_{2}}\hat{K}_{11,n\ell}({\bf k},{\bf k}_{1},{\bf k}_{2})H_{1}({\bf
k}_{1})H_{1}({\bf k}_{2})$
Following [24], we now multiply (Mode-coupling as a Landau theory of the glass
transition) by $H_{1}({\bf k})$ and integrate over ${\bf k}$. The $G_{2}$ part
of the equation vanishes and the remainder yields an equation on $\hat{g}(z)$.
After some algebra and a proper rescaling of $z$ and $\hat{g}$ one finds:
$\frac{1}{z}+\frac{z}{\lambda}\hat{g}^{2}(z)={\cal L}[g^{2}](z)$ (8)
where $\lambda$ is a constant that includes a non-local contribution as
compared to MCT. But the structure of the equation on the scaling function $g$
is exactly the same as in standard MCT. The properties of solution are well
known: $g$ has a singular power law asymptotics at $z\to\infty$:
$\hat{g}(z)\sim z^{a-1}$ and $z\to 0$: $\hat{g}(z)\sim z^{-1-b}$. The small
time exponent $a$ and long time exponent $b$ characterize the decay of $\delta
C({\bf k},\tau)$ to and away from the plateau $f_{\bf k}$. The exponents $a$
and $b$ are related by the famous equation:
$\frac{\Gamma^{2}(1-a)}{\Gamma(1-2a)}=\frac{\Gamma^{2}(1+b)}{\Gamma(1+2b)}=\lambda,$
(9)
which is a genuinely non-trivial and clear-cut prediction of MCT that
constrains the range of values of $a$ and $b$ to: $0\leq a<1/2$ and $0\leq
b\leq 1$, in good agreement with experimental and numerical results. We have
thus found that this relation has a much broader degree of validity and
survives the introduction of an arbitrary number of loop corrections. The
values of $a$ and $b$ are however different from the standard MCT (or
$1$-loop) result, but as alluded to above, the parameter $\lambda$ is usually
taken as an adjustable parameter anyway.
The fact that the form of the scaling function $g$ is the same as at $1$-loop
(MCT) has two consequences. First, it fixes the functional dependence of the
time scales exactly as for MCT:
$\tau_{\beta}=\epsilon^{-1/2a};\qquad\tau_{\alpha}=\epsilon^{-\gamma},$ (10)
$\gamma=1/2a+1/2b$. This is clear from the matching of $C({\bf
k},s\tau_{\beta})$ at both ends of the $\beta$ regime. Second, it can be used
to show that some superficial divergencies encountered in the calculation are
in fact innocuous (see [23] for more details). Note that the only extra
contribution that appear in the generic case to second order in $g(s)$, namely
the non-local term proportional to $\int dug^{\prime}(u)g(s-u)$, does not
modify the basic MCT equation, Eq. (8).
The conclusion, which is the main result of this work, is that although
$T_{d}$ and $\lambda$ are modified by taking into account corrections to MCT,
the relation between the exponent $a$ and $b$, the square root singularity as
well as the scaling function $g$ are truly universal properties. This
universality with respect to higher order local (in time) corrections was of
course already shown by Götze long ago; here we have proven that this result
is robust with respect to general non-local corrections as well, and
suggesting that MCT has the status of a Landau theory of the glass transition
666As a consequence, it has a very different status compared to Mode-Coupling
theories developed to compute critical exponents beyond mean field theory for
standard critical phenomena..
The above schematic arguments can be made precise within the context of
specialized model. We have in particular studied in full details the finite
$N$ corrections to mean-field 3-spin glass model, where the structure of the
perturbation theory can be used to check that the above conclusions hold in
that case, see [23] for details.
It was recently understood how MCT equations should be generalized in the
presence of spatial inhomogeneities, where the correlation function $C$ can be
space dependent: $C({\bf k},\vec{r};\tau)$, where $\vec{r}$ is the average of
the two points $\vec{r}_{1},\vec{r}_{2}$ between which the correlation is
computed, and ${\bf k}$ is the Fourier vector corresponding to
$\vec{r}_{1}-\vec{r}_{2}$. When wavelength of inhomogeneities is large, one
can establish a gradient expansion of the MCT equations. In the schematic
limit where all dependence on ${\bf k}$ is discarded, the self-energy reads,
to the lowest order:
$\Sigma[C](s)=C(\vec{r},s)^{2}+w_{1}C(\vec{r},s)\nabla^{2}C(\vec{r},s)+w_{2}\vec{\nabla}C(\vec{r},s)\cdot\vec{\nabla}C(\vec{r},s),$
where $w_{1}$ and $w_{2}$ are some coefficients [17]. As mentioned in the
introduction, these gradient terms are very important because they show how
the MCT transition is in fact associated with a diverging correlation length,
which corresponds to the scale over which a localized perturbation affects the
surrounding dynamics [17]. The long-ranged critical fluctuations renormalize
the value of the MCT exponents in $d<d_{u}=8$ [18]. The above analysis, which
was done in the homogeneous limit $\nabla\to 0$, should be repeated in the
inhomogeneous case to complete our proof. We expect that the same conclusion
will hold, namely that the results about dynamical correlation obtained within
inhomogeneous MCT [17] are stable against the addition of higher order
corrections.
In conclusion, we have shown that MCT, which describes a specific slowing down
mechanism through the progressive disappearance of unstable directions, has
the status of a Landau theory and is therefore expected to make generic
predictions, albeit polluted by activated events and critical fluctuations in
finite dimensions. The interplay between critical fluctuations and activated
events when $d<d_{u}$, and the crossover to low temperature dynamics is still
largely an exciting open problem [28]. Note also that even for the exact MCT
equations, the critical region where the asymptotic scaling predictions are
valid is unusually narrow [25, 26]. It would be interesting to generalize our
Landau approach to the aging regime and show what are the truly universal
properties of the mean-field and MCT-like description of the aging dynamics
[9].
In constructing the Landau theory, we have assumed that the freezing
transition is discontinuous, with a finite value $f_{\bf k}$ of the plateau at
the transition. A viable alternative is of course that of a continuous
transition of the spin-glass type, which leads to a completely different
phenomenology. This raises the question of the possible realization of this
second scenario in the context of supercooled liquids. All short-range
interacting glasses seem to be characterized by rather small Lindemann
parameters at the transition, meaning that it is hard to maintain any kind of
amorphous long range order when individual molecules move substantially, and
that the glass transition is therefore discontinuous [29]. This argument
suggests that continuous glasses can only exist for long-ranged interacting
particles or quantum systems. In the quantum case, it is imaginable that
amorphous density waves can indeed form with a vanishing modulation amplitude
(see [30]). It would be very interesting to find experimental realizations of
such a scenario.
###### Acknowledgements.
We thank A. Lefèvre for useful discussions. GB and JPB are supported by ANR
Grant DYNHET; AA was supported in part by EPSRC Grant No. EP/D050952/1.
## References
* [1] U. Bengtzelius, W. Götze, A. Sjöilander, J. Phys. C 17, 5915 (1984).
* [2] E. Leuthesser, Phys. Rev. A 29, 2765 (1984).
* [3] S. P. Das, Rev. Mod. Phys. 76, 785 (2004).
* [4] S. P. Das, G. F. Mazenko, S. Ramaswamy, and J. J. Toner, Phys. Rev. Lett. 54, 118 (1985).
* [5] K. Miyazaki, D. R. Reichman, J. Phys. A: Math. Gen. 38, 20 (2005).
* [6] A. Andreanov, G. Biroli, A. Lefèvre, J. Stat. Mech., P07008 (2006).
* [7] B. Kim and K. Kawasaki, J. Stat. Mech. (2008) P02004.
* [8] T. H. Nishino and H. Hayakawa, Phys. Rev. E 78, 061502 (2008).
* [9] JP. Bouchaud, L. Cugliandolo, J. Kurchan, M. Mézard, in Spin glasses and Random Fields, A.P. Young Editor (World Scientific) 1998.
* [10] V. Lubchenko, P. G. Wolynes, Ann. Rev. Phys. Chem. 58 235 (2007).
* [11] G. Szamel, Phys. Rev. Lett. 90, 228301 (2003).
* [12] P. Mayer, K. Miyazaki, D. R. Reichman, Phys. Rev. Lett. 97, 095702 (2006).
* [13] S. N. Majumdar, D. Das, J. Kondev, B. Chakraborty, Phys. Rev. E 70, 060501(R) (2004).
* [14] L.D. Landau, Zh. Eksper. Teor. Fis. 7, 627 (1937).
* [15] J.-C. Tolédano, P. Tolédano, The Landau theory of phase transitions, (World Scientific Publishing Co. Pte Ltd) 1987.
* [16] G. Biroli, J.-P. Bouchaud, Europhys. Lett. 67, 21 (2004).
* [17] G. Biroli, J.-P. Bouchaud, K. Miyazaki, D.R. Reichman, Phys. Rev. Lett. 97, 195701 (2006) .
* [18] G. Biroli, J.-P. Bouchaud, J. Phys.: Condens. Matter 19 205101 (2007).
* [19] T. R. Kirkpatrick and D. Thirumalai, Phys. Rev. A 37, 4439 (1988).
* [20] S. Franz and G. Parisi, J. Phys.: Condens. Matter 12, 6335 (2000).
* [21] J.-P. Bouchaud, G. Biroli, J. Chem. Phys. 121, 7347 (2004); G. Biroli, J.-P. Bouchaud, A. Cavagna, T. S. Grigera, P. Verrocchio, Nature Phys. 4 771 (2008); M. Mézard, A. Montanari, J. Stat. Phys. 124 (2006) 1317.
* [22] D. S. Dean, J. Phys. A: Math. Gen. 29, L613 (1996).
* [23] A. Andreanov, Ph.D. thesis; http://www.imprimerie.polytechnique.fr/Theses/Files/Andreanov.pdf.
* [24] W. Götze, Z. Phys. B - Condensed Matter 59, 195 (1985).
* [25] V. Krakoviack and C. Alba-Simionesco, J. Chem. Phys. 117, 2161 (2002).
* [26] T. Sarlat, A. Billoire, G. Biroli, J.-P. Bouchaud, in preparation.
* [27] W. Götze, Sjörgen, Rep. Prog. Phys. 55, 241 (1992).
* [28] S. M. Bhattacharya, B. Bagchi and P. G. Wolynes, Phys. Rev. E 72, 031509 (2005).
* [29] For a related argument, see M. P. Eastwood and P. G. Wolynes, Europhys. Lett. 60, 587-593 (2002).
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|
arxiv-papers
| 2009-03-26T15:54:39 |
2024-09-04T02:49:01.444104
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Andreanov, G. Biroli and J.-P. Bouchaud",
"submitter": "Alexei Andreanov",
"url": "https://arxiv.org/abs/0903.4619"
}
|
0903.4657
|
# Pure quantum interpretations are not viable
I. Schmelzer mailto:ilja.schmelzer@gmail.comilja.schmelzer@gmail.com
http://ilja-schmelzer.deilja-schmelzer.de
###### Abstract.
Pure interpretations of quantum theory, which throw away the classical part of
the Copenhagen interpretation without adding new structure to its quantum
part, are not viable. This is a consequence of a non-uniqueness result for the
canonical operators.
Berlin, Germany
## 1\. Introduction: The non-uniqueness of the canonical structure
In [1] we have proven two non-uniqueness theorems: For some fixed Hamilton
operator $\hat{h}$, we have constructed for some continuous parameter $s$
different pairs $\hat{q}(s)$, $\hat{p}(s)$ of canonical operators so that
(1)
$\mbox{$\hat{h}$}=\frac{1}{2m}\mbox{$\hat{p}(s)$}^{2}+V(\mbox{$\hat{q}(s)$},s)$
with physically different, but equally nice (smooth, bounded, descreasing in
infinity) potentials $V(q,s)$. In addition, we have constructed different
tensor product structures (or “decompositions into systems”) so that $\hat{h}$
has an equally nice, but physically different representation of type
(2)
$\mbox{$\hat{h}$}=\sum\frac{1}{2m_{i}}\mbox{$\hat{p}$}_{i}(s)^{2}+V(\mbox{$\hat{q}(s)$},s)$
in all or them.
From point of view of canonical quantization, there seems nothing problematic
with this result. It nicely corresponds to the standard way to define
canonical quantum theories: One has to define an irreducible representation of
the canonical operators $\hat{p}$, $\hat{q}$ (with
$[\mbox{$\hat{p}$},\mbox{$\hat{q}$}]=-i\hbar$) and then to define the Hamilton
operator $\hat{h}$ as a function of these operators
(3) $\mbox{$\hat{h}$}=\frac{1}{2m}\mbox{$\hat{p}$}^{2}+V(\mbox{$\hat{q}$}).$
Once the theory is defined in such a way, no non-uniqueness problem appears –
the canonical operators $\hat{p}$, $\hat{q}$ are those used in the definition
of the canonical theory.
The situation is different if we consider interpretations of quantum theory.
The point is that the interpretation has to define the physical meaning not
only for the Hamilton operator $\hat{h}$, but for all physically relevant
parts of the theory. Once different choices of $s$, thus, identifications of
$\hat{p}$, $\hat{q}$ with different $\hat{p}(s)$, $\hat{q}(s)$ would lead to
physically different theories (with different potentials $V(q)$), the
operators $\hat{p}$, $\hat{q}$ are physically important, thus, the
interpretation has to describe their physical meaning too. In [1] we have
already considered the consequences of these non-uniqueness results for
applications of decoherence in fundamental physics: The widely held belief
that decoherence allows to define the classical limit without additional
structure has to be given up. We have also evaluated and rejected the idea to
postulate some fundamental decomposition into systems to derive a preferred
basis. An emergent configuration space $Q$ would lead to a lot of losses
(uncertainty, dependence on dynamics) which are in no way compensated by gains
in explanatory power.
The aim of this paper is to continue the consideration of the consequences of
these non-uniqueness theorems, in particular for various interpretations of
quantum theory.
First, we discuss and reject the proposal to embrace the different
$\hat{q}(s)$, $\hat{p}(s)$ as many different but equally real worlds – an idea
close to but not identical with “many worlds”. Once this proposal is rejected,
an interpretation has to identify the “correct” canonical operators $\hat{q}$,
$\hat{p}$ among the $\hat{q}(s)$, $\hat{p}(s)$. We argue that this requires
more than assigning pure labels. The canonical operators $\hat{p}$, $\hat{q}$
are in some sense different from the alternatives $\hat{p}(s)$, $\hat{q}(s)$,
a difference which is not part of the mathematical formalism of pure quantum
theory and has to be defined by the interpretation. This requires some
additional physical structure which the interpretation has to define.
Some interpretations have adequate structures – pilot wave theories [4, 5],
Nelsonian stochastics [16] and physical collapse theories [11], [10] have a
preferred configuration space $Q$, and the Copenhagen interpretation
associates the $\hat{q}$, $\hat{p}$ with classical measurement procedures. But
there is a whole class of interpretations which does not have such a structure
– a class we name “pure interpretations”. Such interpretations are the result
of a very natural approach: One the one hand, one wants to get rid of the
uncertain, problematic “classical part” of the Copenhagen interpretation. One
the other hand, one does not want to introduce additional structure into the
theory. A reasonable minimalistic approach, and it would be nice if it would
work. But, as a consequence, the Copenhagen solution of our non-uniqueness
problem no longer works, and without any new structure no new solution is
available. Thus, this minimal, pure program fails.
We discuss shortly some important examples of such interpretations: Mermin’s
Ithaca interpretation [15], consistent histories [12], and the Everett
interpretation [9]. The aim is not to give a complete list of interpretations
endangered by our non-uniqueness result. While I think that the problem is
sufficiently general, so that every interpretation deserves a consideration if
and how it solves this non-uniqueness problem, this paper can be only a
starting point. Our examples merely illustrate that the problem appears in
quite different approaches to the interpretation of quantum theory.
If interpretations which have a non-uniqueness problem will be given up or
saved by introducing some additional structure is a decision we of course have
to leave to the proponents of these interpretations. Only they can be expected
to find the optimal solution for their preferred interpretations. The point of
this paper is to clarify some general, common points – that, if one removes
the classical part of the Copenhagen interpretation, one has to introduce
something else, a replacement. This replacement has to be a non-trivial
physical structure, which contains sufficient additional information to
identify the canonical operators. The identification of the canonical
operators with momentum and position measurements, which is postulated in the
Copenhagen interpretation, has to be derived now based on the new physical
structure.
This fact alone already removes one of the main advantages of the Everett
program and similar programs. What can be obtained in this way is no longer a
pure, minimal interpretation, but only an interpretation with some additional
structure. The question is no longer if an interpretation has some additional
structure (with automatic rejection of interpretations which have such “hidden
variables”) but what is the additional structure connected with a given
interpretation, what are their particular advantages and disadvantages (now
with automatic rejection of interpretations without such a structure).
The loss of purity is not the only possible consequence of the additional
structure. Other questions may be influenced too. We consider two examples:
First the popular many world argument “[P]ilot-wave theories are parallel-
universe theories in a state of chronic denial” [8] becomes invalid if (as one
would expect) the additional structure introduced into many worlds is not part
of pilot wave theory. On the other hand, if we introduce one “preferred”
consistent family of histories as the additional structure into the consistent
histories interpretation, this interpretation becomes more compatible with
classical logic and realism, which would remove some arguments against it. But
there are lots of other problems to be considered in future research. In
particular, the symmetry properties of the theory may be heavily influenced by
an additional structure.
## 2\. What is wrong with “many laws”
Anticipating a possible non-uniqueness of the construction of a preferred
basis, Saunders [17] has proposed a solution which avoids the introduction of
new physical structure: One could accept the non-uniqueness and consider all
the different classical limits defined by different $\hat{p}(s)$, $\hat{q}(s)$
as equally real different worlds. Brown and Wallace [7] describe this idea in
the following words:
> Suppose that there were several such decompositions, each supporting
> information-processing systems. Then the fact that we observe one rather
> than another is a fact of purely local significance: we happen to be
> information-processing systems in one set of decoherent histories rather
> than another. [7]
This looks like a many-worlds-like solution of the problem. But it should be
noted that there is an essential difference between usual many worlds and this
proposal. The important point is that, as we have shown in [1], the different
$\hat{p}(s)$, $\hat{q}(s)$ define different physics. They have even different
classical limits $H(p,q,s)=\frac{1}{2m}p^{2}+V(q,s)$. This situation is very
different from the standard “many worlds”, where all worlds follow the same
physical laws, only with different initial values. We have not only “many
worlds”, but these many worlds follow different physical laws, a situation
which is better named “many laws”.
This difference allows some counterargumentation.
### 2.1. Loss of predictive power
The first one is based on Popper’s criterion of empirical content. Its
absolute version is that scientific, empirical theories should be falsifiable,
they should allow the derivation of testable predictions. But we need the
relative version, which allows to compare the empirical content of different
theories. If theory $\mathcal{T}_{1}$ makes at least one prediction which
cannot be made by theory $\mathcal{T}_{2}$, but all predictions of theory
$\mathcal{T}_{2}$ are also predictions of theory $\mathcal{T}_{1}$, then
theory $\mathcal{T}_{1}$ has more predictive power, or higher empirical
content in comparison with $\mathcal{T}_{1}2$. Popper’s criterion tells us
that the theories with higher empirical content should be preferred.
Let’s apply this to our situation, and let’s compare the empirical content of
a single law theory (which has somehow fixed one value of $s$ and chosen the
corresponding set of canonical operators $\hat{p}$, $\hat{q}$) with a many
laws variant which embraces all the $\hat{p}(s)$, $\hat{q}(s)$ as different
really existing worlds. We have shown in [1] that the physical predictions for
at least some experiments depend on $s$. The single law theory where $s$ is
fixed allows to make the usual predictions of canonical quantum theory. The
situation for the many laws version is different. This version cannot exclude
that the value of $s$ in our universe is the same as that chosen by the single
law theory, simply because that single law is one of the possible many laws.
Therefore it is in principle impossible to falsify the many laws version
without falsifying the single law version.
In the other direction this is possible: The single law theory may be
falsified simply because the value of $s$ is wrong. Because different $s$ lead
to different physical predictions, it may happen in principle that we observe
an effect as predicted for a value $s^{\prime}$ which is different from the
$s$ used in the single world theory. This would falsify the single world
theory. But if the correct value $s^{\prime}$ of our universe is among the
allowed values in the many laws theory, many laws is not falsified by this
observation. So many laws theory remains unfalsified – the other, correct
value $s^{\prime}$ is allowed, it is simply the actual value of some other
world.
Thus, the consideration of the predictive power gives a clear answer. A single
world theory has more predictive power, higher empirical content, is able to
make more specific predictions, and, therefore, has to be preferred following
Popper’s criterion. This is a natural consequence of the fact that the
canonical operators $\hat{p}$, $\hat{q}$ of the theory are fixed and well-
defined.
### 2.2. A symmetry argument
Let’s add a completely different argument, which is based on symmetry. Once in
the many laws version all worlds are equally real, have the same ontological
status, the physical properties of our particular world cannot be further
restricted by Ockham’s razor or further symmetry principles. These principles
allow to restrict only theories about what really exists. All the really
existing worlds are already on equal footing.
In this case, one would expect that the law of our actual universe should be a
typical, generic element of the set of laws. Indeed, if our observed law would
be a very special element, say, for the sake of the argument, the one having a
special value $s=0$, then there would be no point of considering all the other
laws. One could simply use Ockham’s razor to cut all the laws with $s\neq 0$
out of the theory.
Is the law of our actual universe a typical, general element of the set of all
possible laws? This is something we cannot decide, given that we have
considered in [1] essentially only the one-dimensional case (a two-dimensional
construction was based on a product of the one-dimensional case), thus, don’t
know nor the real law of our universe, nor the possible modifications of it if
we allow for other choices of the canonical operators. But let’s nonetheless
use the one-dimensional case considered in [1], with Hamiltonians of type (1),
as a toy model of the possible laws for the universe, so that another choice
of the canonical operators corresponds to another choice of $s$ in (1). What
would be, in this toy-many-laws theory, the analogon of the law of our
universe? Given the important role of the potential $V(q)=1/\lvert q\rvert$,
this potential seems to be the only reasonable candidate.
Can the potential $V(q)=1/\lvert q\rvert$ be considered as a typical, generic
element of some class of potentials $V(q,s)$ connected with each other by
different choices of the canonical operators? The answer is a clear no. The
potential $V(q)=1/\lvert q\rvert$ can be characterized among them by an
extreme symmetry property.
To see this symmetry property we have to ask what changes if we change $s$ in
terms of the eigenstates and eigenvalues of the Hamilton operator. The answer
is that the eigenvalues remain unchanged (the operator $\hat{h}$ remains the
same by construction, as an abstract operator in the Hilbert space, and the
eigenvalues $E_{k}$ are completely defined by the operator alone). The
eigenstates $|\psi_{k}\rangle$ themself, as abstract elements of the Hilbert
space, remain unchanged too. But their positions
$q_{k}=\langle\psi_{k}|\mbox{$\hat{q}(s)$}|\psi_{k}\rangle$ change, because
they depend on the operator $\hat{q}(s)$ which changes. This change is not a
simple common shift – this would be the result of a simple shift in the
potential $V(q)\to V(q-q_{0})$. Different eigenstates obtain different shifts,
and explicit formulas for these shifts can be found from the theory of the KdV
equation [2].
Now, the positions of the energy eigenstates for $V(q)=1/\lvert q\rvert$ can
we described in a very simple way: they are simply all zero. Thus, in our toy
model our universe is distinguished by the very special symmetry property that
$\forall k\;q_{k}=0$. This is certainly not a generic element. It would need
high conspiracy. Thus, we have (at least for our toy model) a fine tuning
problem: The “many laws” approach would lead to the expectation that the
$q_{k}$ are a quite arbitrary sequence of real numbers. Instead, our choice of
the potential leads to the very special case $q_{k}=0$ for all $k$.
For a theory with a single law this would be the most natural choice, clearly
preferred by Ockham’s razor. Instead, for a theory of may laws this is an
extremal property of our own universe which requires explanation.
#### 2.2.1. But what about our real world?
A natural objection is that our one-dimensional toy consideration is much too
trivial and our choice of the potential $V(q)=1/\lvert q\rvert$ much too
artificial to be relevant for our universe. So let’s try to find out which
part of the toy argument can be expected to generalize to a more general,
high-dimensional, realistic situation. First, of course, it remains unchanged
that the eigenstates $E_{k}$ of $\hat{h}$ remain unchanged. But once the
eigenstates of $\hat{h}$ are unable to fix the position operator $\hat{q}$ and
the potential $V(\mbox{$\hat{q}$})$ already in the simplest one-dimensional
case, one would not expect that this changes in higher dimensions (even if the
nice exact mathematics of the Korteweg-de Vries equation works only in the
one-dimensional case). A modification of $\hat{q}$ will also change the
positions of the eigenstates
$q_{k}=\langle\psi_{k}|\mbox{$\hat{q}$}|\psi_{k}\rangle$ of $\hat{h}$, since
they obviously depend on $\hat{q}$. The question is how reasonable it is to
assume some symmetries like $\forall k\;q_{k}=0$. But this is quite common
already in the one-dimensional case. All we need in this case is the discrete
symmetry $V(q)=V(-q)$. In this case, for an assumed asymmetric eigenstate
$\psi_{k}(q)$ with $q_{k}\neq 0$ $\psi_{k}(-q)$ would be an eigenstate of the
same energy $E_{k}$, and then their symmetric and antisymmetric combinations
$\psi_{\pm}(q)=\psi_{k}(q)\pm\psi_{k}(-q)$ would define other eigenstates
already with $q_{k}=0$, so that every eigenstate can be represented as a
linear combination of the same eigenvalue with $q_{k}=0$. (In the one-
dimensional case this would be even easier, because in this case there are no
degenerate eigenstates.) One can imagine that almost every symmetry which acts
nontrivially on $\hat{q}$ (generalizing the
$\mbox{$\hat{q}$}\to-\mbox{$\hat{q}$}$ of our example) may have similar
nontrivial consequences for the positions of the eigenstates. Given the large
role of symmetry in modern physics it seems quite reasonable to expect that
there will be some symmetries in the final theory of everything too. Therefore
the key elements of our toy example seem to have at least a chance to carry
over to the situation of our real world.
#### 2.2.2. Maybe the symmetry helps?
But maybe the symmetry we have mentioned here allows to solve the very
problem? We prefer, of course, symmetric theories. And, of course, if we have
to choose between a symmetric theory and one without symmetry we prefer the
symmetric theory. But the very point of a “many laws” theory is that the
theory itself does not make such a choice. The “symmetric theory” is, in this
concept, not a separate symmetric theory, but only a particular universe, with
a particular law which has a particular symmetry. We do not have to make a
choice between theories – there is only one theory, which contains different
worlds with different laws. So, Ockham’s razor or human preferences for
symmetric theories are of no use here. A special symmetry of the laws of our
particular universe is something which requires explanation.
The situation becomes different if we reject the many laws proposal and want
to use this symmetry to find a preferred set of canonical operators. For this
purpose, symmetry properties of some choices of $\hat{p}$, $\hat{q}$ may be
useful, and we will clearly prefer a more symmetric choice.
#### 2.2.3. Maybe anthropic argumentation helps?
The laws of our world may be not a typical element in the set of all possible
laws. Last but not least, our laws allow the existence of human beings. Maybe
anthropic considerations allow to fix the parameters $s$ so that no conspiracy
is needed?
Now, anthropic arguments seem quite weak in their ability to restrict
parameters. The general picture, as defended, for example, by Weinberg [25],
is that some parameters may be restricted in some regions of the parameter
space by anthropic considerations, other parameters not. For some parameters
may exist wide ranges where anthropic considerations are irrelevant, because
these parameters do not seem to influence anything relevant for human
existence. For example, the cosmological constant $\Lambda$ should be small
enough to allow human existence. But if it is below a certain limit
$\Lambda_{0}$, anthropic considerations are unable to tell anything. And if
$\lvert\Lambda\rvert\ll\Lambda_{0}$ the fine tuning problem is not solvable by
anthropic considerations. For other parameters, anthropic considerations may
not exclude anything interesting. For example, the mass of the top quark seems
quite irrelevant for everything related with humans. It could influence
something only if it would be many orders lower than it is. But nobody would
observe any important difference if it would be many orders greater than it
is. Thus, the predictive properties of anthropic considerations seem to be
quite restricted.
Let’s see what would be required. The construction as given in [1] contains
only one parameter $s$, but can be easily extended to an infinite set of
parameters $\\{s_{2k+1}\\}$ where $s=s_{3}$ is related with the KdV equation
itself, $s_{1}$ defines a simple shift, and the other $s_{2k+1}$ are related
with other differential equations of order $2k+1$, so-called higher KdV
equations (see, for example, [2]). And for each of these parameters we have a
similar situation: Different $s_{2k+1}$ define different physics, with a
different potential $V(q,s_{2k+1})$, changing the operators $\hat{p}$,
$\hat{q}$, but leaving $\hat{h}$ unchanged. Thus, the non-uniqueness problem
is a multi-dimensional one, all the parameters $s_{2k+1}$ would have to be
restricted. Similarly, if we consider, instead, the positions of the
eigenstates $q_{k}$ as the parameters to be restricted to $q_{k}\approx 0$, we
also have an infinite set of parameters.
So, even if one can reasonably hope that anthropic considerations may restrict
a few of the $s_{2k+1}$, or some of the $q_{k}$, what would be the base for
the hope that it allows to restrict all of them?
### 2.3. Summary
We have here even two independent arguments against a “many laws” proposal, of
quite different character. A simple methodological one using Popper’s
criterion of empirical content and fine tuning argument based on the thesis
that our laws are more symmetric than the average laws in this construction.
If these two arguments are sufficient to convince proponents of this idea is
another question. Given that the proposal has been made by Saunders [17] in a
situation where the non-uniqueness construction of [1] was yet unknown, and
that Tegmark has proposed an even more radical version of Platonism where
every imaginable mathematical universe really exists [19], the idea to extend
many worlds ruled by a common law into many laws seems to be attractive to
many worlders on its own right, even without the necessity to solve the non-
uniqueness problem.
But in the remaining part of this paper we will ignore the “many laws”
possibility. So in the following we assume that there is only a single law of
physics, which makes certain predictions. Thus, it follows from the experiment
considered in [1] (which proves that the different choices lead to different
physical predictions) that the complete description of physics consists not of
$\hat{h}$ alone but also of the canonical operators $\hat{p}$, $\hat{q}$. The
alternative choices $\hat{p}(s)$, $\hat{q}(s)$ are unphysical.
## 3\. Pure interpretations: The minimal program for replacement of
Copenhagen
So, assume that we don’t want to embrace all $\hat{p}(s)$, $\hat{q}(s)$ as
describing different real worlds. There is only one $\hat{p}$, $\hat{q}$,
which describes the true momentum and position measurements, while all the
other $\hat{p}(s)$, $\hat{q}(s)$ are only mathematical constructions without
any physical meaning. They may describe some other measurements, but nobody
knows which, and nobody cares.
But the $\hat{p}$, $\hat{q}$ are not distinguished among the $\hat{p}(s)$,
$\hat{q}(s)$ by their mathematical properties. Each defines an irreducible
representation of the canonical commutation relations:
$[\mbox{$\hat{p}(s)$},\mbox{$\hat{q}(s)$}]=-i\hbar$. Each of them is connected
with the Hamilton operator $\hat{h}$ in a quite similar way – the Hamilton
operator has the same form (1), and the potential $V(q,s)$ has similar nice
qualitative properties. But something in the interpretation should tell us why
we nonetheless have to use the operators $\hat{p}$, $\hat{q}$ (instead of one
of the $\hat{p}(s)$, $\hat{q}(s)$) if we want to measure the momentum or the
position.
In the Copenhagen interpretation this is done. We have the canonical operators
$\hat{p}$, $\hat{q}$. And these canonical operators are defined as describing
the momentum and position measurements. What do these phrases “momentum
measurement” and “position measurement” mean? The answer is contained in the
classical part of the Copenhagen interpretation. Or at least supposed to be.
In fact, this classical part is not formalized, and there seems to be no hope
that such a notion as “momentum measurement”, which covers lots of very
different macroscopic measurement devices, can be really made precise and
certain.
That’s why this “classial part” of the Copenhagen interpretation has been
widely considered as unsatisfactory, and has motivated attempts to get rid of
it. The ideal solution would be a derivation of the classical part from the
quantum part taken alone. The program to find an interpretation of this ideal
type, which reject the classical part of Copenhagen and start with the pure
quantum part, without introduction of additional structure, we call _pure
program_ , and the resulting interpretations (even if they are in fact not
completely realized) _pure interpretations_.
The most popular example is the Everett interpretation (better named “Everett
program”), described by Everett in the following way:
> “This paper proposes to regard pure wave mechanics …as a complete theory. It
> postulates that a wave function that obeys a linear wave equation everywhere
> and at all times supplies a complete mathematical model for every isolated
> physical system without exception. …The wave function is taken as the basic
> physical entity with no a priori interpretation. Interpretation only comes
> after an investigation of the logical structure of the theory. Here as
> always the theory itself sets the framework for its interpretation. …The new
> theory is not based on any radical departure from the conventional one. The
> special postulates in the old theory which deal with observation are omitted
> in the new theory. The altered theory thereby acquires a new character. It
> has to be analyzed in and for itself before any identification becomes
> possible between the quantities of the theory and the properties of the
> world of experience.” [9]
There are, of course, lots of variants of many worlds interpretations, and not
all of them follow the original pure program. But our point is, of course, not
to criticize non-pure variants of many worlds: Instead, our point is that the
original, pure program is not viable.
Many worlds is not the only such program. Another example is Mermin’s “Ithaca
interpretation” (also more appropriately named “Ithaca program”) [15]: On the
one hand, Mermin tells that “…I would like to have a quantum mechanics that
does not require the existence of a classical domain” and introduces as one of
the desiderata “The concept of measurement should play no fundamental role”.
On the other hand, we read that “…by quantum mechanics I mean quantum
mechanics as it is – not some other theory in which the time evolution is
modified by non-linear or stochastic terms, nor even the old theory augmented
with some new physical entities (like Bohmian particles) which supplement the
conventional formalism without altering any of its observable predictions.”
Thus, the basic idea of a pure quantum interpretation is shared by quite
different approaches. But this is quite natural. The most questionable part of
the Copenhagen interpretation is its classical part, so it is natural that one
wants to get rid of it. On the other hand, one wants to minimize the number of
assumptions one has to make. And the minimum would be reached if nothing would
be added.
Unfortunately, because these pure interpretations reject the classical domain
of the Copenhagen interpretation, the Copenhagen way to solve our non-
uniqueness problem has been lost. The “correct” operators $\hat{p}$ and
$\hat{q}$ can no longer be distinguished among the $\hat{p}(s)$, $\hat{q}(s)$
by a postulated association with specific experimental arrangements described
in classical language – this is part of what has been removed. On the other
hand, once pure interpretations refuse to add some replacement, some new,
additional structure, they seem unable to compensate for the loss.
Thus, the non-uniqueness result of [1] shows that the “pure program” – the
program to develop pure interpretations of quantum theories – cannot be
realized and has to be given up. If one removes the association of the
canonical operators $\hat{p}$, $\hat{q}$ with certain measurement procedures,
which is defined by the classical part of the Copenhagen interpretation, one
has to add something else, something which allows to identify the $\hat{p}$,
$\hat{q}$, in some other way with momentum and position measurements we make.
For this, the $\hat{p}$, $\hat{q}$ have to have some special properties which
distinguish them from all the other $\hat{p}(s)$, $\hat{q}(s)$.
## 4\. “Special” interpretations?
Let’s return now to the hope that some special symmetries of the problem may
be used to distinguish the $\hat{p}$, $\hat{q}$, by their special properties
from the other $\hat{p}(s)$, $\hat{q}(s)$. In our toy example this has been
the property that all the positions of the eigenstates
$q_{k}=\langle\psi_{k}|\mbox{$\hat{q}$}|\psi_{k}\rangle$ are zero.
Let’s note here an important and interesting difference between an imagined
interpretation based on such an idea and existing interpretations. Canonical
quantization works for arbitrary potentials $V(q)$, and the Copenhagen
interpretation does not object and gives all these canonical quantum theories
the same sort of interpretation. But we can do canonical quantization for all
the potentials $V(q,s)$, and the theory defined by $\hat{h}$ as given by (1),
and $\hat{p}$ and $\hat{q}$ as given by $\hat{p}(s)$ and $\hat{q}(s)$ defines
a realization of the canonical quantization for the potential $V(q,s)$. If an
interpretation forgets now about the canonical operators $\hat{p}$, $\hat{q}$
as defined by the canonical quantization procedure, and makes a new choice of
the $\hat{p}$, $\hat{q}$ based some mathematical properties of their relation
to $\hat{h}$ (like the property $q_{k}=0$) then we do not recover in the
classical limit the original theory we have canonically quantized – with
potential $V(q,s)$ – but another one, with the potential $V(q,s^{\prime})$ for
the preferred $s^{\prime}$.
Thus, if we follow this strategy, we obtain a lot of changes in the general
picture of quantization: Only a few potentials, distinguished by some special
properties, allow to be quantized. Canonical quantization of other potentials
gives, of course, a canonical quantum theory as before, but the classical
limit of this theory, as described by an interpretation of this type, leads to
a different classical theory with different potential. For example, if we
would use the property $\forall k\;q_{k}=0$, the corresponding potentials
would have the symmetry $V(q)=V(-q)$, and only potentials of this type could
be obtained as a classical limit of quantum theories in this interpretation.
This is not obviously false – essentially, to be viable, an interpretation
should be able to quantize only a single potential, the one we observe in our
universe. Then, such a restriction of the allowed potentials is a testable,
falsifiable prediction, also something nice. And if the potentials preferred
by this interpretation have, moreover, nice additional symmetry properties, as
the $V(q)=V(-q)$ symmetry of our toy example, this gives the interpretation
some advantage in beauty.
So this may be an interesting way to meet the non-uniqueness problem of [1].
But it does not meet the criteria of the “pure program”, because it adds new
physics, even important new physics, by restricting the class of potentials
$V(q)$ allowed in canonical quantum theories. Thus, an interpretation of this
type is physically very different from the Copenhagen interpretation (which
does not make any such restrictions). And, in particular, if we start with a
canonical theory which has a “wrong” potential, an interpretation of this type
makes physical predictions different from the Copenhagen interpretation.
Thus, the thesis in the title of our paper is not endangered by
interpretations of this type. Therefore we can ignore them in the remaining
part of the paper and focus our interest on interpretations which are not
“special” in this sense, interpretations which handle all potentials $V(q)$ on
equal foot.
## 5\. The necessity of a new physical structure
That means, we assume that they allow canonical quantization for all
sufficiently well-behaved potentials $V(q)$, and position and momentum
measurements, as described on the base of these interpretations, have some
association with the canonical operators. Note that in interpretations which
do not contain the Copenhagen classical part this association has to be
derived, because it is no longer postulated. And we assume that this
derivation, whatever the potential $V(q)$ used in the theory, recovers at
least approximately the Copenhagen association of the canonical operators with
the canonical measurements and recovers also the classical limit.
The point we want to make in this section is that this requires the
introduction of some additional physical structure.
### 5.1. Pure labels are not sufficient
The problem is that a canonical quantum theory in itself gives only labels.
There is some operator denoted $\hat{p}$ with the label “momentum operator”,
some other operator denoted $\hat{q}$ with the label “position operator”,
which form an irreducible representation of their commutation relations
$[\mbox{$\hat{p}$},\mbox{$\hat{q}$}]=-i\hbar$. The Hamilton operator $\hat{h}$
has the form (3) with some potential $V(q)$. That’s all what is given by the
canonical quantum theory itself.
Now let’s compare this with some other choice of $s$. We have now another
operator, denoted here $\hat{p}$’, which has now the label “momentum
operator”, and also another operator $\hat{q}$’ which has the label “position
operator”. But in all other aspects this is a standard canonical quantum
theory. Thus, there is some (other) potential $V^{\prime}(q)$, but the general
form of the Hamilton operator $\hat{h}$ in terms of the $\hat{p}$’, $\hat{q}$’
is the same canonical form (3). This other theory is simply equivalent to
standard canonical quantum theory for a different potential $V^{\prime}(q)$.
But as a consequence of our assumptions, momentum and position measurements
for these two theories have to be different. In the first canonical theory, we
obtain the predictions for momentum and position for the potential $V(q)$, in
the second theory those for the potential $V^{\prime}(q)$. But the canonical
theories themself have distinguished the operators $\hat{p}$ and $\hat{q}$
only by giving them different labels. The operator $\hat{h}$ which defines the
time evolution was the same, the operators $\hat{p}$ and $\hat{q}$ define an
unitarily equivalent representation of the same commutation relation
$[\mbox{$\hat{p}$},\mbox{$\hat{q}$}]=[\mbox{$\hat{p}$}^{\prime},\mbox{$\hat{q}$}^{\prime}]=-i\hbar$.
Pure labels don’t change anything. Thus, no physical predictions should differ
simply because we have decided to name $\hat{p}$’ instead of $\hat{p}$ the
“momentum operator”.
Thus, there should be also something else which changes together with the
label “momentum operator”. Something physical, because it leads to differences
in the physical predictions, in particular for momentum measurements. There
should be some connection between the label “momentum operator” and physics, a
connection which is not covered by the mathematics of canonical quantum
theories (that means, by the irreducible representation of the canonical
commutation relations and the general form (3) of the Hamilton operator), but
which allows to identify the expectation value
$\langle\psi|\mbox{$\hat{p}$}|\psi\rangle$ of the operator $\hat{p}$ (the one
with the label “momentum operator”) with the expectation value of some real
physical experiment which measures momentum.
In the Copenhagen interpretation, such a connection exists – the association
between the label “momentum operator” and the momentum measurement is simply
postulated in the classical part of this interpretation. Removing this part of
the Copenhagen interpretation would remove this association, reducing
“momentum operator” to a pure label without association to measurement
procedures. But we have to recover this association, because this is what the
theory predicts. So there should be something else, some physics defined by
the interpretation, which allows to establish such an association.
To clarify what is meant with a new physical structure, let’s consider a few
examples.
### 5.2. Theories of pilot wave type
With “theories of pilot wave type” we mean a large number of quite different
interpretations. First, there are of course de Broglie-Bohm pilot wave
theories [4, 5] with different choices of the configuration space. But we
include here also some stochastic theories like Bell-type field theories [3]
and Nelsonian stochastics [16]. To combine them all into a single class is
justified only because for the question discussed here their differences do
not matter. They use, essentially, the same type of additional physical
structure – an explicit trajectory $q(t)\in Q$ in the configuration space $Q$.
This trajectory may be deterministic in pilot wave theories theories,
stochastic in Nelsonian stochastics, and even discontinuous stochastic in the
case of discrete configuration spaces $Q$. But in all these theories we have a
new physical law, a variant of the “guiding equation” of pilot wave theory,
which defines the evolution of the configuration $q(t)$.
Then it is postulated what our own state is described by the configuration
$q(t)$ instead of the wave function $\psi(q)$. This postulate allows to
identify measurements as something which has to change the physical state of
our brain, thus, as something which changes the value of some particular
configuration variables $q_{brain}(t)$. This is a sufficient base for the
development of a measurement theory. In particular, in the classical limit we
obtain the classical trajectory $q(t)$ simply as the limit of the quantum
trajectory $q(t)$.
The new physical structure, therefore, consists of the following elements: The
identification of the configuration space $Q$, or, in other words, of the
operator $\hat{q}$, a new equation for the evolution of $q(t)$, and the
identification of the state of the universe with the configuration of the
universe $q$. The canonical operator $\hat{q}$ has therefore a direct
connection with the new structure, which does not exist for the other
$\hat{q}(s)$. The canonical theory based on the other operator $\hat{q}$’
leads to a very different pilot wave theory, with another configuration space
$Q^{\prime}$. The trajectory $q(t)$ of the first theory and the trajectory
$q^{\prime}(t)$ of the second theory have nothing to do with each other – we
cannot even compare them because they live in different spaces.
### 5.3. Physical collapse theories
Another class of interpretations which have introduced a new physical
structure are physical collapse theories [11, 10]. In these theories the new
physics consists of a modification of the Schrödinger equation. Some
additional physical collapse mechanism disturbs the unitary Schrödinger
evolution and leads to a localization of the wave function. This localization
happens in the position representation
$\psi(q)\in\mathcal{L}^{2}(Q,\mbox{$\mathbb{C}$})$.
Given the collapse mechanism, we do not have to consider all wave functions
$\psi(q)$ in the classical limit, but only a small subclass of localized wave
functions $\psi(q,t)\approx\delta(q-q(t))$ which are localized around the
classical trajectory.
The new physical structure in these theories is defined by the terms which
modify the Schrödinger equation. These terms depend on something which
explicitly depends on the canonical operator $\hat{q}$. Thus, the wave
functions obtained in different canonical quantum theories follow different
equations, and the classical trajectories $q(t)\in Q$ and $q^{\prime}(t)\in
Q^{\prime}$ approximating them have nothing to do with each other.
The author prefers theories of pilot wave type, considering de Broglie’s old
argument that “[I]t seems a little paradoxical to construct a configuration
space with the coordinates of points which do not exist” [5] as sufficiently
strong. But this preference is irrelevant for the question considered in this
paper. What is interesting here is that above classes of theories have a
sufficient additional physical structure and therefore no non-uniqueness
problem.
### 5.4. Predefined subsystems
In the previous examples, the configuration space $Q$ itself has already
played a special physical role. But in principle the new structure may be of a
quite different type. A nice example to illustrate this is a predefined
subdivision $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ of the Hilbert
space into physically different subspaces, for example into a bosonic and a
fermionic part, as considered, for example, by Kent [14]. Then we can apply
techniques like decoherence or the Schmidt decomposition to derive a preferred
basis in one of them, defined, say, by some operator $\mbox{$\hat{q}$}_{A}$ on
$\mathcal{H}_{A}$.
Here, the additional structure is defined by the subdivision
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ itself and the connection
between the Hamilton operator and this subdivision. In the simplest case, one
could, for example, imagine a Hamilton operator
(4)
$\mbox{$\hat{h}$}=\frac{1}{2m_{A}}\mbox{$\hat{p}$}_{A}^{2}+\frac{1}{2m_{B}}\mbox{$\hat{p}$}_{B}^{2}+V(\mbox{$\hat{q}$}_{A},\mbox{$\hat{q}$}_{B})$
which restricts observations of $A$ made by the $B$-part to measurements of
$\mbox{$\hat{q}$}_{A}$.
In [1] we have presented some arguments against interpretations of this type.
But these arguments are irrelevant for the point of this paper, which is that
we need an additional physical structure. This additional physical structure
is present, and it is also sufficient to identify the canonical operator, even
if it is only the operator $\mbox{$\hat{q}$}_{A}$ of some part of the theory
$\mathcal{H}_{A}$.
### 5.5. Summary: What we need
Thus, to solve the non-uniqueness problem, we need an additional physical
structure. The pure labels “position operator” and “momentum operator” which
remain if we remove the classical part of the Copenhagen interpretation (which
have given them an explicit, even if only postulated, connection with position
and momentum measurements) are not sufficient. We need something which gives
different predictions for these measurements for different choices of the
canonical operators among the $\hat{p}(s)$, $\hat{q}(s)$.
This requirement is not too strong, as we have seen in some examples of
interpretations which have such additional structures. To solve the non-
uniqueness problem, it is important that, first, we have an additional
structure, and, second, that this additional structure is sufficient to make a
choice among the candidates for the canonical operators $\hat{p}(s)$,
$\hat{q}(s)$.
## 6\. Consistent histories
And interesting example where the question if the additional structure is
sufficient is problematic is the “consistent histories interpretation”. An
implicit reference to it we have already cited – our quote from Brown and
Wallace [7] which describes what we have named the “many laws” solution has
used the language of consistent histories. And this seems not to be an
accident. It seems quite reasonable to expect that in consistent histories
different choices of $s$ will be associated with different consistent families
of histories..
In its intentions, the “consistent histories interpretation” seems close to a
pure interpretation, despite the fact that it includes some additional
structure. In particular, it rejects the classical Copenhagen part (“The
interpretive scheme which results is applicable to closed (isolated) quantum
systems, …has no need for wave function ‘collapse,’ makes no reference to
processes of measurement (though it can be used to analyze such processes) …”
[12]). What it adds to the quantum formalism is “…an extension of the standard
transition probability formula of nonrelativistic quantum mechanics to certain
situations, we call them ‘consistent histories,’ in which it is possible to
assign joint probability distributions to events occurring at different times
in a closed system without assuming that the corresponding quantum operators
commute.”
Is the additional structure added by consistent histories sufficient to
identify the canonical operators? The answer seems to be negative. Or, more
accurate, I see no reason for hope that the consistency condition for families
alone allows to distinguish between different standard quantum theories having
the same standard form
(5) $\mbox{$\hat{h}$}=\frac{1}{2m}\mbox{$\hat{p}$}^{2}+V(\mbox{$\hat{q}$})$
only with different potentials $V(\mbox{$\hat{q}$},s)$. But this is what would
have to happen if the only thing added – the consistency condition for
families – would allow to distinguish between the different $s$.
Indeed, let’s look what we would need. We have a definition of histories, a
definition of families of histories, and some consistency conditions for these
families. What would solve the problem would be the identification of a
history which has a natural association with the canonical operators
$\hat{p}$, $\hat{q}$. Now, histories are by definition given only at some
discrete times $t_{i}$, and at each moment of time the operators
$E^{\alpha}(t_{i})$ defining the possible events have to commute. But these
seem to be purely technical complications. If there would be a consistent
family of histories
(6) $\mathcal{H}=\\{O_{jk}(t_{0}),O_{jk}(t_{1}),\ldots,O_{jk}(t_{n})\\}$
where each event $O_{jk}$ gives $\langle\mbox{$\hat{p}$}\rangle\approx p_{j}$,
$\langle\mbox{$\hat{q}$}\rangle\approx q_{k}$ with appropriate accuracy
$\Delta p$, $\Delta q$, one could consider this part of the problem as solved.
But the problem we want to solve is not simply that for one of the
$\hat{p}(s)$, $\hat{q}(s)$ such a consistent family should exist. The problem
is that the structure added by the consistent histories approximation should
identify a single $s$. It would not be a very big problem if this is only an
approximate identification modulo some $\Delta s$ such that measurement
results of the operators $\hat{q}(s)$ cannot be distinguished from each other.
But some preferred $s$ should be distinguished at least approximately.
What does it mean? If we forget a moment about $\hat{h}$, then all the
canonical pairs $\hat{p}(s)$, $\hat{q}(s)$ are unitarily equivalent (that’s
how they have been constructed in [1]). Thus, if there is a family of
histories $\mathcal{H}$ associated with one $\hat{p}$, $\hat{q}$, we can
simply use this equivalence to construct families of histories
$\mathcal{H}_{s}$ associated with every $\hat{p}(s)$, $\hat{q}(s)$. The
original $\mathcal{H}$ is consistent for the Hamilton operator $\hat{h}$. The
question is if the $\mathcal{H}_{s}$ are consistent. In principle, they may
not – the question if $\mathcal{H}_{s}$ is consistent given the Hamilton
operator $\hat{h}$ is unitarily equivalent to the question if the original
family $\mathcal{H}$ remains consistent if $V(q)$ will be replaced by
$V(q,s)$.
Of course, the consistency of a family of histories depends on the Hamilton
operator. So, in principle, it may be possible that it among the $V(q,s)$
there is only one value of $s$ such that the family $\mathcal{H}$ is
consistent. But would you bet that this will happen? I will certainly not. The
$V(q,s)$ are solutions of the Korteweg-de Vries equation with $s$ as the
“time” parameter [1]. Their minima are localized at very different positions
$q$, but look qualitatively equally nice and have equally nice formal
properties (they are smooth, decrease at infinity) and even some important
properties like the eigenvalues of $\hat{h}$ are the same. What could be a
base for the hope that a pure consistency requirement allows to prefer, among
them, a single value of $s$? I cannot see anything supporting such a hope.
So, if the additional structure which the consistent histories interpretation
adds to pure quantum theory is some variant of a consistency condition for
families of histories, there seems no reason to hope that the non-uniqueness
problem may be solved.
But is it possible to save consistent histories by adding more structure? This
seems not only possible, there is even a natural candidate – some preferred
consistent family of histories. The various criteria for consistency of
families may have different solutions, different families of histories which
are consistent, nice and beatiful according to various criteria. But the
different families are incompatible with each other. If we hear different
incompatible histories in everyday life, we believe at most one of them, and
even if we have not yet decided which story we believe, we do not doubt that
at most one of the stories can possibly be true. So all we have to do is to
apply this rule of common sense to consistent histories. That means there is
one consistent family which is correct, and everything incompatible with this
family of histories has to be rejected. Now, if this preferred family is
somehow associated with one set of canonical operators $\hat{p}$, $\hat{q}$
but not with others, then everything is fine. This hypothesis seems already
quite natural.
Thus, while consistent histories as it is (with various consistency
conditions, but no definite choice of a preferred family of histories) seems
unable to solve the non-uniqueness problem, it probably may be solved by
introducing a preferred family of consistent histories.
## 7\. Consequences of the loss of purity
Even if one accepts that one needs additional structure to solve the non-
uniqueness problem, one may decide that particular attempts to develop pure
interpretations have their own value and should not be given up, even if the
initial hope to obtain a pure interpretation cannot be realized. This seems
sociologically plausible in particular for the Everett program.
What would be the consequences? Most importantly, the previously pure
interpretation would lose its most attractive property – its purity.
### 7.1. The fate of the denial-argument
An example of an application of purity is the argument that “[P]ilot-wave
theories are parallel-universe theories in a state of chronic denial” [8].
This argument is quite popular in the many worlds community [26, 6, 23]. Given
the counterargumentation by Valentini [20], one would concede far too much if
one would accept this argument as somehow endangering pilot wave theory, even
if it remains popular among the many worlders [21].
But let’s nonetheless assume, simply for the sake of the argument, that the
argument in its original version is not completely invalid. Assume now that,
to meet our non-uniqueness argument, one introduces some additional structure
into many worlds. In this case, it is very probable that the argument becomes
invalid, for the simple reason that the new many worlds structure is not part
of pilot wave theory. Pilot wave theory already has an additional structure –
the trajectory of the configuration $q(t)\in Q$ guided by the guiding equation
– which distinguishs a certain $\hat{q}$, thus, there is no reason to
introduce anything else. But the argument works only if all real, physical
parts of the many worlds interpretation are also real, physical parts of pilot
wave theory and follow the same equations (like the Schrödinger equation for
the wave function).
Thus, the argument depends on the purity of the many worlds interpretation
relative to pilot wave theory: Everything which is physical in the Everett
interpretation should be physical in pilot wave theory too. If we save many
worlds by the introduction of some additional structure, the argument has to
be reconsidered. If the new structure is not part of pilot wave theory too,
the argument becomes invalid.
Of course, one cannot exclude completely that the many worlders use an
additional structure which is also part of pilot wave theory, so that the
denial-argument survives. But this would be a rather strange choice. Last but
not least, the additional structure is the configuration $q(t)\in Q$. This
hidden variable is considered as unnecessary today and does not seem to be the
first candidate to be embraced by the many worlds community. Even more, one
could even question if a theory which gives a $q(t)\in Q$ the status of
reality is yet correctly classified as “many worlds” – it may be better
characterized as a variant of pilot wave theory. It would be much closer to
the current many worlds approach if, instead, some fundamental “decomposition
into systems” would be used as the additional structure. But once no such
“decomposition into systems” is part of pilot wave theory, the denial argument
would be dead in this case.
But this may be not the only loss. One of the major arguments in favour of
many worlds as well as of other pure interpretations is their claimed
compatibility with relativistic symmetry. 111 Given that a preferred frame
allows a simple explanation of the SM fermions and gauge fields in terms of a
condensed matter model [18], relativistic symmetry does no longer seem to have
the fundamental importance which is attributed to it by the relativistic
tradition. But whatever the additional structure, it may restrict the
symmetry group of the theory. In particular, the configuration space itself –
the structure defined by the operator $\hat{q}$ we have to choose among the
$\hat{q}(s)$ – is (at least in its usual form) not covariant. The danger that
some additional structure will destroy relativistic symmetry is recognized,
for example, by Wallace, who notes that “…there seems to be no
relativistically covariant way to define a world …” [23].
### 7.2. Does consistent histories have to modify logic?
On the other hand, the loss of purity may also lead to improvements. The
additional structure may destroy arguments against the interpretation. Here,
our proposal to introduce a preferred family of histories into the consistent
histories interpretation may be an example. This is not only a possibility to
solve the non-uniqueness problem, but removes also another argument against
consistent histories – that it modifies classical logic without necessity.
The problem (discussed for example in [27]) is the incompatibility of
different, separately consistent, families of histories. A situation where we
have different sets of statements which are internally consistent but
incompatible with each other is quite common in everyday life – these are
simply different incompatible theories. Because they are incompatible with
each other, at most one of them can be true. And even if we do not have
sufficient information to identify the true theory, if all of them, taken
separately, are compatible with all we know, it would not change our certainty
that at most one of the theories may be true. This is, essentially, the whole
point of logic.
But in consistent histories this everyday situation is interpreted in a
completely different way: “Note that incompatibility, the fact that the two
families cannot be combined, does not mean that one is ‘wrong’ and the other
is ‘right.’ Seeking some law of nature which ‘chooses’ one rather than the
other is to misunderstand the nature of quantum descriptions.” [13]. In other
words, we have different incompatible theories, but they are somehow all
equally true.
The consistent historians are, of course, aware of the straightforward logical
consequence – if two theories which are incompatible with each other are above
true, one can construct contradictions. They have “solved” this problem by
introducing a new rule of logic – that one is not allowed to combine
statements which belong to different families: “Since both $\mathcal{F}_{a}$
and $\mathcal{F}_{b}$ are consistent families, the conclusions of a
probabilistic analysis applied using just one of them while disregarding the
other will be correct. However, the families are incompatible, and so these
conclusions cannot be combined. One cannot say that at time $t_{2}$ the
particle is both in a superposition state $c$ AND that it is moving on the
upper trajectory $u$, for that would be meaningless in the same way that
‘$S_{x}=+1/2\text{ AND }S_{z}=+1/2$’ makes no sense.” [13].
This “solution” is not accepted as satisfactoy by the critics. It looks like
there are true statements $A$ and $B$, but to combine them into the statement
$A\text{ AND }B$, something which is always allowed in classical logic,
appears to be forbidden. Is this some sort of modification of the rules of
logic, some variant of “quantum logic”? If yes, then it seems extremely
difficult to justify it. Even if the very foundations of the scientific
method, including the logic, are in principle open to discussion and
modification, the justification for a modification of logic should be
extremely strong. And, given that there are viable alternatives which do not
require modifications of logic, this does not seem to be the case. If
classical logic is not changed, then what is modified if the incompatibility
of two families of statements does not mean that at most one of them can be
right?
Whatever, all these problems with incompatibility simply vanish if we
introduce, as an additional physical structure, one consistent family of
histories – the one which contains the histories which are really possible.
Thus, the additional structure which seems necessary to solve the non-
uniqueness problem would, by the way, solve also another weak point of the
consistent histories interpretation.
## 8\. Discussion
We have argued that the only way to handle our non-uniqueness result is to
make a choice among the $\hat{p}(s)$, $\hat{q}(s)$, and to associate the
preferred $\hat{p}$, $\hat{q}$ with some physical structure powerful enough to
distinguish them as associated with momentum and position measurements. This
destroys the minimal program for the improvement of the Copenhagen
interpretation – to throw away the classical part of the Copenhagen
interpretation (which solves this non-uniqueness problem) without adding any
new physical structure to its quantum part. This “pure program” is unable to
give viable interpretations, interpretations which are able to solve our non-
uniqueness problem. A quantum interpretation which does not embrace the
classical part of the Copenhagen interpretation needs an additional physical
structure.
It was not the aim of the paper to present a complete overview over all the
proposed interpretations of quantum theory whose viability is endangered in
the light of our non-uniqueness problem. We have presented some examples – the
Everett, Ithaca, and consistent histories interpretations. These examples
illustrate that the problem is relevant for quite different approaches to the
foundations of quantum theory.
But it was also not the aim of this paper to claim that the particular
interpretations we have considered cannot be saved. Instead, one can save them
by adding some new physical structure. Sometimes reasonable candidates are
already part of the mathematical apparatus, and giving them physical
significance could even improve them. We have argued that this in the case of
consistent histories if one introduces one consistent family of histories as
preferred.
On the other hand, the introduction of an additional physical structure means
also some sort of loss. There is clearly a loss of purity. The interpretation
becomes in some sense more complicate. It possibly decreases the symmetry of
the interpretation and destroys some arguments in favour of it. We have
discussed this for the case of the “pilot wave theory is many worlds in a
state of denial” argument.
The decision if the particular interpretations are worth to be saved, and what
are the best ways to save them, the optimal choices for the additional
structure, is something we leave to the proponents of these interpretations.
The interpretations preferred by the author – theories of pilot wave type
which have a physically preferred configuration space $Q$ with a trajectory
$q(t)\in Q$ – do not have any non-uniqueness problem. This means not only that
the problem is solvable and solved by other interpretations. It means also
that what was a strong argument against pilot wave theory – the existence of
such an additional structure – becomes now a strong argument in its favour.
## References
* [1] Schmelzer, I.: Why the Hamilton operator alone is not enough, Found. Phys. vol. 39, p. 486-498 (2009), http://arxiv.org/abs/arXiv:0901.3262arXiv:0901.3262
* [2] Ablowitz, M. J., Clarkson, P. A.: Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, 149, Cambridge University Press, Cambridge (1991)
* [3] J.S. Bell, Beables for quantum field theory, Phys. Rep. 137, 49-54, 1986
* [4] Bohm, D: A suggested interpretation of the quantum theory in terms of “hidden” variables, Phys. Rev. 85, 166-193 (1952)
* [5] de Broglie, L., La nouvelle dynamique des quanta, in Electrons et Photons: Rapports et Discussions du Cinquieme Conseil de Physique, ed. J. Bordet, Gauthier-Villars, Paris, 105-132 (1928), English translation in: Bacciagaluppi, G., Valentini, A.: “Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference”, Cambridge University Press, and http://arxiv.org/abs/arXiv:quant-ph/0609184arXiv:quant-ph/0609184 (2006)
* [6] Brown, H.R., Comment on Valentini, “De Broglie-Bohm Pilot-Wave Theory: Many Worlds in Denial?”, http://arxiv.org/abs/arXiv:0901.1278arXiv:0901.1278
* [7] Brown, H.R., Wallace, D.: Solving the measurement problem: de Broglie-Bohm loses out to Everett, Foundations of Physics, Vol. 35, No. 4, 517 (2005) http://arxiv.org/abs/arXiv:quant-ph/0403094arXiv:quant-ph/0403094
* [8] D. Deutsch, Comment on Lockwood. British Journal for the Philosophy of Science 47, 222–228 (1996)
* [9] Everett, H.: ”Relative State” Formulation of Quantum Mechanics, Rev. Mod. Phys. vol. 29. n. 3 (1957)
* [10] Ghirardi, G. C. (2002). Collapse theories. In the Stanford Encyclopedia of Philosophy (Summer 2002 edition), Edward N. Zalta (ed.), available online at . http://plato.stanford.edu/archives/spr2002/entries/qm-collapsehttp://plato.stanford.edu/archives/spr2002/entries/qm-collapse
* [11] Ghirardi, G., A. Rimini, and T. Weber: Unified Dynamics for Micro and Macro Systems. Physical Review D 34, 470-491 (1986)
* [12] Griffiths, R. B.: Consistent Histories and the Interpretation of Quantum Mechanics, Journal of Statistical Physics, vol. 36, nr. 1/2, 219-272 (1984)
* [13] Griffiths, R. B.: Quantum mechanics without measurements, http://arxiv.org/abs/arXiv:quant-ph/0612065arXiv:quant-ph/0612065 (2006)
* [14] Kent, A., Real World Interpretations of Quantum Theory, http://arxiv.org/abs/arXiv:0708.3710arXiv:0708.3710 (2007)
* [15] Mermin, N. D.: The Ithaca Interpretation of Quantum Mechanics, Pramana 51, 549-565 (1998) http://arxiv.org/abs/arXiv:quant-ph/9609013arXiv:quant-ph/9609013
* [16] E. Nelson, Derivation of the Schrödinger Equation from Newtonian Mechanics, Phys.Rev. 150, 1079-1085 (1966)
* [17] Saunders, S.: Time, Decoherence and Quantum Mechanics. Synthese 102, 235-266 (1995)
* [18] Schmelzer, I.: A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Foundations of Physics, vol. 39, 1, p. 73, http://arxiv.org/abs/arXiv:0908.0591arXiv:0908.0591 (2009)
* [19] Tegmark, M.: The Mathematical Universe, Found Phys 38: 101-150 (2008)
* [20] Valentini, A.: De Broglie-Bohm Pilot-Wave Theory: Many Worlds in Denial? in [22], http://arxiv.org/abs/arXiv:0811.0810arXiv:0811.0810
* [21] Brown, H.R.: Comment on Valentini, “De Broglie-Bohm Pilot-Wave Theory: Many Worlds in Denial?”, in [22], http://arxiv.org/abs/arXiv:0901.1278arXiv:0901.1278
* [22] Saunders, S., Barrett, J., Kent, A., Wallace, D. (eds.), Many Worlds? Realism, Everett, and quantum mechanics, Oxford University Press (2009)
* [23] Wallace, D.: Worlds in the Everett interpretation, Studies in the History and Philosopy of Modern Physics 33, 637–661, http://arxiv.org/abs/arXiv:quant-ph/0103092arXiv:quant-ph/0103092 (2002)
* [24] Wallace, D.: The quantum measurement problem: state of play, http://arxiv.org/abs/arXiv:0712.0149arXiv:0712.0149 (2007)
* [25] Weinberg, S.: Living in the Multiverse, http://arxiv.org/abs/arXiv:hep-th/0511037v1arXiv:hep-th/0511037v1
* [26] H.D. Zeh, Why Bohm’s Quantum Theory? http://arxiv.org/abs/arXiv:quant-ph/9812059arXiv:quant-ph/9812059
* [27] An Exchange of Letters in Physics Today on “Quantum Theory Without Observers”, February 1999, http://www.math.rutgers.edu/ oldstein/papers/qtwoe/qtwoe.htmlwww.math.rutgers.edu/$\sim$oldstein/papers/qtwoe/qtwoe.html
|
arxiv-papers
| 2009-03-26T16:58:37 |
2024-09-04T02:49:01.452557
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "I. Schmelzer",
"submitter": "Ilja Schmelzer",
"url": "https://arxiv.org/abs/0903.4657"
}
|
0903.5041
|
# Theory of incompressible MHD turbulence with scale-dependent alignment and
cross-helicity
J. J. Podesta and A. Bhattacharjee Center for Integrated Computation and
Analysis of Reconnection and Turbulence,University of New Hampshire, Durham,
New Hampshire, 03824
###### Abstract
A phenomenological anisotropic theory of MHD turbulence with nonvanishing
cross-helicity is constructed based on Boldyrev’s phenomenology and
probabilities $p$ and $q$ for fluctuations $\delta\bm{v}_{\perp}$ and
$\delta\bm{b}_{\perp}$ to be positively or negatively aligned. The positively
aligned fluctuations occupy a fractional volume $p$ and the negatively aligned
fluctuations occupy a fractional volume $q$. Guided by observations suggesting
that the normalized cross-helicity $\sigma_{c}$ and the probabilities $p$ and
$q$ are approximately scale invariant in the inertial range, a generalization
of Boldyrev’s theory is derived that depends on the three ratios
$w^{+}/w^{-}$, $\epsilon^{+}/\epsilon^{-}$, and $p/q$. It is assumed that the
cascade process for positively and negatively aligned fluctuations are both in
a state of critical balance and that the eddy geometries are scale invariant.
The theory reduces to Boldyrev’s original theory when $\sigma_{c}=0$,
$\epsilon^{+}=\epsilon^{-}$, and $p=q$ and extends the theory of Perez and
Boldyrev when $\sigma_{c}\neq 0$. The theory is also an anisotropic
generalization of the theory of Dobrowolny, Mangeney, and Veltri.
## I Introduction
Phenomenological theories of incompressible MHD turbulence that take into
account the anisotropy of the fluctuations with respect to the direction of
the mean magnetic field $\bm{B}_{0}$ were pioneered by Goldreich & Sridhar and
others in the 1990s. The influencial and somewhat controversial theory of
Goldreich & Sridhar (Goldreich and Sridhar, 1995, 1997) established the idea
that the timescale or coherence time for motions of a turbulent eddy parallel
and perpendicular to $\bm{B}_{0}$ must be equal to each other and this unique
timescale then defines the energy cascade time. This concept, called critical
balance, leads immediately to the perpendicular energy spectrum
$E(k_{\perp})\propto k_{\perp}^{-5/3}$ and to the scaling relation
$k_{\parallel}\propto k_{\perp}^{2/3}$ describing the anisotropy of the
turbulent eddies.
The decade following the publication of the paper by Goldreich & Sridhar in
1995 was a time when significant advances in computing power were brought to
bear on computational studies of MHD turbulence. Simulations of incompressible
MHD turbulence during this time showed that when the mean magnetic field is
strong, $B_{0}^{2}\gg(\delta b)^{2}$, the perpendicular energy spectrum
exhibits a power-law scaling closer to $k_{\perp}^{-3/2}$ than
$k_{\perp}^{-5/3}$ (Maron and Goldreich, 2001; Müller et al., 2003; Müller and
Grappin, 2005). Motivated by this result, Boldyrev modified the Goldreich &
Sridhar theory to explain the $k_{\perp}^{-3/2}$ power-law seen in simulations
(Boldyrev, 2005, 2006). A new concept that emerged in Boldyrev’s theory is the
scale dependent alignment of velocity and magnetic field fluctuations whereby
the angle $\theta$ formed by $\delta\bm{v}_{\perp}$ and $\delta\bm{b}_{\perp}$
scales like $\lambda_{\perp}^{1/4}$ in the inertial range. This alignment
effect weakens the nonlinear interactions and yields the perpendicular energy
spectrum $E(k_{\perp})\propto k_{\perp}^{-3/2}$. Evidence for this alignment
effect has been found in numerical simulations of incompressible MHD
turbulence (Mason et al., 2006, 2008) and in studies of solar wind data
(Podesta et al., 2008, 2009).
The phenomenological theory of Galtier et al. (2005) can also be used to
explain the observed $k_{\perp}^{-3/2}$ energy spectrum. Using a slightly
modified critical balance relation that retains the $k_{\parallel}\propto
k_{\perp}^{2/3}$ scaling of the Goldreich & Sridhar theory, their model admits
the $k_{\perp}^{-3/2}$ energy spectrum (as well as other solutions). However,
the theory of Galtier et al. (2005) does not include the scale dependent
alignment that arises in Boldyrev’s theory and, more importantly, is seen in
the solar wind (Podesta et al., 2008, 2009).
The theories discussed so far (Goldreich and Sridhar, 1995, 1997; Boldyrev,
2005, 2006; Galtier et al., 2005) all assume that the cross-helicity vanishes
and, therefore, these theories cannot be applied to solar wind turbulence.
When the cross-helicity of the turbulence is nonzero it is necessary to take
into account the cascades of both energy and cross-helicity. A generalization
of the Goldreich & Sridhar theory to turbulence with nonvanishing cross-
helicity, also called imbalanced turbulence, has been developed by Lithwick,
Goldreich, & Sridhar (Lithwick et al., 2007). Other theories of imbalanced
turbulence have been derived by Beresnyak & Lazarian (Beresnyak and Lazarian,
2008) and Chandran (Chandran, 2008; Chandran et al., 2009). However, none of
these theories contains the scale dependent alignment of velocity and magnetic
field fluctuations seen in the solar wind. Therefore, to develop a theory that
may be applicable to solar wind turbulence it is of interest to generalize
Boldyrev’s theory to incompressible MHD turbulence with non-vanishing cross-
helicity. An extension of Boldyrev’s theory to imbalanced turbulence has been
discussed by Perez & Boldyrev (Perez and Boldyrev, 2009). The purpose of the
present paper is to develop a theoretical framework which generalizes the
results of Perez & Boldyrev and is consistent with solar wind observations.
Our theory is founded, in part, on two new solar wind observations presented
in this paper. The first is the observation that the normalized cross-helicity
$\sigma_{c}$, the ratio of cross-helicity to energy, is _scale invariant_ in
the inertial range. The second is the observation that the probabilities $p$
and $q$ that fluctuations are positively or negatively aligned, respectively,
are also _scale invariant_ , that is, these quantities are approximately
constant in the inertial range. Experimental evidence for the scale invariance
of $\sigma_{c}$ comes from solar wind observations by Marsch and Tu (1990) and
also Figure 1 below, and from numerical simulations (Verma et al., 1996; Perez
and Boldyrev, 2009). Evidence for the scale invariance of $p$ and $q$ is shown
in Figure 2 below. Assuming these quantities are all scale invariant we deduce
expressions for the energy cascade rates and the rms fluctuations that
generalize the results in (Boldyrev, 2006) and (Perez and Boldyrev, 2009) and
are consistent with the concept of scale dependent alignment of velocity and
magnetic field fluctuations, a concept neglected in other phenomenological
theories (Galtier et al., 2005; Lithwick et al., 2007; Beresnyak and Lazarian,
2008; Chandran, 2008). The resulting theory, which is founded on the concept
of scale-invariance and grounded in solar wind observations, contains the
theories of Boldyrev (2006) and Perez and Boldyrev (2009) as special cases,
but opens up a broader range of physical possibilities.
Consistent with numerical simulations and solar wind observations, in our
approach the fluctuations at a given point may assume one of two possible
states referred to as positively aligned
$\delta\bm{v}_{\perp}\cdot\delta\bm{b}_{\perp}>0$ and negatively aligned
$\delta\bm{v}_{\perp}\cdot\delta\bm{b}_{\perp}<0$. Each state is characterized
by its own rms energy $v^{2}$, alignment angle $\theta$, and nonlinear
timescale $\tau$. Positively aligned fluctuations have a characteristic
spatial gradient which determines their nonlinear timescale and negatively
aligned fluctuations have a different spatial gradient which determines their
nonlinear timescale. These timescales are estimated from the nonlinear terms
in the MHD equations as described in sections 2 and 3.
Section 2 describes the geometries of velocity and magnetic field fluctuations
that are either aligned ‘$\uparrow$’ or anti-aligned ‘$\downarrow$’ and these
are used to form estimates of the nonlinear terms in the MHD equations. From
this foundation, estimates of the energy cascade times are constructed in
section 3 and the theory of the energy cascade process is developed in section
4. The summary and conclusions are presented in section 5.
## II Fluctuations in imbalanced turbulence
Consider velocity and magnetic field fluctuations measured between two points
separated by a distance $\lambda_{\perp}$ in the field perpendicular plane.
Let $\bm{v}$ and $\bm{b}$ denote the fluctuations in the plane perpendicular
to the local mean magnetic field, where $\bm{v}$ and $\bm{b}$ are both
measured in velocity units. Suppose that $\bm{v}$ and $\bm{b}$ are aligned
with some small angle $\theta>0$ and assume, as for Alfvén waves, that
$|\bm{v}|=|\bm{b}|$. Then $\bm{w}^{+}=\bm{v}+\bm{b}$ is nearly aligned with
$\bm{v}$ and $\bm{w}^{-}=\bm{v}-\bm{b}$ is nearly perpendicular to $\bm{v}$ as
sketched in Figure 1.
Figure 1: Geometry of the fluctuation vectors $\bm{v}$ and $\bm{b}$ for
positively aligned fluctuations (a) denoted by ‘$\uparrow$’ and negatively
aligned fluctuations (b) denoted by ‘$\downarrow$’. The gradient is
perpendicular to the velocity fluctuation $\bm{v}$. The magnitude of $\bm{v}$
for positively and negatively aligned fluctuations are $v_{\uparrow}$ and
$v_{\downarrow}$, respectively. The angles formed by $\bm{v}$ and $\bm{b}$,
$\theta_{\uparrow}$ and $\theta_{\downarrow}$, are both assumed to be small.
It follows from the identity $\bm{w}^{\pm}\times\bm{v}=\mp\bm{v}\times\bm{b}$
that
$w^{+}\sin\theta^{+}=v\sin\theta=w^{-}\sin\theta^{-},$ (1)
where $\theta^{+}$ is the angle formed by $\bm{w}^{+}$ and $\bm{v}$,
$\theta^{-}$ is the angle formed by $\bm{w}^{-}$ and $\bm{v}$, and $\theta$ is
the angle formed by $\bm{v}$ and $\bm{b}$. In addition,
$\theta^{+}+\theta^{-}=\pi/2$. Following Boldyrev (2006), suppose that the
gradient of the fluctuations is in the direction perpendicular to $\bm{v}$. In
this case,
$(\bm{w}^{-}\cdot\nabla)\simeq\frac{w^{-}\sin\theta^{-}}{\lambda_{\perp}}=\frac{v\sin\theta}{\lambda_{\perp}}$
(2)
and
$(\bm{w}^{+}\cdot\nabla)\simeq\frac{w^{+}\sin\theta^{+}}{\lambda_{\perp}}=\frac{v\sin\theta}{\lambda_{\perp}}.$
(3)
The time rate of change caused by nonlinear interactions is estimated from the
relations
$\frac{\partial}{\partial
t}\frac{|\bm{w}^{+}|^{2}}{2}\simeq\bm{w}^{+}\cdot(\bm{w}^{-}\cdot\nabla)\bm{w}^{+}$
(4)
and
$\frac{\partial}{\partial
t}\frac{|\bm{w}^{-}|^{2}}{2}\simeq\bm{w}^{-}\cdot(\bm{w}^{+}\cdot\nabla)\bm{w}^{-}.$
(5)
If $\bm{v}$ and $\bm{b}$ are aligned with some small angle $\theta$, then the
fluctuations are called “positively aligned” and denoted by ‘$\uparrow$’
(Figure 1a). Similarly, if $\bm{v}$ and $-\bm{b}$ are aligned with some small
angle $\theta$, then the fluctuations are called “negatively aligned” or
“anti-aligned” and denoted by ‘$\downarrow$’ (Figure 1b). For positively
aligned fluctuations, equations (2)–(5) imply
$\frac{\partial}{\partial
t}\frac{(w^{+}_{\uparrow})^{2}}{2}\simeq\frac{(w^{+}_{\uparrow})^{2}w^{-}_{\uparrow}\sin\theta^{-}_{\uparrow}}{\lambda_{\perp}}=\frac{(w^{+}_{\uparrow})^{2}v_{\uparrow}\sin\theta_{\uparrow}}{\lambda_{\perp}}$
(6)
and
$\frac{\partial}{\partial
t}\frac{(w^{-}_{\uparrow})^{2}}{2}\simeq\frac{(w^{-}_{\uparrow})^{2}w^{+}_{\uparrow}\sin\theta^{+}_{\uparrow}}{\lambda_{\perp}}=\frac{(w^{-}_{\uparrow})^{2}v_{\uparrow}\sin\theta_{\uparrow}}{\lambda_{\perp}},$
(7)
where $\theta_{\uparrow}$ is the angle formed by $\bm{v}$ and $\bm{b}$ and
quantities with the subscript $\uparrow$ describe positively aligned
fluctuations. It is clear from the middle term in equation (6) that the time
rate of change of $w^{+}_{\uparrow}$ depends on $w^{-}_{\uparrow}$, consistent
with the nonlinear terms in the MHD equations, although this dependence is not
immediately apparent in the last term in (6). For negatively aligned
fluctuations, equations (2)–(5) imply
$\frac{\partial}{\partial
t}\frac{(w^{+}_{\downarrow})^{2}}{2}\simeq\frac{(w^{+}_{\downarrow})^{2}w^{-}_{\downarrow}\sin\theta^{-}_{\downarrow}}{\lambda_{\perp}}=\frac{(w^{+}_{\downarrow})^{2}v_{\downarrow}\sin\theta_{\downarrow}}{\lambda_{\perp}}$
(8)
and
$\frac{\partial}{\partial
t}\frac{(w^{-}_{\downarrow})^{2}}{2}\simeq\frac{(w^{-}_{\downarrow})^{2}w^{+}_{\downarrow}\sin\theta^{+}_{\downarrow}}{\lambda_{\perp}}=\frac{(w^{-}_{\downarrow})^{2}v_{\downarrow}\sin\theta_{\downarrow}}{\lambda_{\perp}},$
(9)
where $\theta_{\downarrow}$ is the angle formed by $\bm{v}$ and $-\bm{b}$ and
quantities with the subscript $\downarrow$ describe negatively aligned
fluctuations. Here, $0<\theta_{\uparrow}<\pi/2$ and
$0<\theta_{\downarrow}<\pi/2$.
In general, the fluctuations $\bm{v}$ and $\bm{b}$ observed at any point
$(\bm{x},t)$ are either positively or negatively aligned. For a point
$(\bm{x},t)$ picked at random, let $p$ and $q$ be the probabilities the
alignment is positive or negative, respectively ($p+q=1$). Then, on average,
$\frac{\partial}{\partial
t}\frac{(\tilde{w}^{+})^{2}}{2}\simeq\frac{1}{\lambda_{\perp}}\big{[}p(w^{+}_{\uparrow})^{2}v_{\uparrow}\sin\theta_{\uparrow}+q(w^{+}_{\downarrow})^{2}v_{\downarrow}\sin\theta_{\downarrow}\big{]}$
(10)
and
$\frac{\partial}{\partial
t}\frac{(\tilde{w}^{-})^{2}}{2}\simeq\frac{1}{\lambda_{\perp}}\big{[}p(w^{-}_{\uparrow})^{2}v_{\uparrow}\sin\theta_{\uparrow}+q(w^{-}_{\downarrow})^{2}v_{\downarrow}\sin\theta_{\downarrow}\big{]},$
(11)
where the rms values $\tilde{w}^{\pm}$ are defined by
$\displaystyle(\tilde{w}^{+})^{2}$
$\displaystyle=p(w_{\uparrow}^{+})^{2}+q(w_{\downarrow}^{+})^{2},$ (12)
$\displaystyle(\tilde{w}^{-})^{2}$
$\displaystyle=p(w_{\uparrow}^{-})^{2}+q(w_{\downarrow}^{-})^{2}.$ (13)
The following relations also hold. For a positively aligned fluctuation,
assuming $|\bm{v}|=|\bm{b}|$,
$\displaystyle\bm{w}_{\uparrow}^{+}\cdot\bm{w}_{\uparrow}^{+}$
$\displaystyle=2v_{\uparrow}^{2}(1+\cos\theta_{\uparrow}),$ (14)
$\displaystyle\bm{w}_{\uparrow}^{-}\cdot\bm{w}_{\uparrow}^{-}$
$\displaystyle=2v_{\uparrow}^{2}(1-\cos\theta_{\uparrow}),$ (15)
and $w^{+}_{\uparrow}w^{-}_{\uparrow}=2v_{\uparrow}^{2}\sin\theta_{\uparrow}$.
The energy of a positively aligned fluctuation is $v_{\uparrow}^{2}$. For a
negatively aligned fluctuation
$\displaystyle\bm{w}_{\downarrow}^{+}\cdot\bm{w}_{\downarrow}^{+}$
$\displaystyle=2v_{\downarrow}^{2}(1-\cos\theta_{\downarrow}),$ (16)
$\displaystyle\bm{w}_{\downarrow}^{-}\cdot\bm{w}_{\downarrow}^{-}$
$\displaystyle=2v_{\downarrow}^{2}(1+\cos\theta_{\downarrow}),$ (17)
and
$w^{+}_{\downarrow}w^{-}_{\downarrow}=2v_{\downarrow}^{2}\sin\theta_{\downarrow}$.
The energy of a negatively aligned fluctuation is $v_{\downarrow}^{2}$. Thus,
the rms values (12) and (13) are
$\displaystyle(\tilde{w}^{+})^{2}$
$\displaystyle=2\big{[}pv_{\uparrow}^{2}(1+\cos\theta_{\uparrow})+qv_{\downarrow}^{2}(1-\cos\theta_{\downarrow})\big{]},$
(18) $\displaystyle(\tilde{w}^{-})^{2}$
$\displaystyle=2\big{[}pv_{\uparrow}^{2}(1-\cos\theta_{\uparrow})+qv_{\downarrow}^{2}(1+\cos\theta_{\downarrow})\big{]}.$
(19)
If the angles are small, $\theta_{\uparrow}\ll 1$ and $\theta_{\downarrow}\ll
1$, then the small parameter $\theta$ can be used to order the terms in
equations (18) and (19) so that to leading order
$(\tilde{w}^{+})^{2}\simeq
4v_{\uparrow}^{2}p\qquad\mbox{and}\qquad(\tilde{w}^{-})^{2}\simeq
4v_{\downarrow}^{2}q,$ (20)
where $p+q=1$. This may be derived as follows. In equations (18) and (19)
assume that the angles are both small and then substitute $1+\cos\theta\simeq
2$ and $1-\cos\theta=2\sin^{2}(\theta/2)$ to obtain
$(\tilde{w}^{+})^{2}\simeq
4\big{[}pv_{\uparrow}^{2}+qv_{\downarrow}^{2}\sin^{2}(\theta_{\downarrow}/2)\big{]}\\\
$ (21)
and
$(\tilde{w}^{-})^{2}\simeq
4\big{[}pv_{\uparrow}^{2}\sin^{2}(\theta_{\uparrow}/2)+qv_{\downarrow}^{2}\big{]}.$
(22)
As $\lambda_{\perp}\rightarrow 0$, both $\theta_{\uparrow}\rightarrow 0$ and
$\theta_{\downarrow}\rightarrow 0$ and, therefore, to first order, the terms
proportional to $\sin^{2}(\theta)$ may be neglected. Alternatively, note that
$\bigg{(}\frac{\tilde{w}^{+}}{\tilde{w}^{-}}\bigg{)}^{2}\simeq\frac{(pv_{\uparrow}^{2}/qv_{\downarrow}^{2})+\sin^{2}(\theta_{\downarrow}/2)}{(pv_{\uparrow}^{2}/qv_{\downarrow}^{2})\sin^{2}(\theta_{\uparrow}/2)+1}.$
(23)
As discussed below, solar wind observations show that this quantity is
approximately constant in the inertial range. Now, as
$\theta_{\uparrow}\rightarrow 0$ and $\theta_{\downarrow}\rightarrow 0$ the
only way that this can remain constant is if
$pv_{\uparrow}^{2}/qv_{\downarrow}^{2}$ is bounded away from zero and
$\bigg{(}\frac{\tilde{w}^{+}}{\tilde{w}^{-}}\bigg{)}^{2}\simeq\frac{pv_{\uparrow}^{2}}{qv_{\downarrow}^{2}}.$
(24)
This justifies the approximation in Eqn (20).
Equation (20) shows that at a given scale $\lambda_{\perp}$ the total energy
$[(\tilde{w}^{+})^{2}+(\tilde{w}^{-})^{2}]/4$ is partitioned into two parts,
the energy $(\tilde{w}^{+})^{2}/4$ associated with positive alignment and the
energy $(\tilde{w}^{-})^{2}/4$ associated with negative alignment. The
normalized cross-helicity $\sigma_{c}$ is defined as the ratio of the cross-
helicity to the energy at a given scale and can be written
$\sigma_{c}=\frac{(\tilde{w}^{+})^{2}-(\tilde{w}^{-})^{2}}{(\tilde{w}^{+})^{2}+(\tilde{w}^{-})^{2}}.$
(25)
For small angles, equations (10) and (11) become, to leading order,
$\displaystyle\frac{\partial}{\partial t}\frac{(\tilde{w}^{+})^{2}}{2}$
$\displaystyle\simeq\frac{4pv_{\uparrow}^{3}\theta_{\uparrow}}{\lambda_{\perp}},$
(26) $\displaystyle\frac{\partial}{\partial t}\frac{(\tilde{w}^{-})^{2}}{2}$
$\displaystyle\simeq\frac{4qv_{\downarrow}^{3}\theta_{\downarrow}}{\lambda_{\perp}}.$
(27)
To express these in terms of the rms values $\tilde{w}^{\pm}$, eliminate
$v_{\uparrow}$ and $v_{\downarrow}$ using equation (20). This yields
$\displaystyle\frac{\partial}{\partial t}\frac{(\tilde{w}^{+})^{2}}{2}$
$\displaystyle\simeq\frac{(\tilde{w}^{+})^{3}\theta_{\uparrow}}{2\lambda_{\perp}p^{1/2}},$
(28) $\displaystyle\frac{\partial}{\partial t}\frac{(\tilde{w}^{-})^{2}}{2}$
$\displaystyle\simeq\frac{(\tilde{w}^{-})^{3}\theta_{\downarrow}}{2\lambda_{\perp}q^{1/2}}.$
(29)
These estimates shall be used to derive the cascade times.
## III Energy cascade time
When nonlinear interactions are strong and a large number of Fourier modes are
excited, fluctuations occur continuously in time and space. During a time
$\tau$ the fractional change in the quantity $(\tilde{w}^{+})^{2}$ is, from
(28),
$\chi^{+}(\tau)\simeq\frac{(w^{+})^{3}\theta_{\uparrow}}{2\lambda_{\perp}p^{1/2}}\cdot\frac{2\tau}{(w^{+})^{2}}=\frac{w^{+}\theta_{\uparrow}\tau}{\lambda_{\perp}p^{1/2}},\qquad\tau\leq\tau^{+},$
(30)
where $\tau^{+}$ is the cascade time at the lengthscale $\lambda_{\perp}$ and
the tildes have been dropped. Similarly, the fractional change in the quantity
$(\tilde{w}^{-})^{2}$ is, from (29),
$\chi^{-}(\tau)\simeq\frac{(w^{-})^{3}\theta_{\downarrow}}{2\lambda_{\perp}q^{1/2}}\cdot\frac{2\tau}{(w^{-})^{2}}=\frac{w^{-}\theta_{\downarrow}\tau}{\lambda_{\perp}q^{1/2}},\qquad\tau\leq\tau^{-},$
(31)
where $\tau^{-}$ is the cascade time of $\tilde{w}^{-}$ and the tildes have
been dropped for brevity. Hereafter, the tildes will be omitted and $w^{+}$
and $w^{-}$ will always represent the rms values.
According to the definition of the energy cascade time, the fractional change
$\chi^{+}$ is of order unity when the interaction time $\tau$ is equal to the
cascade time $\tau^{+}$. Therefore, the relations (30) and (31) imply
$\tau^{+}\simeq\frac{\lambda_{\perp}p^{1/2}}{w^{+}\theta_{\uparrow}},\qquad\tau^{-}\simeq\frac{\lambda_{\perp}q^{1/2}}{w^{-}\theta_{\downarrow}}.$
(32)
By similar reasoning, equations (6) and (9) imply
$\tau_{\uparrow}\simeq\frac{\lambda_{\perp}}{2v_{\uparrow}\theta_{\uparrow}},\qquad\tau_{\downarrow}\simeq\frac{\lambda_{\perp}}{2v_{\downarrow}\theta_{\downarrow}}.$
(33)
Moreover, equations (32), (33), and (20) imply $\tau^{+}=\tau_{\uparrow}$ and
$\tau^{-}=\tau_{\downarrow}$. Thus, the energy cascade times for the rms
Elsasser amplitudes are equal to the energy cascade times for the positively
and negatively aligned fluctuations.
For balanced turbulence, $\sigma_{c}\rightarrow 0$, $w^{+}/w^{-}\rightarrow
1$, $p=q$, $\theta_{\uparrow}=\theta_{\downarrow}$, and the energy cascade
times (32) reduce to the cascade time in Boldyrev’s original theory (Boldyrev,
2006). For imbalanced turbulence, $\sigma_{c}\neq 0$, the cascade times (32)
are different from the cascade times
$\tau^{\pm}\sim\lambda_{\perp}/w^{\mp}\theta^{\mp}$ in the theory of Perez &
Boldyrev (Perez and Boldyrev, 2009). The theory presented here is different
from the theory of Perez & Boldyrev (Perez and Boldyrev, 2009) because the
latter theory does not take into account the existence of two separate types
of fluctuations, positively and negatively aligned, with separate
probabilities of occurrence $p$ and $q$. Taking this into account and also the
definitions of the rms amplitudes (12) and (13), it follows from the preceding
analysis that the timescales for the rms amplitudes take the form (32).
As pointed out by Kraichnan (Kraichnan, 1965), Dobrowolny, Mangeney, and
Veltri (Dobrowolny et al., 1980), and others, the energy cascade in MHD
turbulence occurs through collisions between Alfvén wavepackets propagating in
opposite directions along the mean magnetic field. In other words, it is the
interaction between $w^{+}$ and $w^{-}$ waves that causes the energy to
cascade to smaller scales in MHD turbulence. Consequently, the cascade time
for $w^{+}$, say, should depend on $w^{-}$. While it may appear from equations
(32)–(33) that the timescale for $w^{+}$ fluctuations depends only on $w^{+}$
and, therefore, the interaction with the $w^{-}$ waves is absent, this is not
true. The interactions are still present in the expressions (32) and (33)
through the dependence on the angles and other parameters as will be shown in
the next section.
## IV Theory of the energy cascade process
Assuming there is no direct injection of energy or cross-helicity within the
inertial range and there is no dissipation of energy or cross-helicity within
the inertial range, the energy cascade rate $\varepsilon$ and the cross-
helicity cascade rate $\varepsilon_{c}$ are scale-invariant in the inertial
range. It follows that the energy cascade rates for the two Elsasser variables
$\varepsilon^{\pm}=\varepsilon\pm\varepsilon_{c}$ are also scale-invariant.
The theory of the energy cascade process is based on Kolmogorov’s relations
$\frac{(w^{+})^{2}}{2\tau^{+}}=\varepsilon^{+}\qquad\mbox{and}\qquad\frac{(w^{-})^{2}}{2\tau^{-}}=\varepsilon^{-},$
(34)
where the non-zero constants $\varepsilon^{+}$ and $\varepsilon^{-}$ are the
energy cascade rates per unit mass for the two Elsasser variables $w^{+}$ and
$w^{-}$, respectively. These equations describe the conservation of energy
flux in $\bm{k}$-space (Fourier space). In addition to Kolmogorov’s relations,
there are two observational constraints that must be taken into account.
Solar wind observations show that the energy and cross-helicity spectra of the
turbulence follow approximately the same power law in the inertial range (Fig.
2)
Figure 2: Typical energy $E$ and cross-helicity spectra $C$ (trace spectra)
obtained using 3-second plasma velocity and magnetic field data from the Wind
spacecraft near the orbit of the earth at 1 AU. (a) An interval of highly
Alfvénic high-speed wind from 3 Jan 1995 09:00 to 8 Jan 1995 00:00, 4.625
days. (b) A weak high-speed stream embedded in low-speed wind; 24 Jul 1996
12:00 to 7 Aug 1996 00:00, 14 days. (c) The normalized cross-helicity
$\sigma_{c}=C/E$ as a function of frequency. The rapid change in $\sigma_{c}$
near the Nyquist frequency is at least partly caused by the FFT processing
techniques and may not be a real physical effect.
which implies that the normalized cross-helicity $\sigma_{c}$ is approximately
constant. In other words, the quantity $\sigma_{c}$ is approximately scale
invariant. Similar results have been found in simulations of incompressible
MHD turbulence (Verma et al., 1996; Perez and Boldyrev, 2009; Beresnyak and
Lazarian, 2008). In particular, the 3D simulations of Perez & Boldyrev (Perez
and Boldyrev, 2009) indicate that the perpendicular Elsasser spectra are
proportional to each other in Fourier space. Solar wind observations also
suggest that the probabilities $p$ and $q$ are approximately scale invariant
as shown in Fig. 3. These observations will now be taken into account in the
theory.
Assuming $\sigma_{c}$ and $p$ are both scale invariant quantities, then
$w^{+}/w^{-}$, $v_{\uparrow}/v_{\downarrow}$, $\tau^{+}/\tau^{-}$, and
$\theta_{\downarrow}/\theta_{\uparrow}$ are scale invariant by equations (25),
(24), (34), and (32), respectively. In all, there are six different scale
invariant ratios in the theory
$\frac{w^{+}}{w^{-}},\quad\frac{\varepsilon^{+}}{\varepsilon^{-}},\quad\frac{p}{q},\quad\frac{v_{\uparrow}}{v_{\downarrow}},\quad\frac{\tau^{+}}{\tau^{-}},\quad\frac{\theta_{\uparrow}}{\theta_{\downarrow}}.$
(35)
At most, only three of these are independent, say, the first three. Equations
(24), (34), and (32) imply
$\displaystyle\frac{v_{\uparrow}}{v_{\downarrow}}=\sqrt{\frac{q}{p}}\cdot\frac{w^{+}}{w^{-}},$
(36)
$\displaystyle\frac{\tau^{+}}{\tau^{-}}=\left(\frac{w^{+}}{w^{-}}\right)^{2}\frac{\varepsilon^{-}}{\varepsilon^{+}},$
(37)
$\displaystyle\frac{\theta_{\downarrow}}{\theta_{\uparrow}}=\sqrt{\frac{q}{p}}\left(\frac{w^{+}}{w^{-}}\right)^{3}\frac{\varepsilon^{-}}{\varepsilon^{+}}.$
(38)
Therefore, the six scale invariant ratios (35) can all be expressed in terms
of the first three $w^{+}/w^{-}$, $\varepsilon^{+}/\varepsilon^{-}$, and
$p/q$.
Figure 3: The probabilities $p$ and $q$ obtained from solar wind data by
integrating the observed probability density function for the angle $\theta$
from 0 to $\pi/2$ and from $\pi/2$ to $\pi$, respectively. The data was
acquired by the Wind spacecraft between 8 Jan 1997 and 9 June 1997 and
analyzed using the techniques described in (Podesta et al., 2009). Examples of
the probability density functions can be found in (Podesta et al., 2009).
To be able to solve Kolmogorov’s relations (34) for $w^{\pm}$ it is necessary
to express the alignment angle $\theta_{\uparrow}$ in terms of $w^{\pm}$. In
general, $\theta_{\uparrow}$ can depend on $w^{+}$, $w^{-}$, the Alfvén speed
$v_{A}$, the lengthscale $\lambda_{\perp}$, the cascade rates
$\varepsilon^{+}$ and $\varepsilon^{-}$, and the probabilities $p$ and $q$. By
dimensional analysis, $\theta_{\uparrow}$ must be a function of the following
six dimensionless quantities
$\frac{w^{+}}{v_{A}},\quad\frac{w^{-}}{v_{A}},\quad\frac{\varepsilon^{+}\lambda_{\perp}}{v_{A}^{3}},\quad\frac{\varepsilon^{-}\lambda_{\perp}}{v_{A}^{3}},\quad
p,\quad q.$ (39)
Moreover, $\theta_{\uparrow}$ must change into $\theta_{\downarrow}$ when
$w^{+}$, $\varepsilon^{+}$, and $p$ are interchanged with $w^{-}$,
$\varepsilon^{-}$, and $q$, respectively, to be consistent with the nonlinear
terms (28) and (29). For a theory composed of power law functions, the only
forms that satisfy all these requirements are
$\displaystyle\theta_{\uparrow}$
$\displaystyle\propto\bigg{(}\frac{w^{+}}{v_{A}}\bigg{)}^{\\!\alpha}\bigg{(}\frac{w^{-}}{v_{A}}\bigg{)}^{\\!\beta}\bigg{(}\frac{\varepsilon^{+}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!\gamma}\bigg{(}\frac{\varepsilon^{-}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!\delta}p^{\mu}q^{\nu},$
(40) $\displaystyle\theta_{\downarrow}$
$\displaystyle\propto\bigg{(}\frac{w^{-}}{v_{A}}\bigg{)}^{\\!\alpha}\bigg{(}\frac{w^{+}}{v_{A}}\bigg{)}^{\\!\beta}\bigg{(}\frac{\varepsilon^{-}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!\gamma}\bigg{(}\frac{\varepsilon^{+}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!\delta}q^{\mu}p^{\nu},$
(41)
where $\alpha$, $\beta$, $\gamma$, $\delta$, $\mu$, and $\nu$ are constants
that must be determined by the theory. In addition, there is a leading
coefficient which is omitted.
The substitution of (40) and (41) into equation (38) yields $\beta=\alpha+3$,
$\delta=\gamma-1$, and $\nu=\mu-1/2$. The parameters are further constrained
by considering the geometry of the “turbulent eddies” associated with the
fluctuations $v_{\uparrow}$ and $v_{\downarrow}$. The parallel correlation
length is defined by $\lambda_{\parallel}^{\uparrow}=v_{A}\tau_{\uparrow}$ and
the correlation length in the direction of the velocity fluctuation is
$\bm{\xi}_{\uparrow}=\bm{v}_{\uparrow}\tau_{\uparrow}$. Similarly, the
correlation lengths for negatively aligned fluctuations are
$\lambda_{\parallel}^{\downarrow}=v_{A}\tau_{\downarrow}$ and
$\xi_{\downarrow}=v_{\downarrow}\tau_{\downarrow}$. In the plane perpendicular
to the local mean magnetic field $\bm{\xi}$ is parallel to $\bm{v}$, the
gradient direction is perpendicular to $\bm{v}$ with lengthscale
$\lambda_{\perp}$, and the eddy dimensions are $\xi\times\lambda_{\perp}$. The
dimension parallel to the mean magnetic field is $\lambda_{\parallel}$. Hence,
in physical space the turbulent eddies can be visualized as three-dimensional
structures with dimensions
$\lambda_{\perp}\times\xi\times\lambda_{\parallel}$.
The coherence times for longitudinal and transverse motions of the eddy must
be equal to each other and also to the cascade time. This is the critical
balance condition of Goldreich and Sridhar which is also implicit in the work
of Higdon Higdon (1984). Equation (33) and the definitions of the correlation
lengths in the last paragraph immediately yield the critical balance condition
$\tau_{\uparrow}=\frac{\lambda_{\parallel}^{\uparrow}}{v_{A}}=\frac{\xi_{\uparrow}}{v_{\uparrow}}\simeq\frac{\lambda_{\perp}}{2v_{\uparrow}\theta_{\uparrow}}$
(42)
with a similar condition for the negatively aligned fluctuations
$\tau_{\downarrow}=\frac{\lambda_{\parallel}^{\downarrow}}{v_{A}}=\frac{\xi_{\downarrow}}{v_{\downarrow}}\simeq\frac{\lambda_{\perp}}{2v_{\downarrow}\theta_{\downarrow}}.$
(43)
Now consider the eddy geometry. When the mean magnetic field is strong enough
that $w^{\pm}/v_{A}<1$, then $\lambda_{\parallel}>\xi>\lambda_{\perp}$ and the
eddies are elongated in the parallel direction. The condition
$w^{\pm}/v_{A}<1$ is assumed hereafter. Equation (42) shows that the aspect
ratio in the field perpendicular plane is
$\phi_{\uparrow}=\lambda_{\perp}/\xi_{\uparrow}=2\theta_{\uparrow}$ and the
aspect ratio in the parallel direction is, from equations (42) and (20),
$\psi_{\uparrow}=\frac{\xi_{\uparrow}}{\lambda_{\parallel}^{\uparrow}}=\frac{v_{\uparrow}}{v_{A}}=\frac{w^{+}}{2v_{A}p^{1/2}}.$
(44)
The two aspect ratios will scale in the same way if
$\phi_{\uparrow}/\psi_{\uparrow}$ is scale invariant. This implies that
$\alpha=-1$ and $\gamma=1/2$. The assumption that the ratio
$\phi_{\uparrow}/\psi_{\uparrow}$ is scale invariant is different from
Boldyrev’s original approach in which he assumed that the alignment angles in
and out of the field perpendicular plane are simultaneously minimized.
Nevertheless, our assumption retains the spirit of Boldyrev’s original theory
which implies the geometry of turbulent fluctuations are scale-invariant.
Solving Kolmogorov’s relation (34) using (32), (40), (41), and the parameter
values obtained so far, one finds
$\frac{w^{\pm}}{v_{A}}\simeq\bigg{(}\frac{w^{+}}{w^{-}}\bigg{)}^{\\!\pm
1/2}\bigg{(}\frac{\varepsilon^{-}}{\varepsilon^{+}}\bigg{)}^{\\!\pm
1/8}\bigg{(}\frac{\varepsilon^{\pm}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!1/4}(pq)^{-\nu/4}$
(45)
and the total energy cascade rate
$\varepsilon=(\varepsilon^{+}+\varepsilon^{-})/2$ is
$\varepsilon=\frac{(w^{+}w^{-})^{2}}{4v_{A}\lambda_{\perp}}\Bigg{(}\sqrt{\frac{\varepsilon^{+}}{\varepsilon^{-}}}+\sqrt{\frac{\varepsilon^{-}}{\varepsilon^{+}}}\Bigg{)}(pq)^{\nu}.$
(46)
The total energy at scale $\lambda_{\perp}$ is
$\frac{(w^{+})^{2}+(w^{-})^{2}}{4}=\frac{w^{+}w^{-}}{4}\bigg{(}\frac{w^{+}}{w^{-}}+\frac{w^{-}}{w^{+}}\bigg{)}\equiv
v^{2}.$ (47)
Therefore, the energy cascade rate can be written
$\epsilon=\frac{4v^{4}}{v_{A}\lambda_{\perp}}\Bigg{(}\sqrt{\frac{\varepsilon^{+}}{\varepsilon^{-}}}+\sqrt{\frac{\varepsilon^{-}}{\varepsilon^{+}}}\Bigg{)}\bigg{(}\frac{w^{+}}{w^{-}}+\frac{w^{-}}{w^{+}}\bigg{)}^{\\!-2}(pq)^{\nu}.$
(48)
Assuming the rms energy $v^{2}$ at scale $\lambda_{\perp}$ is held constant,
the terms on the right-hand side describe the dependence of the energy cascade
rate on the ratios $\varepsilon^{+}/\varepsilon^{-}$ and $w^{+}/w^{-}$. The
value of $\nu$ may be determined by comparison with experiment or possibly by
further physical considerations. This parameter does not affect the inertial
range scaling laws and is left undetermined for the moment.
At this point it is of interest to return to the expressions (32) for the
cascade times and ask: How do the cascade times depend on the rms Elsasser
amplitudes? Using the parameter values obtained previously, equation (40)
becomes
$\theta_{\uparrow}\sim\bigg{(}\frac{w^{+}}{v_{A}}\bigg{)}^{\\!-1}\bigg{(}\frac{w^{-}}{v_{A}}\bigg{)}^{\\!2}\bigg{(}\frac{\varepsilon^{+}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!1/2}\bigg{(}\frac{\varepsilon^{-}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!-1/2}p^{\nu+1/2}q^{\nu}$
(49)
and the substitution of this result into equation (32) yields
$\tau^{+}\simeq\frac{\lambda_{\perp}}{v_{A}}\bigg{(}\frac{v_{A}}{w^{-}}\bigg{)}^{\\!2}\bigg{(}\frac{\varepsilon^{-}}{\varepsilon^{+}}\bigg{)}^{\\!1/2}(pq)^{-\nu}.$
(50)
A similar expression holds for $\tau^{-}$ so that the ratio
$\tau^{+}/\tau^{-}$ satisfies (37). Ignoring scale invariant factors, the
preceding equation shows that
$\tau^{+}\propto\frac{\lambda_{\perp}v_{A}}{(w^{-})^{2}}\qquad\mbox{and}\qquad\tau^{-}\propto\frac{\lambda_{\perp}v_{A}}{(w^{+})^{2}}.$
(51)
In this form, the angle dependence has been eliminated. Note that the simple
estimate $\tau^{+}\sim\lambda_{\perp}/w^{-}$ suggested by the nonlinear term
in the MHD equations is modified by the factor $v_{A}/w^{-}$ which accounts
for the weakening of nonlinear interactions caused by scale dependent
alignment. The presence of this algebraic factor is one of the hallmarks of
Boldyrev’s original (2006) theory which is generalized here to imbalanced
turbulence. Remarkably, the relations (51) are identical to those in the
isotropic theory of imbalanced turbulence developed by Dobrowolny, Mangeney,
and Veltri; see equation (10) in (Dobrowolny et al., 1980). Recall that
Dobrowolny, Mangeney, and Veltri concluded from their expressions for the
cascade times that steady state turbulence with nonvanishing cross-helicity is
impossible. On the contrary, the theory presented here allows such a steady
state because the additional coefficients shown in (50) but not (51) maintain
the relation (37) even when $\varepsilon^{+}\neq\varepsilon^{-}$. Thus, the
theory presented here is also a generalization of the theory of Dobrowolny,
Mangeney, and Veltri (Dobrowolny et al., 1980).
A remark about the timescales in the theory should be mentioned. If
$w^{+}>w^{-}$, then equation (37) implies it is possible that
$\tau^{+}<\tau^{-}$ since there is nothing in the theory that prevents this.
That is, the energy of the more energetic Elsasser species may be transferred
to smaller scales in less time than the energy of the less energetic Elsasser
species. This is not inconsistent with dynamic alignment, a well known effect
seen in simulations of decaying incompressible MHD turbulence where the
minority species usually decays more rapidly than the dominant species causing
the magnitude of the normalized cross-helicity to increase with time
(Dobrowolny et al., 1980; Matthaeus et al., 1983; Matthaeus and Montgomery,
1984; Pouquet et al., 1986).
In freely decaying turbulence, dynamic alignment occurs whenever the total
energy decays more rapidly than the cross-helicity, that is,
$\varepsilon>|\varepsilon_{c}|$, where the cascade rate of cross-helicity
$\varepsilon_{c}$ may be positive or negative. From the relations
$\varepsilon>0$ and $\varepsilon^{\pm}=\varepsilon\pm\varepsilon_{c}$, it
follows that dynamic alignment occurs if and only if $\varepsilon^{+}>0$ and
$\varepsilon^{-}>0$. If $w^{+}>w^{-}$, it is not necessary that
$\tau^{+}>\tau^{-}$, only that
$\frac{\tau^{+}}{\tau^{-}}>\frac{\varepsilon^{-}}{\varepsilon^{+}},$ (52)
as can be seen from equation (37). Therefore, even though the relation
$\tau^{+}<\tau^{-}$ may seem counter-intuitive, it is not inconsistent with
dynamic alignment.
## V Summary and Conclusions
Observations of scale dependent alignment of velocity and magnetic field
fluctuations $\delta\bm{v}_{\perp}$ and $\delta\bm{b}_{\perp}$ in the solar
wind suggest that this effect must be included in any theory of solar wind
turbulence (Podesta et al., 2008, 2009). Perez and Boldyrev (2009) have
recently discussed a theory of imbalanced turbulence that includes scale
dependent alignment of the fluctuations $\delta\bm{v}_{\perp}$ and
$\delta\bm{b}_{\perp}$ in the inertial range. We have extended the Perez-
Boldyrev theory by including the probabilities $p$ and $q$ which solar wind
observations indicate are not necessarily equal. Operationally, the
probabilities $p$ and $q$ may be defined as follows. Suppose space is covered
by a uniform cartesian grid or three dimensional mesh. At each grid-point one
may compute the fluctuations $\delta\bm{v}_{\perp}$ and $\delta\bm{b}_{\perp}$
and the angle between them $\theta$. If the angle lies in the range
$0<\theta<\pi/2$, then the fluctuation is positively aligned and if
$\pi/2<\theta<\pi$, then the fluctuation is negatively aligned. By counting
the number of positively and negatively aligned fluctuations in a large volume
$V$, much larger than the lengthscales of the turbulent eddies, the
probabilities $p$ and $q$ may be defined as the fractional numbers of
positively and negatively aligned fluctuations in the volume $V$.
The phenomenological theory developed in this paper was guided primarily by
two new solar wind observations. It should be noted that both of these solar
wind observations are necessary for the development of the theory. At first
glance, it may seem that the condition $\sigma_{c}=$ const implies that $p$
and $q$ are both constant. Or that these two conditions are somehow
equivalent. However, the relation $(w^{+}/w^{-})^{2}\simeq
pv_{\uparrow}^{2}/qv_{\downarrow}^{2}$, equation (24), shows that $p/q$ can
vary with the lengthscale even if $w^{+}/w^{-}$ is constant. Therefore, it is
essential to have separate observations of the scale invariance of
$\sigma_{c}$ and the scale invariance of $p$ and $q$ to support the
theoretical framework developed here.
In summary, using estimates of the cascade times derived from the nonlinear
terms in the incompressible MHD equations and two new observational
constraints derived from studies of solar wind data, we have constructed a
generalization of Boldyrev’s theory (Boldyrev, 2006) that depends on the three
parameters $w^{+}/w^{-}$, $\varepsilon^{+}/\varepsilon^{-}$, and $p/q$. The
theory reduces to the original theory of Boldyrev (2006) when $w^{+}=w^{-}$,
$\epsilon^{+}=\epsilon^{-}$, and $p=q$ since in this limit
$\theta_{\uparrow}=\theta_{\downarrow}$ and the cascade times (32) become
equal to those of Boldyrev (2006). For imbalanced turbulence $w^{+}\neq
w^{-}$, $p\neq q$, and the theory predicts the scaling laws
$w^{\pm}\propto\lambda_{\perp}^{1/4}$,
$\theta_{\uparrow\downarrow}\propto\lambda_{\perp}^{1/4}$, and
$\lambda_{\parallel}^{\pm}\propto\lambda_{\perp}^{1/2}$. Interestingly, the
scaling laws for balanced and imbalanced turbulence are the same. The
perpendicular energy spectrum defined by $k_{\perp}E^{\pm}\sim|w^{\pm}|^{2}$
has the inertial range scaling $E^{\pm}\propto k_{\perp}^{-3/2}$ with
$\frac{E^{+}}{E^{-}}=\bigg{(}\frac{w^{+}}{w^{-}}\bigg{)}^{\\!2}=\frac{1+\sigma_{c}}{1-\sigma_{c}}=\mbox{const}.$
(53)
The theory assumes that the cascades for positively and negatively aligned
fluctuations are both in a state of critical balance (42), although they are
governed by different timescales, and that the eddy geometry is scale
invariant. The positively aligned fluctuations occupy a fractional volume $p$
and the negatively aligned fluctuations occupy a fractional volume $q$ so that
the energy cascade rate is
$\varepsilon=p\frac{v_{\uparrow}^{2}}{\tau_{\uparrow}}+q\frac{v_{\downarrow}^{2}}{\tau_{\downarrow}}$
(54)
or, equivalently,
$\varepsilon=\frac{(w^{+})^{2}}{4\tau^{+}}+\frac{(w^{-})^{2}}{4\tau^{-}}.$
(55)
In the discussion following equation (35) it was shown that at most three of
the ratios $w^{+}/w^{-}$, $\varepsilon^{+}/\varepsilon^{-}$, and $p/q$ can be
independent. However, the two ratios $w^{+}/w^{-}$ and
$\varepsilon^{+}/\varepsilon^{-}$ cannot be independent since in the case of
homogeneous steady-state turbulence $w^{+}=w^{-}$ implies
$\varepsilon^{+}=\varepsilon^{-}$ and vice versa. This is because the
injection of cross-helicity into the system, $\varepsilon_{c}\neq 0$ or
$\varepsilon^{+}\neq\varepsilon^{-}$, will create a nonzero cross-helicity
spectrum and a cascade of cross-helicity from large to small scales which
implies a net accumulation of cross-helicity within the volume
($\sigma_{c}\neq 0$). Hence, at most two of the ratios and $w^{+}/w^{-}$ and
$p/q$ are independent. Whether $p/q$ can be expressed in terms of
$w^{+}/w^{-}$ and $\varepsilon^{+}/\varepsilon^{-}$ is an open question.
###### Acknowledgements.
We are grateful to S. Boldyrev for valuable comments on an earlier version of
the manuscript and to Pablo Mininni and Jean Perez for helpful discussions.
This research is supported by DOE grant number DE-FG02-07ER46372, NASA grant
number NNX06AC19G, and NSF. Additional support for John Podesta comes from the
NASA Solar and Heliospheric Physics Program and the NSF SHINE Program.
## References
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* Boldyrev (2005) S. Boldyrev, Astrophys. J. Lett. 626, L37 (2005).
* Boldyrev (2006) S. Boldyrev, Phys. Rev. Lett. 96, 115002 (2006).
* Mason et al. (2006) J. Mason, F. Cattaneo, and S. Boldyrev, Phys. Rev. Lett. 97, 255002 (2006).
* Mason et al. (2008) J. Mason, F. Cattaneo, and S. Boldyrev, Phys. Rev. E. 77, 036403 (2008).
* Podesta et al. (2008) J. J. Podesta, A. Bhattacharjee, B. D. G. Chandran, M. L. Goldstein, and D. A. Roberts, in _Particle Acceleration and Transport in the Heliosphere and Beyond_ (2008), vol. 1039 of _AIP Conference Series_ , pp. 81–86.
* Podesta et al. (2009) J. J. Podesta, B. D. G. Chandran, A. Bhattacharjee, D. A. Roberts, and M. L. Goldstein, Journal of Geophysical Research (Space Physics) 114, A01107 (2009).
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* Perez and Boldyrev (2009) J. C. Perez and S. Boldyrev, Phys. Rev. Lett. 102, 025003 (2009), eprint 0807.2635.
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* Dobrowolny et al. (1980) M. Dobrowolny, A. Mangeney, and P. Veltri, Phys. Rev. Lett. 45, 144 (1980).
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|
arxiv-papers
| 2009-03-29T12:36:10 |
2024-09-04T02:49:01.475453
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. J. Podesta and A. Bhattacharjee",
"submitter": "John Podesta",
"url": "https://arxiv.org/abs/0903.5041"
}
|
0903.5072
|
# Chemical Engineering Science 65 (2010) 2310-2324
Asymptotology of Chemical Reaction Networks
A. N. Gorban ag153@le.ac.uk University of Leicester, UK Corresponding author:
University of Leicester, LE1 7RH, UK O. Radulescu ovidiu.radulescu@univ-
rennes1.fr IRMAR, UMR 6625, University of Rennes 1, Campus de Beaulieu, 35042
Rennes, France A. Y. Zinovyev andrei.zinovyev@curie.fr Institut Curie, U900
INSERM/Curie/Mines ParisTech, 26 rue d’Ulm, F75248, Paris, France
###### Abstract
The concept of the limiting step is extended to the asymptotology of
multiscale reaction networks. Complete theory for linear networks with well
separated reaction rate constants is developed. We present algorithms for
explicit approximations of eigenvalues and eigenvectors of kinetic matrix.
Accuracy of estimates is proven. Performance of the algorithms is demonstrated
on simple examples. Application of algorithms to nonlinear systems is
discussed.
###### keywords:
Reaction network , asymptotology , dominant system , limiting step ,
multiscale asymptotic , model reduction
###### PACS:
64.60.aq , 82.40.Qt , 82.39.Fk , 82.39.Rt 87.15.R- , 89.75.Fb
## 1 Introduction
Most of mathematical models that really work are simplifications of the basic
theoretical models and use in the backgrounds an assumption that some terms
are big, and some other terms are small enough to neglect or almost neglect
them. The closer consideration shows that such a simple separation on “small”
and “big” terms should be used with precautions, and special culture was
developed. The name “asymptotology” for this direction of science was proposed
by Kruskal (1963), but fundamental research in this direction are much older,
and many fundamental approaches were developed by I. Newton (Newton
polyhedron, and many other things).
Following Kruskal (1963), asymptotology is “the art of describing the behavior
of a specified solution (or family of solutions) of a system in a limiting
case. … The art of asymptotology lies partly in choosing fruitful limiting
cases to examine … The scientific element in asymptotology resides in the
nonarbitrariness of the asymptotic behavior and of its description, once the
limiting case has been decided upon.”
Asymptotic behavior of rational functions of several positive variables
$k_{i}>0$ gives us a toy-example. Let
$R(k_{1},\ldots k_{n})=P(k_{1},\ldots k_{n})/Q(k_{1},\ldots k_{n})$
be such a function and $P,Q$ be polynomials. To derive fruitful limiting cases
we consider logarithmic straight lines $\ln k_{i}=\theta_{i}\xi$ and study
asymptotical behavior of $R$ for $\xi\to\infty$. In this asymptotics, for
almost every vector $(\theta_{i})$ (outside several hyperplanes) there exists
such a dominant monomial $R_{\infty}(k)=A\prod_{i}k_{i}^{\alpha_{i}}$ that
$R=R_{\infty}+o(R_{\infty})$. The function that associates a monomial with
vector $(\theta_{i})$ is piecewise constant: it is constant inside some
polyhedral cones.
Implicit functions given by equations which depend on parameters provide
plenty of more interesting examples, especially in the case when the implicit
function theorem is not applicable. Some analytical examples are presented by
Andrianov & Manevitch (2002) and White (2006). Introduction of algebraic
backgrounds and special software is provided by Greuel & Pfister (2002).
For a difficult problem, analysis of eigenvalues and eigenvectors of non-
symmetric matrices, Vishik & Ljusternik (1960) studied asymptotic behavior of
spectra and spectral projectors along the logarithmic straight lines in the
space of matrices. This analysis was continued by Lidskii (1965).
We study networks of linear reactions. For a linear system with reaction rate
constants $k_{i}$ all the dynamical information is contained in eigenvalues
and eigenvectors of the kinetic matrix or, more precisely, in its
transformation to the Jordan normal form. It is computationally expensive task
to find this transformation for a non-symmetric matrix which is usually stiff
(Golub & Van Loan (1996)). Moreover, the answer could be very sensitive to the
errors in constants $k_{i}$. Nevertheless, it appears that stiffness can help
us to find a robust approximation, and in the limit when all constants are
very different (well-separated constants) the asymptotical behavior of
eigenvalues and eigenvectors follow simple explicit expressions. Analysis of
this asymptotics is our main goal.
In our approach, we study asymptotic behavior of eigenvalues and eigenvectors
of kinetic matrices along logarithmic straight lines, $\ln
k_{i}=\theta_{i}\xi$ in the space of constants. We significantly use the graph
representation of chemical reaction networks and demonstrate, that for almost
every vector $(\theta_{i})$ there exists a simple reaction network which
describes the dominant term of this asymptotic. Following the asymptotology
terminology (White (2006)), we call this simple network the dominant system.
For these dominant system there are explicit formulas for eigenvalues and
eigenvectors. The topology of dominant systems is rather simple: they are
acyclic networks without branching. This allows us to construct the explicit
asymptotics of eigenvectors and eigenvalues. All algorithms are represented
topologically by transformation of the graph of reaction (labeled by reaction
rate constants). The reaction rate constants for dominant systems may not
coincide with constant of original network. In general, they are monomials of
the original constants.
This result fully supports the observation by Kruskal (1963): “And the answer
quite generally has the form of a new system (well posed problem) for the
solution to satisfy, although this is sometimes obscured because the new
system is so easily solved that one is led directly to the solution without
noticing the intermediate step.”
The dominant systems can be used for direct computation of steady states and
relaxation dynamics, especially when kinetic information is incomplete, for
design of experiments and mining of experimental data, and could serve as a
robust first approximation in perturbation theory or for preconditioning. They
can be used to answer an important question: given a network model, which are
its critical parameters? Many of the parameters of the initial model are no
longer present in the dominant system: these parameters are non-critical.
Parameters of dominant subsystems indicate putative targets to change the
behavior of the large network.
Most of reaction networks are nonlinear, it is nevertheless useful to have an
efficient algorithm for solving linear problems. First, nonlinear systems
often include linear subsystems, containing reactions that are
(pseudo)monomolecular with respect to species internal to the subsystem (at
most one internal species is reactant and at most one is product). Second, for
binary reactions $A+B\to...$, if concentrations of species $A$ and $B$
($c_{A},c_{B}$) are well separated, say $c_{A}\gg c_{B}$ then we can consider
this reaction as $B\to...$ with rate constant proportional to $c_{A}$ which is
practically constant, because its relative changes are small in comparison to
relative changes of $c_{B}$. We can assume that this condition is satisfied
for all but a small fraction of genuinely nonlinear reactions (the set of
nonlinear reactions changes in time but remains small). Under such an
assumption, nonlinear behavior can be approximated as a sequence of such
systems, followed one each other in a sequence of “phase transitions”. In
these transitions, the order relation between some of species concentrations
changes. Some applications of this approach to systems biology are presented
by Radulescu, Gorban, Zinovyev & Lilienbaum (2008). The idea of controllable
linearization “by excess” of some reagents is in the background of the
efficient experimental technique of Temporal Analysis of Products (TAP), which
allows to decipher detailed mechanisms of catalytic reactions (Yablonsky,
Olea, & Marin (2003)).
In chemical kinetics various fundamental ideas about asymptotical analysis
were developed (Klonowski (1983)): quasieqiulibrium asymptotic (QE), quasi
steady-state asymptotic (QSS), lumping, and the idea of limiting step.
Most of the works on nonequilibrium thermodynamics deal with the QE
approximations and corrections to them, or with applications of these
approximations (with or without corrections). There are two basic formulation
of the QE approximation: the thermodynamic approach, based on entropy maximum,
or the kinetic formulation, based on selection of fast reversible reactions.
The very first use of the entropy maximization dates back to the classical
work of Gibbs (1902), but it was first claimed for a principle of
informational statistical thermodynamics by Jaynes (1963). A very general
discussion of the maximum entropy principle with applications to dissipative
kinetics is given in the review by Balian, Alhassid & Reinhardt (1986).
Corrections of QE approximation with applications to physical and chemical
kinetics were developed by Gorban, Karlin, Ilg, & Öttinger (2001); Gorban &
Karlin (2005).
QSS was proposed by Bodenstein (1913) and was elaborated into an important
tool for analysis of chemical reaction mechanism and kinetics (Semenov (1939);
Christiansen (1953); Helfferich (1989)). The classical QSS is based on the
relative smallness of concentrations of some of “active” reagents (radicals,
substrate-enzyme complexes or active components on the catalyst surface) (Aris
(1965); Segel & Slemrod (1989)).
Lumping analysis aims to combine reagents into “quasicomponents” for dimension
reduction (Wei & Kuo (1969); Kuo & Wei (1969); Li & Rabitz (1989); Toth, Li,
Rabitz, & Tomlin (1997).
The concept of limiting step gives the limit simplification: the whole network
behaves as a single step. This is the most popular approach for model
simplification in chemical kinetics and in many areas beyond kinetics. In the
form of a bottleneck approach this approximation is very popular from traffic
management to computer programming and communication networks. The proposed
asymptotic analysis can be considered as a wide extension of the classical
idea of limiting step (Gorban & Radulescu (2008)).
The structure of the paper is as follows. In Sec. 2 we introduce basic notions
and notations. We consider thermodynamic restrictions on the reaction rate
constants and demonstrate how appear systems with arbitrary constants (as
subsystems of more detailed models). For linear networks, the main theorems
which connect ergodic properties with topology of network, are reminded. Four
basic ideas of model reduction in chemical kinetics are described: QE, QSS,
lumping analysis and limiting steps.
In Sec. 3, we introduce the dominant system for a simple irreversible
catalytic cycle with limiting step. This is just a chain of reactions which
appears after deletion the limiting step from the cycle. Even for such simple
examples several new observation are presented:
* •
The relaxation time for a cycle with limiting step is inverse second reaction
rate constant;
* •
For chains of reactions with well separated rate constants left eigenvectors
have coordinates close to 0 or 1, and right eigenvectors have coordinates
close to 0 or $\pm 1$.
For general reaction networks instead of linear chains appear general acyclic
non-branching networks. For them we also provide explicit formulas for
eigenvectors and their 0, $\pm 1$ asymptotics for well-separated constants
(Sec. 4). In (Sec. 5) the main algorithm is presented. Sec. 6 is devoted to a
simple demonstration of the algorithm application. In Sec. 7, we briefly
discuss further corrections to dominant systems. The estimates of accuracy are
given in Appendix.
## 2 Main Asymptotic Ideas in Chemical Kinetics
### 2.1 Chemical Reaction Networks
To define a chemical reaction network, we have to introduce:
* •
a list of components (species);
* •
a list of elementary reactions;
* •
a kinetic law of elementary reactions.
The list of components is just a list of symbols (labels) $A_{1},...A_{n}$.
Each elementary reaction is represented by its stoichiometric equation
$\sum_{i}\alpha_{si}A_{i}\to\sum_{si}\beta_{si}A_{i},$ (1)
where $s$ enumerates the elementary reaction, and the non-negative integers
$\alpha_{si}$, $\beta_{si}$ are the stoichiometric coefficients. A
stoichiomentric vector $\gamma_{s}$ with coordinates
$\gamma_{si}=\beta_{si}-\alpha_{si}$ is associated with each elementary
reaction.
For analysis of closed chemical systems with detailed balance it is usual
practice to group reactions in pairs, direct and inverse reactions together,
but in more general settings this is not convenient.
A non-negative real extensive variable $N_{i}\geq 0$, amount of $A_{i}$, is
associated with each component $A_{i}$. It measures “the number of particles
of that species” (in particles, or in moles). The concentration of $A_{i}$ is
an intensive variable: $c_{i}=N_{i}/V$, where $V$ is volume. It is necessary
to stress, that in many practically important cases the extensive variable $V$
is neither constant, nor the same for all components $A_{i}$. For more details
see, for example the book of Yablonskii, Bykov, Gorban, & Elokhin (1991). For
simplicity, we will consider systems with one constant volume and under
constant temperature, but it is necessary always keep in mind the possibility
to return to general equations. For that conditions, the kinetic equations
have the following form
$\frac{{\mathrm{d}}c}{{\mathrm{d}}t}=\sum_{s}w_{s}(c,T)\gamma_{s}+\upsilon,$
(2)
where $\upsilon$ is the vector of external fluxes normalized to unit volume.
It may be useful to represent external fluxes as elementary reactions by
introduction of new component $\varnothing$ together with incoming and
outgoing reactions $\varnothing\to A_{i}$ and $A_{i}\to\varnothing$.
The most popular kinetic law of elementary reactions is the mass action law
for perfect systems:
$w_{s}(c,T)=k_{s}(T)\prod c_{i}^{\alpha_{si}},$ (3)
where “kinetic constant” $k_{s}(T)$ depends on temperature $T$. More general
kinetic law, which can be used for most of non-ideal (non-perfect) systems is
$w_{s}(c,T)=\varphi_{s}\exp\left(\frac{1}{RT}\sum_{i}\alpha_{si}\mu_{i}\right),$
(4)
where $R$ is the universal gas constant, $\mu_{i}$ is the chemical potential,
$\mu_{i}=\frac{\partial F(N,T,V)}{\partial N_{i}}=\frac{\partial
G(N,T,P)}{\partial N_{i}}$, $F$ is the Helmgoltz free energy, $G$ is the Gibbs
energy (free enthalpy), $P$ is pressure and $\varphi_{s}>0$ is an intensive
variable, kinetic factor, which can depend on any set of intensive variables,
first of all, on $T$.
Chemical thermodynamics (Prigogine & Defay (1954)) provides tools of choice
for stability analysis of reaction networks (Procaccia & Ross (1977)) and
chemical reactors (Aris (1965)). The laws of thermodynamics have been used for
analyzing of structural stability of process systems by Hangos, Bokor, &
Szederkényi (2004). In general reaction network coefficients $k_{s}$ (3) or
$\varphi_{s}$ (4) are not independent. In order to respect the second law of
thermodynamics, they should satisfy some equations and inequalities. The most
famous sufficient condition gives the principle of detailed balance. Let us
group the elementary reactions in pairs, direct and inverse reactions, and
mark the variables for direct reactions by superscript $+$, and for inverse
reactions by $-$. Then the principle of detailed balance for general kinetics
(4) reads:
$\varphi_{s}^{+}=\varphi_{s}^{-}$ (5)
(Feinberg (1972)). For the isothermal mass action law the principle of
detailed balance can be formulated as follows: there exists a strictly
positive point $c^{*}$ of detailed balance, at this point
$w_{s}^{+}(c^{*})=w_{s}^{-}(c^{*})$ (6)
for all $s$. This is, essentially, the same principle: if we substitute in the
general reaction rate (4) the fraction $\mu_{i}/RT$ by $\ln(c_{i}/c_{i}^{*})$,
then we will get the mass action law, and $\varphi_{s}^{+}=\varphi_{s}^{-}$.
The principle of detailed balance is closely related to the microreversibility
and Onsager relations.
More general condition was invented by Stueckelberg (1952) for the Boltzmann
equation. He produced them from the $S$-matrix unitarity (the quantum complete
probability formula). For the general law (4) without direct-inverse reactions
grouping for any state the following identity holds:
$\begin{split}&\sum_{s}\varphi_{s}\exp\left(\frac{1}{RT}\sum_{i}\alpha_{si}\mu_{i}\right)\\\
&\equiv\sum_{s}\varphi_{s}\exp\left(\frac{1}{RT}\sum_{i}\beta_{si}\mu_{i}\right).\end{split}$
(7)
Even more general condition which guarantees the second law and has clear
microscopic sense (the complete probability does not increase) was obtained by
Gorban (1984): for any state
$\begin{split}&\sum_{s}\varphi_{s}\exp\left(\frac{1}{RT}\sum_{i}\alpha_{si}\mu_{i}\right)\\\
&\geq\sum_{s}\varphi_{s}\exp\left(\frac{1}{RT}\sum_{i}\beta_{si}\mu_{i}\right).\end{split}$
(8)
To obtain formulas for the isothermal mass action law, it is sufficient just
to apply the general law (4) with constant $\varphi_{s}$ to the perfect free
energy $F=RT\sum_{i}c_{i}(\ln c_{i}+\mu_{i0})$ with constant $\mu_{i0}$. More
detailed analysis was presented, by Gorban (1984).
In any case, reaction constants are dependent, and this dependence guarantees
stability of equilibrium and existence of global thermodynamic Lyapunov
functions for closed systems (2) with $\upsilon=0$. Nevertheless, we often
study equations for such systems with oscillations, bifurcations, chaos, and
other effects, which are impossible in systems with global Lyapunov function.
Usually this means that we study a subsystem of a large system, where some of
concentrations do not change because they are stabilized by external fluxes or
by a large external reservoir. These constant (or very slow) concentrations
are included into new reaction constants, and after this redefinition they can
loose any thermodynamic property.
### 2.2 Linear Networks and Ergodicity
In this Sec., we consider a general network of linear reactions. This network
is represented as a directed graph (digraph) (Temkin, Zeigarnik, & Bonchev
(1996)): vertices correspond to components $A_{i}$, edges correspond to
reactions $A_{i}\to A_{j}$ with kinetic constants $k_{ji}>0$. For each vertex,
$A_{i}$, a positive real variable $c_{i}$ (concentration) is defined. A basis
vector $e^{i}$ corresponds to $A_{i}$ with components $e^{i}_{j}=\delta_{ij}$,
where $\delta_{ij}$ is the Kronecker delta. The kinetic equation for the
system is
$\frac{{\mathrm{d}}c_{i}}{{\mathrm{d}}t}=\sum_{j}(k_{ij}c_{j}-k_{ji}c_{i}),$
(9)
or in vector form: $\dot{c}=Kc$. We don’t assume any special relation between
constants, and consider them as independent quantities. The thermodynamic
restrictions on constants are not applicable here because, in general, we
study pseudomonomolecular systems which are subsystems of larger nonlinear
systems and don’t represent by themselves closed monomolecular systems.
For any network of linear reactions the matrix of kinetic coefficients $K$ has
the following properties:
* •
non-diagonal elements of $K$ are non-negative;
* •
diagonal elements of $K$ are non-positive;
* •
elements in each column of $K$ have zero sum.
For any $K$ with these properties there exists a network of linear reactions
with kinetic equation $\dot{c}=Kc$. This family of matrices coincide with the
family of generators of finite Markov chains, and this class of kinetic
equations coincide with the class of inverse Kolmogorov’s equations or master
equations for the finite Markov chains in continuous time (Meyn & Tweedie
(2009); Meyn (2007)).
A linear conservation law is a linear function defined on the concentrations
$b(c)=\sum_{i}b_{i}c_{i}$, whose value is preserved by the dynamics (9). The
conservation laws coefficient vectors $b_{i}$ are left eigenvectors of the
matrix $K$ corresponding to the zero eigenvalue. The set of all the
conservation laws forms the left kernel of the matrix $K$. Equation (9) always
has a linear conservation law: $b^{0}(c)=\sum_{i}c_{i}={\rm const}$. If there
is no other independent linear conservation law, then the system is weakly
ergodic.
A set $E$ is positively invariant with respect to kinetic equations (9), if
any solution $c(t)$ that starts in $E$ at time $t_{0}$ ($c(t_{0})\in E$)
belongs to $E$ for $t>t_{0}$ ($c(t)\in E$ if $t>t_{0}$). It is straightforward
to check that the standard simplex $\Sigma=\\{c\,|\,c_{i}\geq
0,\,\sum_{i}c_{i}=1\\}$ is positively invariant set for kinetic equation (9):
just to check that if $c_{i}=0$ for some $i$, and all $c_{j}\geq 0$ then
$\dot{c}_{i}\geq 0$. This simple fact immediately implies the following
properties of ${K}$:
* •
All eigenvalues $\lambda$ of ${K}$ have non-positive real parts,
$Re\lambda\leq 0$, because solutions cannot leave $\Sigma$ in positive time;
* •
If $Re\lambda=0$ then $\lambda=0$, because intersection of $\Sigma$ with any
plane is a polygon, and a polygon cannot be invariant with respect to
rotations to sufficiently small angles;
* •
The Jordan cell of ${K}$ that corresponds to zero eigenvalue is diagonal –
because all solutions should be bounded in $\Sigma$ for positive time.
* •
The shift in time operator $\exp({K}t)$ is a contraction in the $l_{1}$ norm
for $t>0$.
* •
For weakly ergodic systems there exists such a monotonically decreasing
function $\delta(t)$ ($t>0$, $0<\delta(t)<1$, $\delta(t)\to 0$ when
$t\to\infty$) that for any two solutions of (9) $c(t),c^{\prime}(t)\in\Sigma$
$\sum_{i}|c_{i}(t)-c^{\prime}_{i}(t)|\leq\delta(t)\sum_{i}|c_{i}(0)-c^{\prime}_{i}(0)|\
.$ (10)
The ergodicity coefficient $\delta(t)$ was introduced by Dobrushin (1956) (see
also a book by Seneta (1981)). It can be estimated using the structure of the
network graph (Gorban, Bykov & Yablonskii (1986); Meyn (2007)).
Two vertices are called adjacent if they share a common edge. A path is a
sequence of adjacent vertices. A graph is connected if any two of its vertices
are linked by a path. A maximal connected subgraph of graph $G$ is called a
connected component of $G$. Every graph can be decomposed into connected
components.
A directed path is a sequence of adjacent edges where each step goes in
direction of an edge. A vertex $A$ is reachable from a vertex $B$, if there
exists a directed path from $B$ to $A$.
A nonempty set $V$ of graph vertices forms a sink, if there are no directed
edges from $A_{i}\in V$ to any $A_{j}\notin V$. For example, in the reaction
graph $A_{1}\leftarrow A_{2}\rightarrow A_{3}$ the one-vertex sets
$\\{A_{1}\\}$ and $\\{A_{3}\\}$ are sinks. A sink is minimal if it does not
contain a strictly smaller sink. In the previous example, $\\{A_{1}\\}$,
$\\{A_{3}\\}$ are minimal sinks. Minimal sinks are also called ergodic
components.
A digraph is strongly connected, if every vertex $A$ is reachable from any
other vertex $B$. Ergodic components are maximal strongly connected subgraphs
of the graph, but inverse is not true: there may exist maximal strongly
connected subgraphs that have outgoing edges and, therefore, are not sinks.
The weak ergodicity of the network follows from its topological properties.
Theorem 1. The following properties are equivalent (and each one of them can
be used as an alternative definition of weak ergodicity):
1. 1.
There exist the only independent linear conservation law for kinetic equations
(9) (this is $b^{0}(c)=\sum_{i}c_{i}={\rm const}$).
2. 2.
For any normalized initial state $c(0)$ ($b^{0}(c)=1$) there exists a limit
state
$c^{*}=\lim_{t\rightarrow\infty}\exp(Kt)\,c(0)$
that is the same for all normalized initial conditions: For all $c$,
$\lim_{t\rightarrow\infty}\exp(Kt)\,c=b^{0}(c)c^{*}.$
3. 3.
For each two vertices $A_{i},\>A_{j}\>(i\neq j)$ we can find such a vertex
$A_{k}$ that is reachable both from $A_{i}$ and from $A_{j}$. This means that
the following structure exists:
$A_{i}\to\ldots\to A_{k}\leftarrow\ldots\leftarrow A_{j}.$
One of the paths can be degenerated: it may be $i=k$ or $j=k$.
4. 4.
The network has only one minimal sink (one ergodic component).$\square$
The proof of this theorem could be extracted from detailed books about Markov
chains and networks (Meyn (2007); Van Mieghem (2006)). In its present form it
was published by Gorban, Bykov & Yablonskii (1986) with explicit estimations
of ergodicity coefficients.
For every monomolecular kinetic system, the maximal number of independent
linear conservation laws (i.e. the geometric multiplicity of the zero
eigenvalue of the matrix $K$) is equal to the maximal number of disjoint
ergodic components (minimal sinks).
### 2.3 Quasi-equilibrium (QE) or Fast Equilibrium
Quasi-equilibrium approximation uses the assumption that a group of reactions
is much faster than other and goes fast to its equilibrium. We use below
superscripts ‘f’ and ‘s’ to distinguish fast and slow reactions. A small
parameter appears in the following form
$\begin{split}\frac{{\mathrm{d}}c}{{\mathrm{d}}t}=&\sum_{\sigma,\ {\rm
slow}}w_{\sigma}^{\rm s}(c,T)\gamma_{\sigma}^{\rm
s}+\frac{1}{\varepsilon}\sum_{\varsigma,\ {\rm fast}}w^{\rm
f}_{\varsigma}(c,T)\gamma_{\varsigma}^{\rm f},\end{split}$ (11)
To separate variables, we have to study the spaces of linear conservation law
of the initial system (11) and of the fast subsystem
$\frac{{\mathrm{d}}c}{{\mathrm{d}}t}=\frac{1}{\varepsilon}\sum_{\varsigma,\
{\rm fast}}w^{\rm f}_{\varsigma}(c,T)\gamma_{\varsigma}^{\rm f}$
If they coincide, then the fast subsystem just dominates, and there is no
fast-slow separation for variables (all variables are either fast, or
constant). But if there exist additional linearly independent linear
conservation laws for the fast system, then let us introduce new variables:
linear functions $b^{1}(c),...b^{n}(c)$, where $b^{1}(c),...b^{m}(c)$ is the
basis of the linear conservation laws for the initial system, and
$b^{1}(c),...b^{m+l}(c)$ is the basis of the linear conservation laws for the
fast subsystem. Then $b^{m+l+1}(c),...b^{n}(c)$ are fast variables,
$b^{m+1}(c),...b^{m+l}(c)$ are slow variables, and $b^{1}(c),...b^{m}(c)$ are
constant. The quasi-equilibrium manifold is given by the equations
$\sum_{\varsigma}w^{\rm f}_{\varsigma}(c,T)\gamma_{\varsigma}^{\rm f}=0$ and
for small $\varepsilon$ it serves as an approximation to a slow manifold. In
the old and standard approach it is assumed that system (11) as well as system
of fast reactions satisfies the thermodynamic restrictions, and the quasi-
equilibrium is just a partial thermodynamic equilibrium, and could be defined
by conditional extremum of thermodynamic functions. This guarantees global
stability of fast subsystems and all the classical singular perturbation
theory like Tikhonov theorem could be applied.
Recently, Vora & Daoutidis (2001) took notice that this type of reasoning does
not require classical thermodynamic restrictions on constants. For example,
let us consider the mass action law kinetics and group the reactions in pairs,
direct and inverse reactions. If the set of stoichiometric vectors for fast
reactions is linearly independent, then for this system the detailed balance
principle holds (obviously), and it demonstrates the “thermodynamic behaviour”
without connection to classical thermodynamics. This case of
“stoichiometrically independent fast reactions” can be generalized for
irreversible reactions too (Vora & Daoutidis (2001)). For such fast system the
quasiequilbrium manifold has the same nice properties as for thermodynamic
partial equilibrium, and approximates slow dynamics for sufficiently small
$\varepsilon$.
There are other classes of mass action law subsystems with such a “quasi-
thermodynamic” behaviour, which depends on structure, but not on constants.
For example, any system of reactions without interactions has such a property
(Gorban, Bykov, & Yablonskii (1986)). These reactions have the form $\alpha
A_{i}\to\sum...$: any linear reaction are allowed, as well as reactions like
$2A_{i}\to A_{j}+A_{k}$, $3A_{i}\to A_{j}+A_{k}+A_{l}$, etc. All such fast
subsystems can serve for quasi-equilibrium approximation, because for them
dynamics is globally stable.
Quasi-equilibrium manifold approximates exponentially attractive slow manifold
and is used in many areas of kinetics either as initial approximation for slow
motion, or just by itself (more discussion and further references are
presented by Gorban & Karlin (2005)).
### 2.4 Quasi Steady-State (QSS) or Fast Species
The quasi steady-state (or pseudo steady state) assumption was invented in
chemistry for description of systems with radicals or catalysts. In the most
usual version the species are split in two groups with concentration vectors
$c^{\rm s}$ (“slow” or basic components) and $c^{\rm f}$ (“fast
intermediates”). For catalytic reactions there is additional balance for
$c^{\rm f}$, amount of catalyst, usually it is just a sum $b_{\rm
f}=\sum_{i}c^{\rm f}_{i}$. The amount of the fast intermediates is assumed
much smaller than the amount of the basic components, but the reaction rates
are of the same order, or even the same (both intermediates and slow
components participate in the same reactions). This is the source of a small
parameter in the system. Let us scale the concentrations $c^{\rm f}$ and
$c^{\rm s}$ to the compatible amounts. After that, the fast and slow time
appear and we could write $\dot{c}^{\rm s}=W^{\rm s}(c^{\rm s},c^{\rm f})$,
$\dot{c}^{\rm f}=\frac{1}{\varepsilon}W^{\rm f}(c^{\rm s},c^{\rm f})$, where
$\varepsilon$ is small parameter, and functions $W^{\rm s},W^{\rm f}$ are
bounded and have bounded derivatives (are “of the same order”). We can apply
the standard singular perturbation techniques. If dynamics of fast components
under given values of slow concentrations is stable, then the slow attractive
manifold exists, and its zero approximation is given by the system of
equations $W^{\rm f}(c^{\rm s},c^{\rm f})=0$. Bifurcations in fast system
correspond to critical effects, including ignition and explosion.
This scheme was analyzed many times with plenty of details, examples, and some
complications. Exhaustive case study of the simplest enzyme reaction was
provided by Segel & Slemrod (1989) . For heterogenious catalytic reactions,
the book by Yablonskii, Bykov, Gorban, & Elokhin (1991) gives analysis of
scaling of fast intermediates (there are many kinds of possible scaling). In
the context of the Computational Singular Perturbation (CSP) approach, Lam
(1993) and Lam & Goussis (1994) developed concept of the CSP radicals. Gorban
& Karlin (2003, 2005) considered QSS as initial approximation for slow
invariant manifold. Analysis of the error of the QSS was provided by Turanyi,
Tomlin, & Pilling (1993).
The QE approximation is also extremely popular and useful. It has simpler
dynamical properties (respects thermodynamics, for example, and gives no
critical effects in fast subsystems of closed systems). Nevertheless, neither
radicals in combustion, nor intermediates in catalytic kinetics are, in
general, close to quasi-equilibrium. They are just present in much smaller
amount, and when this amount grows, then the QSS approximation fails.
The simplest demonstration of these two approximation gives the simple
reaction: $S+E\leftrightarrow SE\to P+E$ with reaction rate constants
$k^{\pm}_{1}$ and $k_{2}$. The only possible quasi-equilibrium appears when
the first equilibrium is fast: $k^{\pm}_{1}=\kappa^{\pm}/\varepsilon$. The
corresponding slow variable is $C^{s}=c_{S}+c_{SE}$,
$b_{E}=c_{E}+c_{SE}=const$. For the QE manifold we get a quadratic equation
$\frac{k_{1}^{-}}{k_{1}^{+}}c_{SE}=c_{S}c_{E}=(C^{s}-c_{SE})(b_{E}-c_{SE})$.
This equation gives the explicit dependence $c_{SE}(C^{s})$, and the slow
equation reads $\dot{C}^{s}=-k_{2}c_{SE}(C^{s})$, $C^{s}+c_{P}=b_{S}=const$.
For the QSS approximation of this reaction kinetics, under assumption
$b_{E}\ll b_{S}$, we have fast intermediates $E$ and $SE$. For the QSS
manifold there is a linear equation
$k^{+}_{1}c_{S}c_{E}-k_{1}^{-}c_{SE}-k_{2}c_{SE}=0$, which gives us the
explicit expression for $c_{SE}(c_{S})$:
$c_{SE}=k_{1}^{+}c_{S}b_{E}/(k_{1}^{+}c_{S}+k_{1}^{-}+k_{2})$ (the standard
Michaelis–Menten formula). The slow kinetics reads
$\dot{c}_{S}=-k_{1}^{+}c_{S}(b_{E}-c_{SE}(c_{S}))+k_{1}^{-}c_{SE}(c_{S})$. The
difference between the QSS and the QE in this example is obvious.
The terminology is not rigorous, and often QSS is used for all singular
perturbed systems, and QE is applied only for the thermodynamic exclusion of
fast variables by the maximum entropy (or minimum of free energy, or extremum
of another relevant thermodynamic function) principle (MaxEnt). This
terminological convention may be convenient. Nevertheless, without any
relation to terminology, the difference between these two types of
introduction of a small parameter is huge. There exists plenty of
generalizations of these approaches, which aim to construct a slow and
(almost) invariant manifold, and to approximate fast motion as well. The
following references can give a first impression about these methods: Method
of Invariant Manifolds (MIM) (Roussel & Fraser (1991); Gorban & Karlin (2005),
Method of Invariant Grids (MIG), a discrete analogue of invariant manifolds
(Gorban, Karlin, & Zinovyev (2004)), Computational Singular Perturbations
(CSP) (Lam (1993); Lam & Goussis (1994); Zagaris, Kaper, & Kaper (2004))
Intrinsic Low-Dimensional Manifolds (ILDM) by Maas, & Pope (1992), developed
further in series of works by Bykov, Goldfarb, Gol’dshtein, & Maas, U.
(2006)), methods based on the Lyapunov auxiliary theorem (Kazantzis & Kravaris
(2006)).
### 2.5 Lumping Analysis
Wei & Prater (1962) demonstrated that for (pseudo)monomolecular systems there
exist linear combinations of concentrations which evolve in time
independently. These linear combinations (quasicomponents) correspond to the
left eigenvectors of kinetic matrix: if $lK=\lambda l$ then
${\mathrm{d}}(l,c)/{\mathrm{d}}t=(l,c)\lambda$, where the standard inner
product $(l,c)$ is concentration of a quasicomponent. They also demonstrated
how to find these quasicomponents in a properly organized experiment.
This observation gave rise to a question: how to lump components into proper
quasicomponents to guarantee the autonomous dynamics of the quasicomponents
with appropriate accuracy. Wei and Kuo studied conditions for exact (Wei & Kuo
(1969)) and approximate (Kuo & Wei (1969)) lumping in monomolecular and
pseudomonomolecular systems. They demonstrated that under certain conditions
large monomolecular system could be well–modelled by lower–order system.
More recently, sensitivity analysis and Lie group approach were applied to
lumping analysis (Li & Rabitz (1989); Toth, Li, Rabitz, & Tomlin (1997)), and
more general nonlinear forms of lumped concentrations are used (for example,
concentration of quasicomponents could be rational function of $c$).
Hutchinson & Luss (1970) studied lumping-analysis of mixtures with many
parallel first order reactions. Farkas (1999) generalized these results and
characterized those lumping schemes which preserve the kinetic structure of
the original system. Coxson & Bischoff (1987) placed lumping analysis in the
linear systems theory and demonstrated the relationships between lumpability
and the concepts of observability, controllability and minimal realization.
Djouad & Sportisse (2002) considered the lumping procedures as efficient
techniques leading to nonstiff systems and demonstrated efficiency of
developed algorithm on kinetic models of atmospheric chemistry. Lin, Leibovici
& Jorgensen (2008) formulated an optimal lumping problem as a mixed integer
nonlinear programming (MINLP) and demonstrated that it can be efficiently
solved with a stochastic optimization method, Tabu Search (TS) algorithm.
The power of lumping using a time-scale based approach was demonstrated by
Whitehouse, Tomlin, & Pilling (2004). This computationally cheap approach
combines ideas of sensitivity analysis with simple and useful grouping of
species with similar lifetimes and similar topological properties caused by
connections of the species in the reaction networks. The lumped concentrations
in this approach are simply sums of concentrations in groups. For example,
species with similar composition and functionalities could be lumped into one
single representative species (Pepiot-Desjardins & Pitsch (2008)).
Lumping analysis based both on mathematical arguments and fundamental physical
and chemical properties of the components is now one of the main tools for
model reduction in highly multicomponent systems, such as the hydrocarbon
mixture in petroleum chemistry (Zavala & Rodriguez & Vargas-Villamil (2004))
or biochemical networks in systems biology (Maria (2006)). The optimal
solution of lumping problem often requires the exhaustive search, and instead
of them various heuristics are used to avoid combinatorial explosion. For the
lumping analysis of the systems biology models Dokoumetzidis & Aarons (2009)
developed a heuristic greedy search strategy which allowed them to avoid the
exhaustive search of proper lumped components.
Procedures of lumping analysis form a part of general algebra of model
building and model simplification transformations. Hangos & Cameron (2001)
applied formal methods of computer science and artificial intelligence for
analysis of this algebra. In particular, a formal method for defining syntax
and semantics of process models has been proposed.
The modern systems and control theory provides efficient tools for
lumping–analysis. The so-called balanced model reduction was invented in late
1970s (Moore (1981)). For a linear system a set of “target variables” is
selected. The dimension of the system $n$ is large, while the number of the
target variables, for example, inputs $m$ and outputs $p$, usually satisfies
$m,p\ll n$. The balanced model reduction problem can be stated as follows
(Gugercin & Antoulas (2004)): find a reduced order system such that the
following properties are satisfied:
1. 1.
The approximation error in the target variables is small, and there exists a
global error bound.
2. 2.
System properties, like stability and passivity, are preserved.
3. 3.
The procedure is computationally efficient.
In large dimensions, special efforts are needed to resolve the
accuracy/efficiency dilemma and to find efficiently the approximate solution
of the model reduction problem (Antoulas & Sorensen (2002)).
Various methods for balanced truncation are developed: Lyapunov balancing,
stochastic balancing, bounded real balancing, positive real balancing, and
frequency weighted balancing (Gugercin & Antoulas (2004)). Nonlinear
generalizations are proposed as well (Lall, Marsden & Glavaki (2002); Condon &
Ivanov (2004)).
### 2.6 Limiting Steps
In the IUPAC Compendium of Chemical Terminology (2007) one can find a
definition of limiting steps. Rate-controlling step (2007): “A rate-
controlling (rate-determining or rate-limiting) step in a reaction occurring
by a composite reaction sequence is an elementary reaction the rate constant
for which exerts a strong effect – stronger than that of any other rate
constant – on the overall rate.”
Let us complement this definition by additional comment: usually when people
are talking about limiting step they expect significantly more: there exists a
rate constant which exerts such a strong effect on the overall rate that the
effect of all other rate constants together is significantly smaller. For the
IUPAC Compendium definition a rate-controlling step always exists, because
among the control functions generically exists the biggest one. On the
contrary, for the notion of limiting step that is used in practice, there
exists a difference between systems with limiting step and systems without
limiting step.
During XX century, the concept of the limiting step was revised several times.
First simple idea of a “narrow place” (the least conductive step) could be
applied without adaptation only to a simple cycle or a chain of irreversible
steps that are of the first order (see Chap. 16 of the book Johnston (1966) or
the paper by Boyd (1978)). When researchers try to apply this idea in more
general situations they meet various difficulties such as:
* •
Some reactions have to be “pseudomonomolecular.” Their constants depend on
concentrations of outer components, and are constant only under condition that
these outer components are present in constant concentrations, or change
sufficiently slow (i.e. are present in significantly bigger amount).
* •
Even under fixed or slow outer components concentration, the simple “narrow
place” behaviour could be spoiled by branching or by reverse reactions. The
simplest example is given by the cycle: $A_{1}\leftrightarrow A_{2}\to
A_{3}\to A_{1}$. Even if the constant of the last step $A_{3}\to A_{1}$ is the
smallest one, the stationary rate may be much smaller than $k_{3}b$ (where $b$
is the overall balance of concentrations, $b=c_{1}+c_{2}+c_{3}$), if the
constant of the reverse reaction $A_{2}\to A_{1}$ is sufficiently big.
In a series of papers, Northrop (1981, 2001) clearly explained these
difficulties and suggested that the concept of rate–limiting step is
“outmoded”. Nevertheless, the main idea of limiting is so attractive that
Northrop’s arguments stimulated the search for modification and improvement of
the main concept.
Ray (1983) proposed the use of sensitivity analysis. He considered cycles of
reversible reactions and suggested a definition: The rate–limiting step in a
reaction sequence is that forward step for which a change of its rate constant
produces the largest effect on the overall rate.
Ray’s approach was revised by Brown & Cooper (1993) from the system control
analysis point of view (see the book of Cornish-Bowden & Cardenas (1990)).
They stress again that there is no unique rate–limiting step specific for an
enzyme, and this step, even if it exists, depends on substrate, product and
effector concentrations.
Near critical conditions the critical simplification appears, which is also a
type of limitation, because some reactions become critically important
(Yablonsky, Mareels, & Lazman (2003))
Two classical examples of limiting steps demonstrate us the chain of linear
reaction and the linear catalytic cycle, when they include a reaction which is
significantly slower, than other reactions.
A linear chain of reactions, $A_{1}\to A_{2}\to...A_{n}$, with reaction rate
constants $k_{i}$ (for $A_{i}\to A_{i+1}$), gives the first example of
limiting steps. Let the reaction rate constant $k_{q}$ be the smallest one.
Then we expect the following behaviour of the reaction chain in time scale
$\gtrsim 1/k_{q}$: all the components $A_{1},...A_{q-1}$ transform fast into
$A_{q}$, and all the components $A_{q+1},...A_{n-1}$ transform fast into
$A_{n}$, only two components, $A_{q}$ and $A_{n}$ are present (concentrations
of other components are small) , and the whole dynamics in this time scale can
be represented by a single reaction $A_{q}\to A_{n}$ with reaction rate
constant $k_{q}$. This picture becomes more exact when $k_{q}$ becomes smaller
with respect to other constants.
The catalytic cycle is one of the most important substructures that we study
in reaction networks. In the reduced form the catalytic cycle is a set of
linear reactions:
$A_{1}\to A_{2}\to\ldots A_{n}\to A_{1}.$
Reduced form means that in reality some of these reaction are not
monomolecular and include some other components (not from the list
$A_{1},\ldots A_{n}$). But in the study of the isolated cycle dynamics,
concentrations of these components are taken as constant and are included into
kinetic constants of the cycle linear reactions.
For the constant of elementary reaction $A_{i}\to$ we use the simplified
notation $k_{i}$ because the product of this elementary reaction is known, it
is $A_{i+1}$ for $i<n$ and $A_{1}$ for $i=n$. The elementary reaction rate is
$w_{i}=k_{i}c_{i}$, where $c_{i}$ is the concentration of $A_{i}$. The kinetic
equation is:
$\dot{c}_{i}=k_{i-1}c_{i-1}-k_{i}c_{i},$ (12)
where by definition $c_{0}=c_{n}$, $k_{0}=k_{n}$, and $w_{0}=w_{n}$. In the
stationary state ($\dot{c}_{i}=0$), all the $w_{i}$ are equal: $w_{i}=w$. This
common rate $w$ we call the cycle stationary rate, and
$w=\frac{b}{\frac{1}{k_{1}}+\ldots\frac{1}{k_{n}}};\;\;c_{i}=\frac{w}{k_{i}},$
(13)
where $b=\sum_{i}c_{i}$ is the conserved quantity for reactions in constant
volume. Let one of the constants, $k_{\min}$, be much smaller than others (let
it be $k_{\min}=k_{n}$):
$k_{i}\gg k_{\min}\ \ {\rm if}\ \ i\neq n\ .$ (14)
In this case, in linear approximation $w=k_{n}b$,
$c_{n}=b\left(1-\sum_{i<n}\frac{k_{n}}{k_{i}}\right),\ {\rm
and}\;c_{i}=b\frac{k_{n}}{k_{i}}\ {\rm for}\ i\neq n\ .$ (15)
The simplest zero order approximation for the steady state gives
$c_{n}=b,\;c_{i}=0\;(i\neq n).$ (16)
This is trivial: all the concentration is collected at the starting point of
the “narrow place,” but may be useful as an origin point for various
approximation procedures.
So, the stationary rate of a cycle is determined by the smallest constant,
$k_{\min}$, if it is much smaller than the constants of all other reactions
(14):
$w\approx k_{\min}b.$ (17)
In that case we say that the cycle has a limiting step with constant
$k_{\min}$.
## 3 Dynamics of Catalytic Cycle with Limiting Step
### 3.1 Eigenvalues
There is significant difference between the examples of limiting steps for the
chain of reactions and for irreversible cycle. For the chain, the steady state
does not depend on nonzero rate constants. It is just
$c_{n}=b,c_{1}=c_{2}=...=c_{n-1}=0$. The smallest rate constant $k_{q}$ gives
the smallest positive eigenvalue, the relaxation time is $\tau=1/k_{q}$. The
corresponding approximation of eigenmode (right eigenvector) $r^{1}$ has
coordinates: $r^{1}_{1}=...=r^{1}_{q-1}=0$, $r^{1}_{q}=1$,
$r^{1}_{q+1}=...=r^{1}_{n-1}=0$, $r_{n}=-1$. This exactly corresponds to the
statement that the whole dynamics in the time scale $\gtrsim 1/k_{q}$ can be
represented by a single reaction $A_{q}\to A_{n}$ with reaction rate constant
$k_{q}$. The left eigenvector for eigenvalue $k_{q}$ has approximation $l^{1}$
with coordinates $l^{1}_{1}=l^{1}_{2}=...=l^{1}_{q}=1$,
$l^{1}_{q+1}=...=l^{1}_{n}=0$. This vector provides the almost exact lumping
on time scale $\gtrsim 1/k_{q}$. Let us introduce a new variable $c_{\rm
lump}=\sum_{i}l_{i}c_{i}$, i.e. $c_{\rm lump}=c_{1}+c_{2}+...+c_{q}$. For the
time scale $\gtrsim 1/k_{q}$ we can write $c_{\rm lump}+c_{n}\approx b$,
${\mathrm{d}}c_{\rm lump}/{\mathrm{d}}t\approx-k_{q}c_{\rm lump}$,
${\mathrm{d}}c_{n}/{\mathrm{d}}t\approx k_{q}c_{\rm lump}$.
In the example of a cycle, we approximate the steady state, that is, the right
eigenvector $r^{0}$ for zero eigenvalue (the left eigenvector is known and
corresponds to the main linear balance $b$: $l^{0}_{i}\equiv 1$). In the zero-
order approximation, this eigenvector has coordinates
$r^{0}_{1}=...=r^{0}_{n-1}=0$, $r^{0}_{n}=1$.
If ${k_{n}}/{k_{i}}$ is small for all $i<n$, then the kinetic behaviour of the
cycle is determined by a linear chain of $n-1$ reactions $A_{1}\to
A_{2}\to...A_{n}$, which we obtain after cutting the limiting step. The
characteristic equation for an irreversible cycle,
$\prod_{i=1}^{n}(\lambda+k_{i})-\prod_{i=1}^{n}k_{i}=0$, tends to the
characteristic equation for the linear chain,
$\lambda\prod_{i=1}^{n-1}(\lambda+k_{i})=0$, when $k_{n}\to 0$.
The characteristic equation for a cycle with limiting step ($k_{n}/k_{i}\ll
1$) has one simple zero eigenvalue that corresponds to the conservation law
$\sum c_{i}=b$ and $n-1$ nonzero eigenvalues
$\lambda_{i}=-{k_{i}}+\delta_{i}\;(i<n).$ (18)
where $\delta_{i}\to 0$ when $\sum_{i<n}\frac{k_{n}}{k_{i}}\to 0$.
A cycle with limiting step (12) has real eigenspectrum and demonstrates
monotonic relaxation without damped oscillations. Of course, without
limitation such oscillations could exist, for example, when all $k_{i}\equiv
k>0$, ($i=1,...n$).
The relaxation time of a stable linear system (12) is, by definition,
$\tau=1/\min\\{Re(-\lambda_{i})\\}$ ($\lambda\neq 0$). For small $k_{n}$,
$\tau\approx 1/k_{\tau}$, $k_{\tau}=\min\\{k_{i}\\}$, ($i=1,...n-1$). In other
words, for a cycle with limiting step, $k_{\tau}$ is the second slowest rate
constant: $k_{\min}\ll k_{\tau}\leq...$.
### 3.2 Eigenvectors for Reaction Chain and for Catalytic Cycle with Limiting
Step
Let the irreversible cycle include a limiting step: $k_{n}\ll k_{i}$
($i=1,...,n-1$) and, in addition, $k_{n}\ll|k_{i}-k_{j}|$ ($i,j=1,...,n-1$,
$i\neq j$), then the eigenvectors of the kinetic matrix almost coincide with
the eigenvectors for the linear chain of reactions $A_{1}\to
A_{2}\to...A_{n}$, with reaction rate constants $k_{i}$ (for $A_{i}\to
A_{i+1}$) (Gorban & Radulescu (2008)).
The kinetic equation for the linear chain is
$\dot{c_{i}}=k_{i-1}c_{i-1}-k_{i}c_{i},$ (19)
The coefficient matrix $K$ of these equations is very simple. It has nonzero
elements only on the main diagonal, and one position below. The eigenvalues of
$K$ are $-k_{i}$ ($i=1,...n-1$) and 0. The left and right eigenvectors for 0
eigenvalue, $l^{0}$ and $r^{0}$, are:
$l^{0}=(1,1,...1),\;\;r^{0}=(0,0,...0,1),$ (20)
all coordinates of $l^{0}$ are equal to 1, the only nonzero coordinate of
$r^{0}$ is $r^{0}_{n}$ and we represent vector–column $r^{0}$ in row.
Below we use explicit form of $K$ left and right eigenvectors. Let
vector–column $r^{i}$ and vector–row $l^{i}$ be right and left eigenvectors of
$K$ for eigenvalue $-k_{i}$. For coordinates of these eigenvectors we use
notation $r^{i}_{j}$ and $l^{i}_{j}$. Let us choose a normalization condition
$r^{i}_{i}=l^{i}_{i}=1$. It is straightforward to check that $r^{i}_{j}=0$
$(j<i)$ and $l^{i}_{j}=0$ $(j>i)$, $r^{i}_{j+1}=k_{j}r_{j}/(k_{j+1}-k_{i})$
$(j\geq i)$ and $l^{i}_{j-1}=k_{j-1}l_{j}/(k_{j-1}-k_{j})$ $(j\leq i)$, and
$r^{i}_{i+m}=\prod_{j=1}^{m}\frac{k_{i+j-1}}{k_{i+j}-k_{i}};\;l^{i}_{i-m}=\prod_{j=1}^{m}\frac{k_{i-j}}{k_{i-j}-k_{i}}.$
(21)
It is convenient to introduce formally $k_{0}=0$. Under selected normalization
condition, the inner product of eigenvectors is: $l^{i}r^{j}=\delta_{ij}$,
where $\delta_{ij}$ is the Kronecker delta.
If the rate constants any two constants, $k_{i}$, $k_{j}$ are connected by
relation $k_{i}\gg k_{j}$ or $k_{i}\ll k_{j}$ (i.e. they are well separated),
then
$\frac{k_{i-j}}{k_{i-j}-k_{i}}\approx\left\\{\begin{aligned}
&1,\;&\mbox{if}\;k_{i}\ll k_{i-j};\\\ &0,\;&\mbox{if}\;k_{i}\gg
k_{i-j},\end{aligned}\right.$ (22)
Hence, $|l^{i}_{i-m}|\approx 1$ or $|l^{i}_{i-m}|\approx 0$. To demonstrate
that also $|r^{i}_{i+m}|\approx 1$ or $|r^{i}_{i+m}|\approx 0$, we shift
nominators in the product (21) on such a way:
$r^{i}_{i+m}=\frac{k_{i}}{k_{i+m}-k_{i}}\prod_{j=1}^{m-1}\frac{k_{i+j}}{k_{i+j}-k_{i}}.$
Exactly as in (22), each multiplier $\frac{k_{i+j}}{k_{i+j}-k_{i}}$ here is
either almost 1 or almost 0, and $\frac{k_{i}}{k_{i+m}-k_{i}}$ is either
almost 0 or almost $-1$. In this zero-one asymptotics
$\begin{split}l^{i}_{i}=&1,\;l^{i}_{i-m}\approx 1\;\\\
&\mbox{if}\;k_{i-j}>k_{i}\;\mbox{for all}\;j=1,\ldots
m,\;\mbox{else}\;l^{i}_{i-m}\approx 0;\\\
r^{i}_{i}=&1,\;r^{i}_{i+m}\approx-1\;\\\ &\mbox{if}\;k_{i+j}>k_{i}\;\mbox{for
all}\;j=1,\ldots m-1\;\\\
&\mbox{and}\;k_{i+m}<k_{i},\;\mbox{else}\;r^{i}_{i+m}\approx 0.\end{split}$
(23)
In this asymptotic (Fig. 1), only two coordinates of right eigenvector $r^{i}$
can have nonzero values, $r^{i}_{i}=1$ and $r^{i}_{i+m}\approx-1$ where $m$ is
the first such positive integer that $i+m<n$ and $k_{i+m}<k_{i}$. Such $m$
always exists because $k_{n}=0$. For left eigenvector $l^{i}$,
$l^{i}_{i}\approx\ldots l^{i}_{i-q}\approx 1$ and $l^{i}_{i-q-j}\approx 0$
where $j>0$ and $q$ is the first such positive integer that $i-q-1>0$ and
$k_{i-q-1}<k_{i}$. It is possible that such $q$ does not exist. In that case,
all $l^{i}_{i-j}\approx 1$ for $j\geq 0$. It is straightforward to check that
in this asymptotic $l^{i}r^{j}=\delta_{ij}$.
Figure 1: Graphical representation of eigenvectors approximation for the
linear chain of reactions with well separated constants. To find the left
($l$) and right ($r$) eigenvectors for eigenvalue $k$ it is necessary to
delete from the chain all the reactions with the rate constants $<k$ (dashed
lines) and to find the maximal connected interval, where the reaction with
constant $k$ (bold arrow) is situated. The right eigenvector $r$ has
coordinate 1 for the vertex, which is the beginning of the reaction with
constant $k$, and coordinate $-1$ for the vertex, which is end of the interval
in the direction of reactions. The left eigenvector $l$ has coordinate 1 for
the beginning of the reaction with constant $k$ and for all preceding vertices
from the connected interval. All other coordinates of $r$ and $l$ are zero.
The simplest example gives the order $k_{1}\gg k_{2}\gg...\gg k_{n-1}$:
$l^{i}_{i-j}\approx 1$ for $j\geq 0$, $r^{i}_{i}=1$, $r^{i}_{i+1}\approx-1$
and all other coordinates of eigenvectors are close to zero. For the inverse
order, $k_{1}\ll k_{2}\ll...\ll k_{n-1}$, $l^{i}_{i}=1$, $r^{i}_{i}=1$,
$r^{i}_{n}\approx-1$ and all other coordinates of eigenvectors are close to
zero.
For less trivial example, let us find the asymptotic of left and right
eigenvectors for a chain of reactions:
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!5}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{6},$
where the upper index marks the order of rate constants: $k_{4}\gg k_{5}\gg
k_{2}\gg k_{3}\gg k_{1}$ ($k_{i}$ is the rate constant of reaction
$A_{i}\to...$).
For left eigenvectors, rows $l^{i}$, we have the following asymptotics:
$\begin{split}&l^{1}\approx(1,0,0,0,0,0),\;l^{2}\approx(0,1,0,0,0,0),\;\\\
&l^{3}\approx(0,1,1,0,0,0),l^{4}\approx(0,0,0,1,0,0),\;\\\
&l^{5}\approx(0,0,0,1,1,0).\end{split}$ (24)
For right eigenvectors, columns $r^{i}$, we have the following asymptotics (we
write vector-columns in rows):
$\begin{split}&r^{1}\approx(1,0,0,0,0,-1),\;r^{2}\approx(0,1,-1,0,0,0),\;\\\
&r^{3}\approx(0,0,1,0,0,-1),r^{4}\approx(0,0,0,1,-1,0),\;\\\
&r^{5}\approx(0,0,0,0,1,-1).\end{split}$ (25)
The corresponding approximation to the general solution of the kinetic
equations is:
$c(t)=(l^{0},c(0))r^{0}+\sum_{i=1}^{n-1}(l^{i}c(0))r^{i}\exp(-k_{i}t),$ (26)
where $c(0)$ is the initial concentration vector, and for left and right
eigenvectors $l^{i}$ and $r^{i}$ we use their zero-one asymptotic. In other
words, approximation of the left eigenvectors provides us with almost exact
lumping (for analysis of exact lumping see the paper by Li & Rabitz (1989)) .
## 4 Acyclic Non-branching Network: Explicit Formulas for Eigenvectors
So, to analyze asymptotic of eigenvalues and eigenvectors for a irreversible
cycle, we cut the reaction with the smallest constant, get a linear chain, and
analyze the eigenvalues and eigenvectors for this chain. For a general
multiscale reaction network (instead of a cycle) we will come, after some
surgery, to acyclic non-branching reaction networks (instead of a linear
chain).
For any network without branching, we can simplify the notation for the
kinetic constants, by introducing $\kappa_{i}=k_{ji}$ for the only reaction
$A_{i}\to A_{j}$, or $\kappa_{i}=0$, if there is no such a reaction. Also it
is useful to introduce a map $\phi$ on the set of vertices: $\phi(i)=j$, if
there exist reaction $A_{i}\to A_{j}$, and $\phi(i)=i$ if there are no
outgoing reactions from the $A_{i}\to A_{j}$. For iterations of the map $\phi$
we use notation $\phi^{q}$.
For an acyclic non-branching reaction network, for any vertex $A_{i}$ there is
an eigenvalue $-\kappa_{i}$ and the corresponding eigenvector. If $A_{i}$ is a
sink vertex, then this eigenvalue is zero. For left and right eigenvectors of
$K$ that correspond to $A_{i}$ we use notations $l^{i}$ (vector-row) and
$r^{i}$ (vector-column), correspondingly.
Let us suppose that $A_{f}$ is a sink vertex of the network. Its associated
right and left eigenvectors corresponding to the zero eigenvalue are given by:
$r^{i}_{j}=\delta_{ij}$; $l^{i}_{j}=1$ if and only if $\phi^{q}(j)=i$ for some
$q>0$.
Figure 2: Graphical representation of eigenvectors approximation for the
acyclic non-branching reaction network with well separated constants (compare
to Fig. 1). The eigenvalue $-k$ corresponds to the reaction $A_{i}\to
A_{\phi(i)}$ (bold arrow). To the right from $A_{i}$ are vertices
$A_{\phi^{q}(i)}$ and to the left are those $A_{j}$, for which there exists
such $q$ that $\phi^{q}(j)=i$. The reactions with the rate constants $<k$
(dashed lines) are deleted from the network. The right and left eigenvectors
could have nonzero coordinates only for vertices from the maximal connected
subgraph of the presented graph, where the $A_{i}$ is situated. The right
eigenvector $r$ has coordinate 1 for $A_{i}$ (beginning of the bold arrow),
and coordinate $-1$ for the vertex, which is the minimal in that connected
subgraph. The left eigenvector $l$ has coordinate 1 for the beginning of the
reaction with constant $k$ and for all preceding vertices from the subgraph.
All other coordinates of $r$ and $l$ are zero.
For nonzero eigenvalues, right eigenvectors will be constructed by recurrence
starting from the vertex $A_{i}$ and moving in the direction of the flow. The
construction is in opposite direction for left eigenvectors.
For right eigenvector $r^{i}$ only coordinates $r^{i}_{\phi^{k}(i)}$
($k=0,1,\ldots\tau_{i}$) could have nonzero values, and
$\begin{split}r^{i}_{\phi^{k+1}(i)}=\frac{\kappa_{\phi^{k}(i)}}{\kappa_{\phi^{k+1}(i)}-\kappa_{i}}r^{i}_{\phi^{k}(i)}=\prod_{j=0}^{k}\frac{\kappa_{\phi^{j}(i)}}{\kappa_{\phi^{j+1}(i)}-\kappa_{i}}\\\
=\frac{\kappa_{i}}{\kappa_{\phi^{k+1}(i)}-\kappa_{i}}\prod_{j=0}^{k-1}\frac{\kappa_{\phi^{j+1}(i)}}{\kappa_{\phi^{j+1}(i)}-\kappa_{i}}.\end{split}$
(27)
For left eigenvector $l^{i}$ coordinate $l^{i}_{j}$ could have nonzero value
only if there exists such $q\geq 0$ that $\phi^{q}(j)=i$ (this $q$ is unique
because the system is acyclic):
$l^{i}_{j}=\frac{\kappa_{j}}{\kappa_{j}-\kappa_{i}}l^{i}_{\phi(j)}=\prod_{k=0}^{q-1}\frac{\kappa_{\phi^{k}(j)}}{\kappa_{\phi^{k}(j)}-\kappa_{i}}.$
(28)
For well separated constants, we can write the asymptotic representation
explicitly, analogously to (23) (Fig. 2). For left eigenvectors, $l^{i}_{i}=1$
and $l^{i}_{j}=1$ (for $i\neq j$) if there exists such $q$ that
$\phi^{q}(j)=i$, and $\kappa_{\phi^{d}(j)}>\kappa_{i}$ for all $d=0,\ldots
q-1$, else $l^{i}_{j}=0$. For right eigenvectors, $r^{i}_{i}=1$ and
$r^{i}_{\phi^{k}(i)}=-1$ if $\kappa_{\phi^{k}(i)}<\kappa_{i}$ and for all
positive $m<k$ inequality $\kappa_{\phi^{m}(i)}>\kappa_{i}$ holds, i.e. $k$ is
first such positive integer that $\kappa_{\phi^{k}(i)}<\kappa_{i}$ (for fixed
point $A_{p}$ we use $\kappa_{p}=0$). Vector $r^{i}$ has not more than two
nonzero coordinates. It is straightforward to check that in this asymptotic
$l^{i}r^{j}=\delta_{ij}$.
For example, let us find that asymptotic for a branched acyclic system of
reactions:
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!7}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!5}}\,\,A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{8},\;\;A_{6}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{7}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{4}$
where the upper index marks the order of rate constants:
$\kappa_{6}>\kappa_{4}>\kappa_{7}>\kappa_{5}>\kappa_{2}>\kappa_{3}>\kappa_{1}$
($\kappa_{i}$ is the rate constant of reaction $A_{i}\to...$).
For zero eigenvalue, the left and right eigenvectors are
$l^{8}=(1,1,1,1,1,1,1,1,1),\;r^{8}=(0,0,0,0,0,0,0,1).$
For left eigenvectors, rows $l^{i}$, that correspond to nonzero eigenvalues we
have the following asymptotics:
$\begin{split}&l^{1}\approx(1,0,0,0,0,0,0,0),\;l^{2}\approx(0,1,0,0,0,0,0,0),\;\\\
&l^{3}\approx(0,1,1,0,0,0,0,0),l^{4}\approx(0,0,0,1,0,0,0,0),\;\\\
&l^{5}\approx(0,0,0,1,1,1,1,0),\;l^{6}\approx(0,0,0,0,0,1,0,0).\\\
&l^{7}\approx(0,0,0,0,0,1,1,0)\end{split}$ (29)
For the corresponding right eigenvectors, columns $r^{i}$, we have the
following asymptotics (we write vector-columns in rows):
$\begin{split}&r^{1}\\!\approx\\!(1,0,0,0,0,0,0,-1),\,r^{2}\\!\approx\\!(0,1,-1,0,0,0,0,0),\\\
&r^{3}\\!\approx\\!(0,0,1,0,0,0,0,-1),\,r^{4}\\!\approx\\!(0,0,0,1,-1,0,0,0),\\\
&r^{5}\\!\approx\\!(0,0,0,0,1,0,0,-1),\,r^{6}\\!\approx\\!(0,0,0,0,0,1,-1,0),\\\
&r^{7}\\!\approx\\!(0,0,0,0,-1,0,1,0).\end{split}$ (30)
## 5 Calculating the Dominant System for a Linear Multiscale Network
### 5.1 Problem Statement
We study asymptotical behavior of the transformation of the kinetic matrix $K$
to the normal form along the lines $\ln k_{ij}=\theta_{ij}\xi$ when
$\xi\to\infty$. For almost all direction vectors $(\theta_{ij})$ (outside
several hyperplanes) there exists a minimal reaction network which reaction
rate constants are monomials of $k_{ij}$ ($\prod_{ij}k_{ij}^{f_{ij}}$, where
$f_{ij}$ are not obligatory positive numbers) and eigenvectors and eigenvalues
approximate the eigenvectors and eigenvalues when $\xi\to\infty$ with
arbitrary high relative accuracy. We call this minimal system the dominant
system. Existence of dominant systems is proven by direct construction (this
Sec.) and estimates of accuracy of approximations (Appendix).
The dominant systems coincide for vectors $(\theta_{ij})$ from some polyhedral
cones. Therefore, we don’t need to study a given value of $(\theta_{ij})$ but
rather have to build these cones together with the correspondent dominant
systems. The following formal rule (“assumption of well separated constants”)
allows us to simplify this task: if in construction of dominant systems we
need to compare two monomials, $M_{f}=\prod_{ij}k_{ij}^{f_{ij}}$ and
$M_{g}=\prod_{ij}k_{ij}^{g_{ij}}$ then we can always state that either
$M_{f}\gg M_{g}$ or $M_{f}\ll M_{g}$ and consider the logarithmic hyperplane
$M_{f}=M_{g}$ as a boundary between different cones. At the end, we can join
all cones with the same dominant system. We are interested in robust
asymptotic and do not analyze directions $(\theta_{ij})$ which belong to the
boundary hyperplanes. This robust asymptotic with well separated constants and
acyclic dominant systems is typical because the exclusive direction vectors
belon to a finite number of hyperplanes.
There may be other approaches based on (i) the Maslov dequantization and
idempotent algebras (Litvinov & Maslov (2005)), (ii) the limit of log-uniform
distributions in wide boxes of constants under some conditions (Feng,
Hooshangi, Chen, Li, Weiss, & Rabitz (2004); Gorban & Radulescu (2008)), or
(iii) on consideration of all possible orderings of all monomials with integer
exponents and construction of correspondent dominant systems (Robbiano (1985)
proved that there exists only a final number of such orderings and enumerated
all of them, see also the book by Greuel & Pfister (2002)). They give the same
final result but with different intermediate steps.
### 5.2 Auxiliary Operations
#### 5.2.1 From Reaction Network to Auxiliary Dynamical System
Let us consider a reaction network $\mathcal{W}$ with a given structure and
fixed ordering of constants. The set of vertices of $\mathcal{W}$ is
$\mathcal{A}$ and the set of elementary reactions is $\mathcal{R}$. Each
reaction from $\mathcal{R}$ has the form $A_{i}\to A_{j}$,
$A_{i},A_{j}\in\mathcal{A}$. The corresponding constant is $k_{ji}$. For each
$A_{i}\in\mathcal{A}$ we define $\kappa_{i}=\max_{j}\\{k_{ji}\\}$ and
$\phi(i)={\rm arg\,max}_{j}\\{k_{ji}\\}$. In addition, $\phi(i)=i$ if
$k_{ji}=0$ for all $j$.
Figure 3: Construction of the auxiliary reaction network by pruning. For
every vertex, it is necessary to leave the outgoing reaction with maximal
reaction rate constant. Other reactions should be deleted.
The auxiliary discrete dynamical system for the reaction network $\mathcal{W}$
is the dynamical system $\Phi=\Phi_{\mathcal{W}}$ defined by the map $\phi$ on
the finite set $\mathcal{A}$. The auxiliary reaction network (Fig. 3)
$\mathcal{V}=\mathcal{V}_{\mathcal{W}}$ has the same set of vertices
$\mathcal{A}$ and the set of reactions $A_{i}\to A_{\phi(i)}$ with reaction
constants $\kappa_{i}$. Auxiliary kinetics is described by
$\dot{c}=\tilde{K}c$, where
$\tilde{K}_{ij}=-\kappa_{j}\delta_{ij}+\kappa_{j}\delta_{i\,\phi(j)}$.
#### 5.2.2 Decomposition of Discrete Dynamical Systems on Finite Sets
Discrete dynamical system on a finite set $V=\\{A_{1},A_{2},\ldots A_{n}\\}$
is a semigroup $1,\phi,\phi^{2},...$, where $\phi$ is a map $\phi:V\to V$.
$A_{i}\in V$ is a periodic point, if $\phi^{l}(A_{i})=A_{i}$ for some $l>0$;
else $A_{i}$ is a transient point. A cycle of period $l$ is a sequence of $l$
distinct periodic points $A,\phi(A),\phi^{2}(A),\ldots\phi^{l-1}(A)$ with
$\phi^{l}(A)=A$. A cycle of period one consists of one fixed point,
$\phi(A)=A$. Two cycles, $C,C^{\prime}$ either coincide or have empty
intersection.
The set of periodic points, $V^{\rm p}$, is always nonempty. It is a union of
cycles: $V^{\rm p}=\cup_{j}C_{j}$. For each point $A\in V$ there exist such a
positive integer $\tau(A)$ and a cycle $C(A)=C_{j}$ that $\phi^{q}(A)\in
C_{j}$ for $q\geq\tau(A)$. In that case we say that $A$ belongs to basin of
attraction of cycle $C_{j}$ and use notation $Att(C_{j})=\\{A\ |\
C(A)=C_{j}\\}$. Of course, $C_{j}\subset Att(C_{j})$. For different cycles,
$Att(C_{j})\cap Att(C_{l})=\varnothing$. If $A$ is periodic point then
$\tau(A)=0$. For transient points $\tau(A)>0$.
Figure 4: Decomposition of a discrete dynamical system.
So, the phase space $V$ is divided onto subsets $Att(C_{j})$ (Fig. 4). Each of
these subsets includes one cycle (or a fixed point, that is a cycle of length
1). Sets $Att(C_{j})$ are $\phi$-invariant: $\phi(Att(C_{j}))\subset
Att(C_{j})$. The set $Att(C_{j})\setminus C_{j}$ consist of transient points
and there exists such positive integer $\tau$ that
$\phi^{q}(Att(C_{j}))=C_{j}$ if $q\geq\tau$.
Discrete dynamical systems on a finite sets correspond to graphs without
branching points. Notice that for the graph that represents a discrete dynamic
system, attractors are ergodic components, while basins are connected
components.
### 5.3 Algorithm for Calculating the Dominant System
For this general case, the algorithm consists of two main procedures: (i)
cycles gluing and (ii) cycles restoration and cutting.
#### 5.3.1 Cycles Gluing
Let us start from a reaction network $\mathcal{W}$ with a given structure and
fixed ordering of constants. The set of vertices of $\mathcal{W}$ is
$\mathcal{A}$ and the set of elementary reactions is $\mathcal{R}$.
If all attractors of the auxiliary dynamic system $\Phi_{\mathcal{W}}$ are
fixed points $A_{f1},A_{f2},...\in\mathcal{A}$, then the auxiliary reaction
network is acyclic, and the auxiliary kinetics approximates relaxation of the
whole network $\mathcal{W}$.
In general case, let the system $\Phi_{\mathcal{W}}$ have several attractors
that are not fixed points, but cycles $C_{1},C_{2},...$ with periods
$\tau_{1},\tau_{2},...>1$. By gluing these cycles in points, we transform the
reaction network $\mathcal{W}$ into $\mathcal{W}^{1}$. The dynamical system
$\Phi_{\mathcal{W}}$ is transformed into $\Phi^{1}$. For these new system and
network, the connection $\Phi^{1}=\Phi_{\mathcal{W}^{1}}$ persists: $\Phi^{1}$
is the auxiliary discrete dynamical system for $\mathcal{W}^{1}$.
For each cycle, $C_{i}$, we introduce a new vertex $A^{i}$. The new set of
vertices,
$\mathcal{A}^{1}=\mathcal{A}\cup\\{A^{1},A^{2},...\\}\setminus(\cup_{i}C_{i})$
(we delete cycles $C_{i}$ and add vertices $A^{i}$).
Figure 5: Gluing a cycle with rate constants renormalization. $c^{\rm QS}_{l}$
are the quasistationary concentrations on the cycle. After gluing, we have to
leave the outgoing from $A^{1}$ reaction with the maximal renormalized rate
constant, and delete others.
All the reaction $A\to B$ from the initial set $\mathcal{R}$,
($A,B\in\mathcal{A}$) can be separated into 5 groups:
1. 1.
both $A,B\notin\cup_{i}C_{i}$;
2. 2.
$A\notin\cup_{i}C_{i}$, but $B\in C_{i}$;
3. 3.
$A\in C_{i}$, but $B\notin\cup_{i}C_{i}$;
4. 4.
$A\in C_{i}$, $B\in C_{j}$, $i\neq j$;
5. 5.
$A,B\in C_{i}$.
Reactions from the first group do not change. Reaction from the second group
transforms into $A\to A^{i}$ (to the whole glued cycle) with the same
constant. Reaction of the third type changes into $A^{i}\to B$ with the rate
constant renormalization: let the cycle $C^{i}$ be the following sequence of
reactions $A_{1}\to A_{2}\to...A_{\tau_{i}}\to A_{1}$, and the reaction rate
constant for $A_{i}\to A_{i+1}$ is $k_{i}$ ($k_{\tau_{i}}$ for
$A_{\tau_{i}}\to A_{1}$). For the limiting reaction of the cycle $C_{i}$ we
use notation $k_{\lim\,i}$. If $A=A_{j}$ and $k$ is the rate reaction for
$A\to B$, then the new reaction $A^{i}\to B$ has the rate constant
$kk_{\lim\,i}/k_{j}$. This corresponds to a quasistationary distribution on
the cycle (15). The new rate constant is smaller than the initial one:
$kk_{\lim\,i}/k_{j}<k$, because $k_{\lim\,i}<k_{j}$ due to definition of
limiting constant. The same constant renormalization is necessary for
reactions of the fourth type. These reactions transform into $A^{i}\to A^{j}$.
Finally, reactions of the fifth type vanish.
After we glue all the cycles (Fig. 5) of auxiliary dynamical system in the
reaction network $\mathcal{W}$, we get $\mathcal{W}^{1}$. Let us assign
$\mathcal{W}:=\mathcal{W}^{1}$, $\mathcal{A}:=\mathcal{A}^{1}$ and iterate
until we obtain an acyclic network and exit. This acyclic network is a
“forest” and consists of trees oriented from leafs to a root. The number of
such trees coincide with the number of fixed points in the final network.
After gluing we can identify the reactions, which will be included into the
dominant system. Their constants are the critical parameters of the networks.
The list of these parameters, consists of all reaction rates of the final
acyclic auxiliary network, and of the rate constants of the glued cycles, but
without their limiting steps. Some of these parameters are rate constants of
the initial network, other have the monomial structure. Other constants and
corresponding reactions do not participate in the following operations. To
form the structure of the dominant network, we need one more procedure.
#### 5.3.2 Cycles Restoration and Cutting
We start the reverse process from the glued network $\mathcal{V}^{m}$ on
$\mathcal{A}^{m}$. On a step back, from the set $\mathcal{A}^{m}$ to
$\mathcal{A}^{m-1}$ and so on, some of glued cycles should be restored and
cut. On the $q$th step we build an acyclic reaction network on
$\mathcal{A}^{m-q}$, the final network is defined on the initial vertex set
and approximates relaxation of $\mathcal{W}$.
To make one step back from $\mathcal{V}^{m}$ let us select the vertices of
$\mathcal{A}^{m}$ that are glued cycles from $\mathcal{V}^{m-1}$. Let these
vertices be $A^{m}_{1},A^{m}_{2},...$. Each $A^{m}_{i}$ corresponds to a glued
cycle from $\mathcal{V}^{m-1}$, $A^{m-1}_{i1}\to
A^{m-1}_{i2}\to...A^{m-1}_{i\tau_{i}}\to A^{m-1}_{i1}$, of the length
$\tau_{i}$. We assume that the limiting steps in these cycles are
$A^{m-1}_{i\tau_{i}}\to A^{m-1}_{i1}$. Let us substitute each vertex
$A^{m}_{i}$ in $\mathcal{V}^{m}$ by $\tau_{i}$ vertices
$A^{m-1}_{i1},A^{m-1}_{i2},...A^{m-1}_{i\tau_{i}}$ and add to
$\mathcal{V}^{m}$ reactions $A^{m-1}_{i1}\to
A^{m-1}_{i2}\to...A^{m-1}_{i\tau_{i}}$ (that are the cycle reactions without
the limiting step) with corresponding constants from $\mathcal{V}^{m-1}$.
Figure 6: The main operation of the cycle surgery: on a step back we get a
cycle $A_{1}\to...\to A_{\tau}\to A_{1}$ with the limiting step $A_{\tau}\to
A_{1}$ and one outgoing reaction $A_{i}\to A_{j}$. We should delete the
limiting step, reattach (“recharge”) the outgoing reaction $A_{i}\to A_{j}$
from $A_{i}$ to $A_{\tau}$ and change its rate constant $k$ to the rate
constant $kk_{\lim}/k_{i}$. The new value of reaction rate constant is always
smaller than the initial one: $kk_{\lim}/k_{i}<k$ if $k_{\lim}\neq k_{i}$. For
this operation only one condition $k\ll k_{i}$ is necessary ($k$ should be
small with respect to reaction $A_{i}\to A_{i+1}$ rate constant, and can
exceed any other reaction rate constant).
If there exists an outgoing reaction $A^{m}_{i}\to B$ in $\mathcal{V}^{m}$
then we substitute it by the reaction $A^{m-1}_{i\tau_{i}}\to B$ with the same
constant, i.e. outgoing reactions $A^{m}_{i}\to...$ are reattached to the
heads of the limiting steps (Fig. 6). Let us rearrange reactions from
$\mathcal{V}^{m}$ of the form $B\to A^{m}_{i}$. These reactions have
prototypes in $\mathcal{V}^{m-1}$ (before the last gluing). We simply restore
these reactions. If there exists a reaction $A^{m}_{i}\to A^{m}_{j}$ then we
find the prototype in $\mathcal{V}^{m-1}$, $A\to B$, and substitute the
reaction by $A^{m-1}_{i\tau_{i}}\to B$ with the same constant, as for
$A^{m}_{i}\to A^{m}_{j}$.
After that step is performed, the vertices set is $\mathcal{A}^{m-1}$, but the
reaction set differs from the reactions of the network $\mathcal{V}^{m-1}$:
the limiting steps of cycles are excluded and the outgoing reactions of glued
cycles are included (reattached to the heads of the limiting steps). To make
the next step, we select vertices of $\mathcal{A}^{m-1}$ that are glued cycles
from $\mathcal{V}^{m-2}$, substitute these vertices by vertices of cycles,
delete the limiting steps, attach outgoing reactions to the heads of the
limiting steps, and for incoming reactions restore their prototypes from
$\mathcal{V}^{m-2}$, and so on.
After all, we restore all the glued cycles, and construct an acyclic reaction
network on the set $\mathcal{A}$. This acyclic network approximates relaxation
of the network $\mathcal{W}$. We call this system the dominant system of
$\mathcal{W}$ and use notation ${\rm dom\,mod}(\mathcal{W})$.
In the simplest case, the dominant system is determined by the ordering of
constants. But for sufficiently complex systems we need to introduce auxiliary
elementary reactions. They appear after cycle gluing and have monomial rate
constants of the form $k_{\varsigma}=\prod_{i}k_{i}^{\varsigma_{i}}$, where
${\varsigma_{i}}$ are integers, but not mandatory positive. The dominant
system depends on the place of these monomial values among the ordered
constants. For systems with well separated constants we can also assume that
each of these new constants will be well separated from other constants
(Gorban & Radulescu (2008)).
### 5.4 Example
To demonstrate a possible branching of described algorithm for cycles surgery
(gluing, restoring and cutting) with necessity of additional orderings, let us
consider the following system:
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{3},\;\;A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!5}}\,\,A_{2},\;\;$
(31)
(where the upper index marks the order of rate constants). The auxiliary
discrete dynamical system for reaction network (31) is
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{3}.$
It has only one attractor, a cycle
$A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{3}$.
This cycle is not a sink for the whole network (31) because reaction
$A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!5}}\,\,A_{2}$ leads from that cycle.
After gluing the cycle into a vertex $A^{1}_{3}$ we get the new network
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A^{1}_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!?}}\,\,A_{2}$.
The rate constant for the reaction $A^{1}_{3}{\rightarrow}A_{2}$ is
$k^{1}_{23}=k_{24}k_{35}/k_{54}$, where $k_{ij}$ is the rate constant for the
reaction $A_{j}\to A_{i}$ in the initial network ($k_{35}$ is the cycle
limiting reaction). The new network coincides with its auxiliary system and
has one cycle,
$A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A^{1}_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!?}}\,\,A_{2}$.
This cycle is a sink, hence, we can start the back process of cycles restoring
and cutting. One question arises immediately: which constant is smaller,
$k_{32}$ or $k^{1}_{23}$. The smallest of them is the limiting constant, and
the answer depends on this choice. Let us consider two possibilities
separately: (1) $k_{32}>k^{1}_{23}$ and (2) $k_{32}<k^{1}_{23}$.
(1) Let as assume that $k_{32}>k^{1}_{23}$. The final auxiliary system after
gluing cycles is
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A^{1}_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!?}}\,\,A_{2}$.
Let us delete the limiting reaction
$A^{1}_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!?}}\,\,A_{2}$ from the cycle. We get
an acyclic system
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A^{1}_{3}$.
The component $A^{1}_{3}$ is the glued cycle
$A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{3}$.
Let us restore this cycle and delete the limiting reaction
$A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{3}$. We get the dominant
system
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{5}$.
Relaxation of this system approximates relaxation of the initial network (31)
under additional condition $k_{32}>k^{1}_{23}$.
(2) Let as assume now that $k_{32}<k^{1}_{23}$. The final auxiliary system
after gluing cycles is the same,
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A^{1}_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!?}}\,\,A_{2}$,
but the limiting step in the cycle is different,
$A_{2}{\rightarrow^{\\!\\!\\!\\!\\!\\!6}}\,\,A^{1}_{3}$. After cutting this
step, we get acyclic system
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}{\leftarrow^{\\!\\!\\!\\!?}}\,A^{1}_{3}$,
where the last reaction has rate constant $k^{1}_{23}$.
The component $A^{1}_{3}$ is the glued cycle
$A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{3}\,.$
Let us restore this cycle and delete the limiting reaction
$A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{3}$. The connection from glued
cycle $A^{1}_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!?}}\,\,A_{2}$ with constant
$k^{1}_{23}$ transforms into connection
$A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!?}}\,\,A_{2}$ with the same constant
$k^{1}_{23}$.
We get the dominant system:
$A_{1}{\rightarrow^{\\!\\!\\!\\!\\!\\!1}}\,\,A_{2}\,,\;A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!?}}\,\,A_{2}\,.$
The order of constants is now known: $k_{21}>k_{43}>k_{54}>k^{1}_{23}$, and we
can substitute the sign “?” by “4”:
$A_{3}{\rightarrow^{\\!\\!\\!\\!\\!\\!2}}\,\,A_{4}{\rightarrow^{\\!\\!\\!\\!\\!\\!3}}\,\,A_{5}{\rightarrow^{\\!\\!\\!\\!\\!\\!4}}\,\,A_{2}$.
For both cases, $k_{32}>k^{1}_{23}$ ($k^{1}_{23}=k_{24}k_{35}/k_{54}$) and
$k_{32}<k^{1}_{23}$ it is easy to find the eigenvectors explicitly and to
write the solution to the kinetic equations in explicit form.
## 6 The Reversible Triangle of Reactions
In this section, we illustrate the analysis of dominant systems on a simple
example, the reversible triangle of reactions.
$A_{1}\leftrightarrow A_{2}\leftrightarrow A_{3}\leftrightarrow A_{1}\,$ (32)
This triangle appeared in many works as an ideal object for a case study. Our
favorite example is the work of Wei & Prater (1962). Now in our study the
triangle (32) is not necessarily a closed system. We can assume that it is a
subsystem of a larger system, and any reaction $A_{i}\to A_{j}$ represents a
reaction of the form $\ldots+A_{i}\to A_{j}+\ldots$, where unknown but slow
components are substituted by dots. This means that there are no mandatory
relations between reaction rate constants, and six reaction rate constants are
arbitrary nonnegative numbers.
Let the reaction rate constant $k_{21}$ for the reaction $A_{1}\to A_{2}$ be
the largest.
Figure 7: Four possible auxiliary dynamical systems for the reversible
triangle of reactions with $k_{21}>k_{ij}$ for $(i,j)\neq(2,1)$: (a)
$k_{12}>k_{32}$, $k_{23}>k_{13}$; (b) $k_{12}>k_{32}$, $k_{13}>k_{23}$; (c)
$k_{32}>k_{12}$, $k_{23}>k_{13}$; (d) $k_{32}>k_{12}$, $k_{13}>k_{23}$. For
each vertex the outgoing reaction with the largest rate constant is
represented by the solid bold arrow, and other reactions are represented by
the dashed arrows. The digraphs formed by solid bold arrows are the auxiliary
discrete dynamical systems. Attractors of these systems are isolated in
frames.
Let us describe all possible auxiliary dynamical systems for the triangle
(32). For each vertex, we have to select the fastest outgoing reaction. For
$A_{1}$, it is always $A_{1}\to A_{2}$, because of our choice of enumeration
(the higher scheme in Fig. 7). There exist two choices of the fastest outgoing
reaction for two other vertices and, therefore, only four versions of
auxiliary dynamical systems for (32) (Fig. 7). Let us analyze in detail case
(a). For the cases (b) and (c) the details of computations are similar. The
irreversible cycle (d) is even simpler and was already discussed.
### 6.1 Auxiliary System (a): $A_{1}\leftrightarrow A_{2}\leftarrow A_{3}$;
$k_{12}>k_{32}$, $k_{23}>k_{13}$
#### 6.1.1 Gluing Cycles
The attractor is a cycle (with only two vertices) $A_{1}\leftrightarrow
A_{2}$. This is not a sink, because two outgoing reactions exist: $A_{1}\to
A_{3}$ and $A_{2}\to A_{3}$. They are relatively slow: $k_{31}\ll k_{21}$ and
$k_{32}\ll k_{12}$. The limiting step in this cycle is $A_{2}\to A_{1}$ with
the rate constant $k_{12}$. We have to glue the cycle $A_{1}\leftrightarrow
A_{2}$ into one new component $A_{1}^{1}$ and to add a new reaction
$A_{1}^{1}\rightarrow A_{3}$ with the rate constant (see Fig. 5)
$k_{31}^{1}=\max\\{k_{32},\,k_{31}k_{12}/k_{21}\\}\,.$ (33)
As a result, we get a new system, $A_{1}^{1}\leftrightarrow A_{3}$ with
reaction rate constants $k_{31}^{1}$ (for $A_{1}^{1}\rightarrow A_{3}$) and
initial $k_{23}$ (for $A_{1}^{1}\leftarrow A_{3}$). This cycle is a sink,
because it has no outgoing reactions (the whole system is a trivial example of
a sink).
#### 6.1.2 Dominant System
At the next step, we have to restore and cut the cycles. First cycle to cut is
the result of cycle gluing, $A_{1}^{1}\leftrightarrow A_{3}$. It is necessary
to delete the limiting step, i.e. the reaction with the smallest rate
constant. If $k_{31}^{1}>k_{23}$, then we get $A_{1}^{1}\rightarrow A_{3}$.
If, inverse, $k_{23}>k_{31}^{1}$, then we obtain $A_{1}^{1}\leftarrow A_{3}$.
After that, we have to restore and cut the cycle which was glued into the
vertex $A_{1}^{1}$. This is the two-vertices cycle $A_{1}\leftrightarrow
A_{2}$. The limiting step for this cycle is $A_{1}\leftarrow A_{2}$, because
$k_{21}\gg k_{12}$. If $k_{31}^{1}>k_{23}$, then following the rule visualized
by Fig. 6, we get the dominant system $A_{1}\to A_{2}\to A_{3}$ with reaction
rate constants $k_{21}$ for $A_{1}\to A_{2}$ and $k_{31}^{1}$ for $A_{2}\to
A_{3}$. If $k_{23}>k_{31}^{1}$ then we obtain $A_{1}\to A_{2}\leftarrow A_{3}$
with reaction rate constants $k_{21}$ for $A_{1}\to A_{2}$ and $k_{23}$ for
$A_{2}\leftarrow A_{3}$. All the procedure is illustrated by Fig. 8.
Figure 8: Dominant systems for case (a) (defined in Fig. 7)
#### 6.1.3 Eigenvalues and Eigenvectors
The eigenvalues and the corresponding eigenvectors for dominant systems in
case (a) are represented below in zero-one asymptotic.
1. 1.
$k_{31}^{1}>k_{23}$,
the dominant system $A_{1}\to A_{2}\to A_{3}$,
$\begin{array}[]{lll}\lambda_{0}=0\,,&r^{0}\approx(0,0,1)\,,&l^{0}=(1,1,1)\,;\\\
\lambda_{1}\approx-k_{21}\,,&r^{1}\approx(1,-1,0)\,,&l^{1}\approx(1,0,0)\,;\\\
\lambda_{2}\approx-
k_{31}^{1}\,,&r^{2}\approx(0,1,-1)\,,&l^{2}\approx(1,1,0)\,;\end{array}$ (34)
2. 2.
$k_{23}>k_{31}^{1}$,
the dominant system $A_{1}\to A_{2}\leftarrow A_{3}$,
$\begin{array}[]{lll}\lambda_{0}=0\,,&r^{0}\approx(0,1,0)\,,&l^{0}=(1,1,1)\,;\\\
\lambda_{1}\approx-k_{21}\,,&r^{1}\approx(1,-1,0)\,,&l^{1}\approx(1,0,0)\,;\\\
\lambda_{2}\approx-
k_{23}\,,&r^{2}\approx(0,-1,1)\,,&l^{2}\approx(0,0,1)\,.\end{array}$ (35)
Here, the value of $k_{31}^{1}$ is given by formula (33).
Analysis of examples provided us by an important conclusion: the number of
different dominant systems in examples was less than the number of all
possible orderings. For many pairs of constants $k_{ij},k_{lr}$ it is not
important which of them is larger. There is no need to consider all orderings
of monomials. We have to consider only those inequalities between constants
and monomials that appear in the construction of the dominant systems.
## 7 Corrections to Dominant Dynamics
The hierarchy of systems $\mathcal{W}$, $\mathcal{W}^{1}$, $\mathcal{W}^{2}$,
… can be used for multigrid correction of the dominant dynamics. The simple
example of multigrid approach gives the algorithm of steady state
approximation (Gorban & Radulescu (2008)). For this purpose, on the way up
(cycle restoration and cutting, Sec. 5.3.2) we calculate distribution in
restoring cycles with higher accuracy, by exact formula (13), or in linear
approximation (15) instead of the simplest zero-one asymptotic (16).
Essentially, the way up remains the same.
After termination of the gluing process, we can find all steady state
distributions by restoring cycles in the auxiliary reaction network
$\mathcal{V}^{m}$. Let $A^{m}_{f1},A^{m}_{f2},...$ be fixed points of
$\Phi^{m}$. The set of steady states for $\mathcal{V}^{m}$ is the set of all
distributions on the set of fixed points $\\{A^{m}_{f1},A^{m}_{f2},...\\}$.
Let us take one of the basis distributions, $c^{m}_{fi}=1$, other $c_{i}=0$ on
$\mathcal{V}^{m}$. If the vertex $A^{m}_{fi}$ is a glued cycle, then we
substitute them by all the vertices of this cycle. Redistribute the
concentration $c^{m}_{fi}$ between the vertices of the corresponding cycle by
the rule (13) (or by an approximation). As a result, we get a set of vertices
and a distribution on this set of vertices. If among these vertices there are
glued cycles, then we repeat the procedure of cycle restoration. Terminate
when there is no glued cycles in the support of the distribution.
The resulting distribution is the approximation to a steady state of
$\mathcal{W}$, and the basis of steady states for $\mathcal{W}$ can be
approximated by this method.
For example, for the system Fig. 8 we have, first of all, to compute the
stationary distribution in the cycle $A_{1}^{1}\leftrightarrow A_{3}$,
$c^{1}_{1}$ and $c_{3}$. On the base of the general formula for a simple cycle
(13) we obtain:
$w=\frac{1}{\frac{1}{k_{31}^{1}}+\frac{1}{k_{23}}}\,,\;c^{1}_{1}=\frac{w}{k_{31}^{1}}\,,\;c_{3}=\frac{w}{k_{23}}\,.$
(36)
After that, we have to restore the cycle glued into $A_{1}^{1}$. This means to
calculate the concentrations of $A_{1}$ and $A_{2}$ with normalization
$c_{1}+c_{2}=c^{1}_{1}$. Formula (13) gives:
$w^{\prime}=\frac{c^{1}_{1}}{\frac{1}{k_{21}}+\frac{1}{k_{12}}}\,,\;c_{1}=\frac{w^{\prime}}{k_{21}}\,,\;c_{2}=\frac{w^{\prime}}{k_{12}}\,.$
(37)
For eigenvectors, there appear two operations of corrections: (i) correction
for an acyclic network without branching (43), (45), and (ii) corrections for
a cycle with relatively slow outgoing reactions (49). These corrections are
by-products of the accuracy estimates given in Appendix.
## 8 Conclusion
Now, the idea of limiting step is developed to the asymptotology of multiscale
reaction networks. We found the main terms of eigenvectors and eigenvalues
asymptotic on logarithmic straight lines $\ln k_{ij}=\theta_{ij}\xi$ when
$\xi\to\infty$. These main terms could be represented by acyclic dominant
system which is a piecewise constant function of the direction vectors
$(\theta_{ij})$. This theory gives the analogue of the Vishik & Ljusternik
(1960) theory for chemical reaction networks. We demonstrated also how to
construct the accuracy estimates and the first order corrections to
eigenvalues and eigenvectors.
There are several ways of using the developed theory and algorithms:
* •
For direct computation of steady states and relaxation dynamics; this may be
useful for complex systems because of the simplicity of the algorithm and
resulting formulas and because often we do not know the rate constants for
complex networks, and kinetics that is ruled by orderings rather than by exact
values of rate constants may be very useful in practically frequent situation
when the values of the various reaction constants are unknown or poorly known;
* •
For planning experiments and mining the experimental data – the observable
kinetics is more sensitive to reactions from the dominant network, and much
less sensitive to other reactions, the relaxation spectrum of the dominant
network is explicitly connected with the correspondent reaction rate
constants, and the eigenvectors (“modes”) are sensitive to the constant
ordering, but not to exact values;
* •
The steady states and dynamics of the dominant system could serve as a robust
first approximation in perturbation theory or as a preconditioning in
numerical methods.
The next step should be development of asymptotic estimates for networks with
modular structure and time separations between modules, not between individual
reactions. But now it seems that the most important further development should
be the asymptotology of nonlinear reaction networks. For multiscale nonlinear
reaction networks the expected dynamical behaviour is to be approximated by
the system of dominant networks. These networks may change in time (this is
the significant difference from the linear case) but remain relatively simple.
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* (85)
## Appendix: Mathematical Backgrounds of Accuracy Estimation
### Estimates for Perturbed Acyclic Networks
The famous Gerschgorin theorem (Marcus & Minc (1992), Varga (2004)) gives
estimates of eigenvalues. We need also estimates of eigenvectors. Below
$A=(a_{ij})$ is a complex $n\times n$ matrix, $Q_{i}=\sum_{j,j\neq i}|a_{ji}|$
(sums of non-diagonal elements in columns).
Gerschgorin theorem (Marcus & Minc (1992), p. 146): The characteristic roots
of $A$ lie in the closed region $G^{Q}$ of the $z$-plane
$G^{Q}=\bigcup_{i}G^{Q}_{i}\;\;(G^{Q}_{i}=\\{z\,\bigl{|}\,|z-a_{ii}|\leq
Q_{i}\\}.$ (38)
Areas $G^{Q}_{i}$ are the Gerschgorin discs. (The same estimate are valid for
sums in rows, $P_{i}$. Here and below we don’t duplicate the estimates.)
Gerschgorin disks $G^{Q}_{i}$ ($i=1,\ldots n$) are isolated, if $G^{Q}_{i}\cap
G^{Q}_{j}=\varnothing$ for $i\neq j$. If disks $G^{P}_{i}$ ($i=1,\ldots n$)
are isolated, then the spectrum of $A$ is simple, and each Gerschgorin disk
$G^{Q}_{i}$ contains one and only one eigenvalue of $A$ (Marcus & Minc (1992),
p. 147).
We assume that Gerschgorin disks $G^{Q}_{i}$ ($i=1,\ldots n$) are isolated:
for all $i,j$ ($i\neq j$)
$|a_{ii}-a_{jj}|>Q_{i}+Q_{j}.$ (39)
Let us introduce the following notations:
$\begin{split}&\frac{Q_{i}}{|a_{ii}|}=\varepsilon_{i},\;\varepsilon=\max_{i}\varepsilon_{i},\;\frac{|a_{ij}|}{|a_{jj}|}=\chi_{ij},\;\chi=\max_{i,j,i\neq
j}\chi_{ij},\\\ &g_{i}=\min_{j,j\neq
i}\frac{|a_{ii}-a_{jj}|}{|a_{ii}|},\;g=\min_{i}g_{i}.\end{split}$ (40)
Usually, we consider $\varepsilon_{i}$ and $\chi_{ij}$ as sufficiently small
numbers. In contrary, the diagonal gap $g$ should not be small, (this is the
gap condition). For example, if for any two diagonal elements $a_{ii}$,
$a_{jj}$ either $a_{ii}\gg a_{jj}$ or $a_{ii}\ll a_{jj}$, then $g_{i}\gtrsim
1$ for all $i$.
Let $\lambda_{i}\in G^{Q}_{i}$ be the eigenvalue of $A$
($|\lambda_{i}-a_{11}|<Q_{1}$). Let us estimate the corresponding right
eigenvector $r^{(i)}$. We take $r^{i}_{i}=1$ and for $j\neq i$ introduce a
$(n-1)$-dimensional vector $\tilde{x}^{i}$:
$\tilde{x}^{i}_{j}=r^{i}_{j}(a_{jj}-a_{ii})$ ($i\neq j$). For $\tilde{x}^{i}$
we get equation
$(1-B^{(i)})\tilde{x}^{i}=-\tilde{a}^{i}$ (41)
where $\tilde{a}^{i}$ is a vector of the non-diagonal elements of the $i$th
column of $A$ ($\tilde{a}^{i}_{j}=a_{ij}$, $j\neq i$), and the
$(n-1)\times(n-1)$ matrix $B^{i}$ has matrix elements ($j,l\neq i$)
$b^{(i)}_{jj}=\frac{\lambda_{i}-a_{ii}}{a_{jj}-a_{ii}},\;\;b^{(i)}_{jl}=\frac{a_{jl}}{a_{ll}-a_{ii}}\;(l\neq
j)$ (42)
Due to the Gerschgorin estimate,
$|b^{(i)}_{jj}|<\frac{Q_{i}}{|a_{jj}-a_{ii}|}$. From Eq. (41) we obtain:
$\tilde{x}^{i}=-\tilde{a}^{i}-B^{(i)}(1-B^{(i)})^{-1}\tilde{a}^{i}.$ (43)
From this definition and simple estimates in $l^{1}$ norm, we get the
following estimate of eigenvectors.
Theorem 2. Let the Gerschgoring disks be isolated, and the diagonal gap be big
enough: $g>n\varepsilon$. Then for the $i$th eigenvector of $A$ the following
uniform estimate holds:
$|r^{i}_{j}|\leq\frac{\chi}{g}+\frac{n\varepsilon^{2}}{g(g-n\varepsilon)}\;\;(j\neq
1,\ r^{i}_{i}=1).\;\;\square$ (44)
So, if the matrix $A$ is diagonally dominant and the diagonal gap $g$ is big
enough, then the eigenvectors are proven to be close to the standard basis
vectors with explicit evaluation of accuracy.
The first correction to eigenvectors is also given by Eq. (43). If for the
iteration we use the Gerschgorin estimates for eigenvalue $\lambda_{i}\approx
a_{ii}$, then we can write in the next approximation for eigenvectors
($r^{i}_{i}=1,j\neq i$):
$r^{i}_{j}=-\frac{a_{ji}}{a_{jj}-a_{ii}}-\frac{(B^{(i)}_{\rm
nd}(1-B^{(i)}_{\rm nd})^{-1}\tilde{a}^{i})_{j}}{a_{jj}-a_{ii}}$ (45)
where $B^{(i)}_{\rm nd}$ is the non-diagonal part of $B^{(i)}$: it has the
same non-diagonal elements and zeros on diagonal. There exists plenty of
further simplifications for this iteration formula. For example, one can leave
just the first term, that gives the first order approximation in the power of
$\varepsilon$ ($\chi\leq\varepsilon$).
To apply these estimates to an acyclic network supplemented by additional
reactions, we have to use the eigenbasis of this acyclic network (Sec. 4).
Direct use of this theorem and estimates for a kinetic matrix $K$ in the
standard basis is impossible, the diagonal dominance in this coordinate system
is not large, and sums of elements in columns are zero. To apply this theorem
we need two lemmas.
Let $\mathcal{W}$ be a reaction network without branching (a finite dynamical
system) with $n$ vertices. Then the number of reactions in $\mathcal{W}$ is
$n-f$, where $f$ is the number of fixed points (the vertices without outgoing
reactions). Let $\Gamma$ be the set of stoichiometric vectors for
$\mathcal{W}$.
Lemma 1. $\Gamma$ forms a basis in the subspace $\\{c\,|\,\sum_{i}c_{i}=0\\}$
if and only if the reaction network $\mathcal{W}$ is acyclic and connected
(has only one fixed point). $\square$
Let us consider a general reaction network on the set $A_{1},...A_{n}$. For
stoichiometric vector of reaction $A_{i}\to A_{l}$ we use notation
$\gamma_{li}$. Assume that the auxiliary dynamical system $i\mapsto\phi(i)$
for a given reaction network is acyclic and has only one attractor, a fixed
point. For this auxiliary network, we use notation: $\kappa_{i}=k_{ji}$ for
the only reaction $A_{i}\to A_{j}$, or $\kappa_{i}=0$.
For every reaction of the initial network, $A_{i}\to A_{l}$, a linear
operators $Q_{il}$ can be defined by its action on the basis vectors,
$\gamma_{\phi(i)\,i}$:
$Q_{il}(\gamma_{\phi(i)\,i})=\gamma_{li},\;Q_{il}(\gamma_{\phi(p)\,p})=0\;\mbox{for}\;p\neq
i.$ (46)
Lemma 2. The kinetic equation for the whole reaction network (9) could be
transformed to the form
$\begin{split}\frac{{\mathrm{d}}c}{{\mathrm{d}}t}&=\sum_{i}\left(1+\sum_{l,\,l\neq\phi(i)}\frac{k_{li}}{\kappa_{i}}Q_{il}\right)\gamma_{\phi(i)\,i}\kappa_{i}c_{i}\\\
&=\left(1+\
\sum_{j,l\,(l\neq\phi(j))}\frac{k_{lj}}{\kappa_{j}}Q_{jl}\right)\sum_{i}\gamma_{\phi(i)\,i}\kappa_{i}c_{i}\\\
&=\left(1+\
\sum_{j,l\,(l\neq\phi(j))}\frac{k_{lj}}{\kappa_{j}}Q_{jl}\right)\tilde{K}c,\end{split}$
(47)
where $\tilde{K}$ is kinetic matrix of the kinetic equation for the auxiliary
network. $\square$
By construction of auxiliary dynamical system, ${k_{li}}<{\kappa_{i}}$ if
${l\neq\phi(i)}$, and for reaction networks with well separated constants
${k_{li}}\ll{\kappa_{i}}$. Notice also that the matrix $Q_{jl}$ does not
depend on rate constants values.
For matrix $\tilde{K}$ we have the eigenbasis in explicit form. Let us
represent system (47) in this eigenbasis of $\tilde{K}$. Any matrix $B$ in
this eigenbasis has the form $B=(\tilde{b}_{ij})$,
$\tilde{b}_{ij}=l^{i}Br^{j}=\sum_{qs}l^{i}_{q}b_{qs}r^{j}_{s}$, where
$(b_{qs})$ is matrix $B$ in the initial basis, $l^{i}$ and $r^{j}$ are left
and right eigenvectors of $\tilde{K}$ (27), (28). In eigenbasis of $\tilde{K}$
the estimates of eigenvalues and estimates of eigenvectors are much more
efficient than in original coordinates: the system is strongly diagonally
dominant. Transformation to this basis is an effective preconditioning for the
perturbation theory that uses auxiliary kinetics as a first approximation to
the kinetics of the whole system.
### Estimates for Perturbed Ergodic Systems
Let us consider a strongly connected network with kinetic matrix $K$. The
corresponding kinetics is ergodic and there exists unique normalized steady
state $c_{i}^{*}>0$, $\sum_{i}c_{i}^{*}=1$. For each $i$ we define
$\kappa_{i}=\sum_{j}k_{ji}$. The number $-\kappa_{i}$ is the $ii$th diagonal
element of unperturbed kinetic matrix $K$.
Let this network be perturbed by outgoing reactions $A_{i}\to 0$. The
perturbation has the “loss form”: the perturbed matrix is $K-{\rm
diag}(\varepsilon_{i}\kappa_{i})$, perturbation of each diagonal element is
relatively small (diag is the diagonal matrix).
The perturbations $\varepsilon_{i}\kappa_{i}$ are relatively small with
respect to $\kappa_{i}$, but not obligatory small with respect to other rate
constants.
First, we do not assume anything about value of $\varepsilon_{i}\geq 0$ and
make the following transformation. For an arbitrary normalized vector $r$
($r_{i}\geq 0$, $\sum_{i}r_{i}=1$) we add to the network reactions $A_{i}\to
A_{j}$ with reaction rates $q_{ji}=r_{j}\varepsilon_{i}\kappa_{i}$. We use
$Q(r)$ for the kinetic matrix of this additional network. Simple algebra gives
$\begin{split}Q(r)+{\rm
diag}(\varepsilon_{i}\kappa_{i})&=[\varepsilon_{1}\kappa_{1}r,\varepsilon_{2}\kappa_{2}r,...\varepsilon_{n}\kappa_{n}r]\\\
&=r(\varepsilon_{1}\kappa_{1},\varepsilon_{2}\kappa_{2},...\varepsilon_{n}\kappa_{n}).\end{split}$
(48)
Here, in the right hand side we have a matrix, all columns of which are
proportional to the vector $r$, this is a product of $r$ on the vector-raw of
coefficients. We represent the perturbed matrix in the form $K-{\rm
diag}(\varepsilon_{i}\kappa_{i})=K+Q(r)-(Q(r)+{\rm
diag}(\varepsilon_{i}\kappa_{i}))$.
Theorem 3. There exists such normalized positive $r^{*}$ that
$(K+Q(r^{*}))r^{*}=0$. This $r^{*}$ is an eigenvector of the perturbed network
with the eigenvalue $\lambda=\sum_{i}r^{*}_{i}\varepsilon_{i}\kappa_{i}$, and,
at the same time, it is a steady-state for the network with kinetic matrix
$K+Q(r^{*})$.
To prove existence it is sufficient to mention, that for any $r$ the network
with kinetic matrix $K+Q(r)$ has unique positive normalized steady state
$c^{*}(r)$, which depends continuously on $r$. The map $r\mapsto c^{*}(r)$ has
a fixed point $r^{*}$ (the Brouwer fixed point theorem). $\square$
This representation allows us to produce useful estimates, for example, when
the unperturbed system is a cycle, we find
$|r^{*}_{i}-c^{*}_{i}|<3\varepsilon|c^{*}_{i}|$ under condition
$\varepsilon<0.25$, where $\varepsilon=\sum\varepsilon_{i}$. Formula for the
first correction gives ($r^{*}=c^{*}_{i}+\delta r_{i}$, $w=k_{i}c^{*}_{i}$):
$\begin{split}\delta
r_{i}=\frac{v_{i}}{k_{i}},\;v_{i}=v+w\sum_{j=1}^{i}(\varepsilon
c^{*}_{j}-\varepsilon_{j}),\\\ v=\frac{w}{n}\sum_{i=1}^{n}i(\varepsilon
c^{*}_{i}-\varepsilon_{i}).\end{split}$ (49)
For more complex networks, the explicit formulas for corrections could be
produced on the base of the network graphs, similar to the steady-state
formulas, presented, for example, by Yablonskii, Bykov, Gorban, & Elokhin
(1991).
So, the asymptotic analysis gives good approximation of eigenvectors and
eigenvalues for kinetic matrix. The condition number is big (unbounded) but
these estimates work even better when the constants become more separated.
Nevertheless, some caution is needed: the error is proven to be small, but the
residuals (the values $\|Kr-\lambda r\|$ for approximations of $r$ and
$\lambda$) may be not small (Gorban & Radulescu (2008)).
|
arxiv-papers
| 2009-03-29T17:21:31 |
2024-09-04T02:49:01.484884
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. N. Gorban, O. Radulescu, A. Y. Zinovyev",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/0903.5072"
}
|
0903.5082
|
# Quantum Darwinism
Wojciech Hubert Zurek Theory Division, MS B213, LANL Los Alamos, NM, 87545,
U.S.A.
###### Abstract
Quantum Darwinism describes the proliferation, in the environment, of multiple
records of selected states of a quantum system. It explains how the fragility
of a state of a single quantum system can lead to the classical robustness of
states of their correlated multitude; shows how effective ‘wave-packet
collapse’ arises as a result of proliferation throughout the environment of
imprints of the states of quantum system; and provides a framework for the
derivation of Born’s rule, which relates probability of detecting states to
their amplitude. Taken together, these three advances mark considerable
progress towards settling the quantum measurement problem.
The quantum principle of superposition implies that any combination of quantum
states is also a legal state. This seems to be in conflict with everyday
reality: States we encounter are localized. Classical objects can be either
here or there, but never both here and there. Yet, the principle of
superposition says that localization should be a rare exception and not a rule
for quantum systems.
Fragility of states is the second problem with quantum-classical
correspondence: Upon measurement, a general preexisting quantum state is
erased – it “collapses” into an eigenstate of the measured observable. How is
it then possible that objects we deal with can be safely observed, even though
their basic building blocks are quantum?
To bypass these obstacles Bohr 11 followed Alexander the Great’s example:
Rather than try disentangling the Gordian Knot at the beginning of his
conquest, he cut it. The cut separates the quantum from the classical. Bohr’s
Universe consists of two realms, each governed by its own laws. Fragile
superpositions were banished from the classical realm deemed more fundamental
and indispensable to interpret or even practice quantum theory. Thus, instead
of trying to understand Universe (including “the classical”) in quantum terms
one “quantized” this and that, always starting from the classical base.
This was a brilliant tactical move: Physicists could conquer the quantum realm
without getting distracted by interpretational worries. In those days only
gedankenexperiments like the famous Schrödinger cat 47 were truly disturbing:
Real experiments dealt with electrons, photons, atoms, or other microscopic
systems. Bohr’s rule of thumb – that the macroscopic is classical – was
enough. Moreover, many (including Einstein) believed that quantum physics is
just a step on a way to a deeper theory that will solve or bypass
interpretational conundrums.
That did not happen. Instead, old gedankenexperiments were carried out. They
confirmed validity of quantum laws on scales that have, of recent, begun to
infringe on “the macroscopic”. Quantum theory is here to stay. It is also
increasingly clear that its weirdest predictions – superpositions and
entanglement – are experimental facts, in principle relevant also for
macroscopic objects. Therefore, questions about the origin of “the classical”,
with its restriction to localized states that are robust, unperturbed by
measurements, can no longer be dismissed.
## I Decoherence and einselection
Decoherence turns one of the two problems we noted above – fragility of
quantum states – into a solution of the other. Environment-induced decoherence
recognizes that if a measurement can put a state at risk and re-prepare it, so
can accidental information transfers that happen whenever a system interacts
with its environment.
Decoherence is by now well understood 36 ; 75 ; 52 : Fragility of states makes
quantum systems very difficult to isolate. Transfer of information (which has
no effect on classical states) has dramatic consequences in the quantum realm.
So, while fundamental problems of classical physics were always solved in
isolation (it sufficed to prevent energy loss) this is not so in quantum
physics (leaks of information are much harder to plug).
When a quantum system gives up information, its own state becomes consistent
with the information that was disseminated. “Collapse” in measurements is an
extreme example, but any interaction that leads to a correlation can
contribute to such re-preparation: Interactions that depend on a certain
observable correlate it with the environment, so its eigenstates are singled
out, and phase relations between such pointer states are lost 69 .
Negative selection due to decoherence is the essence of environment-induced
superselection, or einselection 70 : Under scrutiny of the environment, only
pointer states remain unchanged. Other states decohere into mixtures of stable
pointer states that can persist, and, in this sense, exist: They are
einselected.
These ideas can be made precise. The basic tool is the reduced density matrix
$\rho_{{\mathcal{S}}}$. It represents the state of the system that obtains
from the composite state ${\Psi_{{\mathcal{S}}{\mathcal{E}}}}$ of
${\mathcal{S}}$ and ${\mathcal{E}}$ by tracing out the environment
${\mathcal{E}}$:
$\rho_{\mathcal{S}}=Tr_{{\mathcal{E}}}|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle\\!\langle\Psi_{{\mathcal{S}}{\mathcal{E}}}|\
.$ $None$
Evolution of $\rho_{\mathcal{S}}$ reveals preferred states: It is most
predictable when the system starts in a pointer state. To quantify this one
can use (von Neumann) entropy,
$H_{{\mathcal{S}}}=H(\rho_{\mathcal{S}})=-Tr\rho_{\mathcal{S}}\lg\rho_{\mathcal{S}}$,
as a function of time. Pointer states result in smallest entropy increase. By
contrast, their superpositions produce entropy rapidly, at decoherence rates,
especially when ${\mathcal{S}}$ is macroscopic.
When pure states of the system are sorted by predictability, according to
entropy of the evolved $\rho_{\mathcal{S}}$, pointer states are at the top.
This criterion – the predictability sieve 75 ; 45 ; 80 – yields a short list
of candidates for effectively classical states: A cat can persist in one of
the two obvious stable states, but their superposition would deteriorate into
a mixture of $|\tt dead\rangle$ and $|\tt alive\rangle$ when initiated in a
way envisaged by Schrödinger 47 .
The special role of position is traced to the nature of the
${\mathcal{S}}{\mathcal{E}}$ interactions: They tend to depend on distance.
Hence, information about position is most readily passed on to the
environment. This is why localized states survive while nonlocal
superpositions decay into their mixtures. For example, in a weakly damped
harmonic oscillator the minimum uncertainty wavepackets – familiar coherent
states, best quantum approximation of classical points in phase space – are
einselected 80 ; 30 ; 55 .
## II Environment as a witness
Monitoring by the environment means that information about ${\mathcal{S}}$ is
deposited in ${\mathcal{E}}$. What role does it play, and what is its fate?
Decoherence theory ignores it. Environment is “traced out”. Information it
contains is treated as inaccessible and irrelevant: ${\mathcal{E}}$ is a “rug
to sweep under” the data that might endanger classicality.
Quantum Darwinism recognizes that “tracing out” is not what we do: Observers
eavesdrop on the environment. Vast majority of our data comes from fragments
of ${\mathcal{E}}$. Environment is a witness to the state of the system.
For example, this very moment you intercept a fraction of the photon
environment emitted by a screen or scattered by a page. We never access all of
${\mathcal{E}}$. Tiny fractions suffice to reveal the state of various
“systems of interest”.
This insight captures the essence of Quantum Darwinism: Only states that
produce multiple informational offspring – multiple imprints on the
environment – can be found out from small fragments of ${\mathcal{E}}$. The
origin of the emergent classicality is then not just survival of the fittest
states (the idea already captured by einselection), but their ability to
“procreate”, to deposit multiple records – copies of themselves – throughout
${\mathcal{E}}$.
---
Figure 1: _Quantum Darwinism and the structure of the environment_.
Decoherence theory distinguishes between a system ($\mathcal{S}$) and its
environment ($\mathcal{E}$) as in (a), but makes no further recognition of the
structure of ${\mathcal{E}}$; it could as well be monolithic. In Quantum
Darwinism the focus is on redundancy. We recognize the subdivision of
$\mathcal{E}$ into subsystems, as in (b). The only requirement for a subsystem
is that it should be individually accessible to measurements; observables of
different subsystems commute. To obtain information about ${\mathcal{S}}$ from
${\mathcal{E}}$ one can then measure _fragments_ ${\mathcal{F}}$ of the
environment – non-overlapping collections of subsystems of ${\mathcal{E}}$,
(c). ically, there are many copies of the information about ${\mathcal{S}}$ in
${\mathcal{E}}$ – “progeny” of the “fittest observable” that survived
monitoring by ${\mathcal{E}}$ proliferates throughout ${\mathcal{E}}$. This
proliferation of the multiple informational offspring defines Quantum
Darwinism. The environment becomes a witness with redundant copies of
information about the preferred observable. This leads to the objective
existence of pointer states: Many can find out the state of the system
independently, without prior information, and they can do it indirectly,
without perturbing ${\mathcal{S}}$.
Proliferation of records allows information about ${\mathcal{S}}$ to be
extracted from many fragments of ${\mathcal{E}}$ (in the example above, photon
${\mathcal{E}}$). Thus, ${\mathcal{E}}$ acquires redundant records of
${\mathcal{S}}$. Now, many observers can find out the state of ${\mathcal{S}}$
independently, and without perturbing it. This is how preferred states of
${\mathcal{S}}$ become objective. Objective existence – hallmark of
classicality – emerges from the quantum substrate as a consequence of
redundancy.
Decoherence theory was focused on the system. Its aim was to determine what
states survive information leaks to ${\mathcal{E}}$. Now we ask: What
information about the system can be found out from fragments of
${\mathcal{E}}$? This change of focus calls for a more realistic model of the
environment (Fig. 1): Instead of a monolithic ${\mathcal{E}}$ we recognize
that environments consist of subsystems that comprise fragments independently
accessible to observers.
The reduced density matrix $\rho_{\mathcal{S}}$ representing the state of the
system was the basic tool of decoherence. To study Quantum Darwinism we focus
on correlations between fragments of the environment and the system. The
relevant reduced density matrix $\rho_{{\mathcal{S}}{\mathcal{F}}}$ is given
by:
$\rho_{{\mathcal{S}}{\mathcal{F}}}=Tr_{{\mathcal{E}}/{\mathcal{F}}}|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle\\!\langle\Psi_{{\mathcal{S}}{\mathcal{E}}}|\
.$ $None$
Above, trace is over “${\mathcal{E}}$ less ${\mathcal{F}}$”, or
${{\mathcal{E}}/{\mathcal{F}}}$ – all of ${\mathcal{E}}$ except for the
fragment ${\mathcal{F}}$. How much ${\mathcal{F}}$ knows about ${\mathcal{S}}$
can be quantified using mutual information:
$I({\mathcal{S}}:{\mathcal{F}})=H_{{\mathcal{S}}}+H_{{\mathcal{F}}}-H_{{\mathcal{S}},{\mathcal{F}}}\
,$ $None$
defined as the difference between entropies of two systems (here
${\mathcal{S}}$ and ${\mathcal{F}}$) treated separately and jointly. For
example, the mutual information between an original and a perfect copy (of,
say, a book) is equal to the entropy of the original, as either contains the
same text. So, every bit of information in the first copy reveals a bit of
information in the original. However, having extra copies does not increase
the information about the original. Yet, it determines how many can
independently access this information. The number of copies defines
redundancy.
---
Figure 2: _Information about ${\mathcal{S}}$ stored in ${\mathcal{E}}$ and its
redundancy_. Mutual information is monotonic in $f$. When global state of
${\mathcal{S}}{\mathcal{E}}$ is pure, $I({\mathcal{S}}:{\mathcal{F}}_{f})$ in
a typical fraction $f$ of the environment is antisymmetric around $f=0.5$ 8 .
For pure states picked out at random from the combined Hilbert space
${\mathcal{H}}_{{\mathcal{S}}{\mathcal{E}}}$, there is little mutual
information between ${\mathcal{S}}$ and a typical ${\mathcal{F}}$ smaller than
half of ${\mathcal{E}}$. However, once a threshold $f=\frac{1}{2}$ is
attained, nearly all information is in principle at hand. Thus, such random
states (green line) exhibit no redundancy. By contrast, states of
${\mathcal{S}}{\mathcal{E}}$ created by decoherence (where the environment
monitors preferred observable of ${\mathcal{S}}$) contain almost all (all but
$\delta$) of the information about ${\mathcal{S}}$ in small fractions
$f_{\delta}$ of ${\mathcal{E}}$. The corresponding
$I({\mathcal{S}}:{\mathcal{F}}_{f})$ (red line) quickly rises to
$H_{\mathcal{S}}$ (entropy of ${\mathcal{S}}$ due to decoherence), which is
all of the information about ${\mathcal{S}}$ available from either
${\mathcal{E}}$ or ${\mathcal{S}}$. (More, up to $2H_{\mathcal{S}}$, can be
obtained only through global measurements on ${\mathcal{S}}$ and nearly all
${\mathcal{E}}$). $H_{\mathcal{S}}$ is therefore the _classically accessible
information_. As $(1-\delta)H_{\mathcal{S}}$ of information is contained in
$f_{\delta}=1/R_{\delta}$ of ${\mathcal{E}}$, there are $R_{\delta}$ such
fragments in ${\mathcal{E}}$: $R_{\delta}$ is the redundancy of the
information about ${\mathcal{S}}$. Large redundancy implies objectivity: The
state of the system can be found out indirectly and independently by many
observers, who will agree about their conclusions. Thus, _Quantum Darwinism
accounts for the emergence of objective existence_.
Similar ideas apply to the quantum case. Initially, every bit of information
gained from a fraction $f\ll 1$ of ${\mathcal{E}}$ that was pure before it
monitored (and decohered) the system is a bit about ${\mathcal{S}}$. The red
plot in Fig. 2 starts with this steep “bit for bit” slope, but moderates as
$I({\mathcal{S}}:{\mathcal{F}}_{f})$ approaches redundancy plateau at
$H_{{\mathcal{S}}}$, where additional bits only confirm what is already known.
Redundancy is the number of independent fragments of the environment that
supply almost all classical information about ${\mathcal{S}}$, i.e.,
$(1-\delta)H_{\mathcal{S}}$. In other words;
$R_{\delta}=1/f_{\delta}\ .$ $None$
$R_{\delta}$ is the number of times one can acquire $(1-\delta)$ of the
information about ${\mathcal{S}}$ independently (from distinct
${\mathcal{F}}$’s) and indirectly – without perturbing ${\mathcal{S}}$.
Rapid rise and gradual leveling of $I({\mathcal{S}}:{\mathcal{F}}_{f})$, Fig.
2, implies redundancy. The information in ${\mathcal{F}}_{f}$ allows one to
determine the state of ${\mathcal{S}}$ as it reaches redundancy plateau.
Observables of different ${\mathcal{F}}$’s commute – such measurements are
independent. Yet, underlying correlations mean that their outcomes imply the
same state of the system, as if ${\mathcal{S}}$ were classical: The redundancy
plateau is a classical plateau. Its level $H_{{\mathcal{S}}}$ is the classical
information accessible from a small fraction of ${\mathcal{E}}$.
Redundancy allows for objective existence of the state of ${\mathcal{S}}$: It
can be found out indirectly, so there is no danger of perturbing
${\mathcal{S}}$ with a measurement. Error correction allowed by redundancy is
also important: Fragility of quantum states means that copies in
${\mathcal{F}}$’s are damaged by measurements (we destroy photons!), and may
be measured in a “wrong” basis. One cannot access records in ${\mathcal{E}}$
without endangering their existence. But with many ($R_{\delta}$) copies,
state of ${\mathcal{S}}$ can be found out by $\sim R_{\delta}$ observers who
can get their information independently, and without prior knowledge about
${\mathcal{S}}$. Consensus between copies suggests objective existence of the
state of ${\mathcal{S}}$.
The mutual information $I({\mathcal{S}}:{\mathcal{F}}_{f})$ computed in models
of decoherence exhibits behavior illustrated by the red plot of Fig. 2. In the
family of models representing spin ${\mathcal{S}}$ surrounded by environments
of many spins 42 ; 8 ; 9 the same number of spins suffices to reach the
plateau: Adding more spins to ${\mathcal{E}}$ only extends length of the
plateau measured in “absolute units” – in the number of the environment spins.
In this model (that can be viewed as a simplified model of a photon
environment) redundancy is then proportional to the number of the environment
subsystems that interact with the system of interest.
Quantum Brownian motion – harmonic oscillator surrounded by many environmental
oscillators – is the other well known model of decoherence. It is exactly
solvable, and the case of an underdamped oscillator yields surprisingly simple
results 10 ; PR : (i) Mutual information is approximately given by
$I({\mathcal{S}}:{\mathcal{F}})\approx
H_{{\mathcal{S}}}+\frac{1}{2}\ln{\frac{f}{(1-f)}}$, and; (ii) Redundancy for
an initially squeezed state of ${\mathcal{S}}$ reaches $R_{\delta}\approx
s^{2\delta}$, where $s$, the squeeze factor, quantifies delocalization of the
state. Similar equation should hold for more general “Schrödinger cat” states,
with $s$ quantifying the separation of the two localized alternatives.
These results confirm intuitions that originally motivated Quantum Darwinism
75 ; Z2000 : Monitoring of the system by the environment can deposit multiple
records of preferred states of ${\mathcal{S}}$ in ${\mathcal{E}}$. States of
${\mathcal{S}}{\mathcal{E}}$ that arise from decoherence are special 8 ; 9 ,
as $I({\mathcal{S}}:{\mathcal{F}}_{f})$ for a typical pure state selected with
Haar measure in the whole Hilbert space of ${\mathcal{S}}{\mathcal{E}}$ (green
plot in Fig. 2) shows. In such random states small fragments reveal almost
nothing about the rest of the state. Only when half of ${\mathcal{E}}$ is
found out the whole state is suddenly revealed.
States that arise from decoherence are then far from random. Roughly speaking,
they have a branch structure. This is why the rest of such a branch including
the state of the system – the “bud” from which this branch has originated –
can be deduced from its fragment. We shall see how such branches grow in the
next section.
Plots of $I({\mathcal{S}}:{\mathcal{F}}_{f})$ for pure
${\mathcal{S}}{\mathcal{E}}$ are antisymmetric around the point
$\\{H_{{\mathcal{S}}},f=\frac{1}{2}\\}$ for typical fragments of
${\mathcal{E}}$ 8 . Thus, rapid rise for small $f$ must be matched at the
other end, for $f\sim 1$. This is a signature of entanglement that allows
state to be known “as the whole”, while states of subsystems are unknown. The
joint state of ${\mathcal{S}}{\mathcal{E}}$ is then pure, so that
$H_{{\mathcal{S}},{\mathcal{F}}={\mathcal{E}}}=0$, and
$I({\mathcal{S}}:{\mathcal{F}}_{f})$ must rise to
$H_{{\mathcal{S}}}+H_{{\mathcal{E}}}=2H_{{\mathcal{S}}}$ when $f$ approaches
1.
This is a very quantum aspect of information. In classical physics knowing a
composite object implies knowing each of its subsystems. This is not so in
quantum physics, where composite states are given by tensor (rather than
Cartesian) products of their constituents. Thus, one can know perfectly
quantum state of the whole, but know nothing about states of parts. We shall
see in Section IV how this feature can be used to derive Born’s rule 12 that
relates probabilities with wavefunctions.
To reveal this latent quantumness one would have to measure the right global
observable on all of ${\mathcal{S}}{\mathcal{E}}$. For example, when mutual
information, Eq. (3), is defined using Shannon entropy with probabilities
corresponding to optimal observables in ${\mathcal{S}}$ and in
${\mathcal{E}}$, the resulting Shannon $I({\mathcal{S}}:{\mathcal{F}}_{f})$
graph for small $f$ would look very similar to Fig. 2. However, using Shannon
entropy involves local probabilities (precluding global observables), so such
Shannon $I({\mathcal{S}}:{\mathcal{F}}_{f})$ never exceeds
$H_{{\mathcal{S}}}$, antisymmetry is lost, and the plateau continues until the
end at $f\sim 1$.
Effective unattainability of the $f\sim 1$ part of the plot also shows why
decoherence is so hard to undo: Correlations that reveal coherence can be
usually detected only by such global measurements of whole
${\mathcal{S}}{\mathcal{E}}$. We intercept small fractions of ${\mathcal{E}}$,
and never have the luxury of perfect global measurements needed to undo
decoherence. Yet, because of redundancy, we get $\sim H_{{\mathcal{S}}}$
information with “sloppy” measurements of $f\ll 1$.
Quantum Darwinism does not require pure ${\mathcal{E}}$. Mixed environment is
a noisy communication channel: Its initial entropy of $h$ per bit can still
increase after interaction with ${\mathcal{S}}$, reflecting mutual information
buildup. However, now a bit gained from ${\mathcal{E}}$ yields only $1-h$ of a
bit about ${\mathcal{S}}$. So, a completely mixed ${\mathcal{E}}$ ($h=1$) is
useless (even though it can still induce decoherence!). For a partly mixed
${\mathcal{E}}$ mutual information will increase more slowly, pure case “bit
per bit” rate tempered to $\sim 1-h$. Yet, it can still climb the same
redundancy plateau at $H_{{\mathcal{S}}}$ QZZ .
These conclusions apply when ${\mathcal{E}}$ is initially mixed, but are also
relevant when this channel is noisy for other reasons (e.g., imperfect
measurements). In all such cases one can still reach the same redundancy
plateau, although now a proportionally larger fragment of the environment is
needed to get the same information about ${\mathcal{S}}$.
Suitability of the environment as a channel depends on whether it provides a
direct and easy access to the records of the system. This depends on the
structure and evolution of ${\mathcal{E}}$. Photons are ideal in this respect:
They interact with various systems, but, in effect, do not interact with each
other. This is why light delivers most of our information. Moreover, photons
emitted by the usual sources (e.g., sun) are far from equilibrium with our
surroundings. Thus, even when decoherence is dominated by other environments
(e.g., air) photons are much better in passing on information they acquire
while “monitoring the system of interest”: Air molecules scatter from one
another, so that whatever record they may have gathered becomes effectively
undecipherable.
Stability of the level of the redundancy plateau at $H_{{\mathcal{S}}}$, even
for mixed ${\mathcal{E}}$’s, is a compelling reason to think of it as
“classical”. The question we shall now address concerns the nature of that
information – what does the environment know about the system, and why?
## III From Copying to Quantum Jumps
Quantum Darwinism leads to appearance, in the environment, of multiple copies
of the state of the system. However, the no-cloning theorem 24 ; 65 prohibits
copying of unknown quantum states. If cloning is outlawed, how can redundancy
seen in Fig. 2 be possible?
Figure 3: Quantum Darwinism in a simple model of decoherence 42 . The
spin-$\frac{1}{2}$ ${\mathcal{S}}$ interacts with $N=50$ spin-$\frac{1}{2}$
subsystems of ${\mathcal{E}}$ with an Ising Hamiltonian ${\bf
H_{{\mathcal{S}}{\mathcal{E}}}}=\sum_{k=1}^{N}g_{k}\sigma_{z}^{{\mathcal{S}}}\otimes\sigma_{y}^{{\mathcal{E}}_{k}}$.
The initial state of $\mathcal{S}\otimes{\mathcal{E}}$ is
$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes|0\rangle^{{\mathcal{E}}_{1}}\otimes\ldots\otimes|0\rangle^{{\mathcal{E}}_{N}}$.
Couplings $g_{k}$ are distributed randomly in the interval (0,1]. All the
plotted quantities are a function of the observable
$\sigma(\mu)=\cos(\mu)\sigma_{z}+\sin(\mu)\sigma_{x}$, where $\mu$ is the
angle between its eigenstates and the pointer states of ${\mathcal{S}}$ –
eigenstates of $\sigma_{z}^{\mathcal{S}}$. a) Information acquired by the
optimal measurement on the whole environment, $\hat{I}_{N}(\sigma)$, as a
function of the inferred observable $\sigma(\mu)$ and the average interaction
action $\langle g_{k}t\rangle=a$. A lot of information is accessible in the
whole ${\mathcal{E}}$ about any observable $\sigma(\mu)$ except when $a$ is so
small that there was no decoherence. b) Redundancy of the information about
${\mathcal{S}}$ as a function of the inferred observable $\sigma(\mu)$ and the
average action $\langle g_{k}t\rangle=a$. $R_{\delta=0.1}(\sigma)$ counts the
number of times 90% of the total information can be “read off” independently
by measuring distinct fragments of ${\mathcal{E}}$. It is sharply peaked
around the pointer observable: Redundancy is a very selective criterion – the
number of copies of relevant information is high only for the observables
$\sigma(\mu)$ inside the theoretical bound (see Ref.42 ) indicated by the
dashed line. c) Information about $\sigma(\mu)$ extracted by local random
measurements on $m$ environmental subsystems. Because of redundancy, pointer
states – and only pointer states – can be found out through this far-from-
optimal strategy. Information about any other observable $\sigma(\mu)$ is
restricted to what can be inferred from the pointer observable 42 .
Quick answer is that cloning refers to (unknown) quantum states. So, copying
of observables evades the theorem. Nevertheless, the tension between the
prohibition on cloning and the need for copying is revealing: It leads to
breaking of unitary symmetry implied by the superposition principle, accounts
for quantum jumps, and suggests origin of the “wavepacket collapse”, setting
stage for the study of quantum origins of probability in Section IV.
Quantum physics is based on several “textbook” postulates 23 . The first two;
(i) States are represented by vectors in Hilbert space, and; (ii) Evolutions
are unitary – give complete account of mathematics of quantum theory, but make
no connection with physics. For that one needs to relate calculations made
possible by the superposition principle of (i) and unitarity of (ii) to
experiments.
Postulate (iii) Immediate repetition of a measurement yields the same outcome
starts this task. This is the only uncontroversial measurement postulate (even
if it is difficult to approximate in the laboratory): Such repeatability or
predictability is behind the very idea of “a state”.
In contrast to (i)-(iii), collapse postulate (iv) Outcomes correspond to
eigenstates of the measured observable, and only one of them is detected in
any given run of the experiment, is inconsistent with (i) and (ii). Conflict
arises for two reasons: Restriction to a preferred set of outcome states seems
at odds with with the egalitarian principle of superposition, embodied in (i).
This restriction prevents one from finding out unknown quantum states, so it
is responsible for their fragility. And a single outcome per run is at odds
with unitarity (and, hence, linearity) of quantum dynamics that preserves
superpositions.
The last axiom; (v) Probability of an outcome is given by the square of the
associated amplitude, $p_{k}=|\psi_{k}|^{2}$, is known as Born’s rule 12 . It
completes the relation between mathematics of (i) and (ii) and the
experiments.
Bohr bypassed conflict of (i) and (ii) with (iv) by insisting that apparatus
is classical, so unitarity and the principle of superposition need not apply
to measurements. But this is an excuse, not an explanation. We are dealing
with a quantum environment, and redundancy of previous section strengthened
motivation for postulate (iii) – repeatability. Let us see where this demand
takes us in a purely quantum setting of postulates (i), (ii), and (iii).
Suppose there are states of ${\mathcal{S}}$ (say, $|u\rangle$ and $|v\rangle$)
that produce an imprint in a subsystem of ${\mathcal{E}}$ (which plays a role
of an apparatus), but remain unperturbed (so they can produce more imprints).
This repeatability implies:
$|u\rangle|e_{0}\rangle\Rightarrow|u\rangle|e_{u}\rangle$,
$|v\rangle|e_{0}\rangle\Rightarrow|v\rangle|e_{v}\rangle$ in obvious notation.
In a unitary process scalar product is preserved. Thus;
$\langle u|v\rangle=\langle u|v\rangle\langle e_{u}|e_{v}\rangle\ ,$ $None$
where we have set $\langle e_{0}|e_{0}\rangle=1$. This simple equation can be
satisfied only when; (a) either $\langle e_{u}|e_{v}\rangle=1$ (which means
that copying was completely unsuccessful), or; (b) $\langle u|v\rangle=0$,
i.e., they are orthogonal. In that case $\langle e_{u}|e_{v}\rangle$ is
arbitrary – perfect record $\langle e_{u}|e_{v}\rangle=0$ is also possible.
It follows that multiple (perfect or imperfect) copies of $|u\rangle$ and
$|v\rangle$ can be imprinted in disjoint ${\mathcal{F}}$’s. As a consequence
of unitarity, only sets of orthogonal states (that define Hermitean
observables 23 ) can be so copied, explaining selection of a set of outcomes –
terminal points of quantum jumps 79 . Before, they had to be postulated by the
first part of axiom (iv). We emphasize that this result relies on just two
values of the scalar product – 0 and 1 – and, thus, does not appeal to Born’s
rule.
This breaking of unitary symmetry (choice of preferred states in an
egalitarian Hilbert space) is induced by repeatability of the information
transfer. It is a “nonlinear demand”: As in cloning, one asks for “two (or
more) of the same”. Its conflict with linearity of quantum theory can be
resolved only by restricting states that can be copied. Such pointer states
then act as “buds” of branches that grow by reproducing, in ${\mathcal{E}}$,
multiple copies of the original in ${\mathcal{S}}$. Interaction Hamiltonians
do not perturb observables that commute with them. So, buds of branches
coincide with the einselected pointer states.
Evidence of such symmetry breaking is seen in Fig. 3. Mutual information and
redundancy shown there are obtained using Eq. (3), but with Shannon (rather
than von Neumann) entropies of specific observables of ${\mathcal{S}}$ and
${\mathcal{F}}$, i.e., using probabilities of their eigenstates. While von
Neumann-based $I({\mathcal{S}}:{\mathcal{F}}_{f})$ and $R_{\delta}$
characterized total information, Shannon-based counterparts are well suited to
enquire: What observable is this information about?
It turns out that the environment as a whole “knows” many observables of
${\mathcal{S}}$, as is seen in Fig. 3a. By contrast, in Fig. 3b symmetry
breaking is evident: The ridge of redundancy appears abruptly only when test
observable $\sigma(\mu)$ and the preferred pointer observable $\sigma_{z}$
(that remains unperturbed by the environment) nearly coincide.
Why are pointer states favored? Commonsense says that, to be reproduced, state
must survive copying. This leads to a theorem 42 ; 43 that only pointer
states can be discovered from fractions of ${\mathcal{E}}$. Other observables
(such as $\sigma(\mu)$ in Fig. 3) can be deduced only to the extent they are
correlated with the pointer observable. So, fragments of the environment offer
a very narrow, projective point of view. Redundant imprinting of some
observables happens at the expense of their complements.
Structure of branching state betrays its origin and foreshadows “collapse”.
Starting from
$|\psi_{{\mathcal{S}}}\rangle=\sum_{k}^{n}\psi_{k}|s_{k}\rangle$,
$|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle=\sum_{k}^{n}\psi_{k}|s_{k}\rangle|e^{(1)}_{k}\rangle\dots|e^{({\mathcal{N}})}_{k}\rangle=\sum_{k}^{n}\psi_{k}|s_{k}\rangle|\varepsilon_{k}\rangle\
\ (6)$
branches grow to include $\cal N$ subsystems of ${\mathcal{E}}$. Branch
fragments can be nearly orthogonal; $\Pi_{j=1}^{J}\langle
e_{k}^{(j)}|e_{k^{\prime}}^{(j)}\rangle\simeq\delta_{kk^{\prime}}$ for large
enough $J$. This means that a pointer state $|s_{k}\rangle$ of ${\mathcal{S}}$
can be determined (along with the rest of the branch) from a sufficiently long
fragment (which may still be short compared to the length of the branch,
$J\ll\cal N$).
In the huge Hilbert space ${\mathcal{H}}_{{\mathcal{S}}{\mathcal{E}}}$
branching state is a very atypical minimally entangled superposition of only
$n$ product “branches” labelled by the pointer states of the system. This is
tiny compared to the dimension of ${\mathcal{H}}_{{\mathcal{S}}{\mathcal{E}}}$
that exceeds $n$ by a factor exponential in ${\mathcal{N}}$. This is why the
two plots in Fig. 2 are so different: Branching state is, to a good
approximation, a multi-system Schmidt decomposition, with long branch
fragments constituting “systems”. In a Schmidt decomposition, states of
partners are in one-to-one correspondence. Thus, in Eq. (6), $|s_{k}\rangle$
implies $|\varepsilon_{k}\rangle$ (and, vice versa), and measuring a branch
fragment ${\mathcal{F}}$ can reveal the whole branch.
Initial part of $I({\mathcal{S}}:{\mathcal{F}}_{f})$, Fig. 2, represent
buildup of this correlation: When $f=0$, observer is ignorant of what branch
he will find out, but the structure of the correlations within
$|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$ leaves no doubt of what these
branches are. Using Born’s rule one could assign to them probabilities
$p_{k}=|\psi_{k}|^{2}$ and the corresponding entropy $H_{{\mathcal{S}}}$. Next
section shows how one can deduce these probabilities without axiom (v) – how
symmetries of entanglement imply Born’s rule.
When observer measures enough of ${\mathcal{E}}$, he finds out the branch (and
what the state of ${\mathcal{S}}$ is). Additional data are redundant. They
only confirm what is already known. Probabilities associated with
$|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$ are replaced with certainty of a
branch. This transition from uncertainty (initial presence of many branches –
potential for multiple outcomes) to certainty (once a sufficiently long branch
fragment becomes known) accounts for perception of “collapse”. The initial,
steeply rising, part of $I({\mathcal{S}}:{\mathcal{F}}_{f})$ “resolves” it:
Collapse is brief compared to the ensuing period of certainty about the
outcome, as $f_{\delta}\ll 1$, but, nevertheless, not instantaneous.
Assumptions that lead from copying to preferred states can be relaxed. Thus,
${\mathcal{E}}$ need not be initially pure 79 . Moreover, it suffices that the
records (e.g., in the apparatus ${\cal A}$) are “repeatably accessible”.
Transfer of responsibility for repeatability from a quantum ${\mathcal{S}}$ to
a (still quantum) ${\cal A}$ allows one to model non-orthogonal measurement
outcomes (POVM’s): ${\cal A}$ entangles with the system, and then acts as
ancilla. Its orthogonal pointer states $|A_{k}\rangle$ correlate with non-
orthogonal $|\varsigma_{k}\rangle$ of ${{\mathcal{S}}}$,
$\sum_{k}\tilde{\psi}_{k}|\varsigma_{k}\rangle|A_{k}\rangle$. Interaction of
$\cal A$ with the environment results in multiple copies of $|A_{k}\rangle$.
The usual projective measurement implementation of POVM’s (see e.g. NC ) is
now straightforward. Branches are labelled by $|A_{k}\rangle$. Indeed, we
usually experience “quantum jumps” via an apparatus pointer.
Selection of the set of outcomes by the proliferation of information essential
for Quantum Darwinism parallels Bohr’s insistence 11 that a “classical
apparatus” should determine the outcomes. However, it follows from purely
quantum Eq. (5), and is caused by a unitary evolution responsible for the
information transfer. Nevertheless, as classical apparatus would, preferred
pointer states designate possible future outcomes, precluding measurements of
complementary observables or determining preexisting state of the system.
Thus, information acquisition – a copying process – results in preferred
states.
Consensus between records deposited in fragments of ${\mathcal{E}}$ looks like
“collapse”. In this sense we have accounted for postulate (iv) using only very
quantum postulates (i)-(iii). In particular, in deriving and analyzing Eq. (5)
we have not employed Born’s rule, axiom (v). We shall be therefore able to use
our results as a starting point for such a derivation in the next section.
There was nothing nonunitary above – unitarity was the crux of our argument,
and orthogonality of branch seeds our main result. Relative states of Everett
25 ; 26 ; 22 come to mind. One could speculate about reality of branches with
other outcomes. We abstain from this – our discussion is interpretation-free,
and this is a virtue. Indeed, “reality” or “existence” of universal state
vector seems problematic. Quantum states acquire objective existence when
reproduced in many copies. Individual states – one might say with Bohr – are
mostly information, too fragile for objective existence. And there is only one
copy of the Universe. Treating its state as if it really existed 25 ; 26 ; 22
seems unwarranted and “classical”.
## IV Probabilities from Entanglement
---
Figure 4: Probabilities and symmetry: (a) Laplace used subjective ignorance to
define probability. Player who does not know face values of the cards, but
knows that one of them is a spade will infer probability
$p_{\spadesuit}={\frac{1}{2}}$ for the top card. (b) The real physical state
of the system is however altered by the swap, illustrating subjective nature
of Laplace’s approach, and demonstrating its unsuitability for physics. (c)
Perfectly known entangled states have objective symmetries that allow one to
rigorously deduce probabilities. When two systems are maximally entangled as
above, probabilities of Schmidt partners are equal,
$p_{\heartsuit}=p_{\diamondsuit}$, and $p_{\spadesuit}=p_{\clubsuit}$. After a
swap
$u_{{\mathcal{S}}}=|\spadesuit\rangle\langle\heartsuit|+|\heartsuit\rangle\langle\spadesuit|$
in ${\mathcal{S}}$, the resulting state
$|\spadesuit\rangle|\diamondsuit\rangle+|\heartsuit\rangle|\clubsuit\rangle$
must have $p^{\prime}_{\spadesuit}=p_{\diamondsuit}$, and
$p^{\prime}_{\heartsuit}=p_{\clubsuit}$. (We ‘primed’ probabilities in
${\mathcal{S}}$, as it was acted upon by a swap, so they might have changed.)
A counterswap
$u_{{\mathcal{E}}}=|\diamondsuit\rangle\\!\langle\clubsuit|+|\clubsuit\rangle\\!\langle\diamondsuit|$
in ${\mathcal{E}}$ restores the original entangled state, proving that
$p^{\prime}_{\heartsuit}=p_{\heartsuit}$ and
$p^{\prime}_{\spadesuit}=p_{\spadesuit}$, after all (as counterswap
$u_{{\mathcal{E}}}$ leaves ${\mathcal{S}}$ untouched). This sequence of
equalities implies $p_{\spadesuit}=p_{\diamondsuit}=p_{\heartsuit}$, so that
$p_{\spadesuit}=p_{\heartsuit}=\frac{1}{2}$, as probabilities in
${\mathcal{S}}$ must add up to 1.
Observer prepared ${\mathcal{S}}$ in a state $|{\psi}_{{\mathcal{S}}}\rangle$,
but wants to measure observable with eigenstates $\\{|s_{k}\rangle\\}$. This
will lead to entangled $|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$ with branch
structure, Eq. (6). Pointer states $\\{|s_{k}\rangle\\}$ define the outcomes,
but, as yet, observer has not measured ${\mathcal{E}}$, and does not know the
result. Given $|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$, what is the
probability of, say, $|s_{17}\rangle$?
To derive it we cannot use reduced density matrices, Eqs. (1,2). Tracing out
is averaging Landau ; 59 ; NC – it relies on $p_{k}=|\psi_{k}|^{2}$, Born’s
rule we want to derive. We have imposed that ban while deriving and analyzing
Eq. (5), but relaxed it to plot Fig. 3. Now we reimpose it again. So, Born’s
rule and standard tools of decoherence are off limits – using them courts
circularity. Our derivation will rest instead on certainty and symmetry,
cornerstones that mark two extremal cases of probability.
The case of certainty was just settled without Born’s rule using Eq. (5). When
one re-measures an observable, the same outcome will be seen again. Thus, when
$\\{|s_{k}\rangle\\}$ includes $|{\psi}_{{\mathcal{S}}}\rangle$ (e.g.,
$|{\psi}_{{\mathcal{S}}}\rangle=|s_{17}\rangle$), newly added copies just
extend the branch already correlated with observer’s state, and the outcome is
certain; $p_{17}=1$. Certainty of correlations between partners in Schmidt
decomposition, Eq. (6) is another important example.
Certainty seems trivial but is important. Confirmation that a state “is what
it is” – postulate (iii) – is a part of standard quantum lore 23 . We re-
affirmed it, but with a key insight: Redundancy allows observers to discover
(and not just confirm) that ${\mathcal{S}}$ is in a certain pointer state.
We now turn to the opposite case of complete indeterminacy. Its connection
with symmetry was noted by Laplace. He wrote: “The theory of chance consists
in reducing all the events … to a certain number of cases that are equally
possible… The ratio of this number to that of all the cases possible is the
measure of probability ” 40 . Figure 4 illustrates how this classical
intuition yields – far more convincingly — quantum probabilities.
Symmetry is probed by invariance. Transformations that respect it take system
between states that exhibit no measurable differences. For example, change of
phase in the coefficients in the Schmidt decomposition
$|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle=\sum_{k}^{n}\psi_{k}|s_{k}\rangle|\varepsilon_{k}\rangle$
cannot influence the state of ${\mathcal{S}}$: It is induced by
$u_{{\mathcal{S}}}=e^{i\phi_{k}}|s_{k}\rangle\\!\langle s_{k}|$, local unitary
on ${\mathcal{S}}$, that can be “undone” by
$u_{{\mathcal{E}}}=e^{-i\phi_{k}}|\varepsilon_{k}\rangle\\!\langle\varepsilon_{k}|$
on ${\mathcal{E}}$, or;
$u_{{\mathcal{S}}}\otimes{\bf
1}_{{\mathcal{E}}}|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle=|\Phi_{{\mathcal{S}}{\mathcal{E}}}\rangle;\
{\bf 1}_{{\mathcal{S}}}\otimes
u_{{\mathcal{E}}}|\Phi_{{\mathcal{S}}{\mathcal{E}}}\rangle=|\Psi_{{\mathcal{S}}{\mathcal{E}}}\rangle$
$None$
So, phases of $\psi_{k}$ cannot matter for a local state or influence
probabilities in ${\mathcal{S}}$. This symmetry, Eq. (7), is the entanglement-
assisted invariance or envariance 76 ; 78 .
Such loss of phase significance for ${\mathcal{S}}$ entangled with
${\mathcal{E}}$ implies decoherence 78 . We arrived at its essence using
envariance, without reduced density matrices, trace, etc.
We now use phase envariance to show that equal absolute values of the
coefficients $\psi_{k}$ imply equal probabilities. For equal $|\psi_{k}|$ any
orthogonal basis of ${\mathcal{S}}$ is “Schmidt” (i.e., has an orthogonal
partner in ${\mathcal{E}}$). Thus,
$|\bar{\varphi}_{{\mathcal{S}}{\mathcal{E}}}\rangle=\frac{{|0\rangle}_{{\mathcal{S}}}{|0\rangle}_{{\mathcal{E}}}+{|1\rangle}_{{\mathcal{S}}}{|1\rangle}_{{\mathcal{E}}}}{\sqrt{2}}=\frac{{|+\rangle}_{{\mathcal{S}}}{|+\rangle}_{{\mathcal{E}}}+{|-\rangle}_{{\mathcal{S}}}{|-\rangle}_{{\mathcal{E}}}}{\sqrt{2}}$,
where $|\pm\rangle=\frac{|0\rangle\pm|1\rangle}{\sqrt{2}}$. Sign change
induced by $e^{i\pi}|-\rangle\\!\langle-|$ acting on ${\mathcal{S}}$ produces
$|\bar{\eta}_{{\mathcal{S}}{\mathcal{E}}}\rangle=\frac{{|+\rangle}_{{\mathcal{S}}}{|+\rangle}_{{\mathcal{E}}}-{|-\rangle}_{{\mathcal{S}}}{|-\rangle}_{{\mathcal{E}}}}{\sqrt{2}}=\frac{{|1\rangle}_{{\mathcal{S}}}{|0\rangle}_{{\mathcal{E}}}+{|0\rangle}_{{\mathcal{S}}}{|1\rangle}_{{\mathcal{E}}}}{\sqrt{2}}$.
In other words, one can swap ${|0\rangle}_{{\mathcal{S}}}$ with
${|1\rangle}_{{\mathcal{S}}}$ by rotating phase in a $|\pm\rangle$ basis by
$\pi$. Yet, we just saw that phases of Schmidt coefficients do not matter for
the state of ${\mathcal{S}}$, so probabilities of 0 and 1 in ${\mathcal{S}}$
must have remained the same. Moreover, probabilities of paired up Schmidt
states are equal, so that $p_{{\mathcal{S}}}(0)=p_{{\mathcal{E}}}(0)$ in
$|\bar{\varphi}_{{\mathcal{S}}{\mathcal{E}}}\rangle$ and
$p_{{\mathcal{S}}}(1)=p_{{\mathcal{E}}}(0)$ in
$|\bar{\eta}_{{\mathcal{S}}{\mathcal{E}}}\rangle$. Hence,
$p_{{\mathcal{S}}}(0)=p_{{\mathcal{S}}}(1)=\frac{1}{2}$, where we assumed that
probabilities add up to 1.
In contrast to Laplace’s subjective “ignorance-based” approach, we obtained
objective probabilities for a completely known entangled state. Phase
envariance implied equiprobability in ${\mathcal{S}}$. To paraphrase Beatles,
“All you need is phase…”. We rotated phases of the coefficients to induce a
swap in a complementary basis. Another proof (that implements swap more
directly) is given in Fig. 4.
This equiprobability case is the difficult part of the proof. Instead of
subjectivity (that undermined applicability of Laplace’s approach to physics)
we relied on objective symmetries of entangled quantum states. This was made
possible by the nature of quantum states of composite systems. Classically,
pure states have structure of a Cartesian product – knowing the whole implies
knowledge of each subsystem. In quantum theory they are tensor products – one
can know state of the whole, and thus know nothing about parts, as envariance
shows.
This was the basis of our proof of equiprobability. We assumed unitarity.
Moreover, we assumed; (1) When a system is not acted upon by a unitary
transformation, its state remains unaffected. This state is a property of
${\mathcal{S}}$ alone, so; (2) Predictions regarding measurement outcomes on
${\mathcal{S}}$ (including their probabilities) can be inferred from the state
of ${\mathcal{S}}$. Last not least; (3) When ${\mathcal{S}}$ is entangled with
other systems (e.g., the environment) the state of ${\mathcal{S}}$ alone is
determined by the state of the whole ${\mathcal{S}}{\mathcal{E}}$.
These “facts of life” are accepted properties of systems and states, but given
the fundamental nature of our discussion it seems a good idea to make them
explicit 78 .
For instance, to establish independence from phases of the coefficients
$\psi_{k}$ we noted that the state of ${\mathcal{S}}$ is unaffected by the
unitaries $u_{{\mathcal{S}}}$ diagonal in Schmidt basis acting on
${\mathcal{S}}$ (like changes of Schmidt coefficient phases) that would
normally affect isolated ${\mathcal{S}}$: The global state
$\Psi_{{\mathcal{S}}{\mathcal{E}}}$ is restored by $u_{{\mathcal{E}}}$. Thus,
by fact (3), so is local state of ${\mathcal{S}}$. However, this is done by a
unitary “countertransformation” acting solely on ${\mathcal{E}}$. Hence, by
fact (1), state of ${\mathcal{S}}$ must have been unaffected by
$u_{{\mathcal{S}}}$ in the first place. So, by fact (2), phases of $\psi_{k}$
cannot change outcomes of any measurement on ${\mathcal{S}}$. Equiprobability
follows.
One can now derive Born’s rule, $p_{k}=|\psi_{k}|^{2}$, with straightforward
algebra from the above two simple cases of complete certainty ($p_{k}=1$) and
equiprobability ($p_{k}=\frac{1}{n}$): The general case can be always reduced
to the case case of equal coefficients by “finegraining” (see Box).
The origin of probability is a fascinating problem that is older than quantum
measurement problem, and is forgotten primarily because it is so old. We have
seen how quantum physics sheds a new, very fundamental, light on probability.
We cannot do justice to the history of this subject here, but Ref. Aul
provides a basic overview and exhaustive set of references. In particular,
envariant derivation is very different from the classic proof of Gleason Gle
in that it sheds light on the physical significance of the resulting measure.
Moreover, it does not assume probabilities are additive (except to posit that
probability of an event and its complement are certain, i.e., to establish
normalization; see Box and Ref. 78 ; Z07 ). Bypassing additivity of
probabilities is essential when dealing with a theory with another principle
of additivity – the quantum superposition principle – which trumps additivity
of probabilities or at least classical intuitiions about it (e.g., in the
double-slit experiment). Discussion of the implications of envariance has
already started, with SF ; Bar , and 52 providing insightful commentary.
BOX
We show here how “finegraining” reduces the case of arbitrary $\psi_{k}$ to
equiprobability. To illustrate general strategy consider state in a 2D Hilbert
space ${\cal H}_{{\mathcal{S}}}$ of ${\mathcal{S}}$ spanned by orthonormal
$\\{|0\rangle,|2\rangle\\}$ and (at least) 3D ${\cal H}_{{\mathcal{E}}}$:
$\ \ \ \ \ \ |\psi_{\cal SE}\rangle\ \propto\
\sqrt{\frac{2}{3}}~{}|0\rangle_{{\mathcal{S}}}|+\rangle_{{\mathcal{E}}}\ \ +\
\ \sqrt{\frac{1}{3}}~{}|2\rangle_{{\mathcal{S}}}|2\rangle_{{\mathcal{E}}}\ .$
The state
$|+\rangle_{{\mathcal{E}}}=\frac{|0\rangle_{{\mathcal{E}}}+|1\rangle_{{\mathcal{E}}}}{\sqrt{2}}$
exists in (at least 2D) subspace of ${\cal E}$ orthogonal to
$|2\rangle_{{\mathcal{E}}}$, i.e., $\langle 0|1\rangle=\langle
0|2\rangle=\langle 1|2\rangle=\langle+|2\rangle=0$. We know we can ignore
phases.
To reduce $|\psi_{\cal SE}\rangle$ to equal coefficients case we “extend it”
to a state $|\bar{\Psi}_{\cal SEC}\rangle$ by letting ${\mathcal{E}}$ act on
an ancilla ${\cal C}$. (${\mathcal{S}}$ is not acted upon, so, by fact (1),
probabilities for ${\mathcal{S}}$ cannot change.) This can be done by a
generalization of controlled-not acting between ${\cal E}$ (control) and
${\cal C}$ (target), so that (in obvious notation)
$|k\rangle|0^{\prime}\rangle\Rightarrow|k\rangle|k^{\prime}\rangle$, leading
to
$\sqrt{2}|0\rangle|+\rangle|0^{\prime}\rangle+|2\rangle|2\rangle|0^{\prime}\rangle\Rightarrow\sqrt{2}|0\rangle{{|0\rangle|0^{\prime}\rangle+|1\rangle|1^{\prime}\rangle}\over\sqrt{2}}+|2\rangle|2\rangle|2^{\prime}\rangle.$
Above, and from now on we skip subscripts: The state of ${\cal S}$ will be
listed first, and the state of ${\cal C}$ will be primed.
The cancellation of $\sqrt{2}$ yields an equal coefficient state:
$|\bar{\Psi}_{\cal
SCE}\rangle\propto|0,0^{\prime}\rangle|0\rangle+|0,1^{\prime}\rangle|1\rangle+|2,2^{\prime}\rangle|2\rangle\
.$
We have combined ${\mathcal{S}}$ and ${\cal C}$ in a single ket and (below) we
shall swap states of ${\cal SC}$ as if it was a single system.
Clearly, this is a Schmidt decomposition of (${\mathcal{S}}\cal C){\cal E}$.
Three orthonormal product states have coefficients with the same absolute
value. Therefore, they can be envariantly swapped. Thus, the probabilities of
states $|0\rangle|0^{\prime}\rangle,\ |0\rangle|1^{\prime}\rangle,$ and
$|2\rangle|2^{\prime}\rangle$ are all equal. By normalization they are
$\frac{1}{3}$. So, probability of detecting state $|2\rangle$ of
${\mathcal{S}}$ is $\frac{1}{3}$. Moreover, $|0\rangle$ and $|2\rangle$ are
the only two outcome states for ${\mathcal{S}}$. It follows that probability
of $|0\rangle$ must be $\frac{2}{3}$;
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p_{0}=\frac{2}{3};\ \
p_{2}=\frac{1}{3}\ .$
This is Born’s rule. We have just seen why the amplitudes in the initial
$|\psi_{\cal SE}\rangle$ “get squared” to yield probabilities.
Note that we have avoided assuming additivity of probabilities:
$p_{0}=\frac{2}{3}$ not because it is a sum of two fine-grained alternatives
for ${\mathcal{S}}{\mathcal{E}}$, each with probability of $\frac{1}{3}$, but
rather because there are only two (mutually exclusive and exhaustive)
alternatives for ${\mathcal{S}}$; $|0\rangle$ and $|2\rangle$, and
$p_{2}=\frac{1}{3}$. Therefore, by normalization, $p_{0}=1-\frac{1}{3}$.
Probabilities of Schmidt states can be added because of the loss of phase
coherence that follows directly from phase envariance established earlier (see
also Ref. 76 ; 78 ).
Extension of this proof to the case where probabilities are commensurate is
conceptually straightforward but notationally cumbersome. The case of non-
commensurate probabilities is settled with an appeal to continuity. Frequency
of the outcomes can be also deduced, allowing one to establish connection with
the familiar relative frequency approach to probabilities 76 ; 78 ; Z07 , but
in a quantum setting probability arises as a consequence of symmetries of a
single entangled state.
We end by noting that the finegraining discussed above does not need to be
carried out experimentally each time probabilities are discussed: Rather, it
is a way to deduce a measure that is consistent with the geometry of the
Hilbert spaces using entanglement as a tool. Still, given fundamental
implications of envariance experimental tests would be most useful.
## V Discussion
We derived the two controversial quantum postulates from the first three. We
have thus seen how classical domain of the Universe arises from the
superposition principle (postulate (i)) and unitarity (postulate (ii)) as well
as rudimentary assumptions about information flows (postulate (iii)), and a
few basic facts about states of composite quantum systems (including their
tensor nature, often cited as additional “axiom (0)”).
The essence of the measurement problem – accounting for axioms (iv) and (v) –
has been largely settled. It is of course likely one may be able to clarify
assumptions and simplify proofs. Much work remains to be done on Quantum
Darwinism and envariance. Nevertheless, nature of the quantum-classical
correspondence has been clarified.
Physicists take it for granted that even hard problems are solved by a single
good idea. Therefore, when a single idea does not do the whole job, often our
first instinct is to dismiss it. Measurement problem does not fall into this
“single idea” category. Several ideas, applied in the right order, led to
advances described here. Logically, we may well have started with the
derivation of Eq. (5) and the analysis of quantum jumps. Their randomness
leads to probabilities. And symmetries of entangled states (that arise in
decoherence and Quantum Darwinism) allow one to derive Born’s rule. As we have
seen, phase envariance is (nearly) “all you need”. With probabilities at hand
one has then every right to use reduced density matrices to analyze Quantum
Darwinism and decoherence.
Our presentation was “historical”. We started with decoherence, and used it to
introduce Quantum Darwinism. Analysis of copying essential to information
flows in both of these phenomena led to quantum jumps. This in turn motivated
entangelment-based derivation of Born’s rule. Quantum Darwinism – upgrade of
${\mathcal{E}}$ to a communication channel from a mundane role it played in
decoherence – tied together all of the other developments. This order had the
advantage of making motivations clear, but it is different from more logical
presentation where postulates (i)-(iii) are the starting point (strategy
followed in Z07 ).
The collection of ideas discussed here allows one to understand how “the
classical” emerges from the quantum substrate staring from more basic
assumptions than decoherence. We have bypassed a related question of why is
our Universe quantum to the core. The nature of quantum state vectors is a
part of this larger mystery. Our focus was not on what quantum states are, but
on what they do. Our results encourage a view one might describe (with
apologies to Bohr) as “complementary”. Thus, $|\psi\rangle$ is in part
information (as, indeed, Bohr thought), but also the obvious quantum object to
explain “existence”. We have seen how Quantum Darwinism accounts for the
transition from quantum fragility (of information) to the effectively
classical robustness. One can think of this transition as “It from bit” of
John Wheeler JAW .
In the end one might ask: “How Darwinian is Quantum Darwinism?”. Clearly,
there is survival of the fittest, and fitness is defined as in natural
selection – through the ability to procreate. The no-cloning theorem implies
competition for resources – space in ${\mathcal{E}}$ – so that only pointer
states can multiply (at the expense of their complementary competition). There
is also another aspect of this competition: Huge memory available in the
Universe as a whole is nevertheless limited. So the question arises: What
systems get to be “of interest”, and imprint their state on their obliging
environments, and what are the environments? Moreover, as the Universe has a
finite memory, old events will be eventually “overwritten” by new ones, so
that some of the past will gradually cease to be reflected in the present
record. And if there is no record of an event, has it really happened? These
questions seem far more interesting than deciding closeness of the analogy
with natural selection Darwin . They suggest one more question: Is Quantum
Darwinism (a process of multiplication of information about certain favored
states that seems to be a “fact of quantum life”) in some way behind the
familiar natural selection? I cannot answer this question, but neither can I
resist raising it.
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Acknowledgments: I am grateful to Robin Blume-Kohout, Fernando Cucchietti,
Juan Pablo Paz, David Poulin, Hai-Tao Quan, Michael Zwolak for stimulating
discussions. This research was supported by an LDRD grant at Los Alamos and,
in part, by FQXi.
|
arxiv-papers
| 2009-03-29T19:08:30 |
2024-09-04T02:49:01.500354
|
{
"license": "Public Domain",
"authors": "Wojciech Hubert Zurek",
"submitter": "W. H. Zurek",
"url": "https://arxiv.org/abs/0903.5082"
}
|
0903.5115
|
# The average value inequality in sequential effect algebras††thanks: This
project is supported by Natural Science Found of China (10771191 and
10471124).
Shen Jun1,2, Wu Junde1 E-mail: wjd@zju.edu.cn
###### Abstract
A sequential effect algebra $(E,0,1,\oplus,\circ)$ is an effect algebra on
which a sequential product $\circ$ with certain physics properties is defined,
in particular, sequential effect algebra is an important model for studying
quantum measurement theory. In 2005, Gudder asked the following problem: If
$a,b\in(E,0,1,\oplus,\circ)$ and $a\bot b$ and $a\circ b\bot a\circ b$, is it
the case that $2(a\circ b)\leq a^{2}\oplus b^{2}$ ? In this paper, we
construct an example to answer the problem negatively.
1Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
2Department of Mathematics, Anhui Normal University, Wuhu 241003, P. R. China
Key Words. Effect algebras, Sequential effect algebras, Average value
inequality.
MR(2000) Subject Classification. 81P15
Effect algebra was introduced in 1994 to model the quantum logic which may be
fuzzy or unsharp, to be precise, an effect algebra is a system
$(E,0,1,\oplus)$, where 0 and 1 are distinct elements of $E$ and $\oplus$ is a
partial binary operation on $E$ satisfying [1]:
(EA1) If $a\oplus b$ is defined, then $b\oplus a$ is defined and $b\oplus
a=a\oplus b$.
(EA2) If $a\oplus(b\oplus c)$ is defined, then $(a\oplus b)\oplus c$ is
defined and
$(a\oplus b)\oplus c=a\oplus(b\oplus c).$
(EA3) For each $a\in E$, there exists a unique element $b\in E$ such that
$a\oplus b=1$.
(EA4) If $a\oplus 1$ is defined, then $a=0$.
In an effect algebra $(E,0,1,\oplus)$, if $a\oplus b$ is defined, we write
$a\bot b$. If $a\bot a$, we denote $a\oplus a$ by $2a$. For each
$a\in(E,0,1,\oplus)$, it follows from (EA3) that there exists a unique element
$b\in E$ such that $a\oplus b=1$, we denote $b$ by $a^{\prime}$. If $a\wedge
a^{\prime}=0$, we say that $a$ is a sharp element of $(E,0,1,\oplus)$ (see
[2]). Let $a,b\in(E,0,1,\oplus)$, if there exists a $c\in E$ such that $a\bot
c$ and $a\oplus c=b$, then we say that $a\leq b$. It follows from [1] that
$\leq$ is a partial order of $(E,0,1,\oplus)$ and satisfies that for each
$a\in E$, $0\leq a\leq 1$, $a\bot b$ iff $a\leq b^{\prime}$.
In 2001, in order to study quantum measurement theory, Gudder began to
consider the sequential product of two measurements $A$ and $B$ (see [3]). In
2002, Professors Gudder and Greechie introduced the abstract sequential effect
algebra structure, that is:
A sequential effect algebra is an effect algebra $(E,0,1,\oplus)$ and another
binary operation $\circ$ defined on $(E,0,1,\oplus)$ satisfying [4]:
(SEA1) The map $b\mapsto a\circ b$ is additive for each $a\in E$, that is, if
$b\bot c$, then $a\circ b\bot a\circ c$ and $a\circ(b\oplus c)=a\circ b\oplus
a\circ c$.
(SEA2) $1\circ a=a$ for each $a\in E$.
(SEA3) If $a\circ b=0$, then $a\circ b=b\circ a$.
(SEA4) If $a\circ b=b\circ a$, then $a\circ b^{\prime}=b^{\prime}\circ a$ and
$a\circ(b\circ c)=(a\circ b)\circ c$ for each $c\in E$.
(SEA5) If $c\circ a=a\circ c$ and $c\circ b=b\circ c$, then $c\circ(a\circ
b)=(a\circ b)\circ c$ and $c\circ(a\oplus b)=(a\oplus b)\circ c$ whenever
$a\bot b$.
Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra. Then the operation
$\circ$ is said to be a sequential product on $(E,0,1,\oplus,\circ)$. If
$a,b\in(E,0,1,\oplus,\circ)$ and $a\circ b=b\circ a$, then we say that $a$ and
$b$ is sequentially independent and denoted by $a|b$ (see [4]). If
$a\in(E,0,1,\oplus,\circ)$, we denote $a\circ a$ by $a^{2}$, it follows from
([4, Lemma 3.2]) that $a$ is a sharp element of $(E,0,1,\oplus,\circ)$ iff
$a^{2}=a$. We denote the set of all sharp elements in $(E,0,1,\oplus,\circ)$
by $E_{s}$.
In 2005, in order to motivate the study of sequential effect algebra theory,
Professor Gudder presented 25 important and interesting problems, the 23th
problem asked ([5]): If $a,b\in(E,0,1,\oplus,\circ)$ and $a\bot b$ and $a\circ
b\bot a\circ b$, is it the case that $2(a\circ b)\leq a^{2}\oplus b^{2}$ ? In
this paper, we construct an example to answer the problem negatively.
At first, we show that the above average value inequality does hold in the
underlying sequential effect algebras under some additional conditions. That
is:
Proposition 1. If $(E,0,1,\oplus,\circ)$ is a sequential effect algebra,
$a,b\in E$, $a^{2}\perp b^{2}$(a sufficient condition for this is $a\perp b$),
$a\leq b$(or $b\leq a$) and $a|b$, then $(a\circ b)\perp(a\circ b)$ and
$2(a\circ b)\leq a^{2}\oplus b^{2}$.
Proof. Since $a\leq b$, there exists a $c\in E$ such that $a\oplus c=b$. Since
$a|b$, it follows that $c|b$ (see [4] Lemma 3.1(v)).
$c\circ b=c\circ(a\oplus c)=(c\circ a)\oplus c^{2}$.
$b^{2}=b\circ(a\oplus c)=(b\circ a)\oplus(b\circ c)=(a\circ b)\oplus(c\circ
b)=(a\circ b)\oplus(c\circ a)\oplus c^{2}$.
Since $a^{2}\perp b^{2}$, $a^{2}\oplus b^{2}=a^{2}\oplus(a\circ
b)\oplus(c\circ a)\oplus c^{2}$.
While $a\circ b=a\circ(a\oplus c)=a^{2}\oplus(a\circ c)=a^{2}\oplus(c\circ
a)$, so $a^{2}\oplus b^{2}=(a\circ b)\oplus(a\circ b)\oplus c^{2}$.
It follows that $(a\circ b)\perp(a\circ b)$ and $2(a\circ b)\leq a^{2}\oplus
b^{2}$.
Finally, if $a\perp b$, it follows from $a^{2}\leq a$ and $b^{2}\leq b$ that
$a^{2}\perp b^{2}$. The proposition is proved.
Proposition 2. If $(E,0,1,\oplus,\circ)$ is a sequential effect algebra,
$a,b\in E$, $a\perp b$, $a\in E_{s}$(or $b\in E_{s}$), then $(a\circ
b)\perp(a\circ b)$ and $2(a\circ b)\leq a^{2}\oplus b^{2}$.
Proof. Since $a\perp b$ and $a\in E_{s}$, it follows that $a\circ b=0$ (see
[4] Lemma 3.3(ii)), so $2(a\circ b)\leq a^{2}\oplus b^{2}$.
Now, we construct a sequential effect algebra to show that the above average
value inequality does not always hold.
In this paper, we denote ${\mathbf{Z}}$ the integer set, ${\mathbf{N}}$ the
nonnegative integer set and ${\mathbf{N}}^{+}$ the positive integer set.
Let $E_{0}=\\{0,1,a_{n},b_{n},c_{i,k,m},d_{i,k,m}|\
n\in{\mathbf{N}}^{+},i,k\in{\mathbf{N}}\ and\ i^{2}+k^{2}\neq
0,m\in{\mathbf{Z}}\\}$. For simplicity, in the sequel, unless specified, the
subindex of respective elements will always take values in the corresponding
default sets. To be accurately, when we write $a_{n},b_{n}$, $n$ always take
values in ${\mathbf{N}}^{+}$, when we write $c_{i,k,m},d_{i,k,m}$, $i,k$
always take values in ${\mathbf{N}}$ and $i^{2}+k^{2}\neq 0$ and $m$ always
take values in ${\mathbf{Z}}$.
We define a partial binary operation $\oplus$ on $E_{0}$ as follows(when we
write $x\oplus y=z$, we always mean $x\oplus y=z=y\oplus x$):
For each $x\in E_{0}$, $0\oplus x=x$,
$a_{n}\oplus a_{m}=a_{n+m}$, $a_{n}\oplus c_{i,k,m}=c_{i,k,n+m}$, $a_{n}\oplus
d_{i,k,m}=d_{i,k,m-n}$, $c_{i,k,m}\oplus c_{r,s,t}=c_{i+r,k+s,m+t}$.
For $n<m$, $a_{n}\oplus b_{m}=b_{m-n}$, $a_{n}\oplus b_{n}=1$.
For $i\leq r\ and\ k\leq s\ and\ (r-i)^{2}+(s-k)^{2}\neq 0$. $c_{i,k,m}\oplus
d_{r,s,t}=d_{r-i,s-k,t-m}$.
For $i=r\ and\ k=s\ and\ m<t$, $c_{i,k,m}\oplus d_{r,s,t}=b_{t-m}$.
For $i=r\ and\ k=s\ and\ m=t$, $c_{i,k,m}\oplus d_{r,s,t}=1$.
No other $\oplus$ operation is defined.
Next, we define a binary operation $\circ$ on $E_{0}$ as follows(when we write
$x\circ y=z$, we always mean $x\circ y=z=y\circ x$):
For each $x\in E_{0}$, $0\circ x=0$, $1\circ x=x$,
$a_{n}\circ a_{m}=0$, $a_{n}\circ b_{m}=a_{n}$, $b_{n}\circ b_{m}=b_{m+n}$,
$a_{n}\circ c_{i,k,m}=0$, $c_{i,k,m}\circ b_{n}=c_{i,k,m}$, $a_{n}\circ
d_{i,k,m}=a_{n}$, $b_{n}\circ d_{i,k,m}=d_{i,k,m+n}$, $d_{i,k,m}\circ
d_{r,s,t}=d_{i+r,k+s,m+t-is-kr}$, $c_{i,k,m}\circ d_{r,s,t}=c_{i,k,m-is-kr}$,
$c_{i,k,m}\circ c_{r,s,t}=a_{is+kr}(when\ is+kr\neq 0)\ or\ 0(when\ is+kr=0)$.
Proposition 3. $(E_{0},0,1,\oplus,\circ)$ is a sequential effect algebra.
Proof. First we verify that $(E_{0},0,1,\oplus)$ is an effect algebra.
(EA1) and (EA4) are trivial.
We verify (EA2), we omit the trivial cases about 0,1:
$a_{n}\oplus(a_{m}\oplus a_{k})=(a_{n}\oplus a_{m})\oplus a_{k}=a_{k+m+n}$.
$a_{n}\oplus(a_{m}\oplus c_{i,j,k})=(a_{n}\oplus a_{m})\oplus
c_{i,j,k}=c_{i,j,k+m+n}$.
$a_{n}\oplus(a_{m}\oplus d_{i,j,k})=(a_{n}\oplus a_{m})\oplus
d_{i,j,k}=d_{i,j,k-m-n}$.
$a_{n}\oplus(c_{r,s,t}\oplus c_{i,j,k})=(a_{n}\oplus c_{r,s,t})\oplus
c_{i,j,k}=c_{i+r,s+j,k+t+n}$.
$c_{l,m,n}\oplus(c_{r,s,t}\oplus c_{i,j,k})=(c_{l,m,n}\oplus c_{i,j,k})\oplus
c_{r,s,t}=c_{i+l+r,j+m+s,k+n+t}$.
Each $a_{n}\oplus(a_{m}\oplus b_{k})$ or $(a_{n}\oplus a_{m})\oplus b_{k}$ is
defined iff $n+m\leq k$, at this case, $a_{n}\oplus(a_{m}\oplus
b_{k})=(a_{n}\oplus a_{m})\oplus b_{k}=b_{k-m-n}(when\ m+n<k)\ or\ 1(when\
m+n=k)$.
Each $a_{n}\oplus(c_{r,s,t}\oplus d_{i,j,k})$ or $(a_{n}\oplus
c_{r,s,t})\oplus d_{i,j,k}$ or $(a_{n}\oplus d_{i,j,k})\oplus c_{r,s,t}$ is
defined iff one of the following two conditions is satisfied:
(1) $r\leq i\ and\ s\leq j\ and\ (i-r)^{2}+(j-s)^{2}\neq 0$, at this case,
$a_{n}\oplus(c_{r,s,t}\oplus d_{i,j,k})=(a_{n}\oplus c_{r,s,t})\oplus
d_{i,j,k}=(a_{n}\oplus d_{i,j,k})\oplus c_{r,s,t}=d_{i-r,j-s,k-t-n}$;
(2) $r=i\ and\ s=j\ and\ n+t\leq k$, at this case,
$a_{n}\oplus(c_{r,s,t}\oplus d_{i,j,k})=(a_{n}\oplus c_{r,s,t})\oplus
d_{i,j,k}=(a_{n}\oplus d_{i,j,k})\oplus c_{r,s,t}=b_{k-t-n}(when\ n+t<k)\ or\
1(when\ n+t=k)$.
Each $c_{l,m,n}\oplus(c_{r,s,t}\oplus d_{i,j,k})$ or $(c_{l,m,n}\oplus
c_{r,s,t})\oplus d_{i,j,k}$ is defined iff one of the following two conditions
is satisfied:
(1) $l+r\leq i\ and\ m+s\leq j\ and\ (i-l-r)^{2}+(j-m-s)^{2}\neq 0$, at this
case, $c_{l,m,n}\oplus(c_{r,s,t}\oplus d_{i,j,k})=(c_{l,m,n}\oplus
c_{r,s,t})\oplus d_{i,j,k}=d_{i-l-r,j-m-s,k-t-n}$;
(2) $l+r=i\ and\ m+s=j\ and\ n+t\leq k$, at this case,
$c_{l,m,n}\oplus(c_{r,s,t}\oplus d_{i,j,k})=(c_{l,m,n}\oplus c_{r,s,t})\oplus
d_{i,j,k}=b_{k-t-n}(when\ n+t<k)\ or\ 1(when\ n+t=k)$.
We verify (EA3): $a_{n}\oplus b_{n}=1$, $c_{i,k,m}\oplus d_{i,k,m}=1$.
So $(E_{0},0,1,\oplus)$ is an effect algebra.
We now verify that $(E_{0},0,1,\oplus,\circ)$ is a sequential effect algebra.
(SEA2) and (SEA3) and (SEA5) are trivial.
We verify (SEA1), we omit the trivial cases about 0,1:
$a_{n}\circ(a_{m}\oplus a_{k})=a_{n}\circ a_{m}\oplus a_{n}\circ a_{k}=0$,
$b_{n}\circ(a_{m}\oplus a_{k})=b_{n}\circ a_{m}\oplus b_{n}\circ
a_{k}=a_{m+k}$,
$c_{r,s,t}\circ(a_{m}\oplus a_{k})=c_{r,s,t}\circ a_{m}\oplus c_{r,s,t}\circ
a_{k}=0$,
$d_{r,s,t}\circ(a_{m}\oplus a_{k})=d_{r,s,t}\circ a_{m}\oplus d_{r,s,t}\circ
a_{k}=a_{m+k}$.
$a_{n}\circ(a_{m}\oplus c_{r,s,t})=a_{n}\circ a_{m}\oplus a_{n}\circ
c_{r,s,t}=0$,
$b_{n}\circ(a_{m}\oplus c_{r,s,t})=b_{n}\circ a_{m}\oplus b_{n}\circ
c_{r,s,t}=c_{r,s,m+t}$,
$c_{x,y,z}\circ(a_{m}\oplus c_{r,s,t})=c_{x,y,z}\circ a_{m}\oplus
c_{x,y,z}\circ c_{r,s,t}=a_{xs+yr}(when\ xs+yr\neq\ 0)\ or\ 0(when\ xs+yr=0)$,
$d_{x,y,z}\circ(a_{m}\oplus c_{r,s,t})=d_{x,y,z}\circ a_{m}\oplus
d_{x,y,z}\circ c_{r,s,t}=c_{r,s,m+t-xs-yr}$.
$a_{n}\circ(a_{m}\oplus d_{r,s,t})=a_{n}\circ a_{m}\oplus a_{n}\circ
d_{r,s,t}=a_{n}$,
$b_{n}\circ(a_{m}\oplus d_{r,s,t})=b_{n}\circ a_{m}\oplus b_{n}\circ
d_{r,s,t}=d_{r,s,n+t-m}$,
$c_{x,y,z}\circ(a_{m}\oplus d_{r,s,t})=c_{x,y,z}\circ a_{m}\oplus
c_{x,y,z}\circ d_{r,s,t}=c_{x,y,z-xs-yr}$,
$d_{x,y,z}\circ(a_{m}\oplus d_{r,s,t})=d_{x,y,z}\circ a_{m}\oplus
d_{x,y,z}\circ d_{r,s,t}=d_{x+r,y+s,z+t-m-xs-yr}$.
$a_{n}\circ(c_{x,y,z}\oplus c_{r,s,t})=a_{n}\circ c_{x,y,z}\oplus a_{n}\circ
c_{r,s,t}=0$,
$b_{n}\circ(c_{x,y,z}\oplus c_{r,s,t})=b_{n}\circ c_{x,y,z}\oplus b_{n}\circ
c_{r,s,t}=c_{x+r,y+s,z+t}$,
$c_{i,k,m}\circ(c_{x,y,z}\oplus c_{r,s,t})=c_{i,k,m}\circ c_{x,y,z}\oplus
c_{i,k,m}\circ c_{r,s,t}=a_{i(y+s)+k(x+r)}(when\ i(y+s)+k(x+r)\neq 0)\ or\
0(when\ i(y+s)+k(x+r)=0)$,
$d_{i,k,m}\circ(c_{x,y,z}\oplus c_{r,s,t})=d_{i,k,m}\circ c_{x,y,z}\oplus
d_{i,k,m}\circ c_{r,s,t}=c_{x+r,y+s,z+t-i(y+s)-k(x+r)}$.
For $m\leq k$,
$a_{n}\circ(a_{m}\oplus b_{k})=a_{n}\circ a_{m}\oplus a_{n}\circ b_{k}=a_{n}$,
$b_{n}\circ(a_{m}\oplus b_{k})=b_{n}\circ a_{m}\oplus b_{n}\circ
b_{k}=b_{n+k-m}$,
$c_{x,y,z}\circ(a_{m}\oplus b_{k})=c_{x,y,z}\circ a_{m}\oplus c_{x,y,z}\circ
b_{k}=c_{x,y,z}$,
$d_{x,y,z}\circ(a_{m}\oplus b_{k})=d_{x,y,z}\circ a_{m}\oplus d_{x,y,z}\circ
b_{k}=d_{x,y,z+k-m}$.
For $i\leq r\ and\ k\leq s\ and\ (r-i)^{2}+(s-k)^{2}\neq 0$,
$a_{n}\circ(c_{i,k,m}\oplus d_{r,s,t})=a_{n}\circ c_{i,k,m}\oplus a_{n}\circ
d_{r,s,t}=a_{n}$,
$b_{n}\circ(c_{i,k,m}\oplus d_{r,s,t})=b_{n}\circ c_{i,k,m}\oplus b_{n}\circ
d_{r,s,t}=d_{r-i,s-k,n+t-m}$,
$c_{x,y,z}\circ(c_{i,k,m}\oplus d_{r,s,t})=c_{x,y,z}\circ c_{i,k,m}\oplus
c_{x,y,z}\circ d_{r,s,t}=c_{x,y,z-x(s-k)-y(r-i)}$,
$d_{x,y,z}\circ(c_{i,k,m}\oplus d_{r,s,t})=d_{x,y,z}\circ c_{i,k,m}\oplus
d_{x,y,z}\circ d_{r,s,t}=d_{x+r-i,y+s-k,z+t-m-x(s-k)-y(r-i)}$.
For $i=r\ and\ k=s\ and\ m\leq t$,
$a_{n}\circ(c_{i,k,m}\oplus d_{r,s,t})=a_{n}\circ c_{i,k,m}\oplus a_{n}\circ
d_{r,s,t}=a_{n}$,
$b_{n}\circ(c_{i,k,m}\oplus d_{r,s,t})=b_{n}\circ c_{i,k,m}\oplus b_{n}\circ
d_{r,s,t}=b_{n+t-m}$,
$c_{x,y,z}\circ(c_{i,k,m}\oplus d_{r,s,t})=c_{x,y,z}\circ c_{i,k,m}\oplus
c_{x,y,z}\circ d_{r,s,t}=c_{x,y,z}$,
$d_{x,y,z}\circ(c_{i,k,m}\oplus d_{r,s,t})=d_{x,y,z}\circ c_{i,k,m}\oplus
d_{x,y,z}\circ d_{r,s,t}=d_{x,y,z+t-m}$.
We verify (SEA4), we omit the trivial cases about 0,1:
$a_{n}\circ(a_{m}\circ a_{k})=(a_{n}\circ a_{m})\circ a_{k}=0$,
$a_{n}\circ(a_{m}\circ b_{k})=b_{k}\circ(a_{n}\circ
a_{m})=a_{m}\circ(a_{n}\circ b_{k})=0$,
$a_{n}\circ(a_{m}\circ c_{r,s,t})=c_{r,s,t}\circ(a_{n}\circ
a_{m})=a_{m}\circ(a_{n}\circ c_{r,s,t})=0$,
$a_{n}\circ(a_{m}\circ d_{r,s,t})=d_{r,s,t}\circ(a_{n}\circ
a_{m})=a_{m}\circ(a_{n}\circ d_{r,s,t})=0$,
$a_{n}\circ(b_{m}\circ b_{k})=b_{k}\circ(a_{n}\circ
b_{m})=b_{m}\circ(a_{n}\circ b_{k})=a_{n}$,
$a_{n}\circ(b_{m}\circ c_{r,s,t})=c_{r,s,t}\circ(a_{n}\circ
b_{m})=b_{m}\circ(a_{n}\circ c_{r,s,t})=0$,
$a_{n}\circ(b_{m}\circ d_{r,s,t})=d_{r,s,t}\circ(a_{n}\circ
b_{m})=b_{m}\circ(a_{n}\circ d_{r,s,t})=a_{n}$,
$a_{n}\circ(c_{i,k,m}\circ c_{r,s,t})=c_{r,s,t}\circ(a_{n}\circ
c_{i,k,m})=c_{i,k,m}\circ(a_{n}\circ c_{r,s,t})=0$,
$a_{n}\circ(c_{i,k,m}\circ d_{r,s,t})=d_{r,s,t}\circ(a_{n}\circ
c_{i,k,m})=c_{i,k,m}\circ(a_{n}\circ d_{r,s,t})=0$,
$a_{n}\circ(d_{i,k,m}\circ d_{r,s,t})=d_{r,s,t}\circ(a_{n}\circ
d_{i,k,m})=d_{i,k,m}\circ(a_{n}\circ d_{r,s,t})=a_{n}$,
$b_{n}\circ(b_{m}\circ b_{k})=b_{k}\circ(b_{n}\circ b_{m})=b_{m+n+k}$,
$b_{n}\circ(b_{m}\circ c_{r,s,t})=c_{r,s,t}\circ(b_{n}\circ
b_{m})=b_{m}\circ(b_{n}\circ c_{r,s,t})=c_{r,s,t}$,
$b_{n}\circ(b_{m}\circ d_{r,s,t})=d_{r,s,t}\circ(b_{n}\circ
b_{m})=b_{m}\circ(b_{n}\circ d_{r,s,t})=d_{r,s,n+m+t}$,
$b_{n}\circ(c_{i,k,m}\circ c_{r,s,t})=c_{r,s,t}\circ(b_{n}\circ
c_{i,k,m})=c_{i,k,m}\circ(b_{n}\circ c_{r,s,t})=a_{is+kr}(when\ is+kr\neq 0)\
or\ 0(when\ is+kr=0)$,
$b_{n}\circ(c_{i,k,m}\circ d_{r,s,t})=d_{r,s,t}\circ(b_{n}\circ
c_{i,k,m})=c_{i,k,m}\circ(b_{n}\circ d_{r,s,t})=c_{i,k,m-is-kr}$,
$b_{n}\circ(d_{i,k,m}\circ d_{r,s,t})=d_{r,s,t}\circ(b_{n}\circ
d_{i,k,m})=d_{i,k,m}\circ(b_{n}\circ d_{r,s,t})=d_{i+r,k+s,n+m-t-is-kr}$,
$c_{x,y,z}\circ(c_{i,k,m}\circ c_{r,s,t})=c_{r,s,t}\circ(c_{x,y,z}\circ
c_{i,k,m})=0$,
$c_{x,y,z}\circ(c_{i,k,m}\circ d_{r,s,t})=d_{r,s,t}\circ(c_{x,y,z}\circ
c_{i,k,m})=c_{i,k,m}\circ(c_{x,y,z}\circ d_{r,s,t})=a_{xk+yi}(when\ xk+yi\neq
0)\ or\ 0(when\ xk+yi=0)$,
$c_{x,y,z}\circ(d_{i,k,m}\circ d_{r,s,t})=d_{r,s,t}\circ(c_{x,y,z}\circ
d_{i,k,m})=d_{i,k,m}\circ(c_{x,y,z}\circ d_{r,s,t})=c_{x,y,z-x(k+s)-y(i+r)}$,
$d_{x,y,z}\circ(d_{i,k,m}\circ d_{r,s,t})=d_{r,s,t}\circ(d_{x,y,z}\circ
d_{i,k,m})$$=d_{x+i+r,y+k+s,z+m+t-(is+kr+xk+xs+yi+yr)}$.
So $(E_{0},0,1,\oplus,\circ)$ is a sequential effect algebra.
Our main result is:
Theorem 1. The average value inequality does not always hold in sequential
effect algebras.
Proof. In fact, in $(E_{0},0,1,\oplus,\circ)$, $c_{1,0,0}\perp c_{0,1,0}$,
$c_{1,0,0}\oplus c_{0,1,0}=c_{1,1,0}$. $c_{1,0,0}\circ c_{0,1,0}=a_{1}$,
$a_{1}\perp a_{1}$, $a_{1}\oplus a_{1}=a_{2}$. But $2(c_{1,0,0}\circ
c_{0,1,0})=2a_{1}=a_{1}\oplus a_{1}=a_{2}$, $(c_{1,0,0})^{2}=c_{1,0,0}\circ
c_{1,0,0}=0$, $(c_{0,1,0})^{2}=c_{0,1,0}\circ c_{0,1,0}=0$, so
$2(c_{1,0,0}\circ c_{0,1,0})\not\leq(c_{1,0,0})^{2}\oplus(c_{0,1,0})^{2}$.
Remarks. Recently, the 2th problem, the 3th problem, the 17th problem and the
20th problem of Gudder have also been answered ([6-9]).
References
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[2]. Gudder, S. Sharply dominating effect algebras. Tatra Mt. Math. Publ.,
15(1998), 23-30.
[3]. Gudder, S, Nagy, G. Sequential quantum measurements. J. Math. Phys.
42(2001), 5212-5222.
[4]. Gudder, S, Greechie, R. Sequential products on effect algebras. Rep.
Math. Phys. 49(2002), 87-111.
[5]. Gudder, S. Open problems for sequential effect algebras. Inter. J.
Theory. Phys. 44 (2005), 2219-2230.
[6]. Weihua Liu, Junde Wu. The Uniqueness Problem of Sequence Product on
Operator Effect Algebra ${\cal E}(H)$. J. Physi. A (Accepted to appear).
[7]. Jun Shen, Junde Wu. Not each sequential effect algebra is sharply
dominating. Physics Letter A (Accepted to appear).
[8]. Jun Shen, Junde Wu. Remarks on the sequential effect algebras. Report
Math. Physi. (Accepted to appear).
[9]. Jun Shen, Junde Wu. The square root is not unique in sequential effect
algebras. (To appear).
|
arxiv-papers
| 2009-03-30T02:46:40 |
2024-09-04T02:49:01.510454
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shen Jun and Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0903.5115"
}
|
0903.5116
|
# Remarks on the sequential effect algebras††thanks: This project is supported
by Natural Science Found of China (10771191 and 10471124).
Shen Jun1,2, Wu Junde1 Corresponding author: wjd@zju.edu.cn
###### Abstract
In this paper, first, we answer affirmatively an open problem which was
presented in 2005 by professor Gudder on the sub-sequential effect algebras.
That is, we prove that if $(E,0,1,\oplus,\circ)$ is a sequential effect
algebra and $A$ is a commutative subset of $E$, then the sub-sequential effect
algebra $\overline{A}$ generated by $A$ is also commutative. Next, we also
study the following uniqueness problem: If $na=nb=c$ for some positive integer
$n\geq 2$, then under what conditions $a=b$ hold? We prove that if $c$ is a
sharp element of $E$ and $a|b$, then $a=b$. We give also two examples to show
that neither of the above two conditions can be discarded.
1Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
2Department of Mathematics, Anhui Normal University, Wuhu 241003, P. R. China
Key Words. Sub-sequential effect algebras, commutative, uniqueness.
1\. Introduction
Effect algebra is an important logic model for studying quantum effects or
observations which may be fuzzy or unsharp (see [1]), to be precise, an effect
algebra is a system $(E,0,1,\oplus)$, where 0 and 1 are distinct elements of
$E$ and $\oplus$ is a partial binary operation on $E$ satisfying:
(EA1) If $a\oplus b$ is defined, then $b\oplus a$ is defined and $b\oplus
a=a\oplus b$.
(EA2) If $a\oplus(b\oplus c)$ is defined, then $(a\oplus b)\oplus c$ is
defined and
$(a\oplus b)\oplus c=a\oplus(b\oplus c).$
(EA3) For each $a\in E$, there exists a unique element $b\in E$ such that
$a\oplus b=1$.
(EA4) If $a\oplus 1$ is defined, then $a=0$.
In an effect algebra $(E,0,1,\oplus)$, if $a\oplus b$ is defined, we write
$a\bot b$. For each $a\in E$, it follows from (EA3) that there exists a unique
element $b\in E$ such that $a\oplus b=1$, we denote $b$ by $a^{\prime}$. Let
$a,b\in E$, if there exists an element $c\in E$ such that $a\bot c$ and
$a\oplus c=b$, then we say that $a\leq b$ and write $c=b\ominus a$. It follows
from [1] that $\leq$ is a partial order of $(E,0,1,\oplus)$ and satisfies that
for each $a\in E$, $0\leq a\leq 1$, $a\bot b$ if and only if $a\leq
b^{\prime}$.
Let $(E,0,1,\oplus)$ be an effect algebra and $a\in E$. If $a\wedge
a^{\prime}=0$, then $a$ is said to be a sharp element of $E$. The set
$E_{s}=\\{x\in E|\ x\wedge x^{\prime}=0\\}$ is called the set of all sharp
elements of $E$ (see [2-3]).
As we knew, two measurements $a$ and $b$ cannot be performed simultaneously in
general, so they are frequently executed sequentially ([4]). We denote by
$a\circ b$ a sequential measurement in which $a$ is performed first and $b$
second and call $a\circ b$ a sequential product of $a$ and $b$. Thus, it is an
important and interesting project to study effect algebras which have a
sequential product $\circ$ with some nature properties. To be precise:
A sequential effect algebra (SEA) is an effect algebra $(E,0,1,\oplus)$ and
another binary operation $\circ$ defined on $(E,0,1,\oplus)$ satisfying [5]:
(SEA1) The map $b\mapsto a\circ b$ is additive for each $a\in E$, that is, if
$b\bot c$, then $a\circ b\bot a\circ c$ and $a\circ(b\oplus c)=a\circ b\oplus
a\circ c$.
(SEA2) $1\circ a=a$ for each $a\in E$.
(SEA3) If $a\circ b=0$, then $a\circ b=b\circ a$.
(SEA4) If $a\circ b=b\circ a$, then $a\circ b^{\prime}=b^{\prime}\circ a$ and
for each $c\in E$, $a\circ(b\circ c)=(a\circ b)\circ c$.
(SEA5) If $c\circ a=a\circ c$ and $c\circ b=b\circ c$, then $c\circ(a\circ
b)=(a\circ b)\circ c$ and $c\circ(a\oplus b)=(a\oplus b)\circ c$ whenever
$a\bot b$.
Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra. If $a,b\in E$ and
$a\circ b=b\circ a$, then we say $a$ and $b$ is sequentially independent and
denoted by $a|b$.
Lemma 1 ([1, 5]). If $(E,0,1,\oplus,\circ)$ is a sequential effect algebra and
$a,b,c\in E$, then
(1) $a\perp b$, $a\perp c$ and $a\oplus b=a\oplus c$ implies that $b=c$.
(2) $a\in E_{s}$ if and only if $a\circ a=a$.
(3) If $c\in E_{s}$, then $a\leq c$ if and only if $a=a\circ c=c\circ a$.
2\. Sub-sequential effect algebra generated by a subset
Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra and $F$ a nonempty
subset of $E$. We call $F$ a sub-sequential effect algebra of
$(E,0,1,\oplus,\circ)$ if $0,1\in F$ and $(F,0,1,\oplus,\circ)$ itself is a
sequential effect algebra. From the definition of sub-sequential effect
algebra, it is easy to see that a nonempty subset $F$ of
$(E,0,1,\oplus,\circ)$ is a sub-sequential effect algebra if and only if $F$
is closed under all the three operations $\oplus$, $\circ$ and ′. Moreover, if
$A$ is a nonempty subset of $E$, it is easy to see that there exists a
smallest sub-sequential effect algebra $\overline{A}$ of $E$ which contains
$A$ (That is, the intersection of all sub-sequential effect algebras
containing $A$). We call $\overline{A}$ the sub-sequential effect algebra
generated by $A$. In 2005, Professor Gudder presented the following open
problem (see [6, Problem 17]):
Problem 1. If $(E,0,1,\oplus,\circ)$ is a sequential effect algebra and $A$ a
commutative subset of $E$ (That is, $a|b$ for all $a,b\in A$), is
$\overline{A}$ commutative ?
In this paper, we answer the problem affirmatively. That is:
Theorem 1. Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra and $A$ a
commutative subset of $(E,0,1,\oplus,\circ)$. Then $\overline{A}$ is also
commutative.
Proof. Let $\bigwedge=\\{F|\ F\ be\ a$ commutative subset of $E$ containing
$A$}. We order $\bigwedge$ by including. Using Zorn’s Lemma, it is easy to see
that there exists a maximal element $F_{0}$ in $\bigwedge$. That is, $F_{0}$
is a maximal commutative subset of $E$ containing $A$.
We now prove that $F_{0}$ is a sub-sequential effect algebra of $E$:
If $a\in F_{0}$, then for each $c\in F_{0}$, $c|a$, so $c|a^{\prime}$ by
(SEA4). By maximality, we have $a^{\prime}\in F_{0}$.
If $a,b\in F_{0}$, then for each $c\in F_{0}$, $c|a$, $c|b$, so $c|(a\circ b)$
by (SEA5). By maximality, we have $(a\circ b)\in F_{0}$.
If $a,b\in F_{0}$ and $a\perp b$, then for each $c\in F_{0}$, $c|a$, $c|b$, so
$c|(a\oplus b)$ by (SEA5). By maximality, we have $(a\oplus b)\in F_{0}$.
So $F_{0}$ is closed under all the three operations $\oplus$, $\circ$ and ′.
Thus, $F_{0}$ is a sub-sequential effect algebra of $(E,0,1,\oplus,\circ)$
containing $A$. Since $\overline{A}$ is the smallest sub-sequential effect
algebra of $(E,0,1,\oplus,\circ)$ containing $A$, we have
$\overline{A}\subseteq F_{0}$ and $\overline{A}$ is also commutative.
Moreover, for general subset $A$ of $E$, we can describe the structure of
$\overline{A}$, that is
Theorem 2. Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra and $A$ a
subset of $E$. If we denote
$A_{1}=A\bigcup(\bigcup\limits_{a\in
A}a^{\prime})\bigcup(\bigcup\limits_{a,b\in A}a\circ
b)\bigcup(\bigcup\limits_{a,b\in A\ and\ a\perp b}a\oplus b)$,
$A_{2}=A_{1}\bigcup(\bigcup\limits_{a\in
A_{1}}a^{\prime})\bigcup(\bigcup\limits_{a,b\in A_{1}}a\circ
b)\bigcup(\bigcup\limits_{a,b\in A_{1}\ and\ a\perp b}a\oplus b)$,
$\cdots$
$A_{n}=A_{n-1}\bigcup(\bigcup\limits_{a\in
A_{n-1}}a^{\prime})\bigcup(\bigcup\limits_{a,b\in A_{n-1}}a\circ
b)\bigcup(\bigcup\limits_{a,b\in A_{n-1}\ and\ a\perp b}a\oplus b)$,
$\cdots$
$\Gamma=\bigcup\limits_{n=1}\limits^{\infty}A_{n}$.
Then $\overline{A}=\Gamma$.
Proof. First we prove that $\Gamma$ is a sub-sequential effect algebra of
$(E,0,1,\oplus,\circ)$.
If $a\in\Gamma$, then $a\in A_{n}$ for some $n$, so $a^{\prime}\in
A_{n+1}\subseteq\Gamma$.
If $a,b\in\Gamma$, then $a,b\in A_{n}$ for some $n$, so $(a\circ b)\in
A_{n+1}\subseteq\Gamma$.
If $a,b\in\Gamma$ and $a\perp b$, then $a,b\in A_{n}$ for some $n$, so
$(a\oplus b)\in A_{n+1}\subseteq\Gamma$.
Thus, $\Gamma$ is closed under all the three operations $\oplus$, $\circ$ and
′. So $\Gamma$ is a sub-sequential effect algebra of $(E,0,1,\oplus,\circ)$.
Of course $A\subseteq\Gamma$. Since $\overline{A}$ is the smallest sub-
sequential effect algebra of $(E,0,1,\oplus,\circ)$ containing $A$, we have
$\overline{A}\subseteq\Gamma$. On the other hand, by induction, it is easy to
see that $A_{n}\subseteq\overline{A}$ for all $n$. Thus
$\Gamma\subseteq\overline{A}$. So $\Gamma=\overline{A}$.
Note that by using Theorem 2 we can also answer professor Gudder’s problem by
a constructive way, we omit the process.
3\. An addition property of sequential effect algebras
Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra, $a,b\in E$. If
$\underbrace{a\oplus a\cdots\oplus a}\limits_{the\ number\ is\ n}$ is defined,
we denote it by $na$. Now, we are interested in the following uniqueness
problem: If for some positive integer $n_{0}\geq 2$, $n_{0}a=n_{0}b$, then
under what conditions $a=b$ hold? We have
Theorem 3. Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra, $a,b\in
E$ and for some positive integer $n_{0}\geq 2$, $n_{0}a=n_{0}b=c$. If $c\in
E_{s}$ and $a|b$, then $a=b$.
Proof. Since $a\leq c$, by Lemma 1, $a=a\circ c$, similarly $b=b\circ c$.
By (SEA1), we have $a\circ c=a\circ(n_{0}b)=n_{0}(a\circ b)$, $b\circ
c=b\circ(n_{0}a)=n_{0}(b\circ a)$.
Note that $a|b$, so $a\circ b=b\circ a$ and $a\circ c=b\circ c$. Thus $a=b$.
Now, we show that neither of the two conditions in Theorem 3 can be discarded.
Example 1. Let $I_{1}=[0,1]$, $I_{2}=[0,1]$, $E=HS(I_{1},I_{2})$ be the
horizontal sum of $I_{1},I_{2}$ (see [5, Section 8, the Example in
$P_{109}$]). For each $t\in[0,1]$, if it is in $I_{1}$, we denote it by
$\hat{t}$; if it is in $I_{2}$, we denote it by $\check{t}$. Let
$a=\hat{\frac{1}{n_{0}}}$, $b=\check{\frac{1}{n_{0}}}$. Then
$n_{0}a=1=n_{0}b$, $1\in E_{s}$, $a\neq b$, $a\circ b\neq b\circ a$. So the
condition $a|b$ in Theorem 3 can not be discarded.
Example 2. Let ${\mathbf{N}}$ be the nonnegative integer set, $n_{0}$ be a
positive integer and $n_{0}\geq 2$, $E_{0}=\\{0,1,a_{n,m},b_{n,m}|\
n,m\in{\mathbf{N}},\ n_{0}-1\geq m,\ n^{2}+m^{2}\neq 0\\}$.
First, we define a partial binary operation $\oplus$ on $E_{0}$ as follows
(when we write $x\oplus y=z$, we always mean $x\oplus y=z=y\oplus x$):
For each $x\in E_{0}$, $0\oplus x=x$,
$a_{n,m}\oplus a_{r,s}=\left\\{\begin{array}[]{ll}a_{n+r,m+s}\ ,&\hbox{$if\
m+s<n_{0}$;}\\\ a_{n+r+n_{0},m+s-n_{0}}\ ,&\hbox{$if\ m+s\geq
n_{0}$.}\end{array}\right.$
$a_{n,m}\oplus b_{r,s}=\left\\{\begin{array}[]{ll}b_{r-n,s-m}\ ,&\hbox{$if\
n\leq r,\ m\leq s,\ (r-n)^{2}+(s-m)^{2}\neq 0$;}\\\ 1\ ,&\hbox{$if\ n=r,\
m=s$;}\\\ b_{r-n-n_{0},s-m+n_{0}}\ ,&\hbox{$if\ n+n_{0}\leq r,\
m>s$.}\end{array}\right.$
No other $\oplus$ operation is defined.
Next, we define a binary operation $\circ$ on $E_{0}$ as follows (when we
write $x\circ y=z$, we always mean $x\circ y=z=y\circ x$):
For each $x\in E_{0}$, $0\circ x=0$, $1\circ x=x$,
$a_{n,m}\circ a_{r,s}=0$, $a_{n,m}\circ b_{r,s}=a_{n,m}$,
$b_{n,m}\circ b_{r,s}=\left\\{\begin{array}[]{ll}b_{n+r,m+s}\ ,&\hbox{$if\
m+s<n_{0}$;}\\\ b_{n+r+n_{0},m+s-n_{0}}\ ,&\hbox{$if\ m+s\geq
n_{0}$.}\end{array}\right.$
Now, we prove that $E_{0}$ is a sequential effect algebra.
In fact, (EA1) and (EA4) are trivial.
We verify (EA2), for simplicity, we omit the trivial cases about 0,1:
$a_{k,j}\oplus(a_{n,m}\oplus a_{r,s})=(a_{k,j}\oplus a_{n,m})\oplus a_{r,s}$
$=\left\\{\begin{array}[]{ll}a_{k+r+n,s+j+m}\ ,&\hbox{$if\ s+j+m<n_{0}$;}\\\
a_{k+r+n+n_{0},s+j+m-n_{0}}\ ,&\hbox{$if\ n_{0}\leq s+j+m<2n_{0}$;}\\\
a_{k+r+n+2n_{0},s+j+m-2n_{0}}\ ,&\hbox{$if\ s+j+m\geq
2n_{0}$.}\end{array}\right.$
Each $a_{k,j}\oplus(a_{n,m}\oplus b_{r,s})$ or $(a_{k,j}\oplus a_{n,m})\oplus
b_{r,s}$ is defined if and only if one of the following four conditions is
satisfied, at this case,
$a_{k,j}\oplus(a_{n,m}\oplus b_{r,s})=(a_{k,j}\oplus a_{n,m})\oplus b_{r,s}$
$=\left\\{\begin{array}[]{ll}b_{r-k-n,s-j-m}\ ,&\hbox{$if\ k+n\leq r,\ j+m\leq
s,\ (r-k-n)^{2}+(s-j-m)^{2}\neq 0$;}\\\ b_{r-k-n-n_{0},s-j-m+n_{0}}\
,&\hbox{$if\ k+n+n_{0}\leq r,\ s<j+m\leq n_{0}+s,$}\\\
&\hbox{~{}~{}~{}~{}~{}~{}~{}~{}$(r-k-n-n_{0})^{2}+(s-j-m+n_{0})^{2}\neq
0$;}\\\ b_{r-k-n-2n_{0},s-j-m+2n_{0}}\ ,&\hbox{$if\ k+n+2n_{0}\leq r,\
n_{0}+s<j+m$;}\\\ 1\ ,&\hbox{$if\ (r-k-n)^{2}+(s-j-m)^{2}=0\ or$}\\\
&\hbox{~{}~{}~{}~{}~{}~{}~{}~{}$(r-k-n-
n_{0})^{2}+(s-j-m+n_{0})^{2}=0$.}\end{array}\right.$
Thus, (EA2) is hold.
(EA3) is clear since $a_{n,m}\oplus b_{n,m}=1$. Thus, $(E_{0},0,1,\oplus)$ is
an effect algebra.
Moreover, we verify that $(E_{0},0,1,\oplus,\circ)$ is a sequential effect
algebra.
(SEA2) and (SEA3) and (SEA5) are trivial.
We verify (SEA1), for simplicity, we omit the trivial cases about 0,1:
$a_{k,j}\circ(a_{n,m}\oplus a_{r,s})=a_{k,j}\circ a_{n,m}\oplus a_{k,j}\circ
a_{r,s}=0$.
$b_{k,j}\circ(a_{n,m}\oplus a_{r,s})=b_{k,j}\circ a_{n,m}\oplus b_{k,j}\circ
a_{r,s}=\left\\{\begin{array}[]{ll}a_{n+r,m+s}\ ,&\hbox{$if\ m+s<n_{0}$;}\\\
a_{n+r+n_{0},m+s-n_{0}}\ ,&\hbox{$if\ m+s\geq n_{0}$.}\end{array}\right.$
When $a_{n,m}\oplus b_{r,s}$ is defined,
$a_{k,j}\circ(a_{n,m}\oplus b_{r,s})=a_{k,j}\circ a_{n,m}\oplus a_{k,j}\circ
b_{r,s}=a_{k,j}$,
$b_{k,j}\circ(a_{n,m}\oplus b_{r,s})=b_{k,j}\circ a_{n,m}\oplus b_{k,j}\circ
b_{r,s}$
$=\left\\{\begin{array}[]{ll}b_{r+k-n,s+j-m}\ ,&\hbox{$if\ n\leq r,\ m\leq s,\
j+s<n_{0}+m$;}\\\ b_{r+k-n,s+j-m}\ ,&\hbox{$if\ n+n_{0}\leq r,\ s<m\leq
j+s$;}\\\ b_{r+k-n+n_{0},s+j-m-n_{0}}\ ,&\hbox{$if\ n\leq r,\ n_{0}+m\leq
j+s$;}\\\ b_{r+k-n-n_{0},s+j-m+n_{0}}\ ,&\hbox{$if\ n+n_{0}\leq r,\
j+s<m$.}\end{array}\right.$
Thus, (SEA1) is true.
We verify (SEA4), for simplicity, we omit also the trivial cases about 0,1:
$a_{k,j}\circ(a_{n,m}\circ a_{r,s})=(a_{k,j}\circ a_{n,m})\circ a_{r,s}=0$.
$a_{k,j}\circ(a_{n,m}\circ b_{r,s})=(a_{k,j}\circ a_{n,m})\circ b_{r,s}=0$.
$a_{k,j}\circ(b_{n,m}\circ b_{r,s})=(a_{k,j}\circ b_{n,m})\circ
b_{r,s}=a_{k,j}$.
$b_{k,j}\circ(b_{n,m}\circ b_{r,s})=(b_{k,j}\circ b_{n,m})\circ b_{r,s}$
$=\left\\{\begin{array}[]{ll}b_{k+r+n,s+j+m}\ ,&\hbox{$if\ s+j+m<n_{0}$;}\\\
b_{k+r+n+n_{0},s+j+m-n_{0}}\ ,&\hbox{$if\ n_{0}\leq s+j+m<2n_{0}$;}\\\
b_{k+r+n+2n_{0},s+j+m-2n_{0}}\ ,&\hbox{$if\ s+j+m\geq
2n_{0}$.}\end{array}\right.$
Thus (SEA4) is hold and $(E_{0},0,1,\oplus,\circ)$ is a sequential effect
algebra.
Finally, we show that the condition $c\in E_{s}$ in Theorem 3 can not be
discarded.
Indeed, since $a_{n,0}\oplus a_{r,0}=a_{n+r,0}$, so
$n_{0}a_{1,0}=a_{n_{0},0}$. Note that
$a_{0,m}\oplus a_{0,s}=\left\\{\begin{array}[]{ll}a_{0,m+s}\ ,&\hbox{$if\
m+s<n_{0}$;}\\\ a_{n_{0},m+s-n_{0}}\ ,&\hbox{$if\ m+s\geq
n_{0}$.}\end{array}\right.$
Thus, $(n_{0}-1)a_{0,1}=a_{0,n_{0}-1}$, $n_{0}a_{0,1}=(n_{0}-1)a_{0,1}\oplus
a_{0,1}=a_{0,n_{0}-1}\oplus a_{0,1}=a_{n_{0},0}$, that is,
$n_{0}a_{1,0}=a_{n_{0},0}=n_{0}a_{0,1}$. Note that $a_{n_{0},0}\circ
a_{n_{0},0}=0$, so $a_{n_{0},0}\not\in(E_{0})_{s}$.
References
[1]. D. J. Foulis and M. K. Bennett: Effect algebras and unsharp quantum
logics. Found. Phys. 24, 1331(1994).
[2]. S. Gudder: Sharply dominating effect algebras. Tatra Mt. Math. Publ. 15,
23(1998).
[3]. Z. Riecanova and J. D. Wu: States on sharply dominating effect algebras.
Sci. in China A: Math. 51, 907(2008).
[4]. S. Gudder and G. Nagy: Sequential quantum measurements. J. Math. Phys.
42, 5212(2001).
[5]. S. Gudder and R. Greechie: Sequential products on effect algebras. Rep.
Math. Phys. 49, 87(2002).
[6]. S. Gudder: Open problems for sequential effect algebras. Inter. J.
Theory. Phys. 44, 2219(2005).
|
arxiv-papers
| 2009-03-30T02:55:59 |
2024-09-04T02:49:01.515545
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shen Jun and Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0903.5116"
}
|
0903.5120
|
# The n-th root of sequential effect algebras††thanks: This project is
supported by Natural Science Foundation of China (10771191 and 10471124) and
Natural Science Foundation of Zhejiang Province of China (Y6090105).
Shen Jun1,2, Wu Junde1 Tel: 86-571-87951609-8111, E-mail: wjd@zju.edu.cn
###### Abstract
In 2005, Professor Gudder presented 25 open problems of sequential effect
algebras, the 20th problem asked: In a sequential effect algebra, if the
square root of some element exists, is it unique ? We can strengthen the
problem as following: For each given positive integer $n>1$, is there a
sequential effect algebra such that the n-th root of its some element $c$ is
not unique and the n-th root of $c$ is not the k-th root of $c$ ($k<n$) ? In
this paper, we answer the strengthened problem affirmatively.
1Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
2Department of Mathematics, Anhui Normal University, Wuhu 241003, P. R. China
Keywords. Effect algebra, sequential effect algebra, root.
PACS numbers: 02.10-v, 02.30.Tb, 03.65.Ta.
Let $H$ be a complex Hilbert space and ${\cal D}(H)$ the set of density
operators on $H$, i.e., the trace class positive operators on $H$ of unit
trace, which represent the states of quantum system. A self-adjoint operator
$A$ on $H$ such that $0\leq A\leq I$ is called a quantum effect ([1, 2]), the
set of quantum effects on $H$ is denoted by ${\cal E}(H)$. The set of
orthogonal projection operators on $H$ is denoted by ${\cal P}(H)$. For each
$P\in{\cal P}(H)$ is associated a so-called Lüders transformation
$\Phi_{L}^{P}:{\cal D}(H)\rightarrow{\cal D}(H)$ such that for each $T\in{\cal
D}(H)$, $\Phi_{L}^{P}(T)=PTP$. Moreover, each quantum effect $B\in{\cal E}(H)$
gives also to a general Lüders transformation $\Phi_{L}^{B}$ such that for
each $T\in{\cal D}(H)$, $\Phi_{L}^{B}(T)=B^{\frac{1}{2}}TB^{\frac{1}{2}}$
([3-4]).
Let $B,C\in{\cal E}(H)$ be two quantum effects. It is easy to prove that the
composition $\Phi_{L}^{B}\circ\Phi_{L}^{C}$ satisfies that for each $T\in{\cal
D}(H)$,
($\Phi_{L}^{B}\circ\Phi_{L}^{C})(T)=(B^{\frac{1}{2}}CB^{\frac{1}{2}})^{\frac{1}{2}}T(B^{\frac{1}{2}}CB^{\frac{1}{2}})^{\frac{1}{2}}$
([4]). Professor Gudder called $B^{\frac{1}{2}}CB^{\frac{1}{2}}$ the
sequential product of $B$ and $C$, and denoted it by $B\circ C$ ([5-7]). This
sequential product has been generalized to an algebraic structure called a
sequential effect algebra ([8]). Now, we state the basic definitions and
results of sequential effect algebras.
An effect algebra is a system $(E,0,1,\oplus)$, where 0 and 1 are distinct
elements of $E$ and $\oplus$ is a partial binary operation on $E$ satisfying
that [9]:
(EA1). If $a\oplus b$ is defined, then $b\oplus a$ is defined and $b\oplus
a=a\oplus b$.
(EA2). If $a\oplus(b\oplus c)$ is defined, then $(a\oplus b)\oplus c$ is
defined and
$(a\oplus b)\oplus c=a\oplus(b\oplus c).$
(EA3). For each $a\in E$, there exists a unique element $b\in E$ such that
$a\oplus b=1$.
(EA4). If $a\oplus 1$ is defined, then $a=0$.
In an effect algebra $(E,0,1,\oplus)$, if $a\oplus b$ is defined, we write
$a\bot b$. For each $a\in(E,0,1,\oplus)$, it follows from (EA3) that there
exists a unique element $b\in E$ such that $a\oplus b=1$, we denote $b$ by
$a^{\prime}$. Let $a,b\in(E,0,1,\oplus)$, if there exists a $c\in E$ such that
$a\bot c$ and $a\oplus c=b$, then we say that $a\leq b$, if in addition,
$a\neq b$, then we write $a<b$. It follows from [9] that $\leq$ is a partial
order of $(E,0,1,\oplus)$ and satisfies that for each $a\in E$, $0\leq a\leq
1$, $a\bot b$ if and only if $a\leq b^{\prime}$.
A sequential effect algebra is an effect algebra $(E,0,1,\oplus)$ and another
binary operation $\circ$ defined on $(E,0,1,\oplus)$ satisfying that [8]:
(SEA1). The map $b\mapsto a\circ b$ is additive for each $a\in E$, that is, if
$b\bot c$, then $a\circ b\bot a\circ c$ and $a\circ(b\oplus c)=a\circ b\oplus
a\circ c$.
(SEA2). $1\circ a=a$ for each $a\in E$.
(SEA3). If $a\circ b=0$, then $a\circ b=b\circ a$.
(SEA4). If $a\circ b=b\circ a$, then $a\circ b^{\prime}=b^{\prime}\circ a$ and
$a\circ(b\circ c)=(a\circ b)\circ c$ for each $c\in E$.
(SEA5). If $c\circ a=a\circ c$ and $c\circ b=b\circ c$, then $c\circ(a\circ
b)=(a\circ b)\circ c$ and $c\circ(a\oplus b)=(a\oplus b)\circ c$ whenever
$a\bot b$.
Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra. Then the operation
$\circ$ is said to be a sequential product on $(E,0,1,\oplus,\circ)$. If
$a,b\in(E,0,1,\oplus,\circ)$ and $a\circ b=b\circ a$, then $a$ and $b$ is said
to be sequentially independent and write $a|b$ ([8]). Let
$a\in(E,0,1,\oplus,\circ)$. If there exists an element
$b\in(E,0,1,\oplus,\circ)$ such that $\underbrace{b\circ b\circ\cdots\circ
b}\limits_{the\ number\ is\ n}=a$, then we write $b^{n}=a$ and $b$ is said to
be a n-th root of $a$. Note that $b$ is a n-th root of $a$ implies that $a$
can be obtained by measuring $b$ n-times repeatedly.
The sequential effect algebra is an important and interesting mathematical
model for studying the quantum measurement theory [5-8]. In [10], Professor
Gudder presented 25 open problems to motivate the study of sequential effect
algebra theory. The 20th problem asked:
Problem 1 ([10]). In a sequential effect algebra $(E,0,1,\oplus,\circ)$, if
the square root of some element exists, is it unique ?
Now, we can strengthen Problem 1 as following:
Problem 2. For each given positive integer $n>1$, is there a sequential effect
algebra $(E,0,1,\oplus,\circ)$ such that the n-th root of its some element $c$
is not unique and the n-th root of $c$ is not the k-th root of $c$ ($k<n$) ?
i.e., are there $a,b\in E$, such that $a\neq b$, $a^{n}=c=b^{n}$ and
$a^{k}\neq c$, $b^{k}\neq c$ for $k<n$ ?
In this paper, we present an example to answer Problem 2 affirmatively.
Actually, we will construct a sequential effect algebra $E_{0}$, such that
there are elements $a,b,c\in E_{0}$ having the relations
$a>a^{2}>\cdots>a^{n},$ $b>b^{2}>\cdots>b^{n},$ $a^{k}\neq b^{k}\ for\ k<n\ ,\
a^{n}=b^{n}=c\neq 0.$
In order to construct our example, we need some preliminary steps:
Suppose $Z$ be the integer set, $n>1$ be a given positive integer.
Let $p(x)=\sum\limits_{i=1}^{n-1}k_{i}x^{i}$, where $k_{i}\in Z$, $k_{i}\equiv
0$ or the first nonzero $k_{i}>0$, we denote all the polynomials characterized
above by $I_{0}$ .
Suppose $p_{1},p_{2}\in I_{0}$ and
$p_{1}(x)=\sum\limits_{i=1}^{n-1}k_{1,i}x^{i}$ ,
$p_{2}(x)=\sum\limits_{i=1}^{n-1}k_{2,i}x^{i}$ , let
$F(p_{1},p_{2})(x)=\sum\limits_{i+j\leq n-1}k_{1,i}k_{2,j}x^{i+j}$ ,
$G(p_{1},p_{2})=\sum\limits_{i+j=n}k_{1,i}k_{2,j}$ . Then it is easy to see
that $F(p_{1},p_{2})\in I_{0}$ and $G(p_{1},p_{2})\in Z$ .
Thus we defined mappings
$F:I_{0}\times I_{0}\longrightarrow I_{0}$ and $G:I_{0}\times
I_{0}\longrightarrow Z$ .
Moreover, suppose $p_{1},p_{2},p_{3}\in I_{0}$ and
$p_{1}(x)=\sum\limits_{i=1}^{n-1}k_{1,i}x^{i}$ ,
$p_{2}(x)=\sum\limits_{i=1}^{n-1}k_{2,i}x^{i}$ ,
$p_{3}(x)=\sum\limits_{i=1}^{n-1}k_{3,i}x^{i}$ , let
$\overline{F}(p_{1},p_{2},p_{3})(x)=\sum\limits_{i+j+m\leq
n-1}k_{1,i}k_{2,j}k_{3,m}x^{i+j+m}$ ,
$\overline{G}(p_{1},p_{2},p_{3})=\sum\limits_{i+j+m=n}k_{1,i}k_{2,j}k_{3,m}$ .
Then it is also easy to see that $\overline{F}(p_{1},p_{2},p_{3})\in I_{0}$
and $\overline{G}(p_{1},p_{2},p_{3})\in Z$ . Thus we defined mappings
$\overline{F}:I_{0}\times I_{0}\times I_{0}\longrightarrow I_{0}$ and
$\overline{G}:I_{0}\times I_{0}\times I_{0}\longrightarrow Z$ .
Lemma 1. Suppose $p,p_{1},p_{2},p_{3}\in I_{0}$, we have
(1). $F(p_{1},p_{2})=F(p_{2},p_{1})$, $G(p_{1},p_{2})=G(p_{2},p_{1})$;
(2). $F(p_{1},p_{2}+p_{3})=F(p_{1},p_{2})+F(p_{1},p_{3})$,
$G(p_{1},p_{2}+p_{3})=G(p_{1},p_{2})+G(p_{1},p_{3})$;
(3). $F(0,p)=0$, $G(0,p)=0$;
(4). if $F(p_{1},p_{2})=0$, then $G(p_{1},p_{2})\geq 0$;
(5). $p_{1}-F(p_{1},p_{2})\in I_{0}$, and
$p_{1}=F(p_{1},p_{2})\Longleftrightarrow p_{1}=0$;
(6). $F(F(p_{1},p_{2}),p_{3})=\overline{F}(p_{1},p_{2},p_{3})$,
$G(F(p_{1},p_{2}),p_{3})=\overline{G}(p_{1},p_{2},p_{3})$;
(7). $p_{1}+p_{2}\in I_{0}$, and $p_{1}+p_{2}=0\Longleftrightarrow
p_{1}=p_{2}=0$.
Proof. (1),(2),(3),(6) and (7) are trivial.
(4). Except for the trivial cases, we may suppose
$p_{1}(x)=\sum\limits_{i=n_{1}}^{n-1}k_{1,i}x^{i}$,
$p_{2}(x)=\sum\limits_{i=n_{2}}^{n-1}k_{2,i}x^{i}$, with $k_{1,n_{1}}>0$ and
$k_{2,n_{2}}>0$. Then from $F(p_{1},p_{2})=0$ we have $n_{1}+n_{2}\geq n$. If
$n_{1}+n_{2}=n$, then $G(p_{1},p_{2})=k_{1,n_{1}}k_{2,n_{2}}>0$; otherwise
$n_{1}+n_{2}>n$ and $G(p_{1},p_{2})=0$.
(5). Except for the trivial cases, we may suppose
$p_{1}(x)=\sum\limits_{i=n_{1}}^{n-1}k_{1,i}x^{i}$,
$p_{2}(x)=\sum\limits_{i=n_{2}}^{n-1}k_{2,i}x^{i}$, with $k_{1,n_{1}}>0$ and
$k_{2,n_{2}}>0$. Then the first item of $p_{1}-F(p_{1},p_{2})$ is
$k_{1,n_{1}}x^{n_{1}}$, so $p_{1}-F(p_{1},p_{2})\in I_{0}$. If $p_{1}\neq 0$,
then from the above reason we know that $p_{1}-F(p_{1},p_{2})\neq 0$. Thus,
the lemma is proved.
Now, we take two infinite sets $U$ and $V$ such that $U\cap V=\emptyset$. Let
$f:I_{0}\times I_{0}\times Z\rightarrow U$ and $g:I_{0}\times I_{0}\times
Z\rightarrow V$ be two one to one maps. Then, we construct our example as
following:
Let $E_{0}=\\{f(p,q,m),g(p,q,m)|p,q\in I_{0},m\in Z\ and\ satisfy\ that\ m\geq
0\ whenever\ p=q=0\\}$.
First, we define a partial binary operation $\oplus$ on $E_{0}$ as follows
(when we write $x\oplus y=z$, we always mean that $x\oplus y=z=y\oplus x$):
(i). $f(p_{1},q_{1},m_{1})\oplus
f(p_{2},q_{2},m_{2})=f(p_{1}+p_{2},q_{1}+q_{2},m_{1}+m_{2})$ (the right side
is well-defined, see Lemma 1(7));
(ii). for $p_{2}-p_{1}\in I_{0}$, $q_{2}-q_{1}\in I_{0}$, and satisfy that
$m_{2}\geq m_{1}$ when $p_{2}=p_{1}$ and $q_{2}=q_{1}$,
$f(p_{1},q_{1},m_{1})\oplus
g(p_{2},q_{2},m_{2})=g(p_{2}-p_{1},q_{2}-q_{1},m_{2}-m_{1})$ .
No other $\oplus$ operation is defined.
Next, we define a binary operation $\circ$ on $E_{0}$ as follows (when we
write $x\circ y=z$, we always mean that $x\circ y=z=y\circ x$):
(i). $f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})=f\Big{(}F(p_{1},p_{2}),F(q_{1},q_{2}),G(p_{1},p_{2})+G(q_{1},q_{2})\Big{)}$
(the right side is well-defined, see Lemma 1(4));
(ii). $f(p_{1},q_{1},m_{1})\circ
g(p_{2},q_{2},m_{2})=f\Big{(}p_{1}-F(p_{1},p_{2}),q_{1}-F(q_{1},q_{2}),m_{1}-G(p_{1},p_{2})-G(q_{1},q_{2})\Big{)}$
(the right side is well-defined, see Lemma 1(3), (5));
(iii). $g(p_{1},q_{1},m_{1})\circ
g(p_{2},q_{2},m_{2})=g\Big{(}p_{1}+p_{2}-F(p_{1},p_{2}),q_{1}+q_{2}-F(q_{1},q_{2}),m_{1}+m_{2}-G(p_{1},p_{2})-G(q_{1},q_{2})\Big{)}$
(the right side is well-defined, see Lemma 1(3), (5), (7)).
We denote $f(0,0,0)$ by $0$, $g(0,0,0)$ by $1$.
Proposition 1. $(E_{0},0,1,\oplus,\circ)$ is a sequential effect algebra.
Proof. In the proof below, we will use Lemma 1 frequently without annotation.
First, we verify that $(E_{0},0,1,\oplus)$ is an effect algebra.
(EA1) is obvious. We verify (EA2) as follows:
(i). $f(p_{1},q_{1},m_{1})\oplus\Big{(}f(p_{2},q_{2},m_{2})\oplus
f(p_{3},q_{3},m_{3})\Big{)}=\Big{(}f(p_{1},q_{1},m_{1})\oplus
f(p_{2},q_{2},m_{2})\Big{)}\oplus
f(p_{3},q_{3},m_{3})=f(p_{1}+p_{2}+p_{3},q_{1}+q_{2}+q_{3},m_{1}+m_{2}+m_{3})$;
(ii). $f(p_{1},q_{1},m_{1})\oplus\Big{(}f(p_{2},q_{2},m_{2})\oplus
g(p_{3},q_{3},m_{3})\Big{)}$ or $\Big{(}f(p_{1},q_{1},m_{1})\oplus
f(p_{2},q_{2},m_{2})\Big{)}\oplus g(p_{3},q_{3},m_{3})$ is defined iff
$p_{3}-p_{1}-p_{2}\in I_{0}$, $q_{3}-q_{1}-q_{2}\in I_{0}$ and satisfy that
$m_{3}\geq m_{1}+m_{2}$ when $p_{3}=p_{1}+p_{2}$ and $q_{3}=q_{1}+q_{2}$, at
this point, they all equal to
$g(p_{3}-p_{1}-p_{2},q_{3}-q_{1}-q_{2},m_{3}-m_{1}-m_{2})$.
Note that $f(p,q,m)\oplus g(p,q,m)=g(0,0,0)=1$, we verified (EA3).
For (EA4), we note from our construction that the unique element orthogonal to
$g(0,0,0)(=1)$ is $f(0,0,0)(=0)$, that is, $f(0,0,0)\bot g(0,0,0)$ and
$f(0,0,0)\oplus g(0,0,0)=g(0,0,0)$.
So far, we have proved that $(E_{0},0,1,\oplus)$ is an effect algebra.
Next, we verify that $(E_{0},0,1,\oplus,\circ)$ is a sequential effect
algebra.
(SEA3) and (SEA5) are obvious.
We verify (SEA1) as follows:
(i). $f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\oplus
f(p_{3},q_{3},m_{3})\Big{)}=f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\oplus f(p_{1},q_{1},m_{1})\circ
f(p_{3},q_{3},m_{3})=f\Big{(}F(p_{1},p_{2}+p_{3}),F(q_{1},q_{2}+q_{3}),G(p_{1},p_{2}+p_{3})+G(q_{1},q_{2}+q_{3})\Big{)}$,
$g(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\oplus
f(p_{3},q_{3},m_{3})\Big{)}=g(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\oplus g(p_{1},q_{1},m_{1})\circ
f(p_{3},q_{3},m_{3})=f\Big{(}p_{2}+p_{3}-F(p_{1},p_{2}+p_{3}),q_{2}+q_{3}-F(q_{1},q_{2}+q_{3}),m_{2}+m_{3}-G(p_{1},p_{2}+p_{3})-G(q_{1},q_{2}+q_{3})\Big{)}$;
(ii). when $f(p_{2},q_{2},m_{2})\oplus g(p_{3},q_{3},m_{3})$ is defined, i.e.,
when $p_{3}-p_{2}\in I_{0}$, $q_{3}-q_{2}\in I_{0}$, and satisfy that
$m_{3}\geq m_{2}$ if $p_{3}=p_{2}$ and $q_{3}=q_{2}$ ,
$f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\oplus
g(p_{3},q_{3},m_{3})\Big{)}=f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\oplus f(p_{1},q_{1},m_{1})\circ
g(p_{3},q_{3},m_{3})=f\Big{(}p_{1}-F(p_{1},p_{3}-p_{2}),q_{1}-F(q_{1},q_{3}-q_{2}),m_{1}-G(p_{1},p_{3}-p_{2})-G(q_{1},q_{3}-q_{2})\Big{)}$,
$g(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\oplus
g(p_{3},q_{3},m_{3})\Big{)}=g(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\oplus g(p_{1},q_{1},m_{1})\circ
g(p_{3},q_{3},m_{3})=g\Big{(}p_{1}+p_{3}-p_{2}-F(p_{1},p_{3}-p_{2}),q_{1}+q_{3}-q_{2}-F(q_{1},q_{3}-q_{2}),m_{1}+m_{3}-m_{2}-G(p_{1},p_{3}-p_{2})-G(q_{1},q_{3}-q_{2})\Big{)}$.
We verify (SEA2) as follows:
$1\circ f(p,q,m)=g(0,0,0)\circ f(p,q,m)=f(p,q,m);$ $1\circ
g(p,q,m)=g(0,0,0)\circ g(p,q,m)=g(p,q,m).$
We verify (SEA4) as follows:
(i). $f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\circ
f(p_{3},q_{3},m_{3})\Big{)}$
$=f(p_{1},q_{1},m_{1})\circ
f\Big{(}F(p_{2},p_{3}),F(q_{2},q_{3}),G(p_{2},p_{3})+G(q_{2},q_{3})\Big{)}$
$=f\Big{(}F(p_{1},F(p_{2},p_{3})),F(q_{1},F(q_{2},q_{3})),G(p_{1},F(p_{2},p_{3}))+G(q_{1},F(q_{2},q_{3}))\Big{)}$
$=f\Big{(}\overline{F}(p_{1},p_{2},p_{3}),\overline{F}(q_{1},q_{2},q_{3}),\overline{G}(p_{1},p_{2},p_{3})+\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
by symmetry,
$\Big{(}f(p_{1},q_{1},m_{1})\circ f(p_{2},q_{2},m_{2})\Big{)}\circ
f(p_{3},q_{3},m_{3})$
$=f(p_{3},q_{3},m_{3})\circ\Big{(}f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\Big{)}$
$=f\Big{(}\overline{F}(p_{1},p_{2},p_{3}),\overline{F}(q_{1},q_{2},q_{3}),\overline{G}(p_{1},p_{2},p_{3})+\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
so we have
$f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\circ
f(p_{3},q_{3},m_{3})\Big{)}=\Big{(}f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\Big{)}\circ f(p_{3},q_{3},m_{3})$.
(ii). $f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}$
$=f(p_{1},q_{1},m_{1})\circ
f\Big{(}p_{2}-F(p_{2},p_{3}),q_{2}-F(q_{2},q_{3}),m_{2}-G(p_{2},p_{3})-G(q_{2},q_{3})\Big{)}$
$=f\Big{(}F(p_{1},p_{2}-F(p_{2},p_{3})),F(q_{1},q_{2}-F(q_{2},q_{3})),G(p_{1},p_{2}-F(p_{2},p_{3}))+G(q_{1},q_{2}-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}F(q_{2},q_{3}))\Big{)}$
$=f\Big{(}F(p_{1},p_{2})-F(p_{1},F(p_{2},p_{3})),F(q_{1},q_{2})-F(q_{1},F(q_{2},q_{3})),G(p_{1},p_{2})-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(p_{1},F(p_{2},p_{3}))+G(q_{1},q_{2})-G(q_{1},F(q_{2},q_{3}))\Big{)}$
$=f\Big{(}F(p_{1},p_{2})-\overline{F}(p_{1},p_{2},p_{3}),F(q_{1},q_{2})-\overline{F}(q_{1},q_{2},q_{3}),G(p_{1},p_{2})-\overline{G}(p_{1},p_{2},p_{3})+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(q_{1},q_{2})-\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
$\Big{(}f(p_{1},q_{1},m_{1})\circ f(p_{2},q_{2},m_{2})\Big{)}\circ
g(p_{3},q_{3},m_{3})$
$=f\Big{(}F(p_{1},p_{2}),F(q_{1},q_{2}),G(p_{1},p_{2})+G(q_{1},q_{2})\Big{)}\circ
g(p_{3},q_{3},m_{3})$
$=f\Big{(}F(p_{1},p_{2})-F(F(p_{1},p_{2}),p_{3}),F(q_{1},q_{2})-F(F(q_{1},q_{2}),q_{3}),G(p_{1},p_{2})+G(q_{1},q_{2})-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(F(p_{1},p_{2}),p_{3})-G(F(q_{1},q_{2}),q_{3})\Big{)}$
$=f\Big{(}F(p_{1},p_{2})-\overline{F}(p_{1},p_{2},p_{3}),F(q_{1},q_{2})-\overline{F}(q_{1},q_{2},q_{3}),G(p_{1},p_{2})-\overline{G}(p_{1},p_{2},p_{3})+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(q_{1},q_{2})-\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
so we have
$f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}=\Big{(}f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\Big{)}\circ g(p_{3},q_{3},m_{3})$ .
(iii). $f(p_{1},q_{1},m_{1})\circ\Big{(}g(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}$
$=f(p_{1},q_{1},m_{1})\circ
g\Big{(}p_{2}+p_{3}-F(p_{2},p_{3}),q_{2}+q_{3}-F(q_{2},q_{3}),m_{2}+m_{3}-G(p_{2},p_{3})-G(q_{2},q_{3})\Big{)}$
$=f\Big{(}p_{1}-F(p_{1},p_{2}+p_{3}-F(p_{2},p_{3})),q_{1}-F(q_{1},q_{2}+q_{3}-F(q_{2},q_{3})),m_{1}-G(p_{1},p_{2}+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{3}-F(p_{2},p_{3}))-G(q_{1},q_{2}+q_{3}-F(q_{2},q_{3}))\Big{)}$
$=f\Big{(}p_{1}-F(p_{1},p_{2}+p_{3})+\overline{F}(p_{1},p_{2},p_{3}),q_{1}-F(q_{1},q_{2}+q_{3})+\overline{F}(q_{1},q_{2},q_{3}),m_{1}-G(p_{1},p_{2}+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{3})+\overline{G}(p_{1},p_{2},p_{3})-G(q_{1},q_{2}+q_{3})+\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
$\Big{(}f(p_{1},q_{1},m_{1})\circ g(p_{2},q_{2},m_{2})\Big{)}\circ
g(p_{3},q_{3},m_{3})$
$=f\Big{(}p_{1}-F(p_{1},p_{2}),q_{1}-F(q_{1},q_{2}),m_{1}-G(p_{1},p_{2})-G(q_{1},q_{2})\Big{)}\circ
g(p_{3},q_{3},m_{3})$
$=f\Big{(}p_{1}-F(p_{1},p_{2})-F(p_{1}-F(p_{1},p_{2}),p_{3}),q_{1}-F(q_{1},q_{2})-F(q_{1}-F(q_{1},q_{2}),q_{3}),m_{1}-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(p_{1},p_{2})-G(q_{1},q_{2})-G(p_{1}-F(p_{1},p_{2}),p_{3})-G(q_{1}-F(q_{1},q_{2}),q_{3})\Big{)}$
$=f\Big{(}p_{1}-F(p_{1},p_{2}+p_{3})+\overline{F}(p_{1},p_{2},p_{3}),q_{1}-F(q_{1},q_{2}+q_{3})+\overline{F}(q_{1},q_{2},q_{3}),m_{1}-G(p_{1},p_{2}+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{3})+\overline{G}(p_{1},p_{2},p_{3})-G(q_{1},q_{2}+q_{3})+\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
so we have
$f(p_{1},q_{1},m_{1})\circ\Big{(}g(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}=\Big{(}f(p_{1},q_{1},m_{1})\circ
g(p_{2},q_{2},m_{2})\Big{)}\circ g(p_{3},q_{3},m_{3})$.
(iv). $g(p_{1},q_{1},m_{1})\circ\Big{(}g(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}$
$=g(p_{1},q_{1},m_{1})\circ
g\Big{(}p_{2}+p_{3}-F(p_{2},p_{3}),q_{2}+q_{3}-F(q_{2},q_{3}),m_{2}+m_{3}-G(p_{2},p_{3})-G(q_{2},q_{3})\Big{)}$
$=g\Big{(}p_{1}+p_{2}+p_{3}-F(p_{2},p_{3})-F(p_{1},p_{2}+p_{3}-F(p_{2},p_{3})),q_{1}+q_{2}+q_{3}-F(q_{2},q_{3})-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F(q_{1},q_{2}+q_{3}-F(q_{2},q_{3})),m_{1}+m_{2}+m_{3}-G(p_{2},p_{3})-G(q_{2},q_{3})-G(p_{1},p_{2}+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{3}-F(p_{2},p_{3}))-G(q_{1},q_{2}+q_{3}-F(q_{2},q_{3}))\Big{)}$
$=g\Big{(}p_{1}+p_{2}+p_{3}-F(p_{2},p_{3})-F(p_{1},p_{2})-F(p_{1},p_{3})+\overline{F}(p_{1},p_{2},p_{3}),q_{1}+q_{2}+q_{3}-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F(q_{2},q_{3})-F(q_{1},q_{2})-F(q_{1},q_{3})+\overline{F}(q_{1},q_{2},q_{3}),m_{1}+m_{2}+m_{3}-G(p_{2},p_{3})-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(p_{1},p_{2})-G(p_{1},p_{3})+\overline{G}(p_{1},p_{2},p_{3})-G(q_{2},q_{3})-G(q_{1},q_{2})-G(q_{1},q_{3})+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
by symmetry, we have
$g(p_{1},q_{1},m_{1})\circ\Big{(}g(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}=\Big{(}g(p_{1},q_{1},m_{1})\circ
g(p_{2},q_{2},m_{2})\Big{)}\circ g(p_{3},q_{3},m_{3})$.
Thus, we proved that $(E_{0},0,1,\oplus,\circ)$ is a sequential effect algebra
and the theorem is proved.
Now, let $P_{i}(x)=x^{i}$. Then it is easy to see that
$F(P_{1},P_{j})=\left\\{\begin{array}[]{ll}P_{1+j}\ ,&\hbox{$if\ j<n-1$;}\\\
0\ ,&\hbox{$if\
j=n-1$.}\end{array}\right.~{}and~{}~{}G(P_{1},P_{j})=\left\\{\begin{array}[]{ll}0\
,&\hbox{$if\ j<n-1$;}\\\ 1\ ,&\hbox{$if\ j=n-1$.}\end{array}\right.$
Thus we have
$[f(P_{1},0,0)]^{k}=f(P_{1},0,0)\circ f(P_{k-1},0,0)=f(P_{k},0,0)$ for $k<n$,
$[f(P_{1},0,0)]^{n}=f(P_{1},0,0)\circ f(P_{n-1},0,0)=f(0,0,1)$,
$[f(P_{1},0,0)]^{n+1}=f(P_{1},0,0)\circ f(0,0,1)=0$,
and
$[f(0,P_{1},0)]^{k}=f(0,P_{1},0)\circ f(0,P_{k-1},0)=f(0,P_{k},0)$ for $k<n$,
$[f(0,P_{1},0)]^{n}=f(0,P_{1},0)\circ f(0,P_{n-1},0)=f(0,0,1)$,
$[f(0,P_{1},0)]^{n+1}=f(0,P_{1},0)\circ f(0,0,1)=0$.
If we denote $f(P_{1},0,0)$ by $a$, $f(0,P_{1},0)$ by $b$, $f(0,0,1)$ by $c$,
then it is easy to get the relations
$a>a^{2}>\cdots>a^{n}>a^{n+1},$ $b>b^{2}>\cdots>b^{n}>b^{n+1},$ $a^{k}\neq
b^{k}\ for\ k<n\ ,\ a^{n}=b^{n}=c\neq 0\ and\ a^{n+1}=b^{n+1}=0.$
That is, $a,b$ are the n-th root of $c$, but $a,b$ are not the k-th root of
$c$, where $k=2,3,\cdots,n-1$, moreover, $a,b$ are also the n+1-th root of
$0$, so, the Problem 2 is answered affirmatively.
Finally, we would like to point out that for the advances of sequential effect
algebras, see [11-16].
Acknowledgement
The authors wish to express their thanks to the referee for his valuable
comments and suggestions.
References
[1]. Ludwig, G. Foundations of Quantum Mechanics (I-II), Springer, New York,
1983.
[2]. Ludwig, G. An Axiomatic Basis for Quantum Mechanics (II), Springer, New
York, 1986.
[3]. Davies, E. B. Quantum Theory of Open Systems, Academic Press, London,
1976.
[4]. Busch, P, Grabowski, M and Lahti P. J, Operational Quantum Physics,
Springer-Verlag, Beijing Word Publishing Corporation, 1999.
[5]. Gudder, S, Nagy, G. Sequential quantum measurements. J. Math. Phys.
42(2001), 5212-5222.
[6]. Gheondea, A, Gudder, S. Sequential product of quantum effects. Proc.
Amer. Math. Soc. 132 (2004), 503-512.
[7]. Gudder, S, Latr moli re, F. Characterization of the sequential product on
quantum effects. J. Math. Phys. 49 (2008), 052106-052112.
[8]. Gudder, S, Greechie, R. Sequential products on effect algebras. Rep.
Math. Phys. 49(2002), 87-111.
[9]. Foulis, D J, Bennett, M K. Effect algebras and unsharp quantum logics.
Found Phys 24 (1994), 1331-1352.
[10]. Gudder, S. Open problems for sequential effect algebras. Inter. J.
Theory. Physi. 44 (2005), 2219-2230.
[11] Shen Jun and Wu Junde. Not each sequential effect algebra is sharply
dominating. Phys. Letter A. 373, 1708-1712, (2009)
[12] Shen Jun and Wu Junde. Remarks on the sequential effect algebras. Report.
Math. Phys. 63, 441-446, (2009)
[13] Shen Jun and Wu Junde. Sequential product on standard effect algebra
${\cal E}(H)$. J. Phys. A: Math. Theor. 44, 345203-345214, (2009)
[14] Shen Jun and Wu Junde. The Average Value Inequality in Sequential Effect
Algebras. Acta Math. Sinica, English Series. 25, 1330-1336, (2009)
[15] Liu Weihua and Wu Junde. A uniqueness problem of the sequence product on
operator effect algebra ${\cal E}(H)$. J. Phys. A: Math. Theor. 42,
185206-185215, (2009)
[16] Liu Weihua and Wu Junde. On fixed points of Lüders operation. J. Math.
Phys. 50, 103531-103532, (2009)
|
arxiv-papers
| 2009-03-30T03:40:45 |
2024-09-04T02:49:01.520375
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shen Jun and Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0903.5120"
}
|
0903.5376
|
# Lifshitz tails for the Interband Light Absorption Coefficient
W Kirsch
Facultät für Mathematik und Informatik
Fern Universität in Hagen
58084 Hagen, Germany
and
M Krishna
Institute of Mathematical Sciences
Taramani, Chennai 600113, India
(31 March 2009)
###### Abstract
In this paper we consider the Interband Light Absorption Coefficient for
various models. We show that at the lower and upper edges of the spectrum the
Lifshitz tails behaviour of the density of states implies similar behaviour
for the ILAC at appropriate energies. The Lifshitz tails property is also
exhibited at some points corresponding to the internal band edges of the
density of states.
## 1 Introduction
In this work we look for Lifshitz tails behaviour of the Interband Light
Absorption Coefficient (ILAC) defined in equation (eqnl4). The standard
definition of the ILAC involves considering a pair of operators of the form
$H_{\omega}^{\pm}=\Delta\pm V^{\omega}$, with $\Delta$ the Laplacian on either
$\ell^{2}(\mathbb{Z}^{d})$, in the discrete case or on $L^{2}(\mathbb{R}^{d})$
in the continuous case, and taking a random potential $V^{\omega}$.
Restricting these operators $H_{\omega}^{\pm}$ to boxes $\Lambda$ gives
operators with discrete spectra so that in any finite region of energy these
operators have only finitely many eigenvalues. Using this fact one can define
the quantity
$\frac{1}{Vol(\Lambda)}\sum_{\lambda_{\omega}^{-}+\lambda_{\omega}^{+}\leq
E}|\langle\phi_{\omega,\lambda_{\omega}^{-}},\psi_{\omega,\lambda_{\omega}^{+}}\rangle|^{2}$
where $\phi_{\omega,\lambda_{\omega}^{-}},\psi_{\omega,\lambda_{\omega}^{+}}$
are the eigen functions of the operators $H_{\omega}^{\mp}$ restricted to the
box $\Lambda$, corresponding to the eigenvalues
$\lambda_{\omega}^{-},\lambda_{\omega}^{+}$ respectively.
The limit of the above quantity, when it exits, gives the ILAC.
We consider a correlation measure (mentioned also in [12]) $\rho$ and identify
the ILAC as the distribution function of a marginal of the measure $\rho$ in a
diagonal direction. This identification enables us to prove theorems on the
Lifshitz tails behaviour of the ILAC more easily since it involves only
comparing the marginal of $\rho$ with the density of states of either of the
operators $H_{\omega}^{\pm}$. We also do not need to approximate to define the
ILAC, but can obtain the function directly.
In the next section, we present an abstract version of the correlation measure
$\rho$ and the density of states $n$ for a pair of random covariant operators
and obtain relations between the two.
## 2 General Covariant Operators
We start with a definition of a random family of self adjoint operators which
are covariant under a group action.
###### Hypotheses 1.
1. 1.
$\mathcal{H}$ is a (separable, complex) Hilbert space,
$(\Omega,\mathcal{F},\mathbb{P})$ a probability space.
2. 2.
There is a locally compact abelian group $G$ and $\\{U_{x}\\}_{x\in G}$ is a
group of unitary operators on $\mathcal{H}$, i.e. the $U_{x}$ are unitary and
$U_{x+y}=U_{x}\,U_{y}$, $U_{0}=\textnormal{Id}$, $U_{-x}=U_{x}^{-1}=U_{x}^{*}$
3. 3.
There is a discrete subgroup $L$ of $G$ and an orthogonal projection $P$ on
$\mathcal{H}$ such that $\\{U_{n}^{*}PU_{n}\\}_{n\in L}$,
$\\{U_{n}PU_{n}^{*}\\}_{n\in L}$ are orthogonal partitions of unity on
$\mathcal{H}$. We set $P_{n}=U_{n}^{*}PU_{n},\tilde{P}_{n}=U_{n}PU_{n}^{*}.$
4. 4.
$\\{T_{n}\\}_{n\in L}$ is a group of probability preserving transformations on
$\Omega$.
###### Definition 1.
A family $\\{A_{\omega}\\}_{\omega\in\Omega}$ of self-adjoint operators on
$\mathcal{H}$ is called measurable if the family
$\\{(A_{\omega}+i)^{-1}\\}_{\omega\in\Omega}$ is measurable
It is known (see [3], [2] and section 2.4 of [25]) that a family of _bounded_
self-adjoint operators is measurable iff it’s weakly measurable.
Moreover, if $\\{A_{\omega}\\}$ is a measurable family of self-adjoint
operators then for any bounded measurable function $f$ the operator family
$f(A_{\omega})$ is weakly measurable. (also in [3], [2], section 2.4 [25]).
Finally, the product of weakly measurable families is weakly measurable (see
[2]).
###### Definition 2.
A weakly measurable family $A_{\omega}$ of bounded operators is called
_covariant_ (with respect to $U_{x},T_{x}$) if
$A_{T_{x}\omega}=U_{x}^{*}\,A_{\omega}\,U_{x}\qquad\textnormal{for all $x\in
G$}$
Also, a measurable family $A_{\omega}$ of self adjoint operators is called
_covariant_ (with respect to $U_{x},T_{x}$) if
$A_{T_{x}\omega}=U_{x}^{*}\,A_{\omega}\,U_{x}\qquad\textnormal{for all $x\in
G$}$
If $A_{\omega}$ is a covariant family of self-adjoint operators and $f$ is a
bounded measurable function, then the family $f(A_{\omega})$ is covariant
(also in [3], [2]). Moreover, if both $A_{\omega}$ and $B_{\omega}$ are
covariant families of bounded operators, then $A_{\omega}\,B_{\omega}$ is a
covariant family. We denote by $\|B\|_{1}$ the trace norm of a trace class
operator $B$.
###### Proposition 1.
Let $A_{\omega}$ and $B_{\omega}$ be covariant families of bounded operators
and assume that $A_{\omega}P$ and $B_{\omega}P$ are trace class and
$\mathbb{E}(\|A_{\omega}P\|_{1})<\infty\leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ \mathbb{E}(\|B_{\omega}P\|_{1})<\infty.$ (1)
Then:
$\mathbb{E}(Tr(PA_{\omega}B_{\omega}P))\leavevmode\nobreak\
=\leavevmode\nobreak\ \mathbb{E}(Tr(PB_{\omega}A_{\omega}P))$ (2)
###### Proof.
$\displaystyle Tr(PA_{\omega}B_{\omega}P)\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ Tr(PA_{\omega}B_{\omega}P)$ (3)
$\displaystyle=\leavevmode\nobreak\ \sum_{n}\;Tr(PA_{\omega}P_{n}B_{\omega}P)$
(4) since $P_{n}$ is a partition of unity of orthogonal projections.
$\displaystyle=\leavevmode\nobreak\
\sum_{n}\;Tr(P_{n}B_{\omega}PA_{\omega}P_{n})$ (5) using the $Tr(AB)=Tr(BA)$
and the invariance of trace $Tr(U^{*}CU)=Tr(C)$,
$\displaystyle=\leavevmode\nobreak\
\sum_{n}\;Tr(PA_{T_{n}^{-1}\omega}\tilde{P}_{n}B_{T_{n}^{-1}\omega}P)$ (6)
from the covariance of $A_{\omega}$ and $B_{\omega}$
$\displaystyle=\leavevmode\nobreak\
Tr(PB_{T_{n}^{-1}\omega}A_{T_{n}^{-1}\omega}P)\rangle$ (7)
In the last step we used the fact that $\tilde{P}_{n}$ is a partition of unity
also. Now we take expectations of either side of the above equation and obtain
$\displaystyle\mathbb{E}(Tr(PA_{\omega}B_{\omega}P)\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\mathbb{E}(\sum_{n}\;Tr(PB_{T_{n}^{-1}\omega}\tilde{P}_{n}A_{T_{n}^{-1}\omega}P))$
(8) $\displaystyle=\leavevmode\nobreak\
\sum_{n}\;\mathbb{E}(Tr(PB_{T_{n}^{-1}\omega}\tilde{P}_{n}A_{T_{n}^{-1}\omega}P))$
(9) We have used Fubini’s theorem to interchange expectation and sum, allowed
because of (2) $\displaystyle=\leavevmode\nobreak\
\sum_{n}\;\mathbb{E}(Tr(PB_{\omega}\tilde{P}_{n}A_{\omega}P))$ (10) since
$T_{n}^{-1}$ is probability preserving $\displaystyle=\leavevmode\nobreak\
\mathbb{E}(\sum_{n}\;Tr(PB_{\omega}\tilde{P}_{n}A_{\omega}P))$ (11)
$\tilde{P}_{n}$ is a partition of unity. $\displaystyle=\leavevmode\nobreak\
\mathbb{E}(Tr(B_{\omega}A_{\omega}P))\leavevmode\nobreak\
=\leavevmode\nobreak\ \mathbb{E}(Tr(PB_{\omega}A_{\omega}P))$ (12)
∎
###### Corollary 2.
1. 1.
If $A_{\omega},B_{\omega},C_{\omega}$ are covariant families of bounded
operators satisfying the condition (1) then:
$\mathbb{E}(Tr(PA_{\omega}B_{\omega}C_{\omega}P))\leavevmode\nobreak\
=\leavevmode\nobreak\ \mathbb{E}(Tr(PC_{\omega}A_{\omega}B_{\omega}P))$ (13)
2. 2.
If $A_{\omega},B_{\omega}$ are covariant families of bounded, positive (i.e.
$\geq 0$) operators satisfying the conditions (1) then
$\mathbb{E}(TrPA_{\omega}B_{\omega}P))\leavevmode\nobreak\
\geq\leavevmode\nobreak\ 0$ (14)
###### Proof.
The first assertion is clear as we can apply the proposition to the covariant
families $A_{\omega}B_{\omega}$ and $C_{\omega}$.
For the second claim we observe that $B_{\omega}=C_{\omega}C_{\omega}$ with a
$C_{\omega}=\sqrt{B_{\omega}}$. $C_{\omega}$ as a function of the covariant
family $B_{\omega}$ is covariant as well. Moreover, since $A_{\omega}$ is
positive (and the Hilbert space is complex), $A_{\omega}$ is self-adjoint and
so is $C_{\omega}$.
By part (i) of the corollary we have:
$\displaystyle\mathbb{E}(Tr(PA_{\omega}B_{\omega}P))\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\mathbb{E}(PA_{\omega}C_{\omega}C_{\omega}P))$ (15)
$\displaystyle=\leavevmode\nobreak\
\mathbb{E}(Tr(PC_{\omega}A_{\omega}C_{\omega}P))$ (16)
$\displaystyle\geq\leavevmode\nobreak\ 0\qquad\textnormal{since $A_{\omega}$
is positive}$ (17)
∎
###### Hypotheses 2.
Let $H_{\omega}$ be family of self-adjoint operators, which are bounded below,
on a Hilbert space $\mathcal{H}$. Let $E_{{}_{H_{\omega}}}(\cdot)$ be the
(projection-valued) spectral measure of $H_{\omega}$ such that for any bounded
borel set $A$, the operators $PE_{{}_{H_{\omega}}}(A),E_{{}_{H_{\omega}}}(A)P$
are trace class for a.e. $\omega$ and form a covariant family of operators.
For operators $H_{\omega}$ satisfying the above hypothesis, it is clear that
for any finite $x$, the spectral measure
$E_{{}_{H_{\omega}}}((-\infty,x])=E_{{}_{H_{\omega}}}([c,x])$, with $c$ finite
and smaller than the infimum of the spectrum of $H_{\omega}$. Therefore the
hypothesis implies that for any finite $x$, the operators
$PE_{{}_{H_{\omega}}}((-\infty,x]),E_{{}_{H_{\omega}}}((-\infty,x])P$ are
trace class. Therefore we can now define the density of states for such
operators.
###### Definition 3.
Let $H_{\omega}$ be a family of self adjoint operators satisfying Hypothesis
2. Then the _density of states_ of this family is defined to be the unique
$\sigma$-finite measure $n$ associated with the monotone right continuous
function $F$,
$F(x)=\mathbb{E}\left(Tr(PE_{{}_{H_{\omega}}}((-\infty,x])P)\right),$
via $n((a,b])=F(b)-F(a),\leavevmode\nobreak\ a,b\in\mathbb{R}.$
Thus for any bounded borel set $A$, $n(A)$ agrees with the right hand side of
the above relation with $A$ replacing $(-\infty,x]$.
In the above framework we define another measure that is used to define the
Interband Light Absorption Coefficient (ILAC). To do this we need a pair
$H_{\omega}^{\pm}$ of self-adjoint operators as in the Hypothesis 2 and
consider the associated projection valued measures
$E_{H_{\omega}^{\pm}}(\cdot)$. We then define the density of states of these
operators by,
$n_{\pm}(A)=\mathbb{E}\left(Tr(PE_{H_{\omega}^{\pm}}(A)P)\right).$ (18)
Consider the semi algebra $\mathcal{I}\times\mathcal{I}$ of subsets of
$\mathbb{R}^{2}$ where
$\mathcal{I}=\mathbb{R}\cup\\{(a,b]:a,b\in\mathbb{R}\\}\cup\\{(a,\infty):a\in\mathbb{R}\\}\cup\\{(-\infty,a]:a\in\mathbb{R}\\}.$
We define the correlation measure $\rho$ on $\mathcal{I}\times\mathcal{I}$ as
$\rho(A\times
B)=\mathbb{E}\left(Tr(PE_{H_{\omega}^{+}}(A)E_{H_{\omega}^{-}}(B)P)\right),$
(19)
where $\rho$ is set to be $\infty$ if either $A$ or $B$ is an unbounded
element of $\mathcal{I}$.
This set function takes values in $[0,1]$ if $P$ is trace class and in
$[0,\infty]$, if $PE_{H_{\omega}^{\pm}((a,b])}$ are trace class only for
bounded intervals $(a,b]$, in view of Proposition 3. We set
$\rho(A)=\sum_{i=1}^{\infty}\rho(A_{i}\times B_{i}),\leavevmode\nobreak\
\mathrm{if}\leavevmode\nobreak\
A=\sqcup_{i=1}^{\infty}A_{i},A_{i}\in\mathcal{I}.$
It is a simple exercise to see that this $\rho$ is well defined on
$\mathcal{I}\times\mathcal{I}$ and via standard measure theory extends as a
$\sigma$-finite measure to the whole borel $\sigma$-algebra of
$\mathbb{R}^{2}$..
Using the Hypothesis 2, and Proposition 1 we see that the following is valid.
###### Proposition 3.
Consider the operators $H_{\omega}^{\pm}$ satisfying Hypothesis 2 and let
$n_{\pm}$ and $\rho$ be as in equation (19). Then for any $B,C\in\mathcal{I}$
bounded,
1. 1.
$\rho(B\times
C)=\mathbb{E}\left(Tr(PE_{H_{\omega}^{-}}(C)E_{H_{\omega}^{+}}(B)E_{H_{\omega}^{-}}(C)P)\right)$
2. 2.
$\rho(B\times
C)=\mathbb{E}\left(Tr(PE_{H_{\omega}^{+}}(B)E_{H_{\omega}^{-}}(C)E_{H_{\omega}^{+}}(B)P)\right)$
3. 3.
The following inequalities are valid
$\rho(B\times C)\leq n_{+}(B),\leavevmode\nobreak\ \rho(B\times C)\leq
n_{-}(C).$
Proof: Since the subsets $B,C$ are bounded the operators
$PE_{H_{\omega}^{-}}(C),PE_{H_{\omega}^{+}}(B)$ are covariant trace class
operators satisfying the inequality (1). Therefore the result follows by an
application of Proposition 1 and Corollary 2. ∎
We collect the arguments about $\rho$ in a proposition.
###### Proposition 4.
Consider a pair of covariant operators $H_{\omega}^{\pm}$ satisfying the
Hypothesis 2 and consider the correlation measures $\rho$ extended to the
borel $\sigma$-algebra on $\mathbb{R}^{2}$ from that given by equation (19).
Then the following are valid.
1. 1.
If $P$ is trace class, then $\rho$ is a probability measure on
$\mathbb{R}^{2}$, with support in the closure of
$\cup_{\omega}\sigma(H_{\omega}^{+})\times\sigma(H_{\omega}^{-})$.
2. 2.
If $P$ is not trace class but, $PE_{H_{\omega}^{\pm}}((a,b])P$ is trace class,
for bounded intervals $(a,b]$, then $\rho$ is a positive $\sigma$-finite
measure on $\mathbb{R}^{2}$, with support in the closure of
$\cup_{\omega}\sigma(H_{\omega}^{+})\times\sigma(H_{\omega}^{-})$.
###### Remark 5.
Typically the first case occurs for operators on $\ell^{2}(\mathbb{Z}^{d})$
and the second case occurs in $L^{2}(\mathbb{R}^{d})$.
We take the transformation $T$ on $\mathbb{R}^{2}$ given by
$T\left(\begin{matrix}\lambda_{1}\\\
\lambda_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{\lambda_{1}+\lambda_{2}}{\sqrt{2}}\\\
\frac{\lambda_{1}-\lambda_{2}}{\sqrt{2}}\end{matrix}\right).$
Using this $T$ we define the Interband Light Absorption Coefficient (ILAC) $A$
as the distribution function,
$A(\lambda)-A(\lambda^{\prime})=\nu\left(\frac{1}{\sqrt{2}}(\lambda^{\prime},\lambda]\right),\leavevmode\nobreak\
\mathrm{where}\leavevmode\nobreak\ \nu(B)=\rho\circ T^{-1}(B\times\mathbb{R})$
(20)
In the above equation the factor $\frac{1}{\sqrt{2}}$ is because of the
normalisation we used for $T$, so that this definition of ILAC agrees with the
standard one in the case of finite box operators. We also note that since the
operators $H_{\omega}^{\pm}$ are assumed to be bounded below $A(-\infty)=0$.
In the case when $\mathbb{P}$ in Hypothesis 1 is ergodic with respect to the
action of $G$ on $\Omega$, then, the spectra $\sigma(H_{\omega}^{\pm})$ of
covariant families of operators $H_{\omega}^{\pm}$ are almost everywhere
constant sets. In such a case we can talk about the infimum of spectra of
$H_{\omega}^{\pm}$ without reference to $\omega$. In this context we have the
following theorem.
###### Theorem 2.1.
Suppose $H_{\omega}^{\pm}$ are a pair of random families of self-adjoint
operators satisfying Hypothesis 1. Assume further that $\mathbb{P}$ is ergodic
with respect to the action of $G$ on $\Omega$.
1. 1.
Let $E_{\pm}=\inf\sigma(H_{\omega}^{\pm})$. Then
$A(E_{+}+E_{-}+a)-A(E_{+}+E_{-}-a)\leq
n_{\pm}((E_{\pm}-2a,E_{\pm}+2a)),\leavevmode\nobreak\ a>0$.
2. 2.
Let $E_{\pm}^{\prime}=\sup\sigma(H_{\omega}^{\pm})$. Then
$A(E_{+}^{\prime}+E_{-}^{\prime}+a)-A(E_{+}^{\prime}+E_{-}^{\prime}-a)\leq
n_{\pm}((E_{\pm}^{\prime}-2a,E_{\pm}^{\prime}+2a)),\leavevmode\nobreak\ a>0$.
Proof: We shall prove the first case, the other proof is similar (where one
has to use the fact that $\lambda_{1}\leq E_{+}^{\prime},\lambda_{2}\leq
E_{-}^{\prime}$ respectively for the other case and work it out). Let
$E_{+},E_{-}$ to be the infima of the spectra
$\sigma(H_{\omega}^{+}),\sigma(H_{\omega}^{-})$ of
$H_{\omega}^{+},H_{\omega}^{-}$. We consider the closure of the Cartesian
product $\Sigma=\sigma(H_{\omega}^{+})\times\sigma(H_{\omega}^{-})$ of the
spectra of $H_{\omega}^{\pm}$, which is the support of the measure $\rho$.
Therefore if we denotes points of $\Sigma$ by $(\lambda_{1},\lambda_{2})$, so
that $\lambda_{1}\geq E_{+},\lambda_{2}\geq E_{-}$, then the possible values
of $\lambda_{1}+\lambda_{2}$ have a lower bound $E_{-}+E_{+}$, so
$\lambda_{1}+\lambda_{2}\in(E_{-}+E_{+},E_{-}+E_{+}+a)$ implies
$\lambda_{1}\in(E_{+}-2a,E_{+}+2a)\leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ \lambda_{2}\in(E_{-}-2a,E_{-}+2a)$, (see
Figure 2). This immediately implies the inclusions (the first inclusion is
clear and the second one uses the above):
$\displaystyle\\{(\lambda_{1},\lambda_{2}):\lambda_{2}\in(E_{-},E_{-}+(a/2))\leavevmode\nobreak\
and\leavevmode\nobreak\ \lambda_{1}\in(E_{+},E_{+}+(a/2))\\}$
$\displaystyle\subset\\{(\lambda_{1},\lambda_{2}):\lambda_{1}+\lambda_{2}\in(E_{-}+E_{+}-a,E_{-}+E_{+}+a)\\}$
$\displaystyle\subset\\{(\lambda_{1},\lambda_{2}):\lambda_{2}\in(E_{-},E_{-}+2a)\leavevmode\nobreak\
and\leavevmode\nobreak\ \lambda_{1}\in(E_{+},E_{+}+2a)\\}.$
This then would lead to the inequalities that
$\displaystyle A(E_{+}+E_{-}+a)-A(E_{+}+E_{-})$ $\displaystyle=\rho\circ
T^{-1}(\frac{1}{\sqrt{2}}(E_{+}+E_{-},E_{+}+E_{-}+a]\times\mathbb{R})$
$\displaystyle=\rho\left(\\{(\lambda_{1},\lambda_{2}):E_{-}+E_{+}\leq\lambda_{1}+\lambda_{2}\leq
E_{-}+E_{+}+a\\}\right)$
$\displaystyle\leq\rho\left((E_{-},E_{-}+2a)\times(E_{+},E_{+}+2a)\right)$
$\displaystyle\leq
min\\{\rho\left((E_{-},E_{-}+2a)\times\mathbb{R}\right),\rho\left((E_{+},E_{+}+2a)\times\mathbb{R}\right)\\}$
$\displaystyle\leq
min\\{n_{-}\left((E_{-}-2a,E_{-}+2a)\right),n_{+}\left((E_{+}-2a,E_{+}+2a)\right)\\},$
where the last inequality comes from Proposition 3(3) and enlarging the
intervals slightly, which only increases the bound since $n_{\pm}$ are
measures. ∎
###### Remark 6.
If the density of states $n_{\pm}$ have Lifshitz tails behaviour
$n_{\pm}((E_{\pm}-a,E_{\pm}+a))\approx e^{-Ca^{\alpha}}$ as $a$ goes to zero,
for an appropriate $\alpha$, at $E_{\pm}$ respectively, then we have
$\limsup_{a>0}\frac{1}{h(2a)}n_{\pm}\left((E_{-}-2a,E_{-}+2a)\right)<\infty,$
for $h(a)=e^{-Ca^{\alpha}}$ for some $\alpha$, so, using the above
inequalities,
$\displaystyle\displaystyle{\limsup_{a>0}}\frac{1}{h(2a)}(A(E_{-}+E_{+}+a)-A(E_{-}+E_{+}-a))$
$\displaystyle\leq\displaystyle{\limsup_{a>0}}\frac{1}{h(2a)}n_{+}\left((E_{+}-a,E_{+}+2a)\right)<\infty.$
In the case when the density of states $n_{\pm}$ have Lifshitz tails behaviour
at other internal band edges, the same behaviour is valid for ILAC under some
conditions. Suppose the spectra of $H_{\omega}^{\pm}$ consist of bands
$\cup_{i=1}^{N}[a_{i}^{\pm},b_{i}^{\pm}]$. Then the product of the spectra is
$\cup_{i=1,j}^{N}[a_{i}^{+},b_{i}^{+}]\times[a_{j}^{-},b_{j}^{-}]$. Let us
denote $R_{ij}=[a_{i}^{+},b_{i}^{+}]\times[a_{j}^{-},b_{j}^{-}]$. Then, the
measure $\rho$ is supported on the set $\cup_{i=1,j}^{N}R_{ij}$.
We index the pairs $(ij)$ by $\beta$ and use $R_{\beta}$ to denote a rectangle
forming part of $\Sigma$ henceforth. So we have
$\Sigma=\cup_{\beta}R_{\beta}$.
The central point in the proof of Theorem 2.1 is that if $(c,d)$ is a corner
of the rectangle formed by the lowest bands of the spectra of
$H_{\omega}^{\pm}$, then the strip
$\\{(\lambda_{1},\lambda_{2}):c+d\leq\lambda_{1}+\lambda_{2}\leq c+d+a\\}$
intersected with the support of $\rho$ is a triangle of side length
$\sqrt{2}a$, (see Figure 2 ), hence its $\rho$ measure is smaller than that of
the square with the corner $(c,d)$ and side length $2a$, as can be seen in the
Figure 2. As we see in Figure 1, there may be some rectangles in the support
of $\rho$, with this property. Those rectangles in Figure 1, where this is not
true are marked by $X$ and the solid lines are those lines
$\lambda_{1}+\lambda_{2}=const$ for which this feature is valid and the dashed
lines are those for which this is not true.
In the definition below the sets $R_{\beta}\subset\mathbb{R}^{2}$ and we
denote the coordinates of $\mathbb{R}^{2}$ by $(\lambda_{1},\lambda_{2})$.
###### Definition 4.
Let the support of $\rho$ be $\Sigma=\cup_{\beta}R_{\beta}$, with
$R_{\beta}=[a_{i}^{+},b_{i}^{+}]\times[a_{j}^{-},b_{j}^{-}],\leavevmode\nobreak\
\beta=(ij)$. Then we call a corner $(c,d)$ of a rectangle $R_{\beta}$ good, if
the intersection of the line $\lambda_{1}+\lambda_{2}=c+d$ with $\Sigma$
consists of finitely many points and all of them are corners of rectangles
forming $\Sigma$. Given a corner $(c,d)$ in $\Sigma$ we shall denote by
$K_{c,d}$ the set of corners that lie on the line
$\lambda_{1}+\lambda_{2}=c+d$.
###### Theorem 2.2.
Let spectra of $H_{\omega}^{\pm}$ be as in theorem 2.1 and let $\Sigma$ be the
support of the measure $\rho$ given in equation 19. Let $A$, as given in
equation (20) be the corresponding ILAC. If $(c,d)$ is a good corner in
$\Sigma$. Denote the elements of $K_{c,d}$ by $\\{(c_{\gamma},d_{\gamma})\\}$.
Then we have
$\displaystyle A(c+d+a)-A(c+d-a)$
$\displaystyle\leq\sum_{(c_{\gamma},d_{\gamma})\in
K_{c,d}}\mathrm{min}\left\\{n_{+}((c_{\gamma}-2a,c_{\gamma}+2a)),n_{-}((d_{\gamma}-2a,d_{\gamma}+2a))\right\\}.$
Proof: Firstly we note that if we take a rectangle, $R_{\beta}$, then only
the lower-left and the top-right corners are candidates of being _good_
corners, since for the other two corners, the line
$\lambda_{1}+\lambda_{2}=const$ that contains the said corner will pass
through the rectangle and hence has infinitely many points. We will prove the
theorem for a good corner $(c,d)$ which is a lower left corner of a rectangle,
the proof for the case of a top-right good corner is similar. In this case we
see immediately that if $(c,d)$ is a good corner in $\Sigma$, then the
intersection of the strip
$S_{a}\left((c,d)\right)=\\{(\lambda_{1},\lambda_{2}):c+d\leq\lambda_{1}+\lambda_{2}\leq
c+d+a\\}$ with $\Sigma$ is contained in finitely many rectangles $R_{\beta}$
forming $\Sigma$. Further $S_{a}\left((c,d)\right)\cap R_{\beta}$ is contained
in a square of side length $2a$ contained in $R_{\beta}$ and having one corner
common with a corner of $R_{\beta}$. Given a good corner $(c,d)$ and the
associated strip $S_{a}\left((c,d)\right)$, let $(c_{\gamma},d_{\gamma})\in
K_{c,d}$ denote the corner of rectangle $R_{\gamma}$ that has nonempty
intersection with it. (Note that this corner satisfies
$c_{\gamma}+d_{\gamma}=c+d$).
Then whenever $(c,d)$ is a good corner we have the inequality, with $\gamma$
ranging over a finite set,
$\displaystyle
S_{a}\left((c,d)\right)\cap\Sigma\subset\cup_{(c_{\gamma},d_{\gamma})\in
K_{c,d}}[c_{\gamma},c_{\gamma}+2a]\times[d_{\gamma},d_{\gamma}+2a].$ (21)
This inequality implies immediately that:
$\displaystyle A(c+d+a)-A(c+d-a)$ $\displaystyle\leq
A(c+d+a)-A(c+d)=\rho(S_{a}\left((c,d)\right)\cap\Sigma)$
$\displaystyle\leq\sum_{(c_{\gamma},d_{\gamma})\in
K_{c,d}}\rho\left([c_{\gamma},c_{\gamma}+2a)\times[d_{\gamma},d_{\gamma}+2a)\right)$
$\displaystyle\leq\sum_{(c_{\gamma},d_{\gamma})\in
K_{c,d}}\mathrm{min}\left\\{n_{+}\left((c_{\gamma}-2a,c_{\gamma}+2a)\right),n_{-}\left((d_{\gamma}-2a,d_{\gamma}+2a)\right)\right\\},$
(22)
where in the last inequality we enlarged the sets using the fact that
$n_{\pm}$ are measures.
This shows that at ILAC has the same continuity property as the density of
states at the band edges. ∎
In the theorem below we identify good corners for a simple case of spectra
having two bands.
###### Theorem 2.3.
Consider a pair of self adjoint operators $H_{\omega}^{\pm}$ as in Theorem
2.1. Suppose a.e. $\omega$, the spectra of $H_{\omega}^{+},H_{\omega}^{-}$ are
given by $\cup_{i=1}^{2}[a_{i}^{+},b_{i}^{+}]$ and
$\cup_{i=1}^{2}[a_{i}^{-},b_{i}^{-}]$, respectively, where
$a_{i}^{\pm},b_{j}^{\pm}$ are listed in the increasing order. Then the corners
$\\{(a_{1}^{+},a_{1}^{-}),(b_{1}^{+},b_{1}^{-}),(a_{2}^{+},a_{2}^{-}),(b_{2}^{+},b_{2}^{-})\\}$
are _good_ whenever $a_{i}^{\pm},b_{i}^{\pm}$ satisfy,
$\displaystyle
a_{1}^{+}+a_{1}^{-}<b_{1}^{+}+b_{1}^{-}<max(a_{2}^{+}+a_{1}^{-},a_{1}^{+}+a_{2}^{-})$
$\displaystyle<max(b_{2}^{+}+b_{1}^{-},b_{1}^{+}+b_{2}^{-})<a_{2}^{+}+a_{2}^{-}<b_{2}^{+}+b_{2}^{-}.$
In the case $a_{i}^{+}=a_{i}^{-},b_{i}^{+}=b_{i}^{-},i=1,2$, even the corners
$\\{(a_{2}^{+},a_{1}^{-}),(a_{1}^{+},a_{2}^{-}),(b_{2}^{+},b_{1}^{-}),(b_{1}^{+},b_{2}^{-})\\}$
are good.
Proof: This is direct verification to see that the diagonal lines
$\lambda_{1}+\lambda_{2}=const$ passing through the respective corners do not
intersect any other rectangle. In the latter case when the spectra are the
same, we have $a_{2}^{+}+a_{1}^{-}=a_{1}^{+}+a_{2}^{-}$ and
$b_{2}^{+}+b_{1}^{-}=b_{1}^{+}+b_{2}^{-}$, hence the stated result. ∎
###### Remark 7.
In the symmetric case $a_{i}^{\pm}=a_{i},b_{j}^{\pm}=b_{j}$, however, the
rectangles $R_{\beta},R_{\gamma}\subset S_{\beta}$ if
$\beta=(ij),\gamma=(ji)$. In this case the above assumption still ensures that
the lower-left and top-right corners of the rectangles are good.
$a_{1}$$b_{1}$$a_{2}$$b_{2}$$a_{3}$$b_{3}$$a_{4}$$b_{4}$$a_{1}$$b_{1}$$a_{2}$$b_{2}$$a_{3}$$b_{3}$$a_{4}$$b_{4}$XXXXXB
Figure 1: Products of Spectra
$\lambda_{2}$$\lambda_{1}+\lambda_{2}=c$$\lambda_{1}+\lambda_{2}=c+a$$a$$\sqrt{2}a$$\sqrt{2}a$$\lambda_{1}$
Figure 2: A corner of a rectangle
## 3 Discrete Models:
Consider $\ell^{2}(\mathbb{Z}^{d})$ and the discrete Laplacian $(\Delta
u)(n)=\sum_{|n-i|=1}u(i)$. Consider real valued i.i.d random variables
$\\{q(n)\\}$ with common distribution $\mu$. Let $V_{\omega}$ denote the
operator of multiplication by the sequence $q_{\omega}(n)$. Consider the
operators
$H_{\omega}^{\pm}=\Delta\pm q_{\omega}.$
Taking $G=L=\mathbb{Z}^{d}$, it is known that operators $E_{H_{\omega}}(A)$
are covariant. The projection $P$ is taken to be the projection
$|\delta_{0}\rangle\langle\delta_{0}|$ onto the subspace generated by the
vector $\delta_{0}$, which is an element of the standard basis for
$\ell^{2}(\mathbb{Z}^{d})$.
Then the density of states in these models are given by
$n_{\pm}((a,b))=\mathbb{E}\left(Tr(PE_{H_{\omega}^{\pm}}((a,b))\right)=\mathbb{E}\left(\langle\delta_{0},E_{H_{\omega}^{\pm}}((a,b))\delta_{0}\rangle\right)$
and the correlation measure $\rho$ is given by
$\rho((a,b)\times(c,d))=\mathbb{E}\left(\langle\delta_{0},E_{H_{\omega}^{+}}((a,b))E_{H_{\omega}^{-}}((c,d))\delta_{0}\rangle\right)$
and is a probability measure as per Proposition 4 (1), since $P$ is trace
class in this case.
In this model the density of states of $H_{\omega}^{\pm}$ are shown to have
Lifshitz tails behaviour at the bottom of the spectra [21], under the
condition that $\mu$ satisfies $\mu((a,a+\epsilon))\geq C\epsilon^{N}$, where
$a$ is the infimum of the support of $\mu$.
In the case when the support of $\mu$ has two closed intervals
$[a_{1},b_{1}]\cup[a_{2},b_{2}]$,($a_{i},b_{i}$ arranged in an increasing
order so that $a_{i+1}>a_{i}$ for all $i$) and such that
$b_{i}+2d<a_{i+1}-2d$, Simon [22] proves the Lifshitz tails behaviour at the
internal band edges, if $\mu$ satisfies $\mu((a_{i},a_{i}+\epsilon))\geq
C\epsilon^{N}$ and $\mu((b_{i}-\epsilon,b_{i}))\geq C\epsilon^{N}$ for all $i$
. When $[a_{1}-2d,b_{1}+2d]$ and $[a_{2}-2d,b_{2}+2d]$ are disjoint, Lifshitz
tails behaviour at the band edges is also shown for the associated density of
states $n$. That is at any of the band edges one has
$n((E-\delta,E+\delta))=O(e^{-C\delta^{-\frac{d}{2}}})$ as $\delta\rightarrow
0$.
An application of Theorems 2.1, 2.2 shows that the results are true for the
ILAC $A$, namely
###### Theorem 3.1.
Consider the Anderson models as above on $\ell^{2}(\mathbb{Z}^{d})$. If
$[a_{1}^{\pm},b_{1}^{\pm}]\cup[a_{2}^{\pm},b_{2}^{\pm}]$, are
$\pm(supp(\mu))$. Then, for some $C>0$,
* •
(External band edge case) For
$E\in\\{a_{1}^{+}+a_{1}^{-},b_{2}^{+}+b_{2}^{-}\\}$, one has
$A(E+\delta)-A(E-\delta)=o(e^{-C\delta^{-\frac{d}{2}}}),\leavevmode\nobreak\
\mathrm{as}\leavevmode\nobreak\ \delta\rightarrow 0.$
* •
(Internal band edge case) If the gap between the intervals
$[a_{1}^{\pm}-2d,b_{1}^{\pm}+2d]$ and $[a_{2}^{\pm}-2d,b_{2}^{\pm}+2d]$ is
large enough, then
$A(E+\delta)-A(E-\delta)=o(e^{-C\delta^{-\frac{d}{2}}}),\leavevmode\nobreak\
\mathrm{as}\leavevmode\nobreak\ \delta\rightarrow 0,$
for any
$E\in\\{b_{1}^{+}+a_{1}^{-},a_{2}^{+}+a_{1}^{-},b_{2}^{+}+a_{1}^{-},a_{1}^{+}+a_{2}^{-},b_{1}^{+}+b_{2}^{-},a_{2}^{+}+a_{2}^{-}\\}$.
###### Remark 8.
We gave a simple example of a discrete model, however there are many more,
those with periodic backgrounds [11], those which are unbounded [16] and so
on. We refer to the review [7] for the numerous cases where the Lifshitz tails
for the density of states is proved and for which our theorem applies to the
ILAC.
## 4 Continuous Models:
Let us start by stating a theorem which is essentially a very weak version of
the uncertainty principle.
We take
$H_{0}=-\Delta=-\sum_{i=1}^{d}D_{j}^{2},D_{j}=i\frac{\partial}{\partial
x_{j}}$, is self adjoint on its natural domain in $L^{2}(\mathbb{R}^{d})$ and
its spectrum is $[0,\infty)$.
We start with a couple of lemmas. We recall the definition of the trace ideal
${\mathscr{I}}_{p}$ to be those bounded operators $A$ with the property
$|A|^{p}$ is trace class. Recall that elements of $\mathscr{I}_{1}$ are called
Trace class operators.
###### Lemma 9.
Consider $L^{2}(\mathbb{R}^{d})$ and the operator $M=|-i\nabla|$. Then the
operator $(|x|+i)^{-1}(M+i)^{-1}\in\mathscr{I}_{d+1}$.
Proof: Since the function $f(x)=(|x|+i)^{-1}$ is $L^{d+1}(\mathbb{R}^{d})$
and the operator in question is just $f(x)f(-i\nabla)$, the result follows by
an application of Theorem 4.1 in [23], which gives an estimate
$\|f(x)g(-i\nabla)\|_{\mathscr{I}_{p}}\leq
2\pi^{-\frac{d}{p}}\|f\|_{p}\|g\|_{p}.$
∎
Let $V$ be an operator of multiplication by a function $V(x)$ on
$L^{2}(\mathbb{R}^{d})$ on its natural domain and such that $V$ is bounded
with respect to $H_{0}$ having relative bound smaller than $1$. This means the
operator $V(H+i)^{-1}$ is bounded. Then $H=H_{0}+V$ is also self adjoint
(Kato-Rellich theorem ) on the domain of $H_{0}$ and its spectrum is also
bounded below. Writing $(H_{0}+i)(H+i)^{-1}=I-V(H+i)^{-1}$, we see that
$(H_{0}+i)(H+i)^{-1}$ is also bounded. Let $P$ denote the operator of
multiplication by the indicator function $\chi_{\Lambda}$ of a bounded region
$\Lambda\subset\mathbb{R}^{d}$ on $L^{2}(\mathbb{R}^{d})$. Let $E_{H}(A)$
denote the spectral measure of a bounded borel set $A$, with respect to the
(projection valued) spectral measure of $H$. Then,
###### Theorem 4.1.
Consider $L^{2}(\mathbb{R}^{d})$ and the operator $H_{0}=-\Delta$. Let $V$ be
an operator of multiplication by a function $V(x)$, such that $V$ is
relatively bounded w.r.t. $H_{0}$ with relative bound $c<1$ and consider
$H=H_{0}+V$. Suppose either
1. 1.
$d\leq 3$, then $PE_{H}(A)$ and $E_{H}P$ are Hilbert-Schmidt, so $PE_{H}(A)P$
is trace class for any bounded borel set $A$.
2. 2.
Suppose $d\geq 1$ and suppose $V$ is bounded or $\frac{\partial}{\partial
x_{j}}V,j=1,\dots,d$ are relatively bounded w.r.t $H_{0}$. Then $PE_{H}(A)$
and $E_{H}(A)P$ are trace class.
Proof: (1) Writing
$PE_{H}(A)=P(|x|+i)^{d}(|x|+i)^{-d}(H_{0}+i)^{-1}(H_{0}+i)(H+i)^{-1}(H+i)E_{h}(A)$,
we see that since all the factors are bounded, it is enough to show that
$(|x|+i)^{-d}(H_{0}+i)^{-1}$ is Hilbert-Schmidt. The operator $(H_{0}+i)^{-1}$
is multiplication by $(|\xi|^{2}+i)^{-1}$ after taking Fourier transforms and
hence is in $L^{2}(\mathbb{R}^{d}),d\leq 3$. Therefore an application of Lemma
9, shows that the product is Hilbert-Schmidt.
(2) We will prove that $PE_{H}(A)\in\mathscr{I}_{1}$, the proof for
$E_{H}(A)P$ is similar. By taking a compactly supported smooth function $\phi$
which is value $1$ on the closure of $A$, we have $\phi(H)E_{H}(A)=E_{H}(A)$.
We will therefore show that $P\phi(H)$ is trace class for any compactly
supported smooth function $\phi$. We also note that the function $H\phi(H)$ is
again a function of the same type as $\phi$.
Further since $P$ is multiplication by compactly supported function of $x$,
$P(x^{2}+i)^{d}$ is bounded. Therefore we will show that
$(x^{2}+i)^{-d}\phi(H)\in{\mathscr{I}}_{1}$.
We prove this by induction. Before we start, we note that if
$M\in\mathscr{I}_{p}$ and $N$ is a bounded operator then
$MN\in\mathscr{I}_{p}$.
First consider $(x^{2}+i)^{-1}\phi(H)$. We write this product as
$(x^{2}+i)^{-1}(H+i)^{-1}(H+i)\phi(H)$ and consider (recalling
$M=|-i\nabla|$),
$(x^{2}+i)^{-1}(H+i)^{-1}=(x^{2}+i)^{-1}(M+i)^{-1}(M+i)(H_{0}+i)^{-1}(H_{0}+i)(H+i)^{-1}.$
(23)
The product of the first two factors is in $\mathscr{I}_{d+1}$ (since
$(|\xi|+i){-d-1}$ is integrable), by Lemma 9, the next two factors form a
bounded operator (which can be seen by taking Fourier transforms). The final
two factors form a bounded operator as argued before the lemma. Therefore the
entire product is in $\mathscr{I}_{d+1}$. Since $(H+i)\phi(H)$ is bounded
also, we get that $(x^{2}+i)^{-1}\phi(H)\in\mathscr{I}_{d+1}$.
Now assume that $(x^{2}+i)^{-n}\phi(H)\in\mathscr{I}_{\frac{d+1}{n}}$, and
show that $(x^{2}+i)^{-n-1}\phi(H)\in\mathscr{I}_{\frac{d+1}{n+1}}$. We write,
$\psi(H)=(H+i)\phi(H)$, then
$\displaystyle(x^{2}+i)^{-d-1}\phi(H)$ (24)
$\displaystyle=(x^{2}+i)^{-d-1}(H+i)^{-1}(H+i)\phi(H)$
$\displaystyle(x^{2}+i)^{-1}[(x^{2}+i)^{-n},(H+i)^{-1}]\psi(H)+(x^{2}+i)^{-1}(H+i)^{-1}(x^{2}+i)^{-n}\psi(H).$
Using Theorem 2.8 (2.5b) in [23], (which says
$M\in\mathscr{I}_{q},N\in\mathscr{I}_{r}\implies MN\in\mathscr{I}_{p}$ with
$\frac{1}{p}=\frac{1}{q}+\frac{1}{r}$), so using the induction hypothesis and
the already proved fact that $(x^{2}+i)^{-1}(H+i)^{-1}\in\mathscr{I}_{d+1}$,
the last term is seen to be in $\mathscr{I}_{\frac{d+1}{n+1}}$.
So we concentrate on the first term.
$\displaystyle(x^{2}+i)^{-1}[(x^{2}+i)^{-n},(H+i)^{-1}]\psi(H)$ (25)
$\displaystyle=(x^{2}+i)^{-1}(H+i)^{-1}[H,(x^{2}+i)^{-n}](H+i)^{-1}\psi(H)$
$\displaystyle=(x^{2}+i)^{-1}(H+i)^{-1}[H_{0},(x^{2}+i)^{-n}](H+i)^{-1}\psi(H)$
$\displaystyle=(x^{2}+i)^{-1}(H+i)^{-1}\times$
$\displaystyle\left(-4ni\sum_{j=1}^{d}P_{j}x_{j}(x^{2}+i)^{-1}-2d(x^{2}+i)^{-1}+4d(n+1)x^{2}(x^{2}+i)^{-2}\right)$
$\displaystyle\times(x^{2}+i)^{-n}\psi_{1}(H)$
where we set $(H+i)^{-1}\psi(H)=\psi_{1}(H)$, where $P_{j}=-i\nabla_{j}$.
$\displaystyle(x^{2}+i)^{-1}(H+i)^{-1}P_{j}$
$\displaystyle=(x^{2}+i)^{-1}(H_{0}+1)^{-1/2}(H_{0}+1)^{1\over
2}(H+i)^{-1}(H_{0}+1)^{{1\over 2}}(H_{0}+1)^{-{1\over 2}}P_{j},$
and using Lemma 9, Lemma 10 below, we see that this expression is in
$\mathscr{I}_{d+1}$. Induction hypothesis gives
$(x^{2}+i)^{-n}\psi_{1}(H)\in\mathscr{I}_{\frac{d+1}{n}}$. Therefore combining
these two facts we see that
$(x^{2}+i)^{-n-1}\phi(H)\in\mathscr{I}_{\frac{d+1}{n+1}}$. ∎
###### Lemma 10.
Suppose either $V$ is bounded or $(\frac{\partial}{\partial
x_{j}}V)(H+i)^{-1},j=1,\dots,d$ are bounded. Then $(H_{0}+1)^{1\over
2}(H+i)^{-1}P_{j}$ is a bounded operator for each $j=1,\dots d$.
Proof: Consider the case when $\frac{\partial}{\partial x_{j}}V(H+i)^{-1}$ is
bounded for each $j$. Then writing the expression using commutators
$\displaystyle(H_{0}+1)^{{1\over 2}}(H+i)^{-1}P_{j}=(H_{0}+1)^{1\over
2}P_{j}(H+i)^{-1}$ $\displaystyle+(H_{0}+1)^{1\over
2}(H+i)^{-1}[P_{j},H](H+i)^{-1}$ $\displaystyle=(H_{0}+1)^{1\over
2}P_{j}(H+i)^{-1}+(H_{0}+1)^{1\over 2}(H+i)^{-1}(\frac{\partial}{\partial
x_{j}}V)(H+i)^{-1}.$
The boundedness of the first term was seen before since $(H_{0}+1)^{1\over
2}P_{j}(H_{0}+1)^{-1}$ and $(H_{0}+1)(H+1)^{-1}$ are bounded. second term is
bounded by the assumption on $V$ and the boundedness of $(H_{0}+1)^{-{1\over
2}}$.
Now consider the case when $V$ is bounded, then taking $f,g$ in the domain of
$H_{0}$, we have
$\langle f,(H_{0}+1)f\rangle=\langle g,(H_{0}+V+c+1)f\rangle-\langle
g,(V+c)f\rangle,$
where $c$ is a positive constant such that $H+c+1$ is a positive operator
(which is possible since $H$ is bounded below). Since $H+c+1$ is positive it
has a unique square root, so using the boundedness of $V$ and the above
inequality, we obtain, for some finite $C$,
$\|(H_{0}+1)^{1\over 2}f\|^{2}\leq\|(H+c)^{{1\over 2}}f\|^{2}+C\|f\|\leq
D\|(H+c)^{{1\over 2}}f\|^{2}.$
Taking $f=(H_{0}+1)^{1\over 2}g,\|g\|=1$, for a set of $g$ coming from
$C_{0}^{\infty}(\mathbb{R}^{d})$, we see that
$K\leq\|(H+c)^{1\over 2}(H_{0}+1)^{-{1\over 2}}g\|^{2},\leavevmode\nobreak\
K>0,$
$K$ independent of $g$. This shows that $(H+c)^{1\over 2}(H_{0}+1)^{-{1\over
2}}$ has a bounded inverse and that its inverse $(H_{0}+1)^{-{1\over
2}}(H+c)^{1\over 2}$ and $(H+c)^{1\over 2}(H_{0}+1)^{-{1\over 2}}$ are both
bounded (since $M$ bounded implies $M^{*}$ is also bounded). Therefore writing
$\displaystyle(H_{0}+1)^{1\over 2}(H+i)^{-1}P_{j}$
$\displaystyle=(H_{0}+1)^{1\over 2}(H+c)^{-{1\over
2}}(H+c)(H+i)^{-1}(H+c)^{-{1\over 2}}(H_{0}+1)^{1\over 2}(H_{0}+1)^{-{1\over
2}}P_{j},$
we see that the left hand side is bounded. ∎
We are now ready to present examples where the theorems of the previous
section are applicable. We fist give a few examples of models on the lattice.
Consider $L^{2}(\mathbb{R}^{d})$, $H_{0}=\Delta$, $q(n),n\in\mathbb{Z}^{d}$,
i.i.d random variables with distribution $\mu$ having compact support. Let
$\Lambda$ denote the unit cube centred at $0\in\mathbb{R}^{d}$ and
$\Lambda(n)$ denote the unit cube centred at the point $n\in\mathbb{Z}^{d}$.
Let $V_{\omega}=\sum_{n\in\mathbb{Z}^{d}}q^{\omega}(n)\chi_{\Lambda(n)}$,
where $\chi_{A}$ is the operator of multiplication by the indicator function
of $A$. Then taking
$H_{\omega}^{\pm}=\Delta\pm V_{\omega},$
we see that, since $V_{\omega}$ is bounded for each $\omega$, the conditions
of Theorem 4.1 are satisfied. Further taking $G=\mathbb{R}^{d}$ and
$L=\mathbb{Z}^{d}$, $(U_{x}f)(y)=f(y-x),on\leavevmode\nobreak\
L^{2}(\mathbb{R}^{d})$, $q_{{}_{T^{m}\omega}}(n)=q_{\omega}(n+m)$, in
Hypothesis 1, the spectral projections $E_{H_{\omega}^{\pm}}((a,b))$ are
covariant families in the sense of Definition 1. The Theorem 4.1 shows that
$\chi_{\Lambda(0)}E_{H_{\omega}^{\pm}}((a,b))\chi_{\Lambda(0)}$ is trace class
whenever $(a,b)$ is a bounded interval. Hence we can define the density of
states and the ILAC as in equations (18) and (19), by taking $P$ to be
multiplication by $\chi_{{}_{\Lambda(0)}}$. Therefore the Theorems 2.1, 2.2
are valid in this case.
Our theorem covers models where the random potential has the following forms.
* •
$V^{\omega}(x)=\sum_{n\in\mathbb{Z}^{d}}q^{\omega}(n)u(x-n)$, $\\{q(n)\\}$
i.i.d.random variables whose distribution has compact support and $u$ a nice
function with $u(x-n)$ summable.
* •
An addition of a periodic background potential $W$ to the random potential
above.
* •
Addition of magnetic fields.
If in all these cases the density of states have Lifshitz tails behaviour at
the band edges the same is acquired by the ILAC at an appropriate energy
level.
###### Remark 11.
Let us remark that in the above examples we can even replace the Laplacian
$-\Delta$ with a real polynomial function $Q$ of $-i\nabla$ and the results go
through, if for some $R>0$, the polynomial satisfies
$c_{1}\|\xi\|^{2n}\leq Q(\xi)\leq c_{2}\|\xi\|^{2n},\leavevmode\nobreak\
|\xi|>R,c_{1},c_{2}>0.$
## References
* [1] M.S. Atoyan, E.M. Kazaryan, H.A. Sarkisyan: Interband light absorption in parabolic quantum dot in the presence of electrical and magnetic fields, Physica E: Low-dimensional Systems and Nanostructures Vol 31, 83-85 (2006).
* [2] R. Carmona, J. Lacroix: Spectral Theory of Random Schrödinger Operators, (Birkhäuser Verlag, Boston 1990)
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|
arxiv-papers
| 2009-03-31T06:38:19 |
2024-09-04T02:49:01.534057
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "W. Kirsch and M. Krishna",
"submitter": "M. Krishna",
"url": "https://arxiv.org/abs/0903.5376"
}
|
0903.5444
|
# Stochastic receding horizon control with bounded control inputs: a vector
space approach
Debasish Chatterjee , Peter Hokayem and John Lygeros Automatic Control
Laboratory
Physikstrasse 3
ETH Zürich
8092 Zürich
Switzerland {chatterjee,hokayem,lygeros}@control.ee.ethz.ch
###### Abstract.
We design receding horizon control strategies for stochastic discrete-time
linear systems with additive (possibly) unbounded disturbances, while obeying
hard bounds on the control inputs. We pose the problem of selecting an
appropriate optimal controller on vector spaces of functions and show that the
resulting optimization problem has a tractable convex solution. Under the
assumption that the zero-input and zero-noise system is asymptotically stable,
we show that the variance of the state is bounded when enforcing hard bounds
on the control inputs, for any receding horizon implementation. Throughout the
article we provide several examples that illustrate how quantities needed in
the formulation of the resulting optimization problems can be calculated off-
line, as well as comparative examples that illustrate the effectiveness of our
control strategies.
This research was partially supported by the Swiss National Science Foundation
under grant 200021-122072, and the FeedNetBack project FP7-ICT-223866
(www.feednetback.eu).
###### Contents
1. 1 Introduction
2. 2 Problem Statement
3. 3 Main Result
4. 4 Various Cases of Constrained Controls
1. 4.1 Bounded controls, unbounded noise, and $p=\infty$
2. 4.2 Bounded controls, bounded noise, and $p=2$
3. 4.3 Constraints on control energy
5. 5 Stability Analysis
1. 5.1 Mean-square boundedness
2. 5.2 Input-to-state stability
6. 6 Numerical Examples
7. 7 Conclusion and Future Directions
8. A.1 Some identities
9. A.2 Proof of mean-square boundedness
## 1\. Introduction
Receding horizon control is a popular paradigm for designing control policies.
In the context of deterministic systems it has received a considerable amount
of attention over the last two decades, and significant advancements have been
made in terms of its theoretical foundations as well as industrial
applications. The motivation comes primarily from the fact that receding
horizon control yields tractabile control laws for deterministic systems in
the presence of constraints, and has consequently become popular in the
industry. The counterpart in the context of stochastic systems, however, is a
relatively recent development. In this article we solve the problem of
stochastic receding horizon control for linear systems subject to additive
(possibly) unbounded disturbances and hard norm bounds on the control inputs,
over a class of feedback policies. Methods for guaranteeing hard bounds on the
control inputs, within our context, while ensuring tractability of the
underlying optimization problem are, to our knowledge, not available in the
current literature. Preliminary results in this direction were reported in
[HCL09].
In the deterministic setting, the receding horizon control scheme is dominated
by worst-case analysis relying on robust control and robust optimization
methods, see, for example, [Ber05, MRRS00, BM99, LHBW07, Mac01, Bla99, FB05,
YB09, RH05] and the references therein. The central idea is to synthesize a
controller based on the bounds of the noise such that a certain target set
becomes invariant with respect to the closed-loop dynamics. However, such an
approach tends to yield rather conservative controllers and large
infeasibility regions. Moreover, assigning an a priori bound to the noise
seems to demand considerable insight. A stochastic model of the noise is a
natural alternative approach to this problem: the conservativeness of worst-
case controllers may be reduced, and one may not need to impose any a priori
bounds on the maximum magnitude of the noise. In [BB07], the authors
reformulate the stochastic programming problem as a deterministic one with
bounded noise and solve a robust optimization problem over a finite horizon,
followed by estimating the performance when the noise is unbounded but takes
high values with low probability (as in the Gaussian case). In [PS09] a
slightly different problem is addressed in which the noise enters in a
multiplicative manner, and hard constraints on the states and control inputs
are relaxed to constraints resembling the integrated chance constraints of
[Han83] or risk measures in mathematical finance. Similar relaxations of hard
constraints to soft probabilistic ones have also appeared in [CKW08] for both
multiplicative and additive noise inputs, as well as in [OJM08] for additive
noise inputs. There are also other approaches, e.g., those employing
randomized algorithms as in [BW07, Bat04, MLL05]. Related lines of research
can be found in [vHB03, vHB06] dealing with constrained model predictive
control (MPC) for stochastic systems motivated by industrial applications, in
[RCMA+09, BSW02, SSW06] dealing with stochastic stability, in [SB09b] dealing
with Q-design, in, e.g., [LH07, LHC03] dealing with alternative approaches to
control under actuator constraints and neural-network approximation. The
articles [ACCL09, CACL09] deal with a formulation that allows probabilistic
state constraints but not hard input constraints, and is hence complementary
to the approach in the present article, and [HCCL10] treats the case of output
feedback. . Finally, note that probabilistic constraints on the controllers
naturally raise difficult questions on what actions to take when such
constraints are violated, see [CCCL08] and [CP09] for partial solutions to
these issues.
The main contributions of the article are as follows: We give a tractable,
convex, and globally feasible solution to the finite-horizon stochastic linear
quadratic (LQ) problem for linear systems with possibly unbounded additive
noise and hard constraints on the elements of the control policy. Within this
framework one has two directions to pursue in terms of controller design,
namely, a posteriori bounding the standard LQG controller, or employing
certainty-equivalent receding horizon controller. While the former direction
explicitly incorporates some aspects of feedback, the synthesis of the latter
involves control constraints and implicitly incorporates the notion of
feedback. Our choice of feedback policies explores the middle ground between
these two choices: we explicitly incorporate both the control bounds and
feedback at the design phase. More specifically, we adopt a policy that is
affine in certain bounded functions of the past noise inputs. The optimal
control problem is lifted onto general vector spaces of candidate control
functions from which the controller can be selected algorithmically by solving
a convex optimization problem. Our novel approach does not require
artificially relaxing the hard constraints on the control input to soft
probabilistic ones (to ensure large feasible sets), and still provides a
globally feasible solution to the problem. Minimal assumptions of the noise
sequence being i.i.d and having finite second moment are imposed. The effect
of the noise appears in the convex optimization problem as certain fixed
cross-covariance matrices, which may be computed off-line and stored.
Once tractability of the optimization problem is ensured, we employ the
resulting control policy in a receding horizon scheme. Under our policies the
closed-loop system is in general not necessarily Markovian, and as a result
stability of the closed-loop system is not immediate. In fact, we can no
longer appeal directly to standard Foster-Lyapunov methods. We establish that
our receding horizon control scheme provides stability under the assumption
that the zero-input and zero-noise system is asymptotically stable. We provide
examples that demonstrate the effectiveness of our policies with respect to
standard methods such as certainty-equivalent MPC, standard unconstrained LQG
and saturated LQG control. These examples show that our policies perform no
worse than the standard unconstrained LQG controller in the absence of control
constraints, and outperform the certainty-equivalent MPC as well as the
saturated LQG control by a significant margin.
Our mechanism for selection of a policy consists of two steps: The first
concerns the structure of our policies, and is motivated by preceding work in
robust optimization and MPC [Löf03, BTGGN04, GKM06]. The second concerns the
procedure for selection of an optimal policy from a general vector space of
candidate control functions, and is inspired by approximate dynamic
programming techniques [BT96, LR06, SS85, dFR03, Pow07]. With respect to the
first step, our policies are more general compared to those in [Löf03,
BTGGN04, GKM06]. With respect to the second, the selection procedure of our
policies consists of a one-step tractable static optimization program.
The rest of this article is organized as follows. In Section 2 we state the
main problem to be solved in the most general form. In Section 3 we provide a
tractable solution to the finite horizon optimization problem on general
vector spaces. This result is specialized to various classes of noise and
input constraint sets in Section 4. Stability of receding horizon
implementations of the obtained closed-loop policy is shown in Section 5, and
input-to-state stability properties are discussed in Section 5.2. We provide a
host of numerical examples that illustrate the effectiveness of our approach
in Section 6. Finally, we conclude in Section 7 with a discussion on future
research directions.
### Notation
Hereafter, $\mathbb{N}\coloneqq\\{1,2,\ldots\\}$ is the set of natural
numbers, $\mathbb{N}_{0}\coloneqq\mathbb{N}\cup\\{0\\}$, $\mathbb{Z}$ is the
set of all the integers, $\mathbb{R}_{\geqslant 0}$ is the set of nonnegative
real numbers, and $\mathbb{C}$ denotes the set of complex numbers. We let
$\mathbf{1}_{A}(\cdot)$ denote the indicator function of a set $A$, and
$\mathbf{I}_{n\times n}$ and $\mathbf{0}_{n\times m}$ denote the
$n$-dimensional identity matrix and $n\times m$-dimensional zeros matrix,
respectively. Let $\left\lVert{\cdot}\right\rVert$ denote the standard
Euclidean norm, and $\left\lVert{\cdot}\right\rVert_{p}$ denote the usual
$\ell_{p}$ norms. Also, let $\mathbb{E}_{x_{0}}[\cdot]$ denote the expected
value given $x_{0}$, and $\mathbf{tr}\\!\left(\cdot\right)$ denote the trace
of a matrix. If $M_{1}$ and $M_{2}$ are two matrices with the same number of
rows, we employ the standard notation $[M_{1}\mid M_{2}]$ for the matrix
obtained by stacking the columns of $M_{1}$ followed by the columns of
$M_{2}$. For a given symmetric $n$-dimensional matrix $M$ with real entries,
let $\\{\lambda_{i}(M)\mid i=1,\ldots,n\\}$ be the set of eigenvalues of $M$,
and let $\lambda_{\rm max}(M)\coloneqq\max_{i}\lambda_{i}(M)$ and
$\lambda_{\text{min}}(M)\coloneqq\min_{i}\lambda_{i}(M)$. Finally, for a
random vector $X$ let $\Sigma_{X}$ denote the matrix
$\mathbb{E}\bigl{[}XX^{\mathsf{T}}\bigr{]}$ and $\mu_{X}$ denote the vector
$\mathbb{E}\bigl{[}X\bigr{]}$.
## 2\. Problem Statement
Consider the following discrete-time stochastic dynamical system:
(2.1) $x_{t+1}=\bar{A}x_{t}+\bar{B}u_{t}+w_{t},\qquad t\in\mathbb{N}_{0},$
where $x_{t}\in\mathbb{R}^{n}$ is the state, $u_{t}$ is the control input
taking values in some given control set
$\bar{\mathbb{U}}\subseteq\mathbb{R}^{m}$ to be defined later,
$\bar{A}\in\mathbb{R}^{n\times n}$, $\bar{B}\in\mathbb{R}^{n\times m}$, and
$(w_{t})_{t\in\mathbb{N}_{0}}$ is a sequence of stochastic noise vectors with
$w_{t}\in\mathbb{W}\subseteq\mathbb{R}^{n}$. We assume that the initial
condition $x_{0}$ is given and that, at any time $t$, $x_{t}$ is observed
perfectly. We do not assume that the components of the noise $w_{t}$ are
uncorrelated, nor that they have zero mean; this effectively means that
$w_{t}$ may be of the form $\bar{F}w_{t}^{\prime}+b$ for some noise
$w_{t}^{\prime}\in\mathbb{R}^{p}$ whose components are uncorrelated or
mutually independent, $F\in\mathbb{R}^{n\times p}$, and $b\in\mathbb{R}^{n}$.
Without loss of generality we shall stick to the simpler notation of (2.1)
throughout this article. The results readily extend to the general case of
$w_{t}=\bar{F}w_{t}^{\prime}+b$, as can be seen in [HCL09].
Generally, a _control policy_ $\pi$ is a sequence
$(\pi_{0},\pi_{1},\pi_{2},\ldots)$ of Borel measurable maps
$\pi_{t}:\underbrace{\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}}_{k(t)-\text{
times}}\to\bar{\mathbb{U}},\;t\in\mathbb{N}_{0}$. Policies of finite length
such as $(\pi_{t},\pi_{t+1},\ldots,\pi_{t+N-1})$ will be denoted in the sequel
by $\pi_{t:t+N-1}$.
Fix an optimization horizon $N\in\mathbb{N}$ and let us consider the following
objective function at time $t$ given the state $x_{t}$:
(2.2)
$V_{t}\coloneqq\mathbb{E}\Biggl{[}\sum_{k=0}^{N-1}\bigl{(}x_{t+k}^{\mathsf{T}}Q_{k}x_{t+k}+u_{t+k}^{\mathsf{T}}R_{k}u_{t+k}\bigr{)}+x_{t+N}^{\mathsf{T}}Q_{N}x_{t+N}\,\Bigg{|}\,x_{t}\Biggr{]},$
where $Q_{t}>0,R_{t}>0,Q_{N}>0$ are some given symmetric matrices of
appropriate dimension. At each time instant $t$, we are interested in
minimizing (2.2) over the class of causal state feedback strategies $\Pi$
defined as:
(2.3) $\left[\begin{matrix}u_{t}\\\ u_{t+1}\\\ \vdots\\\
u_{t+N-1}\end{matrix}\right]=\left[\begin{array}[]{l}\pi_{t}(x_{t})\\\
\pi_{t+1}(x_{t},x_{t+1})\\\ \vdots\\\
\pi_{t+N-1}(x_{t},x_{t+1},\cdots,x_{t+N-1})\end{array}\right],$
for some measurable functions
$\pi_{t:t+N-1}\coloneqq\\{\pi_{t},\cdots,\pi_{t+N-1}\\}\in\Pi$, while
satisfying $u_{t}\in\bar{\mathbb{U}}$ for each $t$. The _receding horizon
control_ procedure for a given control horizon $N_{c}\in\\{1,\ldots,N\\}$ and
time $t$ can be described as follows:
* (a)
measure the state $x_{t}$;
* (b)
determine an admissible optimal feedback control policy, say
$\pi^{*}_{t:t+N-1}\in\Pi$, that minimizes the $N$-stage cost function (2.2)
starting from time $t$, given the measured initial condition $x_{t}$;
* (c)
apply the first $N_{c}$ elements $\pi^{*}_{t:t+N_{c}-1}$ of the policy
$\pi^{*}_{t:t+N-1}$;
* (d)
increase $t$ to $t+N_{c}$, and go back to step (a).
In this context, if $N_{c}=1$ then this is usual MPC, and if $N_{c}=N$, then
it is usually known as rolling horizon control.
Since both the system (2.1) and cost (2.2) are time-invariant, it is enough to
consider the problem of minimizing the cost for $t=0$. In view of the above we
consider the problem:
(2.4)
$\displaystyle\min_{\pi_{0:N-1}\in\Pi}\bigl{\\{}V_{0}\,\big{|}\,\text{dynamics
\eqref{eq:system}},\text{ and }u_{t}\in\bar{\mathbb{U}}\text{ for each
}t\bigr{\\}}.$
If feasible, the problem (2.4) generates an optimal sequence of feedback
control laws $\pi^{*}=\left\\{\pi^{*}_{0},\cdots,\pi^{*}_{N-1}\right\\}$.
The evolution of the system (2.1) over a single optimization horizon $N$ can
be described in a compact form as follows:
(2.5) $x=Ax_{0}+Bu+Dw,$
where
$x\coloneqq\begin{bmatrix}x_{0}\\\ x_{1}\\\ \vdots\\\
x_{N}\end{bmatrix},\qquad u\coloneqq\begin{bmatrix}u_{0}\\\ u_{1}\\\ \vdots\\\
u_{N-1}\end{bmatrix},\qquad w\coloneqq\begin{bmatrix}w_{0}\\\ w_{1}\\\
\vdots\\\ w_{N-1}\end{bmatrix},\qquad
A\coloneqq\begin{bmatrix}\mathbf{I}_{n\times n}\\\ \bar{A}\\\ \vdots\\\
\bar{A}^{N}\end{bmatrix},$ $B\coloneqq\begin{bmatrix}\mathbf{0}_{n\times
m}&\cdots&\cdots&\mathbf{0}_{n\times m}\\\ \bar{B}&\ddots&&\vdots\\\
\bar{A}\bar{B}&\bar{B}&\ddots&\vdots\\\ \vdots&&\ddots&\mathbf{0}_{n\times
m}\\\ \bar{A}^{N-1}\bar{B}&\cdots&\bar{A}\bar{B}&\bar{B}\end{bmatrix},\qquad
D\coloneqq\begin{bmatrix}\mathbf{0}_{n\times
n}&\cdots&\cdots&\mathbf{0}_{n\times n}\\\ \mathbf{I}_{n\times
n}&\ddots&&\vdots\\\ \bar{A}&\mathbf{I}_{n\times n}&\ddots&\vdots\\\
\vdots&&\ddots&\mathbf{0}_{n\times n}\\\
\bar{A}^{N-1}&\cdots&\bar{A}&\mathbf{I}_{n\times n}\end{bmatrix}.$
Using the compact notation above, the optimization Problem (2.4) can be
rewritten as follows:
(2.6) $\displaystyle\min_{\pi_{0:N-1}\in\Pi}$
$\displaystyle\bigl{\\{}\mathbb{E}_{x_{0}}\bigl{[}x^{\mathsf{T}}Qx+u^{\mathsf{T}}Ru\bigr{]}\,\big{|}\,\text{dynamics
\eqref{eq:compactdyn}},u\in\mathbb{U}\bigr{\\}},$
where $Q=\operatorname{diag}\\{Q_{0},\ldots,Q_{N}\\}$,
$R=\operatorname{diag}\\{R_{0},\ldots,R_{N-1}\\}$, and
$\mathbb{U}\coloneqq\underbrace{\bar{\mathbb{U}}\times\ldots\times\bar{\mathbb{U}}}_{N-\text{times}}$.
## 3\. Main Result
We require that our controller is selected from a vector space of candidate
controllers spanned by a given set of “simple” basis functions. The precise
algorithmic selection procedure is based on the solution to an optimization
problem. The basis functions may represent particular types of control
functions that are easy or inexpensive to implement, e.g., minimum attention
control [Bro97], or may be the only ones available for a specific application.
For instance, piecewise constant policy elements with finitely many elements
in their range may be viewed as controllers that can provide only finitely
many values; this may be viewed as an extended version of a bang-bang
controller, or as a hybrid controller with a finite control alphabet.
More formally, let $\mathcal{H}$ be a nonempty separable vector space of
functions with the control set $\mathbb{U}$ as their range, i.e.,
$\mathcal{H}$ is the linear span of measurable functions
${\mathfrak{e}}^{\nu}:\mathbb{W}\to\mathbb{U}$, where $\nu\in\mathcal{I}$ \-
an ordered countable index set (see [Lue69] for more details). As mentioned
above, the elements of $\mathcal{H}$ may be linear combinations of typical
“simple” controller functions for $t=0,1,\ldots,N-1$. We are interested in
policies of the form $u_{t}=\eta_{t}+\sum_{i=0}^{t-1}\psi_{t,i}(w_{i})$, where
$\eta_{t}$ is an $m$-dimensional vector and each component of the
$m$-dimensional vector-valued function $\psi_{t,i}$ is a member of
$\mathcal{H}$. Although this feedback function is directly from the noise,
since the state is assumed to be perfectly measured, from the system dynamics
(2.1) it follows at once that this controller $u_{t}$ is actually a feedback
from all the states $x_{0},\ldots,x_{t}$. Indeed, in the spirit of [Löf03,
BTGGN04, GKM06, SB09a] we have
$\displaystyle u_{0}$ $\displaystyle=\eta_{0},$ $\displaystyle u_{1}$
$\displaystyle=\eta_{1}+\psi_{1,0}(x_{1}-\bar{A}x_{0}-\bar{B}\eta_{0}),$
$\displaystyle u_{2}$
$\displaystyle=\eta_{2}+\psi_{2,0}(x_{1}-\bar{A}x_{0}-\bar{B}\eta_{0})+\psi_{2,1}\bigl{(}x_{2}-\bar{A}x_{1}-\bar{B}\bigl{(}\eta_{1}+\psi_{1,0}(x_{1}-\bar{A}x_{0}-\bar{B}\eta_{0})\bigr{)}\bigr{)},$
$\displaystyle\vdots$
In other words, by construction, $u_{t}$ is generally a nonlinear feedback
controller depending on the past $t$ states.111Note that the controller input
at time $t$ is non-Markovian as it is a function of the state vectors at all
the previous times and not just on $x_{t-1}$. Also by construction, it is
causal.
Our general control policy can now be expressed as the vector
(3.1) $u=\eta+\varphi(w)\coloneqq\begin{bmatrix}\eta_{0}\\\ \eta_{1}\\\
\vdots\\\ \eta_{N-1}\end{bmatrix}+\begin{bmatrix}\varphi_{0}\\\
\varphi_{1}(w_{0})\\\ \vdots\\\
\varphi_{N-1}(w_{0},w_{1},\ldots,w_{N-2})\end{bmatrix},$
where,
* •
$\varphi_{0}=0$,
* •
$w_{t}$ for $t=0,\ldots,N-1$ is the $t$-th random noise vector,
* •
$\eta_{t}$ is an $m$-dimensional vector for $t=0,\ldots,N-1$,
* •
$\varphi_{t}(w_{0},\ldots,w_{t-1})=\sum_{i=0}^{t-1}\varphi_{t,i}(w_{i})$ for
$t=1,\ldots,N-1$ is an $m$-dimensional vector, and
* •
each function $\varphi_{t,i}$ belongs to the linear span of the basis elements
$({\mathfrak{e}}^{\nu})_{\nu\in\mathcal{I}}$, and thus has a representation as
a linear combination
$\varphi_{t,i}(\cdot)=\sum_{\nu\in\mathcal{I}}\theta_{t,i}^{\nu}{\mathfrak{e}}^{\nu}(\cdot)$,
$t=1,\ldots,N-1$, $i=0,\ldots,t-1$, where $\theta_{t,i}^{\nu}$ are matrices of
coefficients of appropriate dimension.
Analogous to Fourier coefficients in harmonic analysis, we call the
$\theta_{t,i}^{\nu}$ the $\nu$-th Fourier coefficient of the function
$\varphi_{t,i}$. Therefore, whenever $|\mathcal{I}|<\infty$ for every
$t=1,\ldots,N-1$, we have the finite representation
(3.2)
$\varphi_{t}(w_{0},\ldots,w_{t-1})=\begin{bmatrix}\theta_{t,0}&\theta_{t,1}&\ldots&\theta_{t,t-1}&\boldsymbol{0}&\ldots&\boldsymbol{0}\end{bmatrix}_{R^{m\times
n|\mathcal{I}|(N-1)}}\begin{bmatrix}{\mathfrak{e}}(w_{0})\\\
{\mathfrak{e}}(w_{1})\\\ \vdots\\\
{\mathfrak{e}}(w_{N-2})\end{bmatrix}_{\mathbb{R}^{n|\mathcal{I}|(N-1)\times
1}},$
where $\theta_{t,i}\in\mathbb{R}^{m\times n|\mathcal{I}|}$,
$\boldsymbol{0}\in\mathbb{R}^{m\times n|\mathcal{I}|}$,
$\theta_{t,i}\coloneqq\left[\begin{matrix}\theta^{1}_{t,i}\,\;\cdots\,\,\theta^{|\mathcal{I}|}_{t,i}\end{matrix}\right],\;\;\theta_{t,i}^{\nu}\in\mathbb{R}^{m\times
n},\quad{\rm
and}\quad{\mathfrak{e}}(w_{i})\coloneqq\left[\begin{matrix}{\mathfrak{e}}^{1}(w_{i})\\\
\vdots\\\
{\mathfrak{e}}^{|\mathcal{I}|}(w_{i})\end{matrix}\right],\quad\forall\,i=0,1,\cdots,N-2.$
In this notation the policy (3.1) can be written as
(3.3)
$u=\eta+\varphi(w)=\eta+\begin{bmatrix}\boldsymbol{0}&\boldsymbol{0}&\cdots&\boldsymbol{0}\\\
\theta_{1,0}&\boldsymbol{0}&\cdots&\boldsymbol{0}\\\
\theta_{2,0}&\theta_{2,1}&\cdots&\boldsymbol{0}\\\
\vdots&\vdots&\ddots&\vdots\\\
\theta_{N-1,0}&\theta_{N-1,1}&\cdots&\theta_{N-1,N-2}\end{bmatrix}\begin{bmatrix}{\mathfrak{e}}(w_{0})\\\
{\mathfrak{e}}(w_{1})\\\ \vdots\\\
{\mathfrak{e}}(w_{N-2})\end{bmatrix}\eqqcolon\eta+\Theta{\mathfrak{e}}(w),$
where $\Theta$ is now the matrix of Fourier coefficients having dimension
$Nm\times\bigl{(}n(N-1)|\mathcal{I}|\bigr{)}$. This Fourier coefficient matrix
$\Theta$ and the vector $\eta$ play the role of the optimization parameters in
our search for an optimal policy. Note that ${\mathfrak{e}}(w)$ does not
include the noise vector $w_{N-1}$, and that $\Theta$ is strictly lower block
triangular to enforce causality. In what follows, as a matter of notation, by
$\Theta_{t}$ we shall denote the formal $t$-th block-row of the matrix
$\Theta$ in (3.3), i.e.,
$\Theta_{t}\coloneqq\begin{bmatrix}\theta_{t,0}&\cdots&\theta_{t,t-1}&0&\cdots&0\end{bmatrix}$,
for $t=0,\cdots,N-1$, with $\Theta_{0}$ being the identically $0$ row. We make
the following assumption:
###### Assumption 3.1.
The sequence $(w_{t})_{t\in\mathbb{N}_{0}}$ of noise vectors is i.i.d with
$\Sigma=\mathbb{E}\bigl{[}w_{t}w_{t}^{\mathsf{T}}\bigr{]}$.$\diamondsuit$
So far we have not stipulated any boundedness properties on the elements of
the vector space $\mathcal{H}$. This means that the control policy elements
may be unbounded maps. First we stipulate the following structure on the
control sets:
For a given $p\in[1,\infty]$, the control input vector $u_{t}$ is bounded in
$p$-norm at each instant of time $t$, i.e., for $p\in[1,\infty]$ let
$U_{\max}^{(p)}>0$ be given, with
(3.4) $\displaystyle u_{t}\in\bar{\mathbb{U}}_{p}$
$\displaystyle\coloneqq\bigl{\\{}\xi\in\mathbb{R}^{m}\big{|}\left\lVert{\xi}\right\rVert_{p}\leqslant
U_{\max}^{(p)}\bigr{\\}}\quad\forall\,t\in\mathbb{N}_{0},\quad\text{and}$
$\displaystyle\mathbb{U}_{p}$
$\displaystyle\coloneqq\underbrace{\bar{\mathbb{U}}_{p}\times\ldots\times\bar{\mathbb{U}}_{p}}_{N-\text{times}}.$
One could easily include more general constraint sets $\mathbb{U}_{p}$, for
instance, to capture bounds on the rate of change of inputs.
Our basic result is the next Theorem.
###### Theorem 3.2.
Consider the system (2.1). Suppose that Assumption 3.1 holds, $\mathcal{H}$ is
finite-dimensional ($|\mathcal{I}|<\infty$), and every component of the basis
functions ${\mathfrak{e}}^{\nu}$ is bounded by $\mathcal{E}>0$ in absolute
value. Then the problem (2.6) under the policy (3.1) and control sets (3.4)
for $p\in[1,\infty]$ is convex with respect to the decision variables
$(\eta,\Theta)$ defined in (3.3). For $p=1,2$, and $\infty$ it admits convex
tractable versions with tighter domains of $(\eta,\Theta)$, given by
(3.5) $\displaystyle\underset{(\eta,\Theta)}{\text{minimize}}$
$\displaystyle\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\Theta\Sigma_{{\mathfrak{e}}}\Bigr{)}+2\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}{B}^{\mathsf{T}}QD\Sigma_{{\mathfrak{e}}}^{\prime}\Bigr{)}+\eta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\eta$
$\displaystyle\quad+2\bigl{(}x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QB\eta+\eta^{\mathsf{T}}{B}^{\mathsf{T}}QD\mu_{w}+x_{0}^{\mathsf{T}}A^{\mathsf{T}}QB\Theta\mu_{\mathfrak{e}}\bigr{)}$
$\displaystyle\quad+2\eta^{\mathsf{T}}\bigl{(}R+B^{\mathsf{T}}QB\bigr{)}\Theta\mu_{\mathfrak{e}}+c$
subject to $\displaystyle\text{$\Theta$ strictly lower block triangular as
in~{}\eqref{e:Thetadef}},$
$\displaystyle\begin{cases}p=1:&\left\lVert{\eta_{t}}\right\rVert_{1}+\mathcal{E}t\left\lVert{\Theta_{t}}\right\rVert_{1}\leqslant
U_{\max}^{(1)},\quad\forall\,t=0,1,\ldots,N-1,\\\
p=\infty:&\left\lVert{\eta_{t}}\right\rVert_{\infty}+\mathcal{E}\left\lVert{\Theta_{t}}\right\rVert_{\infty}\leqslant
U_{\max}^{(\infty)},,\quad\forall\,t=0,1,\ldots,N-1,\\\
p=2:&\left\lVert{\begin{bmatrix}\eta_{t}&\Theta_{t}\end{bmatrix}}\right\rVert_{2}\mathchoice{{\hbox{$\displaystyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}\leqslant
U_{\max}^{(2)},\quad\forall\,t=0,1,\ldots,N-1,\end{cases}$
where
$\displaystyle\Sigma_{{\mathfrak{e}}}$
$\displaystyle\coloneqq\mathbb{E}\bigl{[}{\mathfrak{e}}(w){\mathfrak{e}}(w)^{\mathsf{T}}\bigr{]},$
$\displaystyle\Sigma_{{\mathfrak{e}}}^{\prime}$
$\displaystyle\coloneqq\mathbb{E}\bigl{[}w{\mathfrak{e}}(w)^{\mathsf{T}}\bigr{]},$
$\displaystyle\mu_{w}$
$\displaystyle\coloneqq\mathbb{E}[w],\quad\quad\mu_{\mathfrak{e}}\coloneqq\mathbb{E}[{\mathfrak{e}}(w)],$
$\displaystyle c$ $\displaystyle\coloneqq
x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QAx_{0}+2x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QD\mu+\operatorname{\mathbf{tr}}\bigl{(}{D}^{\mathsf{T}}QD\Sigma_{w}\bigr{)}.$
###### Proof of Theorem 3.2.
It is easy to see that $x^{\mathsf{T}}Qx+u^{\mathsf{T}}Ru$ is convex
nondecreasing, and both $x$ and $u$ are affine functions of the design
parameters $(\eta,\Theta)$ for any realization of the noise $w$. Hence,
$V_{0}$ is convex in $(\eta,\Theta)$ since taking expectations of a convex
function retains convexity [BV04, Section 3.2]. Moreover, the control
constraint sets in (3.4) are convex in $(\eta,\Theta)$. This settles the first
claim.
The objective function (2.2) is given by
$\displaystyle\mathbb{E}_{x_{0}}$
$\displaystyle\bigl{[}\bigl{(}Ax_{0}+Bu+Dw\bigr{)}^{\mathsf{T}}Q\bigl{(}Ax_{0}+Bu+Dw\bigr{)}\bigr{]}+\mathbb{E}_{x_{0}}\bigl{[}{u}^{\mathsf{T}}Ru\bigr{]}$
$\displaystyle=\mathbb{E}_{x_{0}}\bigl{[}\bigl{(}Ax_{0}+B(\eta+\Theta{\mathfrak{e}}(w))+Dw\bigr{)}^{\mathsf{T}}Q\bigl{(}Ax_{0}+B(\eta+\Theta{\mathfrak{e}}(w))+Dw\bigr{)}\bigr{]}$
$\displaystyle\quad+\mathbb{E}_{x_{0}}\bigl{[}(\eta+\Theta{\mathfrak{e}}(w))^{\mathsf{T}}R(\eta+\Theta{\mathfrak{e}}(w))\bigr{]}$
$\displaystyle=x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QAx_{0}+2x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QB\eta+\eta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\eta$
$\displaystyle\quad+2\bigl{(}Ax_{0}+B\eta\bigr{)}^{\mathsf{T}}Q\bigl{(}B\Theta\mathbb{E}_{x_{0}}[{\mathfrak{e}}(w)]+D\mathbb{E}_{x_{0}}[w]\bigr{)}$
$\displaystyle\quad+\mathbb{E}_{x_{0}}\bigl{[}\bigl{(}B\Theta{\mathfrak{e}}(w)+Dw\bigr{)}^{\mathsf{T}}Q\bigl{(}B\Theta{\mathfrak{e}}(w)+Dw\bigr{)}\bigr{]}+\mathbb{E}_{x_{0}}\bigl{[}(\Theta{\mathfrak{e}}(w))^{\mathsf{T}}R\Theta{\mathfrak{e}}(w)\bigr{]}$
$\displaystyle=x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QAx_{0}+2x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QB\eta+\eta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\eta+2\eta^{\mathsf{T}}R\Theta\mathbb{E}[{\mathfrak{e}}(w)]$
$\displaystyle\quad+2\bigl{(}Ax_{0}+B\eta\bigr{)}^{\mathsf{T}}Q\bigl{(}D\mathbb{E}_{x_{0}}[w]+B\Theta\mathbb{E}[{\mathfrak{e}}(w)]\bigr{)}+\operatorname{\mathbf{tr}}\Bigl{(}{D}^{\mathsf{T}}QD\mathbb{E}_{x_{0}}\bigl{[}w^{\mathsf{T}}\bigr{]}\Bigr{)}$
$\displaystyle\quad+\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\Theta\mathbb{E}_{x_{0}}\bigl{[}{\mathfrak{e}}(w){\mathfrak{e}}(w)^{\mathsf{T}}\bigr{]}\Bigr{)}+2\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}{B}^{\mathsf{T}}QD\mathbb{E}_{x_{0}}\bigl{[}w{\mathfrak{e}}(w)^{\mathsf{T}}\bigr{]}\Bigr{)}.$
Incorporating the definitions $\Sigma_{{\mathfrak{e}}}$,
$\Sigma_{{\mathfrak{e}}}^{\prime}$, $\mu_{w}$, $\mu_{\mathfrak{e}}$, and $c$,
the right-hand side above equals
$\displaystyle\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\Theta\Sigma_{{\mathfrak{e}}}\Bigr{)}+2\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}{B}^{\mathsf{T}}QD\Sigma_{{\mathfrak{e}}}^{\prime}\Bigr{)}+\eta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\eta$
$\displaystyle\quad+2\bigl{(}x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QB\eta+\eta^{\mathsf{T}}{B}^{\mathsf{T}}QD\mu_{w}+x_{0}^{\mathsf{T}}A^{\mathsf{T}}QB\Theta\mu_{\mathfrak{e}}\bigr{)}+2\eta^{\mathsf{T}}\bigl{(}R+B^{\mathsf{T}}QB\bigr{)}\Theta\mu_{\mathfrak{e}}$
$\displaystyle\quad+\bigl{(}x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QAx_{0}+2x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QD\mu_{w}+\operatorname{\mathbf{tr}}\bigl{(}{D}^{\mathsf{T}}QD\Sigma_{w}\bigr{)}\bigr{)}$
$\displaystyle=\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\Theta\Sigma_{{\mathfrak{e}}}\Bigr{)}+2\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}{B}^{\mathsf{T}}QD\Sigma_{{\mathfrak{e}}}^{\prime}\Bigr{)}+\eta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\eta$
$\displaystyle\quad+2\bigl{(}x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QB\eta+\eta^{\mathsf{T}}{B}^{\mathsf{T}}QD\mu_{w}+x_{0}^{\mathsf{T}}A^{\mathsf{T}}QB\Theta\mu_{\mathfrak{e}}\bigr{)}+2\eta^{\mathsf{T}}\bigl{(}R+B^{\mathsf{T}}QB\bigr{)}\Theta\mu_{\mathfrak{e}}+c.$
Since the matrix $\Sigma_{{\mathfrak{e}}}$ is positive semidefinite, it can be
expressed as a finite nonnegative linear combination of matrices of the type
$\sigma\sigma^{\mathsf{T}}$, for vectors $\sigma$ of appropriate dimension
[BSM03, Theorem 1.10]. Accordingly, if
$\Sigma_{{\mathfrak{e}}}=\sum_{i=1}^{k}\sigma_{i}\sigma_{i}^{\mathsf{T}}$,
then
$\displaystyle\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\Theta\Sigma_{{\mathfrak{e}}}\Bigr{)}$
$\displaystyle=\sum_{i=1}^{k}\operatorname{\mathbf{tr}}\bigl{(}\Theta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\Theta\sigma_{i}\sigma_{i}^{\mathsf{T}}\bigr{)}$
$\displaystyle=\sum_{i=1}^{k}\Bigl{(}\sigma_{i}^{\mathsf{T}}\Theta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\Theta\sigma_{i}\Bigr{)}.$
Defining $\widehat{\Theta}_{i}\coloneqq\Theta\sigma_{i}$ and adjoining these
equalities to the constraints of the optimization program (3.5), we arrive at
the optimization program
(3.6)
$\displaystyle\underset{(\Theta,\widehat{\Theta}_{1},\ldots,\widehat{\Theta}_{k})}{\text{minimize}}$
$\displaystyle\sum_{i=1}^{k}\widehat{\Theta}_{i}^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\widehat{\Theta}_{i}+2\operatorname{\mathbf{tr}}\Bigl{(}\Theta^{\mathsf{T}}{B}^{\mathsf{T}}QD\Sigma_{{\mathfrak{e}}}^{\prime}\Bigr{)}+\eta^{\mathsf{T}}\bigl{(}{B}^{\mathsf{T}}QB+R\bigr{)}\eta$
$\displaystyle\qquad+2\bigl{(}x_{0}^{\mathsf{T}}{A}^{\mathsf{T}}QB\eta+\eta^{\mathsf{T}}{B}^{\mathsf{T}}QD\mu+x_{0}^{\mathsf{T}}A^{\mathsf{T}}QB\Theta\mu^{\mathfrak{e}}\bigr{)}$
$\displaystyle\qquad+2\eta^{\mathsf{T}}\bigl{(}R+B^{\mathsf{T}}QB\bigr{)}\Theta\mu^{\mathfrak{e}}+c$
subject to $\displaystyle\Theta\text{ strictly lower block triangular as
in~{}\eqref{e:Thetadef}},$
$\displaystyle\widehat{\Theta}_{i}=\Theta\sigma_{i}\quad\text{for all
}i=1,\ldots,k.$
We see immediately that (3.6) is a convex program in the parameters $\eta$,
$\Theta$ and $\widehat{\Theta}_{i}$, and is equivalent to the cost in (3.5).
It only remains to consider the last constraint in (3.5). First we consider
the cases of $p=1,\infty$. Using the notation above, an application of the
triangle inequality immediately shows that the constraints can be written as
(3.7)
$\displaystyle\begin{cases}p=1:&\left\lVert{\eta_{t}}\right\rVert_{1}+\mathcal{E}t\left\lVert{\Theta_{t}}\right\rVert_{1}\leqslant
U_{\max}^{(1)},\quad\forall\,t=0,1,\ldots,N-1,\\\
p=\infty:&\left\lVert{\eta_{t}}\right\rVert_{\infty}+\mathcal{E}\left\lVert{\Theta_{t}}\right\rVert_{\infty}\leqslant
U_{\max}^{(\infty)},\quad\forall\,t=0,1,\ldots,N-1.\end{cases}$
It follows that the objective function in (3.6) is quadratic and the
constraints in (3.6)-(3.7) are affine in the optimization parameters $\eta$,
$\Theta$, and $\widehat{\Theta}$. As such, for $p=1,\infty$ our problem is a
quadratic program.
For the case of $p=2$, note that
$\eta_{t}+\Theta_{t}{\mathfrak{e}}(w)=\begin{bmatrix}\eta_{t}&\Theta_{t}\end{bmatrix}\begin{bmatrix}1\\\
{\mathfrak{e}}(w)\end{bmatrix}$, and by definition of $\mathcal{E}$ it is
clear that
$\left\lVert{\begin{bmatrix}\eta_{t}&\Theta_{t}\end{bmatrix}\begin{bmatrix}1\\\
{\mathfrak{e}}(w)\end{bmatrix}}\right\rVert_{2}\leqslant\left\lVert{\begin{bmatrix}\eta_{t}&\Theta_{t}\end{bmatrix}}\right\rVert_{2}\mathchoice{{\hbox{$\displaystyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}$. This immediately
translates to
$\left\lVert{\begin{bmatrix}\eta_{t}&\Theta_{t}\end{bmatrix}}\right\rVert_{2}\mathchoice{{\hbox{$\displaystyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+\mathcal{E}t\,}$}\lower
0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}\leqslant
U_{\max}^{(2)}$, which is the third constraint in Problem 3.5 and it is a
quadratic constraint in the optimization parameters $(\eta,\Theta)$.
Therefore, for $p=2$ our problem is a quadratically constrained quadratic
program. ∎
The optimization problem (3.5) simplifies if we assume that
$\mu^{\mathfrak{e}}=\mathbb{E}[{\mathfrak{e}}(w)]=0$. Note that
$\mathbb{E}[{\mathfrak{e}}(w)]=0$ if and only if
$\mathbb{E}\bigl{[}{\mathfrak{e}}_{t,i}^{\nu}(w_{t,i})\bigr{]}=0$ for all
$\nu\in\mathcal{I}$. At an intuitive level this translates to the condition
that the functions ${\mathfrak{e}}_{t,i}^{\nu}\in\mathcal{H}$ should be
“centered” with respect to the random variables $w_{t,i}$. In particular, this
simply means that for noise distributions that are symmetric about $0$, the
functions ${\mathfrak{e}}^{\nu}$ should be centered at $0$ and be
antisymmetric. For example, if the noise is Gaussian with mean $0$ and
diagonal covariance matrix (uncorrelated components), each component of the
functions ${\mathfrak{e}}^{\nu}$ should be an odd function.
The matrices $\Sigma_{{\mathfrak{e}}}$, $\Sigma_{{\mathfrak{e}}}^{\prime}$,
the vector $v$, and the number $c$ in Theorem 3.2 are all constants
independent of $x_{0}$, and can be computed off-line. As such, even if closed-
form expressions for the entries of the matrices do not exist, they can be
numerically computed to desired precision. The optimization problem (3.5) is a
quadratic program [BV04, p. 152] for $p=1,\infty$, and a quadratically
constrained quadratic program [BV04, p. 152] for $p=2$, in the optimization
parameters
$\bigl{\\{}\eta,\Theta,\bigl{\\{}\widehat{\Theta}_{i},i=1,\ldots,k\bigr{\\}}\bigr{\\}}$,
and can be easily coded in standard software packages such as cvx [GB00] or
YALMIP [Löf04]. Note that the optimization problem (3.5) is always feasible
(simply set $\Theta=0$ and $\eta=0$ to see this). This is not a surprise,
since there are no constraints on the state, and by construction
$0\in\mathbb{U}$. Finally, note that the third constraint in Problem (3.5) for
various values of $p$, is a result of robustly satisfying the constraints
posed by the various control sets (3.4) for any realization of the noise $w$.
In general, the total number of decision variables in the optimization program
(3.5) is $mN\bigl{(}1+\tfrac{1}{2}n(N-1)|\mathcal{I}|\bigr{)}$. The number of
decision variables can be substantially reduced, e.g., by choosing
$\mathcal{H}$ to be $1$-dimensional, or by fixing certain (block) elements of
the Fourier coefficient matrix $\Theta$ to $0$.
## 4\. Various Cases of Constrained Controls
We examine in this section several special cases of Theorem 3.2 under various
restrictions on the classes of noise and control inputs.
### 4.1. Bounded controls, unbounded noise, and $p=\infty$
Let the noise take values in $\mathbb{R}^{n}$. We provide tractable convex
programs to design a policy that by construction respects the control
constraint sets (3.4), with $p=\infty$. Starting from (3.1) let
(4.1) $u=\eta+\Theta\varphi(w),$
where
* •
$\varphi(w)\coloneqq\left[\begin{matrix}\varphi_{0}\\\ \varphi_{1}(w_{0})\\\
\vdots\\\ \varphi_{N-1}(w_{0},\ldots,{w}_{N-2})\end{matrix}\right]$,
* •
$\varphi_{0}=0$,
$\varphi_{t}(w_{0},\ldots,w_{t-1})=\sum_{j=0}^{t-1}\theta_{t}^{j}\varphi_{t,j}(w_{j})$
for $t=1,\ldots,N-2$, and
* •
$\varphi_{t,j}(w_{j})=\bigl{[}\widetilde{\varphi}(w_{j,1}),\ldots,\widetilde{\varphi}(w_{j,n})\bigr{]}^{\mathsf{T}}$
for some function $\widetilde{\varphi}$ such that
$\sup\limits_{s\in\mathbb{R}}\widetilde{\varphi}(s)=\phi_{\max}<\infty$, and
$\varphi_{t,j}:\mathbb{W}\to\mathbb{U}_{\infty}$.
In other words, we saturate the measurements that we obtain from the noise
input vector before inserting them into our control vector. This way we allow
that the noise distribution is supported over the entire $\mathbb{R}^{n}$,
which is an advantage over other approaches [BB07, GKM06]. Moreover, the
choice of the component saturation function $\widetilde{\varphi}$ is left open
as long as the noise sequence satisfies Assumption 3.1. For example, we can
accommodate standard saturation, piecewise linear, and sigmoidal functions to
name a few.
Our choice of saturating the measurement from the noise vectors, as we shall
see below, renders the resulting optimization problem tractable as opposed to
calculating the entire control input vector $u$ and then saturating it a
posteriori; one can see that the latter approach tends to lead to an
intractable optimization problem. Note also that the choice of control inputs
in (4.1) yields a possibly non-Markovian feedback.
###### Corollary 4.1.
Consider the system (2.1). Suppose that Assumption 3.1 holds, and
$\mathbb{E}[{\mathfrak{e}}(w)]=0$ with ${\mathfrak{e}}(w)=\varphi(w)$, where
$\varphi$ is defined in (4.1). Then for $p=\infty$ the problem (2.6) under the
control policy (4.1) is a convex optimization program with respect to the
decision variables $(\eta,\Theta)$, given by
(4.2) $\displaystyle\underset{(\eta,\Theta)}{\text{minimize}}$
$\displaystyle\mathbf{tr}\\!\left(\Theta^{\mathsf{T}}\bigl{(}R+B^{\mathsf{T}}QB\bigr{)}\Theta\Gamma_{1}\right)+2\mathbf{tr}\\!\left(DQB\Theta\Gamma_{2}\right)$
$\displaystyle+\eta^{\mathsf{T}}\bigl{(}R+B^{\mathsf{T}}QB\bigr{)}\eta+b^{\mathsf{T}}\eta+c$
subject to
$\displaystyle\max\limits_{i=1,\cdots,m}\left(|\eta_{t,i}|+\left\lVert{\Theta_{t,i}}\right\rVert_{1}\phi_{\rm
max}\right)\leqslant U_{\max}^{(\infty)},\quad t=0,\ldots,N-1,$
$\displaystyle\text{and $\Theta$ strictly lower block triangular as
in~{}\eqref{e:Thetadef}},$
where $\eta_{t,i}$ and $\Theta_{t,i}$ are the $i$-th rows of $\eta_{t}$ and
$\Theta_{t}$, respectively,
$\displaystyle c$
$\displaystyle=x_{0}^{\mathsf{T}}AQAx_{0}+\mathbf{tr}\\!\left(D^{\mathsf{T}}QD\Sigma_{\bar{w}}\right),$
$\displaystyle b$ $\displaystyle=2B^{\mathsf{T}}QAx_{0},$
$\displaystyle\Gamma_{1}$
$\displaystyle=\mathrm{diag}\bigl{\\{}\mathbb{E}\bigl{[}\varphi_{0}(w_{0})\varphi_{0}(w_{0})^{\mathsf{T}}\bigr{]},\cdots,\mathbb{E}\bigl{[}\varphi_{N-1}(w_{N-1})\varphi_{N-1}(w_{N-1})^{\mathsf{T}}\bigr{]}\bigr{\\}},$
$\displaystyle\Gamma_{2}$
$\displaystyle=\mathrm{diag}\bigl{\\{}\mathbb{E}\bigl{[}\varphi_{0}(w_{0})w_{0}^{\mathsf{T}}\bigr{]},\cdots,\mathbb{E}\bigl{[}\varphi_{N-1}(w_{N-1})w_{N-1}^{\mathsf{T}}\bigr{]}\bigr{\\}}.$
The resulting policy is guaranteed to satisfy the control constraint set (3.4)
for $p=\infty$.
A complete proof may be found in [HCL09]; it proceeds along the lines of the
proof of Theorem 3.2. Note that the program (4.2) exactly solves (2.6) under
the policy (4.1) and is neither a restriction nor a relaxation.
Problem (4.2) is a quadratic program in the optimization parameters
$(\eta,\Theta)$ (see the discussion following Theorem 3.2). The matrices
$\Gamma_{1}$ and $\Gamma_{2}$ capture the statistics of the noise in the
presence of the functions $\varphi$ and can be computed numerically _off-line_
using Monte Carlo techniques [RC04, Section 3.2]. This method will be utilized
in the examples in Section 6. However, in some instances it is actually
possible to compute these matrices in closed form; this is shown in the next
three examples.
###### Example 4.2.
Let us consider (2.1) when the noise process $(w_{t})_{t\in\mathbb{N}_{0}}$ is
an i.i.d sequence of Gaussian random vectors of mean $0$ and covariance
$\Sigma$ and standard sigmoidal policy functions $\widetilde{\varphi}$, i.e.,
$\widetilde{\varphi}(t)\coloneqq
t/\mathchoice{{\hbox{$\displaystyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\textstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\scriptstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=5.59444pt,depth=-4.47557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule height=4.36427pt,depth=-3.49144pt}}}$. Assume further that
the components of $w_{t}$ are mutually independent, which implies that
$\Sigma$ is a diagonal matrix
$\operatorname{diag}\\{\sigma_{1}^{2},\ldots,\sigma_{n}^{2}\\}$. Then from the
identities in Fact 1 in §A.1, we have for $i=1,\ldots,n$ and $j=0,\ldots,N-1$,
$\displaystyle\mathbb{E}\bigl{[}\widetilde{\varphi}(w_{j,i})^{2}\bigr{]}$
$\displaystyle=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{-\infty}^{\infty}\widetilde{\varphi}(t)^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t=2\cdot\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{0}^{\infty}\frac{t^{2}}{1+t^{2}}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}$
$\displaystyle=\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}-\pi\mathrm{e}^{-\frac{1}{2\sigma_{i}^{2}}}\operatorname{erfc}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\Bigr{)}.$
This shows that the matrix $\Gamma_{1}$ in Corollary 4.1 is
$\operatorname{diag}\\{\Sigma^{\prime},\ldots,\Sigma^{\prime}\\}$, where
$\Sigma^{\prime}\coloneqq\operatorname{diag}\left\\{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{1}-\pi\mathrm{e}^{-\frac{1}{2\sigma_{1}^{2}}}\operatorname{erfc}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{1}}\Bigr{)},\ldots,\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{n}-\pi\mathrm{e}^{-\frac{1}{2\sigma_{n}^{2}}}\operatorname{erfc}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{n}}\Bigr{)}\right\\}.$
Similarly, since
$\displaystyle\mathbb{E}\bigl{[}\widetilde{\varphi}(w_{j,i})w_{j,i}\bigr{]}$
$\displaystyle=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{-\infty}^{\infty}t\widetilde{\varphi}(t)\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t=2\cdot\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{-\infty}^{\infty}\frac{t^{2}}{\mathchoice{{\hbox{$\displaystyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\textstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\scriptstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=5.59444pt,depth=-4.47557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=4.36427pt,depth=-3.49144pt}}}}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}}}\mathrm{d}t$
$\displaystyle=\frac{\sigma_{i}}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma_{i}^{2}}\Bigr{)},$
where $U$ is the confluent hypergeometric function (defined in the Appendix),
the matrix $\Gamma_{2}$ in Corollary 4.1 is
$\operatorname{diag}\\{\Sigma^{\prime\prime},\ldots,\Sigma^{\prime\prime}\\}$,
where
$\Sigma^{\prime\prime}\coloneqq\operatorname{diag}\left\\{\frac{\sigma_{1}}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma_{1}^{2}}\Bigr{)},\ldots,\frac{\sigma_{n}}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma_{n}^{2}}\Bigr{)}\right\\}.$
Therefore, given the system (2.1), the control policy (4.6), and the
description of the noise input as above, the matrices $\Gamma_{1}$ and
$\Gamma_{2}$ derived above complete the set of hypotheses of Corollary 4.1.
The problem (2.4) can now be solved as the quadratic program (4.2).$\triangle$
###### Example 4.3.
Consider the setting of Example 4.2 (with $\widetilde{\varphi}$ a standard
sigmoid) under the assumption that $\Sigma$ is a not necessarily diagonal
matrix. To wit, the components of $w_{t}$ may be correlated at each time
$t\in\mathbb{N}_{0}$; however, the random vector sequence
$(w_{t})_{t\in\mathbb{N}_{0}}$ is assumed to be i.i.d. This is equivalent to
the knowledge of the correlations between the random variables
$\bigl{\\{}w_{t,i}\big{|}i=1,\ldots,n\bigr{\\}}$, which are constant over $t$.
Then $\mathbb{E}[\varphi(\bar{w})\varphi(\bar{w})^{\mathsf{T}}]$ is a block
diagonal matrix. Indeed, we have with
$\Sigma_{i,j}\coloneqq\begin{bmatrix}\sigma_{i}^{2}&\rho_{i,j}^{2}\\\
\rho_{i,j}^{2}&\sigma_{j}^{2}\end{bmatrix}$,
$\displaystyle\mathbb{E}\bigl{[}$
$\displaystyle\widetilde{\varphi}(w_{t,i})\widetilde{\varphi}(w_{t,j})\bigr{]}$
$\displaystyle=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\det{\Sigma_{i,j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\textstyle\sqrt{2\pi\det{\Sigma_{i,j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\det{\Sigma_{i,j}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.8611pt,depth=-3.8889pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\det{\Sigma_{i,j}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.47221pt,depth=-2.77779pt}}}}\iint_{\mathbb{R}^{2}}\frac{t_{1}t_{2}}{\mathchoice{{\hbox{$\displaystyle\sqrt{(1+t_{1}^{2})(1+t_{2}^{2})\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\textstyle\sqrt{(1+t_{1}^{2})(1+t_{2}^{2})\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\scriptstyle\sqrt{(1+t_{1}^{2})(1+t_{2}^{2})\,}$}\lower
0.4pt\hbox{\vrule
height=5.59444pt,depth=-4.47557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{(1+t_{1}^{2})(1+t_{2}^{2})\,}$}\lower
0.4pt\hbox{\vrule
height=4.36427pt,depth=-3.49144pt}}}}\exp\biggl{(}-\frac{1}{2}\begin{bmatrix}t_{1}&t_{2}\end{bmatrix}\Sigma_{i,j}^{-1}\begin{bmatrix}t_{1}\\\
t_{2}\end{bmatrix}\biggr{)}\;\mathrm{d}t_{1}\mathrm{d}t_{2},$
and
$\displaystyle\mathbb{E}\bigl{[}$
$\displaystyle\widetilde{\varphi}(w_{t,i})w_{t,j}\bigr{]}$
$\displaystyle=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\det\Sigma_{i,j}\,}$}\lower
0.4pt\hbox{\vrule
height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\textstyle\sqrt{2\pi\det\Sigma_{i,j}\,}$}\lower
0.4pt\hbox{\vrule
height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\det\Sigma_{i,j}\,}$}\lower
0.4pt\hbox{\vrule
height=4.8611pt,depth=-3.8889pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\det\Sigma_{i,j}\,}$}\lower
0.4pt\hbox{\vrule
height=3.47221pt,depth=-2.77779pt}}}}\iint_{\mathbb{R}^{2}}\frac{t_{1}t_{2}}{\mathchoice{{\hbox{$\displaystyle\sqrt{1+t_{1}^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\textstyle\sqrt{1+t_{1}^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\scriptstyle\sqrt{1+t_{1}^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=5.59444pt,depth=-4.47557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+t_{1}^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=4.36427pt,depth=-3.49144pt}}}}\exp\biggl{(}-\frac{1}{2}\begin{bmatrix}t_{1}&t_{2}\end{bmatrix}\Sigma_{i,j}^{-1}\begin{bmatrix}t_{1}\\\
t_{2}\end{bmatrix}\biggr{)}\;\mathrm{d}t_{1}\mathrm{d}t_{2}.$
Note that the computations of the integrals above can be carried out off-line.
We define the matrices $\Sigma_{t}$ and $\Sigma_{t}^{\prime}$ with the
$(i,j)$-th entry of $\Sigma_{t}$ being
$\mathbb{E}\bigl{[}\widetilde{\varphi}(w_{t,i})\widetilde{\varphi}(w_{t,j})\bigr{]}$
and the $(i,j)$-th entry of $\Sigma_{t}^{\prime}$ being
$\mathbb{E}\bigl{[}\widetilde{\varphi}(w_{t,i})w_{t,j}\bigr{]}$, and it
follows that the matrices
$\Gamma_{1}=\operatorname{diag}\bigl{\\{}\Sigma_{0},\ldots,\Sigma_{N-2}\bigr{\\}}$,
and
$\Gamma_{2}=\operatorname{diag}\bigl{\\{}\Sigma_{0}^{\prime},\ldots,\Sigma_{N-2}^{\prime}\bigr{\\}}$.
$\triangle$
###### Example 4.4.
Consider the system (2.1) as in Example 4.2, and with $\widetilde{\varphi}$
the standard saturation function defined as
$\widetilde{\varphi}(t)=\operatorname{sgn}(t)\min\\{|t|,1\\}$. From Corollary
4.1 we have for $i=1,\ldots,n$ and $j=0,\ldots,N-1$, using the identities in
Fact 1 in §A.1,
$\displaystyle\mathbb{E}\bigl{[}\widetilde{\varphi}(w_{j,i})^{2}\bigr{]}$
$\displaystyle=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{-\infty}^{\infty}\widetilde{\varphi}(t)^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t$
$\displaystyle=\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{0}^{1}t^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t+\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{1}^{\infty}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t$
$\displaystyle=\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}^{3}\operatorname{erf}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\Bigr{)}-2\sigma_{i}^{2}\mathrm{e}^{-\frac{1}{2\sigma_{i}^{2}}}+1+\operatorname{erf}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\Bigr{)}$
$\displaystyle\eqqcolon\xi_{i}^{\prime}\text{ (say)},$
and
$\displaystyle\mathbb{E}\bigl{[}\widetilde{\varphi}(w_{j,i})w_{j,i}\bigr{]}$
$\displaystyle=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{-\infty}^{\infty}t\widetilde{\varphi}(t)\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t$
$\displaystyle=\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{0}^{1}t^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t+\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{1}^{\infty}t\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t$
$\displaystyle=\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}^{3}\operatorname{erf}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\Bigr{)}-2\sigma_{i}^{2}\mathrm{e}^{-\frac{1}{2\sigma_{i}^{2}}}+\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{2}{\pi}\,}$}\lower
0.4pt\hbox{\vrule
height=8.59721pt,depth=-6.8778pt}}}{{\hbox{$\textstyle\sqrt{\frac{2}{\pi}\,}$}\lower
0.4pt\hbox{\vrule
height=6.01805pt,depth=-4.81447pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{2}{\pi}\,}$}\lower
0.4pt\hbox{\vrule
height=4.2986pt,depth=-3.4389pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{2}{\pi}\,}$}\lower
0.4pt\hbox{\vrule
height=4.2986pt,depth=-3.4389pt}}}\sigma_{i}\operatorname{Gamma}(2\sigma_{i}^{2},1)$
$\displaystyle\eqqcolon\xi_{i}^{\prime\prime}\text{ (say)}.$
Therefore, in this case the matrix $\Gamma_{1}$ in Corollary 4.1 is
$\operatorname{diag}\\{\Sigma^{\prime},\ldots,\Sigma^{\prime}\\}$ with
$\Sigma^{\prime}\coloneqq\operatorname{diag}\\{\xi_{1}^{\prime},\ldots,\xi_{n}^{\prime}\\}$,
and the matrix $\Gamma_{2}$ is
$\operatorname{diag}\\{\Sigma^{\prime\prime},\ldots,\Sigma^{\prime\prime}\\}$
with
$\Sigma^{\prime\prime}\coloneqq\operatorname{diag}\\{\xi_{1}^{\prime\prime},\ldots,\xi_{n}^{\prime\prime}\\}$.
These information complete the set of hypotheses of Corollary 4.1, and the
problem (2.4) can now be solved as a quadratic program (4.2).$\triangle$
### 4.2. Bounded controls, bounded noise, and $p=2$
In this subsection we specialize to the case of the noise being drawn from a
compact subset of $\mathbb{R}^{n}$, and the control inputs set
$\mathbb{U}_{2}$. We make the following assumption:
###### Assumption 4.5.
The noise takes values in a compact set
$\mathbb{W}\subseteq\mathbb{R}^{n}$.$\diamondsuit$
Under Assumption 4.5 Hilbert space techniques may be effectively employed in
our basic controller synthesis framework of Section 3 in the following way.
Let $(\mathcal{H},\left\langle{\cdot},{\cdot}\right\rangle_{\mathcal{H}})$ be
a separable Hilbert space of measurable maps
${\mathfrak{e}}:\mathbb{W}\to\mathbb{U}_{2}$ supported on the compact set
$\mathbb{W}$. The inner product is defined as
$\left\langle{\varphi_{1}},{\varphi_{2}}\right\rangle_{\mathcal{H}}\coloneqq\sum_{i=1}^{n}\left\langle{\varphi_{1,i}},{\varphi_{2,i}}\right\rangle$
where $\left\langle{\cdot},{\cdot}\right\rangle$ is the standard inner product
on real-valued functions on $\mathbb{W}$. Fix a complete orthonormal basis
$({\mathfrak{e}}^{\nu})_{\nu\in\mathcal{I}}\subseteq\mathcal{H}$. Since
$\mathcal{H}$ is separable, the set $\mathcal{I}$ is at most countable. Just
as in (3.1) we let our candidate control policies be of the form
(4.3) $u=\begin{bmatrix}\eta_{0}\\\ \eta_{1}\\\ \vdots\\\
\eta_{N-1}\end{bmatrix}+\begin{bmatrix}\mathbf{0}&\mathbf{0}&\cdots&\mathbf{0}\\\
\theta_{1,0}&\mathbf{0}&\cdots&\mathbf{0}\\\
\theta_{2,0}&\theta_{2,1}&\cdots&\mathbf{0}\\\ \vdots&\vdots&\ddots&\vdots\\\
\theta_{N-1,0}&\theta_{N-1,1}&\cdots&\theta_{N-1,N-2}\end{bmatrix}\begin{bmatrix}{\mathfrak{e}}(w_{0})\\\
{\mathfrak{e}}(w_{1})\\\ \vdots\\\
{\mathfrak{e}}(w_{N-2})\end{bmatrix}\eqqcolon\eta+\Theta{\mathfrak{e}}(w),$
where the vector ${\mathfrak{e}}(\cdot)$ is the formal vector formed by
concatenating the (ordered) basis elements
$({\mathfrak{e}}^{\nu})_{\nu\in\mathcal{I}}$, the various $\theta$-s are
formal matrices as in Section 3, and $\eta_{t}$ is an $m$-dimensional vector
for $t=0,\ldots,N-1$. This takes us back to the setting of Section 3.
The following Corollary illustrates the technique explained above; its proof
will only be sketched—it is similar to the proof of Theorem 3.2. Note that for
finite-dimensional Hilbert spaces, depending on the choice of the orthonormal
basis, the matrix $\Theta$ may have complex or real entries.
###### Corollary 4.6.
Consider the system (2.1). Suppose that Assumptions 3.1 and 4.5 hold. Then for
$p=2$ and corresponding control set $\mathbb{U}_{2}$ problem (2.6) under the
policy (4.3) admits convex tractable reformulation with tighter domains of the
decision variables $(\eta,\Theta)$ defined in (4.3), and is equivalent to the
following program:
(4.4) the minimization problem (3.5) $\displaystyle\text{subject
to}\quad\left\lVert{\eta_{t}}\right\rVert+\mathchoice{{\hbox{$\displaystyle\sqrt{N-1\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{N-1\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{N-1\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{N-1\,}$}\lower
0.4pt\hbox{\vrule
height=3.41666pt,depth=-2.73334pt}}}\left\lVert{\Theta_{t}}\right\rVert\leqslant
U_{\max}^{(2)},\quad\text{for }t=0,\ldots,N-1,$
$\displaystyle\qquad\qquad\qquad\text{and $\Theta$ strictly lower block
triangular as in~{}\eqref{e:Thetadef}}.$
Moreover, if $\hat{\mathcal{H}}$ is a finite-dimensional subspace of
$\mathcal{H}$ spanned by $({\mathfrak{e}}^{\nu})_{\nu\in\mathcal{J}}$ for some
finite $\mathcal{J}\subseteq\mathcal{I}$, then the problem (4.4) admits a
reformulation as a quadratically constrained quadratic program with respect to
the new decision variables $\bigl{(}\eta,\hat{\Theta}\bigr{)}$ corresponding
to $\hat{\mathcal{H}}$, given by
(4.5) the minimization problem (3.5) $\displaystyle\text{subject
to}\quad\left\lVert{\begin{bmatrix}\eta_{t}&\hat{\Theta}_{t}\end{bmatrix}}\right\rVert\leqslant
U_{\max}^{(2)}/\mathchoice{{\hbox{$\displaystyle\sqrt{N\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{N\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{N\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{N\,}$}\lower
0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}\quad\text{for
}t=0,\ldots,N-1,$ $\displaystyle\qquad\qquad\qquad\text{and $\Theta$ strictly
lower block triangular as in~{}\eqref{e:Thetadef}},$
where the vector ${\hat{\mathfrak{e}}}(\cdot)$ is the vector formed by
concatenating the (ordered) basis elements
$({\mathfrak{e}}^{\nu})_{\nu\in\mathcal{J}}$,
${\hat{\mathfrak{e}}}(w)\coloneqq\bigl{[}{\hat{\mathfrak{e}}}(w_{0})^{\mathsf{T}},\ldots,{\hat{\mathfrak{e}}}(w_{N-2})^{\mathsf{T}}\bigr{]}^{\mathsf{T}}$,
$\hat{\Sigma}_{{\mathfrak{e}}}\coloneqq\mathbb{E}\bigl{[}{\hat{\mathfrak{e}}}(w){\hat{\mathfrak{e}}}(w)^{\mathsf{T}}\bigr{]}$,
$\hat{\Sigma}_{{\mathfrak{e}}}^{\prime}\coloneqq\mathbb{E}\bigl{[}w{\hat{\mathfrak{e}}}(w)^{\mathsf{T}}\bigr{]}$.
In both the above cases the resulting policies are guaranteed to satisfy the
control constraint set (3.4) for $p=2$.
###### Proof.
(Sketch.) Evaluating the objective function in (2.6) gives the objective
function in (3.5). Recall that $\Theta_{t}$ is the $t$-th block row of the
formal matrix $\Theta$, and $\Theta_{t,i}$ is the $i$th sub-row of the block
row $\Theta_{t}$, where $t=0,\ldots,N-1$ and $i=1,\ldots,n$. Applying the
triangle inequality for any $t=0,\ldots,N-1$, we get
$\displaystyle\left\lVert{\eta_{t}+\Theta_{t}{\mathfrak{e}}(w)}\right\rVert$
$\displaystyle\leqslant\left\lVert{\eta_{t}}\right\rVert+\left\lVert{\Theta_{t}{\mathfrak{e}}(w)}\right\rVert=\left\lVert{\eta_{t}}\right\rVert+\mathchoice{{\hbox{$\displaystyle\sqrt{\left\langle{\Theta_{t}{\mathfrak{e}}(w)},{\Theta_{t}{\mathfrak{e}}(w)}\right\rangle_{\mathcal{H}}\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{\left\langle{\Theta_{t}{\mathfrak{e}}(w)},{\Theta_{t}{\mathfrak{e}}(w)}\right\rangle_{\mathcal{H}}\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{\left\langle{\Theta_{t}{\mathfrak{e}}(w)},{\Theta_{t}{\mathfrak{e}}(w)}\right\rangle_{\mathcal{H}}\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\left\langle{\Theta_{t}{\mathfrak{e}}(w)},{\Theta_{t}{\mathfrak{e}}(w)}\right\rangle_{\mathcal{H}}\,}$}\lower
0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}$
$\displaystyle=\left\lVert{\eta_{t}}\right\rVert+\mathchoice{{\hbox{$\displaystyle\sqrt{\sum_{i=1}^{n}\left\langle{\Theta_{t,i}{\mathfrak{e}}(w)},{\Theta_{t,i}{\mathfrak{e}}(w)}\right\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{\sum_{i=1}^{n}\left\langle{\Theta_{t,i}{\mathfrak{e}}(w)},{\Theta_{t,i}{\mathfrak{e}}(w)}\right\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{\sum_{i=1}^{n}\left\langle{\Theta_{t,i}{\mathfrak{e}}(w)},{\Theta_{t,i}{\mathfrak{e}}(w)}\right\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\sum_{i=1}^{n}\left\langle{\Theta_{t,i}{\mathfrak{e}}(w)},{\Theta_{t,i}{\mathfrak{e}}(w)}\right\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}=\left\lVert{\eta_{t}}\right\rVert+\mathchoice{{\hbox{$\displaystyle\sqrt{(N-1)\sum_{i=1}^{n}\left\lVert{\Theta_{t,i}}\right\rVert^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=9.30444pt,depth=-7.44359pt}}}{{\hbox{$\textstyle\sqrt{(N-1)\sum_{i=1}^{n}\left\lVert{\Theta_{t,i}}\right\rVert^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=9.30444pt,depth=-7.44359pt}}}{{\hbox{$\scriptstyle\sqrt{(N-1)\sum_{i=1}^{n}\left\lVert{\Theta_{t,i}}\right\rVert^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=8.78888pt,depth=-7.03114pt}}}{{\hbox{$\scriptscriptstyle\sqrt{(N-1)\sum_{i=1}^{n}\left\lVert{\Theta_{t,i}}\right\rVert^{2}\,}$}\lower
0.4pt\hbox{\vrule height=8.78888pt,depth=-7.03114pt}}}$
$\displaystyle=\left\lVert{\eta_{t}}\right\rVert+\mathchoice{{\hbox{$\displaystyle\sqrt{N-1\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{N-1\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{N-1\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{N-1\,}$}\lower
0.4pt\hbox{\vrule
height=3.41666pt,depth=-2.73334pt}}}\left\lVert{\Theta_{t}}\right\rVert$
by orthogonality of the basis elements
$({\mathfrak{e}}^{\nu})_{\nu\in\mathcal{I}}$. The right-hand side of the last
equality appears as the constraint in (4.4).
For the finite-dimensional case (4.5), we note that the objective function is
identical to the one in (4.4), and the constraint in (4.5) follows from the
fact that
$\left\lVert{\eta_{t}+\hat{\Theta}_{t}{\hat{\mathfrak{e}}}(w)}\right\rVert=\left\lVert{\begin{bmatrix}\eta_{t}&\hat{\Theta}_{t}\end{bmatrix}\begin{bmatrix}1\\\
{\hat{\mathfrak{e}}}(w)\end{bmatrix}}\right\rVert$, and
$\left\lVert{\begin{bmatrix}1\\\
{\hat{\mathfrak{e}}}(w)\end{bmatrix}}\right\rVert=\mathchoice{{\hbox{$\displaystyle\sqrt{1+\sum_{i=0}^{N-2}\left\langle{{\hat{\mathfrak{e}}}(w_{i})},{{\hat{\mathfrak{e}}}(w_{i})}\right\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{1+\sum_{i=0}^{N-2}\left\langle{{\hat{\mathfrak{e}}}(w_{i})},{{\hat{\mathfrak{e}}}(w_{i})}\right\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{1+\sum_{i=0}^{N-2}\left\langle{{\hat{\mathfrak{e}}}(w_{i})},{{\hat{\mathfrak{e}}}(w_{i})}\right\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+\sum_{i=0}^{N-2}\left\langle{{\hat{\mathfrak{e}}}(w_{i})},{{\hat{\mathfrak{e}}}(w_{i})}\right\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}=\mathchoice{{\hbox{$\displaystyle\sqrt{N\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{N\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{N\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{N\,}$}\lower
0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}$. This leads to a
quadratically constrained quadratic program in the finite- dimensional
decision variables $\bigl{(}\eta,\hat{\Theta}\bigr{)}$. ∎
Let us illustrate the usage of Corollary 4.6 through the following example.
###### Example 4.7.
Consider the system (2.1), and suppose that the $n$ components of the noise
vector $w_{t}$ are independent uniform random variables taking values in
$[-a,a]$ for some $a>1$. Therefore, $\mathbb{W}=[-a,a]^{n}$. It is a standard
fact in Fourier analysis that the system
$\bigl{\\{}\mathrm{e}^{2\pi\mathrm{i}\nu(t/(2a))}\,\big{|}\,\nu\in\mathbb{Z}\bigr{\\}}$
is an orthonormal basis for the Hilbert space of square-integrable functions
on $[-a,a]$ equipped with the standard inner product
$\left\langle{f},{g}\right\rangle\coloneqq\frac{1}{2a}\int_{-a}^{a}f(t)g(t){\mathrm{d}t}$.
We let
$\displaystyle\hat{\mathcal{H}}$
$\displaystyle\coloneqq\operatorname{span}\Biggl{\\{}\biggl{[}\frac{\sin(\pi\nu
t_{1}/a)}{\mathchoice{{\hbox{$\displaystyle\sqrt{n\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{n\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{n\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{n\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}},\ldots,\frac{\sin(\pi\nu
t_{n}/2)}{\mathchoice{{\hbox{$\displaystyle\sqrt{n\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{n\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{n\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{n\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}}\biggr{]}^{\mathsf{T}}\,\Bigg{|}\,t_{i}\in[-a,a],i=1,\ldots,n,\nu=1,\ldots,M\Biggr{\\}}.$
Let
${\mathfrak{e}}^{\nu}(t_{1},\ldots,t_{n})\coloneqq\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{2}{n}\,}$}\lower
0.4pt\hbox{\vrule
height=8.59721pt,depth=-6.8778pt}}}{{\hbox{$\textstyle\sqrt{\frac{2}{n}\,}$}\lower
0.4pt\hbox{\vrule
height=6.01805pt,depth=-4.81447pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{2}{n}\,}$}\lower
0.4pt\hbox{\vrule
height=4.2986pt,depth=-3.4389pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{2}{n}\,}$}\lower
0.4pt\hbox{\vrule height=4.2986pt,depth=-3.4389pt}}}\bigl{[}\sin(\pi\nu
t_{1}/a),\ldots,\sin(\pi\nu t_{n}/a)\bigr{]}^{\mathsf{T}}$, $t_{i}\in[-a,a]$.
It is clear that the $\mathbb{R}^{n}$-valued functions
$\bigl{\\{}{\mathfrak{e}}^{\nu},\;\nu=1,\ldots,M\bigr{\\}}$ form an
orthonormal set. Indeed,
$\displaystyle\left\langle{{\mathfrak{e}}_{\nu_{1}}},{{\mathfrak{e}}_{\nu_{2}}}\right\rangle_{\hat{\mathcal{H}}}$
$\displaystyle=\sum_{i=1}^{n}\left\langle{{\mathfrak{e}}_{\nu_{1},i}},{{\mathfrak{e}}_{\nu_{2},i}}\right\rangle=\frac{2}{n}\sum_{i=1}^{n}\frac{1}{2a}\int_{-a}^{a}\sin(\pi\nu_{1}t_{i}/a)\sin(\pi\nu_{2}t_{i}/a)\mathrm{d}t_{i}$
$\displaystyle=\frac{2}{n}\sum_{i=1}^{n}\frac{1}{4}\int_{-1}^{1}\bigl{(}\cos((\nu_{1}-\nu_{2})\pi
s_{i})-\cos((\nu_{1}+\nu_{2})\pi s_{i})\bigr{)}\mathrm{d}s_{i}$
$\displaystyle=\begin{cases}\frac{1}{2n}\sum_{i=1}^{n}2=1&\text{if
}\nu_{1}=\nu_{2},\\\ 0&\text{otherwise}.\end{cases}$
We define
$u_{t}\coloneqq\eta_{t}+\Theta_{t}{\mathfrak{e}}(w)=\eta_{t}+\sum_{j=0}^{t-1}\theta_{t,j}{\mathfrak{e}}(w_{j})=\eta_{t}+\sum_{j=0}^{t-1}\sum_{\nu=1}^{M}\theta_{t,j}^{\nu}{\mathfrak{e}}^{\nu}(w_{j})$
for appropriate matrices $\theta_{t,j}^{\nu}$. Now finding policies of the
form (4.3) that minimize the objective function in (2.6) becomes
straightforward in the setting of Corollary 4.6. The matrices
$\Sigma_{{\mathfrak{e}}}$ and $\Sigma_{{\mathfrak{e}}}^{\prime}$ in Corollary
4.6 are now easy to derive from Euler’s identity
$\mathrm{e}^{\mathrm{i}\theta}=\cos\theta+\mathrm{i}\sin\theta$, and the fact
that the characteristic function of a uniform random variable $\zeta$
supported on $[-a,a]$ is given by
$\mathbb{E}\bigl{[}\mathrm{e}^{2\pi\mathrm{i}v\zeta}\bigr{]}=\frac{1}{2a}\int_{-a}^{a}\mathrm{e}^{2\pi\mathrm{i}vt}\,\mathrm{d}t=\operatorname{sinc}(2\pi
va)$ for some $v\in\mathbb{R}$, where the function $\operatorname{sinc}$ is
defined as $\operatorname{sinc}(\xi)\coloneqq\sin(\xi)/\xi$ if $\xi\neq 0$ and
$1$ otherwise.
An alternative representation of the various matrices may be obtained by
looking at each component of the policy elements separately. In this approach
we define
$\displaystyle{\hat{\mathfrak{e}}}(w_{t,i})$
$\displaystyle\coloneqq\begin{bmatrix}{\mathfrak{e}}_{0}(w_{t,i})&{\mathfrak{e}}_{1}(w_{t,i})&\cdots&{\mathfrak{e}}_{M}(w_{t,i})\end{bmatrix}^{\mathsf{T}},$
$\displaystyle{\hat{\mathfrak{e}}}(w_{t})$
$\displaystyle\coloneqq\begin{bmatrix}{\hat{\mathfrak{e}}}(w_{t,1})^{\mathsf{T}}&\cdots&{\hat{\mathfrak{e}}}(w_{t,n})^{\mathsf{T}}\end{bmatrix}^{\mathsf{T}},\quad{\hat{\mathfrak{e}}}(w)\coloneqq\begin{bmatrix}{\hat{\mathfrak{e}}}(w_{0})^{\mathsf{T}}&\cdots&{\hat{\mathfrak{e}}}(w_{N-2})^{\mathsf{T}}\end{bmatrix}^{\mathsf{T}}.$
In the above notation
$\eta_{t,i}+\sum_{j=0}^{t-1}\theta_{j,i}{\hat{\mathfrak{e}}}(w_{j,i})$ is of
course the $i$-th entry of the input $u_{t}$ at time $t$, where
$t=0,\ldots,N-1$ and $i=1,\ldots,n$. $\triangle$
### 4.3. Constraints on control energy
Some applications require constraints on the total control energy expended
over a finite horizon. In the framework that we have established so far, such
constraints are easy to incorporate. Indeed, if we require that
$u^{\mathsf{T}}Su\leqslant\beta^{2}$ for some preassigned $\beta>0$ and
positive definite matrix $S$, then in the setting of Theorem 3.2 this can be
ensured by adjoining the condition
$\left\lVert{\eta}\right\rVert_{S}+\left\lVert{\Theta}\right\rVert_{S}\left\lVert{S}\right\rVert_{\infty}\mathcal{E}\leqslant\beta$
to the constraints, where
$\left\lVert{\eta}\right\rVert_{M}\coloneqq\mathchoice{{\hbox{$\displaystyle\sqrt{\eta^{\mathsf{T}}M\eta\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\eta^{\mathsf{T}}M\eta\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\eta^{\mathsf{T}}M\eta\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\eta^{\mathsf{T}}M\eta\,}$}\lower
0.4pt\hbox{\vrule height=3.51944pt,depth=-2.81557pt}}}$ is the standard
weighted $2$-norm for a positive definite matrix $M$.
### Comparison with affine policies
As pointed out earlier affine feedback policies from the noise have been
previously treated in [Löf03, BTGGN04, GKM06, GK08], where the following
feedback policy was considered:
(4.6) $u_{t}=\sum_{i=0}^{t-1}\theta_{t,i}w_{i}+\eta_{t}.$
In the deterministic setting it was shown in [GKM06] that there exists a one-
to-one (nonlinear) mapping between control policies in the form (4.6) and the
class of affine state feedback policies. That is, provided one is interested
in affine state feedback policies, the parametrization (4.6) constitutes no
loss of generality. In fact, we shall illustrate in the examples, in the
unconstrained inputs case, that the performance of this strategy with
${\mathfrak{e}}(w_{i})$ in place of $w_{i}$ is almost as good as the standard
LQG controller if not equally good. However, in the constrained inputs case
this choice is suboptimal in the class of measurable control policies, but it
ensures tractability of a large class of optimal control problems. It can be
seen that the solution to the optimization problem (2.4) is tractable with
this parametrization [GKM06]. However, if the elements of the noise vector $w$
are unbounded, the control input (4.6) does not have an upper bound. For the
case of bounded inputs, the control policy (4.6) under unbounded noise will in
general not satisfy the control constraint sets (3.4). This unboundedness is a
potential problem in practical applications, and has been usually circumvented
by assuming that the noise input lies within a compact set [BB07, GKM06] and
designing a worst-case min-max type controller under this assumption.
It is important to point out that our result in Section 4.2 differs from that
in [GKM06] in two aspects. First, we are solving the problem on finite-
dimensional Hilbert spaces with general basis functions as opposed to a finite
collection of affine functions in [GKM06]. Second, the feasibility of our
problem is maintained for any bound on the elements of $\mathbb{W}$, as our
constraint in (4.5) could still produce a feedback gain matrix $\Theta$ that
has norm substantially different that $0$, whereas if there are elements in
$\mathbb{W}$ with large enough norm and we take the control input to be
$u=\eta+\Theta w$, the constraints produce always a solution $\Theta$ with
norm very close to $0$, hence practically only the open-loop term remains in
the case of [GKM06].
## 5\. Stability Analysis
The main result in Theorem 3.2 asserts that the finite horizon optimization
problem (2.6) is convex and tractable using the policy (3.1). To apply this
result in a receding horizon fashion, it is imperative to further study some
qualitative stability properties of the proposed policy. Under this policy the
closed-loop system is not necessarily Markovian, and as such, standard Foster-
Lyapunov methods cannot be directly applied. In what follows, we treat the
stability problem for $p=\infty$ and $U_{\max}\coloneqq U_{\max}^{(\infty)}$.
However, this is without any loss of generality, for the same results hold
(with minor modifications in the proofs) for $p=1,2$ as well. We impose the
following assumption:
###### Assumption 5.1.
The matrix $A$ in (2.1) is Schur stable, i.e., the absolute value of the
eigenvalues of $A$ are all strictly less than $1$.$\diamondsuit$
At a first glance this assumption on $A$ might seem restrictive. Indeed, in
the deterministic setting we know [YSS97] that for discrete-time controlled
systems it is possible to achieve global asymptotic stability with bounded
control inputs if and only if the pair $(A,B)$ is stabilizable with arbitrary
controls, and the spectral radius of $A$ is at most $1$. However, the problem
of ensuring bounded variance of linear stochastic systems with bounded control
inputs is to our knowledge still largely open; see, however, the recent
manuscript [RCMA+09] for partial results as well as in [BSW02, SSW06].
### 5.1. Mean-square boundedness
We shall show that the variance of the state is uniformly bounded under
receding horizon application of the strategy (3.1), for any control horizon
$N_{c}\leqslant N$. The receding horizon implementation is iterative in
nature: the optimization problem is solved every $kN_{c}$ steps, where
$k\in\mathbb{N}_{0}$. The resulting optimal control policy (applied over a
horizon $N_{c}$) is given by
$\pi_{kN_{c}:(k+1)N_{c}-1}^{*}(x_{kN_{c}})\coloneqq\left[\begin{matrix}\pi^{*}_{kN_{c}}(x_{kN_{c}})\\\
\pi^{*}_{kN_{c}+1}(x_{kN_{c}})\\\ \vdots\\\
\pi^{*}_{(k+1)N_{c}-1}(x_{kN_{c}})\end{matrix}\right]=\left[\begin{matrix}\eta_{0}^{*}(x_{kN_{c}})\\\
\eta_{1}^{*}(x_{kN_{c}})+\Theta_{1}^{*}(x_{kN_{c}}){\mathfrak{e}}(w)\\\
\vdots\\\
\eta_{N_{c}-1}^{*}(x_{kN_{c}})+\Theta^{*}_{N_{c}-1}(x_{kN_{c}}){\mathfrak{e}}(w)\end{matrix}\right]$
where the control gains depend explicitly on the initial condition
$x_{kN_{c}}$. For $\ell=1,\cdots,N_{c}$, the resulting closed-loop system over
horizon $N_{c}$ is given by:
(5.1)
$x_{kN_{c}+\ell}=A^{\ell}x_{kN_{c}}+B_{\ell}\pi_{kN_{c}:kN_{c}+\ell-1}^{*}(x_{kN_{c}})+D_{\ell}\tilde{w}_{kN_{c}:kN_{c}+\ell-1},\qquad
k\in\mathbb{N}_{0},$
where
$B_{\ell}\coloneqq\left[\begin{matrix}{\bar{A}}^{\ell-1}\bar{B}&\cdots&\bar{A}\bar{B}&\bar{B}\end{matrix}\right]$,
$D_{\ell}\coloneqq\left[\begin{matrix}{\bar{A}}^{\ell-1}&\cdots&\bar{A}&\mathbf{I}_{n\times
n}\end{matrix}\right]$, and
$\tilde{w}_{kN_{c}:kN_{c}+\ell-1}\coloneqq\left[\begin{matrix}w_{kN_{c}}^{\mathsf{T}}&\cdots&w_{kN_{c}+\ell-1}^{\mathsf{T}}\end{matrix}\right]^{\mathsf{T}}$.
Suppose that the above $N_{c}$-horizon optimal policy is computed as in
Corollary 4.1. We define the receding horizon policy corresponding to the
consecutive concatenation of this $N_{c}$-horizon optimal policy as
(5.2)
$\pi^{*}\coloneqq\bigl{(}\pi_{0:N_{c}-1}^{*}(x_{0}),\;\pi_{N_{c}:2N_{c}-1}^{*}(x_{N_{c}}),\;\pi_{2N_{c}:3N_{c}-1}^{*}(x_{2N_{c}}),\cdots\bigr{)}.$
###### Proposition 5.2.
Consider the system (2.1), and suppose that Assumptions 3.1 and 5.1 hold. For
$p=\infty$ and any control horizon $1\leqslant N_{c}\leqslant N$ the receding
horizon control policy $\pi^{*}$ renders the closed loop system (5.1) mean-
square bounded, i.e.,
$\sup_{t\in\mathbb{N}_{0}}\mathbb{E}_{x_{0}}\bigl{[}\left\lVert{x_{t}}\right\rVert^{2}\bigr{]}<\infty$
for every initial condition $x_{0}\in\mathbb{R}^{n}$.
The proof of this Proposition is postponed to §A.2 in the Appendix.
### 5.2. Input-to-state stability
Input-to-state stability (iss) is an interesting and important qualitative
property of input-output behavior of dynamical systems. In the deterministic
discrete-time setting [JW01], iss generalizes the well-known bounded-input
bounded-output (BIBO) property of linear systems [AM06, p. 490] to the setting
of nonlinear systems. iss provides a description of the behavior of a system
subjected to bounded inputs, and as such it may be viewed as an
$\mathcal{L}_{\infty}$ to $\mathcal{L}_{\infty}$ gain of a given nonlinear
system. In this section we are interested in a useful stochastic variant of
input-to-state stability; see e.g., [Bor00, ST03] for other possible
definitions and ideas (primarily in continuous-time).
###### Definition 5.3.
The system (2.1) is _input-to-state stable in $\mathcal{L}_{1}$_ if there
exist functions $\beta\in\mathcal{KL}$ and
$\alpha,\gamma_{1},\gamma_{2}\in\mathcal{K}_{\infty}$ such that for every
initial condition $x_{0}\in\mathbb{R}^{n}$ we have
(5.3)
$\mathbb{E}_{x_{0}}\bigl{[}\alpha(\left\lVert{x_{t}}\right\rVert)\bigr{]}\leqslant\beta(\left\lVert{x_{0}}\right\rVert,t)+\gamma_{1}\Bigl{(}\sup_{s\in\mathbb{N}_{0}}\left\lVert{u_{s}}\right\rVert_{\infty}\Bigr{)}+\gamma_{2}\bigl{(}\left\lVert{\Sigma}\right\rVert^{\prime}\bigr{)}\qquad\forall\,t\in\mathbb{N}_{0},$
where $\left\lVert{\cdot}\right\rVert^{\prime}$ is an appropriate matrix
norm.$\Diamond$
One evident difference of iss in $\mathcal{L}_{1}$ with the deterministic
definition of iss is the presence of the function $\alpha$ inside the
expectation in (5.3). It turns out that often it is more natural to arrive at
an estimate of $\mathbb{E}_{x_{0}}[\alpha(\left\lVert{x_{t}}\right\rVert)]$
for some $\alpha\in\mathcal{K}_{\infty}$ than an estimate of
$\mathbb{E}_{x_{0}}[\left\lVert{x_{t}}\right\rVert]$. Moreover, in case
$\alpha$ is convex, Jensen’s inequality [Dud02, p. 348] implies that such an
estimate is stronger than an estimate of
$\mathbb{E}_{x_{0}}[\left\lVert{x_{t}}\right\rVert]$.
The property expressed by (5.3) is one possible iss-type property for
stochastic systems. One can come up with alternative stochastic analogs of the
iss property, such as the following: $\forall\,\varepsilon\in\;]0,1[$
$\exists\,\beta\in\mathcal{KL}$ and
$\exists\,\gamma_{1},\gamma_{2}\in\mathcal{K}_{\infty}$ such that
$\mathbb{P}\bigl{(}\left\lVert{x_{t}}\right\rVert\leqslant\beta(\left\lVert{x_{0}}\right\rVert,t)+\gamma(\sup_{s\in\mathbb{N}_{0}}\left\lVert{u_{s}}\right\rVert)+\gamma_{2}(\left\lVert{\Sigma}\right\rVert^{\prime})\;\forall\,t\in\mathbb{N}_{0}\bigr{)}\geqslant
1-\varepsilon$. Intuitively this means that for $1-\varepsilon$ proportion of
the sample paths the deterministic iss property holds uniformly. However, in
an additive i.i.d unbounded noise setting as in (2.1), this property fails to
hold because almost surely the states undergo excursions outside any bounded
set infinitely often; in this case the weaker version:
$\forall\,\varepsilon\in\;]0,1[$ $\exists\,\beta\in\mathcal{KL}$ and
$\exists\,\gamma_{1},\gamma_{2}\in\mathcal{K}_{\infty}$ such that
$\mathbb{P}\bigl{(}\left\lVert{x_{t}}\right\rVert\leqslant\beta(\left\lVert{x_{0}}\right\rVert,t)+\gamma(\sup_{s\in\mathbb{N}_{0}}\left\lVert{u_{s}}\right\rVert)+\gamma_{2}(\left\lVert{\Sigma}\right\rVert^{\prime})\bigr{)}\geqslant
1-\varepsilon\;\forall\,t\in\mathbb{N}_{0}$ is comparatively better suited. We
shall however stick with the iss in $\mathcal{L}_{1}$ property in this
article.
The following Proposition can be established with the aid of Proposition 5.2
for $p=\infty$; the proofs for $p=1$ and $2$ are also similar in spirit.
###### Proposition 5.4.
Consider the system (2.1), and suppose that Assumptions 3.1 and 5.1 hold. Then
the closed-loop system (5.1) is iss in $\mathcal{L}_{1}$ under the policy
$\pi^{*}$ in (5.2) for any $1\leqslant N_{c}\leqslant N$.
## 6\. Numerical Examples
In this section we present several numerical examples to illustrate the
theoretical results in the preceding sections. We start in Example 6.1 by
comparing the performance of our policy (3.3) to that of the standard finite
horizon LQG controller whenever the control inputs set
$\bar{\mathbb{U}}\equiv\mathbb{R}^{m}$, i.e., there are no bounds on the norm
of the inputs. Then we compare the performance of our policy (3.3) against a
saturated LQG controller in Example 6.2. Finally, in Example 6.3 we illustrate
the effectiveness of our policy (3.3) compared to the certainty-equivalent
receding horizon control.
###### Example 6.1 (Unconstrained Inputs).
A natural question that may arise whenever the control inputs in our setup are
not constrained, i.e., $\bar{\mathbb{U}}\equiv\mathbb{R}^{m}$, is the
following: How does the policy (3.3) compare to the globally optimal
controller, which in this case is the standard finite-horizon LQG controller?
One would expect our policy to perform worse on the average since we restrict
to a class of feedback policies that may not contain the globally optimal
controller.
We compared our policy against that of the LQG problem in simulation for two
controllable $3$-dimensional single-input linear systems. In each case we
solved an unconstrained finite-horizon LQ optimal control problem
corresponding to state and control weights $Q_{t}=3\,\mathbf{I}_{3\times 3}$
and $R_{t}=1$ for every $t$. We selected an optimization horizon $N=50$, and
simulated the system responses starting from $10^{3}$ different initial
conditions $x_{0}$ selected at random uniformly from the cube
$[-100,100]^{3}$, and noise sequences $w_{t}$ corresponding to i.i.d Gaussian
noise of mean $0$ and (randomly chosen) variance
$\Sigma_{w}=\begin{bmatrix}2.830399255&5.491512606&3.612257417\\\
5.491512606&11.554870229&6.896706270\\\
3.612257417&6.896706270&4.625993264\end{bmatrix}.$
We selected the nonlinear bounded term ${\mathfrak{e}}(w)$ in our policy
$u=\eta+\Theta{\mathfrak{e}}(w)$ to be a vector of scalar sigmoidal functions
$\varphi(\xi)\coloneqq
0.2\xi/\mathchoice{{\hbox{$\displaystyle\sqrt{1+0.04\xi^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\textstyle\sqrt{1+0.04\xi^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\scriptstyle\sqrt{1+0.04\xi^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=6.14998pt,depth=-4.92001pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+0.04\xi^{2}\,}$}\lower
0.4pt\hbox{\vrule height=4.7611pt,depth=-3.8089pt}}}$ applied to each
coordinate of the vector $w$. The covariance matrices
$\Sigma_{{\mathfrak{e}}}$ and $\Sigma_{{\mathfrak{e}}^{\prime}}$ that are
required to solve the optimization problem (3.3) were computed empirically via
classical Monte Carlo methods [RC04, Section 3.2] using $10^{6}$ i.i.d
samples.
The first system is described by:
(6.1) $x_{k+1}=\begin{bmatrix}0&1&0\\\ 0&0&1\\\
0.4&0.5&-0.25\end{bmatrix}x_{k}+\begin{bmatrix}0\\\ 0\\\
1\end{bmatrix}u_{k}+w_{k}.$
The system pair $(A,B)$ is in Brunovsky canonical form, and $A$ has
eigenvalues at $0.8642$, and $-0.5571\pm\mathrm{i}0.3905$. The test results
showed that the mean of the ratio of the cost corresponding to LQG to the cost
corresponding to our policy is $0.99916$, and the standard deviation of this
ratio is $0.003619$.
The second system is described by:
(6.2) $x_{k+1}=\begin{bmatrix}1&1&0\\\ 0&1&1\\\
0&0&1\end{bmatrix}x_{k}+\begin{bmatrix}0\\\ 0\\\ 1\end{bmatrix}u_{k}+w_{k}.$
This particular system matrix $A$ is in Jordan canonical form and has three
eigenvalues at $1$. The test results showed that the mean of the ratio of the
cost of LQG against the cost of our policy is $0.99673$ and the corresponding
standard deviation is $0.008045$.
Computations for determining our policy in the above two cases were carried
out in the MATLAB-based software package cvx. In the case of the system (6.2)
the solver utilized by cvx reported numerical problems in five different runs,
for which it gave values of the aforementioned ratio below $0.96$. Note that
we have not discarded these five cases from the mean and variance figures
reported above.
The close-to-optimal performance of our policy is surprising in view of the
fact that the vector-space $\mathcal{H}$ is the linear span of one bounded
function, and does not contain the theoretically optimal linear (in the
current state) controller. We conjecture that this is due to injectivity of
the mapping ${\mathfrak{e}}$, due to which ${\mathfrak{e}}(w_{t})$ retains all
information generated by $w_{t}$. Of course, in the absence of control
constraints our solution is much more computationally demanding than the LQG
controller, and would not be used in practice in this case.$\triangle$
###### Example 6.2 (Saturated LQG and Receding Horizon).
We compare the performance of saturated LQG against our policy (3.3) for the
system (6.2) in this example. We fixed the optimization horizon $N=2$, the
control horizon $N_{c}=1$, and the weight matrices for the states and the
control to be $Q_{t}=\mathbf{I}_{3\times 3}$ and $R_{t}=0.01$ for all $t$,
respectively. The control bounds in both cases was $[-2,2]$, the nonlinear
bounded term ${\mathfrak{e}}(w_{t})$ in our policy
$u=\eta+\Theta{\mathfrak{e}}(w)$ was a vector of scalar standard saturation
functions applied to each coordinate of the vector $w_{t}$, and the LQG
control input was saturated at $\pm 2$. The covariance matrices
$\Sigma_{{\mathfrak{e}}}$ and $\Sigma_{{\mathfrak{e}}^{\prime}}$ required to
solve the optimization problem (3.5) were computed empirically via classical
Monte Carlo integration methods [RC04, Section 3.2] using $10^{6}$ i.i.d
samples.
We simulated the system (6.2) starting from the same initial condition
$x_{0}=\left[\begin{matrix}0&0&0\end{matrix}\right]^{\mathsf{T}}$ for $100$
different independent realizations of the noise sequence $w_{t}$ over a
horizon of $200$. The behavior of the average (over the $100$ realizations)
cost corresponding to the two scenarios is shown in Figure 1. The simulations
were coded in MATLAB and the optimization programs were coded in the software
package cvx. The average total cost incurred at the end of the simulation
horizon when using the saturated LQG scheme above was $1.790\times 10^{12}$
units, whereas the average total cost incurred at the end of the simulation
horizon ($t=200$) using our policy (3.3) in a receding horizon fashion was
$4.486\times 10^{8}$ units.$\triangle$
Figure 1. Plots of average costs corresponding to saturated LQG and our
receding horizon scheme for $N_{c}=1$ in Example 6.2.
###### Example 6.3 (Constrained Inputs).
Consider the 2-dimensional linear stochastic system:
(6.3) $x_{t+1}=\begin{bmatrix}1.23&-0.15\\\
0.25&1\end{bmatrix}x_{t}+\begin{bmatrix}0.14\\\ 0.12\end{bmatrix}u_{t}+w_{t},$
where $(w_{t})_{t\in\mathbb{N}_{0}}$ is a sequence of i.i.d Gaussian noise
with zero mean and (randomly generated) variance
$\begin{bmatrix}2.722030613&4.975999693\\\
4.975999693&9.102559685\end{bmatrix}$. Let the weight matrices corresponding
to the states and control be $Q_{t}=\mathbf{I}_{2\times 2}$ and $R_{t}=0.8$
for each $t$. The covariance matrices $\Sigma_{{\mathfrak{e}}}$ and
$\Sigma_{{\mathfrak{e}}^{\prime}}$ that are required to solve the optimization
problem (3.3) were computed empirically via classical Monte Carlo integration
methods [RC04, Section 3.2] using $10^{6}$ samples.
We fixed the optimization horizon $N=7$, the nonlinear saturation
${\mathfrak{e}}(w_{t})$ to be a vector of scalar sigmoidal functions
$\varphi(\xi)\coloneqq
0.2\xi/\mathchoice{{\hbox{$\displaystyle\sqrt{1+0.04\xi^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\textstyle\sqrt{1+0.04\xi^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\scriptstyle\sqrt{1+0.04\xi^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=6.14998pt,depth=-4.92001pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+0.04\xi^{2}\,}$}\lower
0.4pt\hbox{\vrule height=4.7611pt,depth=-3.8089pt}}}$ applied to each
coordinate of the vector $w_{t}$, and compared the certainty-equivalent MPC
strategy ($N_{c}=1$, $\Theta\equiv 0$, $w_{t}\equiv 0$) against our receding
horizon strategy (3.3) with control horizon $N_{c}=4$. The control constraints
in both cases were $u_{t}\in[-200,200]$. We simulated the system in both cases
starting from the same initial condition
$x_{0}=\left[\begin{matrix}0&0\end{matrix}\right]^{\mathsf{T}}$, for $60$
different realizations of the noise sequence $w_{t}$; plots of states, average
cost, and standard deviation are shown in Figures 2 and 3. The average cost
incurred when using the certainty-equivalent MPC scheme was $7.893\times
10^{5}$ units, whereas the average cost incurred when using our policy (3.3)
in a receding horizon fashion was $3.141\times 10^{5}$ units. Therefore,
applying our policy in a receding horizon fashion one saves $60.2\%$ of the
cost corresponding to the certainty-equivalent MPC controller on the average.
This example illustrates that there may be cases where open-loop certainty-
equivalent MPC, in the absence of state-constraints, is outperformed by a
large margin by a judiciously selected receding-horizon strategy. The
simulations were coded in YALMIP and were solved using SDPT-3; the solver-time
statistics (in sec.) for the certainty-equivalent MPC and receding horizon
schemes were as follows:
| certainty-equivalent MPC | receding horizon
---|---|---
Mean | $32.127$ | $59.615$
Standard deviation | $4.610$ | $21.675$
Maximum | $50.590$ | $90.036$
Minimum | $20.240$ | $20.466$
These statistics correspond to the above simulations carried out on an
$\text{x}86\\_64$ octa-core machine with 24GB RAM, each processor of which was
an Intel${}^{\text{\textregistered}}$ Xeon${}^{\text{\textregistered}}$ CPU
E5540 2.53GHz with cache size 8192 KB, running GNU/Linux.
Figure 2. Plots of states corresponding to: certainty-equivalent MPC with
$N_{c}=1$ (left) and our receding horizon control scheme with $N_{c}=4$
(right) in Example 6.3.
(a) Plot of average costs (b) Plot of standard deviations
Figure 3. Plots of average cost (left) and standard deviations (right)
corresponding to: certainty-equivalent MPC with $N_{c}=1$ and our receding
horizon control scheme with $N_{c}=4$ in Example 6.3.
We also applied the first four control values of the certainty-equivalent
scheme and compared it against our receding horizon scheme using policy (3.3),
i.e., $N_{c}=4$ for both controllers. We simulated the system in both cases
starting from the same initial condition
$x_{0}=\left[\begin{matrix}0&0\end{matrix}\right]^{\mathsf{T}}$, for $60$
different realizations of the noise sequence $w_{t}$; plots of the states,
average cost, and standard deviation are shown in Figures 4 and 5. The average
cost incurred when using the certainty-equivalent with control horizon
$N_{c}=4$ was $4.211\times 10^{5}$ units, whereas the average cost incurred
when using our policy (3.3) in a receding horizon fashion was $3.295\times
10^{5}$ units. We see that by applying our policy in a receding horizon
fashion one saves $21.7\%$ of the cost corresponding to the certainty
equivalence controller on the average. The simulations were coded in YALMIP
and were solved using SDPT-3; the solver-time statistics (in sec.) for the
certainty-equivalent and receding horizon schemes were as follows:
| certainty-equivalent | receding horizon
---|---|---
Mean | $7.537$ | $67.494$
Standard deviation | $0.812$ | $11.845$
Maximum | $9.776$ | $85.232$
Minimum | $6.101$ | $43.601$
These statistics correspond to the above simulations carried out on an
$\text{x}86\\_64$ octa-core machine with 24GB RAM, each processor of which was
an Intel${}^{\text{\textregistered}}$ Xeon${}^{\text{\textregistered}}$ CPU
E5540 2.53GHz with cache size 8192 KB, running GNU/Linux.$\triangle$
Figure 4. Plots of states corresponding to: certainty-equivalent with
$N_{c}=4$ (left) and our receding horizon control scheme with $N_{c}=4$
(right) in Example 6.3.
(a) Plot of average costs (b) Plot of standard deviations
Figure 5. Plots of average cost (left) and standard deviations (right)
corresponding to: certainty-equivalent with $N_{c}=4$ and our receding horizon
control scheme with $N_{c}=4$ in Example 6.3.
## 7\. Conclusion and Future Directions
We provided tractable solutions to a variety of finite-horizon stochastic
optimal control problems with quadratic cost, hard control constraints, and
unbounded additive noise. These problems arise as parts of solutions to the
stochastic receding horizon problems (2.4). The control policy obtained as a
result of the finite-horizon optimal control sub-problems may be nonlinear
with respect to the previous states, and the policy elements are chosen from a
vector space that is largely up to the designer. One of the key features of
our approach is that the variance-like matrices employed in the finite-horizon
optimal control sub-problems may be computed off-line, and we illustrated this
feature with several examples. We demonstrated that applying our obtained
policies in a receding horizon fashion results in bounded state variance.
Finally, we provided several numerical examples that illustrate the
effectiveness of our method with respect to the commonly used certainty-
equivalent MPC controllers.
The development in this article affords extensions in several directions. One
is the incorporation of state constraints. As discussed in §1, hard state
constraints do not make sense in the stochastic with additive unbounded noise
setting unless one is prepared to artificially relax them once infeasibility
is encountered. Probabilistic constraints and integrated chance constraints
[Han83] constitute popular alternative methods to impose constraints on the
state that are more probabilistic in nature. It will be interesting to see how
the approach introduced in this article reacts to state-constraints. A second
direction is to consider specific kinds of nonlinear models, particularly
those which involve multiplicative noise, in our framework, and a third is to
consider different objective functions such as affine functions given by the
$\ell_{\infty}$ and the $\ell_{1}$ norms.
## Acknowledgments
We are indebted to Soumik Pal for pointing out the possibility of representing
policies as elements of a vector space. We thank Colin Jones for some useful
discussions on convexity of some of the optimization programs, and the three
anonymous reviewers for their valuable suggestions that have led to
substantial improvements of the original manuscript.
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## Appendix
### A.1. Some identities
Recall the following standard special mathematical functions: the _standard
error function_
$\operatorname{erf}(z)\coloneqq\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\textstyle\sqrt{\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptstyle\sqrt{\pi\,}$}\lower
0.4pt\hbox{\vrule
height=2.10971pt,depth=-1.68779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\pi\,}$}\lower
0.4pt\hbox{\vrule
height=1.50694pt,depth=-1.20557pt}}}}\int_{0}^{z}\mathrm{e}^{-\frac{t^{2}}{2}}\mathrm{d}t$
and the _complementary error function_ [AS64, p. 297] defined by
$\operatorname{erfc}(z)\coloneqq 1-\operatorname{erf}(z)$ for
$z\in\mathbb{R}$, the _incomplete Gamma function_ [AS64, p. 260] defined by
$\Gamma(a,z)\coloneqq\int_{z}^{\infty}t^{a-1}\mathrm{e}^{-t}\mathrm{d}t$ for
$z,a>0$, the _confluent hypergeometric function_ [AS64, p. 505] defined by
$U(a,b,z)\coloneqq\frac{1}{\Gamma(a)}\int_{0}^{\infty}\mathrm{e}^{-zt}t^{a-1}(1+t)^{b-a-1}\mathrm{d}t$
for $a,b,z>0$, and $\Gamma$ is the standard Gamma function. All of these are
implemented as standard functions in Mathematica. The following facts can be
found in [AS64] and are collected here for completeness.
###### Facts about Special Functions.
For $\sigma^{2}>0$ we have
* •
$\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{z}^{\infty}\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t=\frac{1}{2}\Bigl{(}1+\operatorname{erf}\Bigl{(}\frac{z}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma}\Bigr{)}\Bigr{)}}$
* •
$\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{0}^{\infty}\frac{t^{2}}{1+t^{2}}\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t=\frac{1}{2}\Bigl{(}\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma-\pi\mathrm{e}^{-\frac{1}{2\sigma^{2}}}\operatorname{erfc}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma}\Bigr{)}\Bigr{)}}$
* •
$\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{0}^{1}t^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t=\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{\pi}{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.52776pt,depth=-6.02223pt}}}{{\hbox{$\textstyle\sqrt{\frac{\pi}{2}\,}$}\lower
0.4pt\hbox{\vrule
height=5.26944pt,depth=-4.21558pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{\pi}{2}\,}$}\lower
0.4pt\hbox{\vrule
height=3.76387pt,depth=-3.01111pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{\pi}{2}\,}$}\lower
0.4pt\hbox{\vrule
height=3.76387pt,depth=-3.01111pt}}}\sigma^{3}\operatorname{erf}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma}\Bigr{)}-\sigma^{2}\mathrm{e}^{-\frac{1}{2\sigma^{2}}}}$;
* •
$\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{1}^{\infty}t\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t=\frac{\sigma}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}}\operatorname{Gamma}(2\sigma^{2},1)}$
* •
$\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{0}^{\infty}\frac{t^{2}}{\mathchoice{{\hbox{$\displaystyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\textstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\scriptstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=5.59444pt,depth=-4.47557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+t^{2}\,}$}\lower
0.4pt\hbox{\vrule
height=4.36427pt,depth=-3.49144pt}}}}\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t=\frac{\sigma}{2\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma^{2}}\Bigr{)}}$.
### A.2. Proof of mean-square boundedness
###### Proof of Proposition 5.2.
Fix $x_{0}\in\mathbb{R}^{n}$. For any $n\times n$ matrix $P=P^{\mathsf{T}}>0$,
using (5.1) and the fact that $\mathbb{E}\left[{\mathfrak{e}}(w)\right]=0$, we
see that for every $\ell=1,\cdots,N_{c}$
$\displaystyle\mathbb{E}_{x_{kN_{c}}}\bigl{[}x_{kN_{c}+\ell}^{\mathsf{T}}Px_{kN_{c}+l}\bigr{]}$
$\displaystyle=x_{kN_{c}}^{\mathsf{T}}(A^{\ell})^{\mathsf{T}}PA^{\ell}x_{kN_{c}}+2x_{kN_{c}}^{\mathsf{T}}(A^{\ell})^{\mathsf{T}}PB_{\ell}\mathbb{E}_{x_{kN_{c}}}\bigl{[}\pi^{*}_{kN_{c}:kN_{c}+\ell-1}(x_{kN_{c}})\bigr{]}$
$\displaystyle\quad+\mathbb{E}_{x_{kN_{c}}}\bigl{[}\left\lVert{B_{\ell}\pi^{*}_{kN_{c}:kN_{c}+\ell-1}(x_{kN_{c}})+D_{\ell}\tilde{w}_{kN_{c}:kN_{c}+\ell-1}}\right\rVert_{P}^{2}\bigr{]},$
where
$\left\lVert{\xi}\right\rVert_{P}\coloneqq\mathchoice{{\hbox{$\displaystyle\sqrt{\xi^{\mathsf{T}}P\xi\,}$}\lower
0.4pt\hbox{\vrule
height=8.85777pt,depth=-7.08626pt}}}{{\hbox{$\textstyle\sqrt{\xi^{\mathsf{T}}P\xi\,}$}\lower
0.4pt\hbox{\vrule
height=8.85777pt,depth=-7.08626pt}}}{{\hbox{$\scriptstyle\sqrt{\xi^{\mathsf{T}}P\xi\,}$}\lower
0.4pt\hbox{\vrule
height=6.22777pt,depth=-4.98224pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\xi^{\mathsf{T}}P\xi\,}$}\lower
0.4pt\hbox{\vrule height=4.83888pt,depth=-3.87112pt}}}$. Using the fact that
$\left\lVert{\pi^{*}_{kN_{c}:kN_{c}+\ell-1}(x_{kN_{c}})}\right\rVert_{\infty}\leqslant
U_{\rm max}$ by construction, we obtain the following bound:
(A.1)
$\displaystyle\mathbb{E}_{x_{kN_{c}}}\bigl{[}x_{kN_{c}+\ell}^{\mathsf{T}}Px_{kN_{c}+\ell}\bigr{]}\leqslant
x_{kN_{c}}^{\mathsf{T}}(A^{\ell})^{\mathsf{T}}PA^{\ell}x_{kN_{c}}+2c_{1\ell}\left\lVert{x_{kN_{c}}}\right\rVert_{\infty}+c_{2\ell},$
where
$\displaystyle c_{1\ell}$ $\displaystyle\coloneqq
m\left\lVert{(A^{\ell})^{\mathsf{T}}PB_{\ell}}\right\rVert_{\infty}U_{\rm
max},$ $\displaystyle c_{2\ell}$ $\displaystyle\coloneqq
m\left\lVert{B_{\ell}^{\mathsf{T}}PB_{\ell}}\right\rVert_{\infty}U_{\max}^{2}+\mathbf{tr}\\!\left(D_{\ell}^{\mathsf{T}}PD_{\ell}\Sigma_{w}\right)$
$\displaystyle\quad+\max_{\tiny{\left\lVert{\Upsilon(x_{kN_{c}})}\right\rVert_{\infty}\leqslant
U_{\max}/\phi_{\max}}}\big{[}\mathbf{tr}\\!\left(\Upsilon(x_{kN_{c}})^{\mathsf{T}}B_{\ell}^{\mathsf{T}}PB_{\ell}\Upsilon(x_{kN_{c}})\Lambda_{1}\right)+2\mathbf{tr}\\!\left(\Upsilon(x_{kN_{c}})^{\mathsf{T}}B_{\ell}^{\mathsf{T}}PD_{\ell}\Lambda_{2}\right)\big{]},$
$\displaystyle\text{and}\quad\Upsilon(x_{kN_{c}})\coloneqq\left[\begin{matrix}\Theta^{*}_{1}(x_{kN_{c}})\\\
\cdots\\\ \Theta^{*}_{N_{c}-1}(x_{kN_{c}})\end{matrix}\right].$
Since $A$ is a Schur stable matrix (and hence so is $A^{\ell}$) there exists
[Ber09, Proposition 11.10.5] a matrix $P_{\ell}=P_{\ell}^{\mathsf{T}}>0$ with
real-valued entries that satisfies
$(A^{\ell})^{\mathsf{T}}P_{\ell}A^{\ell}-P_{\ell}=-\mathbf{I}_{n\times n}$; in
particular, we have
$x_{kN_{c}}^{\mathsf{T}}(A^{\ell})^{\mathsf{T}}P_{\ell}A^{\ell}x_{kN_{c}}\leqslant
x_{kN_{c}}^{\mathsf{T}}P_{\ell}x_{kN_{c}}-x_{kN_{c}}^{\mathsf{T}}x_{kN_{c}}$.
Therefore, with $P=P_{\ell}$ in (A.1) we arrive at
(A.2)
$\displaystyle\mathbb{E}_{x_{kN_{c}}}\bigl{[}x_{kN_{c}+\ell}^{\mathsf{T}}P_{\ell}x_{kN_{c}+\ell}\bigr{]}\leqslant
x_{kN_{c}}^{\mathsf{T}}P_{\ell}x_{kN_{c}}-\left\lVert{x_{kN_{c}}}\right\rVert^{2}+2c_{1\ell}\left\lVert{x_{kN_{c}}}\right\rVert_{\infty}+c_{2\ell}.$
For $\zeta_{\ell}\in\;]\max\\{0,1-\lambda_{\max}(P_{\ell})\\},1[$ let
$r_{\ell}\coloneqq\frac{1}{\zeta_{\ell}}\bigl{(}c_{1\ell}+\mathchoice{{\hbox{$\displaystyle\sqrt{c_{1\ell}^{2}+c_{2\ell}\zeta_{\ell}\,}$}\lower
0.4pt\hbox{\vrule
height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\textstyle\sqrt{c_{1\ell}^{2}+c_{2\ell}\zeta_{\ell}\,}$}\lower
0.4pt\hbox{\vrule
height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\scriptstyle\sqrt{c_{1\ell}^{2}+c_{2\ell}\zeta_{\ell}\,}$}\lower
0.4pt\hbox{\vrule
height=4.8611pt,depth=-3.8889pt}}}{{\hbox{$\scriptscriptstyle\sqrt{c_{1\ell}^{2}+c_{2\ell}\zeta_{\ell}\,}$}\lower
0.4pt\hbox{\vrule height=3.47221pt,depth=-2.77779pt}}}\bigr{)}$. Then
elementary properties of the quadratic function
$g(y)\coloneqq-\zeta_{\ell}y^{2}+2c_{1\ell}y+c_{2\ell}$ show that
$\displaystyle-\zeta_{\ell}\left\lVert{x_{kN_{c}}}\right\rVert_{\infty}^{2}+2c_{1\ell}\left\lVert{x_{kN_{c}}}\right\rVert_{\infty}+c_{2\ell}\leqslant
0\quad\text{whenever }\left\lVert{x_{kN_{c}}}\right\rVert_{\infty}>r_{\ell},$
In view of the above fact, simple manipulations in (A.2) lead to
$\displaystyle\mathbb{E}_{x_{kN_{c}}}\bigl{[}x_{kN_{c}+\ell}^{\mathsf{T}}P_{\ell}x_{kN_{c}+\ell}\bigr{]}\leqslant
x_{kN_{c}}^{\mathsf{T}}P_{\ell}x_{kN_{c}}-(1-\zeta_{\ell})\left\lVert{x_{kN_{c}}}\right\rVert^{2}\quad\text{whenever
}\left\lVert{x_{kN_{c}}}\right\rVert_{\infty}>r_{\ell},$
from which, letting
$\rho_{\ell}\coloneqq\Bigl{(}1-\frac{1-\zeta_{\ell}}{\lambda_{\text{max}}(P_{\ell})}\Bigr{)}$,
we arrive at
(A.3)
$\displaystyle\mathbb{E}_{x_{kN_{c}}}\bigl{[}x_{kN_{c}+\ell}^{\mathsf{T}}P_{\ell}x_{kN_{c}+\ell}\bigr{]}\leqslant\rho_{\ell}x_{kN_{c}}^{\mathsf{T}}P_{\ell}x_{kN_{c}}\quad\text{whenever
}\left\lVert{x_{kN_{c}}}\right\rVert_{\infty}>r_{\ell}.$
Let us define
$\displaystyle\rho$
$\displaystyle\coloneqq\max\limits_{\ell=1,\cdots,N_{c}}\rho_{\ell},$
$\displaystyle r^{\prime}$
$\displaystyle\coloneqq\max\limits_{\ell=1,\cdots,N_{c}}r_{\ell},$
$\displaystyle\overline{\lambda}$
$\displaystyle\coloneqq\max\limits_{\ell=1,\dots,N_{c}}\lambda_{\max}(P_{\ell}),$
$\displaystyle\underline{\lambda}$
$\displaystyle\coloneqq\min\limits_{\ell=1,\dots,N_{c}}\lambda_{\min}(P_{\ell}).$
Then we can obtain using (A.3) the conservative bound for every
$\ell=1,\ldots,N_{c}$:
$\mathbb{E}_{x_{kN_{c}}}\bigl{[}x_{kN_{c}+\ell}^{\mathsf{T}}P_{N_{c}}x_{kN_{c}+\ell}\bigr{]}\leqslant\rho^{\prime}x_{kN_{c}}^{\mathsf{T}}P_{N_{c}}x_{kN_{c}}\quad\text{whenever
}\left\lVert{x_{kN_{c}}}\right\rVert_{\infty}>r^{\prime},$
where
$\rho^{\prime}\coloneqq\rho\frac{\overline{\lambda}\lambda_{\max}(P_{N_{c}})}{\underline{\lambda}\lambda_{\min}(P_{N_{c}})}$.
It follows immediately that
(A.4)
$\mathbb{E}_{x_{kN_{c}}}\bigl{[}x_{kN_{c}+\ell}^{\mathsf{T}}P_{N_{c}}x_{kN_{c}+\ell}\bigr{]}\leqslant\rho^{\prime}x_{kN_{c}}^{\mathsf{T}}P_{N_{c}}x_{kN_{c}}+b^{\prime}\mathbf{1}_{K^{\prime}}(x_{kN_{c}}),$
where
$K^{\prime}\coloneqq\bigl{\\{}\xi\in\mathbb{R}^{n}\big{|}\left\lVert{\xi}\right\rVert_{\infty}\leqslant
r^{\prime}\bigr{\\}}$.
Let us define the function $V(\xi)\coloneqq\xi^{\mathsf{T}}P_{N_{c}}\xi$, and
fix $k\in\mathbb{N}$ and $\ell=1,\dots,N_{c}$. Let
$K_{N_{c}}\coloneqq\bigl{\\{}\xi\in\mathbb{R}^{n}\big{|}\left\lVert{\xi}\right\rVert_{\infty}\leqslant
r_{N_{c}}\bigr{\\}}$, $b\coloneqq\sup\limits_{x\in
K}\mathbb{E}_{x}\bigl{[}V(x_{N_{c}})\bigr{]}$, and
$b^{\prime}\coloneqq\max\limits_{\ell=1,\ldots,N_{c}}\sup\limits_{x\in
K^{\prime}}\mathbb{E}_{x}\bigl{[}V(x_{\ell})\bigr{]}$. From (A.4) we get
$\displaystyle\mathbb{E}_{x_{0}}\bigl{[}V(x_{kN_{c}+\ell})\bigr{]}$
$\displaystyle=\mathbb{E}_{x_{0}}\bigl{[}\mathbb{E}\bigl{[}V(x_{kN_{c}+\ell})\,\big{|}\,x_{kN_{c}}\bigr{]}\bigr{]}\leqslant\mathbb{E}_{x_{0}}\bigl{[}\rho^{\prime}V(x_{kN_{c}})+b^{\prime}\mathbf{1}_{K^{\prime}}(x_{kN_{c}})\bigr{]}$
$\displaystyle\leqslant\mathbb{E}_{x_{0}}\bigl{[}\rho^{\prime}\mathbb{E}\bigl{[}V(x_{kN_{c}})\,\big{|}\,x_{(k-1)N_{c}}\bigr{]}+b^{\prime}\mathbf{1}_{K^{\prime}}(x_{kN_{c}})\bigr{]}$
$\displaystyle\leqslant\mathbb{E}_{x_{0}}\bigl{[}\rho^{\prime}\rho_{N_{c}}V(x_{(k-1)N_{c}})+b\mathbf{1}_{K_{N_{c}}}(x_{(k-1)N_{c}})+b^{\prime}\mathbf{1}_{K^{\prime}}(x_{kN_{c}})\bigr{]}$
$\displaystyle\cdots$
$\displaystyle\leqslant\rho^{\prime}\rho_{N_{c}}^{k}V(x)+\sum_{i=0}^{k-1}b\rho_{N_{c}}^{k-1-i}\mathbb{E}_{x_{0}}\bigl{[}\mathbf{1}_{K_{N_{c}}}(x_{iN_{c}})\bigr{]}+b^{\prime}\mathbb{E}_{x_{0}}\bigl{[}\mathbf{1}_{K^{\prime}}(x_{kN_{c}})\bigr{]}$
(A.5)
$\displaystyle\leqslant\rho^{\prime}\rho_{N_{c}}^{k}V(x)+\frac{b\bigl{(}1-\rho_{N_{c}}^{k}\bigr{)}}{1-\rho_{N_{c}}}+b^{\prime}.$
Note that the conditioning in the first few steps of (A.5) is well-defined
because it is performed every $N_{c}$ steps starting from $0$, and the
structure of our policy $\pi^{*}$ makes the process
$(x_{tN_{c}})_{t\in\mathbb{N}_{0}}$ Markovian. Therefore, it follows from
(A.5) that for all $t\coloneqq kN_{c}+\ell$,
$\displaystyle\sup\limits_{t\in\mathbb{N}_{0}}\mathbb{E}_{x_{0}}\bigl{[}\left\lVert{x_{t}}\right\rVert^{2}\bigr{]}$
$\displaystyle\leqslant\frac{1}{\lambda_{\min}(P_{N_{c}})}\sup\limits_{t\in\mathbb{N}_{0}}\mathbb{E}_{x_{0}}\bigl{[}V(x_{kN_{c}+\ell})\bigr{]}$
$\displaystyle\leqslant\frac{1}{\lambda_{\min}(P_{N_{c}})}\left(\rho^{\prime}\rho_{N_{c}}^{k}V(x)+\frac{b}{1-\rho_{N_{c}}}+b^{\prime}\right)$
$\displaystyle<\infty,$
where the last step follows from the fact that $\rho_{N_{c}}<1$. This
completes the proof. ∎
|
arxiv-papers
| 2009-03-31T16:53:46 |
2024-09-04T02:49:01.545011
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Debasish Chatterjee, Peter Hokayem and John Lygeros",
"submitter": "Debasish Chatterjee",
"url": "https://arxiv.org/abs/0903.5444"
}
|
0904.0027
|
11institutetext: Center for Nonlinear Studies
Los Alamos National Laboratory, Los Alamos NM 87545, USA
11email: marko@lanl.gov 22institutetext: International and Applied Technology
Los Alamos National Laboratory, Los Alamos NM 87545, USA
22email: jhw@lanl.gov
# Faith in the Algorithm, Part 2:
Computational Eudaemonics
Marko A. Rodriguez 11 Jennifer H. Watkins 22
###### Abstract
Eudaemonics is the study of the nature, causes, and conditions of human well-
being. According to the ethical theory of eudaemonia, reaping satisfaction and
fulfillment from life is not only a desirable end, but a moral responsibility.
However, in modern society, many individuals struggle to meet this
responsibility. Computational mechanisms could better enable individuals to
achieve eudaemonia by yielding practical real-world systems that embody
algorithms that promote human flourishing. This article presents eudaemonic
systems as the evolutionary goal of the present day recommender system.
> [Those who condemn individualism] slur over the chief problems—that of
> remaking society to serve the growth of a new type of individual.
>
> John Dewey, “Individualism Old and New”
## 1 Introduction
Eudaemonia is the theory that the highest ethical goal is personal happiness
and well-being [1]. This theory holds that an ethical life is one filled with
the meaning and satisfaction that arises from living according to one’s
values—where everything one does is of great importance to their character.
Eudaemonia parallels the notion of Abraham Maslow’s self-actualization [2] and
Mihály Csíkszentmihályi’s flow state [3] except that, as an ethical theory, it
argues that it is a personal responsibility to strive for this state. As a
social theory, eudaemonia holds that the purpose of society is to promote this
state in all of its people. The ethical foundation of personal flourishing is
grounded in the contention that the purpose of life is to reap satisfaction
and fulfillment from an engagement in the world and that such a state is
objectively good for society. Thus, learning how to flourish is a form of
moral development.
Moral development, when used in this sense, extends beyond civility, honesty,
and other facets of rectitude. It refers to a personal onus to achieve well-
being. One proponent of the ethical theory of eudaemonia, David L. Norton,
states that “[…] the broader eudaimonistic thesis is that all virtues subsist
in potentia in every person; thus to be a human being is to be capable of
manifesting virtues, and the problem of moral development is the problem of
discovering the conditions of their manifestation” [4]. Typically, the
discovery of the conditions that will manifest virtues in the individual is
guided by the recommendations of family, friends, and community—those who know
the individual well and the options available to them. Despite this guidance,
the achievement of eudaemonia remains elusive for most. Maslow notes that a
very small group of people achieve self-actualization and Csíkszentmihályi has
shown that very few are able to control their consciousness well enough to
reliably reach the state of flow. Given the individual moral imperative to
achieve eudaemonia and the resulting societal benefits, resources should be
dedicated to guaranteeing this realization for as many people as possible.
Eudaemonics is the study of the nature, causes, and conditions of eudaemonia
[5]. For Owen Flanagan, the domains of moral and political philosophy,
neuroethics, neuroeconomics, and positive psychology are the sources from
which a developed understanding of human well-being will spring. In this
article, it is posited that computational eudaemonics will make advances to
bring eudaemonia to more than a select few in society. Computer and
information science can greatly contribute to the eudaemonic endeavor by
yielding practical real-world systems that embody algorithms that promote
human flourishing. Systems that promote eudaemonia are called eudaemonic
systems. Such systems would foster eudaemonia by providing the right
conditions for the manifestation of virtues. This article presents a vision of
eudaemonic systems as the evolutionary goal of the present day recommender
system.
## 2 From Recommender to Eudaemonic Systems
The purpose of a eudaemonic system is to produce societies in which the
individuals experience satisfaction through a deep engagement in the world.
This engagement can be fostered by uniting individuals with those resources
that resonate with their nature. Resources can take many forms, a few of which
are itemized below.
* •
activities: vocations, hobbies, gatherings, projects.
* •
education: universities, lectures, areas of study.
* •
entertainment: books, movies, music, shows.
* •
people: friends, work associates, life partners.
* •
places: to live, to vacation, to dine.
There are many ways a eudaemonic system could contribute to individual well-
being. Perhaps the most ambitious eudaemonic system is one that supplies the
satisfaction of the need for a resource before the need is even felt. For
Thomas Hobbes, eudaemonia is encumbered by conation—goals, plans, and desires
[6]. Practically speaking, humans seek books and movies to stimulate their
cognitive faculties, friends and partners to fulfill their social affinities,
art to entice their affective natures, and sports to satiate their physical
needs. While every individual longs for varying degrees of these requirements,
in general, a flourishing life is one where all these requirements are met
through the active process of enacting them [7]. Thus, a Hobbesian eudaemonic
system would be one that satisfied requirements before they were felt (pre-
conation), so that the experience of need could not disrupt a life of
contentment. Through computational mechanisms, it may be possible to produce
pre-conate eudaemonic systems. A pre-conate system is one that makes use of
indicators of coming discontent and provides avenues to rectify the situation
prior to its actualization.
Recommender systems [8], when viewed within the context of the eudaemonic
thesis, could evolve to become such systems. A recommender system is an
information filtering tool that matches individuals to resources of potential
interest. Such systems are commonly employed by businesses in an attempt to
sell more products. However, this conceptualization of the recommender system
trivializes their potential role.
The satisfaction one reaps from the world can be represented in terms of one’s
interactions with resources. These interactions need not be extraordinary, but
are the stuff of everyday life. Norton articulates the importance of everyday
activities when he states that “if the development of character is the moral
objective, it is obvious that […] the choices of vocation and avocations to
pursue, of friends to cultivate, of books to read are moral for they clearly
influence such development” [4]. For the techno-social society, this
development of character is driven every day, to some extent, by the use of
recommender systems. Thus, to the extent that recommender systems influence
choices, they already influence moral development. By purposely designing
these systems to orient individuals toward life optima, recommender systems
can evolve to become eudaemonic systems.
The current generation of recommender systems are limited to a particular
representational slice of the world (such as movies). This is represented in
Figure 1a, where there exists a tight coupling between the data and the
application which operates on that data. A eudaemonic system must account not
for a single aspect of an individual’s life, but for the multitude of domains
in which that individual exists. The emerging Web of Data provides a
distributed data structure that cleanly separates the data providers from the
application developers. This is represented in Figure 1b. The remainder of
this section will discuss recommender systems and their transition to
eudaemonic systems through the exploitation of the Web of Data.
Figure 1: a.) The current paradigm in which the application and the data upon
which it operates are tightly coupled both technically and proprietarily—§2.1.
b.) The emerging Web of Data provides a collectively generated, publicly
accessible world model that can be leveraged by independent application
developers—§2.2.
### 2.1 Recommender Systems
Most recommender systems model individual users, resources, and their
relationships to one another [8]. For example, in an online store, users may
have an ex:hasPurchased relationship to some of the store’s products. If the
purchasing behavior of user $x$ and user $y$ has a strong, positive
correlation, then any products purchased by only one can be recommended to the
other. Purchasing behavior is not the only way in which resources are deemed
similar. It is possible to relate resources by shared metadata properties [9].
For example, an online movie rental service can represent movie $a$ as having
an ex:directedBy relationship to director $b$ and director $b$ can maintain an
ex:directed relationship to movie $c$. The similarity that exists between
movies $a$ and $c$ is determined, not by user behavior, but by similarity of
metadata—the same person directed both. By building a graph of typed
relationships between resources, it is possible to identify different forms of
relatedness and utilize these forms to aid an individual in their decision
making process regarding the use of such resources.
The power of recommender systems is currently limited because they rely on a
single silo of data that must be generated before they can provide useful
recommendations (see Figure 1a). Due to the data acquisition hurdle,
application designers must focus on a particular niche in which to provide
recommendations. For example, services either provide recommendations for
books,111For example: Amazon.com, Feedbooks.com or for music,222For example:
Pandora.com, Last.fm, or for partners,333For example: Match.com,
Chemistry.com, eHarmony.com etc. With such a limited worldview, these services
do not respect the multi-faceted nature of human beings. If a system only has
access to data on movies, then it can never recommend the perfect beach novel.
Eudaemonia requires a complete representation of the domains in which one
conducts life in order to recommend the right resource at the right time.
Therefore, eudaemonic systems require an integrated representation of the
world’s resources and the individual’s place within them.
### 2.2 Eudaemonic Systems
The recommender system data structure described previously can be conveniently
represented as a multi-relational network. The most prevalent multi-relational
data model is the Resource Description Framework (RDF) of the Semantic Web
initiative. The Semantic Web’s Linked Data community is dedicated to the
development of the emerging RDF-based Web of Data. On the Web of Data, all
data is represented in the URI address space and interlinked to form a single,
global data structure that can be used by both man and machine for various
application scenarios (see Figure 1b) [10].444The public exposure of data has
stimulated interest in the development of the legal structures for the use of
such data. Much like the Open Source movement, the Linked Data community is
actively involved in the Open Data movement [11]. The Web of Data provides two
significant benefits over the data silos used by recommender systems. First,
application developers need not focus on data acquisition and instead can
focus directly on algorithm development. This feature ultimately reduces the
labor involved in web service deployment. Second, the application developer
can create algorithms that make use of a rich world model that incorporates
the various ways in which resources relate to each other. Thus, these
algorithms have a larger knowledge-base with which to understand the world and
the individual’s place within it.
Figure 2 presents a visualization of the linking structure of the $89$ data
sets currently in the Linked Data cloud.555The Linked Data cloud is a subset
of the larger Web of Data that includes those data sets that are directly or
indirectly connected to DBpedia and are maintained by the Linked Data
community. Each vertex represents a unique data set that exists on an Internet
server. The directed relationships denote that the source data set references
resources in the sink data set. The current Linked Data cloud maintains
approximately $4.5$ billion relationships on data from various domains of
interest. Table 1 indicates the domain of interest for each data set.
Figure 2: A representation of the $89$ RDF data sets currently in the Linked Data cloud. Table 1: The domains of the $89$ data sets currently in the Linked Data cloud. data set | domain | data set | domain | data set | domain
---|---|---|---|---|---
acm | computer | geospecies | biology | pubchem | biology
audioscrobbler | music | govtrack | government | pubguide | books
bbcjohnpeel | music | hgnc | biology | pubmed | medical
bbclatertotp | music | homologene | biology | qdos | social
bbcplaycountdata | music | ibm | computer | rae2001 | computer
bbcprogrammes | media | ieee | computer | rdfbookmashup | books
budapestbme | computer | interpro | biology | rdfohloh | social
cas | biology | irittoulouse | computer | reactome | biology
chebi | biology | jamendo | music | resex | computer
citeseer | computer | kegg | biology | revyu | reference
crunchbase | business | laascnrs | computer | riese | government
dailymed | medical | libris | books | semanticweborg | computer
dblpberlin | computer | lingvoj | reference | semwebcentral | social
dblphannover | computer | linkedct | medical | siocsites | social
dblprkbexplorer | computer | linkedmdb | movie | surgeradio | music
dbpedia | general | magnatune | music | swconferencecorpus | computer
diseasome | medical | mgi | biology | symbol | medical
doapspace | social | musicbrainz | music | taxonomy | reference
drugbank | medical | myspacewrapper | social | umbel | general
ecssouthampton | computer | newcastle | computer | uniparc | biology
eprints | computer | omim | biology | uniprot | biology
eurecom | computer | opencalais | reference | uniref | biology
eurostat | government | opencyc | general | unists | biology
flickrexporter | images | openguides | reference | uscensusdata | government
flickrwrappr | images | pdb | biology | virtuososponger | reference
foafprofiles | social | pfam | biology | w3cwordnet | reference
freebase | general | pisa | computer | wikicompany | business
geneid | biology | prodom | biology | worldfactbook | government
geneontology | biology | projectgutenberg | books | yago | general
geonames | geographic | prosite | biology | |
By publicly exposing data sets such as Amazon.com’s RDF book mashup,
MusicBrainz.org’s metadata archive, the Internet Movie Database’s (IMDB)
collection of movie facts, Revyu.com’s user ratings, and the publishing and
conference behavior of scholars, the Web of Data hosts a rich model of the
world that is not built by a single provider, but by many providers
collaboratively integrating their data. Such a massive public data structure
can be exploited by a community of developers focused on ensuring that the
right resource reaches the right person at the right time. Ultimately, an
orchestration of this magnitude could yield virtuous individuals whose lives
are filled with experiences tailored to their nature.
The Web of Data already includes data sets that are pertinent to modeling
individuals and resources; however, the success of a eudaemonic system depends
on the availability of data regarding the individual and their past, current,
and predicted responses to resources. At the societal level, research has
demonstrated that resources relevant to flourishing are those that support
life expectancy, nutrition, purchasing power, freedom, equality, education,
literacy, access to information, and mental health [12].666The World Database
of Happiness provides data concerning the study of well-being worldwide and is
available at http://worlddatabaseofhappiness.eur.nl. At the individual level,
gathering and maintaining data regarding fluctuations in an individual’s well-
being in relation to resources would support the automatic determination of
optimal future states for that individual.
While the Linked Data community is providing a distributed data structure,
they are not providing a distributed process infrastructure [13]. Currently,
the Linked Data practice is to mint http-based URIs. These http-based URIs are
dereferenced in order to retrieve a collection of RDF statements associated
with that URI. The problem with this model is that it relegates the Web of
Data to use primarily by man. For a machine to traverse parts of the larger
Web of Data, the pull-based mechanism of HTTP greatly reduces the speed of
processing. It would be unfortunate to limit the sophistication of the
algorithms that can reasonably process this data due to an infrastructure
issue that can be solved using distributed computing.
Ultimately, once these computational hurdles are overcome, what can emerge is
a “society of algorithms” that leverages the Web of Data to support
individuals in ways that are not possible given the current recommender system
architectures. Through such an undertaking, the niche recommender system is
transformed into a eudaemonic system, one that fosters a society of
individuals where the vocation one takes, the person one dates, the books one
reads, the restaurants one frequents, and so on are chosen not through the
advice of one’s family, friends, and community, but through a deep
computational understanding of what is required for that individual to live an
optimal life.
## 3 Conclusion
The evolution of the recommender system to the eudaemonic system will be
driven by the public exposure of massive-scale, interlinked, heterogenous data
and algorithms that can effectively and efficiently process such data. The
goal of a eudaemonic system is to orient people towards those resources that
will produce a life that is devoid of pretense, doubt, and ultimately, fear.
That is, a eudaemonic system will aid the individual in situating themselves
within that area of the world that makes sense to them. A pre-conate
eudaemonic system would direct the individual to choose need-mitigating
options before the individual becomes aware of their need. In other words, the
individual would choose options that they do not perceive as necessary.
Without the perception of need, the individual would take on faith that the
algorithm knows what is best for them in a resource complex world. Thus, the
perfect life is not an aspiration, but a well-computed path.
## Note
Faith in the Algorithm is a series of articles that focuses on the
intersection of political philosophy, ethics, and computation.
## References
* [1] Aristotle: Nicomachean Ethics. (350 B.C.)
* [2] Maslow, A.H.: A theory of human motivation. Psychological Review (50) (1943) 370–396
* [3] Csíkszentmihályi, M.: Flow: The Psychology of Optimal Experience. Harper and Row, New York, NY (1990)
* [4] Norton, D.L.: Democracy and Moral Development: A Politics of Virtue. University of California Press (1995)
* [5] Flanagan, O.: The Really Hard Problem: Meaning in a Material World. MIT Press (2007)
* [6] Hobbes, T.: Leviathan. (1651)
* [7] Kraut, R.: What is Good and Why: The Ethics of Well-Being. Harvard University Press (2007)
* [8] Resnick, P., Varian, H.R.: Recommender systems. Communications of the ACM 40(3) (1997) 56–58
* [9] Pazzani, M., Billsus, D.: Content-Based Recommendation Systems. Lecture Notes in Computer Science. In: The Adaptive Web. Springer (2007) 325–341
* [10] Bizer, C., Heath, T., Idehen, K., Berners-Lee, T.: Linked data on the web. In: Proceedings of the International World Wide Web Conference. Linked Data Workshop, Beijing, China (April 2008)
* [11] Miller, P., Styles, R., Heath, T.: Open data commons: A license for open data. In: Workshop on Linked Data on the Web, New York, NY, ACM Press (April 2008)
* [12] Heylighen, F., Bernheim, J.: Global progress I: empirical evidence for increasing quality of life. Journal of Happiness Studies 1(3) (200) 323–349
* [13] Rodriguez, M.A.: A distributed process infrastructure for a distributed data structure. Semantic Web and Information Systems Bulletin (2008)
|
arxiv-papers
| 2009-04-01T16:28:20 |
2024-09-04T02:49:01.564476
|
{
"license": "Public Domain",
"authors": "Marko A. Rodriguez and Jennifer H. Watkins",
"submitter": "Marko A. Rodriguez",
"url": "https://arxiv.org/abs/0904.0027"
}
|
0904.0093
|
# Electromagnetic response in kinetic energy driven cuprate superconductors:
Linear response approach
Mateusz Krzyzosiak Department of Physics, Beijing Normal University, Beijing
100875, China Institute of Physics, Wrocław University of Technology,
Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland Zheyu Huang and Shiping
Feng∗ Department of Physics, Beijing Normal University, Beijing 100875, China
Ryszard Gonczarek Institute of Physics, Wrocław University of Technology,
Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
###### Abstract
Within the framework of the kinetic energy driven superconductivity, the
electromagnetic response in cuprate superconductors is studied in the linear
response approach. The kernel of the response function is evaluated and
employed to calculate the local magnetic field profile, the magnetic field
penetration depth, and the superfluid density, based on the specular
reflection model for a purely transverse vector potential. It is shown that
the low temperature magnetic field profile follows an exponential decay at the
surface, while the magnetic field penetration depth depends linearly on
temperature, except for the strong deviation from the linear characteristics
at extremely low temperatures. The superfluid density is found to decrease
linearly with decreasing doping concentration in the underdoped regime. The
problem of gauge invariance is addressed and an approximation for the dressed
current vertex, which does not violate local charge conservation is proposed
and discussed.
###### pacs:
74.25.Ha, 74.25.Nf, 74.20.Mn
Keywords: Electromagnetic response; Magnetic field penetration depth; Cuprate
superconductors
## I Introduction
Observation of superconductor’s response to a weak external electromagnetic
stimulus allows us to collect a number of subtle characteristics schrieffer83
. The way the magnetic field is expelled from a superconducting (SC) sample in
the spectacular Meissner effect can be used to infer about many fundamental
features of the system. Therefore the phenomena at the length scale of the
magnetic field penetration depth $\lambda$, i.e. in the region at the edge of
the sample where the induced supercurrents effectively screen the external
magnetic field, are subject to intensive studies both on the theoretical and
the experimental fronts of the research in cuprate superconductors bonn96 ;
tsuei00 . In particular, the magnetic field penetration depth can be used as a
probe of the pairing symmetry since it can distinguish between a fully gapped
and a nodal quasiparticle excitation spectrum bonn96 ; tsuei00 . The former
results in the thermally activated (exponential) temperature dependence of the
magnetic field penetration depth, whereas the latter one implies a power law
behavior.
The magnetic field penetration depth is a basic parameter of superconductors,
closely related to the superfluid density schrieffer83 . Earlier on, the
linear temperature dependence of the magnetic field penetration depth
$\lambda(T)$ was observed for the cuprate superconductor YBa2Cu3O7-y at low
temperatures ($T=4$K$\sim$20K) hardy93 , which first provided a strong
experimental support for the nodes in the d-wave SC gap function of cuprate
superconductors, then confirmed by the angle-resolved photoemission
spectroscopy (ARPES) experiments ding9495 ; damascelli03 . Later, this linear
temperature dependence of the magnetic field penetration depth has been
observed in different families of cuprate superconductors kamal98 ; jackson00
; panagopoulos99 ; pereg07 . However, at extremely low temperatures ($T<4$K),
the linear temperature dependence of the magnetic field penetration depth is
modified, and a nonlinearity emerges khasanov04 ; suter04 ; sonier99 .
Moreover, some indications of nonlocal effects giving rise to the nonlinearity
have been reported in the field dependence of the effective magnetic field
penetration depth in cuprate superconductors sonier99 . Furthermore, the
doping dependence of the electromagnetic response in cuprate superconductors
has been studied in terms of the zero-temperature superfluid density. The
superfluid density is proportional to the squared amplitude of the coherent
macroscopic wave function describing the SC charge carriers, and therefore it
is an important physical quantity and can provide significant information
about the SC state. In particular, the superfluid density of cuprate
superconductors in the underdoped regime vanishes more or less linearly with
decreasing doping concentration uemura8991 ; broun07 ; bernhard01 . This in
turn gives rise to the linear relation between the critical temperature
$T_{\rm{c}}$ and the superfluid density observed in the underdoped regime
uemura8991 .
Theoretically, the electromagnetic response in cuprate superconductors has
been extensively studied based on the the phenomenological Bardeen-Cooper-
Schrieffer (BCS) formalism with the d-wave SC gap function yip92 ; kosztin97 ;
franz97 ; li00 ; sheehy04 . It has been shown tsuei00 ; kosztin97 that for a
d-wave superconductor in the local limit ($\zeta\ll\lambda$, where $\zeta$ is
the coherence length), the simple d-wave pairing state (assuming a tetragonal
symmetry and ignoring the dispersion in the c-axis direction) gives the
magnetic field penetration depth $\lambda(T)\propto T/\Delta_{0}$, where
$\Delta_{0}$ is the zero-temperature value of the d-wave gap amplitude. In
particular, it has been argued that this linear temperature dependence of the
magnetic field penetration depth is attributed to the excitation of
quasiparticles out of the condensate at the nodes of the SC gap function.
Furthermore, the fact kamal98 ; jackson00 ; panagopoulos99 ; pereg07 that
this linear relation holds down to very low doping concentrations suggests
that near the nodes these quasiparticle excitations are well described by a
simple BCS-like formalism with the d-wave SC gap function, even for the doping
concentration $\delta\to 0$ sheehy04 . This is also consistent with the ARPES
experiments matsui . However, this depends sensitively on the quasiparticle
scattering. In particular, at extremely low temperatures, the coherence length
may diverge at the nodes. This may imply that the local condition no longer
holds, and the electromagnetic field varies significantly over the size of a
Cooper pair. Consequently, the nonlocal effect emerges suter04 and then plays
an important role in the electromagnetic response of cuprate superconductors
yip92 ; kosztin97 ; franz97 ; li00 ; sheehy04 . It has been suggested yip92 ;
kosztin97 ; franz97 ; li00 ; sheehy04 that nonlocal effects can imply a
crossover from the linear temperature dependence of the magnetic field
penetration depth at low temperatures to a nonlinear one in the extremely low
temperature range. To the best of our knowledge, the electromagnetic response
in cuprate superconductors has not been treated starting from a microscopic SC
theory, and no explicit calculations of the doping dependence of the
superfluid density in the underdoped regime have been made so far.
Recently, a kinetic energy driven SC mechanism has been developed feng0306 ,
where the charge carrier-spin interaction from the kinetic energy term induces
a charge carrier pairing state with the d-wave symmetry by exchanging spin
excitations. Then the electron Cooper pairs originating from the charge
carrier pairing state are due to charge-spin recombination, and their
condensation reveals the d-wave SC ground-state. In particular, this SC-state
is controlled by both the SC gap function and the quasiparticle coherence,
then the maximal SC transition temperature occurs around the optimal doping,
and decreases in both underdoped and overdoped regimes. The unique feature of
this kinetic energy driven SC mechanism is that the pairing comes out from the
kinetic energy by exchanging spin excitations and is not driven by the
magnetic superexchange interaction as in the resonant valence bond type
theories anderson87 . Within the framework of the kinetic energy driven
superconductivity, we have discussed the low energy electronic structure
feng07 ; guo07 of cuprate superconductors and the spin response feng0306 ;
cheng08 , and qualitatively reproduced some main features of ARPES experiments
ding9495 ; damascelli03 and inelastic neutron scattering dai01 ; arai99
measurements on cuprate superconductors.
The layered crystal structure gives rise to a strong anisotropy of cuprate
superconductors, and it is possible to observe both in-plane and inter-plane
electromagnetic responses. The former one is characterized by the ab-plane
magnetic field penetration depth, whereas the latter one is related to the
magnetic field penetration in the c-axis direction. In this paper we
concentrate on the in-plane electromagnetic response based on the kinetic
energy driven superconductivity and do not consider c-axis properties, which
can be discussed, e.g., by taking into account hopping between adjacent
copper-oxides layers within the tunneling Hamiltonian approach sheehy04 .
The paper is organized as follows. Within the framework of the kinetic energy
driven d-wave superconductivity feng0306 , we discuss the electromagnetic
response of cuprate superconductors in Section II, deriving the kernel of the
linear response with a purely transverse vector potential. In Section III,
based on the specular reflection model landau80 ; tinkham96 , we calculate the
temperature and doping dependence of quantitative characteristics of the
electromagnetic response, such as the local magnetic field profile, the
magnetic field penetration depth, and the superfluid density. Our results show
that the electromagnetic response in cuprate superconductors can be understood
within the framework of the kinetic energy driven d-wave SC mechanism in the
presence of a weak external magnetic field. We conclude the paper with a brief
summary in Section IV. In Appendix A we present a method to generalize the
approach in order to obtain gauge invariant results.
## II Electromagnetic response in cuprate superconductors
A common feature of cuprate superconductors is the presence of two-dimensional
CuO2 planes, and it is believed that the unconventional physical properties of
these systems are closely related to the doped CuO2 plane damascelli03 . It
has been argued that the essential physics of the doped CuO2 plane anderson87
; damascelli03 is captured by the _t–J_ model on a square lattice. However,
for discussions of the electromagnetic response in cuprate superconductors,
the _t–J_ model can be extended by including the exponential Peierls factors
as,
$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\sum_{l\hat{\eta}\sigma}e^{-i({e}/{\hbar}){\bf{A}}(l)\cdot\hat{\eta}}C^{\dagger}_{l\sigma}C_{l+\hat{\eta}\sigma}+\mu\sum_{l\sigma}C^{\dagger}_{l\sigma}C_{l\sigma}$
(1) $\displaystyle+$ $\displaystyle J\sum_{l\hat{\eta}}{\bf S}_{l}\cdot{\bf
S}_{l+\hat{\eta}},$
where $\hat{\eta}=\pm\hat{x},\pm\hat{y}$, $C^{\dagger}_{l\sigma}$
($C_{l\sigma}$) is the electron creation (annihilation) operator, ${\bf
S}_{l}=(S^{x}_{l},S^{y}_{l},S^{z}_{l})$ are spin operators, $\mu$ is the
chemical potential, and the exponential Peierls factors account for the
coupling of electrons to the weak external magnetic field in terms of the
vector potential ${\bf{A}}(l)$ hirsch92 ; misawa94 . This $t$-$J$ model is
subject to an important local constraint
$\sum_{\sigma}C^{\dagger}_{l\sigma}C_{l\sigma}\leq 1$ in order to avoid the
double occupancy. The strong electron correlation in the $t$-$J$ model
manifests itself by this local constraint anderson87 , which can be treated
properly in analytical calculations within the charge-spin separation (CSS)
fermion-spin theory feng0304 , where the constrained electron operators are
decoupled as $C_{l\uparrow}=h^{\dagger}_{l\uparrow}S^{-}_{l}$ and
$C_{l\downarrow}=h^{\dagger}_{l\downarrow}S^{+}_{l}$, with the spinful fermion
operator $h_{l\sigma}=e^{-i\Phi_{l\sigma}}h_{l}$ representing the charge
degree of freedom together with some effects of spin configuration
rearrangements due to the presence of the doped hole itself (charge carrier),
while the spin operator $S_{l}$ represents the spin degree of freedom. In
particular, it has been shown that under the decoupling scheme, this CSS
fermion-spin representation is a natural representation of the constrained
electron defined in the Hilbert subspace without double electron occupancy
feng07 . The advantage of this CSS fermion-spin approach is that the electron
single occupancy local constraint is satisfied in analytical calculations.
Furthermore, these charge carriers and spins are gauge invariant, and in this
sense, they are real and can be interpreted as the physical excitations
laughlin97 . In this CSS fermion-spin representation, the _t–J_ model (1) can
be expressed as,
$\displaystyle H$ $\displaystyle=$ $\displaystyle
t\sum_{l\hat{\eta}}e^{-i({e}/{\hbar}){\bf{A}}(l)\cdot\hat{\eta}}(h^{\dagger}_{l+\hat{\eta}\uparrow}h_{l\uparrow}S^{+}_{l}S^{-}_{l+\hat{\eta}}$
(2) $\displaystyle+$ $\displaystyle
h^{\dagger}_{l+\hat{\eta}\downarrow}h_{l\downarrow}S^{-}_{l}S^{+}_{l+\hat{\eta}})$
$\displaystyle-$
$\displaystyle\mu\sum_{l\sigma}h^{\dagger}_{l\sigma}h_{l\sigma}+J_{{\rm
eff}}\sum_{l\hat{\eta}}{\bf S}_{l}\cdot{\bf S}_{l+\hat{\eta}},$
where $J_{{\rm eff}}=(1-\delta)^{2}J$, and $\delta=\langle
h^{\dagger}_{l\sigma}h_{l\sigma}\rangle=\langle h^{\dagger}_{l}h_{l}\rangle$
is the charge carrier doping concentration. As an important consequence, the
kinetic energy term in the _t–J_ model has been transferred as the charge
carrier-spin interaction, which reflects that even the kinetic energy term in
the _t–J_ Hamiltonian has strong Coulomb contribution due to the restriction
of no double occupancy of a given site.
In the case of zero magnetic field, we feng0306 have shown in terms of
Eliashberg’s strong coupling theory mahan00 that the charge carrier-spin
interaction from the kinetic energy term in the _t–J_ model (2) induces a
charge carrier pairing state with the d-wave symmetry by exchanging spin
excitations in the higher power of the charge carrier doping concentration
$\delta$, then the SC transition temperature is identical to the charge
carrier pair transition temperature. Moreover, it has been shown that this SC
state is the conventional BCS-like with the d-wave symmetry feng07 ; guo07 ,
so that the basic BCS formalism with the d-wave SC gap function is still valid
in quantitatively reproducing all main low energy features of the SC coherence
of the quasiparticle peaks in cuprate superconductors, although the pairing
mechanism is driven by the kinetic energy by exchanging spin excitations, and
other exotic magnetic scattering dai01 ; arai99 is beyond the BCS formalism.
Following the previous discussions feng0306 ; feng07 ; guo07 , the full charge
carrier diagonal and off-diagonal Green’s functions in the SC state can be
obtained explicitly in the Nambu representation as,
$\displaystyle\mathbb{G}({\bf{k}},i\omega_{n})=Z_{\rm{hF}}\,\frac{i\omega_{n}\tau_{0}+\bar{\xi}_{\bf{k}}\tau_{3}-\bar{\Delta}_{\rm{hZ}}({\bf{k}})\tau_{1}}{(i\omega_{n})^{2}-E_{{\rm{h}}{\bf{k}}}^{2}},$
(3)
where $\tau_{0}$ is the unit matrix, $\tau_{1}$ and $\tau_{3}$ are Pauli
matrices, the renormalized charge carrier excitation spectrum $\bar{\xi}_{{\bf
k}}=Z_{\rm hF}\xi_{\bf k}$, with the mean-field (MF) charge carrier excitation
spectrum $\xi_{{\bf k}}=Zt\chi\gamma_{{\bf k}}-\mu$, the spin correlation
function $\chi=\langle S_{i}^{+}S_{i+\hat{\eta}}^{-}\rangle$, $\gamma_{{\bf
k}}=(1/Z)\sum_{\hat{\eta}}e^{i{\bf k}\cdot\hat{\eta}}$, $Z$ is the number of
the nearest neighbor sites, the renormalized charge carrier d-wave pair gap
function $\bar{\Delta}_{\rm hZ}({\bf k})=Z_{\rm hF}\bar{\Delta}_{\rm h}({\bf
k})$, where the effective charge carrier d-wave pair gap function
$\bar{\Delta}_{\rm h}({\bf k})=\bar{\Delta}_{\rm h}\gamma^{(d)}_{{\bf k}}$
with $\gamma^{(d)}_{{\bf k}}=({\rm cos}k_{x}-{\rm cos}k_{y})/2$, and the
charge carrier quasiparticle spectrum $E_{{\rm{h}}{\bf
k}}=\sqrt{\bar{\xi}^{2}_{{\bf k}}+|\bar{\Delta}_{\rm hZ}({\bf k})|^{2}}$,
while the charge carrier quasiparticle coherent weight $Z_{\rm hF}$ and the
effective charge carrier gap parameter $\bar{\Delta}_{\rm h}$ have been
determined self-consistently along with another seven quantities and
correlation functions feng0306 ; feng07 ; guo07 . Let us emphasize that the
quasiparticle coherent weight renormalizing the physical quantities naturally
emerges in our formalism (3), and then both the SC gap function and the
quasiparticle coherence control the SC state. Therefore in our approach there
is no need to introduce any phenomenological charge renormalization factors in
order to describe the electromagnetic response sheehy04 .
Now we turn to the discussion of the electromagnetic response in the kinetic
energy driven cuprate superconductors. The weak external magnetic field
applied to the system usually represents a weak perturbation, but the induced
field generated by supercurrents cancels this weak external field over most of
the volume of the sample. Consequently, the net field acts only very near the
surface on a scale of the magnetic field penetration depth and so it can be
treated as a weak perturbation on the system as a whole. Therefore the
electromagnetic response can be successfully studied within the linear
response approach fetter71 ; fukuyama69 , where the averaged value ${\bf{J}}$
of the induced microscopic screening current ${\bf{j}}$ in the presence of the
vector potential ${\bf{A}}$ is found as,
$J_{\mu}({\bf{q}},\omega)=-\sum\limits_{\nu=1}^{3}K_{\mu\nu}({\bf{q}},\omega)A_{\nu}({\bf{q}},\omega),$
(4)
where $K_{\mu\nu}$ is the kernel of the response function and the Greek
indices label the axes of the Cartesian coordinate system. Recall that, as
always in the linear response method, the thermal average of the supercurrent
is calculated with the unperturbed Hamiltonian, i.e. for ${\bf{A}}\equiv 0$ in
Eq. (2). Let us also note that the relation (4), which is local in the
reciprocal space, in general implies a nonlocal response in the coordinate
space.
The kernel, which plays a central role in the description of the
electromagnetic response, and once known allows us to calculate quantitative
characteristics of the electromagnetic response, can be separated into two
parts:
$K_{\mu\nu}({\bf{q}},\omega)=K^{({\rm{d}})}_{\mu\nu}({\bf{q}},\omega)+K^{({\rm{p}})}_{\mu\nu}({\bf{q}},\omega),$
(5)
a diamagnetic part $K^{({\rm{d}})}_{\mu\nu}$ and a paramagnetic one
$K^{({\rm{p}})}_{\mu\nu}$. The evaluation of the diamagnetic contribution
usually poses no difficulties since it is known almost immediately from the
form of the diamagnetic current operator: it turns out to be diagonal and
proportional to the average kinetic term. However, the paramagnetic part can
only be calculated approximately since it involves evaluation of a retarded
current-current correlation function (polarization bubble). As the retarded
function is inconvenient for perturbation analysis one usually proceeds with
the corresponding imaginary-time-ordered Matsubara function,
$P_{\mu\nu}({\bf{q}},\tau)=-\langle
T_{\tau}\\{j^{({\rm{p}})}_{\mu}({\bf{q}},\tau)j_{\nu}^{({\rm{p}})}(-{\bf{q}},0)\\}\rangle,$
(6)
where the paramagnetic current operator is defined in the imaginary time
$\tau$ Heisenberg picture. Hence, the main problem is reduced to the
evaluation of a retarded current commutator for the unperturbed system. The
retarded current-current correlation function is then obtained in a standard
way from the imaginary time Fourier transform
$P_{\mu\nu}({\bf{q}},i\omega_{n})$ of the Matsubara function (6) by analytic
continuation to real frequencies mahan00 .
The vector potential ${\bf{A}}$ (then the weak external magnetic field
$B=rot{\bf{A}}$) has been coupled to the electrons, which are now represented
by $C_{l\uparrow}=h^{\dagger}_{l\uparrow}S^{-}_{l}$ and
$C_{l\downarrow}=h^{\dagger}_{l\downarrow}S^{+}_{l}$ in the CSS fermion-spin
representation. However, in the CSS framework, the vector potential ${\bf{A}}$
is coupled to $h^{\dagger}_{l\sigma}$, while the corresponding weak external
magnetic field ${\bf B}=rot{\bf{A}}$ is coupled to ${\bf S}_{l}$ by including
the Zeeman term zhang09 in the Hamiltonian (1). For cuprate superconductors,
the upper critical magnetic field is 50 Tesla or greater around the optimal
doping. In this paper, we mainly focus on the case where the applied external
magnetic field $B<10$ mT is much less than the upper critical magnetic field.
In this case, the Zeeman term zhang09 in the Hamiltonian (1) has been
dropped, and then the electron current operator
$j_{\mu}=j_{\mu}^{(\rm{d})}+j_{\mu}^{(\rm{p})}$ can be obtained by
differentiating the Hamiltonian (2) with respect to the vector potential as,
$\displaystyle j_{\mu}^{(\rm{d})}$ $\displaystyle=$ $\displaystyle\frac{\chi
e^{2}t}{2\hbar^{2}}\sum\limits_{l\sigma}\left(h_{l+\hat{\mu}\,\sigma}^{\dagger}h_{l\,\sigma}+h_{l\,\sigma}^{\dagger}h_{l+\hat{\mu}\,\sigma}\right)A_{\mu}(l),~{}~{}~{}~{}$
(7a) $\displaystyle j_{\mu}^{(\rm{p})}$ $\displaystyle=$
$\displaystyle-\frac{i\chi
et}{2\hbar}\sum\limits_{l\sigma}\left(h_{l+\hat{\mu}\,\sigma}^{\dagger}h_{l\,\sigma}-h_{l\,\sigma}^{\dagger}h_{l+\hat{\mu}\,\sigma}\right),$
(7b)
being the diamagnetic and paramagnetic contributions, respectively.
Since the diamagnetic current is explicitly proportional to the vector
potential, it is straightforward to find the diamagnetic part of the response
kernel as,
$\displaystyle K_{\mu\nu}^{(\rm{d})}({\bf{q}})$ $\displaystyle=$
$\displaystyle-\frac{Z_{\rm{hF}}\chi e^{2}t}{\hbar^{2}}{1\over
N}\sum\limits_{{\bf{k}}}\delta_{\mu\nu}\cos k_{\mu}$ (8) $\displaystyle\times$
$\displaystyle\left(1-\frac{\bar{\xi}_{\bf{k}}}{E_{{\rm{h}}\bf{k}}}\tanh{\frac{\beta
E_{{\rm{h}}\bf{k}}}{2}}\right)$ $\displaystyle=$
$\displaystyle-\frac{2\chi\phi
e^{2}t}{\hbar^{2}}\,\delta_{\mu\nu}.~{}~{}~{}~{}~{}~{}$
The paramagnetic part of the response kernel is more complicated to calculate,
as it involves evaluation of the current-current correlation function (6). In
particular, if we want to keep the theory gauge invariant, it is crucial to
approximate the correlation function in a way maintaining local charge
conservation fukuyama69 ; schrieffer83 ; misawa94 ; arseev06 . Since in the
following calculations we will work with a fixed gauge of the vector
potential, we postpone the detailed discussion of this problem until Appendix
A. Starting with the paramagnetic current operator (7b), we can rewrite its
Fourier transform in the notation of Nambu fields
$\Psi^{\dagger}_{\bf{k}}=\left(h_{{\bf{k}}\,\uparrow}^{\dagger},h_{-{\bf{k}}\,\downarrow}\right)$
and
$\Psi_{{\bf{k}}+{\bf{q}}}=\left(h_{{\bf{k}}+{\bf{q}}\,\uparrow},h_{-{\bf{k}}-{\bf{q}}\,\downarrow}^{\dagger}\right)^{T}$
as
$j^{(\rm{p})}_{\mu}({\bf{q}})={1\over
N}\sum\limits_{{\bf{k}}}\Psi_{{\bf{k}}}^{\dagger}\left[-\frac{\chi
et}{\hbar}\,e^{i\frac{q_{\mu}}{2}}\sin\left(k_{\mu}+\frac{q_{\mu}}{2}\right)\tau_{0}\right]\Psi_{{\bf{k}}+{\bf{q}}}.$
(9)
For the purpose of the discussion addressing the gauge invariance problem,
presented in Appendix A, it is convenient to find the charge density in the
Nambu notation as well. Within the CSS fermion-spin scheme, we first find
$\rho({\bf{q}})\approx-({e}/{2N})\sum_{{\bf{k}}}(\delta_{{\bf{q}},0}-h_{{\bf{k}}\,\uparrow}^{\dagger}h_{{\bf{k}}+{\bf{q}}\,\uparrow}-h_{{\bf{k}}\,\downarrow}^{\dagger}h_{{\bf{k}}+{\bf{q}}\,\downarrow})$.
Then the paramagnetic four-current operator can be represented in the Nambu
form as
$j_{\mu}^{\rm{(p)}}({\bf{q}})=\sum\limits_{{\bf{k}}}\Psi_{{\bf{k}}}^{\dagger}\gamma_{\mu}({\bf{k+q}},{\bf{k}})\Psi_{{\bf{k}}+{\bf{q}}}$,
where the bare current vertex,
${\mathbf{\gamma}}_{\mu}({\bf{k}}+{\bf{q}},{\bf{k}})=\left\\{\begin{array}[]{ll}-\frac{\chi
et}{\hbar}\,e^{i\frac{q_{\mu}}{2}}\sin\left(k_{\mu}+\frac{q_{\mu}}{2}\right)\tau_{0}&{\rm{for}}\
\mu\neq 0\\\ -\frac{e}{2}\,\tau_{3}&{\rm{for}}\ \mu=0.\\\ \end{array}\right.$
(10)
It is necessary to be aware that we are calculating the polarization bubble
with the paramagnetic current operator (9), i.e., bare current vertices (10),
but charge carrier Green functions. Consequently, as in this scenario we do
not take into account longitudinal excitations properly schrieffer83 ;
misawa94 , the obtained results are valid only in the gauge, where the vector
potential is purely transverse, e.g. in the Coulomb gauge. In this case, we
can obtain the correlation function (6) in the Matsubara representation as,
$\displaystyle P_{\mu\nu}({\bf{q}},i\omega_{n})$ $\displaystyle=$
$\displaystyle\left(\frac{\chi
et}{\hbar}\right)^{2}e^{\frac{i}{2}(q_{\mu}-q_{\nu})}{1\over
N}\sum\limits_{{\bf{k}}}\sin\left(k_{\mu}+\frac{q_{\mu}}{2}\right)\sin\left(k_{\nu}+\frac{q_{\nu}}{2}\right)\frac{1}{\beta}\sum\limits_{i\nu_{m}}{\rm{Tr}}\,\left[{\mathbb{G}}({\bf{k+q}},i\omega_{n}+i\nu_{m}){\mathbb{G}}({\bf{k}},i\nu_{m})\right].~{}~{}~{}~{}$
(11)
Restricting the discussion to the static limit ($\omega\sim 0$) and completing
the summation over Matsubara frequencies, we obtain the bare vertex current-
current correlation function, and hence the paramagnetic part of the response
kernel as,
$\displaystyle K_{\mu\nu}^{(\rm{p})}({\bf{q}},0)$ $\displaystyle=$
$\displaystyle-\left(\frac{\chi
etZ_{\rm{hF}}}{\hbar^{2}}\right)^{2}e^{\frac{i}{2}(q_{\mu}-q_{\nu})}{1\over
N}\sum\limits_{{\bf{k}}}\sin\left(k_{\mu}+\frac{q_{\mu}}{2}\right)\sin\left(k_{\nu}+\frac{q_{\nu}}{2}\right)$
(12) $\displaystyle\times$
$\displaystyle\left\\{\frac{1}{E_{{\rm{h}}{\bf{k}}}+E_{{\rm{h}}{\bf{k+q}}}}\left[1-\frac{\bar{\xi}_{{\bf{k+q}}}\bar{\xi}_{{\bf{k}}}+\bar{\Delta}_{{\rm{hZ}}}({\bf{k+q}})\bar{\Delta}_{{\rm{hZ}}}({\bf{k}})}{E_{{\rm{h}}{\bf{k}}}E_{{\rm{h}}{\bf{k+q}}}}\right]\left[1-n_{\rm{F}}(E_{{\rm{h}}{\bf{k}}})-n_{\rm{F}}(E_{{\rm{h}}{\bf{k+q}}})\right]\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{E_{{\rm{h}}{\bf{k}}}-E_{{\rm{h}}{\bf{k+q}}}}\left[1+\frac{\bar{\xi}_{{\bf{k+q}}}\bar{\xi}_{{\bf{k}}}+\bar{\Delta}_{{\rm{hZ}}}({\bf{k+q}})\bar{\Delta}_{{\rm{hZ}}}({\bf{k}})}{E_{{\rm{h}}{\bf{k}}}E_{{\rm{h}}{\bf{k+q}}}}\right]\left[n_{\rm{F}}(E_{{\rm{h}}{\bf{k+q}}})-n_{\rm{F}}(E_{{\rm{h}}{\bf{k}}})\right]\right\\}.~{}~{}~{}~{}~{}$
Note that in the long wavelength limit, when $|{\bf{q}}|\to 0$, the former
term in Eq. (12) vanishes, and the latter turns into $-2\left({\chi
etZ_{\rm{hF}}}/{N\hbar^{2}}\right)^{2}\sum\limits_{{\bf{k}}}\sin k_{\mu}\sin
k_{\nu}\,n_{\rm{F}}(E_{{\rm{h}}{\bf{k}}})[1-n_{\rm{F}}(E_{{\rm{h}}{\bf{k}}})]$,
which is equal to zero in the zero-temperature limit. Hence, in this case, the
long wavelength electromagnetic response at zero temperature is determined by
the diamagnetic part of the kernel only.
## III Quantitative characteristics
The way the system reacts to a weak electromagnetic stimulus is entirely
described by the linear response kernel, which is calculated within a
microscopic model. Once the kernel is known, the effect of a weak external
magnetic field can be quantitatively characterized by experimentally
measurable quantities such as the magnetic field penetration depth and the
local magnetic field profile. Technically, we need to combine one of the
Maxwell equations with the relation (4) describing the response of the system
and solve them together for the vector potential. This is the step in which a
particular gauge of the vector potential—usually implied by the geometry of
the system—is set. However, the kernel function derived within the linear
response theory describes the response of an _infinite_ system. In order to
take into account the confined geometry of cuprate superconductors it is
necessary to introduce a surface being the boundary between the environment
and the sample. This can be done within the standard specular reflection model
landau80 ; tinkham96 with a two-dimensional geometry of the SC plane, in the
configuration with external magnetic field perpendicular to the ab plane, as
shown in Fig. 1. In the present paper we study magnetic field penetration
effects within the ab plane only, so our primary goal is to find and discuss
the magnetic field in-plane penetration depth.
Figure 1: Geometry of the specular reflection model. The current
${\bf{J}}_{\rm{ext}}$ simulates external magnetic field at the edge of the
sample ($x=0$), whereas the induced supercurrent ${\bf{J}}_{\rm{int}}$ is the
(linear) reaction of the system.
In order to simulate an external magnetic field at the surface of a two-
dimensional sample, we introduce an external current sheet
$J_{y,\rm{ext}}(x)=-2B_{0}\delta(x)/\mu_{0}$ at the edge $x=0$, where
$\mu_{0}$ is the magnetic permeability and $B_{0}$ is the amplitude of the
weak external magnetic field at the surface ($x=0$). From the Maxwell equation
for the curl of the local magnetic field
${\rm{rot}}\,{{\bf{h}}}=\mu_{0}({\bf{J}}_{\rm{int}}+{\bf{J}}_{\rm{ext}})=\mu_{0}{\bf{J}}_{\rm{int}}+[0,-2B_{0}\delta(x),0]$
and the fact, that the induced supercurrent ${\bf{J}}_{\rm{int}}$ flows along
the $y$ axis, we can state that the local magnetic field is of the form
${\bf{h}}({\bf{r}})=[0,0,h_{z}(x)]$. In order to discuss the magnetic field
penetration effect, spatial dependence of the local magnetic field has to be
found. Let us begin with the identity
$\rm{rot}\,\rm{rot}\,{\bf{A}}={\rm{grad}\,\rm{div}\,{\bf{A}}}-\nabla^{2}{\bf{A}}$
and choose the vector potential as ${\bf{A}}({\bf{r}})=[0,A_{y}(x),0]$ setting
the Coulomb gauge. In this case,
$q_{x}^{2}A_{y}({\bf{q}})=\mu_{0}\left[J_{y,\rm{int}}({\bf{q}})+J_{y,\rm{ext}}({\bf{q}})\right],$
because the vector potential has only non-zero $y$ component. Finally,
including the form of the external current, the linear relation (4) between
the induced supercurrent and the vector potential
$J_{y,\rm{int}}({\bf{q}})=-K_{yy}({\bf{q}})A_{y}({\bf{q}})$, and solving for
the vector potential we obtain,
$A_{y}({\bf{q}})=-8\pi^{2}B_{0}\,\frac{\delta(q_{y})\delta(q_{z})}{\mu_{0}K_{yy}({\bf{q}})+q_{x}^{2}}.$
(13)
Since the vector potential has only the $y$ component, the only non-zero
component of the local magnetic field ${\bf{h}}=\rm{rot}\,{\bf{A}}$ is that
along the $z$ axis and $h_{z}({\bf{q}})=iq_{x}A_{y}({\bf{q}})$. Substituting
the derived form of the vector potential (13), and taking the inverse Fourier
transform, the local magnetic field profile can be obtained as,
$h_{z}(x)=\frac{B_{0}}{\pi}\int\limits_{-\infty}^{\infty}{\rm{d}}q_{x}\,\frac{q_{x}\sin
q_{x}x}{\mu_{0}K_{yy}(q_{x},0,0)+q_{x}^{2}}.$ (14)
Local magnetic field profiles can be measured experimentally, e.g. using the
muon-spin rotation technique khasanov04 ; suter04 , providing an important
tool to investigate the details of magnetic field screening inside the sample.
In cuprate superconductors the screening is found to be of exponential
character khasanov04 ; suter04 , in support of a local (London-type) nature of
the electrodynamics schrieffer83 . For the convenience of the following
discussions, we introduce a characteristic length scale
$a_{0}=\sqrt{\hbar^{2}a/\mu_{0}e^{2}J}$. Using a reasonably estimative value
of $J/k_{\rm{B}}\approx 1000$K and $a\approx 0.383$nm, which is the lattice
parameter for the cuprate superconductor YBa2Cu3O7-y, we obtain $a_{0}\approx
97.8$nm. In Fig. 2, we plot the local magnetic field profile (14) as a
function of the distance from the surface at temperature $T=0.02J$ for the
doping concentration $\delta=0.150$ (solid line), $\delta=0.147$ (dashed
line), and $\delta=0.144$ (dotted line) with parameter $t=2.5J$. For
comparison, the corresponding experimental result suter04 of the local
magnetic field profiles for the high quality YBa2Cu3O7-y sample is also shown
in Fig. 2 (inset, bottom). If a weak external field $B_{0}\approx 10$ mT is
applied to the system just as it has been done in the experimental measurement
suter04 , then the experimental result suter04 for YBa2Cu3O7-y is well
reproduced. In particular, our theoretical results perfectly follow an
exponential law as expected for the local electrodynamic response.
Figure 2: The local magnetic field profile as a function of the distance from
the surface at temperature $T=0.02J$ for doping concentration $\delta=0.150$
(solid line), $\delta=0.147$ (dashed line), and $\delta=0.144$ (dotted line)
with parameter $t=2.5J$. Inset (top): zoom into the intermediate range of the
local magnetic field profile. Inset (bottom): the corresponding experimental
result for YBa2Cu3O7-y taken from Ref. suter04, . Figure 3: Temperature
dependence of the magnetic field in-plane penetration depth $\Delta\lambda(T)$
for the doping concentration $\delta=0.150$ (solid line), $\delta=0.149$
(dashed line), and $\delta=0.148$ (dotted line) with parameter $t/J=2.5$.
Inset: the corresponding experimental data for YBa2Cu3O7-y taken from Ref.
kamal98, .
The above obtained local magnetic field profile $h_{z}(x)$ allows us to
determine the magnetic field in-plane penetration depth $\lambda(T)$ in a
straightforward way. According to the definition
$\lambda(T)=B_{0}^{-1}\int_{0}^{\infty}h_{z}(x)\,{\rm{d}}x$, the magnetic
field in-plane penetration depth can be evaluated as,
$\lambda(T)=\frac{2}{\pi}\int\limits_{0}^{\infty}\frac{{\rm{d}}q_{x}}{\mu_{0}K_{yy}(q_{x},0,0)+q_{x}^{2}}.$
(15)
In this case, we obtain the zero-temperature magnetic field in-plane
penetration depth $\lambda(0)\approx 380.8$nm for the doping concentration
$\delta=0.150$ with parameter $t/J=2.5$. This anticipated value is very close
to the values of the magnetic field in-plane penetration depth $\lambda\approx
156$nm $\sim 400$nm observed in different families of cuprate superconductors
bernhard01 ; khasanov04 ; uemura93 . Furthermore,
$\Delta\lambda(T)=\lambda(T)-\lambda(0)$ as a function of temperature $T$ for
the doping concentration $\delta=0.150$ (solid line), $\delta=0.149$ (dashed
line), and $\delta=0.148$ (dotted line) with parameter $t/J=2.5$ is plotted in
Fig. 3 in comparison with the corresponding experimental results kamal98 of
YBa2Cu3O7-y (inset). Our theoretical results show linear characteristics of
the magnetic field in-plane penetration depth $\Delta\lambda(T)$, except for
extremely low temperatures where a strong deviation from the linear
characteristics (a nonlinear effect) appears. In particular, this crossover
from the linear temperature dependence in the low temperature regime into the
nonlinear one at extremely low temperatures is observed experimentally in
nominally clean crystals of cuprate superconductors bonn96 ; kamal98 ;
jackson00 ; panagopoulos99 ; pereg07 ; khasanov04 ; suter04 ; sonier99 .
Apparently, there is a substantial difference between theory and experiment,
namely, the value of the difference between $\lambda(T)$ and $\lambda(0)$
calculated theoretically is much smaller than the corresponding value measured
in the experiment. However, upon a closer examination one can see immediately
that the main difference is due to fact that the calculated $\lambda(T)$
increases slowly with temperature. As for a qualitative discussion in this
paper, the overall tendency seen in the theoretical result is consistent with
that in the experiment kamal98 . In cuprate superconductors, the values of $J$
and $t$ are believed to vary somewhat from compound to compound damascelli03 .
Therefore the quantitative agreement can be reached by adjustments of theory’s
parameters $t$ and $J$, or by introducing the next neighbor hopping
$t^{\prime}$.
Figure 4: Doping dependence of the zero-temperature in-plane superfluid
density in the underdoped regime with $t/J=2.5$. Inset: the corresponding
experimental result for YBa2Cu3O7-y taken from Ref. broun07, .
A quantity which is closely related to the magnetic field in-plane penetration
depth $\lambda(T)$ is the in-plane superfluid density $\rho_{\rm
s}(T)\equiv\lambda^{-2}(T)$. For a better understanding of the physical
properties of cuprate superconductors, we have calculated the doping
dependence of the zero-temperature in-plane superfluid density $\rho_{\rm
s}(0)$ in the underdoped regime. The result for parameter $t/J=2.5$ is plotted
in Fig. 4 in comparison with the corresponding experimental data broun07 for
YBa2Cu3O7-y (inset). It is shown that the in-plane superfluid density
$\rho_{\rm s}(0)$ in the underdoped regime vanishes more or less linearly with
decreasing doping concentration $\delta$, in qualitative agreement with
experimental results of cuprate superconductors uemura8991 ; broun07 ;
bernhard01 . This result also is a natural consequence of the linear doping
dependence of the SC transition temperature $T_{c}\propto\delta$ in the
underdoped regime in the framework of the kinetic energy driven SC mechanism
feng0306 , where the SC transition temperature $T_{c}$ is set by the charge
carrier doping concentration, and then the density of the charge carriers
directly determines the in-plane superfluid density in the underdoped regime.
The appearance of the nonlinearity in the temperature dependence of the
magnetic field in-plane penetration depth in cuprate superconductors at
extremely low temperatures, as shown in Fig. 3, can be attributed to the
nonlocal effects, which in the case of a pure d-wave cuprate superconductor
with nodes in the gap become significant for the electromagnetic response
yip92 ; kosztin97 ; franz97 ; li00 ; sheehy04 . In general, the relation
between the supercurrent and the vector potential (4) is nonlocal in the
coordinate space due to the finite size of charge carrier Cooper pairs. In the
framework of the kinetic energy driven d-wave SC mechanism, the size of charge
carrier pairs in the clean limit is of the order of the coherence length
$\zeta({\bf{k}})=\hbar v_{\rm F}/\pi\Delta_{\rm h}({\bf{k}})$, where
$v_{\rm{F}}=\hbar^{-1}\partial\xi_{\bf k}/\partial{\bf k}|_{k_{F}}$ is the
charge carrier velocity at the Fermi surface, and therefore the size of charge
carrier pairs is momentum dependent. Although the weak external magnetic field
decays exponentially on the scale of the magnetic field in-plane penetration
length $\lambda(T)$, any nonlocal contributions to measurable quantities are
of the order of $\kappa^{-2}$, where $\kappa$, known as the Ginzburg–Landau
parameter, is the ratio of the magnetic field in-plane penetration depth
$\lambda$ and the coherence length $\zeta$. However, in the d-wave cuprate
superconductors, the characteristic feature is the existence of four nodal
points $[\pm\pi/2,\pm\pi/2]$ in the Brillouin zone, where the charge carrier
gap function vanishes $\Delta_{\rm
h}({\bf{k}})|_{[\pm\pi/2,\pm\pi/2]}=\Delta_{\rm h}({\rm cos}k_{x}-{\rm
cos}k_{y})/2|_{[\pm\pi/2,\pm\pi/2]}=0$. As a consequence, the coherence length
$\zeta({\bf{k}})$ diverges around the nodes. In particular, at extremely low
temperatures, the quasiparticles selectively populate the nodal region, and
the major contribution to measurable quantities comes from these
quasiparticles. In this case, the Ginzburg–Landau ratio $\kappa({\bf{k}})$
around the nodal region is no longer large enough for the system to belong to
the class of type-II superconductors, and the condition of the local limit is
not fulfilled kosztin97 . On contrary, the system falls then into the extreme
nonlocal limit, and therefore the nonlinear characteristic in the temperature
dependence of the magnetic field in-plane penetration depth can be observed
experimentally in cuprate superconductors at sufficiently low temperatures
bonn96 ; khasanov04 ; suter04 ; sonier99 . However, with increasing
temperature, the quasiparticles around the nodal region become excited out of
the condensate, and the nonlocal effect fades away. In this case, the momentum
dependent coherence length $\zeta({\bf{k}})$ can be replaced approximately
with the isotropic one $\zeta_{0}=\hbar v_{\rm F}/\pi\Delta_{\rm h}$. Then the
Ginzburg–Landau parameter $\kappa_{0}\approx\lambda(0)/\zeta_{0}\approx 180$,
and the condition for the local limit is satisfied. This anticipated value of
the Ginzburg–Landau parameter $\kappa_{0}\approx 180$ is not too far from the
range $\kappa_{0}\approx 150\sim 400$ estimated experimentally for different
families of cuprate superconductors bernhard01 ; khasanov04 ; uemura93 .
Consequently, the cuprate superconductors at moderately low temperatures turn
out to be type-II superconductors, where nonlocal effects are negligible, the
electrodynamics is purely local and the magnetic field decays exponentially
over a length of the order of a few hundreds nm. In this local limit, the pure
d-wave pairing state in the kinetic energy driven SC mechanism gives the
magnetic field penetration depth $\lambda(T)\propto T$ tsuei00 ; kosztin97 .
This is why the linear temperature dependence of the magnetic field in-plane
penetration depth $\lambda(T)$ is observed experimentally bonn96 ; hardy93 ;
kamal98 ; jackson00 ; panagopoulos99 ; pereg07 in cuprate superconductors at
moderately low temperatures.
Finally, we have to note that a deviation from the linear Uemura relation
between the in-plane superfluid density $\rho_{\rm s}(0)$ and doping
concentration $\delta$ has been observed recently in the underdoped regime
broun07 ; pereg07 ; hardy04 . This deviation from the linear Uemura relation
suggests a sublinear dependence of the critical temperature $T_{\rm{c}}$ and
the superfluid density $\rho_{\rm s}(0)$, since $T_{\rm{c}}$ must fall to zero
when $\rho_{\rm s}(0)$ does pereg07 ; hardy04 . The parent compound of doped
cuprate superconductors is a Mott insulator with an antiferromagnetic long-
range order and superconductivity occurs when the antiferromagnetic long-range
order state is suppressed by doped charge carriers. Since these doped charge
carriers in cuprate superconductors are induced by the replacement of some
ions by other ones with different valences, or the addition of excess oxygens
in the block layer, therefore, in principle, all cuprate superconductors have
natural impurities damascelli03 . Therefore the impurities play an important
role in the electromagnetic response and lead to some subtle differences in
the electromagnetic response for different families of cuprate superconductors
bonn96 . In this case, the impurity effect on the SC state of cuprate
superconductors is also a possible source for the deviation from the linear
Uemura relation. In this context we wang08 have discussed the effect of the
extended impurity scatterers on the quasiparticle transport of cuprate
superconductors in the SC state based on the nodal approximation of the
quasiparticle excitations and scattering processes, and predicted that in
contrast with the dome shape of the doping dependent SC gap parameter, the
minimum of the microwave conductivity occurs around the optimal doping, and
then increases in both underdoped and overdoped regimes. However, in this
paper we are primarily interested in exploring the general notion of the
electromagnetic response in cuprate superconductors in the SC state. The
qualitative agreement between the present theoretical results in the clean
limit and experimental data for different families of cuprate superconductors
provides an important confirmation of the nature of the SC phase of cuprate
superconductors as a d-wave BCS-like SC state within the kinetic energy driven
SC mechanism.
## IV Conclusions
In this paper we have discussed the electromagnetic response in cuprate
superconductors within the framework of kinetic energy driven d-wave
superconductivity. Following the linear response theory and taking into
account the two-dimensional geometry of cuprate superconductors within the
specular reflection model, we have reproduced some main features of the
electromagnetic response experiments on cuprate superconductors, including the
exponential local magnetic field profile, the linear temperature dependence of
the in-plane penetration depth in the low temperature range and its nonlinear
temperature dependence at extremely low temperatures. Moreover, the linear
doping dependence of the zero-temperature in-plane superfluid density in the
underdoped regime has been reproduced. In particular, we have clearly
identified the limitations of the used approximations, especially with respect
to the problem of gauge invariance. Furthermore, we have proposed a method to
generalize the discussions in order to make them independent of a particular
choice of the vector potential.
###### Acknowledgements.
The authors would like to thank Dr. Zhi Wang and Dr. Yu Lan for helpful
discussions. This work was supported by the National Natural Science
Foundation of China under Grant No. 10774015, and the funds from the Ministry
of Science and Technology of China under Grant Nos. 2006CB601002 and
2006CB921300. MK gratefully acknowledges support from a research scholarship
funded by Institute of Physics, Wrocław University of Technology.
## Appendix A Gauge-invariant electromagnetic response
It is well known that gauge invariance is a direct consequence of local charge
conservation fukuyama69 ; schrieffer83 , which is mathematically expressed by
the charge density-current continuity equation or its Green function analogue
called the generalized Ward identity (GWI) fukuyama69 ; schrieffer83 ;
misawa94 ; arseev06
$-2N\sum\limits_{\mu=0}^{3}q_{\mu}\Gamma_{\mu}(k+q,k)=\tau_{3}\mathbb{G}^{-1}(k)-\mathbb{G}^{-1}(k+q)\tau_{3}.$
(16)
Here $\Gamma_{\mu}$ is a dressed version of the four-current vertex function,
and the four-vector notation $q=({\bf{q}},q_{0}=i\omega)$ along with the
metric $(1,1,1,-1)$ has been introduced.
Since the local charge conservation requirement is quite universal and
fundamental, it should be inherent to any theory of the electromagnetic
response which is expected to be gauge invariant. The purpose of this appendix
is to propose—within the framework of the kinetic energy driven
superconductivity—a method to dress the current vertex in a way, which does
not violate the GWI. Once such a method is found, the bare polarization bubble
(6) can be replaced with its dressed version presented in Fig. 5, and the
resulting kernel of the response function will provide correct results for any
gauge of the vector potential.
Figure 5: Dressed polarization bubble (Nambu notation). Here both the Green
function and the current vertex are dressed with the pairing interaction due
to the spin bubble.
In the first step we will note that
$-2N\sum\limits_{\mu=0}^{3}q_{\mu}\gamma_{\mu}\left(k+q,k\right)=\tau_{3}\mathbb{G}^{(0)-1}(k)-\mathbb{G}^{(0)-1}(k+q)\tau_{3},$
(17)
i.e. that the GWI for the bare current vertex is satisfied with the MF charge
carrier Green function
$\mathbb{G}^{(0)}(k)=[(i\omega_{n})^{2}-\xi_{\bf{k}}^{2}]^{-1}(i\omega_{n}\tau_{0}+\xi_{\bf{k}}\tau_{3})$.
Substituting the MF charge carrier Green function, the rhs of Eq. (17) turns
into
$\tau_{3}\mathbb{G}^{(0)-1}(k)-\mathbb{G}^{(0)-1}(k+q)\tau_{3}=\left(\xi_{\bf{k+q}}-\xi_{{\bf{k}}}\right)\tau_{0}-q_{0}\tau_{3}$.
Moreover, in the long wavelength limit, after including the explicit form of
the MF charge carrier dispersion relation found within the framework of the
kinetic energy driven superconductivity feng07 , it further simplifies to
$\tau_{3}\mathbb{G}^{(0)-1}(k)-\mathbb{G}^{(0)-1}(k+q)\tau_{3}\approx\left[-2t\chi(q_{x}\sin
k_{x}+q_{y}\sin k_{y})\right]\tau_{0}-q_{0}\tau_{3}.$ Now, recalling the form
of the bare vertex (10), we can notice that in the long wavelength limit the
scalar product on the left-hand side of Eq. (17)
$-q_{0}\gamma_{0}+{\bf{q}}{\bf{\gamma}}=(2N)^{-1}\left(\tau_{3}q_{0}-\tau_{0}\nabla_{\bf{k}}\xi_{\bf{k}}\cdot{\bf{q}}\right)$,
which proves the equality (17).
It is well known that in order to obtain a dressed vertex function, which does
not violate the GWI, a ladder-type approximation can be adapted schrieffer83 ;
fukuyama69 ; misawa94 . The nature of the pairing mechanism feng0306 ; feng07
, which originates from the spin bubble, suggests a ladder-like approximation
of the form,
$\displaystyle\Gamma_{\mu}(k+q,k)$ $\displaystyle=$
$\displaystyle\gamma_{\mu}(k+q,k)+\frac{1}{N}\,\frac{1}{\beta}\sum\limits_{p}\tau_{3}\mathbb{G}(k+p+q)\Gamma_{\mu}(k+p+q,k+p){\mathbb{G}}(k+p)\tau_{3}$
(18) $\displaystyle\times$
$\displaystyle\frac{1}{N}\sum\limits_{{\bf{p}}^{\prime}}\Lambda^{2}_{{\bf{p}}+{\bf{p}}^{\prime}+{\bf{k}}}\Pi({\bf{p}},{\bf{p}}^{\prime};ip_{m}),~{}~{}~{}~{}~{}$
which is graphically presented in Fig. 6.
Figure 6: Ladder-type approximation for the dressed vertex.
In order to prove that the approximation (18) for the dressed vertex in fact
implies a gauge invariant description of the electromagnetic response, it is
necessary and sufficient to check whether it does not violate the GWI (16). In
order to prove it, we insert the dressed vertex function (18) into the left-
hand side of Eq. (16) and use the identity
$-2N\sum_{\mu=0}^{3}q_{\mu}\Gamma_{\mu}(s+q,s)=\tau_{3}{\mathbb{G}}^{-1}(s)-{\mathbb{G}}^{-1}(s+q)\tau_{3}$
to obtain
$\displaystyle\sum\limits_{\mu=0}^{3}q_{\mu}\Gamma_{\mu}(k+q,k)$
$\displaystyle=$
$\displaystyle\sum\limits_{\mu=0}^{3}q_{\mu}\gamma_{\mu}(k+q,k)+\frac{1}{N}\,\frac{1}{\beta}\sum\limits_{p}\left(-\frac{1}{2N}\right)\left[\tau_{3}{\mathbb{G}}(k+p+q)\tau_{3}-\tau_{3}{\mathbb{G}}(k+p)\tau_{3}\right]$
(19) $\displaystyle\times$
$\displaystyle\frac{1}{N}\sum\limits_{{\bf{p}}^{\prime}}\Lambda^{2}_{{\bf{p}}+{\bf{p}}^{\prime}+{\bf{k}}}\Pi({\bf{p}},{\bf{p}}^{\prime};ip_{m}).~{}~{}~{}~{}~{}$
In the long wavelength limit we use the approximation
$\Lambda^{2}_{{\bf{p}}+{\bf{p}}^{\prime}+{\bf{k}}}\approx\Lambda^{2}_{{\bf{p}}+{\bf{p}}^{\prime}+{\bf{k}}+{\bf{q}}}$.
Then we can simplify Eq. (19) into
$\sum_{\mu=0}^{3}q_{\mu}\Gamma_{\mu}(k+q,k)\approx\sum_{\mu=0}^{3}q_{\mu}\gamma_{\mu}(k+q,k)-\left[\Sigma(k+q)\tau_{3}-\tau_{3}\Sigma(k)\right]/2N.$
Using the fact that the free vertex satisfies the GWI with the MF Green
function, as stated in Eq. (17), and arranging the terms with respect to the
Pauli matrices, we have
$\displaystyle-2N\sum\limits_{\mu=0}^{3}q_{\mu}\Gamma_{\mu}(k+q,k)$
$\displaystyle\approx$
$\displaystyle\tau_{3}[\mathbb{G}^{(0)-1}(k)-\Sigma(k)]$ $\displaystyle-$
$\displaystyle[\mathbb{G}^{(0)-1}(k+q)-\Sigma(k+q)]\tau_{3}.$
Hence, identifying the terms in the square brackets as dressed charge carrier
Green functions, we eventually obtain the GWI (16), what proves that the
ladder-type approximation (18) for the vertex function in the dressed
polarization bubble in Fig. 5 is consistent with the GWI. Consequently, the
kernel of the linear response calculated with the dressed polarization bubble
is gauge invariant.
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|
arxiv-papers
| 2009-04-01T08:41:51 |
2024-09-04T02:49:01.571715
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mateusz Krzyzosiak, Zheyu Huang, Shiping Feng, and Ryszard Gonczarek",
"submitter": "Shiping Feng",
"url": "https://arxiv.org/abs/0904.0093"
}
|
0904.0096
|
# Weibel instability and associated strong fields
in a fully 3D simulation of a relativistic shock
K.-I. Nishikawa11affiliation: Center for Space Plasma and Aeronomic Research,
University of Alabama in Huntsville, NSSTC, 320 Sparkman Drive, Huntsville, AL
35805; ken-ichi.nishikawa-1@nasa.gov , J. Niemiec22affiliation: Institute of
Nuclear Physics PAN, ul. Radzikowskiego 152, 31-342 Kraków, Poland , P.E.
Hardee33affiliation: Department of Physics and Astronomy, The University of
Alabama, Tuscaloosa, AL 35487 , M. Medvedev44affiliation: Department of
Physics and Astronomy, University of Kansas, KS 66045 ,
H. Sol55affiliation: LUTH, Observatore de Paris-Meudon, 5 place Jules Jansen,
92195 Meudon Cedex, France , Y. Mizuno11affiliation: Center for Space Plasma
and Aeronomic Research, University of Alabama in Huntsville, NSSTC, 320
Sparkman Drive, Huntsville, AL 35805; ken-ichi.nishikawa-1@nasa.gov , B.
Zhang66affiliation: Department of Physics, University of Nevada, Las Vegas, NV
89154 , M. Pohl77affiliation: Department of Physics and Astronomy, Iowa State
University, Ames, IA 50011 , M. Oka11affiliation: Center for Space Plasma and
Aeronomic Research, University of Alabama in Huntsville, NSSTC, 320 Sparkman
Drive, Huntsville, AL 35805; ken-ichi.nishikawa-1@nasa.gov , D. H.
Hartmann88affiliation: Department of Physics and Astronomy, Clemson
University, Clemson, SC 29634
###### Abstract
Plasma instabilities (e.g., Buneman, Weibel and other two-stream
instabilities) excited in collisionless shocks are responsible for particle
(electron, positron, and ion) acceleration. Using a new 3-D relativistic
particle-in-cell code, we have investigated the particle acceleration and
shock structure associated with an unmagnetized relativistic electron-positron
jet propagating into an unmagnetized electron-positron plasma. The simulation
has been performed using a long simulation system in order to study the
nonlinear stages of the Weibel instability, the particle acceleration
mechanism, and the shock structure. Cold jet electrons are thermalized and
slowed while the ambient electrons are swept up to create a partially
developed hydrodynamic (HD) like shock structure. In the leading shock,
electron density increases by a factor of $\lesssim 3.5$ in the simulation
frame. Strong electromagnetic fields are generated in the trailing shock and
provide an emission site. We discuss the possible implication of our
simulation results within the AGN and GRB context.
###### Subject headings:
relativistic jets: Weibel instability - shock formation - electron-positron
plasma, particle acceleration, magnetic field generation - particle-in-cell
††slugcomment: submitted to ApJL
## 1\. Introduction
Particle-in-cell (PIC) simulations can shed light on the microphysics within
relativistic shocks. Recent PIC simulations show that particle acceleration
occurs within the downstream jet (e.g., Frederiksen et al. 2004; Nishikawa et
al. 2003, 2005, 2006, 2008, 2009; Hededal et al. 2004; Hededal & Nishikawa
2005; Silva et al. 2003; Jaroschek et al. 2005; Chang, Spitkovsky & Arons
2008; Dieckmann, Shukla, & Drury 2008; Spitkovsky 2008a,b; Martins et al.
2009). In general, these simulations confirm that a relativistic shock in
weakly or non magnetized plasma is dominated by the Weibel instability (Weibel
1959). The associated current filaments and magnetic fields (e.g., Medvedev &
Loeb 1999) accelerate electrons (e.g., Nishikawa et al. 2006) and cosmic rays,
which affect the pre-shock medium (Medvedev & Zakutnyaya 2009).
In this paper we present new three-dimensional simulation results for an
electron-positron jet injected into an electron-positron plasma using a long
simulation grid. A leading and trailing shock system develops with strong
electromagnetic fields accompanying the trailing shock.
## 2\. Simulation Setup
The code used in this study is an MPI-based parallel version of the
relativistic electromagnetic particle (REMP) code TRISTAN (Buneman 1993;
Nishikawa et al. 2003, Niemiec et al. 2008). The simulations have been
performed using a grid with ($L_{\rm x},L_{\rm y},L_{\rm z})=(4005,131,131)$
cells and a total of $\sim 1$ billion particles (12 particles$/$cell$/$species
for the ambient plasma) in the active grid. The electron skin depth,
$\lambda_{\rm s}=c/\omega_{\rm pe}=10.0\Delta$, where $\omega_{\rm
pe}=(e^{2}n_{\rm a}/\epsilon_{0}m_{\rm e})^{1/2}$ is the electron plasma
frequency and the electron Debye length $\lambda_{\rm D}$ is half of the cell
size, $\Delta$. This computational domain is six times longer than in our
previous simulations (Nishikawa et al. 2006; Ramirez-Ruiz, Nishikawa & Hededal
2007). The jet-electron number density in the simulation reference frame is
$0.676~{}n_{\rm a}$, where $n_{\rm a}$ is the ambient electron density, and
the jet Lorentz factor is $\gamma_{j}=15$. The jet-electron/positron thermal
velocity is $v_{\rm j,th}=0.014~{}c$ in the jet reference frame, where $c=1$
is the speed of light. The electron/positron thermal velocity in the ambient
plasma is $v_{\rm a,th}=0.05~{}c$. As in our previous work (e.g., Nishikawa et
al. 2006) the jet is injected in a plane across the computational grid located
at $x=25\Delta$ in order to eliminate effects associated with the boundary at
$x=x_{\min}$. Radiating boundary conditions are used on the planes at
$x=x_{\min}$ and $x=x_{\max}$ and periodic boundary conditions on all
transverse boundaries (Buneman 1993).
The jet makes contact with the ambient plasma at a 2D interface spanning the
computational domain. Here the formation and dynamics of a small portion of a
much larger shock are studied in a spatial and temporal way that includes the
spatial development of nonlinear saturation and dissipation from the injection
point to the jet front defined by the fastest moving jet particles.
## 3\. Simulation Results
Figure 1a & b show the averaged (in the $y-z$ plane) (a) jet (red), ambient
(blue), and total (black) electron density and (b) electromagnetic field
energy divided by the total jet kinetic energy ($E^{\rm j}_{\rm
t}=\sum_{i=e,p}m_{\rm i}c^{2}(\gamma_{\rm j}-1)$) at $t=3250~{}\omega_{\rm
pe}^{-1}$. Here, “e” and “p” denote electron and positron. Positron density
profiles are similar to electron profiles.
Figure 1.— Averaged values of (a): jet (red), ambient (blue), and total
(black) electron density, and (b): electric (red) and magnetic (blue) field
energy divided by the jet kinetic energy at $t=3250~{}\omega_{\rm pe}^{-1}$.
Panel (c) shows the evolution of the total electron density in time intervals
of $\delta t=250~{}\omega_{\rm pe}^{-1}$. Diagonal lines indicate motion of
the jet front (blue: $\lesssim c$), predicted contact discontinuity speed
(green: $\sim 0.76~{}c$), and trailing density jump (red: $\sim 0.56~{}c$).
Ambient particles become swept up after jet electrons pass $x/\Delta\sim 500$.
By $t=3250~{}\omega_{\rm pe}^{-1}$, the density has evolved into a two-step
plateau behind the jet front. The maximum density in this shocked region is
about three times the initial ambient density. The jet-particle density
remains nearly constant up to near the jet front.
Current filaments and strong electromagnetic fields accompany growth of the
Weibel instability in the trailing shock region. The electromagnetic fields
are about four times larger than that seen previously using a much shorter
grid system ($L_{\rm x}=640\Delta$). At $t=3250~{}\omega_{\rm pe}^{-1}$, the
electromagnetic fields are largest at $x/\Delta\sim 1700$, and decline by
about one order of magnitude beyond $x/\Delta=2300$ in the shocked region
(Nishikawa 2006; Ramirez-Ruiz, Nishikawa & Hededal 2007).
Figure 1c shows the total electron density plotted at time intervals of
$\delta t=250~{}\omega_{\rm pe}^{-1}$. The jet front propagates with the
initial jet speed ($\lesssim c$). Sharp RMHD-simulation shock surfaces are not
created (e.g., Mizuno et al. 2009). A leading shock region (linear density
increase) moves with a speed between the fastest moving jet particles
$\lesssim c$ and a predicted contact discontinuity speed of $\sim 0.76~{}c$
(see §4). A contact-discontinuity region consisting of mixed ambient and jet
particles moves at a speed between $\sim 0.76~{}c$ and the trailing density
jump speed $\sim 0.56~{}c$. A trailing shock region moves with speed $\lesssim
0.56~{}c$, note the modest density increase just behind the large trailing
density jump.
Figure 2.— Phase-space distribution of jet (red) and ambient (blue) electrons
at $t=3250~{}\omega_{\rm pe}^{-1}$. About 18,600 electrons of both species are
selected randomly.
Figure 2 shows the phase-space distribution of jet (red) and ambient (blue)
electrons at $t=3250~{}\omega_{\rm pe}^{-1}$ and confirms our shock-structure
interpretation. The electrons injected with $\gamma_{j}v_{\rm x}\sim 15$
become thermalized due to Weibel instabililty-induced interactions. The swept-
up ambient electrons (blue) are heated by interaction with jet electrons. Some
ambient electrons are strongly accelerated.
Figure 3.— Velocity distributions at $t=3250~{}\omega_{\rm pe}^{-1}$. All jet
(red) and all ambient (blue), and at $x/\Delta>2300$ jet (orange) and ambient
(green) electrons are also plotted. The small (red) peak indicates jet
electrons injected at $\gamma_{j}=15$.
Figure 3 shows the velocity distribution of all jet and ambient electrons in
the simulation frame. The small peak indicates electrons injected at
$\gamma_{j}=15$. Jet electrons are accelerated to a non-thermal distribution.
Ambient electrons are also accelerated to speeds above the jet injection
velocity. The velocity distributions of jet and ambient electrons near the jet
front (at $x/\Delta>2300$) are also plotted. The fastest jet electrons,
$\gamma>20$, are located near the jet front. On the other hand, the fastest
ambient electrons are located farther behind the jet front (at
$x/\Delta<2300$). Thus, strong acceleration of the ambient electrons
accompanies the strong fields associated with the Weibel instability.
## 4\. Discussion
Our collisionless-shock structure can be compared to 1-D hydrodynamic (HD)
shock predictions (e.g., Blandford & McKee 1976; Zhang & Kobayashi 2005). The
speed of the contact discontinuity (CD) is given by ram pressure balance in
the CD frame. Our initial conditions allow us to set the total energy density
$e\equiv\rho c^{2}+p/(\Gamma-1)=\rho c^{2}$ and pressure $p=0$, so that the
speed in the ambient frame becomes (Rosen et al. 1999)
$\beta_{\rm cd}=[(\gamma_{\rm j}\eta^{1/2})/(\gamma_{\rm
j}\eta^{1/2}+1)]\beta_{\rm j},$ (1)
where $\eta\equiv\rho_{\rm j}/\rho_{\rm a}(=m_{\rm e}n_{\rm j}/m_{\rm e}n_{\rm
a})$ and mass densities are determined in the “jet” and “ambient” proper
frames. In the simulation $n_{\rm j}=0.0451n_{\rm a}$ and $\gamma_{\rm j}=15$,
and $\beta_{\rm cd}=0.759$ ($\gamma_{\rm cd}=1.54$) is the predicted CD speed.
Formally this should represent the average speed of particles in the CD
region.
The leading shock moves at a speed given by
$\gamma_{\rm ls}^{2}={{(\gamma_{\rm cd}+1)[\Gamma_{\rm sa}(\gamma_{\rm
cd}-1)+1]^{2}}\over{\Gamma_{\rm sa}(2-\Gamma_{\rm sa})(\gamma_{\rm cd}-1)+2}}$
(2)
where $5/3>\Gamma_{\rm sa}>4/3$ is the shocked ambient adiabatic index. Thus
the leading shock speed is predicted to be $0.865>\beta_{\rm ls}>0.783$
($2>\gamma_{\rm ls}>1.6$) where upper and lower limits correspond to upper and
lower limits of $\Gamma_{\rm sa}$, respectively.
The jump condition at the leading shock is
${n_{\rm sa}\over n_{\rm a}}={\Gamma_{\rm sa}\gamma_{\rm cd}+1\over\Gamma_{\rm
sa}-1},$ (3)
where $n_{\rm sa}$ is the shocked ambient density in the proper (CD) frame and
we find $5.34~{}n_{\rm a}<n_{\rm sa}<9.15~{}n_{a}$, where the lower and upper
limits correspond to the upper and lower limits to $\Gamma_{\rm sa}$,
respectively. Measured in the ambient (simulation) frame the shocked ambient
density should be $8.2~{}n_{\rm a}<\gamma_{\rm cd}n_{\rm sa}<14.1~{}n_{\rm
a}$. Formally this should represent the total density of particles in the
shocked-ambient region.
Computations associated with the trailing shock are most easily performed in
the jet rest frame designated below as the “primed” frame. In this frame the
CD moves with speed $\beta^{\prime}_{\rm cd}=-(\beta_{\rm j}-\beta_{\rm
cd})/(1-\beta_{\rm j}\beta_{\rm cd})=-0.984$ and $\gamma^{\prime}_{\rm
cd}=5.60$. The speed of the trailing shock in the jet frame,
$\gamma^{\prime}_{\rm ts}$ is given by eq. (2) but with $\gamma_{\rm
cd}\rightarrow\gamma^{\prime}_{\rm cd}$ and $\Gamma_{\rm
sa}\rightarrow\Gamma_{\rm sj}$ where $\Gamma_{\rm sj}$ is the shocked-jet
adiabatic index. In the jet frame $10.4>\gamma^{\prime}_{\rm ts}>7.4$ and
$0.995>-\beta^{\prime}_{\rm ts}>0.991$, where upper and lower limits
correspond to upper $\Gamma_{\rm sj}=5/3$ and lower $\Gamma_{\rm sj}=4/3$
limits to $\Gamma_{\rm sj}$, respectively. The trailing shock speed in the
ambient (simulation) frame is $0.35<\beta_{\rm ts}=(\beta_{\rm
j}-\beta^{\prime}_{\rm ts})/(1-\beta_{\rm j}\beta^{\prime}_{\rm ts})<0.61$
where the lower and upper limits correspond to the upper and lower limits of
$\Gamma_{\rm sj}$, respectively.
The density jump at the trailing shock is given by eq. (3) but with
$\gamma_{\rm cd}\rightarrow\gamma^{\prime}_{\rm cd}$ and $\Gamma_{\rm
sa}\rightarrow\Gamma_{\rm sj}$ where now $n_{\rm sa}/n_{a}\rightarrow n_{\rm
sj}/n_{\rm j}$ where $n_{\rm j}=0.0451~{}n_{\rm a}$ with result that the
proper density of shocked jet material is $0.70~{}n_{a}<n_{\rm
sj}<1.15~{}n_{\rm a}$ where lower and upper limits correspond to upper and
lower limits to $\Gamma_{\rm sj}$, respectively. In the ambient (simulation)
frame the shocked jet density should be $1.08~{}n_{\rm a}<\gamma_{\rm
cd}n_{\rm sj}<1.76~{}n_{\rm a}$. Formally this should represent the total
density of particles in the shocked jet region.
In the simulation the speed of the trailing density jump is $\sim 0.56~{}c$,
which is in the predicted range $0.35<\beta_{\rm ts}<0.61$, a typical speed
within the density-plateau region, $\sim 0.75~{}c$, is close to $\beta_{\rm
cd}=0.76$. The poorly defined leading shock structure moves at a speed between
$\sim 0.76~{}c$ and $\lesssim c$, consistent with the predicted
$0.78<\beta_{\rm ls}<0.86$.
In the simulation the maximum density increase observed in the ambient
(simulation) frame is $\gamma_{\rm cd}n_{\rm sa}/n_{\rm a}\sim 3.5$ behind the
leading shock (see Fig. 1a). This is about a factor of $\sim 3$ smaller than
the predicted increase, $8.2<\gamma_{\rm cd}n_{\rm sa}/n_{\rm a}<14.1$, for a
fully-developed leading shock. On the other hand, the density increase
observed in the ambient (simulation) frame of $\gamma_{\rm cd}n_{\rm
sj}/n_{\rm a}\gtrsim 1$ just before the trailing large density jump is
comparable to that predicted, $1.08<\gamma_{\rm cd}n_{\rm sj}/n_{\rm a}<1.76$,
for a fully developed trailing shock.
Our present results can be compared to those found in the 2-D simulations of
Chang et al. (2008) (see also Spitkovsky 2008a). Their simulations were
performed in the CD frame, and material with proper density, n, moved into the
contact discontinuity with a Lorentz factor $\gamma=15$. A shock moved away
from the CD with the predicted speed
$\beta_{\rm s}=(\Gamma_{\rm
s}-1)\left[{\gamma-1\over\gamma+1}\right]^{1/2}=0.47~{},$ (4)
and predicted density jump
${n_{\rm s}\over\gamma n}={1\over\gamma}{\Gamma_{\rm
s}\gamma+1\over\Gamma_{\rm s}-1}=3.13~{},$ (5)
for a shocked adiabatic index of $\Gamma_{\rm s}=3/2$.
In our simulation we have two shocks that move away from the CD. For our
leading shock, the ambient plasma moves relative to the CD at a speed equal to
$\beta_{\rm cd}=0.759$ and $\gamma=\gamma_{\rm cd}=1.54$ in eqs. 4 & 5\. In
the CD frame $\beta_{\rm s}=0.23$ and the observed density jump becomes
$n_{\rm sa}/\gamma_{\rm cd}n_{a}=4.3$ for $\Gamma_{\rm s}=3/2$. So we see that
our leading shock speed would be about 50% less than that in Chang et al.
(2008) and our density increase would be about 50% larger for a fully-
developed leading shock in the CD frame. For the trailing shock, the jet moves
toward the CD at a speed equal to $-\beta^{\prime}_{\rm cd}=0.984$ and
$\gamma=\gamma^{\prime}_{\rm cd}=5.60$ in eqs. 4 & 5\. In the CD frame
$\beta_{\rm s}=0.417$ and the observed density increase becomes
$n_{sj}/\gamma^{\prime}_{\rm cd}n_{j}=3.36$ for $\Gamma_{\rm s}=3/2$. So we
see that our trailing shock speed would be about 11% less than that in Chang
et al. (2008) and our density increase would be about 7% larger for the fully
developed trailing shock in the CD frame. The parameters associated with our
trailing shock are similar to those found in Chang et al. (2008), and the
Weibel filamentation structures are comparable but now studied in full 3-D.
## 5\. Conclusion
The present simulation finds for the first time a relativistic shock system
comparable to a predicted relativistic HD shock system consisting of leading
and trailing shocks separated by a contact discontinuity, albeit not yet fully
developed. One remarkable aspect of this shock system lies in the generation
of large electromagnetic fields, up to 30% of the kinetic energy density,
associated with the trailing shock. Electromagnetic fields in the leading
shock and contact-discontinuity region are over one order of magnitude lower.
The large value for $\epsilon_{B}\sim 0.3$ in our trailing shock hints that
Poynting-flux-dominated ejecta may not be required to explain some GRB
observations (McMahon et al. 2006).
Visualization of our dual shock system in the ambient (simulation) frame
provides a picture of the shock structure that should exist at the head of a
relativistic astrophysical jet, $\gamma_{\rm jt}=15$, that is less dense than
the surrounding medium, $n_{\rm jt}/n_{\rm am}=0.045$. Within the AGN context,
here we identify our trailing shock with the “jet” shock that decelerates the
relativistic jet and we would expect synchrotron emission to originate from
the strongly magnetized structure. Little synchrotron emission would originate
from the weakly magnetized “bow” shock in front of the contact discontinuity.
This in fact is what is observed at the leading edge of extra-galactic jets
where synchrotron emission from the bow shock is not typically observed.
Visualization of our dual shock system in the “jet” frame provides a picture
of the shock structure that would accompany a relativistic blast wave driven
by relativistic ejecta. Within the GRB context, here we identify the ambient
medium as representing relativistic ejecta moving at $\gamma_{\rm ej}=15$ into
a much less dense ISM, $n_{\rm ej}/n_{\rm ism}=22$. Our trailing shock is now
identified with the “forward” shock and we would expect synchrotron emission
from this strongly magnetized structure. Little synchrotron emission would
originate from the low Lorentz factor, weakly-magnetized “reverse” shock
moving back into the ejecta.
Our present simulation involves an electron-positron jet and ambient medium.
We might expect similar shock-structure development in electron-ion
simulations, albeit on much longer temporal and spatial scales.
This work is supported by AST-0506719, AST-0506666, NASA-NNG05GK73G,
NNX07AJ88G, NNX08AG83G, NNX08AL39G, and NNX09AD16G. JN is supported by MNiSW
research project N N203 393034, and The Foundation for Polish Science through
the HOMING program, which is supported by a grant from Iceland, Liechtenstein,
and Norway through the EEA Financial Mechanism. Simulations were performed at
the Columbia facility at the NASA Advanced Supercomputing (NAS) and Cobalt at
the National Center for Supercomputing Applications (NCSA) which is supported
by the NSF. Part of this work was done while K.-I. N. was visiting The
Observatoire de Paris, Meudon in summer of 2008. Support from the French
Natural Science Research Council is gratefully acknowledged.
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|
arxiv-papers
| 2009-04-01T08:45:28 |
2024-09-04T02:49:01.580553
|
{
"license": "Public Domain",
"authors": "K.-I. Nishikawa, J. Niemiec, P.E. Hardee, M. Medvedev, H. Sol, Y.\n Mizuno, B. Zhang, M. Pohl, M. Oka, D. H. Hartmann",
"submitter": "Ken-Ichi Nishikawa",
"url": "https://arxiv.org/abs/0904.0096"
}
|
0904.0130
|
# Morphological characterization of shocked porous material
Aiguo Xu, Guangcai Zhang, X. F. Pan, Ping Zhang, and Jianshi Zhu National Key
Laboratory of Computational Physics,
Institute of Applied Physics and Computational Mathematics, P. O. Box 8009-26,
Beijing 100088, P.R.China Xu_Aiguo@iapcm.ac.cn
###### Abstract
Morphological measures are introduced to probe the complex procedure of shock
wave reaction on porous material. They characterize the geometry and topology
of the pixelized map of a state variable like the temperature. Relevance of
them to thermodynamical properties of material is revealed and various
experimental conditions are simulated. Numerical results indicate that, the
shock wave reaction results in a complicated sequence of compressions and
rarefactions in porous material. The increasing rate of the total fractional
white area $A$ roughly gives the velocity $D$ of a compressive-wave-series.
When a velocity $D$ is mentioned, the corresponding threshold contour-level of
the state variable, like the temperature, should also be stated. When the
threshold contour-level increases, $D$ becomes smaller. The area $A$ increases
parabolically with time $t$ during the initial period. The $A(t)$ curve goes
back to be linear in the following three cases: (i) when the porosity $\delta$
approaches 1, (ii) when the initial shock becomes stronger, (iii) when the
contour-level approaches the minimum value of the state variable. The area
with high-temperature may continue to increase even after the early
compressive-waves have arrived at the downstream free surface and some
rarefactive-waves have come back into the target body. In the case of
energetic material needing a higher temperature for initiation, a higher
porosity is preferred and the material may be initiated after the precursory
compressive-waves have scanned all the target body. One may desire the
fabrication of a porous body and choose appropriate shock strength according
to what needed is scattered or connected hot-spots. With the Minkowski
measures, the dependence on experimental conditions is reflected simply by a
few coefficients. They may be used as order parameters to classify the maps of
physical variables in a similar way like thermodynamic phase transitions.
††: J. Phys. D: Appl. Phys.
## 1 Introduction
A porous material contains voids or tunnels of different shapes and sizes.
Such materials are commonly found in nature and as industrial materials such
as wood, carbon, foams, ceramics, bricks, metals and explosives. They have
also been used in surgical implant design to fabricate devices to replace or
augment soft and hard tissues, etc. In order to use them effectively, their
mechanical and thermodynamical properties must be understood in relation to
their mesoscopic structures[1, 2].
In this work we focus on porous materials under shock wave reaction. When a
porous material is shocked, the cavities inside the sample may result in jets
and influence its back velocity[3]. Cavity nucleation due to tension waves
controls the spallation behavior of the material[4]. Cavity collapse plays a
prominent role in the initiation of energetic reactions in explosives[5]. In
this side, most of previous studies concerned the Hugoniots[6, 7, 8, 9, 10,
11, 12, 13] and the equation of state[14, 15, 16]. It is known that, under
strong shocks, the porous material is globally in a nonequilibrium state and
show complex dissipative structures. How to describe and pick up information
from such a system is still an open problem. In this work we introduce the
Minkowski functionals to measure the morphological behaviors of the map of
state variable and use them to probe the procedure of shock wave reaction on
porous material.
This study needs also a powerful simulation tool. The molecular dynamics can
discover some atomistic mechanisms of shock-induced void collapse[17, 18], but
the spatial and temporal scales it may cover are far from those comparable
with experiments. To overcome this scale limitation, we resort to a newly
developed mesoscopic particle method, the material-point method(MPM)[19, 20,
21, 22, 23, 24]. The MPM was originally introduced in fluid dynamics by
Harlow, et al[19] and extended to solid mechanics by Burgess, et al[20], then
developed by various researchers, including us[25, 26, 27]. The other reason
for using the MPM is related to the severe difficulties of the traditional
Eulerian and Lagrangian methods in treating with shocked porous materials. The
material under investigation is generally highly distorted during the
collapsing of cavities. The Eulerian description is not convenient for
tracking interfaces. When the Lagrangian formulation is used, the original
element mesh becomes distorted so significantly that the mesh has to be re-
zoned to restore proper shapes of elements. The state fields of mass density,
velocities and stresses must be mapped from the distorted mesh to the newly
generated one. This mapping procedure is not a straightforward task, and
introduces errors. The MPM not only takes advantages of both the Lagrangian
and Eulerian algorithms but makes it possible to avoid their drawbacks as
well. At each time step, calculations consist of two parts: a Lagrangian part
and a convective one. Firstly, the computational mesh deforms with the body,
and is used to determine the strain increment, and the stresses in the sequel.
Then, the new position of the computational mesh is chosen (particularly, it
may be the previous one), and the velocity field is mapped from the particles
to the mesh nodes. Nodal velocities are determined using the equivalence of
momentum calculated for the particles and for the computational grid.
The following part of the paper is planned as follows. Section 2 briefly
reviews the Minkowski descriptions. Section 3 presents the theoretical model
of the material under consideration. Simulation results are shown and analyzed
in section 4. Section 5 makes the conclusion.
## 2 Brief review of morphological characterization
A variety of techniques can be used to describe the complex spatial
distribution and time evolution of state variables in the shocked porous
material. In this study we concentrate on the set of statistics known as
Minkowski functionals[28]. A general theorem of integral geometry states that
all properties of a $d$-dimensional convex set (or more generally, a finite
union of convex sets) which satisfy translational invariance and additivity
(called morphological properties) are contained in $d+1$ numerical values
[29]. For a pixelized map $\psi(\mathbf{x})$, we consider the excursion sets
of the map, defined as the set of all map pixels with value of $\psi$ greater
than some threshold $\psi_{th}$ (see, e.g., Refs. [30, 31]), where
$\mathbf{x}$ is the position, $\psi$ can be a state variable like temperature
$T$, density $\rho$ or pressure $P$; $\psi$ can also be the velocity
$\mathbf{v}$ or its components, some specific stress, etc. Then the $d+1$
functionals of these excursion sets completely describe the morphological
properties of the underlying map $\psi(\mathbf{x})$. In the case of two or
three dimensions, the Minkowski functionals have intuitive geometric
interpretations.
For a two-dimensional map, the three Minkowski functionals correspond
geometrically to the total fractional area $A$ of the excursion set, the
boundary length $L$ of the excursion set per unit area, and the Euler
characteristic $\chi$ per unit area (equivalent to the topological genus).
Such a description has been successfully used to describe patterns in
reaction-diffusion system[32], the cosmic microwave background temperature
fluctuations[33], and patterns in phase separation of complex fluids[34, 35,
36, 37], etc.
In this work we probe the shocked porous material via checking the temperature
map $T(\mathbf{x},t)$, where the time $t$ is explicitly denoted. The maps of
other physical variables can be analyzed in a similar way. When the
temperature $T(\mathbf{x})$ is beyond the threshold value $T_{th}$, the grid
node at position $\mathbf{x}$ is regarded as a white (or hot) vertex, else it
is regarded as a black (or cold) one. For the square lattice, a pixel
possesses four vertices. A region with connected white (hot) or black (cold)
pixels is defined as a white (hot) or black (cold) domain. Two neighboring
white and black domains present a clear interface or boundary. When we
increase the threshold contour-level $T_{th}$ from the lowest temperature to
the highest one in the system, the white area $A$ will decrease from $1$ to
$0$; the boundary length $L$ first increases from $0$, then arrives at a
maximum value, finally decreases to $0$ again. There are several ways to
define the Euler characteristic $\chi$. Two simplest ones are
$\chi=N_{W}-N_{B}\mathtt{,}$ (1)
or
$\chi=\frac{N_{W}-N_{B}}{N}\mathtt{,}$ (2)
where $N_{W}$ ($N_{B}$) is the number of connected white (black) domains, $N$
is the total number of pixels. The only difference of the two definitions is
that the first keeps $\chi$ an integer. In contrast to the white area $A$ and
boundary length $L$, the Euler characteristic $\chi$ describes the
connectivity of the domains in the lattice. It describes the pattern in a
purely topological way, i.e., without referring to any kind of metric. It is
negative (positive) if many disconnected black (white) regions dominate the
image. A vanishing Euler characteristic indicates a highly connected structure
with equal amount of black and white domains. Specifically, for the definition
(1), the integer $\chi$ equals $-1$ when one has a black drop in a large white
lattice, and $+1$ vice versa, since the surrounding white (black) region does
conventionally not count. In this paper, we use the second definition without
making any ambiguity. The ratio
$\kappa=\frac{N_{W}-N_{B}}{NL}$ (3)
describes the mean curvature of the boundary line separating black and white
domains. Despite having global meaning, the Euler characteristic $\chi$ can be
calculated in a local way using the additivity relation[32].
Figure 1: (in JPG format) Configurations with temperature contours. $\delta=2$
and $v_{init}=1000$m/s. From left to right, t=500ns, 1500ns, 2000ns, and
2500ns, respectively. The length unit here is 10 $\mu$m.
Figure 2: (Color online) Minkowski measures for the procedure shown in Fig.1.
The contour levels of the temperature increment are shown in the legend.
## 3 Theoretical model of the material
In this study the material is assumed to follow an associative von Mises
plasticity model with linear kinematic and isotropic hardening[38].
Introducing a linear isotropic elastic relation, the volumetric plastic strain
is zero, leading to a deviatoric-volumetric decoupling. So, it is convenient
to split the stress and strain tensors, $\boldsymbol{\sigma}$ and
$\boldsymbol{\varepsilon}$, as
$\displaystyle\boldsymbol{\sigma}$ $\displaystyle=$
$\displaystyle\mathbf{s}-P\mathbf{I},P=-\frac{1}{3}\verb|Tr|(\boldsymbol{\sigma})\mathtt{,}$
(4) $\displaystyle\boldsymbol{\varepsilon}$ $\displaystyle=$
$\displaystyle\mathbf{e}+\frac{1}{3}\theta\mathbf{I},\theta=\frac{1}{3}\verb|Tr|(\boldsymbol{\varepsilon})\mathtt{,}$
(5)
where $P$ is the pressure scalar, $\mathbf{s}$ the deviatoric stress tensor,
and $\mathbf{e}$ the deviatoric strain. The strain $\mathbf{e}$ is generally
decomposed as $\mathbf{e}=\mathbf{e}^{e}+\mathbf{e}^{p}$, where
$\mathbf{e}^{e}$ and $\mathbf{e}^{p}$ are the traceless elastic and plastic
components, respectively. The material shows a linear elastic response until
the von Mises yield criterion,
$\sqrt{\frac{3}{2}}\left\|\mathbf{s}\right\|=\sigma_{Y}\mathtt{,}$ (6)
is reached, where $\sigma_{Y}$ is the plastic yield stress. The yield
$\sigma_{Y}$ increases linearly with the second invariant of the plastic
strain tensor $\mathbf{e}^{p}$, i.e.,
$\sigma_{Y}=\sigma_{Y0}+E_{\tan}\left\|\mathbf{e}^{p}\right\|\mathtt{,}$ (7)
where $\sigma_{Y0}$ is the initial yield stress and $E_{\tan}$ the tangential
module. The deviatoric stress $\mathbf{s}$ is calculated by
$\mathbf{s}=\frac{E}{1+\nu}\mathbf{e}^{e}\mathtt{,}$ (8)
where $E$ is the Yang’s module and $\nu$ the Poisson’s ratio. Denote the
initial material density and sound speed by $\rho_{0}$ and $c_{0}$,
respectively. The shock speed $U_{s}$ and the particle speed $U_{p}$ after the
shock follows a linear relation, $U_{s}=c_{0}+\lambda U_{p}$, where $\lambda$
is a characteristic coefficient of material. The pressure $P$ is calculated by
using the Mie-Grüneissen state of equation which can be written as
$P-P_{H}=\frac{\gamma(V)}{V}[E-E_{H}(V_{H})]$ (9)
In Eq.(9), $P_{H}$, $V_{H}$ and $E_{H}$ are pressure, specific volume and
energy on the Rankine-Hugoniot curve, respectively. The relation between
$P_{H}$ and $V_{H}$ can be estimated by experiment and can be written as
$P_{H}=\left\\{\begin{array}[]{ll}\frac{\rho_{0}c_{0}^{2}(1-\frac{V_{H}}{V_{0}})}{(\lambda-1)^{2}(\frac{\lambda}{\lambda-1}\times\frac{V_{H}}{V_{0}}-1)^{2}},&V_{H}\leq
V_{0}\\\
\rho_{0}c_{0}^{2}(\frac{V_{H}}{V_{0}}-1),&V_{H}>V_{0}\end{array}\right.$ (10)
In this paper, the transformation of specific internal energy $E-E_{H}(V_{H})$
is taken as the plastic energy. Both the shock compression and the plastic
work cause the increasing of temperature. The increasing of temperature from
shock compression can be calculated as:
$\frac{\mathrm{d}T_{H}}{\mathrm{d}V_{H}}=\frac{c_{0}^{2}\cdot\lambda(V_{0}-V_{H})^{2}}{c_{v}\big{[}(\lambda-1)V_{0}-\lambda
V_{H}\big{]}^{3}}-\frac{\gamma(V)}{V_{H}}T_{H}.$ (11)
where $c_{v}$ is the specific heat. Eq.(11) can be derived from thermal
equation and the Mie-Grüneissen state of equation[39]. The increasing of
temperature from plastic work can be calculated as:
$\mathrm{d}T_{p}=\frac{\mathrm{d}W_{p}}{c_{v}}$ (12)
Both the Eq.(11) and the Eq.(12) can be written as the form of increment.
In this paper we choose aluminum as the sample material. The corresponding
parameters are $\rho_{0}=2700$ kg/m3, $E=69$ Mpa, $\nu=0.33$,
$\sigma_{Y0}=120$ Mpa, $E_{\tan}=384$ MPa, $c_{0}=5.35$ km/s, $\lambda=1.34$,
$c_{v}=880$ J/(Kg$\cdot$K), $k=237$ W/(m$\cdot$K) and $\gamma_{0}=1.96$ when
the pressure is below $270$ GPa. The initial temperature of the material is
300 K.
Figure 3: (in JPG format) Configurations with temperature contours.
$\delta=1.4$ and $v_{init}=1000$m/s. From left to right, t=500ns, 1100ns,
1400ns, and 1700ns, respectively. The length unit here is 10 $\mu$m.
Figure 4: (Color online) Minkowski measures for cases with various porosities.
$T_{th}=400$K. The values of porosity are shown in the legend.
Figure 5: (Color online) Minkowski measures for cases with various porosities.
$T_{th}$=500K. The values of porosity are shown in the legend.
Figure 6: (Color online) Minkowski measures for cases with various porosities.
$T_{th}$=600K. The values of porosity are shown in the legend.
## 4 Simulation results and physical interpretation
In our numerical experiments the porous material is fabricated by a solid
material body with an amount of voids randomly embedded. We denote the mean
density of the porous body as $\rho$ and the density of the solid portion as
$\rho_{0}$. The porosity is defined as $\delta=\rho_{0}/\rho$. The present
work concentrates on two-dimensional case and the porosity $\delta$ is
controlled by the total number $N_{void}$ and mean size $r_{void}$ of voids
embedded. The shock wave reacting on the target porous body is loaded via a
colliding by a rigid wall with the same material. We choose the coordinate
system where the rigid wall is horizontal and keeps static at the position
$y=0$, the target porous body is on the upper side of the rigid wall and moves
towards the rigid wall at a velocity $-v_{init}$. The porous body begins to
touch the rigid wall at the time $t=0$. The simulated porous body is initially
1 mm in width and 5 mm in height, as shown in Fig. 1. Periodic boundary
conditions are set in the horizontal directions, which means the real system
under consideration is composed of many of the simulated ones aligned
periodically in the horizontal direction.
Figure 7: (in JPG format) Configurations with temperature contours.
$\delta=1.4$ and $v_{init}=500$m/s. From left to right, t = 500 ns, 1500 ns,
2000 ns, and 2500 ns, respectively. The length unit here is 10 $\mu$m.
Figure 8: (Color online) Minkowski measures for the case of $\delta=1.4$ and
$v_{init}=500$m/s. The values of contour level are shown in the legend.
Figure 9: (Color online) Minkowski measures for the case of $\delta=1.4$ and
$v_{init}=400$m/s. The values of contour level are shown in the legend.
Figure 10: (Color online) Minkowski measures for the case of $\delta=1.4$ and
$v_{init}=300$m/s. The values of contour level are shown in the legend.
### 4.1 Case with $\delta=2$ and $v_{init}=1000$m/s
Figure 1 shows a set of snapshots for a procedure that a shock wave is
reacting on a porous body, where the contours denote temperature. From blue to
red, the temperature increases. The porosity $\delta=2$, $v_{init}=1000$m/s.
The time t=500ns, 1500ns, 2000ns, 2500ns for the four snapshots from left to
right. It is clear that, different from the case with uniform material, the
original shock wave is scattered and dispersive in the porous body. The first
two snapshots show the loading procedure. When $t=500$ ns, the early
compressive waves arrive at about $y=1$ mm; when $t=1500$ ns, they arrive at
about $y=3.1$ mm. The last two snapshots show the procedure of downloading.
When compressive waves arrive at the upper free surface, rarefactive waves are
reflected back into the target porous body. Under the tension wave, the height
of the porous body increases with time. In fact, before the compressive waves
arrive at the upper free surface, a large number of local downloading
phenomena have occurred within the porous body. When the initial shock wave or
a compressive wave encounters a void, rarefactive waves are reflected back and
propagate within the compressed portion, which destroys the original possible
equilibrium state there. Since the details of wave series are very complex,
when we mention the value of a state variable, for example the density, we
refer to its local mean value.
To perform the Minkowski functional analysis for the temperature map, we can
choose a threshold temperature $T_{th}$ and pixelize the map into white
regions (with $T\geq T_{th}$) and black regions (with $T<T_{th}$ ). Figure 2
shows the Minkowski measures for the same procedure as in Fig.1. “$DT$ ” in
the legend means $T_{th}-300$. The unit of temperature is K. The time unit is
ns. When $DT$ is very small, the wave front is nearly a plane, which is
similar to the case with shock reacting on uniform solid material. When
$DT=10$K, the total fractional white area $A$ increases up to be nearly $1$ at
the time $t=1600$ ns and keeps this value until the time $t=2600$ns, then has
a slight decreasing. This means the early compressive wave arrives at the
upper free surface at about, in fact before, the time $t=1600$ ns, nearly all
material particles in the target body have a temperature beyond $310$ K during
the following $1000$ns. In the downloading procedure the rarefactive waves
make a very small fraction of material particles decrease their temperature to
below $310$ K. With the increase of $DT$, the white area $A$ decreases. For
the case with $DT=100$ K, when $t=1900$ ns, the white area arrives at a steady
value $0.96$, which means $4\%$ of the material particles could not get a
temperature higher than $400$ K in the whole procedure shown here. Compared
with the case of $DT=10$K, we can get another piece of information, the
temperature increase in shocked portion of porous material is much slower than
in shocked uniform solid material. We can find the physical reason for this by
considering the void effects in shocked porous body. When the compressive wave
arrives at a void, it is decomposed of many components. The components in the
solid portion move forwards more quickly, while the portion facing the void
may result in jet phenomenon. When jetted material hit the downstream wall of
the void, new compressive waves are created. At the same time, the void
reflects rarefactive wave back to the compressed region. A large number of
similar processes exist in the shocked porous system. Thus, the shock loading
procedure in the porous body is manifested as successive reactions of many
compressive and rarefactive waves. In the shock-loading procedure, the
compressive waves dominate. Each plastic deformation makes a temperature
increment. The curve for the case of $DT=200$ K can be interpreted in a
similar way. When $DT$ increases from $200$K to $300$K, the curve of white
area has a significant variation. For the case of $DT=400$K, the white area
arrives at $0.2$ at the time $t=3000$ns, which means $80\%$ of material
particles could not get a temperature higher than $700$ K up to this time.
When $DT=500$K, the white area keeps nearly zero during the whole procedure
shown here, which means no local temperature is higher than $800$K in the
system up to the time $t=3000$ns. For cases with $DT=300$K, $330$K, $360$K and
$400$K, after the initial slow increasing period, the white (hot) area has a
quick increasing period. The latter indicates that a large amount of “hot-
spots” in the previously compressed region coalesced during that period. After
that the increasing of $A$ with $t$ shows a slowing-down. The slope of the
$A(t)$ curve approximately corresponds to the mean propagation speed of some
components of the compressive waves. Therefore, the first Minkowski measure
indicates that, in porous material, when a velocity $D$ of the compressive-
wave-series mentioned, the corresponding contour-level of a state variable
like temperature should also be stated. From this figure, it is clear that
$D(T_{th})$ decreases with the increasing of $T_{th}$; The total fractional
white (hot) area $A(t)$ shows a parabolic behavior during the initial period;
When $DT$ approaches $0$, $A(t)$ behavior goes back to be linear.
Now we go to the second Minkowski measure, the boundary length $L$. To
understand this measure, we can consider the three-dimensional plot of
$T(x,y)$ as a mountain. In the case where the mountain has only one peak, when
we increase the contour level $T_{th}$, the white area $A$ decreases, and the
boundary length $L$ decreases, too. But in the case where the mountain has
more than one peaks, the situation will not be so simple: the white area $A$
may decrease while the boundary length $L$ increases. For the case of
$DT=10$K, after the initial increase corresponding to the getting contact of
the target body with the rigid wall, the boundary length $L$ keeps a small
constant for a long time until about $t=2600$ns. The fact that the boundary
length $L$ keeps constant while the white area $A$ increase means also that
the compressive wave is propagating towards the upper free surface and the
wave front is nearly a plane in the pixelized temperature map. The increasing
of boundary length $L$ after the time $t=2600$ns is companying with the
decreasing of white area $A$, which means some small black (cold) spots occur.
The curves for $DT=100$ K and $DT=200$K show similar information. They first
increase with time due to the appearance of more “hot-spots”, then decreases
due to the coalesce of “hot-spot”, finally increase, companied by the slight
decrease of the total fractional white area $A$. When $DT=300$K, during the
period with $1500$ns $<t<2500$ns, the white area $A$ increases, while the
total fractional boundary length $L$ is nearly a constant. Considering that
the wave front has not been a plane any more for this threshold temperature,
this result indicates the following information: during this period, the
compressive waves propagate forwards, more scattered “hot-spots” appeare in
the newly compressed region; at the same time, some previous scattered “hot-
spots” coalesce. From $2500$ns to $3000$ns, the white area $A$ increases very
slowly, but the boundary length $L$ decreases quickly. This result show that
the increasing of white area $A$ is mainly due to coalesce of previous
scattered “hot-spots”. The curves for $DT=330$K and $DT=360$K can be
understood in the similar way. For the present shock strength, only very few
material particles can get a temperature beyond $700K$ before the time
$t=2000$ns. Therefore, the boundary length $L$ for the case with $DT=400$K has
a meaningful increase only after $t=2000$ns.
When $DT$ is small, $T>T_{th}$ in (nearly) all of the compressed portion and
$T<T_{th}$ in the uncompressed part of the material body. The temperature map
shows a highly connected structure with (nearly) equal and very small amount
of black and white domains. So, the Euler characteristic $\chi$ keeps close to
zero in the whole shock-loading procedure and the mean curvature $\kappa$ is
nearly zero. The value of $\chi$ decreases to be evidently less than zero in
the downloading procedure, which indicates that the number of domains with
$T<T_{th}$ increases. (See the $\chi(t)$ curves for cases with $DT=10$,
$DT=100$ and $DT=200$ in Fig.2.) With the increasing of the contour level
$T_{th}$, more regions changes their color from white ($T>T_{th}$) to black
($T<T_{th}$). The pattern evolution in the shock-loading procedure can be
regarded as that scattered white domains appear gradually with time in the
black background. So the Euler characteristic $\chi$ is positive and
increasing with time. (See the $\chi(t)$ curves for cases with $DT=300$,
$DT=330$ and $DT=360$ in Fig.2.) When the contour level $T_{th}$ is further
increased up to $700$K, a meaningful fraction of material particles could not
get a temperature higher than the contour level $T_{th}$. The saturation
phenomenon in the $\chi$ curve during the period, $550$ns $<t<2100$ ns,
indicates that the numbers of connected “hot” and “cold” domains vary with
time in a similar way. The increase of $\chi$ in the period, $2100$ns
$<t<2500$ns, is due to that the rarefactive waves make mean-temperature
decrease, correspondingly, some connected “hot-domains” are disconnected as
scattered “hot-spots” again. For the case of $DT=500$K, the pixelized
temperature map is nearly in black. So, the Euler characterization $\chi$ is
nearly zero.
### 4.2 Effects of porosity
Figure 3 shows a set of snapshots for the case with a lower porosity,
$\delta=1.4$. The other conditions are the same as in Fig.1. From left to
right, the four configurations correspond to the times, $t=500$ns, $1100$ns,
$1400$ns and $1700$ns. Compared with the snapshots in Fig.1, it is clear that
the propagation velocity of compressive wave increases with the decreasing of
porosity. At time $t=500$ns, in the system with $\delta=1.4$, the compressive
wave arrives at about $y=1750\mu$m; while in the system with $\delta=2$, the
compressive wave only arrives at about $y=1000\mu$m. In the case of
$\delta=1.4$, the compressive wave has arrived the top free surface and the
rarefactive wave has been reflected back to the target body before the time
$t=1400$ns; while in the case of $\delta=2$, the shock-loading procedure has
not been finished up to $t=1500$ns.
Figure 4 shows the Minkowski measures for cases with various porosities, where
$T_{th}=400$K and the values of porosity, $\delta=2.45$, $2$, $1.7$, $1.4$,
$1.22$, $1.15$, $1.1$ are shown in the legend. In the subfigure for white area
$A$, the initial shock-loading part presents meaningful information: the
velocity $D$ of the compressive-wave-series is smaller for a higher porosity
$\delta$. The most significant property in the subfigure for boundary length
$L$ is that the largest boundary length $L_{max}$ increases as $\delta$
decreases. When $\delta=1.1$, the total boundary length $L$ gets the maximum
value at about $t=1250$ns. This result indicates that the highest temperature
in shocked porous material decreases when the porosity approaches $1$. The
Euler characteristic $\chi$ becomes more negative when the porosity $\delta$
decreases from $2.45$ to $1.1$, which means the disconnected “cold” domains
with $T<400$K dominate more the image.
Figures 5 and 6 show the Minkowski measures for the same porosities but higher
temperature thresholds, $T_{th}=500$K and $T_{th}=600$K. They present
supplementary information to Fig. 4. For cases with $\delta=1.4$, $1.22$,
$1.15$ and $1.1$, only $88\%$, $55\%$, $36\%$ and $15\%$ of the material
particles get the temperature higher than $500$K. For cases with $\delta=1.4$
and $1.22$, and only $16\%$ and $6\%$ get the temperature higher than $600$K
in the shock-loading procedure. When $T_{th}=500$K, the case with
$\delta=1.15$ has the maximum boundary length and the case with $\delta=1.1$
has the maximum Euler characteristic. When $T_{th}=600$K, the case with
$\delta=1.4$ has the maximum boundary length and maximum Euler characteristic,
which means the “hot-spots” with $T>600$K are scatteredly distributed in the
“cold” background with $T<600$K.
### 4.3 Effects of initial shock-wave-strength
Figure 11: (Color online) Minkowski measures for cases with various shock
strengths. $\delta=1.4$. The values of initial impacting speed $v_{init}$ are
shown in the legend.
We now study the effects of different initial impacting speeds. Figure 7 shows
a set of snapshots for the case with $\delta=1.4$ and $v_{init}=500$m/s. From
left to right, the four configurations are for the times $t=500$ns, $1500$ns,
$2000$ns and $2500$ns. From the first two, we observe the upward propagation
of compressive wave in the target body. From the last two, we observe the
downward rarefactive effects. Compared with Fig.3, it is clear that the
velocity $D$ of compressive-wave-series and the highest temperature $T_{\max}$
decreased. The Minkowski meansures for this procedure is shown in Fig. 8. Such
a shocking procedure could not produce “hot-spot” with $T=500$K. High-
temperature “Hot-area” continue to increase even after some precursory
compressive waves have scanned all the target body and some rarefactive waves
have come into the target body from the upper free surface. Up to the time
$t=3000$ns, the fractional area of “Hot-spots” with $T>400$K reaches $40\%$,
the fractional area for $T>380$K reaches $74\%$, that for $T>360$K reaches
$91\%$. The contour-level with $T=380K$ has the largest boundary length at
about $t=1500$ns when the “hot-spots” mainly distribute scatteredly in the
“cold” background. Figures 9 and 10 show the Minkowski measures for cases with
the same porosity but lower initial impacting speeds. $v_{init}=400$m/s in
Fig.9 and $v_{init}=300$m/s in Fig.10. With the decrease of initial impact
speed, the highest temperature $T_{\max}$ in the system further decreases; the
total fractional white area $A$ for low contour-level, for example $DT=10$K,
increases with time in a more linear way.
We compare Minkowski measures for different initial impacting speeds in Fig.
11, where $\delta=1.4$, $DT=50$K, $v_{init}=1000$ms, $500$m/s, $400$m/s,
$300$m/s, and $200$m/s. It is clear that the higher the initial impacting
speed, the closer to be linear the $A(t)$ curve. The case of $v_{init}=400$m/s
has the longest total boundary separating the “hot” and “cold” domains. For
this case, disconnected “hot” regions dominate the image from the topology
side in the shock-loading procedure; disconnected “cold” regions dominate in
the downloading procedure.
## 5 Conclusions
Under shock wave reaction, the porous material is globally in a nonequilibrium
state and shows complex dissipative structures. We pixelize the map of
temperature into Turing patterns and introduce morphological measures for it.
Relevance of the total fractional white area $A$, boundary length $L$ and the
Euler characteristic $\chi$ to the thermodynamical properties of material is
revealed. Various experimental conditions are simulated via the material-point
method. Numerical results indicate that, the shock wave reaction results in a
complicated sequence of compressions and rarefactions in porous material. The
increasing rate of $A$ roughly gives the velocity $D$ of a compressive-wave-
series. When a velocity $D$ is mentioned, the corresponding threshold contour-
level of the temperature should also be stated. When the threshold contour-
level increases, $D$ becomes smaller. The area $A$ increases parabolically
with time $t$ during the initial period. The $A(t)$ curve goes back to be
linear in the following three cases: (i) when the porosity $\delta$ approaches
1, (ii) when the initial shock becomes stronger, (iii) when the contour-level
approaches the minimum value of the temperature. The area with high-
temperature may continue to increase even after the early compressive-waves
have arrived at the downstream free surface and some rarefactive-waves have
come back into the target body. In the case of energetic material needing a
higher temperature for initiation, a higher porosity is preferred and the
material may be initiated after the precursory compressive-waves have scanned
all the target body. One may desire the fabrication of a porous body and
choose the appropriate shock strength according to what needed is scattered or
connected hot-spots. The same measures can also be used to analyze the maps of
other physical variables, like the density, velocity, or various stresses.
With the Minkowski measures, the dependence on experimental conditions is
reflected simply by a few coefficients. They may be used as order parameters
to classify the maps of state variable in a similar way like thermodynamic
phase transitions.
We warmly thank Jianguo Wang, Hua Li, Yangjun Ying for helpful discussions on
shock waves and porous material. A.Xu is grateful to Drs. G. Gonnella and A.
Lamura for constructive discussions on Minkowski functionals. This work is
supported by Science Foundations of LCP and CAEP, national Science Foundation
of China (under Grant Nos. 10702010,10775018 and 10604010).
## References
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|
arxiv-papers
| 2009-04-01T11:34:23 |
2024-09-04T02:49:01.586936
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Aiguo Xu, Guangcai Zhang, X. F. Pan, Ping Zhang, and Jianshi Zhu",
"submitter": "Aiguo Xu Dr.",
"url": "https://arxiv.org/abs/0904.0130"
}
|
0904.0135
|
# Simulation study of shock reaction on porous material
Aiguo Xu, Guangcai Zhang, X. F. Pan, and Jianshi Zhu National Key Laboratory
of Computational Physics,
Institute of Applied Physics and Computational Mathematics, P. O. Box 8009-26,
Beijing 100088, P.R.China
###### Abstract
Direct modeling of porous materials under shock is a complex issue. We
investigate such a system via the newly developed material-point method. The
effects of shock strength and porosity size are the main concerns. For the
same porosity, the effects of mean-void-size are checked. It is found that,
local turbulence mixing and volume dissipation are two important mechanisms
for transformation of kinetic energy to heat. When the porosity is very small,
the shocked portion may arrive at a dynamical steady state; the voids in the
downstream portion reflect back rarefactive waves and result in slight
oscillations of mean density and pressure; for the same value of porosity, a
larger mean-void-size makes a higher mean temperature. When the porosity
becomes large, hydrodynamic quantities vary with time during the whole shock-
loading procedure: after the initial stage, the mean density and pressure
decrease, but the temperature increases with a higher rate. The distributions
of local density, pressure, temperature and particle-velocity are generally
non-Gaussian and vary with time. The changing rates depend on the porosity
value, mean-void-size and shock strength. The stronger the loaded shock, the
stronger the porosity effects. This work provides a supplement to experiments
for the very quick procedures and reveals more fundamental mechanisms in
energy and momentum transportation.
###### pacs:
05.70.Ln, 05.70.-a, 05.40.-a, 62.50.Ef
## I Introduction
Porous materials have extensive applications in industrial and military fields
as well as in our daily life. For example, people have long been using porous
material to shield delicate objects, to protect things from impact. The
porosity characteristics of the material may significantly influences its
dynamical and thermodynamical behaviors. When a porous material is shocked,
the cavities inside the sample may result in jets and influence its back
velocityn1 . Cavity nucleation due to tension waves controls the spallation
behavior of the materialn3 . Cavity collapse plays a prominent role in the
initiation of energetic reactions in explosivesB2002 . Most studies on shocked
porous materials in literature were experimentalP1 ; Porous1 ; Porous2 ;
Porous4 ; Gray2003 ; Porous8 and theoretical investigationsPastine1970 ;
Porous3 ; Porous5 ; Porous6 ; Porous8 ; WuJing1995 ; WuJing1996 ;
GengWuTanCaiJing . Most of them were focused on the Hugoniots and equations of
state. Due to the inhomogeneities of the material, the underlying
thermodynamical processes in the shocked body are very complex and far from
well-understanding. Understanding these processes plays a fundamental role in
the field and may present helpful information in material preparation.
From the simulation side, molecular dynamics can discover some atomistic
mechanisms of shock-induced void collapsePorous7 ; Yang , but the spatial and
temporal scales it may cover are too small compared with experimentally
measurable ones. When treating with the dynamics of structured and/or porous
materials, traditional simulation methods, both the Eulerian and Lagrangian
ones, encountered severe difficulties. The material under investigation is
generally highly distorted during the collapsing of cavities. The Eulerian
description is not convenient to tracking interfaces. When the Lagrangian
formulation is used, the original element mesh becomes distorted so
significantly that the mesh has to be re-zoned to restore proper shapes of
elements. The state fields of mass density, velocities and stresses must be
mapped from the distorted mesh to the newly generated one. This mapping
procedure is not a straightforward task, and introduces errors. In this study,
we will use a newly developed mixed method, material-point method, to
investigate the shock properties of porous materials.
The material-point method was originally introduced in fluid dynamics by
Harlow, et alH1964 and extended to solid mechanics by Burgess, et al MPM ,
then developed by various researchers, including usJPCM2007 ; CTP2008 ;
JPD2008 . At each time step, calculations consist of two parts: a Lagrangian
part and a convective one. Firstly, the computational mesh deforms with the
body, and is used to determine the strain increment, and the stresses in the
sequel. Then, the new position of the computational mesh is chosen
(particularly, it may be the previous one), and the velocity field is mapped
from the particles to the mesh nodes. Nodal velocities are determined using
the equivalence of momentum calculated for the particles and for the
computational grid. The method not only takes advantages of both the
Lagrangian and Eulerian algorithms but makes it possible to avoid their
drawbacks as well.
The following part of the paper is planned as follows. Section II presents the
theoretical model of the material under consideration. Section III describes
briefly the numerical scheme. Simulation results are shown and analyzed in
section IV. Section V makes the conclusion.
## II Theoretical model of the material
In this study the material is assumed to follow an associative von Mises
plasticity model with linear kinematic and isotropic hardeningCModel .
Introducing a linear isotropic elastic relation, the volumetric plastic strain
is zero, leading to a deviatoric-volumetric decoupling. So, it is convenient
to split the stress and strain tensors, $\bm{\sigma}$ and $\bm{\varepsilon}$,
as
$\displaystyle\bm{\sigma}$ $\displaystyle=$
$\displaystyle\mathbf{s}-P\mathbf{I},P=-\frac{1}{3}\verb|Tr|(\bm{\sigma})\mathtt{,}$
(1) $\displaystyle\bm{\varepsilon}$ $\displaystyle=$
$\displaystyle\mathbf{e}+\frac{1}{3}\theta\mathbf{I},\theta=\frac{1}{3}\verb|Tr|(\bm{\varepsilon})\mathtt{,}$
(2)
where $P$ is the pressure scalar, $\mathbf{s}$ the deviatoric stress tensor,
and $\mathbf{e}$ the deviatoric strain. The strain $\mathbf{e}$ is generally
decomposed as $\mathbf{e}=\mathbf{e}^{e}+\mathbf{e}^{p}$, where
$\mathbf{e}^{e}$ and $\mathbf{e}^{p}$ are the traceless elastic and plastic
components, respectively. The material shows a linear elastic response until
the von Mises yield criterion,
$\sqrt{\frac{3}{2}}\left\|\mathbf{s}\right\|=\sigma_{Y}\mathtt{,}$ (3)
is reached, where $\sigma_{Y}$ is the plastic yield stress. The yield
$\sigma_{Y}$ increases linearly with the second invariant of the plastic
strain tensor $\mathbf{e}^{p}$, i.e.,
$\sigma_{Y}=\sigma_{Y0}+E_{\tan}\left\|\mathbf{e}^{p}\right\|\mathtt{,}$ (4)
where $\sigma_{Y0}$ is the initial yield stress and $E_{\tan}$ the tangential
module. The deviatoric stress $\mathbf{s}$ is calculated by
$\mathbf{s}=\frac{E}{1+\nu}\mathbf{e}^{e}\mathtt{,}$ (5)
where $E$ is the Yang’s module and $\nu$ the Poisson’s ratio. Denote the
initial material density and sound speed by $\rho_{0}$ and $c_{0}$,
respectively. The shock speed $U_{s}$ and the particle speed $U_{p}$ after the
shock follows a linear relation, $U_{s}=c_{0}+\lambda U_{p}$, where $\lambda$
is a characteristic coefficient of material. The pressure $P$ is calculated by
using the Mie-Grüneissen state of equation which can be written as
$P-P_{H}=\frac{\gamma(V)}{V}[E-E_{H}(V_{H})]$ (6)
This description consults the Rankine-Hugoniot curve. In Eq.(6), $P_{H}$,
$V_{H}$ and $E_{H}$ are pressure, specific volume and energy on the Rankine-
Hugoniot curve, respectively. The relation between $P_{H}$ and $V_{H}$ can be
estimated by experiment and can be written as
$P_{H}=\left\\{\begin{array}[]{ll}\frac{\rho_{0}c_{0}^{2}(1-\frac{V_{H}}{V_{0}})}{(\lambda-1)^{2}(\frac{\lambda}{\lambda-1}\times\frac{V_{H}}{V_{0}}-1)^{2}},&V_{H}\leq
V_{0}\\\
\rho_{0}c_{0}^{2}(\frac{V_{H}}{V_{0}}-1),&V_{H}>V_{0}\end{array}\right.$ (7)
In this paper, the transformation of specific internal energy $E-E_{H}(V_{H})$
is taken as the plastic energy. Both the shock compression and the plastic
work cause the increasing of temperature. The increasing of temperature from
shock compression can be calculated as:
$\frac{\mathrm{d}T_{H}}{\mathrm{d}V_{H}}=\frac{c_{0}^{2}\cdot\lambda(V_{0}-V_{H})^{2}}{c_{v}\big{[}(\lambda-1)V_{0}-\lambda
V_{H}\big{]}^{3}}-\frac{\gamma(V)}{V_{H}}T_{H}.$ (8)
where $c_{v}$ is the specific heat. Eq.(8) can be resulted with thermal
equation and the Mie-Grüneissen state of equationexplosion . The increasing of
temperature from plastic work can be calculated as:
$\mathrm{d}T_{p}=\frac{\mathrm{d}W_{p}}{c_{v}}$ (9)
Both the Eq.(8) and the Eq.(9) can be written as the form of increment.
In this paper the sample material is aluminum. The corresponding parameters
are $\rho_{0}=2700$ kg/m3, $E=69$ Mpa, $\nu=0.33$, $\sigma_{Y0}=120$ Mpa,
$E_{\tan}=384$ MPa, $c_{0}=5.35$ km/s, $\lambda=1.34$, $c_{v}=880$
J/(Kg$\cdot$K), $k=237$ W/(m$\cdot$K) and $\gamma_{0}=1.96$ when the pressure
is below $270$ GPa. The initial temperature of the material is 300 K.
## III Outline of the numerical scheme
As a particle method, the material point method discretizes the continuum
bodies with $N_{p}$ material particles. Each material particle carries the
information of position $\mathbf{x}_{p}$, velocity $\mathbf{v}_{p}$, mass
$m_{p}$, density $\rho_{p}$, stress tensor $\bm{\sigma}_{p}$ , strain tensor
$\bm{\varepsilon}_{p}$ and all other internal state variables necessary for
the constitutive model, where $p$ is the index of particle. At each time step,
the mass and velocities of the material particles are mapped onto the
background computational mesh. The mapped momentum at node $i$ is obtained by
$m_{i}\mathbf{v}_{i}=\sum_{p}m_{p}\mathbf{v}_{p}N_{i}(\mathbf{x}_{p})$, where
$N_{i}$ is the element shape function and the nodal mass $m_{i}$ reads
$m_{i}=\sum_{p}m_{p}N_{i}(\mathbf{x}_{p}).$ Suppose that a computational mesh
is constructed of eight-node cells for three-dimensional problems, then the
shape function is defined as
$N_{i}=\frac{1}{8}(1+\xi\xi_{i})(1+\eta\eta_{i})(1+\varsigma\varsigma_{i})\mathtt{,}$
(10)
where $\xi$,$\eta$,$\varsigma$ are the natural coordinates of the material
particle in the cell along the x-, y-, and z-directions, respectively,
$\xi_{i}$,$\eta_{i}$,$\varsigma_{i}$ take corresponding nodal values $\pm 1$.
The mass of each particle is equal and fixed, so the mass conservation
equation, $\mathrm{d}\rho/\mathrm{d}t+\rho\nabla\cdot\mathbf{v}=0$, is
automatically satisfied. The momentum equation reads,
$\rho\mathrm{d}\mathbf{v/}\mathrm{d}t=\nabla\cdot\bm{\sigma}+\rho\mathbf{b}\mathtt{,}$
(11)
where $\rho$ is the mass density, $\mathbf{v}$ the velocity, $\bm{\sigma}$ the
stress tensor and $\mathbf{b}$ the body force. Equation (11) is solved on a
finite element mesh in a lagrangian frame. Its weak form is
$\begin{array}[]{ll}&\int_{\Omega}{\rho\delta\mathbf{v}\cdot\mathrm{d}\mathbf{v/}\mathrm{d}t\mathrm{d}\Omega}+\int_{\Omega}{\delta(\mathbf{v}\nabla)\cdot\bm{\sigma}\mathrm{d}\Omega}-\int_{\Gamma_{t}}{\
\delta\mathbf{v}\cdot\mathbf{t}\mathrm{d}\Gamma}\\\ &-\int_{\Omega}{\
\rho\delta\mathbf{v}\cdot\mathbf{b}\mathrm{d}\Omega}=0\mathtt{.}\end{array}$
(12)
Since the continuum bodies is described with the use of a finite set of
material particles, the mass density can be written as
$\rho(\mathbf{x})=\sum_{p=1}^{N_{p}}{\
m_{p}\delta(\mathbf{x}-\mathbf{x}_{p})}$, where $\delta$ is the Dirac delta
function with dimension of the inverse of volume. The substitution of
$\rho(\mathbf{x})$ into the weak form of the momentum equation converts the
integral to the sums of quantities evaluated at the material particles,
namely,
$m_{i}\mathrm{d}\mathbf{v}_{i}/\mathrm{d}t=(\mathbf{f}_{i})^{\mathrm{int}}+(\mathbf{f}_{i})^{\mathrm{ext}}\mathtt{,}$
(13)
where the internal force vector is given by
$\mathbf{f}_{i}{}^{\mathrm{int}}=-\sum_{p}^{N_{p}}{m_{p}\bm{\sigma}}_{p}{\cdot(\nabla
N_{i})/\rho_{p}}$, and the external force vector reads
$\mathbf{f}_{i}{}^{\mathrm{ext}}=\sum_{p=1}^{N_{p}}{N_{i}\mathbf{b}_{p}+\mathbf{f}_{i}^{c}}$,
where the vector $\mathbf{f}_{i}^{c}$ is the contacting force between two
bodies. In present paper, all colliding bodies are composed of the same
material, and $\mathbf{f}_{i}^{c}$ is treated with in the same way as the
internal force.
The nodal accelerations are calculated by Eq. (13) with an explicit time
integrator. The critical time step satisfying the stability conditions is the
ratio of the smallest cell size to the wave speed. Once the motion equations
are solved on the cell nodes, the new nodal values of acceleration are used to
update the velocity of the material particles. The strain increment for each
material particle is determined using the gradient of nodal basis function
evaluated at the position of the material particle. The corresponding stress
increment can be found from the constitutive model. The internal state
variables can also be completely updated. The computational mesh may be the
original one or a newly defined one, choose for convenience, for the next time
step. More details of the algorithm is referred to JPD2008 ; CTP2008 .
Figure 1: (in JPG format) Snapshots of the shocked porous metal.
$\delta=1.03$, t=250 ns. (a) Contour of pressure, (b) contour of temperature.
The unit of length in this figure is 10 $\mu$m. From blue to red, the contour
value increases. The unit of contour is Mpa in (a) and is K in (b). The
initial velocities of the flyer and target are $\pm v_{init}=\pm 1000$ m/s in
this case. Figure 2: (Color online) Variations of mean density, pressure,
temperature and particle velocity with time. The height of the measured domain
are h= 800 $\mu$m, 400$\mu$m and 100 $\mu$m, respectively, as shown in the
legends. “B" and “T" in the legends means the measured domains are at the
bottom and top of the target body, respectively. The units of density,
pressure, temperature, particle velocity and time are g/cm3, Gpa, K, m/s and
ns, respectively. Figure 3: (in JPG format) Configuration with temperature
contour at time t=1.15 $\mu$s. Other parameters are referred to Fig.1 and
Fig.2. The unit of temperature is K.
## IV Results of numerical experiments
In the present study the porous material is fabricated by a solid material
body with an amount of voids randomly embedded. The porosity $\delta$ is
defined as $\delta=\rho_{0}/\rho$, where $\rho_{0}$ is the original density of
the solid body and $\rho$ is the mean density of porous material. The porosity
$\delta$ in the simulated system is controlled by the total number $N_{void}$
and mean size $r_{void}$ of voids embedded. The shock wave to the target
porous metal is loaded via colliding by a second body. For the convenience of
analysis, we set the configurations and velocities of the two colliding porous
bodies symmetric about their impact interface. The initial velocities of the
two colliding bodies are along the vertical direction and denoted as $\pm
v_{init}$. The impact interface is set at $y=0$. Periodic boundary conditions
are used in the horizontal directions, which means the investigated real
system is composed of many of the simulated ones aligned periodically in the
horizontal direction. We regarded the upper porous body as the target, the
lower one as the flyer. Compared with experiments where the target is
initially static, the initial velocity of the flyer is $2v_{init}$. In this
study we focus on the two-dimensional case. The computational unit is 2 mm in
width, as shown in Fig.1. When we are mainly interested in the loading
procedure of shock wave to porous body, we require that each simulated body
has an enough height so that the rarefactive waves from the upper and lower
free surfaces do not affect the physical procedure within the time scale under
investigation.
Figure 1 shows two snapshots of such a process, where Fig.1(a) shows the
contour of pressure and Fig.1(b) shows the contour of temperature. The
snapshots show clearly that, different from the case with perfect solid
material, there is no stable shock wave in the porous materials. When the
compressive waves arrive at a cavity, rarefactive waves are reflected back and
propagate within the compressed portion, which destroys the original possible
equilibrium state there. Even thus, for the convenience of description, we
still refer the compressive waves to shock waves. Correspondingly, the values
of physical quantities, such as the particle velocity, density, pressure,
temperature, etc, are corresponding mean values calculated in a region
$\Omega$ with $y_{1}\leq y\leq y_{2}$. We will investigate the effects of
initial shock strength and porosity value.
### IV.1 Cases with porosity $\delta$=1.03
We first study the case with $r_{void}$ =50 $\mu$m and the velocity $v_{init}$
= 1000 m/s, which means the flyer velocity relative to the target is 2000 m/s.
The flyer begins to contact the target at the time t = 0. Figure 2 shows the
variations of mean density, pressure, temperature and particle velocity with
time. These values are dynamically measured in a bottom and a top domains,
respectively. The height of the target body is 5 mm in this case. The height
of the measured domains are h=800 $\mu$m, 400 $\mu$m and 100 $\mu$m,
respectively. For the bottom domain, we choose $y_{1}=100\mu$m. For the top
domain, $y_{2}$ takes the y-coordinate of the highest material-particle. The
lines with solid symbols are measured values from the bottom and the lines
with empty symbols are measured values from the top. From the figure, we get
the following information: When the shock waves propagate within the bottom
domain $\Omega_{b}$, the measured mean density, pressure and temperature
increase nearly linearly with time, up to about t= 150 ns for the case of
h=800 $\mu$m, then further to increase with a decreasing changing rate. The
three quantities arrive at their first maximum values, 3.14g/cm3, 16.7GPa, and
432K, at the time t=250 ns. At this time the shock front has passed the
downstream boundary, $y=810\mu$m, of the measured domain. (See Fig. 1.) The
time delay is due to dispersion of shock wave in porous media. The followed
concave in either of the $\rho$-,P-,T-curves at about t = 450 ns shows a
downloading phenomenon. The phenomenon is resulted from rarefactive waves
reflected back from the cavities downstream neighboring to the measured
domain. The values of $\rho$ and P increase and recover to their steady values
after that, but the temperature get a higher value. The secondary loading-
phenomenon is due to the colliding of the upstream and downstream walls during
the collapsing of cavities. Within the following period the density and
pressure keep nearly constants, while the temperature still increases very
slowly. The weak fluctuations in the $\rho$, P, T curves after $t=650$ ns
result from the putting-in of compressive and rarefactive waves from the two
boundaries of the measured domain $\Omega_{b}$. The visco-plastic work by
these wave series makes the temperature increase slowly. Since the
configurations and velocities of the flyer and target are symmetric about the
plane $y=0$, the vertical component of particle velocity, $u_{y}$, is about 0
m/s, the horizontal component $u_{x}$ first increases with time, then
oscillates around a small value which is nearly zero. The lines with empty
symbols show that the shock waves arrive at the top free surface at about t=
800 ns, then rarefactive waves are reflected back into the target body. Within
the time scale shown in the figure, for the cases with h=800 $\mu$m and 400
$\mu$m, the density (or pressure) recovers to a value being slightly larger
than its initial one, but the remained temperature is about 60K higher than
the initial temperature and is still increasing; for the case with h=100
$\mu$m, evident oscillations are found in the curve of density after t=900 ns.
To understand this, we show in Fig.3 the top portion of the configuration with
temperature contour for the time t=1.15 $\mu$s, from which we can find jetting
phenomena at the upper free surface. During the downloading procedure, the top
of the porous body moves upwards with a velocity being about 877 m/s. From the
same data used in Fig.1, we can get the mutual dependence of the
hydrodynamical quantities. The initial transient stage and the final
oscillatory steady state are clearly observable. Due to existence of the
randomly distributed voids, waves with various wave vectors and frequencies
propagate within the shocked sample material. When the measured domain becomes
smaller, more detailed wave structures may be found. Figure 2 shows clearly
this trend.
It is interesting to check more carefully the procedure of approaching steady
state. Figure 4 shows the standard deviations of the above four quantities
versus time measured in the bottom domains. It is found that they increase
quickly with time at the very beginning stage, then decrease nearly
exponentially to their steady values. The standard deviation of $u_{y}$ is
larger than that of $u_{x}$. The finite sizes of these steady values confirm
our analysis above: what the system arrives is a steady state with local
dynamical oscillations. When the height of the measured domain increases, the
standard deviations of measured quantities become larger, at least in the
transient period.
Figure 4: (Color online) Standard deviations(Std) of the local quantities
averaged in various spatial scales. The heights of the measured domains are
shown in the legends where “B" means the measured domains are at the bottom of
the target body. The length and time units are $\mu$m and ns, respectively.
Figure 5: (Color online) Variations of the mean values squared of local
rotation, divergence and strain rate with time. <…> in the legends denote the
mean value of the corresponding quantity and “B" means the measured domains
are at the bottom of the target body. The length and time units are $\mu$m and
ns, respectively. Figure 6: (in JPG format) Configurations with density
contour (a), pressure contour (b), temperature contour (c) and velocity field
(d) at time t=750 ns. The size of particle velocity is denoted by the length
of arrow timed by 50. The units are the same as in Fig.2.
For the case with perfect crystal material, the increase of entropy result
from only from the non-equilibrium procedure of the front of the shock waves.
When cavities exist, the high plastic distortion of the materials surrounding
the collapsed cavities contribute extra entropy increment. So the local
rotation, Rot= $|\nabla\times\mathbf{u}|$, and divergence, Div=
$|\nabla\cdot\mathbf{u}|$, make significance sense in describing shocked
porous media. The local rotation $|\nabla\times\mathbf{u}|$ describes the
circular flow and/or turbulence. The divergence $|\nabla\cdot\mathbf{u}|$
describes the changing rate of volume. They show important mechanisms of
entropy and temperature increase in porous material. The former reflects the
turbulence dissipation and the latter reflects the shock compression. Figure 5
shows the variations of their mean values squared with time. The behavior of
strain rate $\bm{\dot{\varepsilon}}$ is plotted as a comparison. It is found
that all the three quantities decrease nearly exponentially to their steady
state values when shock waves pass the measured domain $\Omega$. The amplitude
of steady strain rate is very close to that of the rotation. The amplitude of
the divergence is a little larger for this case. Cavity collapse and new
cavitation by the rarefactive waves are the main contributors to the local
divergence. To understand better the fluctuations of the local density,
pressure, temperature, particle velocity and the finite values of the
rotation, divergence, we show in Fig.6 a portion of the configuration with
density contour, pressure contour, temperature contour and velocity field at
time $t=750$ns. In this case, there is a void around the position (510$\mu$m,
280$\mu$m).
We now checking the effects of the void size. Results for different void sizes
are compared. There is no evident difference in the steady values of mean
density, pressure and particle velocity. But larger voids contribute to a
higher mean temperature. (See Fig.7.) As for effects on the mean value squared
of the local rotation and divergence, the void size affect only the transient
period, but not the steady values. See Fig.8, where the two cases correspond
to different mean-void-sizes but the same value of porosity, $\delta=1.03$.
Figure 7: (Color online) Effects of the mean void size on the mean
temperature. The mean void size $r$, position and height of the measured
domain are shown in the legend. “B" and “T" means the measured domains are at
the bottom and top of the target body, respectively. The length and time units
are $\mu$m and ns, respectively. Figure 8: (Color online) Effects of mean void
size on the mean values squared of local rotation, divergence. The mean sizes
of void are shown in the legends. The length and time units are $\mu$m and ns,
respectively.
### IV.2 Cases with porosity $\delta=1.4$
In this section we study the case with a higher porosity, $\delta$=1.4. For
this case, the mean void size is r=10 $\mu$m. Figure 9 shows the variations of
mean density, pressure, temperature and particle velocity with time. The
initial velocity of the flyer and the target are $\pm v_{init}=1000$m/s. The
physical quantities are averaged in a bottom and a top domains. Only the case
with h=800 $\mu$m is shown. An evident difference from the low-porosity case
with $\delta=1.03$ is that the mean density and pressure decrease with time
after the initial stage. Correspondingly, the mean temperature increase with a
higher rate. This is due to the rarefactive waves reflected back from the
downstream voids. The reflected rarefactive waves make the shocked material a
little looser and result in a relatively higher local divergence. The latter
transforms more kinetic energy into heat. At the same time, a higher porosity
means more voids embedded in the material, more jetting phenomena occur when
being shocked. The jetting phenomena and the hitting of jetted material to the
downstream walls of the voids make a significant increase of local
temperature, local divergence and local rotation. The mean values squared of
the local rotation, divergence and strain rate are shown in Fig.10. These
quantities are measured in the bottom domain with h=800 $\mu$m. During the
initial transient period, the turbulence dissipation makes the most
significant contribution to temperature-increase in this case. In the later
steady state, the three kinds of dissipation makes nearly the same
contribution.
To understand better the inhomogeneity effects in the shocked portion of the
porous body, we show the distributions of density, pressure, temperature and
particle velocity at times t=1200ns, 1250ns and 1300ns in Fig.11. It is clear
that these distributions generally deviate from the Gaussian distribution and
vary with time. In Fig.12 we study the effects of initial impact velocity on
the mean values of the density, pressure and temperature. It is clear that the
decreasing rate of the mean density and the increasing rate of mean
temperature increase when the initial shock wave becomes stronger. This means
that the porosity effects become more significant when the loaded shock wave
becomes stronger.
We now study porosity effects for a fixed shock strength. Figure 13 shows the
mean density, and temperature versus time for various porosities. The initial
velocity of the flyer and target are $\pm v_{init}$ = 1000m/s. When the
porosity is very small, the decreasing rate with time of the mean density
becomes higher as the porosity increases. But when the porosity becomes large,
the mean density show more complex behavior.
Figure 9: (Color online) Variations of mean density, pressure, temperature and
particle velocity with time. Here the porosity $\delta=1.4$ and initial flyer
velocity relative to the target is $2v_{init}=2000$ m/s. The meanings of “B",
“T" and units are the same as in Fig.2. Figure 10: (Color online) Variations
of the mean values squared of local rotation, divergence and strain rate with
time. The unit of time is ns. Figure 11: (Color online) Distribution of local
density, pressure, temperature, particle velocity at various times. The units
are the same as in Fig.2. Figure 12: (Color online) Mean density and
temperature versus time for various shock strengths. The initial velocity
$v_{init}$ are shown in the legend. The units are the same as in Fig.2.
Figure 13: (Color online) Mean density and temperature versus time for various
porosities. The values of porosity, 1.01,1.02,1.1,1.4,1.7 are shown in the
legends. In the left figure, the lines for cases with $\delta=$1.1,1.4 and 1.7
are moved upwards by 0.01,0.1, and 0.15, respectively. The units are the same
as in Fig.2.
## V Conclusion
Thermodynamic properties of porous material under shock-reaction is studied
via a direct simulation. The effects of shock strength, porosity value and the
mean-void-size are checked carefully. It is found that, when the porosity is
very small, the shocked portion will arrive at a dynamic steady state; the
voids in the downstream portion reflect back rarefactive waves and result in
slight oscillations of mean density and pressure; for the same value of
porosity, a larger mean-void-size makes a higher mean temperature. When the
porosity becomes larger, after the initial stage, the mean density and
pressure decrease significantly with time. The distributions of local density,
pressure, temperature and particle-velocity are generally non-Gaussian and
vary with time. Different from the case with perfect solid material, local
turbulence mixing and volume dissipation exist in the whole loading procedure
and make the system temperature continuously increase. The changing rates
depend on the porosity value, mean-void-size and shock strength. The stronger
the loaded shock, the stronger the porosity effects. This work is
supplementary to experimental investigations for the very quick procedures and
reveals more fundamental mechanisms in energy and momentum transportation.
###### Acknowledgements.
We warmly thank Ping Zhang, Jun Chen, Yangjun Ying for helpful discussions. We
acknowledge support by Science Foundations of Laboratory of Computational
Physics, China Academy of Engineering Physics, and National Science Foundation
of China (under Grant Nos. 10702010 and 10775018).
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|
arxiv-papers
| 2009-04-01T11:55:42 |
2024-09-04T02:49:01.594343
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Aiguo Xu, Guangcai Zhang, X. F. Pan, and Jianshi Zhu",
"submitter": "Aiguo Xu Dr.",
"url": "https://arxiv.org/abs/0904.0135"
}
|
0904.0226
|
# Coding versus ARQ in Fading Channels: How reliable should the PHY be?
Peng Wu and Nihar Jindal
University of Minnesota, Minneapolis, MN 55455
Email: {pengwu,nihar}@umn.edu
###### Abstract
This paper studies the tradeoff between channel coding and ARQ (automatic
repeat request) in Rayleigh block-fading channels. A heavily coded system
corresponds to a low transmission rate with few ARQ re-transmissions, whereas
lighter coding corresponds to a higher transmitted rate but more re-
transmissions. The optimum error probability, where optimum refers to the
maximization of the average successful throughput, is derived and is shown to
be a decreasing function of the average signal-to-noise ratio and of the
channel diversity order. A general conclusion of the work is that the optimum
error probability is quite large (e.g., $10\%$ or larger) for reasonable
channel parameters, and that operating at a very small error probability can
lead to a significantly reduced throughput. This conclusion holds even when a
number of practical ARQ considerations, such as delay constraints and
acknowledgement feedback errors, are taken into account.
## I Introduction
In contemporary wireless communication systems, ARQ (automatic repeat request)
is generally used above the physical layer (PHY) to compensate for packet
errors: incorrectly decoded packets are detected by the receiver, and a
negative acknowledgement is sent back to the transmitter to request a re-
transmission. In such an architecture there is a natural tradeoff between the
transmitted rate and ARQ re-transmissions. A high transmitted rate corresponds
to many packet errors and thus many ARQ re-transmissions, but each
successfully received packet contains many information bits. On the other
hand, a low transmitted rate corresponds to few ARQ re-transmissions, but few
information bits are contained per packet. Thus, a fundamental design
challenge is determining the transmitted rate that maximizes the rate at which
bits are successfully delivered. Since the packet error probability is an
increasing function of the transmitted rate, this is equivalent to determining
the optimal packet error probability, i.e., the optimal PHY reliability level.
We consider a wireless channel where the transmitter chooses the rate based
only on the fading statistics because knowledge of the instantaneous channel
conditions is not available (e.g., high velocity mobiles in cellular systems).
The transmitted rate-ARQ tradeoff is interesting in this setting because the
packet error probability depends on the transmitted rate in a non-trivial
fashion; on the other hand, this tradeoff is somewhat trivial when
instantaneous channel state information at the transmitter (CSIT) is available
(see Remark 1).
We begin by analyzing an idealized system, for which we find that making the
PHY too reliable can lead to a significant penalty in terms of the achieved
goodput (long-term average successful _throughput_), and that the optimal
packet error probability is decreasing in the average SNR and in the fading
selectivity experienced by each transmitted codeword. We also see that for a
large level of system parameters, choosing an error probability of $10\%$
leads to near-optimal performance. We then consider a number of important
practical considerations, such as a limit on the number of ARQ re-
transmissions and unreliable acknowledgement feedback. Even after taking these
issues into account, we find that a relatively unreliable PHY is still
preferred. Because of fading, the PHY can be made reliable only if the
transmitted rate is significantly reduced. However, this reduction in rate is
not made up for by the corresponding reduction in ARQ re-transmissions.
### I-A Prior Work
There has been some recent work on the joint optimization of packet-level
erasure-correction codes (e.g., fountain codes) and PHY-layer error correction
[1, 2, 3, 4]. The fundamental metric with erasure codes is the product of the
transmitted rate and the packet success probability, which is the same as in
the idealized ARQ setting studied in Section III. Even in that idealized
setting, our work differs in a number of ways. References [1, 3, 4] study
multicast (i.e., multiple receivers) while [2] considers unicast assuming no
diversity per transmission, whereas our focus is on the unicast setting with
diversity per transmission. Furthermore, our analysis provides a general
explanation of how the PHY reliability should depend on both the diversity and
the average SNR. In addition, we consider a number of practical issues
specific to ARQ, such as acknowledgement errors (Section IV), as well as
hybrid-ARQ (Section V).
## II System Model
We consider a Rayleigh block-fading channel where the channel remains constant
within each block but changes independently from one block to another. The
$t$-th ($t=1,2,\cdots$) received channel symbol in the $i$-th ($i=1,2,\cdots$)
fading block $y_{t,i}$ is given by
$\displaystyle y_{t,i}=\sqrt{\mbox{\scriptsize\sf
SNR}}~{}h_{i}x_{t,i}+z_{t,i},$ (1)
where $h_{i}\sim\mathcal{CN}(0,1)$ represents the channel gain and is i.i.d.
across fading blocks, $x_{t,i}\sim\mathcal{CN}(0,1)$ denotes the Gaussian
input symbol constrained to have unit average power, and
$z_{t,i}\sim\mathcal{CN}(0,1)$ models the additive Gaussian noise assumed to
be i.i.d. across channel uses and fading blocks. Although we focus on single
antenna systems and Rayleigh fading channel, our model can be easily extended
to multiple-input and multiple-output (MIMO) systems and other fading
distributions as commented upon in Remark 2.
Each transmission (i.e., codeword) is assumed to span $L$ fading blocks, and
thus $L$ represents the time/frequency selectivity experienced by each
codeword. In analyzing ARQ systems, the packet error probability is the key
quantity. If a strong channel code (with suitably long blocklength) is used,
it is well known that the packet error probability is accurately approximated
by the mutual information outage probability [5, 6, 7, 8]. Under this
assumption (which is examined in Section IV-A), the packet error probability
for transmission at rate $R$ bits/symbol is given by [9, eq (5.83)]:
$\displaystyle\varepsilon(\mbox{\scriptsize\sf
SNR},L,R)=\mathbb{P}\left[\frac{1}{L}\sum_{i=1}^{L}\log_{2}(1+\mbox{\scriptsize\sf
SNR}|h_{i}|^{2})\leq R\right].$ (2)
Here we explicitly denote the dependence of the error probability on the
average signal-to-noise ratio SNR, the selectivity order $L$, and the
transmitted rate $R$. We are generally interested in the relationship between
$R$ and $\varepsilon$ for particular (fixed) values of SNR and $L$. When SNR
and $L$ are constant, $R$ can be inversely computed given some $\varepsilon$;
thus, throughout the paper we replace $R$ with $R_{\varepsilon}$ wherever the
relationship between $R$ and $\varepsilon$ needs to be explicitly pointed out.
The focus of the paper is on simple ARQ, in which packets received in error
are re-transmitted and decoding is performed only on the basis of the most
recent transmission.111 _Hybrid_ -ARQ, which is a more sophisticated and
powerful form of ARQ, is considered in Section V. More specifically, whenever
the receiver detects that a codeword has been decoded incorrectly, a NACK is
fed back to the transmitter. On the other hand, if the receiver detects
correct decoding an ACK is fed back. Upon reception of an ACK, the transmitter
moves on to the next packet, whereas reception of a NACK triggers re-
transmission of the previous packet. ARQ transforms the system into a
variable-rate scheme, and the relevant performance metric is the rate at which
packets are successfully received. This quantity is generally referred to as
the long-term average _goodput_ , and is clearly defined in each of the
relevant sections. And consistent with the assumption of no CSIT (and fast
fading), we assume fading is independent across re-transmissions.
## III OPTIMAL PHY Reliability in the Ideal Setting
In this section we investigate the optimal PHY reliability level under a
number of idealized assumptions. Although not entirely realistic, this
idealized model yields important design insights. In particular, we make the
following key assumptions:
* •
Channel codes that operate at the mutual information limit (i.e., packet error
probability is equal to the mutual information outage probability).
* •
Perfect error detection at the receiver.
* •
Unlimited number of ARQ re-transmissions.
* •
Perfect ACK/NACK feedback.
In Section IV we relax these assumptions, and find that the insights from this
idealized setting generally also apply to real systems.
In order to characterize the long-term goodput in this idealized setting. In
order to do so, we must quantify the number of transmission attempts/ARQ
rounds needed for successful transmission of each packet. If we use $X_{i}$ to
denote the number of ARQ rounds for the _i_ -th packet, then a total of
$\sum_{i=1}^{J}X_{i}$ ARQ rounds are used for transmitting $J$ packets; note
that the $X_{i}$’s are i.i.d. due to the independence of fading and noise
across ARQ rounds. Each codeword is assumed to span $n$ channel symbols and to
contain $b$ information bits, corresponding to a transmitted rate of $R=b/n$
bits/symbols. The average rate at which bits are successfully delivered is the
ratio of the bits delivered to the total number of channel symbols required.
The goodput $\eta$ is the long-term average at which bits are successfully
delivered, and by taking $J\rightarrow\infty$ we get [10]:
$\displaystyle\eta=\lim_{J\rightarrow\infty}\frac{Jb}{n\sum_{i=1}^{J}X_{i}}=\lim_{J\rightarrow\infty}\frac{\frac{b}{n}}{\frac{1}{J}\sum_{i=1}^{J}X_{i}}=\frac{R}{\mathbb{E}[X]},$
(3)
where $X$ is the random variable describing the ARQ rounds required for
successful delivery of a packet.
Because each ARQ round is successful with probability $1-\varepsilon$, with
$\varepsilon$ defined in (2), and rounds are independent, $X$ is geometric
with parameter $1-\varepsilon$ and thus $\mathbb{E}[X]=1/(1-\varepsilon)$.
Based upon (3), we have
$\displaystyle\eta\triangleq R_{\varepsilon}(1-\varepsilon),$ (4)
where the transmitted rate is denoted as $R_{\varepsilon}$ to emphasize its
dependence on $\varepsilon$.
Based on this expression, we can immediately see the tradeoff between the
transmitted rate, i.e. the number of bits per packet, and the number of ARQ
re-transmissions per packet: a large $R_{\varepsilon}$ means many bits are
contained in each packet but that many re-transmissions are required, whereas
a small $R_{\varepsilon}$ corresponds to fewer bits per packet and fewer re-
transmissions. Our objective is to find the optimal (i.e., goodput maximizing)
operating point on this tradeoff curve for any given parameters SNR and $L$.
Because $R_{\varepsilon}$ is a function of $\varepsilon$ (for SNR and $L$
fixed), this one-dimensional optimization can be phrased in terms of
$R_{\varepsilon}$ or $\varepsilon$. We find it most insightful to consider
$\varepsilon$, which leads to the following definition:
###### Definition 1
The optimal packet error probability, where optimal refers to goodput
maximization with goodput defined in (3), for average signal-to-noise ratio
SNR and per-codeword selectivity order $L$ is:
$\displaystyle\varepsilon^{\star}(\mbox{\scriptsize\sf
SNR},L)\triangleq\arg\max_{\varepsilon}~{}R_{\varepsilon}(1-\varepsilon).$ (5)
By finding $\varepsilon^{\star}(\mbox{\scriptsize\sf SNR},L)$, we thus
determine the optimal PHY reliability level and how this optimum depends on
channel parameters SNR and $L$, which are generally static over the timescale
of interest.222Note that in this definition we assume all possible code rates
are possible; nonetheless, this formulation provides valuable insight for
systems in which the transmitter must choose from a finite set of code rates.
For $L=1$, a simple calculation shows 333The expression for $L=1$ is also
derived in [2]. However, authors in [2] only consider $L=1$ case rather than
$L>1$ scenarios, which are further investigated in our work.
$\displaystyle\varepsilon^{\star}(\mbox{\scriptsize\sf
SNR},1)=1-e^{\left(1-\mbox{\scriptsize\sf
SNR}\right)/\left(\mbox{\scriptsize\sf SNR}\cdot W(\mbox{\scriptsize\sf
SNR})\right)},$ (6)
where $W(\cdot)$ is the Lambert W function [11]. Unfortunately, for $L>1$ it
does not seem feasible to find an exact analytical solution because a closed-
form expression for the outage probability exists only for $L=1$. However, the
optimization in (5) can be easily solved numerically (for arbitrary $L$). In
addition, an accurate approximation to
$\varepsilon^{\star}(\mbox{\scriptsize\sf SNR},L)$ can be solved analytically,
as we detail in the next subsection.
In order to provide a general understanding of $\varepsilon^{\star}$, Fig. 1
contains a plot of goodput $\eta$ (numerically computed) versus outage
probability $\varepsilon$ for $L=2$ and $L=5$ at $\mbox{\scriptsize\sf SNR}=0$
and $10$ dB. For each curve, the goodput-maximizing value of $\varepsilon$ is
circled. From this figure, we make the following observations:
* •
Making the physical layer too reliable or too unreliable yields poor goodput.
* •
The optimal outage probability decreases with SNR and $L$.
These turn out to be the key behaviors of the coding-ARQ tradeoff, and the
remainder of this section is devoted to analytically explain these behaviors
through a Gaussian approximation.
###### Remark 1
Throughput the paper we consider the setting _without_ channel state
information at the transmitter (CSIT). If there is CSIT, which generally is
the case when the fading is slow relative to the delay in the channel feedback
loop, the optimization problem in _Definition 1_ turns out to be trivial. When
CSIT is available, the channel is essentially AWGN with an instantaneous SNR
that is determined by the fading realization but is known to the TX. If a
capacity-achieving code with infinite codeword block-length is used in the
AWGN channel, the relationship between error and rate is a step-function:
$\displaystyle\varepsilon=$ $\displaystyle 0,$ if
$R<\log_{2}\left(1+\mbox{\scriptsize\sf SNR}|h|^{2}\right)$ (7a)
$\displaystyle\varepsilon=$ $\displaystyle 1,$ if
$R\geq\log_{2}\left(1+\mbox{\scriptsize\sf SNR}|h|^{2}\right)$. (7b)
Thus, it is optimal to choose a rate very slightly below the instantaneous
capacity $\log_{2}\left(1+\mbox{\scriptsize\sf SNR}|h|^{2}\right)$. For
realistic codes with finite blocklength, the $\varepsilon$-$R$ curve is not a
step function but nonetheless is very steep. For example, for turbo codes the
waterfall characteristic of error vs. SNR curves (for fixed rate) translates
to a step-function-like error vs. rate curve for fixed SNR. Therefore, the
transmitted rate should be chosen close to the bottom of the step function.
### III-A Gaussian Approximation
The primary difficulty in finding $\varepsilon^{\star}(\mbox{\scriptsize\sf
SNR},L)$ stems from the fact that the outage probability in (2) can only be
expressed as an $L$-dimensional integral, except for the special case $L=1$.
To circumvent this problem, we utilize a Gaussian approximation to the outage
probability used in prior work [12, 13, 14]. The random variable
$\frac{1}{L}\sum_{i=1}^{L}\log_{2}\left(1+\mbox{\scriptsize\sf
SNR}|h_{i}|^{2}\right)$ is approximated by a
$\mathcal{N}\left(\mu(\mbox{\scriptsize\sf
SNR}),\sigma^{2}(\mbox{\scriptsize\sf SNR})/L\right)$ random variable, where
$\mu(\mbox{\scriptsize\sf SNR})$ and $\sigma^{2}(\mbox{\scriptsize\sf SNR})$
are the mean and the variance of $\log_{2}\left(1+\mbox{\scriptsize\sf
SNR}|h|^{2}\right)$, respectively:
$\displaystyle\mu(\mbox{\scriptsize\sf SNR})$ $\displaystyle=$
$\displaystyle\mathbb{E}_{|h|}\left[\log_{2}(1+\mbox{\scriptsize\sf
SNR}|h|^{2})\right],$ (8) $\displaystyle\sigma^{2}(\mbox{\scriptsize\sf SNR})$
$\displaystyle=$
$\displaystyle\mathbb{E}_{|h|}\left[\log_{2}(1+\mbox{\scriptsize\sf
SNR}|h|^{2})\right]^{2}-\mu^{2}(\mbox{\scriptsize\sf SNR}).$ (9)
Closed forms for these quantities can be found in [15, 16]. Based on this
approximation we have
$\displaystyle\varepsilon$ $\displaystyle\approx$ $\displaystyle
Q\left(\frac{\sqrt{L}}{\sigma(\mbox{\scriptsize\sf
SNR})}(\mu(\mbox{\scriptsize\sf SNR})-R_{\varepsilon})\right),$ (10)
where $Q(\cdot)$ is the tail probability of a standard normal. Solving this
equation for $R_{\varepsilon}$ and plugging into (4) yields the following
approximation for the goodput, which we denote as $\eta_{g}$:
$\displaystyle\eta_{g}=\left(\mu(\mbox{\scriptsize\sf
SNR})-Q^{-1}(\varepsilon)\frac{\sigma(\mbox{\scriptsize\sf
SNR})}{\sqrt{L}}\right)(1-\varepsilon),$ (11)
where $Q^{-1}(\varepsilon)$ is the inverse of the $Q$ function.
### III-B Optimization of Goodput Approximation
The optimization of $\eta_{g}$ turns out to be more tractable. We first
rewrite $\eta_{g}$ as
$\displaystyle\eta_{g}=\mu(\mbox{\scriptsize\sf SNR})\left(1-\kappa\cdot
Q^{-1}(\varepsilon)\right)(1-\varepsilon),$ (12)
where the constant $\kappa\in(0,1)$ is the $\mu$-normalized standard deviation
of the received mutual information:
$\kappa\triangleq\frac{\sigma(\mbox{\scriptsize\sf
SNR})}{\mu(\mbox{\scriptsize\sf SNR})\sqrt{L}}.$ (13)
We can observe that $\kappa$ decreases in SNR and $L$. We now define
$\varepsilon_{g}^{\star}$ as the $\eta_{g}$-maximizing outage probability:
$\displaystyle\varepsilon_{g}^{\star}(\mbox{\scriptsize\sf
SNR},L)\triangleq\arg\max_{\varepsilon}~{}\left(1-\kappa\cdot
Q^{-1}(\varepsilon)\right)(1-\varepsilon),$ (14)
where we have pulled out the constant $\mu(\mbox{\scriptsize\sf SNR})$ from
(12) because it does not affect the maximization.
###### Proposition 1
The PHY reliability level that maximizes the Gaussian approximated goodput is
the unique solution to the following fixed point equation:
$\displaystyle\left(Q^{-1}(\varepsilon_{g}^{\star})-(1-\varepsilon_{g}^{\star})\cdot\left(Q^{-1}(\varepsilon)\right)^{\prime}\mid_{\varepsilon=\varepsilon_{g}^{\star}}\right)^{-1}=\kappa.$
(15)
Furthermore, $\varepsilon_{g}^{\star}$ is increasing in $\kappa$.
###### Proof:
See Appendix A. ∎
We immediately see that $\varepsilon_{g}^{\star}$ depends on the channel
parameters only through $\kappa$. Furthermore, because $\kappa$ is decreasing
in SNR and $L$, we see that $\varepsilon_{g}^{\star}$ decreases in $L$ (i.e.,
the channel selectivity) and SNR. Straightforward analysis shows that
$\varepsilon_{g}^{\star}$ tends to zero as $L$ increases approximately as
$1/\sqrt{L\log L}$, while $\varepsilon_{g}^{\star}$ tends to zero with SNR
approximately as $1/\sqrt{\log\mbox{\scriptsize\sf SNR}}$.
In Fig. 2, the exact optimal $\varepsilon^{\star}$ and the approximate-optimal
$\varepsilon_{g}^{\star}$ are plotted vs. SNR (dB) for $L=2,5,$ and $10$. The
Gaussian approximation is seen to be reasonably accurate, and most
importantly, correctly captures behavior with respect to $L$ and SNR.
In order to gain an intuitive understanding of the optimization, in Fig. 3 the
success probability $1-\varepsilon$ (left) and the goodput
$\eta=R_{\varepsilon}(1-\varepsilon)$ (right) are plotted versus the
transmitted rate $R$ for $\mbox{\scriptsize\sf SNR}=10$ dB. For each $L$ the
goodput-maximizing operating point is circled. First consider the curves for
$L=5$. For $R$ up to approximately $1.5$ bits/symbol the success probability
is nearly one, i.e., $\varepsilon\approx 0$. As a result, the goodput $\eta$
is approximately equal to $R$ for $R$ up to $1.5$. When $R$ is increased
beyond $1.5$ the success probability begins to decrease non-negligibly but the
goodput nonetheless increases with $R$ because the increased transmission rate
makes up for the loss in success probability (i.e., for the ARQ re-
transmissions). However, the goodput peaks at $R=2.3$ because beyond this
point the increase in transmission rate no longer makes up for the increased
re-transmissions; visually, the optimum rate (for each value of $L$)
corresponds to a point beyond which the success probability begins to drop off
sharply with the transmitted rate.
To understand the effect of the selectivity order $L$, notice that increasing
$L$ leads to a steepening of the success probability-rate curve. This has the
effect of moving the goodput curve closer to the transmitted rate, which leads
to a larger optimum rate and a larger optimum success probability
($1-\varepsilon^{\star}$). To understand why $\varepsilon^{\star}$ decreases
with SNR, based upon the rewritten version of $\eta_{g}$ in (12) we see that
the governing relationship is between the success probability $1-\varepsilon$
and the normalized, rather than absolute, transmission rate
$R/\mu(\mbox{\scriptsize\sf SNR})$. Therefore, increasing SNR steepens the
success probability-normalized rate curve (similar to the effect of increasing
$L$) and thus leads to a smaller value of $\varepsilon^{\star}$.
Is is important to notice that the optimum error probabilities in Fig. 2 are
quite large, even for large selectivity and at high SNR levels. This follows
from the earlier explanation that decreasing the error probability (and thus
the rate) beyond a certain point is inefficient because the decrease in ARQ
re-transmissions does not make up for the loss in transmission rate.
To underscore the importance of not operating the PHY too reliably, in Fig. 4
goodput is plotted versus SNR (dB) for $L=2$ and $10$ for the optimum error
probability $\eta(\varepsilon^{\star})$ as well as for $\varepsilon=0.1$,
$0.01$, and $0.001$. Choosing $\varepsilon=0.1$ leads to near-optimal
performance for both selectivity values. On the other hand, there is a
significant penalty if $\varepsilon=0.01$ or $0.001$ when $L=2$; this penalty
is reduced in the highly selective channel ($L=10$) but is still non-
negligible. Indeed, the most important insight from this analysis is that
making the PHY too reliable can lead to a significant performance penalty; for
example, choosing $\varepsilon=0.001$ leads to a power penalty of
approximately $10$ dB for $L=2$ and $2$ dB for $L=10$.
###### Remark 2
_Proposition 1_ shows $\varepsilon_{g}^{\star}$ is only determined by
$\kappa$, which is completely determined by the statistics of the received
mutual information per packet. This implies our results can be easily extended
to different fading distributions and to MIMO by appropriately modifying
$\mu(\mbox{\scriptsize\sf SNR})$ and $\sigma(\mbox{\scriptsize\sf SNR})$.
## IV OPTIMAL PHY Reliability in the Non-ideal Setting
While the previous section illustrated the need to operate the PHY at a
relatively unreliable level under a number of idealized assumptions, a
legitimate question is whether that conclusion still holds when the
idealizations of Section III are removed. Thereby motivated, in this section
we begin to carefully study the following scenarios one by one:
* •
Finite codeword block-length.
* •
Imperfect error detection.
* •
Limited number of ARQ rounds per packet.
* •
Imperfect ACK/NACK feedback.
As we shall see, our basic conclusion is upheld even under more realistic
assumptions.
### IV-A Finite Codeword Block-length
Although in the previous section we assumed operation at the mutual
information of infinite blocklength codes, real systems must use finite
blocklength codes. In order to determine the effect of finite blocklength upon
the optimal PHY reliability, we study the mutual information outage
probability in terms of the information spectrum, which captures the block
error probability for finite blocklength codes. In [17], it was shown that
actual codes perform quite close to the information spectrum-based outage
probability.
By extending the results of [18, 17], the outage probability with blocklength
$n$ (symbols) is
$\displaystyle\varepsilon(n,\mbox{\scriptsize\sf
SNR},L,R)=\mathbb{P}\left[\frac{1}{L}\sum_{i=1}^{L}\log\left(1+|h_{i}|^{2}\mbox{\scriptsize\sf
SNR}\right)+\frac{1}{n}\sum_{i=1}^{L}\left(\sqrt{\frac{|h_{i}|^{2}\mbox{\scriptsize\sf
SNR}}{1+|h_{i}|^{2}\mbox{\scriptsize\sf
SNR}}}\cdot\sum_{j=1}^{n/L}\omega_{ij}\right)\leq R\right],$ (16)
where $R$ is the transmitted rate in nats/symbol, and $\omega_{i,j}$’s are
i.i.d. Laplace random variables [18], each with zero mean and variance two.
The first term in the sum is the standard infinite blocklength mutual
information expression, whereas the second term is due to the finite
blocklength, and in particular captures the effect of atypical noise
realizations. This second term goes to zero as $n\rightarrow\infty$ (i.e.,
atypical noise does not occur in the infinite blocklength limit), but cannot
be ignored for finite $n$.
The sum of i.i.d. Laplace random variables has a Bessel-K distribution, which
is difficult to compute for large $n$ but can be very accurately approximated
by a Gaussian as verified in [17]. Thus, the mutual information conditioned on
the $L$ channel realizations is approximated by a Gaussian random variable:
$\displaystyle\mathcal{N}\left(\frac{1}{L}\sum_{i=1}^{L}\log\left(1+|h_{i}|^{2}\mbox{\scriptsize\sf
SNR}\right),\frac{1}{L}\sum_{i=1}^{L}\frac{2|h_{i}|^{2}\mbox{\scriptsize\sf
SNR}}{n\left(1+|h_{i}|^{2}\mbox{\scriptsize\sf SNR}\right)}\right)$ (17)
(This is different from Section III-A, where the Gaussian approximation is
made with respect to the fading realizations). Therefore, we can approximate
the outage probability with finite block-length $n$ by averaging the
cumulative distribution function (CDF) of (17) over different channel
realizations:
$\displaystyle\varepsilon(n,\mbox{\scriptsize\sf
SNR},L,R)\approx\mathbb{E}_{|h_{1}|,\ldots,|h_{L}|}Q\left(\frac{\frac{1}{L}\sum_{i=1}^{L}\log\left(1+|h_{i}|^{2}\mbox{\scriptsize\sf
SNR}\right)-R}{\sqrt{\frac{1}{L}\sum_{i=1}^{L}\frac{2|h_{i}|^{2}\mbox{\scriptsize\sf
SNR}}{n\left(1+|h_{i}|^{2}\mbox{\scriptsize\sf SNR}\right)}}}\right).$ (18)
In Fig. 5, we compare finite and infinite blocklength codes by plotting
success probability $1-\varepsilon$ vs. $R_{\varepsilon}$ (bits/symbol) for
$L=10$ at $\mbox{\scriptsize\sf SNR}=0$ and $10$ dB. It is clearly seen that
the steepness of the success-rate curve is reduced by the finite blocklength;
this is a consequence of atypical noise realizations.
We can now consider goodput maximization for a given blocklength $n$:
$\displaystyle\varepsilon^{\star}(\mbox{\scriptsize\sf
SNR},L,n)\triangleq~{}\arg\max_{\varepsilon}R_{\varepsilon}(1-\varepsilon),$
(19)
where both $R_{\varepsilon}$ and $\varepsilon$ are computed (numerically) in
the finite codeword block-length regime.
In Fig. 6, the optimal $\varepsilon$ vs. SNR (dB) is plotted for both finite
block-length coding and infinite block-length coding. We see that the optimal
error probability becomes larger, as expected by success-rate curves with
reduced steepness in Fig. 5. At high SNR, the finite block-length coding curve
almost overlaps the infinite block-length coding curve because the unusual
noise term in the mutual information expression is negligible for large values
of SNR. As expected, the optimal reliability level with finite blocklength
codes does not differ significantly from the idealized case.
### IV-B Non-ideal Error Detection
A critical component of ARQ is error detection, which is generally performed
using a cyclic redundancy check (CRC). The standard usage of CRC corresponds
to appending $k$ parity check bits to $b-k$ information bits, yielding a total
of $b$ bits that are then encoded (by the channel encoder) into $n$ channel
symbols. At the receiver, the channel decoder (which is generally agnostic to
CRC) takes the $n$ channel symbols as inputs and produces an estimate of the
$b$ bits, which are in turn passed to the CRC decoder for error detection. A
basic analysis in [19] shows that if the channel decoder is in error (i.e.,
the $b$ bits input to the channel encoder do not match the $b$ decoded bits),
the probability of an undetected error (i.e., the CRC decoder signals correct
even though an error has occurred) is roughly $2^{-k}$. Therefore, the overall
probability of an undetected error is well approximated by $\varepsilon\cdot
2^{-k}$.
Undetected errors can lead to significant problems, whose severity depends
upon higher network layers (e.g., whether or not an additional layer of error
detection is performed at a higher layer) and the application. However, a
general perspective is provided by imposing a constraint $p$ on the undetected
error probability, i.e., $\varepsilon\cdot 2^{-k}\leq p$. Based on this
constraint, we see that the constraint can be met by increasing $k$, which
comes at the cost of overhead, or by reducing the packet error probability
$\varepsilon$, which can significantly reduce goodput (Section III). The
question most relevant to this paper is the following: does the presence of a
stringent constraint on undetected error probability motivate reducing the PHY
packet error probability $\varepsilon$?
The relevant quantity, along with the undetected error probability, is the
rate at which information bits are correctly delivered, which is:
$\displaystyle\eta=\frac{b-k}{n}\cdot(1-\varepsilon)=\left(R_{\varepsilon}-\frac{k}{n}\right)\cdot(1-\varepsilon),$
(20)
where $R_{\varepsilon}-\frac{k}{n}$ is the effective transmitted rate after
accounting for the parity check overhead. It is then relevant to maximize this
rate subject to the constraint on undetected error:444For the sake of
compactness, the dependence of $\varepsilon^{\star}$ and $k^{\star}$ upon SNR,
$L$ and $n$ is suppressed henceforth, except where explicit notation is
required.:
$\displaystyle\left(\varepsilon^{\star},k^{\star}\right)\triangleq$
$\displaystyle\arg\max_{\varepsilon,k}~{}\left(R_{\varepsilon}-\frac{k}{n}\right)\cdot(1-\varepsilon)$
$\displaystyle\text{subject to}~{}~{}\varepsilon\cdot 2^{-k}\leq p$
Although this optimization problem (nor the version based on the Gaussian
approximation) is not analytically tractable, it is easy to see that the
solution corresponds to
$k^{\star}=\lceil-\log_{2}(p/\varepsilon^{\star})\rceil$, where
$\varepsilon^{\star}$ is roughly the optimum packet error probability assuming
perfect error detection (i.e. the solution from Section III). In other words,
the undetected error probability constraint should be satisfied by choosing
$k$ sufficiently large while leaving the PHY transmitted rate nearly
untouched. To better understand this, note that reducing $k$ by a bit requires
reducing $\varepsilon$ by a factor of two. The corresponding reduction in CRC
overhead is very small (roughly $1/n$), while the reduction in the transmitted
rate is much larger. Thus, if we consider the choices of $\varepsilon$ and $k$
that achieve the constraint with equality, i.e., $k=-\log_{2}(p/\varepsilon)$,
goodput decreases as $\varepsilon$ is decreased below the packet error
probability which is optimal under the assumption of perfect error detection.
In other words, operating the PHY at a more reliable point is not worth the
small reduction in CRC overhead.
### IV-C End-to-End Delay Constraint
In certain applications such as Voice-over-IP (VoIP), there is a limit on the
number of re-transmissions per packet as well as a constraint on the fraction
of packets that are not successfully delivered within this limit. If such
constraints are imposed, it may not be clear how aggressively ARQ should be
utilized.
Consider a system where any packet that fails on its $d$-th attempt is
discarded (i.e., at most $d-1$ re-transmissions are allowed), but at most a
fraction $q$ of packets can be discarded, where $q>0$ is a reliability
constraint. Under these conditions, the probability a packet is discarded is
$\varepsilon^{d}$, i.e., the probability of $d$ consecutive decoding failures,
while the long-term average rate at which packets are successfully delivered
still is $R_{\varepsilon}(1-\varepsilon)$. To understand why the goodput
expression is unaffected by the delay limit, note that the number of
successfully delivered packets is equal to the number of transmissions in
which decoding is successful, regardless of which packets are transmitted in
each slot. The delay constraint only affects which packets are delivered in
different slots, and thus does not affect the goodput.555The goodput
expression can alternatively be derived by computing the average number of ARQ
rounds per packet (accounting for the limit $d$), and then applying the
renewal-reward theorem [20].
Since the discarded packet probability is $\varepsilon^{d}$, the reliability
constraint requires $\varepsilon\leq q^{1/d}$. We can thus consider
maximization of goodput $R_{\varepsilon}(1-\varepsilon)$ subject to the
constraint $\varepsilon\leq q^{1/d}$. Because the goodput is observed to be
concave in $\varepsilon$, only two possibilities exist. If $q^{\frac{1}{d}}$
is larger than the optimal value of $\varepsilon$ for the unconstrained
problem, then the optimal value of $\varepsilon$ is unaffected by $q$. In the
more interesting and relevant case where $q^{\frac{1}{d}}$ is smaller than the
optimal unconstrained $\varepsilon$, then goodput is maximized by choosing
$\varepsilon$ equal to the upper bound $q^{\frac{1}{d}}$.
Thus, a strict delay and reliability constraint forces the PHY to be more
reliable than in the unconstrained case. However, amongst all allowed packet
error probabilities, goodput is maximized by choosing the largest. Thus,
although strict constraints do not allow for very aggressive use of ARQ,
nonetheless ARQ should be utilized to the maximum extent possible.
### IV-D Noisy ACK/NACK Feedback
We finally remove the assumption of perfect acknowledgements, and consider the
realistic scenario where ACK/NACK feedback is not perfect and where the
acknowledgement overhead is factored in. The main issue confronted here is the
joint optimization of the reliability level of the forward data channel and of
the reverse acknowledgement (feedback/control) channel. As intuition suggests,
reliable communication is possible only if some combination of the forward and
reverse reliability levels is sufficiently large; thus, it is not clear if
operating the PHY at a relatively unreliable level as suggested in earlier
sections is appropriate. The effects of acknowledgement errors can sometimes
be reduced through higher-layer mechanisms (e.g., sequence number check), but
in order to shed the most light on the issue of forward/reverse reliability,
we focus on an extreme case where acknowledgement errors are most harmful. In
particular, we consider a setting with delay and reliability constraints as in
Section IV-C, and where any NACK to ACK error leads to a packet missing the
delay deadline. We first describe the feedback channel model, and then analyze
performance.
#### IV-D1 Feedback Channel Model
We assume ACK/NACK feedback is performed over a Rayleigh fading channel using
a total of $f$ symbols which are distributed on $L_{\textrm{fb}}$
independently faded subchannels; here $L_{\textrm{fb}}$ is the diversity order
of the feedback channel, which need not be equal to $L$, the forward channel
diversity order. Since the feedback is binary, BPSK is used with the symbol
repeated on each sub-channel $f/L_{\textrm{fb}}$ times. For the sake of
simplicity, we assume that the feedback channel has the same average SNR as
the forward channel, and that the fading on the feedback channel is
independent of the fading on the forward channel.
After maximum ratio combining at the receiver, the effective SNR is
$(f/L_{\textrm{fb}})\cdot\mbox{\scriptsize\sf
SNR}\cdot\sum_{i=1}^{L_{\textrm{fb}}}|h_{i}|^{2}$, where
$h_{1},\cdots,h_{L_{\textrm{fb}}}$ are the feedback channel fading
coefficients. The resulting probability of error (denoted by
$\varepsilon_{\textrm{fb}}$), averaged over the fading realizations, is [21]:
$\displaystyle\varepsilon_{\textrm{fb}}=\left(\frac{1-\nu}{2}\right)^{L_{\textrm{fb}}}\cdot\sum_{j=0}^{L_{\textrm{fb}}-1}{L_{\textrm{fb}}-1+j\choose
j}\left(\frac{1+\nu}{2}\right)^{j},$ (22)
where $\nu=\sqrt{\frac{(f/L_{\textrm{fb}})\cdot\mbox{\scriptsize\sf
SNR}}{1+(f/L_{\textrm{fb}})\cdot\mbox{\scriptsize\sf SNR}}}$. Clearly,
$\varepsilon_{\textrm{fb}}$ is decreasing in $f$ and SNR.666Asymmetric
decision regions can be used, in which case $0\rightarrow 1$ and $1\rightarrow
0$ errors have unequal probabilities. However, this does not significantly
affect performance and thus is not considered.
#### IV-D2 Performance Analysis
In order to analyze performance with non-ideal feedback, we must first specify
the rules by which the transmitter and receiver operate. The transmitter takes
precisely the same actions as in Section IV-C: the transmitter immediately
moves on to the next packet whenever an ACK is received, and after receiving
$d-1$ consecutive NACK’s (for a single packet) it attempts that packet one
last time but then moves on to the next packet regardless of the
acknowledgement received for the last attempt. Of course, the presence of
feedback errors means that the received acknowledgement does not always match
the transmitted acknowledgement. The receiver also operates in the standard
manner, but we do assume that the receiver can always determine whether or not
the packet being received is the same as the packet received in the previous
slot, as can be accomplished by a simple correlation; this reasonable
assumption is equivalent to the receiver having knowledge of acknowledgement
errors.
In this setup an ACK$\rightarrow$NACK error causes the transmitter to re-
transmit the previous packet, instead of moving on to the next packet. The
receiver is able to recognize that an acknowledgement error has occurred
(through correlation of the current and previous received packets), and
because it already decoded the packet correctly it does not attempt to decode
again. Instead, it simply transmits an ACK once again. Thus, each
ACK$\rightarrow$NACK error has the relatively benign effect of wasting one ARQ
round.
On the other hand, NACK$\rightarrow$ACK errors have a considerably more
deleterious effect because upon reception of an ACK, the transmitter
automatically moves on to the next packet. Because we are considering a
stringent delay constraint, we assume that such a NACK$\rightarrow$ACK error
cannot be recovered from and thus we consider it as a lost packet that is
counted towards the reliability constraint. This is, in some sense, a worst-
case assumption that accentuates the effect of NACK$\rightarrow$ACK errors;
some comments related to this point are put forth at the end of this section.
To more clearly illustrate the model, the complete ARQ process is shown in
Fig. 7 for $d=3$. Each branch is labeled with the success/failure of the
transmission as well as the acknowledgement (including errors). Circle nodes
refer to states in which the receiver has yet to successfully decode the
packet, whereas triangles refer to states in which the receiver has decoded
correctly. A packet loss occurs if there is a decoding failure followed by a
NACK$\rightarrow$ACK error in the first two rounds, or if decoding fails in
all three attempts. All other outcomes correspond to cases where the receiver
is able to decode the packet in some round, and thus successful delivery of
the packet. In these cases, however, the number of ARQ rounds depends on the
first time at which the receiver can decode and when the ACK is correctly
delivered. (If an ACK is not successfully delivered, it may take up to $d$
rounds before the transmitter moves on to the next packet.) Notice that after
the $d$-th attempt, the transmitter moves on to the next packet regardless of
what acknowledgement is received; this is due to the delay constraint that the
transmitter follows.
Based on the figure and the independence of decoding and feedback errors
across rounds, the probability that a packet is lost (i.e., it is not
successfully delivered within $d$ rounds) is:
$\displaystyle\xi_{d}=\varepsilon\cdot\varepsilon_{\textrm{fb}}+\varepsilon^{2}(1-\varepsilon_{\textrm{fb}})\varepsilon_{\textrm{fb}}+\cdots+\varepsilon^{d-1}(1-\varepsilon_{\textrm{fb}})^{d-2}\varepsilon_{\textrm{fb}}+\varepsilon^{d}(1-\varepsilon_{\textrm{fb}})^{d-1},$
(23)
where the first $d-1$ terms represent decoding failures followed by a
NACK$\rightarrow$ACK error (more specifically, the $l$-th term corresponds to
$l-1$ decoding failures and $l-1$ correct NACK transmissions, followed by
another decoding failure and a NACK$\rightarrow$ACK error), and the last term
is the probability of $d$ decoding failures and $d-1$ correct NACK
transmissions. If we alternatively compute the success probability, we get the
following different expression for $\xi_{d}$:
$\displaystyle\xi_{d}=1-\sum_{i=1}^{d}(1-\varepsilon)\cdot\varepsilon^{i-1}\cdot(1-\varepsilon_{\textrm{fb}})^{i-1},$
(24)
where the $i$-th summand is the probability that successful forward
transmission occurs in the $i$-th ARQ round. Based upon (23) and (24) we see
that $\xi_{d}$ is increasing in both $\varepsilon$ and
$\varepsilon_{\textrm{fb}}$. Thus, a desired packet loss probability $\xi_{d}$
can be achieved by different combinations of the forward channel reliability
and the feedback channel reliability: a less reliable forward channel requires
a more reliable feedback channel, and vice versa.
As in Section IV-C we impose a reliability constraint $\xi_{d}\leq q$, which
by (23) translates to a joint constraint on $\varepsilon$ and
$\varepsilon_{\textrm{fb}}$. The relatively complicated joint constraint can
be accurately approximated by two much simpler constraints. Since we must
satisfy $\varepsilon\leq q^{\frac{1}{d}}$ even with perfect feedback
($\varepsilon_{\textrm{fb}}=0$), for any $\varepsilon_{\textrm{fb}}>0$ we also
must satisfy $\varepsilon\leq q^{\frac{1}{d}}$ (this ensures that $d$
consecutive decoding failures do not occur too frequently). Furthermore, by
examining (23) it is evident that the first term is dominant in the packet
loss probability expression. Thus the constraint $\xi_{d}\leq q$ essentially
translates to the simplified constraints
$\displaystyle\varepsilon\cdot\varepsilon_{\textrm{fb}}\leq
q\textrm{~{}~{}~{}and~{}~{}~{}}\varepsilon\leq q^{\frac{1}{d}}.$ (25)
These simplified constraints are very accurate for values of $\varepsilon$ not
too close to $q^{\frac{1}{d}}$. On the other hand, as $\varepsilon$ approaches
$q^{\frac{1}{d}}$, $\varepsilon_{\textrm{fb}}$ must go to zero very rapidly
(i.e. much faster than $q/\varepsilon$) in order for $\xi_{d}\leq q$.
The first constraint in (25) reveals a general design principle: the
combination of the forward and feedback channel must be sufficiently reliable.
This is because $\varepsilon\cdot\varepsilon_{\textrm{fb}}$ is precisely the
probability that a packet is lost because the initial transmission is decoded
incorrectly and is followed by a NACK$\rightarrow$ACK error.
Having established the reliability constraint, we now proceed to maximizing
goodput while taking acknowledgement errors and ARQ overhead into account.
With respect to the long-term average goodput, by applying the renewal-reward
theorem again we obtain:
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{n}{n+f}\cdot\frac{R_{\varepsilon}(1-\xi_{d})}{\mathbb{E}[X]}.$
(26)
where random variable $X$ is the number of ARQ rounds per packet, and
$\mathbb{E}[X]$ is derived in Appendix B. Here, $\frac{n}{n+f}$ is the
feedback overhead penalty because each packet spanning $n$ symbols is followed
by $f$ symbols to convey the acknowledgement.
We now maximize goodput with respect to both the forward and feedback channel
error probabilities:
$\displaystyle\left(\varepsilon^{\star},\varepsilon_{\textrm{fb}}^{\star}\right)\triangleq$
$\displaystyle\arg\max_{\varepsilon,\varepsilon_{\textrm{fb}}}~{}~{}~{}\frac{n}{n+f}\cdot\frac{R_{\varepsilon}(1-\xi_{d})}{\mathbb{E}[X]}$
$\displaystyle\text{subject to}~{}~{}\xi_{d}\leq q$
noting that $\varepsilon_{\textrm{fb}}$ is a decreasing function of the number
of feedback symbols $f$, according to (22). This optimization is not
analytically tractable, but can be easily solved numerically and can be
understood through examination of the dominant relationships. The overhead
factor $n/(n+f)$ clearly depends only on $\varepsilon_{\textrm{fb}}$ (i.e.,
$f$). Although the second term $R_{\varepsilon}(1-\xi_{d})/\mathbb{E}[X]$
depends on both $\varepsilon$ and $\varepsilon_{\textrm{fb}}$, the dependence
upon $\varepsilon_{\textrm{fb}}$ is relatively minor as long as
$\varepsilon_{\textrm{fb}}$ is reasonably small (i.e. less than $10\%$). Thus,
it is reasonable to consider the perfect feedback setting, in which case the
second term is $R_{\varepsilon}(1-\varepsilon)$. Therefore, the challenge is
balancing the feedback channel overhead factor $\frac{n}{n+f}$ with the
efficiency of the forward channel, approximately
$R_{\varepsilon}(1-\varepsilon)$, while satisfying the constraint in (25). If
$f$ is chosen small, the feedback errors must be compensated with a very
reliable, and thus inefficient, forward channel; on the other hand, choosing
$f$ large incurs a large feedback overhead penalty but allows for a less
reliable, and thus more efficient, forward channel.
In Fig. 8, the jointly optimal
($\varepsilon^{\star},\varepsilon_{\textrm{fb}}^{\star}$) are plotted for a
conservative set of forward channel parameters ($L=3$ with
$\mbox{\scriptsize\sf SNR}=5$ or $10$ dB, and $n=200$ data symbols per
packet), stringent delay and reliability constraints (up to $d=3$ ARQ rounds
and a reliability constraint $q=10^{-6}$), and different diversity orders
($L_{\textrm{fb}}=1,2$ and $5$) for the feedback channel. Also plotted is the
curve specifying the ($\varepsilon,\varepsilon_{\textrm{fb}}$) pairs that
achieve the reliability constraint $\xi_{d}=q$. As discussed earlier, this
curve has two distinct regions: for $\varepsilon<0.008$ it is essentially the
straight line $\varepsilon\cdot\varepsilon_{\textrm{fb}}=q$, whereas
$\varepsilon_{\textrm{fb}}$ must go to zero very quickly as $\varepsilon$
approaches $q^{1/d}=10^{-2}$.
When $L_{\textrm{fb}}=2$, the optimal point corresponds to the transition
between these two regions. Moving to the right of the optimal corresponds to
making the PHY more reliable while making the control channel less reliable
(i.e., decreasing $\varepsilon$ and $f$), but this is suboptimal because the
overhead savings do not compensate for the loss incurred by a more reliable
PHY. On the other hand, moving to the left is suboptimal because only a very
modest increase in $\varepsilon$ is allowed, and this increase comes at a
large expense in terms of control symbols. If $L_{\textrm{fb}}=5$, the optimal
point is further to the left because the feedback overhead required to achieve
a desired error rate is reduced. However, the behavior is quite different if
there is no diversity on the feedback channel ($L_{\textrm{fb}}=1$). Without
diversity, the feedback error probability decreases extremely slowly with $f$
(at order $1/f$), and thus a very large $f$ is required to achieve a
reasonable feedback error probability. In this extreme case, it is optimal to
sacrifice significant PHY efficiency and choose $\varepsilon$ quite a bit
smaller than $q^{1/d}=10^{-2}$. Notice that increasing SNR moves the optimal
to the left for all values of $L_{\textrm{fb}}$ because a larger SNR improves
the feedback channel reliability while not significantly changing the behavior
of the forward channel.
This behavior is further explained in Fig. 9, where goodput $\eta$ (optimized
with respect to $\varepsilon_{\textrm{fb}}$) is plotted versus forward error
probability $\varepsilon$ for the parameters of the previous figure, with
$\mbox{\scriptsize\sf SNR}=5$ dB and $L_{\textrm{fb}}=1$ and $2$ here. The
figure illustrates the stark contrast with respect to feedback channel
diversity: with diversity (even for $L_{\textrm{fb}}=2$), the goodput
increases monotonically up to a point quite close to $q^{1/d}$, while without
diversity the goodput peaks at a point far below $q^{1/d}$. This is due to the
huge difference in the feedback channel reliability with and without
diversity: in order to achieve $\varepsilon_{\textrm{fb}}=10^{-3}$, at
$\mbox{\scriptsize\sf SNR}=5$ dB without diversity $f=79$ symbols are
required, whereas $f=9$ suffices for $L_{\textrm{fb}}=2$. To more clearly
understand why the optimal point with diversity is so close to $q^{1/d}$, let
us contrast two different choices of $\varepsilon$ for $L_{\textrm{fb}}=2$. At
the optimal $\varepsilon=8\times 10^{-3}$, we require
$\varepsilon_{\textrm{fb}}=6.3\times 10^{-5}$ and thus $f=34$. On the other
hand, at the suboptimal $\varepsilon=10^{-3}$ we require
$\varepsilon_{\textrm{fb}}=10^{-3}$ and thus $f=9$. Reducing the forward error
probability by a factor of $8$ reduces the feedback overhead from
$\frac{34}{234}$ to $\frac{9}{209}$, but reduces the transmitted rate by about
$50\%$.
The takeaway message of this analysis is clear: as long as the feedback
channel has at least some diversity (e.g., through frequency or antennas),
stringent post-ARQ reliability constraints should be satisfied by increasing
the reliability of the feedback channel instead of increasing the forward
channel reliability. This is another consequence of the fact that decreasing
the forward channel error probability requires a huge backoff in terms of
transmitted rate, which in this case is not compensated by the corresponding
decrease in feedback overhead.
## V Hybrid-ARQ
While up to now we have considered simple ARQ, contemporary wireless systems
often utilize more powerful hybrid-ARQ (HARQ) techniques. When incremental
redundancy (IR) HARQ, which is the most powerful type of HARQ, is implemented,
a NACK triggers the transmission of extra parity check bits instead of re-
transmission of the original packet, and the receiver attempts to decode a
packet on the basis of all previous transmissions related to that packet. This
corresponds to accumulation of mutual information across HARQ rounds, and thus
essentially matches the transmitted rate to the instantaneous channel
conditions without requiring CSI at the transmitter [10, 14]. The focus of
this section is understanding how the PHY transmitted rate should be chosen
when HARQ is used.
Unlike simple ARQ, HARQ requires the receiver to keep information from
previous rounds in memory; partly for this reason, HARQ is generally
implemented in a two-layered system (e.g., in 4G cellular networks such as LTE
[22] [23]) in which the HARQ process has to restart (triggered by a higher-
layer simple ARQ re-transmission) if the number of HARQ rounds reaches a
defined maximum. The precise model we study is described as follows. As
before, each HARQ transmission (i.e., round) experiences a diversity order of
$L$. However, a maximum of $M$ HARQ rounds are allowed per packet. If a packet
cannot be decoded after $M$ HARQ rounds, a post-HARQ outage is declared. This
triggers a higher-layer simple ARQ re-transmission, which restarts the HARQ
process for that packet. This two-layered ARQ process continues (indefinitely)
until the packet is successfully received at the receiver. For the sake of
simplicity, we proceed under the ideal assumptions discussed in Section III.
Note that the case $M=1$ reverts to the simple ARQ model discussed in the rest
of the paper.
Given this model, the first-HARQ-round outage probability, denoted
$\varepsilon_{1}$, is exactly the same as the non-HARQ outage probability with
the same SNR, diversity order $L$, and rate $R$ , i.e.,
$\displaystyle\varepsilon_{1}(\mbox{\scriptsize\sf
SNR},L,R)=\mathbb{P}\left[\frac{1}{L}\sum_{i=1}^{L}\log_{2}\left(1+\mbox{\scriptsize\sf
SNR}|h_{i}|^{2}\right)\leq R\right].$ (28)
In this expression $R$ is the transmitted rate during the first HARQ round,
which we refer to as the HARQ initial rate $R_{\textrm{init}}$ hereafter.
Because IR leads to accumulation of mutual information, the number of HARQ
rounds needed to decode a packet is the smallest integer $\mathcal{T}$
($1\leq\mathcal{T}\leq M$) such that
$\displaystyle\sum_{i=1}^{\mathcal{T}}\left(\frac{1}{L}\sum_{j=1}^{L}\log_{2}\left(1+\mbox{\scriptsize\sf
SNR}|h_{i,j}|^{2}\right)\right)>R_{\textrm{init}}.$ (29)
Therefore, the post-HARQ outage, denoted by $\varepsilon$, is:
$\displaystyle\varepsilon(\mbox{\scriptsize\sf SNR},L,M,R_{\textrm{init}})$
$\displaystyle=$
$\displaystyle\mathbb{P}\left[\sum_{i=1}^{M}\left(\frac{1}{L}\sum_{j=1}^{L}\log_{2}\left(1+\mbox{\scriptsize\sf
SNR}|h_{i,j}|^{2}\right)\right)\leq R_{\textrm{init}}\right].$ (30)
This is the probability that a packet fails to be decoded after $M$ HARQ
rounds, and thus is the probability that the HARQ process has to be restarted.
Using the renewal-reward theorem as in [10] yields the following expression
for the long-term average goodput with HARQ:
$\displaystyle\eta=\frac{R_{\textrm{init}}(1-\varepsilon)}{\mathbb{E}[\mathcal{T}]},$
(31)
where the distribution of $\mathcal{T}$ is determined by (29). Our interest is
in finding the initial rate $R_{\textrm{init}}$ that maximizes $\eta$. This
optimization is not analytically tractable, but we can nonetheless provide
some insight.
In Fig. 10, goodput is plotted versus vs. $R_{\textrm{init}}$ for $L=2$ and a
maximum of $M=2$ HARQ rounds, as well as for a system using only simple ARQ
(i.e., $M=1$) with the same $L=2$, at $\mbox{\scriptsize\sf SNR}=5$ and $10$
dB. We immediately observe that goodput with HARQ is maximized at a
considerably higher rate than for the system without HARQ. Although we do not
have analytical proof, we conjecture that the goodput-maximizing initial rate
with HARQ is always larger than the maximizing rate without HARQ (for equal
diversity order per round/transmission). In fact, with HARQ the initial rate
should be chosen such that the first-round outage $\varepsilon_{1}$ is quite
large, and for larger values of $M$ the optimizer actually trends towards one.
If $\varepsilon_{1}$ is small, then HARQ is rarely used which means that the
rate-matching capability provided by HARQ is not exploited. However,
$R_{\textrm{init}}$ should not be chosen so large such that there is
significant probability of post-HARQ outage, because this leads to a simple
ARQ re-transmission and thus forces HARQ to re-start. The following theorem
provides an upper bound on the optimal initial rate:
###### Theorem 1
For any $\mbox{\scriptsize\sf SNR},L$, and $M$, the optimal initial rate with
HARQ is upper bounded by $1/M$ times the optimal transmitted rate for a non-
HARQ system with diversity order $ML$.
###### Proof:
The HARQ goodput can be rewritten as
$\displaystyle\eta=\frac{R_{\textrm{init}}}{M}\cdot(1-\varepsilon)\cdot\frac{M}{\mathbb{E}[\mathcal{T}]}.$
(32)
Based on (30) we see that the post-HARQ outage probability $\varepsilon$ is
precisely the same as the outage probability for a non-HARQ system with
diversity order $ML$ and transmitted rate $R_{\textrm{init}}/M$. Therefore,
the term $(R_{\textrm{init}}/M)(1-\varepsilon)$ in (32) is precisely the
goodput for a non-HARQ system with diversity order $ML$. Based on (29) we can
see that the term $M/\mathbb{E}[\mathcal{T}]$ is decreasing in
$R_{\textrm{init}}/M$, and thus the value of $R_{\textrm{init}}/M$ that
maximizes (32) is smaller than the value that maximizes
$(R_{\textrm{init}}/M)(1-\varepsilon)$. ∎
Notice that $ML$ is the maximum diversity experienced by a packet if HARQ is
used, whereas $ML$ is the precise diversity order experienced by each packet
in the reference system (in the theorem) without HARQ. Combined with our
earlier observation, we see that the initial rate should be chosen large
enough such that HARQ is sufficiently utilized, but not so large such that
simple ARQ is overly used.
## VI Conclusion
In this paper we have conducted a detailed study of the optimum physical layer
reliability when simple ARQ is used to re-transmit incorrectly decoded
packets. Our findings show that when a cross-layer perspective is taken, it is
optimal to use a rather unreliable physical layer (e.g., a packet error
probability of 10% for a wide range of channel parameters). The fundamental
reason for this is that making the physical layer very reliable requires a
very conservative transmitted rate in a fading channel (without instantaneous
channel knowledge at the transmitter).
Our findings are quite general, in the sense that the PHY should not be
operated reliably even in scenarios in which intuition might suggest PHY-level
reliability is necessary. For example, if a smaller packet error mis-detection
probability is desired, it is much more efficient to utilize additional error
detection bits (e.g., CRC) as compared to performing additional error
correction (i.e., making the PHY more reliable). A delay constraint imposes an
upper bound on the number of ARQ re-transmissions and an upper limit on the
PHY error probability, but an optimized system should operate at exactly this
level and no lower. Finally, when acknowledgement errors are taken into
account and high end-to-end reliability is required, such reliability should
be achieved by designing a reliable feedback channel instead of a reliable
data (PHY) channel.
In a broader context, one important message is that traditional diversity
metrics, which characterize how quickly the probability of error can be made
very small, may no longer be appropriate for wireless systems due to the
presence of ARQ. As seen in [24] in the context of multi-antenna
communication, this change can significantly reduce the attractiveness of
transmit diversity techniques that reduce error at the expense of rate.
## Appendix A PROOF of Proposition 1
We first prove the strict concavity of $\eta_{g}$. For any invertible function
$f(\cdot)$, the following holds [25]:
$\displaystyle\left(f^{-1}(a)\right)^{\prime}=\frac{1}{f^{\prime}(f^{-1}(a))}.$
(33)
By combining this with
$Q(x)=\int_{x}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{t^{2}}{2}}dt$, we get
$\displaystyle\left(Q^{-1}(\varepsilon)\right)^{\prime}=-\sqrt{2\pi}e^{\frac{(Q^{-1}(\varepsilon))^{2}}{2}},$
(34)
which is strictly negative. According to this, the second derivative of
$\eta_{g}(\varepsilon)$ is:
$\displaystyle\left(\eta_{g}(\varepsilon)\right)^{\prime\prime}$
$\displaystyle=$
$\displaystyle\kappa\mu\left(Q^{-1}(\varepsilon)\right)^{\prime}\left(2+(1-\varepsilon)\sqrt{2\pi}e^{\frac{(Q^{-1}(\varepsilon))^{2}}{2}}Q^{-1}(\varepsilon)\right).$
(35)
Because $\kappa\left(Q^{-1}(\varepsilon)\right)^{\prime}<0$, in order to prove
$\left(\eta_{g}(\varepsilon)\right)^{\prime\prime}<0$ we only need to show
that the expression inside the parenthesis in (35) is strictly positive. If we
substitute $\varepsilon=Q(x)$ (here we define $x=Q^{-1}(\varepsilon)$) , then
we only need to prove $(Q(x)-1)e^{\frac{x^{2}}{2}}x<\sqrt{\frac{2}{\pi}}$.
Notice when $x\geq 0$, the left hand side is negative (because $Q(x)\leq 1$)
and the inequality holds. When $x<0$, the left hand side becomes
$Q(-x)e^{\frac{x^{2}}{2}}(-x)$. From [26],
$Q(-x)<\frac{1}{\sqrt{2\pi}(-x)}e^{-\frac{x^{2}}{2}}$, so if $x<0$,
$\displaystyle(Q(x)-1)e^{\frac{x^{2}}{2}}x<\frac{1}{\sqrt{2\pi}(-x)}e^{-\frac{x^{2}}{2}}e^{\frac{x^{2}}{2}}(-x)=\frac{1}{\sqrt{2\pi}}<\sqrt{\frac{2}{\pi}}.$
(36)
As a result, the second derivative of $\eta_{g}(\varepsilon)$ is strictly
smaller than zero and thus $\eta_{g}$ is strictly concave in $\varepsilon$.
Since $\eta_{g}$ is strictly concave in $\varepsilon$, we reach the fixed
point equation in (15) by setting the first derivative to zero. The concavity
of $\eta_{g}$ implies $\left(\eta_{g}(\varepsilon)\right)^{\prime}$ is
decreasing in $\varepsilon$, and thus from (15) we see that
$\varepsilon_{g}^{\star}$ is increasing in $\kappa$.
## Appendix B Expected ARQ Rounds with Acknowledgement Errors
If the ARQ process terminates after $i$ rounds ($1\leq i\leq d-1$), the
reasons for that can be:
* •
The first $i$ decoding attempts are unsuccessful, the first $i-1$ NACKs are
received correctly, but a NACK$\rightarrow$ACK error happens in the $i$-th
round, the probability of which is
$\varepsilon^{i}\cdot(1-\varepsilon_{\textrm{fb}})^{i-1}\cdot\varepsilon_{\textrm{fb}}$.
* •
The packet is decoded correctly in the $j$-th round (for $1\leq j\leq i$), but
the ACK is not correctly received until the $i$-th round. This corresponds to
$j-1$ decoding failures with correct acknowledgements, followed by a decoding
success and $i-j$ acknowledgement errors (ACK$\rightarrow$NACK), and then a
correct acknowledgement:
$\sum_{j=1}^{i}\varepsilon^{j-1}(1-\varepsilon_{\textrm{fb}})^{j}(1-\varepsilon)\varepsilon_{\textrm{fb}}^{i-j}$.
These events are all exclusive, and thus we can sum the above probabilities.
For $X=d$, we notice that the ARQ process takes the maximum of $d$ rounds if:
* •
There are $d$ decoding failures with $d-1$ correct NACKs, the probability of
which is $\varepsilon^{d-1}\cdot(1-\varepsilon_{\textrm{fb}})^{d-1}$.
* •
The packet is decoded correctly in the $j$-th round (for $1\leq j\leq d-1$),
but the ACK is never received correctly. This corresponds to $j-1$ decoding
failures with correct NACKs, followed by a decoding success and $d-j$
acknowledgement errors (ACK$\rightarrow$NACK):
$\sum_{j=1}^{d-1}\varepsilon^{j-1}(1-\varepsilon_{\textrm{fb}})^{j-1}(1-\varepsilon)\varepsilon_{\textrm{fb}}^{d-j}$.
These events are again exclusive. Therefore, the expected number of rounds is:
$\displaystyle\mathbb{E}[X]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{d-1}i\cdot\left(\varepsilon^{i}\cdot(1-\varepsilon_{\textrm{fb}})^{i-1}\cdot\varepsilon_{\textrm{fb}}+\sum_{j=1}^{i}\varepsilon^{j-1}(1-\varepsilon_{\textrm{fb}})^{j}(1-\varepsilon)\varepsilon_{\textrm{fb}}^{i-j}\right)$
(37)
$\displaystyle+d\cdot\left(\varepsilon^{d-1}\cdot(1-\varepsilon_{\textrm{fb}})^{d-1}+\sum_{j=1}^{d-1}\varepsilon^{j-1}(1-\varepsilon_{\textrm{fb}})^{j-1}(1-\varepsilon)\varepsilon_{\textrm{fb}}^{d-j}\right).$
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Figure 1: Gooput $\eta$ (bits/symbol) vs. PHY outage probability $\varepsilon$
for $L=2,5$, $\mbox{\scriptsize\sf SNR}=10$ dB Figure 2: Optimal $\varepsilon$
vs. SNR (dB) for $L=2,5,10$
(a) $1-\varepsilon$ vs. $R_{\varepsilon}$ (bits/symbol)
(b) $\eta$ (bits/symbol) vs. $R_{\varepsilon}$ (bits/symbol)
Figure 3: Success probability $1-\varepsilon$ and $\eta$ (bits/symbol) vs.
$R_{\varepsilon}$ (bits/symbol) for $\mbox{\scriptsize\sf SNR}=10$ dB
(a) $L=2$
(b) $L=10$
Figure 4: $\eta$ (bits/symbol) vs. SNR (dB), for $\varepsilon=0.001,0.01,0.1$,
and $\varepsilon^{\star}$ Figure 5: Success probability $1-\varepsilon$ vs.
transmitted rate $R_{\varepsilon}$ (bits/symbol) for $n=50,200,\infty$, $L=10$
at $\mbox{\scriptsize\sf SNR}=0$ and $10$ dB Figure 6: Optimal $\varepsilon$
vs. SNR (dB) for $L=2,5,10$ and $n=200,500$ and $\infty$ Figure 7: The ARQ
process with non-ideal feedback with an end-to-end delay constraint $d=3$.
Figure 8: ($\varepsilon^{\star},\varepsilon_{\textrm{fb}}^{\star}$) with
$L_{\textrm{fb}}=1,2$ and $5$ in Rayleigh fading feedback channel for $n=200$,
$d=3$, $q=10^{-6}$, and $L=3$ at $\mbox{\scriptsize\sf SNR}=5$ and $10$ dB.
The curve specifying the ($\varepsilon,\varepsilon_{\textrm{fb}}$) pairs that
achieve the reliability constraint $\xi_{d}=q$ is also plotted. Figure 9:
Goodput $\eta$ (bits/symbol) vs. PHY outage probability $\varepsilon$ with
$L_{\textrm{fb}}=1$ and $2$ in Rayleigh fading feedback channel for
$\mbox{\scriptsize\sf SNR}=5$ dB, $n=200$, $L=3$, $d=3$ and $q=10^{-6}$.
Figure 10: Goodput (bits/symbol) vs. initial rate (bits/symbol) with HARQ for
$M=2$ and $L=2$ and without HARQ for $M=1$ and $L=2$ at $\mbox{\scriptsize\sf
SNR}=5,10$ dB.
|
arxiv-papers
| 2009-04-01T17:33:03 |
2024-09-04T02:49:01.604907
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Peng Wu and Nihar Jindal",
"submitter": "Peng Wu",
"url": "https://arxiv.org/abs/0904.0226"
}
|
0904.0422
|
# Classical and quantum Cosmology of the Sáez-Ballester theory
J. Socorro socorro@fisica.ugto.mx M. Sabido msabido@fisica.ugto.mx L. Arturo
Ureña-López lurena@fisica.ugto.mx Departamento de Física, DCI, Campus León,
Universidad de Guanajuato, C.P. 37150, Guanajuato, México
###### Abstract
We study the generalization of the Sáez-Ballester theory applied to a flat FRW
cosmological model. Classical exact solutions up to quadratures are easily
obtained using the Hamilton-Jacobi approach. Contrary to claims in the
specialized literature, it is shown that the Sáez-Ballester theory cannot
provide a realistic solution to the dark matter problem of Cosmology.
Furthermore the quantization procedure of the theory can be simplified by
reinterpreting the theory in the Einstein frame, where the scalar field can be
interpreted as part of the matter content of the theory, in this approach,
exact solutions are also found for the Wheeler-DeWitt equation in the quantum
regime.
###### pacs:
04.20.Fy; 04.20.Jb; 04.60.Kz; 98.80.Qc.
## I Introduction
The inclusion of scalar fields into homogeneous cosmologies is a typical
practice to study different scenarios, such as inflation, dark matter, and
dark energyCopeland:2006wr . However, since the early seventies, the problem
exists of finding the appropriate sources of matter and its corresponding
Lagrangian to solve an specific scenarioryan1 ; ryan .
In this respect, Saez and Ballester (SB)s-b formulated a scalar-tensor theory
of gravitation in which the metric is coupled to a dimensionless scalar field
in order to solve the so-called missing matter problem in Cosmology. Some
works about the classical regime are already present in the literaturesingh ;
shri ; mohanty ; singh-shri . In particular, in ref. singh-shri the authors
consider the coupling parameter time-dependent and take a particular ansatz
for mathematical convenience for solving the field equations.
In spite of a the dimensionless character of the scalar field, an antigravity
regime appears, and this fact has been used to suggest a new possible way to
solve the missing matter problem in non-flat FRW cosmologies. On the other
hand, the quantization program of the theory has yet to be made.
In this paper, we shall study a generalization of the SB theory and transform
it into a conventional tensor theory, where the dimensionless scalar field is
interpreted as an exotic matter. We found the general behaviour for the
kinetic scalar field dependent to the scale factor of the universe, but the
behaviour corresponds to stiff matter and not for a dust universe, then the
missing matter problem is not solved.
With respect to the quantization program, in this approach we can construct
the quantization program of the theory using the usual ADM formalismryan1 .
Also, we can in principle quantize the theory following the Loop Quantum
Cosmoloy program.
In this work, we shall use this formulation to obtain classical and quantum
solutions in quadratures, for the flat barotropic FRW cosmology, including a
cosmological term $\lambda$.
The paper is arranged as follows, In section II we write the generalization
Sáez-Ballester formalism in the usual manner, that is, we calculate the
corresponding energy-momentum tensor to the scalar field and give the
equivalent lagrangian density. Next, we proceed to obtain the corresponding
canonical lagrangian ${\cal L}_{can}$ to a flat FRW universe through the
lagrange transformation, we calculate the classical hamiltonian, we also
present solutions to some models. In section III, using the transformation and
the Hamiltonian constraint ${\cal H}$, f we find the Wheeler-DeWitt (WDW)
equation of the corresponding cosmological model under study. Section IV is
devoted to conclusions and outlook.
## II Generalized Saez-Ballester theory
The simplest generalization of the Sáez-Ballester theorys-b with a
cosmological term is
${\cal
L}_{geo}=\left(R-2\lambda-F(\phi)\phi_{,\gamma}\phi^{,\gamma}\right)\,,$ (1)
where $R$ the scalar curvature,
$\phi^{,\gamma}=g^{\gamma\alpha}\phi_{,\alpha}$, and $F(\phi)$ is a
dimensionless and arbitrary functional of the scalar field. According to
common wisdom, the Lagrangian (1) would correspond to a scalar field theory
without scalar potential but with an exotic kinetic term.
The complete action is then
$I=\int_{\Sigma}\sqrt{-g}({\cal L}_{geo}+{\cal L}_{mat})\,d^{4}x\,,$ (2)
where we have included a matter Lagrangian ${\cal L}_{mat}$, and $g$ is the
determinant of metric tensor. The field equations derived from the above
action are
$\displaystyle
G_{\alpha\beta}+g_{\alpha\beta}\lambda-F(\phi)\left(\phi_{,\alpha}\phi_{,\beta}-\frac{1}{2}g_{\alpha\beta}\phi_{,\gamma}\phi^{,\gamma}\right)$
$\displaystyle=$ $\displaystyle 8\pi GT_{\alpha\beta}\,,$ (3a) $\displaystyle
2F(\phi)\phi^{,\alpha}_{\,\,;\alpha}+\frac{dF}{d\phi}\phi_{,\gamma}\phi^{,\gamma}$
$\displaystyle=$ $\displaystyle 0\,,$ (3b)
in which $G$ is the gravitational constant, and a semicolon means covariant
derivative.
The same set of equations(3a,3b) is obtained if we consider the scalar field
$\phi$ as part of the matter budget, i.e. say $\rm{\cal L}_{\phi}=\rm
F(\phi)g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}$. In this new line of
reasoning, action (2) can be rewritten as a geometrical part (Hilbert-Einstein
with $\Lambda$) and matter content (usual matter plus a term that corresponds
to the scalar field component of Sáez-Ballester theory),
$I=\int_{\Sigma}\sqrt{-g}\left(R-2\lambda+{\cal L}_{mat}+{\cal
L}_{\phi}\right)\,d^{4}x\,.$ (4)
Even though the philosophy is different to that of the original SB theory, the
similarity of the latter to a standard scalar field theory at the classical
level will help us to infer the correspondence quantum formulation. We expect
the quantum picture will also be the correct one for the SB theory, as all the
formulation is based upon the same (classical) Hamiltonian constraint.
Using this action we obtain the classical Hamiltonian of the generalized SB
theory for a Friedmann-Robertson-Walker background. Let us start with the line
element for a homogeneous and isotropic universe,
$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-\kappa
r^{2}}+r^{2}d\Omega^{2}\right]\,,$ (5)
where $a(t)$ is the scale factor, $N(t)$ is the lapse function, and $\kappa$
is the curvature constant that can to take the values $0$, $1$ and $-1$, for
flat, closed and open universe, respectively. The total Lagrangian density
then reads
${\cal L}=\frac{6\dot{a}^{2}a}{N}-6\kappa
Na+\frac{F(\phi)a^{3}}{N}\dot{\phi}^{2}+16\pi GNa^{3}\rho-2Na^{3}\lambda\,,$
(6)
where $\rho$ is the matter energy density; we will assume that it complies
with a barotropic equation of state of the form $p=\gamma\rho$, where $\gamma$
is a constant. The conjugate momenta are obtained from
$\displaystyle\Pi_{a}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L}}{\partial\dot{a}}=\frac{12a\dot{a}}{N},\qquad\rightarrow\qquad\dot{a}=\frac{N\Pi_{a}}{12a}\,,$
$\displaystyle\Pi_{\phi}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L}}{\partial\dot{\phi}}=\frac{2Fa^{3}\dot{\phi}}{N}\,,\qquad\rightarrow\qquad\dot{\phi}=\frac{N\Pi_{\phi}}{2Fa^{3}}\,.$
(7)
From the canonical form of the Lagrangian density (6) and the solution for the
barotropic fluid equation of motion we find the Hamiltonian density for this
theory
${\cal
H}=\frac{a^{-3}}{24}\left[a^{2}\Pi_{a}^{2}+\frac{6}{F(\phi)}\Pi_{\phi}^{2}+144\kappa
a^{4}+48a^{6}\lambda-384\pi G\rho_{\gamma}a^{3(1-\gamma)}\right],$ (8)
where $\rho_{\gamma}$ is an integration constant.
### II.1 Classical solutions for flat FRW
Using the transformation $\Pi_{q}=\frac{dS_{q}}{dq}$, the Einstein-Hamilton-
Jacobi corresponding to Eq. (8) is
$a^{2}\left(\frac{dS_{a}}{da}\right)^{2}+\frac{6}{F(\phi)}\left(\frac{dS_{\phi}}{d\phi}\right)^{2}+48a^{6}\lambda-384\pi
G\rho_{\gamma}a^{3(1-\gamma)}=0\,,.$ (9)
The EHJ equation can be further separated in the equations
$\displaystyle\frac{6}{F(\phi)}\left(\frac{dS_{\phi}}{d\phi}\right)^{2}$
$\displaystyle=$ $\displaystyle\mu^{2}\,,$ (10) $\displaystyle
a^{2}\left(\frac{dS_{a}}{da}\right)^{2}+48a^{6}\lambda-384\pi
G\rho_{\gamma}a^{3(1-\gamma)}$ $\displaystyle=$ $\displaystyle-\mu^{2}\,,$
(11)
where $\mu$ is a separation constant. With the help of Eqs. (7), we can obtain
the solution up to quadratures of Eqs. (10) and (11),
$\displaystyle\int\sqrt{F(\phi)}\,d\phi$ $\displaystyle=$
$\displaystyle\frac{\mu}{2\sqrt{6}}\int a^{-3}(\tau)\,d\tau\,,$ (12a)
$\displaystyle\Delta\tau$ $\displaystyle=$
$\displaystyle\int\frac{a^{2}da}{\sqrt{\frac{8}{3}\pi
G\rho_{\gamma}a^{3(1-\gamma)}-\frac{\lambda}{3}a^{6}-\nu^{2}}}\,,$ (12b)
with $\nu=\frac{\mu}{12}$
Eq. (12a) readily indicates that
$F(\phi)\dot{\phi}^{2}=6\nu^{2}a^{-6}(\tau)\,,$ (13)
despite of the particular form of the functional $F(\phi)$. Also, this
structure is directly obtained for this model solving the equation (3b).
Moreover, the matter contribution of the SB scalar field to the rhs of the
Einstein equations would be
$\rho_{\phi}=\frac{1}{2}F(\phi)\dot{\phi}^{2}\propto a^{-6}\,.$ (14)
That is, the contribution of the scalar field is the same as that of stiff
matter with a barotropic equation of state $\gamma=1$.
This is an interesting result, since the original SB theory was thought of as
a form to solve the missing matter problem of Cosmology, now generically
called the dark matter problem; to solve the latter, one needs a fluid
behaving as dust with $\gamma=0$. It is surprising that such a general result
remain unnoticed until now in the literature about SB.
Also, that we have identified the general evolution of the scalar field with
that of a stiff fluid means that the Eq. (12b) can be integrated separately
without a complete solution for the scalar field. For completeness, we give
below a compilation of exact solutions in the case of the original SB theory.
If $F(\phi)=\omega\phi^{m}$, then we have two cases that correspond to $m=-2$
and $m\neq-2$; the general solution for the scalar field is
$\phi=\left\\{\begin{tabular}[]{lr}$Exp\left[\frac{6\nu}{\sqrt{6\omega}}\int
a^{-3}(\tau)d\tau\right]$&\qquad m = -2\\\
$\left[\frac{2\nu(m+2)}{\sqrt{6\omega}}\int
a^{-3}(\tau)d\tau\right]^{\frac{2}{m+2}}$&\qquad$m\not=-2$\\\
\end{tabular}\right.$ (15)
which can be completely integrated once the time dependence of the scale
factor $a$ has been resolved.
* •
Stiff plus a cosmological constant, $\gamma=-1$. The master equation become
$\Delta\tau=\int\frac{a^{2}da}{\sqrt{b_{-1}a^{6}-\nu^{2}}}\,,$ (16)
where $b_{-1}=\frac{8}{3}\pi G\rho_{-1}-\frac{\lambda}{3}$, whose solution is
$\Delta\tau=\frac{1}{3\sqrt{b_{-1}}}\,Ln\left[b_{-1}a^{3}+\sqrt{b_{-1}}\sqrt{b_{-1}a^{6}-\nu^{2}}\right]\,.$
(17)
The volume function is then
$a^{3}=\frac{1}{2b_{-1}}\left(e^{3\sqrt{b_{-1}}\,\Delta\tau}+b_{-1}\nu^{2}e^{-3\sqrt{b_{-1}}\,\Delta\tau}\right)\,,$
(18)
whereas that of the scalar field is
$\phi=\left\\{\begin{tabular}[]{lr}$Exp\left[\frac{4}{\sqrt{6\omega}}\arctan\left(\frac{Exp[3\sqrt{b_{-1}}\Delta\tau]}{\nu\sqrt{b_{-1}}}\right)\right]$&\qquad$m=-2$
\, ;\\\
$\left[\frac{2(m+2)}{\sqrt{6\omega}}\arctan\left(\frac{Exp[3\sqrt{b_{-1}}\Delta\tau]}{\nu\sqrt{b_{-1}}}\right)\right]^{\frac{2}{m+2}}$&\qquad$m\not=-2$
\, .\\\ \end{tabular}\right.$ (19)
For the case $\gamma=1$ the same solutions are found and only a redefinition
of the constants is needed.
* •
Stiff plus a cosmological constant plus dust, $\gamma=0$. In this case the
master equation becomes
$\Delta\tau=\int\frac{a^{2}da}{\sqrt{\frac{8}{3}\pi
G\rho_{0}a^{3}-\frac{\lambda}{3}a^{6}-\nu^{2}}}$ (20)
whose solution is
$\Delta\tau=\frac{1}{\sqrt{3|\lambda|}}\,Ln\left[\frac{b_{0}+\frac{2|\lambda|}{3}a^{3}}{\sqrt{\frac{|\lambda|}{3}}}+2\sqrt{b_{0}a^{3}+\frac{|\lambda|}{3}a^{6}-\nu^{2}}\right]\,,$
(21)
with $|\lambda|>0$ and $\rm b_{0}=\frac{8}{3}\pi G\rho_{0}$. The volume
function is now
$a^{3}=\frac{3}{4\sqrt{3|\lambda|}}e^{-\sqrt{3|\lambda|}\tau}\left[4\nu^{2}+\left(e^{\sqrt{3|\lambda|}\tau}-\frac{3b_{0}}{\sqrt{3|\lambda|}}\right)^{2}\right]\,.$
(22)
In this way, the solution for the field $\phi$ is
$\phi=\left\\{\begin{tabular}[]{lr}$Exp\left[\frac{4}{\sqrt{6\,\omega}}\arctan\left(\frac{\sqrt{3|\lambda|}Exp[\sqrt{3|\lambda|}\Delta\tau]-3b_{0}}{2\nu\sqrt{3|\lambda|}}\right)\right]$&\qquad$m=-2$
\, ;\\\
$\left[\frac{4(m+2)}{\sqrt{6\omega}}\arctan\left(\frac{\sqrt{3|\lambda|}Exp[\sqrt{3|\lambda|}\Delta\tau]-3b_{0}}{2\nu\sqrt{3|\lambda|}}\right)\right]^{\frac{2}{m+2}}$&\qquad$m\not=-2$
\, .\\\ \end{tabular}\right.$ (23)
The classical solution when $\rm F(\phi)=we^{m\phi}$ have the following
structure
$\rm\phi(\tau)=\frac{2}{m}Ln\left[\frac{m}{2}\sqrt{\frac{6\nu^{2}}{w}}\int
a^{-3}(\tau)d\tau+e^{\frac{m}{2}\phi_{0}}\right],$ (24)
where the integration value must be consider the last calculations over the
scale factor.
The solutions above were checked to comply with the Einstein field equations
encoded in equations (3b), using the REDUCE 3.8 package.
## III Quantum FRW cosmological model
One of the open problem of SB is the lack of a quantum model, in this section
using the generalization of the ideas presented in the previos sections we use
canonical quantuization. By the usual representation for the momenta operators
$\rm\Pi_{q}=-i\frac{\partial}{\partial q}$, $(\hbar=1),$ including the factor
ordering problem in the $a$ and $\phi$ variables, we obtain the Wheeler-DeWitt
equation
$\rm\left[-a^{2}\frac{\partial^{2}}{\partial a^{2}}-qa\frac{\partial}{\partial
a}-\frac{6}{F(\phi)}\frac{\partial^{2}}{\partial\phi^{2}}-\frac{6s}{F(\phi)}\phi^{-1}\frac{\partial}{\partial\phi}+144\kappa
a^{4}+48a^{6}\lambda-384\pi G\rho_{\gamma}a^{3(1-\gamma)}\right]\Psi=0,$ (25)
where q and s are real constants that measures the ambiguity in the factor
ordering in the operators $\Pi_{a}$ and $\Pi_{\phi}$, $\Psi$ is the wave
function for this cosmological model. Employing the variables separation
method, $\Psi(a,\phi)={\cal A}(a){\cal B}(\phi)$, (25) gives the set of
equations
$\displaystyle\rm-a^{2}\frac{d^{2}{\cal A}}{da^{2}}-qa\frac{\partial{\cal
A}}{\partial a}+\left(144\kappa a^{4}+48a^{6}\lambda-384\pi
G\rho_{\gamma}a^{3(1-\gamma)}-\mu^{2}\right){\cal A}$ $\displaystyle=$
$\displaystyle 0,$ (26) $\displaystyle\rm\phi\frac{d^{2}{\cal
B}}{d\phi^{2}}+s\frac{d{\cal B}}{d\phi}-\frac{\mu^{2}}{6}\phi F(\phi){\cal B}$
$\displaystyle=$ $\displaystyle 0.$ (27)
The equation (26) does have not a general solution for any $\kappa$, then we
solve for flat case and the particular values in the $\gamma$ parameter. When
$\gamma=-1$, the exact solution is
$\rm{\cal
A}(a)=a^{\frac{1-q}{2}}\,Z_{\nu}\left(\frac{\sqrt{b}}{3}a^{3}\right),$ (28)
where $\nu=\frac{1}{6}\sqrt{(1-q)^{2}-4\mu^{2}}$ and $b=384\pi
G\rho_{-1}-48\lambda$. We can see that when $b>0$, the generic Bessel function
$Z_{\nu}\to J_{\nu}$, and when $b<0$, $Z_{\nu}\to(K_{\nu},I_{\nu})$
Other soluble case is when $\gamma=1$, the solution is the same, and the
changes appear in the constants $\mu^{2}\to 384\pi G\rho_{1}+\mu^{2}$ and
$b=-48\lambda$. In this form, we obtain the exact solution to the wave
function $\Psi(a,\phi)$ in this theory.
For solve the equation (27), we apply this approach at Sáez-Ballester theory.
The case when $m\not=-2$ polyanin is written in term of generic Bessel
function $\rm Z_{\eta}$ as
$\rm
B(\phi)=c\phi^{\frac{1-s}{2}}Z_{\eta}\left(\frac{2\sqrt{-\xi}}{m+2}\phi^{\frac{m+2}{2}}\right),$
(29)
where c is a integration constants, and $\eta=\frac{1-s}{m+2}$,
$\xi=\frac{\mu^{2}\omega}{6}$. Also, we can see that the generic Bessel
function $\rm Z_{\eta}\to J_{\eta}$ when $\omega<0$, or
$\rm(K_{\eta},I_{\eta})$ when $\omega>0$.
We can build the wave packet, introducing the continuum parameters $\eta$ and
$\nu$ as
$\rm\Psi_{\eta\nu}=\int_{\eta}\int_{\nu}{\cal F}(\eta){\cal
G}(\nu)\phi^{\frac{1-s}{2}}Z_{\eta}\left(\frac{2\sqrt{-\xi}}{n+2}\phi^{\frac{n+2}{2}}\right)a^{\frac{1-q}{2}}\,Z_{\nu}\left(\frac{\sqrt{b}}{3}a^{3}\right)d\eta
d\nu$ (30)
For particular values in the constant $m$, the exact solutions are very
simple. For instant when $m=-2$, we have the Euler equation who solution is
$\rm
B(\phi)=\phi^{\frac{1-s}{2}}\left\\{\begin{tabular}[]{lr}$\rm\left[c_{1}\phi^{\alpha}+c_{2}\phi^{-\alpha}\right]$&\qquad$(1-s)^{2}>4b$\\\
$\rm\left[c_{1}+c_{2}Ln\phi\right]$&\qquad$(1-s)^{2}=4b$\\\
$\rm\left[c_{1}sin(\alpha Ln\phi)+c_{2}cos(\alpha
Ln(\phi))\right]$&\qquad$(1-s)^{2}<4b$\end{tabular}\right.$ (31)
with $\alpha=\frac{1}{2}\sqrt{(1-s)^{2}-4b}$ and $b=-\frac{\omega\mu^{2}}{6}$.
When $m=-6$ and $s=-1$, making the transformations $z=\phi^{-2}$ and
$B=\frac{u}{z}$, leads to a constant coefficient linear equation, (27) is
transformed to $4\frac{d^{2}u}{dz^{2}}-\frac{\mu^{2}\omega}{6}u=0$ who exact
solutions becomes
$u(z)=\left\\{\begin{tabular}[]{lr}$\rm
c_{1}\,sinh\left(\sqrt{\frac{\mu^{2}\omega}{24}}z\right)+c_{2}\,cosh\left(\sqrt{\frac{\mu^{2}\omega}{24}}z\right)$&\qquad$\omega>0$\\\
$\rm
c_{1}\,sin\left(\sqrt{\frac{\mu^{2}\omega}{24}}z\right)+c_{2}\,cos\left(\sqrt{\frac{\mu^{2}\omega}{24}}z\right)$&\qquad$\omega<0$\\\
\end{tabular}\right.$ (32)
in the original variables
${\cal B}(\phi)=\phi^{2}\left\\{\begin{tabular}[]{lr}$\rm
c_{1}\,sinh\left(\sqrt{\frac{\mu^{2}\omega}{24}}\frac{1}{\phi^{2}}\right)+c_{2}\,cosh\left(\sqrt{\frac{\mu^{2}\omega}{24}}\frac{1}{\phi^{2}}\right)$&\qquad$\omega>0$\\\
$\rm
c_{1}\,sin\left(\sqrt{\frac{\mu^{2}\omega}{24}}\frac{1}{\phi^{2}}\right)+c_{2}\,cos\left(\sqrt{\frac{\mu^{2}\omega}{24}}\frac{1}{\phi^{2}}\right)$&\qquad$\omega<0$\\\
\end{tabular}\right.$ (33)
## IV conclusions
We studied the generalization of the Sáez-Ballester theory by including a
dimensionless functional of the scalar field $F(\phi)$. The classical dynamics
of the theory were obtained from the corresponding classical Lagragian and
Hamiltonian densities; the solutions were in turn given up to quadratures.
One general result here is that the evolution of the scale factor of the
Universe does not depend upon the particular form of the functional $F(\phi)$;
actually, the contribution of the scalar field in the SB theory is that of
perfect fluid with a stiff (barotropic) equation of state. If any, its
contribution to the matter budget of the Universe is only relevant at early
times.
A separate conclusion is that the SB, whether in its original form as given in
Ref.s-b or in the generalized case studied here, cannot be an answer to the
dark matter riddle of Cosmology.
In the quantum regime was necessary to build one equivalent density lagrangian
in order to apply this, and does not possible to write this solution in closed
form. In this sense, we check this approach using the original Sáez-Ballester
formalism, obtaining the exact solutions in both regimes, classical and
quantum for particular values in the $\gamma$ parameter. This formalism will
be used with anisotropic cosmological models, which will be reported in other
work.
###### Acknowledgements.
This work was partially supported by CONACYT grants 47641, 56946 and 62253,
DINPO 38.07 and PROMEP UGTO-CA-3. This work is part of the collaboration
within the Instituto Avanzado de Cosmología.
## References
* (1) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006) [arXiv:hep-th/0603057].
* (2) M.P. Ryan, Hamiltonian cosmology, (Springer, Berlin, 1972).
* (3) M.P. Ryan and L.C. Shepley, Homogeneous Relativistic Cosmologies, Princeton University Press, Princeton, New Jersey (1975).
* (4) D. Saez and V.J. Ballester, Phys. Lett. A 113, 467 (1986).
* (5) T. Singh and A.K. Agrawal, Astrophys. Space Sci. 182, 289 (1991).
* (6) Shri Ram and J.K. Singh, Astrophys. Space Sci. 234, 325 (1995).
* (7) G. Mohanty and S.K. Pattanaik, Theor. Appl. Mech. 26, 59 (2001).
* (8) C.P. Singh and Shri Ram, Astrophys. Space Sci. 284, 1199 (2003).
* (9) Andrei D. Polyanin and Valentin F. Zaitzev, in: Handbook of exact solutions for ordinary differential equations, second edition, Chapman & Hall/CRC (2003).
|
arxiv-papers
| 2009-04-02T16:27:29 |
2024-09-04T02:49:01.618399
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Socorro, M. Sabido, and L. Arturo Ure\\~na-L\\'opez",
"submitter": "Jose Socorro Garcia",
"url": "https://arxiv.org/abs/0904.0422"
}
|
0904.0430
|
# Sparse NonGaussian Component Analysis111Supported by DFG research center
Matheon ”Mathematics for key technologies” (FZT 86) in Berlin.
$\,\text{\sc Elmar Diederichs}^{1}\,$, $\,\text{\sc Anatoli Juditski}^{3}\,$,
$\,\text{\sc Vladimir Spokoiny}^{2}\,$, $\,\text{\sc Christof
Sch\"{u}tte}^{1}\,$
$\,{}^{1}\text{Institute for Mathematics and Informatics, Free University
Berlin}\,$
Arnimallee 6, 14195 Berlin, Germany
$\,{}^{2}\text{Weierstrass Institute and Humboldt University}\,$
Mohrenstr. 39, 10117 Berlin, Germany
$\,{}^{3}\text{LJK, Universit\'{e} J. Fourier, }\,$
BP 53 38041 GRENOBLE cedex 9, France
###### Abstract
Non-gaussian component analysis (NGCA) introduced in [24] offered a method for
high dimensional data analysis allowing for identifying a low-dimensional non-
Gaussian component of the whole distribution in an iterative and structure
adaptive way. An important step of the NGCA procedure is identification of the
non-Gaussian subspace using Principle Component Analysis (PCA) method. This
article proposes a new approach to NGCA called _sparse NGCA_ which replaces
the PCA-based procedure with a new the algorithm we refer to as _convex
projection_.
keywords: reduction of dimensionality, model reduction, sparsity, variable
selection, principle component analysis, structural adaptation, convex
projection
Mathematical Subject Classification: 62G05, 60G10, 60G35, 62M10, 93E10
## 1 Introduction
Numerous mathematical applications in econometrics or biology are confronted
with high dimensional data. Such data sets present new challenges in data
analysis, since often the data have dimensionality ranging from hundreds to
hundreds of thousands. This means an exponential increase of the computational
burden for many methods. On the other hand the sparsity of the data in high
dimensions entails that data thin out in the local neighborhood of a given
point $\,x\,$. Hence statistical methods are not reliable in high dimensions
if the sample size remains of the same order. This problem is usually referred
to as ”curse of dimensionality” (cf. [8], [27]). The standard approach to deal
with the high dimensional data is to introduce a _structural assumption_ which
allows to reduce the complexity or intrinsic dimension of the data without
significant loss of statistical information [19], [17].
Let a random phenomenon is observed in the high dimensional space
$\,\mathbb{R}^{d}\,$ while the intrinsic dimension of this phenomenon is much
smaller, say $\,m\,$. From a geometrical point of view $\,m\,$ is the
dimension of a linear subspace that approximately contains the structure of
the sample data. Alternatively we can consider this structure as a low
dimensional signal embedded in high dimensional noise. Consequently a lower
dimensional, compact representation that according to some criterion, captures
the interesting information in the original data, is sought. In this paper we
assume that we have a sample of data lying approximately in a $\,m\leq d\,$
dimensional linear (target) subspace $\,\mathcal{I}\subseteq\mathbb{R}^{d}\,$
of $\,\mathbb{R}^{d}\,$. In order to reduce the problem dimension one looks
for a mapping from the original data space onto this subspace.
In the statistical literature the Gaussian components of the data distribution
are often considered as entropy maximizing and consequently as non-informative
noise [5]. It is well known that for high-dimensional clouds of points most
low-dimensional projections are approximately Gaussian [6]. The Non-Gaussian
Component Analysis (NGCA), introduced in [24], is based on the assumption that
the structure of the data is represented by a low dimensional non-Gaussian
component of the observation distribution, as opposed to a full dimensional
Gaussian component, considered as noise. Thus the objective of NGCA is to
”kill the noise” rather than to describe the whole multidimensional
distribution. Note that the suggested way of treating the Gaussian
distribution as a pure nuisance in general exclude the use of the classical
_Principle Component Analysis_ (PCA) which simply searches for the directions
with of largest variance.
In the same way as a number of projection methods of feature extraction (e.g.
Projection Pursuit [10], Partial Least Square Regression [28, 29], Conditional
Minimum Average Variance Estimation [30] or Sliced Inverse Regression [15, 4,
2]), when implementing the NGCA we decompose the problem of dimension
reduction into two tasks: the first one is to extract from data a set of
vectors which are close to the target space $\,\mathcal{I}\,$; the second is
to construct a basis of the target space from these vectors. These
characteristics can also be found in the unsupervised, data driven approach of
SNGCA, presented in this article. When compared to available dimension
reduction methods (e.g. Principal Component Analysis [13], Independent
Component Analysis [11] or Singular Spectrum Analysis [7]) SNGCA does not
assume any a priori knowledge about the density of the original data.
The proposed method, as well as NGCA, is an iterative algorithm which is
structure adaptive in the sense that every new step essentially uses the
result of previous iterations. The main difference between NGCA and SNGCA
algorithms lies in the way the information is extracted from the data. The
algorithm of NGCA heavily relies upon the Euclidean projection and the PCA of
the set of the estimated vectors. In the case when data dimension is important
and the sample size is moderate, computation of the $\,l_{2}\,$-projection can
amplify the noise. Moreover, when most of the estimated vectors do not contain
information about the space $\,\mathcal{I}\,$ but are mainly noise, the
results of using the PCA algorithm to extract the basis of feature space can
be very poor. The reason for that is that the PCA algorithm is known to
accumulate the noise. To address this issue the SNGCA uses convex programming
techniques to estimate elements of the target subspace by “convex projection”,
what allows to bound uniformly the estimation error. Further, another
technique of convex analysis, based on computation of rounding ellipsoids of
the set of estimated vectors, is used to extract the subspace information.
These changes allow the SNGCA algorithm to treat large families of candidate
vectors without increasing significantly the variance of the estimation of the
target subspace.
The paper is organized as follows. First we describe the considered set-up in
Section 2 and discuss the main ideas behind the proposed approach. The formal
description of the algorithm is given in Section 3. A simulation study of the
algorithms is presented in Section 4, where we compare the performance
obtained by SNGCA algorithms and by several other methods of feature
extraction.
## 2 Non-Gaussian Component Analysis
### 2.1 The setup
The following setting is due to [24]. Let $\,X_{1},...,X_{N}\,$ be i.i.d. from
a distribution $\,I\\!\\!P\,$ in $\,\mathbb{R}^{d}\,$. We suppose that
$\,I\\!\\!P\,$ possesses a density $\,\rho\,$ with respect to the Lebesgue
measure on $\,\mathbb{R}^{d}\,$, which can be decomposed as follows:
$\displaystyle\rho(x)=\phi_{\mu=0,\Sigma}(x)q(Tx).$ (2.1)
Here $\,\phi_{\mu,\Sigma}\,$ stands for the density of the multivariate normal
distribution $\,\mathcal{N}(\mu,\Sigma)\,$ with parameters
$\,\mu\in\mathbb{R}^{d}\,$ (expectation) and $\,\Sigma\in\mathbb{R}^{d\times
d}\,$ positive definite (covariance matrix). The function
$\,q:\mathbb{R}^{m}\to\mathbb{R}\,$ with $\,m\leq d\,$ has to be nonlinear and
smooth. $\,T\in\mathbb{R}^{m\times d}\,$ is an unknown linear mapping.
Naturally, we refer to $\,\mathcal{I}={\rm range}\;T\,$ as target or non-
Gaussian subspace. For the sake of simplicity let us assume
$\,I\\!\\!E[X]=0\,$ where $\,I\\!\\!E[X]\,$ stands for the expectation of
$\,X\,$.
Though the representation (2.1) is not uniquely defined, the subspace
$\,\mathcal{I}\subset\mathbb{R}^{d}\,$ is well defined as well as the
Euclidean projector $\,\Pi^{*}\,$ on $\,\mathcal{I}\,$. By analogy with the
regression case [4, 16, 15], we could also call $\,\mathcal{I}\,$ the
effective dimension reduction space (EDR-space). We call $\,m\,$ effective
dimension of the data. In many applications $\,m\,$ is unknown and has to be
recovered from the data. Our task is to recover $\,\Pi^{*}\,$. The model
structure (2.1) allows the following interpretation (cf. [24]) : we can
decompose the random vector $\,X\,$ into two independent components
$\displaystyle X=\Pi^{*}X+(I-\Pi^{*})X=Z+u,$
where $\,Z\,$ is a non-Gaussian $\,m\,$-dimensional signal and $\,u\,$ is
$\,(d-m)\,$-dimensional normal noise.
As we have already noticed in the introduction, SNGCA algorithm relies upon
two basic operations: the first is to construct a set of vectors, say
$\,\beta_{1},...,\beta_{J}\,$, which are ”close” to the target subspace; the
objective of the second is to compute an estimate $\,\widehat{\Pi}\,$ of the
Euclidean projector $\,\Pi^{*}\,$ on $\,\mathcal{I}\,$ using the set
$\,\\{\beta_{j}\\}_{j=1}^{J}\,$.
### 2.2 Estimation of elements of the target subspace
Estimation of elements of $\,\mathcal{I}\,$. The implementation of the first
step of SNGCA is based on the following result (cf. Theorem 1 of [24]):
###### Theorem 1.
Let $\,X\,$ follow the distribution with the density $\,\rho\,$ which
satisfies (2.1) and let $\,I\\!\\!E[X]=0\,$. Suppose that a function
$\,\psi:\,\mathbb{R}^{d}\to\mathbb{R}\,$ is continuously differentiable.
Define
$\displaystyle\beta(\psi):=I\\!\\!E\bigl{[}\nabla\psi(X)\bigr{]}=\int\nabla\psi(x)\,\rho(x)\,dx,$
(2.2)
where $\,\nabla\psi\,$ stands for the gradient of $\,\psi\,$. Then there
exists a vector $\,\beta\in\mathcal{I}\,$ such that
$\displaystyle\|\beta(\psi)-\beta\|_{2}$ $\displaystyle\leq$
$\displaystyle\big{\|}\Sigma^{-1}I\\!\\!E[X\psi(X)]\big{\|}_{2}$
$\displaystyle=$ $\displaystyle\Big{\|}\Sigma^{-1}\int
x\psi(x)\rho(x)\;dx\Big{\|}_{2}.$
In particular, if $\,I\\!\\!E[X\psi(X)]=0\,$, then
$\,\beta(\psi)\in\mathcal{I}\,$.
The bound of Theorem 1 implies that
$\displaystyle\|(I-\Pi^{*})\beta(\psi)\|_{2}\leq\Big{\|}\Sigma^{-1}\int
x\psi(x)\rho(x)\;dx\Big{\|}_{2},$ (2.3)
where $\,I\,$ is the $\,d\,$-dimensional identity matrix and $\,\Pi^{*}\,$ is
the orthogonal projector on $\,\mathcal{I}\,$.
Based on this result, [24] suggested the following way of constructing a set
of vectors $\,\beta\,$ which approximate the target space $\,\mathcal{I}\,$.
Let $\,h_{1},...,h_{L}\,$ be smooth bounded functions on $\,\mathbb{R}^{d}\,$.
Define $\,\gamma_{l}=I\\!\\!E[Xh_{l}(X)]\,$ and $\,\eta_{l}=I\\!\\!E[\nabla
h_{l}(X)]\,$. These vectors are not computable because they rely on the
unknown data distribution, but they can be well estimated from the given data.
Next, for any vector $\,c\in\mathbb{R}^{L}\,$, define the vectors
$\,\beta(c),\gamma(c)\in\mathbb{R}^{d}\,$ with
$\displaystyle\beta(c)=\sum_{l=1}^{L}c_{l}\eta_{l},\qquad\gamma(c)=\sum_{l=1}^{L}c_{l}\gamma_{l}$
Then by Theorem 1, $\,\beta(c)\in\mathcal{I}\,$ conditioned that
$\,\gamma(c)=0\,$. Indeed, if we set $\,\psi(x)=\sum_{l}c_{l}h_{l}(x)\,$, then
$\,I\\!\\!E[X\psi(X)]=0\,$, and by (2.3),
$\gamma(c)=I\\!\\!E[\nabla\psi(X)]\in\mathcal{I}.$
The approach of [24] is to compute the vectors of coefficients
$\,c\in\mathbb{R}^{L}\,$ which ensure $\,\gamma(c)\approx 0\,$ and then to use
the corresponding empirical analogs of $\,\beta(c)\,$ to estimate the target
space. More precisely, given the observations $\,X_{1},...,X_{N}\,$ compute
the set of vectors (empirical counterparts of $\,\eta_{l}\,$ and
$\,\gamma_{l}\,$) according to
$\displaystyle\widehat{\gamma}_{l}=N^{-1}\sum_{i=1}^{N}X_{i}h_{l}(X_{i}),\quad\widehat{\eta}_{l}=N^{-1}\sum_{i=1}^{N}\nabla
h_{l}(X_{i}).\qquad$ (2.4)
Similarly define for $\,c\in\mathbb{R}^{L}\,$
$\displaystyle\widehat{\beta}(c)=\sum_{l=1}^{L}c_{l}\widehat{\eta}_{l},\qquad\widehat{\gamma}(c)=\sum_{l=1}^{L}c_{l}\widehat{\gamma}_{l}\,.$
One can expect that for vectors $\,c\,$ with $\,\widehat{\gamma}(c)=0\,$, the
vectors $\,\widehat{\beta}(c)\,$ are ”close” to $\,\mathcal{I}\,$.
Below we follow a similar way of constructing $\,\widehat{\beta}(c)\,$ with an
additional constraint that the considered vectors of coefficients $\,c\,$
satisfy $\,\|c\|_{1}\leq 1\,$. This constraint allows for both efficient
numerical algorithms and sharp error bounds.
The test functions $\,h_{l}\,$ can be generated as follows: let
$\,\mathcal{B}_{d}\,$ be a unit ball
$\,\mathcal{B}_{d}=\\{x\in\mathbb{R}^{d}|:\,\|x\|_{2}\leq 1\\}\,$ and let
$\,f(x,\omega)\,$, $\,f:\,\mathcal{B}\times\mathbb{R}^{d}\to\mathbb{R}\,$ be a
continuously differentiable function. Consider the functions
$\,h_{l}(x)=f(x,\omega_{l})\,$, for some
$\,\omega_{l}\in\mathcal{B},\;l=1,...,L\,$. The choice of family
$\,f(\cdot,\omega)\,$ is an important parameter of the algorithm design. For
instance, in the simulation examples of Section 4 we consider the following
families:
$\displaystyle f(x,\omega)$ $\displaystyle=$
$\displaystyle\tanh(\omega^{\top}x)e^{-\alpha\|x\|^{2}_{2}/2},$ (2.5)
$\displaystyle f(x,\omega)$ $\displaystyle=$
$\displaystyle[1+(\omega^{\top}x)^{2}]^{-1}\exp^{\omega^{\top}x-\alpha\|x\|^{2}_{2}/2},$
(2.6)
and $\,\omega_{l},\;l=1,...,L\,$ are unit vectors in $\,\mathbb{R}^{d}\,$. The
next result justifies the proposed construction.
###### Theorem 2.
Suppose that $\,f\,$ is continuously differentiable in $\,w\,$ and for some
fixed constant $\,f^{*}_{1}\,$ and any
$\,\omega\in\mathcal{B}_{d},\,x\in\mathbb{R}^{d}\,$
$\displaystyle\operatorname{Var}\bigl{[}X_{j}\,f(X,\omega)\bigr{]}\leq
f^{*}_{1},\quad\operatorname{Cov}\bigl{[}X_{j}\,\nabla_{\omega}f(X,\omega)\bigr{]}\leq
f^{*}_{1}I,$ $\displaystyle\operatorname{Var}\biggl{[}\frac{\partial}{\partial
x_{j}}f(X,\omega)\biggr{]}\leq
f^{*}_{1},\quad\operatorname{Cov}\biggl{[}\nabla_{\omega}\frac{\partial}{\partial
x_{j}}f(X,\omega)\biggr{]}\leq f^{*}_{1}I,$
Consider the (random) set
$\displaystyle\mathscr{C}=\bigl{\\{}c\in\mathbb{R}^{L}:\|c\|_{1}\leq
1,\,\widehat{\gamma}(c)=0\bigr{\\}}.$ (2.7)
Then for any $\,\varepsilon>0\,$ there is a set $\,A\subset\Omega\,$ of
probability at least $\,1-\varepsilon\,$ such that on $\,A\,$ for all
$\,c\in\mathscr{C}\,$,
$\bigl{\|}(I-\Pi^{*})\widehat{\beta}(c)\bigr{\|}_{2}\leq\sqrt{d}\,\delta_{N}\bigl{(}1+\|\Sigma^{-1}\|_{2}\bigr{)},$
where
$\displaystyle\delta_{N}=N^{-1/2}\inf_{\lambda\leq\lambda^{*}_{1}N^{1/2}}\bigl{\\{}5\mathfrak{n}_{0}f^{*}_{1}\lambda+2\lambda^{-1}\bigl{[}\mathfrak{e}_{d}+\log(2d/\varepsilon)\bigr{]}\bigr{\\}}$
and $\,\mathfrak{e}_{d}=4d\log 2\,$.
The proof of the theorem is given in the appendix.
Due to this result, any vector $\,c\in\mathscr{C}\,$ can be used to produce a
vector $\,\widehat{\beta}(c)\,$ which is close to the target subspace
$\,\mathcal{I}\,$. However, such constructed vectors are only informative if
its length is significant relative to the estimation error.
We therefore compute a family of such coefficient vectors $\,c\,$ by solving
the following optimization problems: for a fixed unit vector
$\,\xi\in\mathbb{R}^{d}\,$ called a _probe vector_ , find
$\displaystyle\widehat{c}=\operatorname*{\mathrm{arg\,min}}_{c\in\mathbb{R}^{L}:\,\|c\|_{1}\leq
1}\|\xi-\widehat{\eta}(c)\|_{2},\mbox{ subject to }\widehat{\gamma}(c)=0.$
(2.8)
where $\,\widehat{\eta}(c)=\sum_{l}c_{l}\widehat{\eta}_{l}\,$. This is a
convex optimization problem which can be efficiently solved by some numerical
procedures, e.g. by the interior point method. Then we set
$\displaystyle\widehat{\beta}=\widehat{\beta}(\widehat{c})=\sum_{l}\widehat{c}_{l}\widehat{\eta}_{l}\,.$
(2.9)
It can be easily seen that for $\,\xi\perp\mathcal{I}\,$, the solution
$\,\widehat{\beta}\,$ fulfills $\,\widehat{\beta}\approx 0\,$. On the
contrary, if $\,\xi\in\mathcal{I}\,$, then there is a solution with
significantly positive $\,\|\widehat{c}\|_{1}\,$ and
$\,\|\widehat{\beta}(\widehat{c})\|_{2}\,$. This leads to the following
strategy: In the first step of the algorithm when there is no information
about $\,\mathcal{I}\,$ available, the probe vectors $\,\xi_{1},...,\xi_{J}\,$
in $\,\mathbb{R}^{d}\,$ are generated randomly from $\,\mathcal{B}_{d}\,$. In
the next steps we apply the idea of structural adaptation by generating the
essential part of the vectors $\,\xi_{j}\,$ from the estimated subspace
$\,\widetilde{\mathcal{I}}\,$. For details see Section 3.
We address now the implementation of the second step of SNGCA – inferring the
projector $\,\Pi^{*}\,$ on $\,\mathcal{I}\,$ from estimations
$\,\\{\widehat{\beta}_{j}\\}_{j=1}^{J}\,$ of elements of $\,\mathcal{I}\,$.
Recovering the target subspace. Suppose that we are given vectors
$\,\widehat{\beta}_{1},...,\widehat{\beta}_{J}\,$ which satisfy
$\|\widehat{\beta}_{j}-\beta_{j}\|_{2}\leq\varrho,$
for some $\,\beta_{j}\in\mathcal{I}\,$, $\,j=1,...,J\,$. The problem of
estimating the subspace $\,\mathcal{I}\,$ from $\,\widehat{\beta}_{j}\,$ is a
special case of the so called _Reduced Rank Regression_ (RRR) problem. A
simple and popular PCA estimate of the projector $\,\Pi^{*}\,$ on
$\,\mathcal{I}\,$ is given by solving the quadratic optimization problem
$\displaystyle\widehat{\Pi}=\operatorname*{\mathrm{arg\,min}}_{\Pi_{m}}\,\sum_{j=1}^{J}\|(I-\Pi_{m})\widehat{\beta}_{j}\|_{2}^{2},$
where the minimum is taken over all projectors of rank $\,m\,$. One can easily
verify that $\,\widehat{\Pi}\,$ projects on the subspace in
$\,\mathbb{R}^{d}\,$ generated by the first $\,m\,$ principal eigenvectors of
the matrix $\,\sum_{j}\widehat{\beta}_{j}\widehat{\beta}_{j}^{\top}\,$.
However, if the number of informative vectors $\,\widehat{\beta}_{j}\,$ is
small with respect to $\,J\,$, the quality of estimate $\,\widehat{\Pi}\,$ can
be extremely poor. To address this drawback of the PCA solution we consider a
sparse estimate of $\,\mathcal{I}\,$ which uses rounding ellipsoids for the
set $\,\\{\widehat{\beta}_{j}\\}_{j=1}^{J}\,$.
For a symmetric positive-definite matrix $\,B\,$ and $\,r>0\,$, the ellipsoid
$\,\mathcal{E}_{r}(B)\,$ is defined as
$\mathscr{E}_{r}(B)=\\{x\in\mathbb{R}^{d}\mid x^{\top}Bx\leq r^{2}\\},$
For $\,\alpha\leq 1\,$,$\,\mathcal{E}(B)\equiv\mathscr{E}_{1}(B)\,$ is
$\,\alpha\,$-_rounding_ ellipsoid for a convex set $\,\mathscr{S}\,$ if
$\mathcal{E}_{1/\alpha}(B)\subseteq\mathscr{S}\subseteq\mathcal{E}(B).$
Note that such ellipsoid exists with $\,\alpha=d^{-1/2}\,$ due to the Fritz
John theorem [12]. Furthermore, numerically efficient algorithms for computing
$\,\sqrt{d}\,$-rounding ellipsoids are available, see e.g. [18]. So, for
recovering the spatial information from the vector system
$\,\\{\pm\widehat{\beta}_{j}\\}_{j=1}^{J}\,$ one can look for the
$\,d^{1/2}\,$ rounding ellipsoid for the convex hull $\,\mathscr{S}\,$ of
points $\,\\{\pm\widehat{\beta}_{j}\\}_{j=1}^{J}\,$.
We measure the quality of estimation of the subspace $\,\mathcal{I}\,$ by the
closeness of the estimated projector $\,\widehat{\Pi}\,$ to $\,\Pi^{*}\,$:
$\displaystyle\varepsilon(\mathcal{I},\widehat{\mathcal{I}})=\|\widehat{\Pi}-\Pi^{*}\|_{2}^{2}=\mathrm{Tr}\bigl{[}(\widehat{\Pi}-\Pi^{*})^{2}\bigr{]}.$
(2.10)
The property of the spatial information recovery, based on the idea of
rounding ellipsoids, is described in the following theorem.
###### Theorem 3.
1\. Let $\,\mathscr{S}\,$ be the convex envelope of the set
$\,\\{\pm\widehat{\beta}_{j}\\},\;j=1,...,J\,$, and let
$\,\mathscr{E}_{1}(B)\,$ be an ellipsoid inscribed into $\,\mathscr{S}\,$,
such that $\,\mathscr{E}_{\sqrt{d}}(B)\,$ is $\,\sqrt{d}\,$-rounding ellipsoid
for $\,\mathscr{S}\,$. Then for any unit vector $\,v\perp\mathcal{I}\,$,
$v^{\top}B^{-1}v\leq\varrho^{2}.$
2\. If there is $\,\mu\in\mathbb{R}^{J}\,$ with $\,\mu_{j}\geq 0\,$ and
$\,\sum_{j}\mu_{j}=1\,$ such that
$\lambda_{m}\biggl{(}\sum_{j}\mu_{j}\beta_{j}\beta_{j}^{\top}\biggr{)}\geq\lambda^{*}>2\varrho^{2},$
where $\,\lambda_{m}(A)\,$ stands for the $\,m\,$-th principal eigenvalue of
$\,A\,$, then
$\displaystyle\lambda_{m}(B^{-1})\geq\frac{\lambda^{*}-2\varrho^{2}}{2\sqrt{d}}\,.$
(2.11)
3\. Moreover, let
$\,\widehat{\Pi}=\widehat{\Gamma}_{m}\widehat{\Gamma}_{m}^{\top}\,$ where
$\,\Gamma_{m}\,$ is the matrix of $\,m\,$ principal eigenvectors of
$\,B^{-1}\,$. Then
$\|\widehat{\Pi}-\Pi^{*}\|_{2}^{2}\leq\frac{4\varrho^{2}d\sqrt{d}}{\lambda^{*}-2\varrho^{2}}.$
The proof of the theorem is presented in the appendix.
The results of Theorems 2 and 3 provide a kind of theoretical justification
for the algorithms, presented in the next section. Indeed, suppose that the
test functions $\,h_{1},...,h_{L}\,$ and the vectors $\,\xi_{1},...,\xi_{J}\,$
are chosen in such a way that there are at least $\,m\,$ vectors with
”significant” projection on $\,\mathcal{I}\,$ among
$\,\widehat{\beta}_{1},...,\widehat{\beta}_{J}\,$ as in (2.9). Then the
projector estimate $\,\widehat{\Pi}\,$, computed using the ellipsoid
$\,\mathcal{E}(B)\,$ which is rounding for the set
$\,\\{\pm\widehat{\beta}_{j}\\}\,$, with high probability will be close to
$\,\Pi^{*}\,$.
However, the results about the estimation quality depend critically on the
dimension $\,d\,$. Numerical results also indicate that with growing
dimension, the fraction of non-informative vectors $\,\widehat{\beta}_{j}\,$
increases leading to the situation when some of the longest semi-major axis of
$\,\mathcal{E}_{\sqrt{d}}\,$ are also non-informative and nearly orthogonal to
$\,\mathcal{I}\,$. This enforces us to introduce an additional check of non-
normality for the directions suggested by the estimated ellipsoid
$\,\mathcal{E}\,$.
Identifying the non-Gaussian subspace by statistical tests: Currently the
estimation procedure of the vectors $\,\beta(\psi_{h,c})\,$ itself does not
allow the identification of the semi-axis within the target space. Hence the
basic idea is to apply statistical tests on normality w.r.t. the significance
level $\,\alpha\,$ to the original data from $\,\mathbb{R}^{d}\,$ projected on
every semi-axis of $\,\mathcal{E}_{\sqrt{d}}\,$. If the hypothesis of
normality is rejected w.r.t. the projected data, the corresponding semi-axis
is used as a basis vector for the reduced target space $\,\mathcal{I}\,$.
Structural adaptation: At the beginning of the algorithm, we have no prior
information about $\,\mathcal{I}\,$ and therefore sample the directions
$\,\xi_{j}\,$ and $\,\omega_{l}\,$ randomly from the uniform law. However, the
SNGCA procedure assumes that the obtained estimated structure
$\,\widehat{\mathcal{I}}\,$ delivers some information about $\,\mathcal{I}\,$
which can be used for improving the sample mechanism and therefore, the final
quality of estimation. This leads to the _structurally adaptation_ iterative
procedure [9]: the step of estimating the vectors
$\,\\{\widehat{\beta}_{j}\\}_{j=1}^{J}\,$ and the step of estimating subspace
$\,\mathcal{I}\,$ are iterated, the estimated structural information given by
$\,\widehat{\mathcal{I}}\,$ is used to improve the quality of estimating the
vectors $\,\widehat{\beta}_{j}\,$ in the next iteration of SNGCA. In our
implementation, we sample a fraction of directions $\,\xi_{j}\,$ and
$\,\omega_{l}\,$ due to the previously estimated ellipsoid $\,\widehat{B}\,$
and the other part randomly. However the number of the randomly selected
directions remains constant during iteration. In the next section we present
the formal description of SNGCA.
## 3 Algorithms
This section describes the principal steps of the procedure. The detailed
description is given in the Appendix.
### 3.1 Normalization
As a preprocessing step the SNGCA procedure uses a componentwise normalization
of the data. Let $\,\sigma=(\sigma_{1},\ldots\sigma_{d})\,$ be the standard
deviations of the data components of $\,x_{1},\ldots,x_{d}\,$. For
$\,i=1,\ldots,N\,$ the componentwise normalization of the data is done by
$\,Y_{i}=\mathrm{diag}(\sigma^{-1})X_{i}\,$.
### 3.2 Estimation of the vectors from non-Gaussian subspace:
Let $\,\\{\omega_{jl}\\}\,$, $\,l=1,\ldots,L\,$, and $\,\\{\xi_{j}\\}\,$,
$\,j=1,\ldots,J\,$ be two collections of unit vectors called the measurement
directions. Define for all $\,j=1,\ldots,J\,$ and $\,l\leq L\,$, the functions
$\,h_{jl}(x)=f(x,\omega_{jl})\,$, and compute the vectors
$\,\widehat{\gamma}_{jl}\,$ and $\,\widehat{\eta}_{jl}\,$ due to (2.4). Next,
for every $\,j\leq J\,$, compute the vector $\,\widehat{c}_{j}\,$ by solving
the problem (2.8) with $\,\xi=\xi_{j}\,$ leading to the vector
$\,\widehat{\beta}_{j}\,$ by (2.9).
### 3.3 Computing the estimator $\,\widehat{\Pi}\,$ of the projector
$\,\Pi^{*}\,$
The projector $\,\widehat{\Pi}\,$ is constructed on the base of the first
$\,m\,$ principal eigenvectors of the rounding ellipsoid $\,\mathcal{E}\,$ for
the set $\,\mathscr{S}\,$ spanned by the vectors $\,\pm\widehat{\beta}_{j}\,$,
$\,j=1,\ldots,J\,$. To build the ellipsoid $\,\mathcal{E}\,$ we use the
algorithm in [18] which in fact computes the minimum volume ellipsoid (MVEE)
which covers $\,\mathscr{S}\,$. For convenience we provide the algorithm in
the appendix.
### 3.4 Building the subspace $\,\widehat{\mathcal{E}}\,$ using statistical
tests
In order to construct the projector $\,\widehat{\Pi}\,$ the identification of
the $\,m\,$ principal eigenvectors of $\,\mathcal{E}\,$ that approximate
$\,\mathcal{I}\,$ is required. In projecting the data onto the semi-axis of
$\,\mathcal{E}\,$ and testing the projected data on normality the projective
approach from the estimation step is repeated.
Since statistical tests specialized for a certain deviation from the normal
distribution, are more powerful, we use different tests inside of SNGCA in
order to cope with different deviations from normality of the projected data.
To be more precise we use the $\,K^{2}\,$-test according to D’Agostino-Pearson
[31] to identify a significant asymmetry in the projected distribution and the
EDF-test according to Anderson-Darling [1] with the modification of Stephens
[25], which is sensitive to the tails of the projected distribution. In order
to confirm these test results from above we use the Shapiro-Wilks test [22]
based on a regression strategy in the version given by Royston [20, 21]. Once
we have classified the semi-axis of $\,\mathcal{E}_{\sqrt{d}}\,$ as being
close to the target space we can use the identified subset of axis in the
structural adaptation step.
### 3.5 Structural Adaptation
The first step of the algorithm assumes that the measurement directions
$\,\omega_{jl}\,$ and $\,\xi_{j}\,$ are drawn randomly from the unit sphere in
$\,\mathbb{R}^{d}\,$. At each further step of the algorithm we can use the
result of the previous iterations of SNGCA in order to accumulate information
about $\,\mathcal{I}\,$ in a sequence
$\,\widehat{\mathcal{I}}_{1},\widehat{\mathcal{I}}_{2},\ldots\,$ of estimators
of the target space. This information is used to draw a fraction of the
measurement directions from the estimated subspaces and the other part of such
direction is selected randomly. The procedure is described in detail in
algorithm 7.
### 3.6 The stopping criterion
Suppose that $\,\mathcal{I}\,$ is a priori given. Then the convergence of
SNGCA can be measured according to the criterion (2.10). More precisely we
assume convergence if the improvement of the error measured by (2.10) from one
iteration to the next one is less than $\,\delta\,$ percent of the error in
the former iteration. To this end the maximum angle $\,\theta\,$ between the
subspaces specified by the matrix of eigenvectors
$\,V^{(k)}=\big{[}\widehat{v}_{1}^{(k)},\widehat{v}_{2}^{(k)},\ldots\big{]}\,$
and
$\,V^{(k+1)}=\big{[}\widehat{v}_{1}^{(k+1)},\widehat{v}_{2}^{(k+1)},\ldots\big{]}\,$
given by
$\displaystyle\cos(\theta)=\max_{x,y}\frac{|x^{\top}V^{(k)^{\top}}V^{(k+1)}y|}{\|V^{(k)}x\|_{2}\;\|V^{(k+1)}y\|_{2}}$
is computed. In the next section we demonstrate the improvement of the
estimation error between subsequent iterations of SNGCA.
## 4 Numerical results
The aim of this section is to compare SNGCA with other statistical methods of
dimension reduction. The reported results from Projection Pursuit (PP) and
NGCA were already published in [24].
### 4.1 Synthetic Data
Each of the following test data sets includes $\,1000\,$ samples in $\,10\,$
dimension and each sample consists of $\,8\,$-dimensional independent,
standard and homogeneous Gaussian distributions. The other $\,2\,$ components
of each sample are non-Gaussian with variance unity. The densities of the non-
Gaussian components are chosen as follows:
* (A)
Gaussian mixture: $\,2\,$-dimensional independent Gaussian mixtures with
density of each component given by
$\,0.5\;\phi_{-3,1}(x)+0.5\;\phi_{3,1}(x)\,$.
* (B)
Dependent super-Gaussian: $\,2\,$-dimensional isotropic distribution with
density proportional to $\,\exp(-\|x\|)\,$.
* (C)
Dependent sub-Gaussian: $\,2\,$-dimensional isotropic uniform with constant
positive density for $\,\|x\|_{2}\leq 1\,$ and $\,0\,$ otherwise.
* (D)
Dependent super- and sub-Gaussian: $\,1\,$-dimensional Laplacian with density
proportional to $\,\exp(-|x_{Lap}|)\,$ and $\,1\,$-dimensional dependent
uniform $\,\mathcal{U}(c,c+1)\,$, where $\,c=0\,$ for
$\,|x_{Lap}|\leq\log(2)\,$ and $\,c=-1\,$ otherwise.
* (E)
Dependent sub-Gaussian:$\,2\,$-dimensional isotropic Cauchy distribution with
density proportional to $\,\lambda(\lambda^{2}-x^{2})^{-1}\,$ where
$\,\lambda=1\,$.
That means, that the non-normal distributed data are located in a linear
subspace.
In the sequel we compare SNGCA with PP and NGCA using the test data sets from
above and the estimation error defined in (2.10). Each simulation is repeated
$\,100\,$ times. All simulations are done with the hyperbolic tangent index as
in (2.5). Since the speed of convergence varies with the type of non-Gaussian
components we use the maximum number $\,maxIter=3\log(d)\,$ of allowed
iterations to stop SNGCA. In the experiments the error measure
$\,\epsilon(\mathcal{I},\widehat{\mathcal{I}})\,$ is used only to determine
the final estimation error. All simulations other than whose w.r.t. model (C)
are computed with a componentwise pre-normalization.
|
---|---
(A) | (B)
|
(C) | (D)
|
(E) |
Figure 4.1: densities of the non-Gaussian components: (A) $\,2\,$d independent
Gaussian mixtures, (B) $\,2\,$d isotropic super-Gaussian, (C) $\,2\,$d
isotropic uniform and (D) dependent $\,1\,$d Laplacian with additive $\,1\,$d
uniform, (E) $\,2\,$d isotropic sub-Gaussian
Figure 4.1 illustrates the densities of the non-Gaussian components of the
test data. For all numerical experiments reported in this article the
dimension of the target space $\,\mathcal{I}\,$ is a priori given as a tuning
parameter for the algorithm.
Since the optimizer used in PP tends to trap in a local minima in each of the
100 simulations, PP is 10 times restarted with random starting points. The
best result w.r.t. (2.10) is reported as the result of each PP-simulation. In
all PP-simulations the number of non-Gaussian dimensions is a priori given. In
the next figure 4.2 we present boxplots of the error (2.10) obtained from the
methods PP, NGCA and SNGCA.
|
---|---
(A) | (B)
|
(C) | (D)
|
(E) |
Figure 4.2: performance comparison in $\,10\,$ dimensions of PP and NGCA
versus SNGCA (wrt. the error criterion
$\,\mathcal{E}(\widehat{\mathcal{I}},\mathcal{I})\,$ ) using the index
$\,tanh(x)\,$. The doted line denotes the mean, the solid lines the variance
of (2.10).
Concerning the results of SNGCA on the data sets (A) and (D) we observe a
slightly inferior performance compared to NGCA. In case of model (A) this is
due to the fact that most of the data projections have almost a Gaussian
density. Consequently the decrease of the estimation error is slow with
increasing number of iterations. In case of the model (D) the higher variance
of the results indicate that the initial sampling of the data sets gives a
poor result. Consequently more iterations are needed to get an estimation
error that is comparable to the result of NGCA. In order to illustrate this
interpretation we report in table (4.1) the progress of SNGCA w.r.t.
estimation error $\,\varepsilon(\mathcal{I},\widehat{\mathcal{I}})\,$ in each
iteration for every test model.
| $\,j\,$ | $\,\mu_{\epsilon}\,$ | $\,\sigma^{2}_{\epsilon}\,$
---|---|---
1 | 0.232504 | 0.045787
2 | 0.163022 | 0.072263
3 | 0.066537 | 0.032436
4 | 0.009380 | 0.021975
5 | 0.002359 | 0.000853
| $\,j\,$ | $\,\mu_{\epsilon}\,$ | $\,\sigma^{2}_{\epsilon}\,$
---|---|---
1 | 0.30350 | 0.175313
2 | 0.144430 | 0.057856
3 | 0.088142 | 0.015168
4 | 0.041420 | 0.008197
5 | 0.026436 | 0.000917
(A) | (B)
| $\,j\,$ | $\,\mu_{\epsilon}\,$ | $\,\sigma^{2}_{\epsilon}\,$
---|---|---
1 | 0.040556 | 0.004215
2 | 0.016012 | 0.002441
3 | 0.012427 | 0.001105
4 | 0.008874 | 0.000169
5 | 0.003770 | 0.000125
| $\,j\,$ | $\,\mu_{\epsilon}\,$ | $\,\sigma^{2}_{\epsilon}\,$
---|---|---
1 | 0.203419 | 0.044672
2 | 0.023023 | 0.000314
3 | 0.019960 | 0.000211
4 | 0.012709 | 0.000197
5 | 0.009343 | 0.000127
(C) | (D)
| $\,j\,$ | $\,\mu_{\epsilon}\,$ | $\,\sigma^{2}_{\epsilon}\,$
---|---|---
1 | 0.2762e-3 | 0.1371e-6
2 | 0.0450e-3 | 0.0031e-6
3 | 0.0416e-3 | 0.0033e-6
4 | 0.0360e-3 | 0.0014e-6
5 | 0.0287e-3 | 0.0024e-6
(E) |
Table 4.1: Progress of SNGCA for test models in $\,10\,$ dimensions with
increasing number $\,j\,$ of iterations. The empirical mean of the error
$\,\mathcal{E}(\widehat{\mathcal{I}},\mathcal{I})\,$ defined in (2.10) is
denoted by $\,\mu_{\epsilon}\,$ and $\,\sigma^{2}_{\epsilon}\,$ is its
empirical variance.
Illustration of one-step-improvement: We shall now illustrate the iterative
gain of information about the EDR space. To this end we use the projection of
$\,\widehat{\beta}_{j}\,$ to the EDR-space in order to demonstrate, how the
algorithm works. Figure 4.3 shows that
$\,dist(\widehat{\beta},\widehat{\mathcal{I}})\,$ decreases with increasing
number of iterations. We observe, that estimators $\,\widehat{\beta}\,$ with
higher norm tend to be close to $\,\mathcal{I}\,$. Nevertheless, this can not
be assured for much higher dimensions. Moreover the improvement in each
iteration depends on the size of the sampling of measurement directions.
Figure 4.3: illustrative plots of SNGCA applied to toy 20 dimensional data of type (C) (see section 4): We show $\,\|\widehat{\beta}\|\,$ vs. $\,\cos(\theta(\widehat{\beta},\mathcal{I}))\,$ for different iterations of the algorithm where $\,\mathcal{I}\,$ is the a priori known EDR-space. |
---|---
(A) | (B)
|
(C) | (D)
|
(E) |
Figure 4.4: results wrt. $\,\mathcal{E}(\widehat{\mathcal{I}},\mathcal{I})\,$
with deviations of Gaussian components following a geometrical progression on
$\,[10^{-r},10^{r}]\,$ where $\,r\,$ is the parameter on the abscissa) .
Now let us switch to the question of robustness of the estimation procedure
with respect to a bad conditioning of the covariance matrix $\,\Sigma\,$ of
the data. In figure 4.4 we consider the same test data sets as above. The non-
Gaussian coordinates always have unity variance, but the standard deviation of
the $\,8\,$ Gaussian dimensions now follows the geometrical progression
$\,10^{-r},10^{-r+2r/7},\ldots,10^{r}\,$ where $\,r=1,\ldots,8\,$. Again we
apply a componentwise normalization procedure to the data from the models (A),
(B), (D), (E). We observe that the condition of the covariance matrix heavily
influences the estimation error for the methods NGCA and PP(tanh). In
comparison SNGCA is independent of differences in the noise variance along
different directions in most cases. Only the detection of the uniform
distribution by SNGCA is influenced by the condition of $\,\Sigma\,$.
|
---|---
(A) | (B)
|
(C) | (D)
|
(E) |
Figure 4.5: results wrt. $\,\mathcal{E}(\widehat{\mathcal{I}},\mathcal{I})\,$
with increasing number of gaussian components.
Figure 4.5 compares the behavior of SNGCA with PP and NGCA as the number of
standard and homogeneous Gaussian dimensions increases. As described above we
use the test models with $\,2\,$-dimensional non-Gaussian components with
unity variance. We plot the mean of errors
$\,\varepsilon(\widehat{\mathcal{I}},\mathcal{I})\,$ over $\,100\,$
simulations w.r.t. the test models (A) to (E).
Again concerning the mean of errors
$\,\varepsilon(\widehat{\mathcal{I}},\mathcal{I})\,$ over $\,100\,$
simulations of PP and NGCA we find a transition in the error criterion to a
failure mode for the test models (A), (C) between $\,d=30\,$ and $\,d=40\,$
and between $\,d=20\,$ and $\,d=30\,$ respectively. For the test models
(B),(D) and (E) we found a relative continuous increase in
$\,\varepsilon(\widehat{\mathcal{I}},\mathcal{I})\,$ for the methods PP and
NGCA. In comparison SNGCA fails to analyze test model (A) independently from
the size of the sampling, if the dimension exceeds $\,d=12\,$. Concerning test
model (B) there is a sharp transition in the simulation result between
$\,d=35\,$ and $\,d=40\,$.
Failure modes: In order to provide a better insight into the details of the
failure modes we present box plots of
$\,\varepsilon(\widehat{\mathcal{I}},\mathcal{I})\,$ in the transition phases
w.r.t. the models (A) and (B).
Figure 4.6: failure modes of SNGCA - upper figure: model (A) - lower figure:
model(B)
Figure 4.6 demonstrates the differences in the transition phases of model (A)
and (B) respectively. The transition phase is characterized by a high variance
of the estimation error. For model (A) the increase of the variance
$\,\sigma^{2}_{\varepsilon}\,$ of
$\,\varepsilon(\widehat{\mathcal{I}},\mathcal{I})\,$ beginning at dimensions
$\,13\,$ and its decrease beginning at dimension $\,15\,$ indicates that a
sharp transition phase happens in the interval $\,[13,15]\,$. For higher
dimensions more iterations of SNGCA have a decreasing effect on the estimation
result. This indicates that by the sampling of the measurement directions, we
can not detect the non-Gaussian components of the data density. For model (B)
the transition phase starts at dimension $\,35\,$ and ends at dimension
$\,43\,$.
Moreover the decrease of $\,\sigma^{2}_{\varepsilon}\,$ towards higher
dimensions and the increase of the mean of
$\,\varepsilon(\widehat{\mathcal{I}},\mathcal{I})\,$ is much slower. This
indicates that the non-Gaussian density components might be detectable if we
would allow much more iterations of SNGCA and an enlarged size of the set of
measurement directions. This observation motivates the interpretation that the
Monte-Carlo sampling of the measurement directions is a very poor strategy
that fails to provide sufficient information about the Laplace distribution in
high dimensions. Currently the SNGCA performance is limited by the sampling
strategy.
### 4.2 Application to real life examples
We consider a simulating of a mixture of oil and gas flowing under high
pressure through a pipeline. Under these physical conditions different phases
of the oil-gas-mixture may exist at the same time in the phase space
$\,\Gamma\,$. Only some of these phase configurations in $\,\Gamma\,$ are
stable over long periods of time. Consequently one expects some clusters of
points in $\,\Gamma\,$ indicating the physical state of the mixture. The
$\,12\,$-dimensional data set, obtained by numerical simulations of a
stationary physical model, was already used before for testing techniques of
dimension reduction [3]. The data set comes with a subset of training data and
a subset of test data. The length of the time series is $\,1000\,$ in each
dimension.
The task with this data is to find the clusters representing the stable
configurations in the training data set. It is not known a priori if some
dimensions are more relevant than others. However it is known a priori that
the data is divided into $\,3\,$ classes, indicated by different shapes of the
data points. The cluster information is not used in finding the EDR-space.
Again we compare SNGCA with NGCA and PP using the hyperbolic tangent index
(2.5). For PP and NGCA the results are shown in figure 4.7. They were already
published in [24].
|
---|---
Figure 4.7: left: 2D projection of the ”oil flow” data manually chosen from 3D
projection obtained from by vanilla FastICA methods using the tanh index -
right: projection obtained by NGCA using a combination of Fourier, tanh,
Gauss-pow3 indices
Figure 4.7 shows a slice through $\,\Gamma\,$ such that the structure in the
data set becomes visible: Using NGCA we can distinguish $\,10-11\,$ clusters
versus at most $\,5\,$ for PP with index (2.5).
For the SNGCA method the results are shown in the figure 4.8. SNGCA identifies
3 non-Gaussian dimensions. All figures are rotated by hand such that the
separation of the cluster is illustrated at best. The next figure 4.8 shows
the result of the oil-flow data obtained from SNGCA using a combination of the
indices (2.5) and (2.6).
Figure 4.8: phase configurations of the ”oil flow” data with apriori cluster
mapping induced by crosses, circles and triangles obtained by SNGCA using a
combination of asymmetric-Gauss and the tanh index
In this case we can distinguish $\,10-11\,$ clusters versus at most $\,5\,$
for PP. Moreover we confirm the result of NGCA on the data set. The clusters
are clearly separated from each other on the SNGCA projection. Only on the PP
projection they are partially confounded in one single cluster. By applying
the projection obtained from SNGCA to the test data, we found the cluster
structure to be relevant. We conclude that SNGCA gives a more relevant
estimation of $\,\mathcal{I}\,$ than PP. However it is found that the family
of functions $\,h_{\omega}(x)\,$ is an important tuning parameter in SNGCA: If
we use only the tanh-index, we found only 6-7 cluster are identified and they
are partially confounded. Hence a combination should be used in order to cope
with symmetric data distributions.
## 5 Conclusion
We propose a new improved methodology for the non-Gaussian component analysis,
as proposed in [24]. As well as NGCA the suggested method is based on a semi-
parametric framework for separation an uninteresting multivariate Gaussian
noise subspace from a linear subspace, where the data are non-Gaussian
distributed. Both methods assume that the non-Gaussian contribution to the
data density contains the structure in a given data set. The combined strategy
of convex projection and structural adaptation provides promising results of
SNGCA. Moreover SNGCA provides an estimate for the dimension of the non-
Gaussian subspace. On the other hand, the quality and the numerical complexity
of Monte-Carlo sampling of the measurement directions is the main limitation
of the proposed technique.
## Appendix A Statistical tests
In this section we shortly report the statistical tests on normality used the
dimension reduction step of SNGCA.
In order to detect a significant asymmetry in the distribution of the original
data projected on the semi-axis of the numerical approximation of the rounding
ellipsoid $\,\mathcal{E}_{\sqrt{d}}\,$ we use the $\,K^{2}\,$-test according
to D’Agostino-Pearson [31]. The D’Agostino-Pearson test computes how far the
empirical skewness and kurtosis of the given data distribution differs from
the value expected with a Gaussian distribution. The test statistic is
approximately distributed according to the $\,\chi^{2}_{2}\,$-distribution and
its empirical data counterpart is given by
$\displaystyle\widehat{K}^{2}$ $\displaystyle=$
$\displaystyle\mathcal{Z}^{2}(\sqrt{b_{1}})+\mathcal{Z}^{2}(b_{2})$
$\displaystyle\sqrt{b_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}[\sigma^{-1}(X_{i}-\mu)]^{3}$
$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}[\sigma^{-1}(X_{i}-\mu)]^{4}$
Here $\,\mu\,$ denotes the empirical mean, $\,\sigma\,$ the empirical standard
deviation of the data and $\,\mathcal{Z}(\cdot)\,$ denotes a normalizing
transformations of skewness and kurtosis. The test is more powerful w.r.t. an
asymmetry of a distribution.
Furthermore we use the EDF-test according to Anderson-Darling [1] with the
modification of Stephens [25]: Let $\,F_{N}\,$ be the empirical cumulative
distribution function and $\,F\,$ the assumed theoretical cumulative
distribution function. The test statistics $\,\mathcal{T}\,$ measures the
quadratic deviations between $\,F_{N}\,$ and $\,F\,$:
$\displaystyle\mathcal{T}=\int_{\mathbb{R}}[F_{N}(x)-F(x)]^{2}\nu(x)\;dF$
where $\,\nu(x)\,$ is the weighting function
$\,\nu(x)=[F_{N}(x)(1-F_{N}(x))]^{-1}\,$. In sum the data counterpart of
$\,\mathcal{T}\,$ is given by
$\displaystyle\widehat{\mathcal{T}}=$ $\displaystyle-
cN-c\sum_{i=1}^{N}N^{-1}(2i-1)[\log(F(\sigma^{-1}(X_{i}-\mu))$
$\displaystyle+\log(1-F(\sigma^{-1}(X_{N-i+1}-\mu))]$
where $\,c=1+0.75N^{-1}+2.25N^{-2}\,$. Again $\,\mu\,$ is the empirical mean
and $\,s\,$ the empirical standard deviation of the data. We compute
$\,\widehat{\mathcal{T}}\,$ to detect deviations from normality in the tails
of the projected distributions. The test is rejected if
$\,\widehat{\mathcal{T}}\,$ exceeds a critical value $\,cv\,$ specific for a
given level of significance:
$\,\alpha\,$ : | $\,0.10\,$ | $\,0.05\,$ | $\,0.025\,$ | $\,0.01\,$ | $\,0.005\,$
---|---|---|---|---|---
$\,cv\,$ : | $\,0.631\,$ | $\,0.752\,$ | $\,0.873\,$ | $\,1.035\,$ | $\,1.159\,$
The last test, applied to the projected data is the Shapiro-Wilks test [22]
based on a regression strategy in the version given by Royston [20, 21]:
$\displaystyle W$ $\displaystyle=$
$\displaystyle\sigma^{-1}[1-b^{2}(\sigma^{2}(N-1))^{-1}]^{\lambda}\sim\mathcal{N}(\mu,1)$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N/2}a_{N-i+1}(X_{N-i+1}-x_{i})$
$\displaystyle(a_{1},\dots,a_{N})$ $\displaystyle=$
$\displaystyle{m^{\top}\Sigma^{-1}\over(m^{\top}\Sigma^{-1^{\top}}\Sigma^{-1}m)^{1/2}}$
In this test $\,m=(m_{1},\ldots,m_{n})\,$ denotes the expected values of
standard normal order statistics for a sample of size $\,N\,$ and $\,\Sigma\,$
is the corresponding covariance matrix.
## Appendix B Proofs
### B.1 Proof of Theorem 2
We use the following result from the empirical process theory (similar
statements under slightly different assumptions can be found e.g. in [26]).
Let $\,\mathcal{B}\,$ stand for the unit Euclidean ball, centered at the
origin. Similarly,
$\,B(\mu,\omega^{\circ})=\\{\omega:\|\omega-\omega^{\circ}\|_{2}\leq\mu\\}\,$
is a ball of radius $\,\mu\,$ centered at $\,\omega^{\circ}\,$. For a function
$\,q(\omega,x)\,$, denote
$\,I\\!\\!E_{N}[q(\omega,X)]=N^{-1}\sum_{i=1}^{N}q(\omega,X_{i})\,$.
###### Lemma 1.
Let $\,q(\omega,x)\,$ be a continuously differentiable function of
$\,\omega\in\mathcal{B}_{d}\,$ and $\,x\in\mathbb{R}^{d}\,$ such that for
every $\,\omega\in\mathcal{B}_{d}\,$
$\displaystyle\operatorname{Var}\bigl{[}q(\omega,X)\bigr{]}\leq
q^{*},\quad\operatorname{Cov}\bigl{[}\nabla_{\omega}q(\omega,X)\bigr{]}\leq
q^{*}I,$ (B.12)
with some $\,q^{*},q^{*}>0\,$. Define
$\displaystyle\zeta(\omega)=N^{1/2}\bigl{\\{}I\\!\\!E_{N}[q(\omega,X)]-I\\!\\!E[q(\omega,X)]\bigr{\\}}$
and $\,\zeta(\omega,\omega^{\prime})=\zeta(\omega)-\zeta(\omega^{\prime})\,$.
Then for any $\,\mathfrak{n}_{0}>1\,$, there is
$\,\lambda^{*}_{1}=\lambda^{*}_{1}(\mathfrak{n}_{0})>0\,$ such that for any
$\,\omega^{\circ}\in\mathcal{B}_{d}\,$, $\,\mu\leq 1\,$, and
$\,\lambda\leq\lambda^{*}_{1}N^{1/2}\,$
$\displaystyle\log
I\\!\\!E\exp\bigl{[}\lambda\zeta(\omega^{\circ})\bigr{]}\\!\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!\mathfrak{n}_{0}q^{*}\lambda^{2}/2,$ (B.13)
$\displaystyle\log I\\!\\!E\exp\Bigl{[}\frac{\lambda}{\mu}\sup_{\omega\in
B(\mu,\omega^{\circ})}\zeta(\omega,\omega^{\circ})\Bigr{]}\\!\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!2\mathfrak{n}_{0}q^{*}\lambda^{2}+\mathfrak{e}_{d}\,,\qquad$
(B.14)
where $\,\mathfrak{e}_{d}=\sum_{k=1}^{\infty}2^{-k}\log(2^{kd})=4d\log 2\,$.
Moreover, define
$\displaystyle\mathfrak{z}(\lambda)=\mathfrak{n}_{0}\bigl{(}q^{*}/2+2q^{*}\bigr{)}\lambda^{2}+\mathfrak{e}_{d}.$
Then for any $\,\varepsilon>0\,$
$\displaystyle
I\\!\\!P\biggl{(}\sup_{\omega\in\mathcal{B}_{d}}\zeta(\omega)\geq
2\lambda^{-1}\bigl{[}\mathfrak{z}(\lambda)+\log\varepsilon^{-1}\bigr{]}\biggr{)}\leq\varepsilon.$
###### Proof.
Define for $\,\omega\in\mathcal{B}_{d}\,$
$\displaystyle g_{0}(\lambda;\omega)=\log
I\\!\\!E\exp\Big{[}\frac{\lambda}{\sqrt{\mathfrak{n}_{0}q^{*}}}\bigl{\\{}q(\omega,X_{1})-I\\!\\!E[q(\omega,X_{1})]\bigr{\\}}\Big{]}.$
Then $\,g_{0}(\lambda;\omega)\,$ is analytic in $\,\lambda\,$ and satisfies
$\,g_{0}(0;\omega)=g^{\prime}_{0}(0;\omega)=0\,$. Moreover, the condition
(B.12) implies $\,g^{\prime\prime}_{0}(0;\omega)<1\,$. Therefore, there is
some $\,\lambda^{*}_{1}>0\,$ such that for any
$\,\lambda_{1}\leq\lambda^{*}_{1}\,$ and any unit vector $\,\omega\,$, it
holds $\,g_{0}(\lambda_{1};\omega)\leq\lambda_{1}^{2}/2\,$. Independence of
the $\,X_{i}\,$’s implies (B.13) for
$\,\lambda\leq\lambda^{*}_{1}N^{1/2}(\mathfrak{n}_{0}q^{*})^{-1/2}\,$. In the
same way, for $\,\omega,u\in\mathcal{B}_{d}\,$ define
$\,\zeta(\omega,X)=\nabla_{\omega}q(\omega,X_{1})-I\\!\\!E[\nabla_{\omega}q(\omega,X_{1})]\,$
and
$\displaystyle g(\lambda;\omega,u)=\log I\\!\\!E\exp\big{[}\frac{2\lambda
u^{\top}}{\sqrt{\mathfrak{n}_{0}q^{*}}}\zeta(\omega,X_{1})\big{]}.$
Then similarly to the above, the function $\,g(\lambda;\omega,u)\,$ is
analytic in $\,\lambda\,$ and satisfies with some $\,\lambda^{*}_{1}>0\,$, any
$\,\lambda_{1}\leq\lambda^{*}_{1}\,$ and any unit vectors $\,u\,$ and
$\,\omega\,$
$\displaystyle g(\lambda_{1};\omega,u)\leq 2\lambda_{1}^{2}.$
The bound (B.14) is derived from [23], Lemma 5.1. Independence of the
$\,X_{i}\,$’s yields for
$\,\lambda\leq\lambda^{*}_{1}N^{1/2}(\mathfrak{n}_{0}q^{*})^{-1/2}\,$
$\displaystyle\log
I\\!\\!E\exp\biggl{\\{}\frac{2\lambda}{\sqrt{\mathfrak{n}_{0}q^{*}}}u^{\top}\nabla\zeta(\omega)\biggr{\\}}\leq
2\lambda^{2}.$
This means that the condition $\,(\mathscr{E}D)\,$ of [23] is verified and the
result (B.14) follows from [23], Lemma 5.1. Introduce a random set
$\,A=\\{(\lambda/2)\sup_{\omega}\zeta(\omega)>\mathfrak{z}(\lambda)+\log\varepsilon^{-1}\\}\,$.
and $\,A^{c}\,$ is its complement. By the Cauchy-Schwartz inequality
$\displaystyle I\\!\\!P(A^{c})\\!\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!\\!I\\!\\!E\exp\biggl{\\{}\frac{\lambda}{2}\sup_{\omega}\zeta(\omega)-\mathfrak{z}(\lambda)-\log\varepsilon^{-1}\biggr{\\}}$
$\displaystyle\leq$ $\displaystyle\\!\\!\\!\varepsilon
I\\!\\!E^{1/2}\exp\bigl{\\{}\lambda\zeta(\omega^{\circ})-\mathfrak{n}_{0}q^{*}\lambda^{2}/2\bigr{\\}}$
$\displaystyle\\!\\!\\!\times\,I\\!\\!E^{1/2}\exp\bigl{\\{}\lambda\sup_{\omega}\zeta(\omega,\omega^{\circ})-2\mathfrak{n}_{0}q^{*}\lambda^{2}-\mathfrak{e}_{d}\bigr{\\}}\leq\varepsilon$
and the last result follows.
∎
The result of Lemma 1 can be easily extended to the case of a vector function
$\,q(\omega,x)\in\mathbb{R}^{d}\,$:
$\displaystyle
I\\!\\!P\biggl{(}\sup_{\omega\in\mathcal{B}_{d}}\|\zeta(\omega)\|_{\infty}\geq
2\lambda^{-1}\bigl{[}\mathfrak{z}(\lambda)+\log(d/\varepsilon)\bigr{]}\biggr{)}\leq\varepsilon.$
This fact can be obtained by applying Lemma 1 to each component of the vector
$\,\zeta(\omega)\,$. The term $\,\log(d/\varepsilon)\,$ is responsible for the
overall deviation probability.
Let now $\,f(x,\omega)\,$ be a twice continuously differentiable function of
$\,\omega\in\mathcal{B}_{d}\,$ and $\,x\in\mathbb{R}^{d}\,$ such that for
every $\,j\leq d\,$, $\,\omega\in\mathcal{B}_{d}\,$, and
$\,x\in\mathbb{R}^{d}\,$, it holds
$\displaystyle\operatorname{Var}\bigl{[}X_{j}\,f(X,\omega)\bigr{]}\leq
f^{*}_{1},\quad\operatorname{Cov}\bigl{[}X_{j}\,\nabla_{\omega}f(X,\omega)\bigr{]}\leq
f^{*}_{1}I,$ $\displaystyle\operatorname{Var}\biggl{[}\frac{\partial}{\partial
x_{j}}f(X,\omega)\biggr{]}\leq
f^{*}_{1},\quad\operatorname{Cov}\biggl{[}\nabla_{\omega}\frac{\partial}{\partial
x_{j}}f(X,\omega)\biggr{]}\leq f^{*}_{1}I,$
Then for any $\,\mathfrak{n}_{0}>1\,$, there is
$\,\lambda^{*}_{1}=\lambda^{*}_{1}(\mathfrak{n}_{0})>0\,$ and for any
$\,\varepsilon>0\,$, a random set $\,A\,$ with $\,I\\!\\!P(A)\geq
1-\varepsilon\,$ such that on $\,A\,$ it holds by Lemma 1
$\displaystyle\sup_{\omega\in\mathcal{B}_{d}}\big{\|}I\\!\\!E_{N}[Xf(X,\omega)]-I\\!\\!E[Xf(X,\omega)]\big{\|}_{\infty}\leq\delta_{N},$
$\displaystyle\sup_{\omega\in\mathcal{B}_{d}}\big{\|}I\\!\\!E_{N}[\nabla_{x}f(X,\omega)]-I\\!\\!E[\nabla_{x}f(X,\omega)]\big{\|}_{\infty}\leq\delta_{N},$
where
$\displaystyle\delta_{N}=N^{-1/2}\inf_{\lambda\leq\lambda^{*}_{1}N^{1/2}}\bigl{\\{}5\mathfrak{n}_{0}f^{*}_{1}\lambda+2\lambda^{-1}\bigl{[}\mathfrak{e}_{d}+\log(2d/\varepsilon)\bigr{]}\bigr{\\}}.$
By construction of vectors $\,\widehat{\gamma}_{l}\,$ and
$\,\widehat{\eta}_{l}\,$, it holds on $\,A\,$
$\displaystyle\max_{1\leq l\leq
L}\|\widehat{\gamma}_{l}-\gamma_{l}\|_{\infty}\leq\delta_{N},\quad\max_{1\leq
l\leq L}\|\widehat{\eta}_{l}-\eta_{l}\|_{\infty}\leq\delta_{N}\,.$
This implies for any $\,\|c\|_{1}\leq 1\,$
$\displaystyle\|\widehat{\gamma}(c)-\gamma(c)\|_{\infty}\leq\delta_{N},\;\;\;\|\widehat{\eta}(c)-\eta(c)\|_{\infty}\leq\delta_{N}.$
The constraint $\,\widehat{\gamma}(\widehat{c})=0\,$ implies
$\,\|\gamma(\widehat{c})\|_{\infty}\leq\delta_{N}\,$, thus
$\|\gamma(\widehat{c})\|_{2}\leq\sqrt{d}\,\delta_{N},$
and by (2.3)
$\displaystyle\bigl{\|}(I-\Pi^{*})\widehat{\eta}(\widehat{c})\bigr{\|}_{2}$
$\displaystyle\leq$
$\displaystyle\bigl{\|}(I-\Pi^{*})\\{\widehat{\eta}(\widehat{c})-{\eta}(\widehat{c})\\}\bigr{\|}_{2}+\bigl{\|}(I-\Pi^{*})\eta(\widehat{c})\bigr{\|}_{2}$
$\displaystyle\leq$
$\displaystyle\bigl{\|}\widehat{\eta}(\widehat{c})-\eta(\widehat{c})\bigr{\|}_{2}+\bigl{\|}\Sigma^{-1}\gamma(\widehat{c})\bigr{\|}_{2}$
$\displaystyle\leq$
$\displaystyle\sqrt{d}\bigl{(}\delta_{N}+\bigl{\|}\Sigma^{-1}\bigr{\|}_{2}\delta_{N}\bigr{)}.$
### B.2 Proof of Theorem 3
Let $\,\mathscr{S}\,$ stand for the convex envelope of
$\,\\{\pm\widehat{\beta}_{j}\\}_{j=1}^{J}\,$. As $\,\mathscr{E}_{1}(B)\,$ is
inscribed in $\,\mathscr{S}\,$, its support function
$\,\xi_{\mathscr{E}_{1}(B)}(x)=\max_{s\in\mathscr{E}_{1}(B)}s^{\top}x\,$ is
majorated by that of $\,\mathscr{S}\,$:
$\xi_{\mathscr{E}_{1}(B)}(v)\leq\xi_{\mathscr{S}}(v)=\max_{j=1,...,J}|v^{\top}\widehat{\beta}_{j}|,\;\;\mbox{for
any}\;\;v\in\mathbb{R}^{d}.$
Next, the support function of the ellipsoid $\,\mathscr{E}_{1}(B)\,$ is
$\xi_{\mathscr{E}_{1}(B)}(v)=(v^{\top}B^{-1}v)^{1/2},$
so that the condition $\,\|\widehat{\beta}_{j}-\beta_{j}\|_{2}\leq\varrho\,$
implies
$v^{\top}B^{-1}v\leq\max_{j=1,...,J}|v^{\top}\widehat{\beta}_{j}|^{2}\leq\varrho^{2},$
for any $\,v\perp\mathcal{I}\,$.
Let us prove the second claim of the proposition. Let $\,\Pi^{*}\,$ be a
projector onto $\,\mathcal{I}\,$. By the assumption of the proposition there
exist coefficients $\,\mu_{j}\,$ with $\,\sum_{j}\mu_{j}\leq 1\,$ such that
$\displaystyle
S\stackrel{{\scriptstyle\operatorname{def}}}{{=}}\frac{1}{2}\biggl{[}\sum_{j}\mu_{j}\beta_{j}\beta_{j}^{\top}-2\varrho^{2}\Pi^{*}\biggr{]}\succeq
0.$
This implies (2.11). Now, for any such $\,S\,$ and its pseudo-inverse
$\,S^{+}\,$, the ellipsoid, $\,\mathscr{E}^{f}_{1}(S^{+})\,$ with
$\mathscr{E}^{f}_{1}(S^{+})=\\{x\in\mathcal{I}\mid x^{\top}S^{+}x\leq 1\\}$
is inscribed into $\,\mathscr{S}\,$. Indeed, the support function
$\,\xi_{\mathscr{E}^{f}_{1}(S^{+})}(x)=(x^{\top}Sx)^{1/2}\,$ of this ellipsoid
fulfills for $\,x\in\mathcal{B}_{d}\,$
$\displaystyle\xi_{\mathscr{E}^{f}_{1}(S^{+})}(x)$ $\displaystyle\leq$
$\displaystyle\biggl{(}\sum_{j}\mu_{j}\Bigl{[}\frac{1}{2}(x^{\top}\beta_{j})^{2}-\varrho^{2}\Bigr{]}\biggr{)}^{1/2}$
$\displaystyle\leq$
$\displaystyle\biggl{(}\sum_{j}\mu_{j}\bigl{|}x^{\top}\widehat{\beta}_{j}\bigr{|}^{2}\biggr{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle\max_{1\leq j\leq
J}|x^{\top}\widehat{\beta}_{j}|=\xi_{\mathscr{S}}(x),$
Now we are done: as the ellipsoid $\,\mathscr{E}^{f}_{1}(S^{+})\,$ is
inscribed into $\,\mathscr{S}\,$, it is contained in the concentric to
$\,\mathscr{E}_{1}(B)\,$ ellipsoid $\,\mathscr{E}_{\sqrt{d}}(B)\,$ which
covers $\,\mathscr{S}\,$.
To show the last statement of the theorem, observe that
$\displaystyle\mathrm{Tr}\bigl{[}(\widehat{\Pi}-\Pi^{*})^{2}\bigr{]}=2(m-\mathrm{Tr}[\Pi^{*}\widehat{\Pi}])=2\mathrm{Tr}\bigl{[}(I-\Pi^{*})\widehat{\Pi}\bigr{]}.$
On the other hand, using the second claim one gets
$\displaystyle\mathrm{Tr}\bigl{[}(I-\Pi^{*})\widehat{\Pi}\bigr{]}$
$\displaystyle\leq$
$\displaystyle(d-m)\sup_{v\perp\mathcal{I}}v^{\top}\widehat{\Pi}v$
$\displaystyle\leq$
$\displaystyle(d-m)\sup_{v\perp\mathcal{I}}\frac{v^{\top}B^{-1}v}{\lambda_{m}(B^{-1})}$
$\displaystyle\leq$
$\displaystyle\frac{2d^{3/2}\varrho^{2}}{\lambda^{*}-2\varrho^{2}}.$
## Appendix C The algorithm
Here we present the full algorithmic description of the SNGCA procedure. We
start with the linear estimation subprocedure:
Data: $\,Y\,$,$\,L\,$,$\,J\,$
Result: $\,\\{\widehat{\beta}_{j}\\}_{j=1}^{J}\,$
Sampling: choice of measurement directions
for _j=1 to J_ do for _l=1 to L_ do Compute:
$\,\widehat{\eta}_{jl}=N^{-1}\sum_{i=1}^{N}\nabla h_{\omega_{jl}}(Y_{i})\,$
$\,\widehat{\gamma}_{jl}=N^{-1}\sum_{i=1}^{N}Y_{i}h_{\omega_{jl}}(Y_{i})\,$
end Compute $\,\widehat{c}_{j}\,$ as in (2.8) and
$\,\widehat{\beta}_{j}=\sum_{l=1}^{L}\widehat{c}_{j}\widehat{\eta}_{jl}\,$.
end
Algorithm 4 (linear estimation of $\,\beta(\psi_{h,c})\,$).
The following subprocedure reports the computation of the $\sqrt{d}$-rounding
ellipsoid based on a proposal in [18]:
Data: $\,\\{\widehat{\beta}_{j}\\}_{j=1}^{J}\,$
Result: $\,\widehat{B}\,$,
Let $\,\delta_{i}^{k^{\ast}}=\max_{1\leq j\leq
J}\;\langle\widehat{\beta}_{j},\widehat{B}_{i}\widehat{\beta}_{j}\rangle\,$
and set $\,\nu_{i}=\delta_{i}^{k^{\ast}}d^{-1}\,$.
Let $\,\widehat{B}_{0}\,$ be the inverse empirical covariance matrix of the
$\,\widehat{\beta}_{j}\,$ and set
$\,t_{i}=\nu_{i}(\delta_{i}^{k^{\ast}}d^{-1}-1)^{-1}\,$.
Moreover let $\,i\,$ be the loop index.
repeat $\,x_{i}=\widehat{B}_{i}\widehat{\beta}_{k^{\ast}}\,$
$\,\widehat{B}_{i+1}=(1-t_{i})^{-1}\Big{(}\widehat{B}_{i}-t_{i}(1+\nu_{i})^{-1}x_{i}x_{i}^{\top}\Big{)}\,$
$\,\delta_{i+1}^{k^{\ast}}=(1-t_{i})^{-1}\Big{(}\delta_{i}^{k^{\ast}}-t_{i}(1+\nu_{i})^{-1}\langle\widehat{\beta}_{k^{\ast}},x_{i}\rangle^{2}\Big{)}\,$
until _$\,\delta_{i}^{k^{\ast}}\leq C\cdot d\,$ where $\,C\,$ is a tuning
parameter._
Algorithm 5 (Compute of the $\,\sqrt{d}\,$-rounding of the MVEE).
The next algorithm 6 reports the pseudocode for constructing a reduced basis
of the target space from the estimated elements by means of algorithm 5:
Data: $\,\widehat{B}\,$
Result: $\,\big{\langle}\text{first }m\text{ eigenvectors of
}\widehat{B}\big{\rangle}\,$
Let $\,\widehat{V}\,$ be the matrix of eigenvectors $\,\widehat{v}_{i}\,$ from
$\,\widehat{B}\,$
computed according to algorithm 5.
for _i=1 to d_ do Project the data orthogonal on $\,\widehat{v}_{i}\,$.
Compute tests on normality of the projected data.
end Discard every eigenvector with associated normal
distributed projected data.
Algorithm 6 (Dimension Reduction).
In algorithm 4 we start with a random initialization of the non-parametric
estimator $\,\widehat{\beta}_{j}\,$ by means of a Monte-Carlo sampling of the
directions $\,\omega_{jl}\,$ and $\,\xi_{j}\,$. However we can use the result
of the first iteration $\,k=1\,$ of SNGCA in order to accumulate information
about $\,\mathcal{I}\,$ in a sequence
$\,\widehat{\mathcal{I}}_{1},\widehat{\mathcal{I}}_{2},\ldots\,$ of estimators
of the target space. The procedure is described in detail in algorithm 7.
Data: $\,\big{\langle}\text{first }m\text{ eigenvectors of
}\widehat{B}\big{\rangle}\,$
Let $\,\\{\widehat{v_{i}}\\}_{i=1}^{m}\,$ denote the reduced set of
eigenvectors from
$\,\widehat{B}\,$ and let $\,k\,$ iterations be completed. To initialize
iteration $\,k+1\,$ choose random numbers $\,z_{j1},\ldots,z_{jm}\,$
and $\,u_{l1},\ldots,u_{lm}\,$ from $\,\mathcal{U}_{[-1,1]}\,$ and set
$\,\quad\quad\quad\xi_{j}:=\sum_{s=1}^{m}z_{js}\widehat{v}_{i_{s}}\mbox{ for
}1\leq j\leq n_{1}<J\,$
$\,\quad\quad\quad\omega_{l}:=\sum_{s=1}^{m}u_{ls}\widehat{v}_{i_{s}}\mbox{
for }1\leq l\leq n_{2}<L\,$
Then define $\,\omega_{L-n_{2}},\ldots,\omega_{L}\,$ and
$\,\xi_{J-n_{1}},\ldots,\xi_{J}\,$ analogous to the case $\,k=1\,$. Now
compose the sets
$\,\quad\quad\quad\\{\xi_{1}^{(k)},\ldots,\xi_{n_{1}}^{(k)},\xi_{n_{1}+1}^{(k)},\ldots,\xi_{J}^{(k)}\\}\,$
$\,\quad\quad\quad\\{\omega_{1}^{(k)},\ldots,\omega_{n_{2}}^{(k)},\omega_{n_{2}+1}^{(k)},\ldots,\omega_{L}^{(k)}\\}\,$
For the initialization in the case $\,k=k+1\,$. Moreover we choose
$\,n_{1}=kd\,$ and $\,n_{2}=kd\,$ until $\,n_{1}>J-d\,$ or $\,n_{2}>L-d\,$.
Otherwise set $\,n_{1}=J-d\,$ or $\,n_{2}=L-d\,$.
Algorithm 7 (structural adaptation of the linear estimation ).
Choice of parameters: One of the advantages of the algorithm proposed above is
the fact that there are only a few tuning parameters.
* i)
Suppose now that $\,\omega_{i}\,$ is an absolute continuous random variable
with $\,\omega_{i}\sim\mathcal{U}_{[-1,1]}\,$. Without loss of generality we
set $\,e=(1,0,\ldots,0)\,$. Due to the normalization of
$\,(\omega_{1},\ldots,\omega_{d})\,$, it holds:
$\displaystyle I\\!\\!P\big{(}|(\omega_{1},\ldots,\omega_{d})^{\top}e|\geq
0.5\big{)}=\big{(}\sqrt{d}\big{)}^{-1}$
However the choice of $\,J\,$ and $\,L\,$ heavily depends on the non-gaussian
components. In the experiments we use $\,7d\leq J\leq 18d\,$ and $\,6d\leq
L\leq 16d\,$.
* ii)
Set the parameter of the stopping rule to $\,\delta=0.05\,$.
* iii)
Set the constant in the stopping rule for the computation of the MVEE to
$\,C=2\,$.
* iv)
Set the significance level of the statistical tests to $\,\alpha=0.05\,$.
Finally we give a description of the complete algorithm.
Data: $\,\\{X_{i}\\}_{i=1}^{N}\,$,$\,L\,$,$\,J\,$,$\,\alpha\,$
Result: $\,\widehat{\mathcal{I}}\,$
Normalization: The data $\,(X_{i})_{i=1}^{N}\,$ are recentered. Let
$\,\sigma=(\sigma_{1},\ldots\sigma_{d})\,$ be the standard deviations of the
components of $\,X_{i}\,$. Then $\,Y_{i}=\mathrm{diag}(\sigma^{-1})X_{i}\,$
denotes the
componentwise empirically normalized data.
Main Procedure:; // loop on $\,k\,$
while _$\,\sim StoppingCriterion(\mathcal{I},\widehat{\mathcal{I}})\,$_ do
Sampling: The components of the Monte-Carlo-parts
of $\,\xi_{j}^{(k)}\,$ and $\,\omega_{jl}^{(k)}\,$ are randomly chosen from
$\,\mathcal{U}_{[-1,1]}\,$.
The other part of the measurement directions are
initialized according to the structural adaptation
approach described in algorithm 7. Then $\,\xi_{j}^{(k)}\,$ and
$\,\omega_{jl}^{(k)}\,$ are normalized to unit length.
Linear Estimation Procedure:
for _j=1 to J_ do for _l=1 to L_ do
$\,\widehat{\eta}_{jl}^{(k)}=N^{-1}\sum_{i=1}^{N}\nabla
h_{\omega_{jl}^{(k)}}(Y_{i})\,$
$\,\widehat{\gamma}_{jl}^{(k)}=N^{-1}\sum_{i=1}^{N}Y_{i}h_{\omega_{jl}^{(k)}}(Y_{i})\,$
endCompute the coefficients $\,\\{c_{l}\\}_{l=1}^{L}\,$ by solving the
second-order conic optimization problem (2.8):
$\,\quad\quad\quad\quad\quad\min\;q\qquad\mbox{s.t.}\,$
$\,\quad\quad\quad\quad\quad\quad\frac{1}{2}\|z\|_{2}\leq q\,$
$\,\quad\quad\sum_{l=1}^{L}(c_{l}^{+}-c_{l}^{-})\widehat{\eta}_{jl}^{(k)}-z=\xi_{j}^{(k)}\,$
$\,\quad\quad\quad\sum_{l=1}^{L}(c_{l}^{+}-c_{l}^{-})\widehat{\gamma}_{jl}^{(k)}=0\,$
$\,\sum_{l=1}^{L}(c_{l}^{+}-c_{l}^{-})\leq 1,\quad 0\leq
c_{l}^{+},c_{l}^{-}\quad\forall l\,$
Compute
$\,\widehat{\beta}_{j}^{(k)}=\sum_{l=1}^{L}(\widehat{c}_{l}^{+}-\widehat{c}_{l}^{-})\widehat{\eta}_{jl}^{(k)}\,$
end Dimension Reduction:
Compute the symmetric matrix $\,\widehat{B}^{(k)}\,$ defining the
approximation of $\,\mathcal{E}\,$ according to algorithm 5. Reduce the basis
of $\,\mathcal{X}\,$ according to algorithm 6.
end
Algorithm 8 (full procedure of SNGCA).
Complexity: We restrict ourselves to the leading polynomial terms of the
arithmetical complexity of corresponding computations counting only the
multiplications.
* 1.
The numerical effort to compute $\,\eta_{jl}\,$ and $\,\gamma_{jl}\,$ in
algorithm 4 heavily depends on the choice of $\,h(\omega^{\top}x)\,$. Let
$\,h(\omega^{\top}x)=\tanh(\omega^{\top}x)\,$. Then this step takes
$\,\mathcal{O}(J(\log N)^{2}N^{2})\,$ operations.
* 2.
Algorithm 5 takes $\,\mathcal{O}(d^{2}J\log(J))\,$ operations [18].
* 3.
For the optimization step in 4 we use a commercial
solver222http://www.mosek.com based on an interior point method. The
constrained convex projection solved as an SOCP takes
$\,\mathcal{O}(d^{2}n^{3})\,$ operations there $\,n\,$ is the number of
constraints.
* 4.
Computation of the statistical tests in one dimension: Let $\,N\,$ denote the
number of samples. D’Agostino-Pearson-test needs $\,\mathcal{O}(N^{3}\log
N)\,$ and the Anderson-Darling-test $\,\mathcal{O}((\log N)^{2}N^{2})\,$
operations. The test of Shapiro-Wilks takes $\,\mathcal{O}(N^{2})\,$. In order
to avoid robustness problems [14] the number of samples is limited to $\,N\leq
1000\,$. For larger data sets, $\,N=1000\,$ points are randomly chosen.
Hence without tests $\,\widehat{\mathcal{I}}\,$ is computed in
$\,\mathcal{O}(J(\log N)^{2}N^{2}+d^{2}J\log(J)+d^{2}n^{3})\,$ arithmetical
operations per iteration.
## Acknowledgment
We are grateful to Yuri Nesterov from the CORE, Louvain-la-Neuve for helpful
discussions and Gilles Blanchard from the FIRST.IDA Fraunhofer Institute
Berlin for the permission to republish the results of NGCA.
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|
arxiv-papers
| 2009-04-02T17:05:23 |
2024-09-04T02:49:01.625287
|
{
"license": "Public Domain",
"authors": "Elmar Diederichs, Anatoli Juditsky, Vladimir Spokoiny, Christof\n Schuette",
"submitter": "Elmar Diederichs",
"url": "https://arxiv.org/abs/0904.0430"
}
|
0904.0439
|
STAR Collaboration
# $J/\psi$ production at high transverse momenta in $p$+$p$ and Cu+Cu
collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 200 GeV
B. I. Abelev University of Illinois at Chicago, Chicago, Illinois 60607, USA
M. M. Aggarwal Panjab University, Chandigarh 160014, India Z. Ahammed
Variable Energy Cyclotron Centre, Kolkata 700064, India B. D. Anderson Kent
State University, Kent, Ohio 44242, USA D. Arkhipkin Particle Physics
Laboratory (JINR), Dubna, Russia G. S. Averichev Laboratory for High Energy
(JINR), Dubna, Russia J. Balewski Massachusetts Institute of Technology,
Cambridge, MA 02139-4307, USA O. Barannikova University of Illinois at
Chicago, Chicago, Illinois 60607, USA L. S. Barnby University of Birmingham,
Birmingham, United Kingdom J. Baudot Institut de Recherches Subatomiques,
Strasbourg, France S. Baumgart Yale University, New Haven, Connecticut
06520, USA D. R. Beavis Brookhaven National Laboratory, Upton, New York
11973, USA R. Bellwied Wayne State University, Detroit, Michigan 48201, USA
F. Benedosso NIKHEF and Utrecht University, Amsterdam, The Netherlands M. J.
Betancourt Massachusetts Institute of Technology, Cambridge, MA 02139-4307,
USA R. R. Betts University of Illinois at Chicago, Chicago, Illinois 60607,
USA A. Bhasin University of Jammu, Jammu 180001, India A. K. Bhati Panjab
University, Chandigarh 160014, India H. Bichsel University of Washington,
Seattle, Washington 98195, USA J. Bielcik Nuclear Physics Institute AS CR,
250 68 Řež/Prague, Czech Republic J. Bielcikova Nuclear Physics Institute AS
CR, 250 68 Řež/Prague, Czech Republic B. Biritz University of California,
Los Angeles, California 90095, USA L. C. Bland Brookhaven National
Laboratory, Upton, New York 11973, USA M. Bombara University of Birmingham,
Birmingham, United Kingdom B. E. Bonner Rice University, Houston, Texas
77251, USA M. Botje NIKHEF and Utrecht University, Amsterdam, The
Netherlands J. Bouchet Kent State University, Kent, Ohio 44242, USA E.
Braidot NIKHEF and Utrecht University, Amsterdam, The Netherlands A. V.
Brandin Moscow Engineering Physics Institute, Moscow Russia E. Bruna Yale
University, New Haven, Connecticut 06520, USA S. Bueltmann Old Dominion
University, Norfolk, VA, 23529, USA T. P. Burton University of Birmingham,
Birmingham, United Kingdom M. Bystersky Nuclear Physics Institute AS CR, 250
68 Řež/Prague, Czech Republic X. Z. Cai Shanghai Institute of Applied
Physics, Shanghai 201800, China H. Caines Yale University, New Haven,
Connecticut 06520, USA M. Calderón de la Barca Sánchez University of
California, Davis, California 95616, USA O. Catu Yale University, New Haven,
Connecticut 06520, USA D. Cebra University of California, Davis, California
95616, USA R. Cendejas University of California, Los Angeles, California
90095, USA M. C. Cervantes Texas A&M University, College Station, Texas
77843, USA Z. Chajecki Ohio State University, Columbus, Ohio 43210, USA P.
Chaloupka Nuclear Physics Institute AS CR, 250 68 Řež/Prague, Czech Republic
S. Chattopadhyay Variable Energy Cyclotron Centre, Kolkata 700064, India H.
F. Chen University of Science & Technology of China, Hefei 230026, China J.
H. Chen Kent State University, Kent, Ohio 44242, USA J. Y. Chen Institute
of Particle Physics, CCNU (HZNU), Wuhan 430079, China J. Cheng Tsinghua
University, Beijing 100084, China M. Cherney Creighton University, Omaha,
Nebraska 68178, USA A. Chikanian Yale University, New Haven, Connecticut
06520, USA K. E. Choi Pusan National University, Pusan, Republic of Korea
W. Christie Brookhaven National Laboratory, Upton, New York 11973, USA R. F.
Clarke Texas A&M University, College Station, Texas 77843, USA M. J. M.
Codrington Texas A&M University, College Station, Texas 77843, USA R.
Corliss Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
T. M. Cormier Wayne State University, Detroit, Michigan 48201, USA M. R.
Cosentino Universidade de Sao Paulo, Sao Paulo, Brazil J. G. Cramer
University of Washington, Seattle, Washington 98195, USA H. J. Crawford
University of California, Berkeley, California 94720, USA D. Das University
of California, Davis, California 95616, USA S. Dash Institute of Physics,
Bhubaneswar 751005, India M. Daugherity University of Texas, Austin, Texas
78712, USA L. C. De Silva Wayne State University, Detroit, Michigan 48201,
USA T. G. Dedovich Laboratory for High Energy (JINR), Dubna, Russia M.
DePhillips Brookhaven National Laboratory, Upton, New York 11973, USA A. A.
Derevschikov Institute of High Energy Physics, Protvino, Russia R. Derradi
de Souza Universidade Estadual de Campinas, Sao Paulo, Brazil L. Didenko
Brookhaven National Laboratory, Upton, New York 11973, USA P. Djawotho Texas
A&M University, College Station, Texas 77843, USA S. M. Dogra University of
Jammu, Jammu 180001, India X. Dong Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA J. L. Drachenberg Texas A&M University,
College Station, Texas 77843, USA J. E. Draper University of California,
Davis, California 95616, USA J. C. Dunlop Brookhaven National Laboratory,
Upton, New York 11973, USA M. R. Dutta Mazumdar Variable Energy Cyclotron
Centre, Kolkata 700064, India W. R. Edwards Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA L. G. Efimov Laboratory for High
Energy (JINR), Dubna, Russia E. Elhalhuli University of Birmingham,
Birmingham, United Kingdom M. Elnimr Wayne State University, Detroit,
Michigan 48201, USA V. Emelianov Moscow Engineering Physics Institute,
Moscow Russia J. Engelage University of California, Berkeley, California
94720, USA G. Eppley Rice University, Houston, Texas 77251, USA B. Erazmus
SUBATECH, Nantes, France M. Estienne SUBATECH, Nantes, France L. Eun
Pennsylvania State University, University Park, Pennsylvania 16802, USA P.
Fachini Brookhaven National Laboratory, Upton, New York 11973, USA R. Fatemi
University of Kentucky, Lexington, Kentucky, 40506-0055, USA J. Fedorisin
Laboratory for High Energy (JINR), Dubna, Russia A. Feng Institute of
Particle Physics, CCNU (HZNU), Wuhan 430079, China P. Filip Particle Physics
Laboratory (JINR), Dubna, Russia E. Finch Yale University, New Haven,
Connecticut 06520, USA V. Fine Brookhaven National Laboratory, Upton, New
York 11973, USA Y. Fisyak Brookhaven National Laboratory, Upton, New York
11973, USA C. A. Gagliardi Texas A&M University, College Station, Texas
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D. R. Gangadharan University of California, Los Angeles, California 90095,
USA M. S. Ganti Variable Energy Cyclotron Centre, Kolkata 700064, India E.
J. Garcia-Solis University of Illinois at Chicago, Chicago, Illinois 60607,
USA A. Geromitsos SUBATECH, Nantes, France F. Geurts Rice University,
Houston, Texas 77251, USA V. Ghazikhanian University of California, Los
Angeles, California 90095, USA P. Ghosh Variable Energy Cyclotron Centre,
Kolkata 700064, India Y. N. Gorbunov Creighton University, Omaha, Nebraska
68178, USA A. Gordon Brookhaven National Laboratory, Upton, New York 11973,
USA O. Grebenyuk Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA D. Grosnick Valparaiso University, Valparaiso, Indiana 46383, USA
B. Grube Pusan National University, Pusan, Republic of Korea S. M. Guertin
University of California, Los Angeles, California 90095, USA K. S. F. F.
Guimaraes Universidade de Sao Paulo, Sao Paulo, Brazil A. Gupta University
of Jammu, Jammu 180001, India N. Gupta University of Jammu, Jammu 180001,
India W. Guryn Brookhaven National Laboratory, Upton, New York 11973, USA
B. Haag University of California, Davis, California 95616, USA T. J. Hallman
Brookhaven National Laboratory, Upton, New York 11973, USA A. Hamed Texas
A&M University, College Station, Texas 77843, USA J. W. Harris Yale
University, New Haven, Connecticut 06520, USA W. He Indiana University,
Bloomington, Indiana 47408, USA M. Heinz Yale University, New Haven,
Connecticut 06520, USA S. Heppelmann Pennsylvania State University,
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Subatomiques, Strasbourg, France A. Hirsch Purdue University, West
Lafayette, Indiana 47907, USA E. Hjort Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA A. M. Hoffman Massachusetts
Institute of Technology, Cambridge, MA 02139-4307, USA G. W. Hoffmann
University of Texas, Austin, Texas 78712, USA D. J. Hofman University of
Illinois at Chicago, Chicago, Illinois 60607, USA R. S. Hollis University of
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California, Los Angeles, California 90095, USA T. J. Humanic Ohio State
University, Columbus, Ohio 43210, USA L. Huo Texas A&M University, College
Station, Texas 77843, USA G. Igo University of California, Los Angeles,
California 90095, USA A. Iordanova University of Illinois at Chicago,
Chicago, Illinois 60607, USA P. Jacobs Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA W. W. Jacobs Indiana University,
Bloomington, Indiana 47408, USA P. Jakl Nuclear Physics Institute AS CR, 250
68 Řež/Prague, Czech Republic C. Jena Institute of Physics, Bhubaneswar
751005, India F. Jin Shanghai Institute of Applied Physics, Shanghai 201800,
China C. L. Jones Massachusetts Institute of Technology, Cambridge, MA
02139-4307, USA P. G. Jones University of Birmingham, Birmingham, United
Kingdom J. Joseph Kent State University, Kent, Ohio 44242, USA E. G. Judd
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SUBATECH, Nantes, France K. Kajimoto University of Texas, Austin, Texas
78712, USA K. Kang Tsinghua University, Beijing 100084, China J. Kapitan
Nuclear Physics Institute AS CR, 250 68 Řež/Prague, Czech Republic D. Keane
Kent State University, Kent, Ohio 44242, USA A. Kechechyan Laboratory for
High Energy (JINR), Dubna, Russia D. Kettler University of Washington,
Seattle, Washington 98195, USA V. Yu. Khodyrev Institute of High Energy
Physics, Protvino, Russia D. P. Kikola Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA J. Kiryluk Lawrence Berkeley
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University, Columbus, Ohio 43210, USA S. R. Klein Lawrence Berkeley National
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University, Valparaiso, Indiana 46383, USA M. Kopytine Kent State
University, Kent, Ohio 44242, USA W. Korsch University of Kentucky,
Lexington, Kentucky, 40506-0055, USA L. Kotchenda Moscow Engineering Physics
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Řež/Prague, Czech Republic P. Kravtsov Moscow Engineering Physics Institute,
Moscow Russia V. I. Kravtsov Institute of High Energy Physics, Protvino,
Russia K. Krueger Argonne National Laboratory, Argonne, Illinois 60439, USA
M. Krus Nuclear Physics Institute AS CR, 250 68 Řež/Prague, Czech Republic
C. Kuhn Institut de Recherches Subatomiques, Strasbourg, France L. Kumar
Panjab University, Chandigarh 160014, India P. Kurnadi University of
California, Los Angeles, California 90095, USA M. A. C. Lamont Brookhaven
National Laboratory, Upton, New York 11973, USA J. M. Landgraf Brookhaven
National Laboratory, Upton, New York 11973, USA S. LaPointe Wayne State
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Laboratory, Upton, New York 11973, USA R. Lednicky Particle Physics
Laboratory (JINR), Dubna, Russia C-H. Lee Pusan National University, Pusan,
Republic of Korea J. H. Lee Brookhaven National Laboratory, Upton, New York
11973, USA W. Leight Massachusetts Institute of Technology, Cambridge, MA
02139-4307, USA M. J. LeVine Brookhaven National Laboratory, Upton, New York
11973, USA C. Li University of Science & Technology of China, Hefei 230026,
China N. Li Institute of Particle Physics, CCNU (HZNU), Wuhan 430079, China
Y. Li Tsinghua University, Beijing 100084, China G. Lin Yale University,
New Haven, Connecticut 06520, USA S. J. Lindenbaum City College of New York,
New York City, New York 10031, USA M. A. Lisa Ohio State University,
Columbus, Ohio 43210, USA F. Liu Institute of Particle Physics, CCNU (HZNU),
Wuhan 430079, China J. Liu Rice University, Houston, Texas 77251, USA L.
Liu Institute of Particle Physics, CCNU (HZNU), Wuhan 430079, China T.
Ljubicic Brookhaven National Laboratory, Upton, New York 11973, USA W. J.
Llope Rice University, Houston, Texas 77251, USA R. S. Longacre Brookhaven
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National Laboratory, Upton, New York 11973, USA Y. Lu University of Science
& Technology of China, Hefei 230026, China T. Ludlam Brookhaven National
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Applied Physics, Shanghai 201800, China Y. G. Ma Shanghai Institute of
Applied Physics, Shanghai 201800, China D. P. Mahapatra Institute of
Physics, Bhubaneswar 751005, India R. Majka Yale University, New Haven,
Connecticut 06520, USA O. I. Mall University of California, Davis,
California 95616, USA L. K. Mangotra University of Jammu, Jammu 180001,
India R. Manweiler Valparaiso University, Valparaiso, Indiana 46383, USA S.
Margetis Kent State University, Kent, Ohio 44242, USA C. Markert University
of Texas, Austin, Texas 78712, USA H. S. Matis Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA Yu. A. Matulenko Institute of
High Energy Physics, Protvino, Russia D. McDonald Rice University, Houston,
Texas 77251, USA T. S. McShane Creighton University, Omaha, Nebraska 68178,
USA A. Meschanin Institute of High Energy Physics, Protvino, Russia R.
Milner Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
N. G. Minaev Institute of High Energy Physics, Protvino, Russia S.
Mioduszewski Texas A&M University, College Station, Texas 77843, USA A.
Mischke NIKHEF and Utrecht University, Amsterdam, The Netherlands B. Mohanty
Variable Energy Cyclotron Centre, Kolkata 700064, India D. A. Morozov
Institute of High Energy Physics, Protvino, Russia M. G. Munhoz Universidade
de Sao Paulo, Sao Paulo, Brazil B. K. Nandi Indian Institute of Technology,
Mumbai, India C. Nattrass Yale University, New Haven, Connecticut 06520, USA
T. K. Nayak Variable Energy Cyclotron Centre, Kolkata 700064, India J. M.
Nelson University of Birmingham, Birmingham, United Kingdom P. K. Netrakanti
Purdue University, West Lafayette, Indiana 47907, USA M. J. Ng University of
California, Berkeley, California 94720, USA L. V. Nogach Institute of High
Energy Physics, Protvino, Russia S. B. Nurushev Institute of High Energy
Physics, Protvino, Russia G. Odyniec Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA A. Ogawa Brookhaven National Laboratory,
Upton, New York 11973, USA H. Okada Brookhaven National Laboratory, Upton,
New York 11973, USA V. Okorokov Moscow Engineering Physics Institute, Moscow
Russia D. Olson Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA M. Pachr Nuclear Physics Institute AS CR, 250 68 Řež/Prague,
Czech Republic B. S. Page Indiana University, Bloomington, Indiana 47408,
USA S. K. Pal Variable Energy Cyclotron Centre, Kolkata 700064, India Y.
Pandit Kent State University, Kent, Ohio 44242, USA Y. Panebratsev
Laboratory for High Energy (JINR), Dubna, Russia T. Pawlak Warsaw University
of Technology, Warsaw, Poland T. Peitzmann NIKHEF and Utrecht University,
Amsterdam, The Netherlands V. Perevoztchikov Brookhaven National Laboratory,
Upton, New York 11973, USA C. Perkins University of California, Berkeley,
California 94720, USA W. Peryt Warsaw University of Technology, Warsaw,
Poland S. C. Phatak Institute of Physics, Bhubaneswar 751005, India P. Pile
Brookhaven National Laboratory, Upton, New York 11973, USA M. Planinic
University of Zagreb, Zagreb, HR-10002, Croatia J. Pluta Warsaw University
of Technology, Warsaw, Poland D. Plyku Old Dominion University, Norfolk, VA,
23529, USA N. Poljak University of Zagreb, Zagreb, HR-10002, Croatia A. M.
Poskanzer Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA B. V. K. S. Potukuchi University of Jammu, Jammu 180001, India D.
Prindle University of Washington, Seattle, Washington 98195, USA C. Pruneau
Wayne State University, Detroit, Michigan 48201, USA N. K. Pruthi Panjab
University, Chandigarh 160014, India P. R. Pujahari Indian Institute of
Technology, Mumbai, India J. Putschke Yale University, New Haven,
Connecticut 06520, USA R. Raniwala University of Rajasthan, Jaipur 302004,
India S. Raniwala University of Rajasthan, Jaipur 302004, India R. L. Ray
University of Texas, Austin, Texas 78712, USA R. Redwine Massachusetts
Institute of Technology, Cambridge, MA 02139-4307, USA R. Reed University of
California, Davis, California 95616, USA A. Ridiger Moscow Engineering
Physics Institute, Moscow Russia H. G. Ritter Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA J. B. Roberts Rice University,
Houston, Texas 77251, USA O. V. Rogachevskiy Laboratory for High Energy
(JINR), Dubna, Russia J. L. Romero University of California, Davis,
California 95616, USA A. Rose Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA C. Roy SUBATECH, Nantes, France L. Ruan
Brookhaven National Laboratory, Upton, New York 11973, USA M. J. Russcher
NIKHEF and Utrecht University, Amsterdam, The Netherlands R. Sahoo SUBATECH,
Nantes, France I. Sakrejda Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA T. Sakuma Massachusetts Institute of Technology,
Cambridge, MA 02139-4307, USA S. Salur Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA J. Sandweiss Yale University,
New Haven, Connecticut 06520, USA M. Sarsour Texas A&M University, College
Station, Texas 77843, USA J. Schambach University of Texas, Austin, Texas
78712, USA R. P. Scharenberg Purdue University, West Lafayette, Indiana
47907, USA N. Schmitz Max-Planck-Institut für Physik, Munich, Germany J.
Seger Creighton University, Omaha, Nebraska 68178, USA I. Selyuzhenkov
Indiana University, Bloomington, Indiana 47408, USA P. Seyboth Max-Planck-
Institut für Physik, Munich, Germany A. Shabetai Institut de Recherches
Subatomiques, Strasbourg, France E. Shahaliev Laboratory for High Energy
(JINR), Dubna, Russia M. Shao University of Science & Technology of China,
Hefei 230026, China M. Sharma Wayne State University, Detroit, Michigan
48201, USA S. S. Shi Institute of Particle Physics, CCNU (HZNU), Wuhan
430079, China X-H. Shi Shanghai Institute of Applied Physics, Shanghai
201800, China E. P. Sichtermann Lawrence Berkeley National Laboratory,
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Munich, Germany R. N. Singaraju Variable Energy Cyclotron Centre, Kolkata
700064, India M. J. Skoby Purdue University, West Lafayette, Indiana 47907,
USA N. Smirnov Yale University, New Haven, Connecticut 06520, USA R.
Snellings NIKHEF and Utrecht University, Amsterdam, The Netherlands P.
Sorensen Brookhaven National Laboratory, Upton, New York 11973, USA J.
Sowinski Indiana University, Bloomington, Indiana 47408, USA H. M. Spinka
Argonne National Laboratory, Argonne, Illinois 60439, USA B. Srivastava
Purdue University, West Lafayette, Indiana 47907, USA A. Stadnik Laboratory
for High Energy (JINR), Dubna, Russia T. D. S. Stanislaus Valparaiso
University, Valparaiso, Indiana 46383, USA D. Staszak University of
California, Los Angeles, California 90095, USA M. Strikhanov Moscow
Engineering Physics Institute, Moscow Russia B. Stringfellow Purdue
University, West Lafayette, Indiana 47907, USA A. A. P. Suaide Universidade
de Sao Paulo, Sao Paulo, Brazil M. C. Suarez University of Illinois at
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China, Hefei 230026, China Z. Sun Institute of Modern Physics, Lanzhou,
China B. Surrow Massachusetts Institute of Technology, Cambridge, MA
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Berkeley, California 94720, USA A. Szanto de Toledo Universidade de Sao
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230026, China L. H. Tarini Wayne State University, Detroit, Michigan 48201,
USA T. Tarnowsky Michigan State University, East Lansing, Michigan 48824,
USA D. Thein University of Texas, Austin, Texas 78712, USA J. H. Thomas
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA J.
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Timmins Wayne State University, Detroit, Michigan 48201, USA S. Timoshenko
Moscow Engineering Physics Institute, Moscow Russia D. Tlusty Nuclear
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Laboratory for High Energy (JINR), Dubna, Russia T. A. Trainor University of
Washington, Seattle, Washington 98195, USA V. N. Tram Lawrence Berkeley
National Laboratory, Berkeley, California 94720, USA A. L. Trattner
University of California, Berkeley, California 94720, USA S. Trentalange
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National Laboratory, Upton, New York 11973, USA D. G. Underwood Argonne
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Utrecht University, Amsterdam, The Netherlands A. M. Vander Molen Michigan
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Campinas, Sao Paulo, Brazil I. M. Vasilevski Particle Physics Laboratory
(JINR), Dubna, Russia A. N. Vasiliev Institute of High Energy Physics,
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Y. P. Viyogi Institute of Physics, Bhubaneswar 751005, India S. Vokal
Laboratory for High Energy (JINR), Dubna, Russia S. A. Voloshin Wayne State
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Austin, Texas 78712, USA M. Walker Massachusetts Institute of Technology,
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USA G. D. Westfall Michigan State University, East Lansing, Michigan 48824,
USA C. Whitten Jr University of California, Los Angeles, California 90095,
USA H. Wieman Lawrence Berkeley National Laboratory, Berkeley, California
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Berkeley National Laboratory, Berkeley, California 94720, USA Q. H. Xu
Shandong University, Jinan, Shandong 250100, China Y. Xu University of
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Lanzhou, China P. Yepes Rice University, Houston, Texas 77251, USA K. Yip
Brookhaven National Laboratory, Upton, New York 11973, USA I-K. Yoo Pusan
National University, Pusan, Republic of Korea Q. Yue Tsinghua University,
Beijing 100084, China M. Zawisza Warsaw University of Technology, Warsaw,
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Zhan Institute of Modern Physics, Lanzhou, China S. Zhang Shanghai
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of Science & Technology of China, Hefei 230026, China Y. Zhao University of
Science & Technology of China, Hefei 230026, China C. Zhong Shanghai
Institute of Applied Physics, Shanghai 201800, China J. Zhou Rice
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Laboratory (JINR), Dubna, Russia Y. Zoulkarneeva Particle Physics Laboratory
(JINR), Dubna, Russia J. X. Zuo Shanghai Institute of Applied Physics,
Shanghai 201800, China
###### Abstract
The STAR collaboration at RHIC presents measurements of $J/\psi$
$\rightarrow{e^{+}e^{-}}$ at mid-rapidity and high transverse momentum
($p_{T}>5$ GeV/$c$) in $p$+$p$ and central Cu+Cu collisions at
$\sqrt{s_{{}_{\mathrm{NN}}}}$ = 200 GeV. The inclusive $J/\psi$ production
cross section for Cu+Cu collisions is found to be consistent at high $p_{T}$
with the binary collision-scaled cross section for $p$+$p$ collisions, in
contrast to previous measurements at lower $p_{T}$, where a suppression of
$J/\psi$ production is observed relative to the expectation from binary
scaling. Azimuthal correlations of $J/\psi$ with charged hadrons in $p$+$p$
collisions provide an estimate of the contribution of $B$-meson decays to
$J/\psi$ production of $13\%\pm 5\%$.
###### pacs:
12.38.Mh, 14.40.Gx, 25.75.Dw, 25.75.Nq
Suppression of the $c\bar{c}$ bound state $J/\psi$ meson production in
relativistic heavy-ion collisions arising from $J/\psi$ dissociation due to
screening of the $c\bar{c}$ binding potential in the deconfined medium has
been proposed as a signature of Quark-Gluon Plasma (QGP) formation Matsui and
Satz (1986). Measurements at $\sqrt{s_{{}_{\mathrm{NN}}}}$ $=17.3$ GeV at the
CERN-SPS observed a strong suppression of $J/\psi$ production in heavy-ion
collisions Abreu et al. (2001), although the magnitude of the suppression
decreases with increasing $J/\psi$ $p_{T}$. This systematic dependence may be
explained by initial state scattering (Cronin effect Zhao and Rapp (2007)), as
well as the combined effects of finite $J/\psi$ formation time and the finite
space-time extent of the hot, dense volume where the dissociation can occur
Karsch and Petronzio (1988).
At higher beam energy ($\sqrt{s_{{}_{\mathrm{NN}}}}$ $=200$ GeV), the PHENIX
collaboration at RHIC has measured $J/\psi$ suppression for $p_{T}<5$ GeV/$c$
in central (small impact parameter) Au+Au and Cu+Cu collisions Adare et al.
(2007a) that is similar in magnitude to that observed at the CERN-SPS. This
similarity is surprising in light of the expectation that the energy density
is significantly higher at larger collision energy. It may be due to the
counterbalancing of larger dissociation with recombination of unassociated $c$
and $\bar{c}$ in the medium, which are more abundant at higher energy Braun-
Munzinger and Stachel (2000); Grandchamp and Rapp (2001); Gorenstein et al.
(2002); Thews et al. (2001); Frawley et al. (2008).
Measurements of open heavy-flavor production may also shed light on $J/\psi$
suppression mechanisms. Non-photonic electrons from the semi-leptonic decay of
heavy flavor mesons are found to be strongly suppressed in heavy-ion relative
to p+p collisions at RHIC Abelev et al. (2007); Adare et al. (2007b), an
effect that has been attributed to partonic energy loss in dense matter
Dokshitzer and Kharzeev (2001). This process may also contribute to
high-$p_{T}$ $J/\psi$ suppression, if $J/\psi$ formation proceeds through a
channel carrying color.
The medium generated in RHIC heavy-ion collisions is thought to be strongly
coupled Adams et al. (2005a), making accurate QCD calculations of quarkonium
propagation difficult. The AdS/CFT duality for QCD-like theories may provide
insight into heavy fermion pair propagation in a strongly coupled liquid. One
such calculation predicts that the dissociation temperature decreases with
increasing $J/\psi$ $p_{T}$ (or velocity) Liu et al. (2007). The temperature
achieved at RHIC ($\sim 1.5$ Tc) Adams et al. (2005a) is below this
dissociation temperature at low $J/\psi$ $p_{T}$, and above it at
$p_{T}\gtrsim 5$ GeV/$c$. Consequently, $J/\psi$ production is predicted to be
more suppressed at high $p_{T}$, in contrast to the standard suppression
mechanism. This prediction can be tested with measurements of $J/\psi$ over a
broad kinematic range, in both $p$+$p$ and nuclear collisions.
The interpretation of $J/\psi$ suppression observed at the SPS and by the
PHENIX collaboration requires understanding of the quarkonium production
mechanism in hadronic collisions, which include direct production via gluon
fusion and color-octet (CO) and color-singlet (CS) transitions, as described
by Non-Relativistic Quantum ChromoDynamics (NRQCD) Bodwin et al. (1995);
parton fragmentation; and feeddown from higher charmonium states ($\chi_{c}$,
$\psi(2S)$) and $B$ meson decays. No model at present fully explains the
$J/\psi$ systematics observed in elementary collisions Brambilla et al.
(2004). $J/\psi$ measurements at high-$p_{T}$ both in $p$+$p$ and nuclear
collisions may provide additional insights into the basic processes underlying
quarkonium production.
This letter reports new measurements by the STAR collaboration at RHIC of
$J/\psi$ production at high transverse momentum in $p$+$p$ and Cu+Cu
collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 200 GeV Ackermann et al. (2003).
The inclusive cross section and semi-inclusive $J/\psi$-hadron correlations
are presented.
The Cu+Cu data are from the RHIC 2005 run, while the $p$+$p$ data are from
2005 and 2006. The online trigger, utilizing the STAR Barrel Electromagnetic
Calorimeter (BEMC) Beddo et al. (2003) as well as other trigger detectors,
required one BEMC tower with an energy deposition above a given threshold in
coincidence with a minimum bias (MB) collision trigger Abelev et al. (2008a).
The online trigger threshold, MB trigger condition, and sampled integrated
luminosity for each dataset are listed in Tab. 1. In Cu+Cu data, the most
central 0-20% and 0-60% of the total hadronic cross section were selected as
in Abelev et al. (2008a, b).
In this analysis, $J/\psi$ $\rightarrow{e^{+}e^{-}}$ (Branching Ratio
(B)=5.9%) was reconstructed using the STAR Time Projection Chamber (TPC)
Anderson et al. (2003) and BEMC, with acceptance $|\eta|<1$ and full azimuthal
coverage. Hadron rejection was achieved through the combination of BEMC shower
energy, shower shape measured in the embedded Shower-Maximum Detector (SMD),
and ionization loss ($dE/dx$) in the TPC Abelev et al. (2007); Xu et al.
(2008). Electron purity is $>70\%$ with high efficiency. At moderate $p_{T}$,
the TPC alone can measure electrons with efficiency $>90\%$ and sufficient
hadron rejection ($\sim 10^{3}$) Abelev et al. (2007); Adams et al. (2005b).
Table 1: Trigger conditions, off-line cuts and $J/\psi$ signal statistics. $E_{T}$ is the BEMC trigger threshold. $p_{T1}$ and $p_{T2}$ are the lower bounds for the two electron candidates. BBC (ZDC) means the coincidence of Beam Beam Counters (Zero Degree Calorimeters). S/B is the ratio of signal to background. | $p$+$p$ (2005) | $p$+$p$ (2006) | Cu+Cu
---|---|---|---
MB trigger | BBC | BBC | ZDC
$E_{T}$ (GeV) | $>$ 3.5 | $>$ 5.4 | $>$ 3.75
Sampled int. lumi | 2.8 $pb^{-1}$ | 11.3 $pb^{-1}$ | 860 $\mu b^{-1}$
$p_{T1}$ (GeV/$c$) | $>$ 2.5 | $>$ 4.0 | $>$ 3.5
$p_{T2}$ (GeV/$c$) | $>$ 1.2 | $>$ 1.2 | $>$ 1.5
$J/\psi$ $p_{T}$ (GeV/c) | 5-8 | 5-14 | 5-8
$J/\psi$ counts | 32 $\pm$ 6 | 51 $\pm$ 10 | 23 $\pm$ 8
S/B | 9:1 | 2:1 | 1:4
Figure 1: (Color online.) Left: invariant dielectron mass distribution in (a)
$p$+$p$ and (b) Cu+Cu collisions, for opposite sign (solid red) and same sign
pairs (grey band) from data, and simulated $J/\psi$ peak for $p$+$p$ (dashed).
Right: $J/\psi$ $p_{T}$ distributions in $p$+$p$ and Cu+Cu collisions at
$\sqrt{s_{{}_{\mathrm{NN}}}}$ = 200 GeV. Horizontal brackets show bin limits.
Also shown are perturbative calculations for LO CS+CO (solid line) and NNLO*
CS (band) direct yields, without feeddown contributions.
Figure 1 shows di-electron invariant mass distributions for (a) $p$+$p$ and
(b) Cu+Cu collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 200 GeV. The like-sign
distribution measures random pair background from Dalitz decays and photon
conversions. The $J/\psi$ mass window is $2.7<M_{inv}^{ee}<3.2$ GeV/$c^{2}$.
Other correlated $e^{+}e^{-}$ background is estimated to be $<10\%$ Adare et
al. (2007c); Abe et al. (1997); Acosta et al. (2005). Table 1 lists the
offline cuts and $J/\psi$ signal statistics. Different thresholds were used
for the two electron candidates, corresponding to different online trigger
thresholds.
The $J/\psi$ detection efficiency was calculated by two complementary methods.
The first method was to determine the electron trigger efficiency by comparing
triggered electron yield to the measured inclusive electron spectrum Abelev et
al. (2007). The non-triggered electron efficiency depends only on the TPC
tracking efficiency, which was determined by embedding simulated electron
tracks into real events Abelev et al. (2008a), and $dE/dx$ efficiencies,
determined from the distributions in real data Xu et al. (2008). The second
method was to simulate $J/\psi$ events in PYTHIA Sjostrand et al. (2006),
embed them into real events, and reconstruct the hybrid event to determine the
$J/\psi$ trigger and detection efficiencies. The difference in estimated
efficiency between the two methods is $<10\%$ for all datasets and is included
into the systematic uncertainties of the inclusive spectra. This systematic
uncertainty is correlated in $p$+$p$ and Cu+Cu. A log-likelihood method is
used to correct the $J/\psi$ efficiency and calculate the yields Tang (2009).
Figure 1 (c) shows the measured $J/\psi\rightarrow{e^{+}e^{-}}$ $p_{T}$
spectra. The systematic uncertainties are dominated by kinematic cuts, trigger
efficiency (9%) and reconstruction efficiency (8%), and are similar and
correlated in $p$+$p$ and Cu+Cu. The normalization uncertainty for the
inclusive non-singly diffractive $p$+$p$ cross section is 14% Adams et al.
(2003). Theoretical calculations shown in the figure are NRQCD from CO and CS
transitions for direct $J/\psi$’s in $p+p$ collisions Nayak et al. (2003)
(solid line) and NNLO⋆ CS result Artoisenet et al. (2008) (gray band). Neither
calculation includes feeddown contributions. The band for NNLO⋆ gives the
uncertainty due to scale parameters and the charm quark mass. The CS+CO
calculation describes the data well and leaves little room for feeddown from
$\psi^{\prime}$, $\chi_{c}$ and $B$, estimated to be a factor of $\sim 1.5$.
NNLO⋆ CS predicts a steeper $p_{T}$ dependence.
Figure 2: $x_{T}$ distributions of pions and protons Banner et al. (1982);
Adams et al. (2006, 2005c); Alper et al. (1975); Antreasyan et al. (1979) and
$J/\psi$ (CDF Acosta et al. (2005); Abe et al. (1997), UA1 Albajar et al.
(1991), PHENIX Adare et al. (2007c), and ISR Kourkoumelis et al. (1980)).
Proton and pion inclusive production cross sections in high energy $p$+$p$
collisions have been found to follow $x_{T}$ scaling Clark et al. (1978);
Angelis et al. (1978); Adler et al. (2004):
$E\frac{d^{3}\sigma}{dp^{3}}=g(x_{T})/s^{n/2}$, where $x_{T}=2p_{T}/\sqrt{s}$.
In the parton model, $n$ reflects the number of constituents taking an active
role in hadron production. Figure 2 shows the $x_{T}$ distributions of this
data and previous $J/\psi$, pion and proton data, from $p$+$p$ collisions. The
$J/\psi$ data Acosta et al. (2005); Abe et al. (1997); Albajar et al. (1991);
Adare et al. (2007c); Kourkoumelis et al. (1980) cover the range $\sqrt{s}$
=30 GeV to $\sqrt{s}$ =1.96 TeV. The $J/\psi$ exhibits $x_{T}$ scaling
($n=5.6\pm 0.2$) at high $p_{T}$, similar to the trend for pions and protons
($n=6.6\pm 0.1$) Adams et al. (2006, 2005c). While low $p_{T}$ $J/\psi$
production originates in a hard process due to the mass scale, subsequent soft
processes could cause violation of $x_{T}$ scaling. At high $p_{T}$, the power
parameter $n=5.6\pm 0.2$ is closer to the predictions from CO and Color-
Evaporation production ($n\simeq 6$) Nayak et al. (2003); Bedjidian et al.
(2004) and much smaller than that from next-to-next-to leading order (NNLO*)
CS production ($n\simeq 8$) Artoisenet et al. (2008). This is also evident
from Fig. 1 (c).
Figure 3: (Color online). $J/\psi$ $R_{AA}$ vs. $p_{T}$. STAR data points
have statistical (bars) and systematic (caps) uncertainties. The box about
unity on the left shows $R_{AA}$ normalization uncertainty, which is the
quadrature sum of p+p normalization and binary collision scaling
uncertainties. The solid line and band show the average and uncertainty of the
two 0-20% data points. The curves are model calculations described in the
text. The uncertainty band of 10% for the dotted curve is not shown.
The nuclear modification factor $R_{AA}(p_{T}$) Adler et al. (2002), defined
as the ratio of the inclusive hadron yield in nuclear collisions to that in
$p$+$p$ collisions scaled by the underlying number of binary nucleon-nucleon
collisions, measures medium-induced effects on inclusive particle production.
In the absence of such effects, $R_{AA}$ is unity for hard processes.
Figure 3 shows $R_{AA}$ for $J/\psi$ vs $p_{T}$, in 0-20% Cu+Cu collisions
from PHENIX Adare et al. (2008) and STAR, and 0-60% Cu+Cu from STAR. Cu+Cu and
$p$+$p$ data with $p_{T}>5$ GeV/$c$ are from STAR. The $R_{AA}$ systematic
uncertainty takes into account the correlated efficiencies of the Cu+Cu and
$p$+$p$ datasets. $R_{AA}$ for $J/\psi$ is seen to increase with increasing
$p_{T}$. The average of the two STAR 0-20% data points at high-$p_{T}$ is
$R_{AA}=1.4\pm 0.4~{}(stat.)\pm 0.2~{}(syst.)$. Utilizing the STAR Cu+Cu and
$p$+$p$ data reported here and PHENIX Cu+Cu data at high-$p_{T}$ Adare et al.
(2008) gives $R_{AA}=1.1\pm 0.3~{}(stat.)\pm 0.2~{}(syst.)$ for $p_{T}>5$
GeV/$c$. Both results are consistent with unity and differ by two standard
deviations from a PHENIX measurement at lower $p_{T}$ ($R_{AA}=0.52\pm 0.05$
Adare et al. (2008)). A notable conclusion from these data is that $J/\psi$ is
the only hadron measured in RHIC heavy-ion collisions that does not exhibit
significant high $p_{T}$ suppression. However, for the $J/\psi$ population
reported here, the initial scattered partons have average momentum fraction
$\sim 0.1$ (see also Fig. 2), where initial state effects such as anti-
shadowing may lead to increasing $R_{AA}$ with increasing $p_{T}$.
The dashed curve in Fig. 3 shows the prediction of an AdS/CFT-based
calculation, in which the $J/\psi$ is embedded in a hydrodynamic model Gunji
(2008) and the $J/\psi$ dissociation temperature decreases with increasing
velocity according to Liu et al. (2007). Its $p_{T}$ dependence is at variance
with that of the data. The dotted line shows the prediction of a two-component
model including color screening, hadronic phase dissociation, statistical
$c\bar{c}$ coalescence at the hadronization transition, $J/\psi$ formation
time effects, and $B$-meson feeddown Zhao and Rapp (2007). This calculation
describes the overall trend of the data.
The other calculations in Fig. 3 provide a comparison to open charm $R_{AA}$.
The solid line is based on the WHDG model for charm quark energy loss, with
assumed medium gluon density $dN_{g}/dy=254$ for 0-20% Cu+Cu Wicks et al.
(2007). The dash-dotted line shows a GLV model calculation for D-meson energy
loss, with $dN_{g}/dy=275$ Adil and Vitev (2007). Both models, which correctly
describe heavy-flavor suppression in Au+Au collisions, predict charm meson
suppression of a factor $\sim 2$ at $p_{T}>5$ GeV/$c$. This is in contrast to
the $J/\psi$ $R_{AA}$. This comparison suggests that high-$p_{T}$ $J/\psi$
production does not proceed dominantly via a channel carrying color. However,
other effects Zhao and Rapp (2007); Xu (2002) may compensate for the predicted
loss in this $p_{T}$ range.
Figure 4: (Color online). $J/\psi$-hadron azimuthal correlations. Lines show
PYTHIA calculation of prompt (dashed) and $B$-meson (dot-dashed) feeddown
contributions, and their sum (solid).
Figure 4 shows the azimuthal correlation between high-$p_{T}$ $J/\psi$
($p_{T}>5$ GeV/$c$) and charged hadrons with $p_{T}>0.5$ GeV/$c$ in 200 GeV
p+p collisions. The $J/\psi$ mass window is narrowed to 2.9-3.2 GeV/$c^{2}$ to
increase the S/B ratio. There is no significant correlated yield in the near-
side ($\Delta\phi\sim 0$), in contrast to dihadron correlation measurements
Adams et al. (2005d). The lines show the result of a PYTHIA calculation
Sjostrand et al. (2006), which exhibits a near-side correlation due dominantly
to $B\rightarrow J/\psi+X$. A $\chi^{2}$ fit to the data of the summed
distribution (directly produced $J/\psi$ , feeddown from $\chi_{c}$,
$\psi(2S)$ and $B$-meson) gives a contribution from $B$-meson feeddown to
inclusive $J/\psi$ production of $13\%\pm 5\%$ at $p_{T}>5$ GeV/$c$.
In summary, we report new measurements of $J/\psi$ production in $\sqrt{s}$
=200 GeV $p$+$p$ and Cu+Cu collisions at high $p_{T}$ ($p_{T}>5$ GeV/$c$) at
RHIC. The $J/\psi$ inclusive cross section was found to obey $x_{T}$ scaling
for $p_{T}$ $\gtrsim 5$ GeV/c, in contrast to lower $p_{T}$ $J/\psi$
production. The $J/\psi$ nuclear modification factor $R_{AA}$ in Cu+Cu
increases from low to high $p_{T}$ and is consistent with no $J/\psi$
suppression for $p_{T}$ $>$5 GeV/c, in contrast to the prediction from a
theoretical model of quarkonium dissociation in a strongly coupled liquid
using an AdS/CFT approach. The two-component model with finite $J/\psi$
formation time describes the increasing trend of the $J/\psi$ $R_{AA}$. Based
on the measurement of azimuthal correlations and the comparison to model
calculations, we estimate the fraction of $J/\psi$ from $B$-meson decay to be
$13\pm 5\%$ at $p_{T}>5$ GeV/$c$.
The authors thank G.C. Nayak, J.P. Lansberg, W.A. Horowitz and I. Vitev for
providing calculations and discussion. We thank the RHIC Operations Group and
RCF at BNL, and the NERSC Center at LBNL and the resources provided by the
Open Science Grid consortium for their support. This work was supported in
part by the Offices of NP and HEP within the U.S. DOE Office of Science, the
U.S. NSF, the Sloan Foundation, the DFG Excellence Cluster EXC153 of Germany,
CNRS/IN2P3, RA, RPL, and EMN of France, STFC and EPSRC of the United Kingdom,
FAPESP of Brazil, the Russian Ministry of Sci. and Tech., the NNSFC, CAS,
MoST, and MoE of China, IRP and GA of the Czech Republic, FOM of the
Netherlands, DAE, DST, and CSIR of the Government of India, the Polish State
Committee for Scientific Research, and the Korea Sci. & Eng. Foundation.
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|
arxiv-papers
| 2009-04-02T19:26:22 |
2024-09-04T02:49:01.637604
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "STAR Collaboration: B. I. Abelev, et al",
"submitter": "Zebo Tang",
"url": "https://arxiv.org/abs/0904.0439"
}
|
0904.0538
|
# Cesàro summation for random fields
Allan Gut
Uppsala University Ulrich Stadtmüller
University of Ulm
###### Abstract
Various methods of summation for divergent series of real numbers have been
generalized to analogous results for sums of i.i.d. random variables. The
natural extension of results corresponding to Cesàro summation amounts to
proving almost sure convergence of the Cesàro means. In the present paper we
extend such results as well as weak laws and results on complete convergence
to random fields, more specifically to random variables indexed by
$\mathbb{Z}_{+}^{2}$, the positive two-dimensional integer lattice points.
††footnotetext: AMS 2000 subject classifications. Primary 60F15, 60G50, 60G60,
40G05; Secondary 60F05.
Keywords and phrases. Cesàro summation, sums of i.i.d. random variables,
complete convergence, convergence in probability, almost sure convergence,
strong law of large numbers.
Abbreviated title. Cesàro summation for random fields.
Date.
## 1 Introduction
Various methods of summation for divergent series have been studied in the
literature; see e.g. [10, 21]. Several analogous results have been proved for
sums of independent, identically distributed (i.i.d.) random variables.
The most commonly studied method is _Cesàro_ summation, which is defined as
follows: Let $\\{x_{n},\,n\geq 0\\}$ be a sequence of real numbers and set,
for $\alpha>-1$,
$\displaystyle
A_{n}^{\alpha}=\frac{(\alpha+1)(\alpha+2)\cdots(\alpha+n)}{n!},\quad
n=1,2,\dots,\quad\mbox{ and}\quad A^{\alpha}_{0}=1.$ (1.1)
The sequence $\\{x_{n},\,n\geq 0\\}$ is said to be $(C,\alpha)$-_summable_ iff
$\displaystyle\frac{1}{A_{n}^{\alpha}}\sum_{k=0}^{n}A_{n-k}^{\alpha-1}x_{k}\quad\mbox{
converges as}\quad n\to\infty.$ (1.2)
It is easily checked (with $A_{n}^{-1}=0$ for $n\geq 1$ and $A_{0}^{-1}=1$)
that $(C,0)$-convergence is the same as ordinary convergence, and that
$(C,1)$-convergence is the same as convergence of the arithmetic means.
Now, let $\\{X_{k},\,k\geq 1\\}$ be i.i.d. random variables with partial sums
$\\{S_{n},\,n\geq 1\\}$, and let $X$ be a generic random variable. The
following result is a natural probabilistic analog of (1.2).
###### Theorem 1.1
Let $0<\alpha\leq 1$. The sequence $\\{X_{k},\,k\geq 1\\}$ is _almost surely
(a.s.)_ $(C,\alpha)$-summable iff $E|X|^{1/\alpha}<\infty$. More precisely,
$\frac{1}{A_{n}^{\alpha}}\sum_{k=0}^{n}A_{n-k}^{\alpha-1}X_{k}\stackrel{{\scriptstyle
a.s.}}{{\to}}\mu\quad\mbox{ as}\quad n\to\infty\quad\mbox{
$\Longleftrightarrow$}\quad E|X|^{1/\alpha}<\infty\mbox{ and }E\,X=\mu.$
For $\alpha=1$ this is, of course, the classical Kolmogorov strong law. For
proofs we refer to [14] ($\frac{1}{2}<\alpha<1$), [1] ($0<\alpha<\frac{1}{2}$)
and [2] ($\alpha=\frac{1}{2}$).
Convergence in probability for strongly integrable random variables taking
their values in real separable Banach spaces was establised in [11] under the
assumption of strong integrability. In the real valued case finite mean is not
necessary; for $\alpha=1$ we obtain Feller’s weak law of large numbers for
which a tail condition is both necessary and sufficient; cf. e.g. [8], Section
6.4.1.
Next we present Theorem 2.1 of [7] where complete convergence was obtained.
###### Theorem 1.2
Let $0<\alpha\leq 1$. The sequence $\\{X_{k},\,k\geq 1\\}$ _converges
completely to $\mu$_, i.e.,
$\sum_{n=1}^{\infty}P\big{(}\Big{|}\sum_{k=0}^{n}A_{n-k}^{\alpha-1}X_{k}-\mu\Big{|}>A_{n}^{\alpha}\varepsilon\big{)}<\infty\quad\mbox{
for every}\quad\varepsilon>0\,,$
if and only if
$\begin{cases}E|X|^{1/\alpha}<\infty,&\quad\mbox{ for}\quad
0<\alpha<\frac{1}{2},\\\ E|X|^{2}\log^{+}|X|<\infty,&\quad\mbox{
for}\quad\alpha=\frac{1}{2},\\\ E|X|^{2}<\infty,&\quad\mbox{
for}\quad\frac{1}{2}<\alpha\leq 1,\end{cases}$
and $E\,X=\mu$.
Here and in the following $\log^{+}x=\max\\{\log x,1\\}$.
The aim of the present paper is to generalize these results to random fields.
For simplicity we shall focus on random variables indexed by
$\mathbb{Z}_{+}^{2}$, leaving the corresponding results for the index set
$\mathbb{Z}_{+}^{d}$, $d>2$, to the readers.
The definition of Cesàro summability for arrays extends as follows:
###### Definition 1.1
Let $\alpha,\,\beta>0$. The array $\\{x_{m,n},\,m,n\geq 0\\}$ is said to be
$(C,\alpha,\beta)$-_summable_ iff
$\displaystyle\frac{1}{A_{m}^{\alpha}A_{n}^{\beta}}\sum_{m,n}\,\sum_{k,l=0}^{m,n}A_{n-k}^{\alpha-1}A_{n-l}^{\beta-1}\,x_{k,l}\quad\mbox{
converges as}\quad m,n\to\infty\,.$ (1.3)
Our setup thus is the set $\\{X_{k,l},\,(k,l)\in\mathbb{Z}_{+}^{2}\\}$ with
partial sums $S_{m,n}$, $(m,n)\in\mathbb{Z}_{+}^{2}$. The Kolmogorov and
Marcinkiewicz-Zygmund strong law runs as follows.
###### Theorem 1.3
Let $0<r<2$, and suppose that
$X,\\{X_{\mathbf{k}},\,\mathbf{k}\in\mathbb{Z}^{d}\\}$ are i.i.d. random
variables with partial sums
$S_{\mathbf{n}}=\sum_{\mathbf{k}\leq\mathbf{n}}X_{\mathbf{k}}$,
$\mathbf{n}\in\mathbb{Z}^{d}$. If $E|X|^{r}(\log^{+}|X|^{d-1})<\infty$, and
$E\,X=0$ when $1\leq r<2$, then
$\frac{S_{\mathbf{n}}}{|\mathbf{n}|^{1/r}}\stackrel{{\scriptstyle
a.s.}}{{\to}}0\quad\mbox{ as}\quad\mathbf{n}\to\infty.$
Conversely, if almost sure convergence holds as stated, then
$E|X|^{r}(\log^{+}|X|^{d-1})<\infty$, and $E\,X=0$ when $1\leq r<2$.
Here $|\mathbf{n}|=\prod_{k=1}^{d}n_{i}$ and $\mathbf{n}\to\infty$ means
$\inf_{1\leq k\leq d}n_{i}\to\infty$, that is, all coordinates tend to
infinity. The theorem was proved in [18] for the case $r=1$ and, generally, in
[5].
For the analogous weak laws a finite moment of order $r$ suffices (in fact,
even a little less), since convergence in probability is independent of the
order of the index set.
The central object of investigation in the present paper is
$\displaystyle\frac{1}{A_{m}^{\alpha}A_{n}^{\beta}}\sum_{k,l=0}^{m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}\,,$
(1.4)
for which we shall establish conditions for convergence in probability, almost
sure convergence and complete convergence
Let us already at this point observe that for $\alpha=\beta=1$ the quantity in
(1.4) reduces to that of Theorem 1.3 with $r=1$, that is, to the multiindex
Kolmogorov strong law obtained in [18]. A second thought leads us to
extensions of Theorem 1.3 to the case when we do not normalize the partial
sums with the product of the coordinates raised to some power, but the product
of the coordinates raised to _different_ powers, viz., to, for example
($d=2$),
$\frac{S_{m,n}}{m^{\alpha}n^{\beta}}\,\quad\mbox{ for}\quad 0<\alpha<\beta\leq
1,$
(where thus the case $\alpha=\beta=1/r$ relates to Theorem 1.3). Here we only
mention that some surprises occur depending on the domain of the parameters
$\alpha$ and $\beta$. For details concerning this “asymmetric” Kolmogorov-
Marcinkiewicz-Zygmund extension we refer to [9].
After some preliminaries we present our results for the different modes of
convergence mentioned above. A final appendix contains a collection of so-
called elementary but tedious calculations.
## 2 Preliminaries
Here we collect some facts that will be used on and off, in general without
specific reference.
$\bullet$ The first fact we shall use is that whenever weak forms of
convergence or sums of probabilites are inyvolved we may equivalently compute
sums “backwards”, which, in view of the i.i.d. assumption shows that, for
example
$\displaystyle\sum_{m,n}\,\sum_{k,l=0}^{m,n}P(A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}|X_{k}|>A_{m}^{\alpha}A_{n}^{\beta})<\infty\iff\sum_{m,n}\,\sum_{k,l=1}^{m,n}P(A_{k}^{\alpha-1}A_{l}^{\beta-1}|X|>A_{m}^{\alpha}A_{n}^{\beta})<\infty.$
(2.1)
In the same vein the order of the index set is irrelevant, that is, one-
dimensional results and methods remain valid.
$\bullet$ Secondly we recall from (1.1) that $A^{\alpha}_{0}=1$ and that.
$A_{n}^{\alpha}=\frac{(\alpha+1)(\alpha+2)\cdots(\alpha+n)}{n!},\quad
n=1,2,\dots,$
which behaves asymptotically as
$\displaystyle
A_{n}^{\alpha}\sim\frac{n^{\alpha}}{\Gamma(\alpha+1)}\quad\mbox{ as}\quad
n\to\infty,$ (2.2)
where $\sim$ denotes that the limit as $n\to\infty$ of the ratio between the
members on either side equals 1. Combining the two relations above tells us
that
$\displaystyle\sum_{m,n}\,\sum_{k,l=0}^{m,n}P(A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}|X|>A_{m}^{\alpha}A_{n}^{\beta})<\infty\quad\mbox{
$\Longleftrightarrow$}\quad\sum_{m,n}\,\sum_{k,l=1}^{m,n}P(k^{\alpha-1}l^{\beta-1}|X|>m^{\alpha}n^{\beta})<\infty\,.$
(2.3)
$\bullet$ We shall also make abundant use of the fact that if
$\\{a_{k}\in\mathbb{R}$, $k\geq 1\\}$, then
$\displaystyle a_{n}\to 0\quad\mbox{ as}\quad
n\to\infty\quad\Longrightarrow\quad\frac{1}{n}\sum^{n}_{k=1}a_{k}\to
0\quad\mbox{ as}\quad n\to\infty,$ (2.4)
that if, in addition, $w_{k}\in\mathbb{R}^{+}$, $k\geq 1$, with
$B_{n}=\sum^{n}_{k=1}w_{k}$, $n\geq 1$, where $B_{n}\nearrow\infty$ as
$n\to\infty$, then
$\displaystyle\frac{1}{B_{n}}\sum^{n}_{k=1}w_{k}a_{k}\to 0\quad\mbox{ as}\quad
n\to\infty,$ (2.5)
as well as integral versions of the same.
## 3 Convergence in probability
We thus begin by investigating convergence in probability. We do not aim at
optimal conditions, except that, as will be seen, the weak law does not
require finiteness of the mean (whereas the strong law does so).
###### Theorem 3.1
Let $0<\alpha\leq\beta\leq 1$ and suppose that $\\{X_{k,l},\,k,l\geq 0\\}$ are
i.i.d. random variables. Further, set, for $0\leq k\leq m,\,0\leq l\leq n,$
$Y_{k,l}^{m,n}=A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}I\\{|X_{k,l}|\leq
A_{m}^{\alpha}A_{n}^{\beta}\\},\quad
S_{m,n}^{\prime}=\sum_{k,l=0}^{m,n}Y_{k,l}^{m,n}\quad\mbox{
and}\quad\mu_{m,n}=E\,S_{m,n}^{\prime}.$
Then
$\displaystyle\frac{1}{A_{m}^{\alpha}A_{n}^{\beta}}\Big{(}\sum_{k,l=0}^{m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}-\mu_{m,n}\Big{)}\stackrel{{\scriptstyle
p}}{{\to}}0\quad\mbox{ as}\quad m,n\to\infty$ (3.1)
if
$\displaystyle nP(|X|>n)\to 0\quad\mbox{ as}\quad n\to\infty\,.$ (3.2)
If, in addition,
$\displaystyle\frac{\mu_{m,n}}{A_{m}^{\alpha}A_{n}^{\beta}}\to 0\quad\mbox{
as}\quad m,n\to\infty,$ (3.3)
then
$\displaystyle\frac{1}{A_{m}^{\alpha}A_{n}^{\beta}}\sum_{k,l=0}^{m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}\stackrel{{\scriptstyle
p}}{{\to}}0\quad\mbox{ as}\quad m,n\to\infty\,.$ (3.4)
###### Remark 3.1
_Condition ( 3.2) is short of $E|X|<\infty$, i.e., the theorem extends the
Kolmogorov-Feller weak law [12], [13], and [3], Section VII.7, to a weak law
for weigthed random fields for a class of weights decaying as powers of order
less than 1 in each direction._
###### Corollary 3.1
If, in addition, $E\,X=0$, then (3.4) holds (and if the mean $\mu$ is not
equal to zero the limit in (3.4) equals $\mu$).
###### Corollary 3.2
If, in addition, the distribution of the summands is symmetric, then (3.2)
alone suffices for (3.4) to hold.
Proof of Theorem 3.1. The proof of the theorem amounts to an application of
the so-called degenerate convergence criterion, see e.g. [8], Theorem 6.3.3.
Recalling (2.1) and (2.3) we may, equivalently, prove the theorem for the
respective powers of $k$ and $l$, viz., we redefine the truncated means as
$\displaystyle
Y_{k,l}^{m,n}=k^{\alpha-1}l^{\beta-1}X_{k,l}I\\{k^{\alpha-1}l^{\beta-1}\,|X_{k,l}|\leq
m^{\alpha}n^{\beta}\\},$ (3.5)
with partial sums and means as
$\displaystyle S_{m,n}^{\prime}=\sum_{k,l=1}^{m,n}Y_{k,l}^{m,n}\quad\mbox{
and}\quad\mu_{m,n}=E\,S_{m,n}^{\prime}\,,$ (3.6)
respectively.
In order to check the conditions of the degenerate convergence criterion we
thus wish to show that, if (3.2) is satisfied, then
$\displaystyle\sum_{k,l=1}^{m,n}P(k^{\alpha-1}l^{\beta-1}|X|>m^{\alpha}n^{\beta})\to
0\quad\mbox{ as}\quad m,n\to\infty\,,$ (3.7)
and that
$\displaystyle\frac{1}{m^{\alpha}n^{\beta}}\sum_{k,l=1}^{m,n}\mathrm{Var\,}\big{(}Y_{k,l}^{m,n}\big{)}\to
0\quad\mbox{ as}\quad m,n\to\infty.$ (3.8)
As for (3.7),
$\sum_{k,l=1}^{m,n}P(k^{\alpha-1}l^{\beta-1}|X|>m^{\alpha}n^{\beta})=\frac{1}{m^{\alpha}n^{\beta}}\sum_{k,l=1}^{m,n}k^{\alpha-1}l^{\beta-1}\cdot
m^{\alpha}n^{\beta}k^{1-\alpha}l^{1-\beta}P(k^{\alpha-1}l^{\beta-1}|X|>m^{\alpha}n^{\beta}),$
which converges to 0 as $m,n\to\infty$ via (2.5).
In order to verify (3.8) we apply the usual “slicing device” to obtain
$\displaystyle\hskip-24.0pt\frac{1}{m^{2\alpha}n^{2\beta}}\sum_{k,l=1}^{m,n}\mathrm{Var\,}\big{(}Y_{k,l}^{m,n}\big{)}\leq\frac{1}{m^{2\alpha}n^{2\beta}}\sum_{k,l=1}^{m,n}E\big{(}Y_{k,l}^{m,n}\big{)}^{2}$
$\displaystyle\hskip
24.0pt\leq\frac{1}{m^{2\alpha}n^{2\beta}}\sum_{k,l=1}^{m,n}E\big{(}k^{2(\alpha-1)}l^{2(\beta-1)}X^{2}I\\{k^{\alpha-1}l^{\beta-1}|X|\leq
m^{\alpha}n^{\beta}\\}\big{)}$ $\displaystyle\hskip
24.0pt=\frac{1}{m^{2\alpha}n^{2\beta}}\sum_{k,l=1}^{m,n}k^{2(\alpha-1)}l^{2(\beta-1)}\sum_{j=1}^{mn^{\beta/\alpha}}E\big{(}X^{2}I\\{(j-1)^{\alpha}<k^{\alpha-1}l^{\beta-1}|X|\leq
j^{\alpha}\\}\big{)}$ $\displaystyle\hskip
24.0pt\leq\frac{1}{m^{2\alpha}n^{2\beta}}\sum_{k,l=1}^{m,n}\sum_{j=1}^{mn^{\beta/\alpha}}j^{2\alpha}\,P\big{(}(j-1)^{\alpha}<k^{\alpha-1}l^{\beta-1}|X|\leq
j^{\alpha}\big{)}$ $\displaystyle\hskip
24.0pt\leq\frac{C}{m^{2\alpha}n^{2\beta}}\sum_{k,l=1}^{m,n}\sum_{j=1}^{mn^{\beta/\alpha}}\Big{(}\sum_{i=1}^{j}i^{2\alpha-1}\Big{)}P\big{(}(j-1)^{\alpha}<k^{\alpha-1}l^{\beta-1}|X|\leq
j^{\alpha}\big{)}$ $\displaystyle\hskip
24.0pt\leq\frac{C}{m^{2\alpha}n^{2\beta}}\sum_{k,l=1}^{m,n}\sum_{i=1}^{mn^{\beta/\alpha}}i^{2\alpha-1}\,P(|X|\geq
i^{\alpha}k^{1-\alpha}l^{1-\beta})$ $\displaystyle\hskip
24.0pt=\frac{C}{m^{\alpha}n^{\beta}}\sum_{k,l=1}^{m,n}k^{\alpha-1}l^{\beta-1}\Big{(}\frac{1}{m^{\alpha}n^{\beta}}\sum_{i=1}^{mn^{\beta/\alpha}}i^{\alpha-1}\big{(}i^{\alpha}k^{1-\alpha}l^{1-\beta}\,P(|X|\geq
i^{\alpha}k^{1-\alpha}l^{1-\beta})\big{)}\Big{)},$ $\displaystyle\hskip
24.0pt\to 0\quad\mbox{ as}\quad m,n\to\infty\,,$
by applying (2.5) twice to (3.2). This completes the proof of (3.1), from
which (3.4) is immediate. $\Box$
Proof of Corollary 3.1. In order to conclude that also (3.4) holds we use the
usual method to show that the normalized trruncated means tend to zero, where
w.l.o.g. we assume that $E\,X=0$. Then
$\displaystyle\hskip-48.0pt\Big{|}\frac{1}{m^{\alpha}n^{\beta}}\sum_{k,l=1}^{m,n}E\big{(}k^{(\alpha-1)}l^{(\beta-1)}XI\\{k^{(\alpha-1)}l^{(\beta-1)}|X|\leq
m^{\alpha}n^{\beta}\\}\big{)}\Big{|}$ $\displaystyle=$
$\displaystyle\Big{|}-\frac{1}{m^{\alpha}n^{\beta}}\sum_{k,l=1}^{m,n}E\big{(}k^{(\alpha-1)}l^{(\beta-1)}XI\\{k^{(\alpha-1)}l^{(\beta-1)}|X|>m^{\alpha}n^{\beta}\\}\big{)}\Big{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{m^{\alpha}n^{\beta}}\sum_{k,l=1}^{m,n}E\big{(}k^{(\alpha-1)}l^{(\beta-1)}|X|I\\{k^{(\alpha-1)}l^{(\beta-1)}|X|>m^{\alpha}n^{\beta}\\}\big{)}\to
0\quad\mbox{ as}\quad n,m\to\infty.$
Proof of Corollary 3.2. Immediate, since the truncated means are (also) equal
to zero. $\Box$
## 4 Complete convergence
###### Theorem 4.1
Let $0<\alpha\leq\beta\leq 1$. The field $\\{X_{k,l},\,k,l\geq 0\\}$
_converges completely to $\mu$_, i.e.,
$\sum_{m\,n}P\big{(}\Big{|}\sum_{k,l=0}^{m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}-\mu\Big{|}>A_{m}^{\alpha}A_{n}^{\beta}\varepsilon\big{)}<\infty\quad\mbox{
for every}\quad\varepsilon>0\,,$
if and only if
$\begin{cases}E|X|^{\frac{1}{\alpha}},&\quad\mbox{ for}\quad
0<\alpha<1/2\,,\;\alpha<\beta\leq 1,\\\\[6.0pt]
E|X|^{\frac{1}{\alpha}}\log^{+}|X|,&\quad\mbox{ for}\quad
0<\alpha=\beta<\frac{1}{2},\\\\[6.0pt] E|X|^{2}(\log^{+}|X|)^{3},&\quad\mbox{
for}\quad\alpha=\beta=\frac{1}{2},\\\\[6.0pt]
E|X|^{2}(\log^{+}|X|)^{2},&\quad\mbox{ for}\quad\alpha=\frac{1}{2}<\beta\leq
1,\\\\[6.0pt] E|X|^{2}\log^{+}|X|,&\quad\mbox{
for}\quad\frac{1}{2}<\alpha\leq\beta\leq 1.\end{cases}$
and $E\,X=\mu$.
Proof. For the proof of the sufficiency we refer to the Appendix.
As for the necessity, we argue as in [6], p. 59. We first suppose that the
distribution is symmetric. Now, if complete convergence holds, then, using the
fact that
$\max_{0\leq k,l\leq m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}|X_{k,l}|\leq
2\max_{0\leq\mu,\nu\leq
m,n}\Big{|}\sum_{k,l=0}^{\mu,\nu}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}\Big{|},$
together with the Lévy inequalities we must have, say,
$\sum_{m,n}P\big{(}\max_{0\leq k,l\leq
m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}|X_{k,l}|>A_{m}^{\alpha}A_{n}^{\beta}\big{)}<\infty\,,$
so that, by the first Borel-Cantelli lemma
$P(A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}|X_{k,l}|>A_{m}^{\alpha}A_{n}^{\beta}\quad\mbox{i.o.
for }1\leq k,l\leq m,n\,\,;m,n\geq 1)=0.$
At this point we use a device from [17], p. 379. Namely, if the sums
$\sum_{k,l=1}^{m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}$ were
independent, we would conclude that
$\sum_{m,n}\,\sum_{k,l=1}^{m,n}P(A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}|X|>A_{m}^{\alpha}A_{n}^{\beta})$
were finite. Since, however, finiteness of the sum is only a matter of the
tail probabilities, the sum is also finite in the general case.
An application of (A.6) now tells us that the finiteness of the sum is
equivalent to the moment conditions as given in the statement of the theorem.
This proves the necessity in the symmetric case. The general case follows the
standard desymmetrization procedure, for which we use Theorem 3.1 in order to
take care of the asymptotics for the normalized medians (cf. [5], p. 472 for
analogous details in the multiindex setting of the Marcinkiewicz-Zygmund
strong laws). $\Box$
## 5 Almost sure convergence
###### Theorem 5.1
Let $0<\alpha\leq\beta\leq 1$. The field $\\{X_{k,l},\,k,l\geq 0\\}$ is
_almost surely (a.s.)_ $(C,\alpha,\beta)$-summable, that is,
$\frac{1}{A_{m}^{\alpha}A_{n}^{\beta}}\sum_{k,l=0}^{m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}\stackrel{{\scriptstyle
a.s.}}{{\to}}\mu\quad\mbox{ as}\quad n,m\to\infty$
if and only if
$\begin{cases}E|X|^{\frac{1}{\alpha}},&\quad\mbox{ for}\quad
0<\alpha<\beta\leq 1,\\\\[6.0pt]
E|X|^{\frac{1}{\alpha}}\log^{+}|X|,&\quad\mbox{ for}\quad 0<\alpha=\beta\leq
1.\end{cases}$
and $E\,X=\mu$.
Proof. Since complete convergence always implies almost sure convergence, the
sufficiency follows immediately for the case $\alpha<1/2$.
Thus, let in the following $1/2\leq\alpha\leq\beta\leq 1$. We first consider
the symmetric case (and recall Section 2. In analogy with [11], p. 538, the
moment assumptions permit us to choose an array $\\{\eta_{k,l},\,k,l\geq 1\\}$
of nonincreasing reals in $(0,1)$ converging to 0, and such that
$\sum_{k,l=1}^{\infty}P(|X_{k,l}|>\eta_{k,l}k^{\alpha}l^{\beta})<\infty.$
Defining
$Y_{k,l}=X_{k,l}I\\{|X_{k,l}|\leq\eta_{k,l}k^{\alpha}l^{\beta}\\}\quad\mbox{
and}\quad S_{m,n}^{\prime}=\sum_{k,l=0}^{m,n}Y_{k,l}^{m,n}\,,$
it thus remains to prove the theorem for the truncated sequence.
This will be achieved via the multiindex Kolmogorov convergence criterion (see
e.g [4]) and the multiindex Kronecker lemma (cf. [16]). The first series has
just been taken care of, the second one vanishes since we are in the symmetric
case, so it remains to check the third series.
Toward that end, let, for $k,l\geq 1$,
$Z_{k,l}=\frac{(m-k)^{\alpha-1}(n-l)^{\beta-1}}{m^{\alpha}n^{\beta}}Y_{k,l}\,.$
Then
$\displaystyle|Z_{k,l}|\leq\frac{(m-k)^{\alpha-1}(n-l)^{\beta-1}}{m^{\alpha}n^{\beta}}k^{\alpha}l^{\beta}\eta_{k,l}\leq\eta_{k,l}\leq\eta_{00}.$
(5.1)
Now, for any $\varepsilon>0$, arbitrarily small, we may choose our
$\eta$-sequence such that $\eta_{00}<\varepsilon$, so that an application of
the (iterated) Kahane-Hoffman-Jørgensen inequality (cf. [8], Theorem 3.7.5)
yields
$\displaystyle
P\Big{(}\Big{|}\sum_{k,l=0}^{m,n}Z_{k,l}\Big{|}>3^{j}\varepsilon\Big{)}$
$\displaystyle\leq$ $\displaystyle
C_{j}\bigg{(}P\Big{(}\Big{|}\sum_{k,l=0}^{m,n}Z_{k,l}\Big{|}>\varepsilon\Big{)}\bigg{)}^{2^{j}}$
$\displaystyle\leq$ $\displaystyle
C_{j}\bigg{(}\frac{\sum_{k,l=0}^{m,n}\big{(}(m-k)^{(\alpha-1)}(n-l)^{\beta-1}\big{)}^{1/\alpha}E|X|^{1/\alpha}}{\big{(}\varepsilon
m^{\alpha}n^{\beta}\big{)}^{1/\alpha}}\bigg{)}^{2^{j}}$ $\displaystyle=$
$\displaystyle
C_{j}^{\prime}\bigg{(}\frac{\sum_{k,l=0}^{m,n}k^{(1-1/\alpha)}l^{(\beta-1)/\alpha}}{mn^{\beta/\alpha}}\bigg{)}^{2^{j}}$
$\displaystyle=$
$\displaystyle\begin{cases}C_{j}^{\prime\prime}\Big{(}\frac{1}{(mn)^{\frac{1}{\alpha}-1}}\Big{)}^{2^{j}},&\quad\mbox{
for}\quad\frac{1}{2}<\alpha<\beta<1,\\\\[6.0pt]
C_{j}^{\prime\prime}\Big{(}\frac{\log m}{nm}\Big{)}^{2^{j}},&\quad\mbox{
for}\quad\frac{1}{2}=\alpha<\beta<1,\\\\[6.0pt]
C_{j}^{\prime\prime}\Big{(}\frac{\log m\log n}{nm}\Big{)}^{2^{j}},&\quad\mbox{
for}\quad\frac{1}{2}=\alpha=\beta,\end{cases}$
(since the usual first term in the RHS vanishes in view of (5.1)).
By choosing $j$ sufficiently large it then follows that
$\sum_{m,n}\,\sum_{k,l=0}^{m,n}P\Big{(}\Big{|}\sum_{k,l=0}^{m,n}Z_{k,l}\Big{|}>3^{j}\varepsilon\Big{)}<\infty.$
By replacing $3^{j}\varepsilon$ by $\varepsilon$ we have thus, due to the
arbitrariness of $\varepsilon$, shown that
$\displaystyle P\big{(}|Z_{k,l}|>\varepsilon\mbox{ i.o.}\big{)}=0\quad\mbox{
for any}\quad\varepsilon>0,$ (5.2)
from which the desired almost sure convergence follows via the multiindex
Kronecker lemma referred to above.
This proves the sufficiency in the symmetric case from which the general case
follows by the standard desymmetrization procedure hinted at in the proof of
Theorem 4.1.
Finally, suppose that almost sure convergence holds as stated. It then follows
that
$\frac{A_{0}^{\alpha-1}A_{0}^{\beta-1}X_{m,n}}{A_{m}^{\alpha}A_{n}^{\beta}}\stackrel{{\scriptstyle
a.s.}}{{\to}}0\quad\mbox{ as}\quad m,n\to\infty,$
and, hence, also that
$\frac{X_{m,n}}{m^{\alpha}n^{\beta}}\stackrel{{\scriptstyle
a.s.}}{{\to}}0\quad\mbox{ as}\quad m,n\to\infty,$
which, in view of i.i.d. assumption and the second Borel-Cantelli lemma, tells
us that
$\sum_{m,n}P(|X|>m^{\alpha}n^{\beta})<\infty,$
which, in turn, is equivalent to the given moment conditions.
This concludes the proof of the theorem. $\Box$
## 6 Concluding remarks
We close with some comments on the present and related work.
* •
Convergence in probability has earlier been established in [11] via
approximation with indicator variables, and under the assumption of finite
mean. Our proof is simpler (more elementary) and presupposes only a Feller
condition.
* •
As pointed out above, almost sure convergence was established in three steps
([14], [1] and [2]) with different proofs. Our proof, which also works for the
case $d=1$, takes care of the whole proof in one go (since our proof also
works for the case $\alpha<1/2$).
* •
For simplicity we have confined ourselves to the case $d=2$. The same ideas
can be modified for the case $d>2$ and
$(C,\alpha_{1},\alpha_{2},\ldots,\alpha_{d})$-summability. However, the moment
conditions then depend on the number of $\alpha$:s that are equal to the
smallest one (corresponding to $\alpha<\beta$ or $\alpha=\beta$ in the present
paper); see [9] for Kolmogorov-Marcinkiewicz-Zygmund laws.
* •
Results on complete convergence are special cases of results on convergence
rates. In this vein our results are extendable to results concerning
$\sum_{m,n}n^{r-2}m^{r-2}P\big{(}\Big{|}\sum_{k,l=0}^{m,n}A_{m-k}^{\alpha-1}A_{n-l}^{\beta-1}X_{k,l}-\mu\Big{|}>A_{m}^{\alpha}A_{n}^{\beta}\varepsilon\big{)}<\infty\quad\mbox{
for every}\quad\varepsilon>0$
(cf. [7] for the case $d=1$). For the proofs one would need i.a. extensions of
the relevant computations in the appendix below.
## Appendix A Appendix
In this appendix we collect a number of so-called elementary but tedious
calculations.
First, let $0<\alpha\leq\beta<1$. Then
$\displaystyle\sum_{m,n}\,\sum_{k,l=1}^{m,n}P(k^{\alpha-1}l^{\beta-1}|X|>m^{\alpha}n^{\beta})<\infty\quad\mbox{
$\Longleftrightarrow$}\quad$
$\displaystyle\int_{1}^{\infty}\int_{1}^{\infty}\int_{1}^{x}\int_{1}^{y}P(|X|>u^{1-\alpha}v^{1-\beta}x^{\alpha}y^{\beta})\,dudvdxdy<\infty\quad\mbox{
$\Longleftrightarrow$}\quad$ $\displaystyle\hskip
56.9055pt\Big{[}u^{1-\alpha}x^{\alpha}=z,\qquad v^{1-\beta}y^{\beta}=w\Big{]}$
$\displaystyle\int_{1}^{\infty}\int_{1}^{\infty}\int_{x^{\alpha}}^{x}\int_{y^{\beta}}^{y}\Big{(}\frac{z}{x}\Big{)}^{\frac{\alpha}{1-\alpha}}\Big{(}\frac{w}{y}\Big{)}^{\frac{\beta}{1-\beta}}P(|X|>zw)\,dzdwdxdy<\infty\quad\mbox{
$\Longleftrightarrow$}\quad$
$\displaystyle\int_{1}^{\infty}\int_{1}^{\infty}\bigg{(}\int_{z}^{z^{1/\alpha}}\frac{dx}{x^{\frac{\alpha}{1-\alpha}}}\bigg{)}\bigg{(}\int_{w}^{w^{1/\beta}}\frac{dy}{y^{\frac{\beta}{1-\beta}}}\bigg{)}z^{\frac{\alpha}{1-\alpha}}w^{\frac{\beta}{1-\beta}}P(|X|>zw)\,dzdw<\infty\,.$
(A.1)
In case $0<\alpha<\beta=1$ we have
$\displaystyle\sum_{m,n}\,\sum_{k,l=1}^{m,n}P(k^{\alpha-1}|X|>m^{\alpha}n)<\infty\quad\mbox{
$\Longleftrightarrow$}\quad$
$\displaystyle\int_{1}^{\infty}\int_{1}^{\infty}\bigg{(}\int_{z}^{z^{1/\alpha}}\frac{dx}{x^{\frac{\alpha}{1-\alpha}}}\bigg{)}z^{\frac{\alpha}{1-\alpha}}\,w\,P(|X|>zw)\,dzdw<\infty\,.$
(A.2)
Next we note that
$\displaystyle\int_{y}^{y^{1/\gamma}}\frac{dx}{x^{\frac{\gamma}{1-\gamma}}}\sim
C\,\begin{cases}y^{\frac{1-2\gamma}{\gamma(1-\gamma)}},&\quad\mbox{ for}\quad
0<\gamma<\frac{1}{2},\\\ \log y,&\quad\mbox{ for}\quad\gamma=\frac{1}{2},\\\
y^{\frac{1-2\gamma}{1-\gamma}},&\quad\mbox{
for}\quad\frac{1}{2}<\gamma<1,\end{cases}$ (A.3)
so that
$\displaystyle\hskip-24.0pt\bigg{(}\int_{z}^{z^{1/\alpha}}\frac{dx}{x^{\frac{\alpha}{1-\alpha}}}\bigg{)}\bigg{(}\int_{w}^{w^{1/\beta}}\frac{dy}{y^{\frac{\beta}{1-\beta}}}\bigg{)}z^{\frac{\alpha}{1-\alpha}}w^{\frac{\beta}{1-\beta}}$
$\displaystyle\hskip 24.0pt\sim
C\,\begin{cases}z^{\frac{1-\alpha}{\alpha}}w^{\frac{1-\beta}{\beta}},&\quad\mbox{
for}\quad 0<\alpha,\beta<\frac{1}{2},\\\
(zw)^{\frac{1-\alpha}{\alpha}},&\quad\mbox{ for}\quad
0<\alpha=\beta<\frac{1}{2},\\\ zw\log z\log w=\frac{zw}{2}\big{(}(\log
zw)^{2}&\\\ \hskip 24.0pt-(\log z)^{2}-(\log w)^{2}\big{)},&\quad\mbox{
for}\quad\alpha=\beta=\frac{1}{2},\\\ z^{\frac{1-\alpha}{\alpha}}w\log
w,&\quad\mbox{ for}\quad\alpha<\beta=\frac{1}{2},\\\
z^{\frac{1-\alpha}{\alpha}}w,&\quad\mbox{
for}\quad\alpha<\frac{1}{2}<\beta\leq 1,\\\ zw\log z,&\quad\mbox{
for}\quad\alpha=\frac{1}{2}<\beta\leq 1,\\\ zw,&\quad\mbox{
for}\quad\frac{1}{2}<\alpha\leq\beta\leq 1,\end{cases}$
from which it follows that
$\displaystyle\hskip-24.0pt\int_{1}^{\infty}\int_{1}^{\infty}\bigg{(}\int_{z}^{z^{1/\alpha}}\frac{dx}{x^{\frac{\alpha}{1-\alpha}}}\bigg{)}\bigg{(}\int_{w}^{w^{1/\beta}}\frac{dy}{y^{\frac{\beta}{1-\beta}}}\bigg{)}z^{\frac{\alpha}{1-\alpha}}x^{\frac{\beta}{1-\beta}}P(|X|>zw)\,dzdw$
$\displaystyle\hskip 85.35826pt=\Big{[}x=zw,\qquad y=z\Big{]}$
$\displaystyle=\begin{cases}\int_{1}^{\infty}\int_{1}^{x}x^{\frac{1-\beta}{\beta}}y^{\frac{1}{\alpha}-\frac{1}{\beta}-1}P(|X|>x)\,dydx\\\\[4.0pt]
\hskip
24.0pt=C\int_{1}^{\infty}x^{\frac{1}{\alpha}-1}P(|X|>x)\,dx,&\quad\mbox{
for}\quad 0<\alpha<\beta<\frac{1}{2},\\\\[6.0pt]
\int_{1}^{\infty}\int_{1}^{x}x^{\frac{1-\alpha}{\alpha}}\frac{1}{y}P(|X|>x)\,dydx\\\\[4.0pt]
\hskip 24.0pt=C\int_{1}^{\infty}x^{\frac{1-\alpha}{\alpha}}\log
xP(|X|>x)\,dx,&\quad\mbox{ for}\quad 0<\alpha=\beta<\frac{1}{2},\\\\[6.0pt]
\int_{1}^{\infty}\int_{1}^{x}\big{(}\frac{1}{2}x(\log
x)^{2}\frac{1}{y}-x\frac{(\log y)^{2}}{y}\big{)}P(|X|>x)\,dxdy&\\\\[4.0pt]
\hskip 24.0pt=\frac{1}{6}\int_{1}^{\infty}x(\log
x)^{3}P(|X|>x)\,dx,&\quad\mbox{ for}\quad\alpha=\beta=\frac{1}{2},\\\\[6.0pt]
\int_{1}^{\infty}\int_{1}^{x}xy^{\frac{1}{\alpha}-2}(\log x-\log
y)P(|X|>x)\,dydx&\\\\[4.0pt] \hskip
24.0pt=C\int_{1}^{\infty}x^{\frac{1}{\alpha}-1}P(|X|>x)\,dx,&\quad\mbox{
for}\quad\alpha<\beta=\frac{1}{2},\\\\[6.0pt]
\int_{1}^{\infty}\int_{1}^{x}xy^{\frac{1}{\alpha}-2}P(|X|>x)\,dydx\\\\[4.0pt]
\hskip
24.0pt=C\int_{1}^{\infty}x^{\frac{1}{\alpha}-1}P(|X|>x)\,dx,&\quad\mbox{
for}\quad\alpha<\frac{1}{2}<\beta\leq 1,\\\\[6.0pt]
\int_{1}^{\infty}\int_{1}^{x}x\frac{\log y}{y}P(|X|>x)\,dydx\\\\[4.0pt] \hskip
24.0pt=\frac{1}{2}\int_{1}^{\infty}x(\log x)^{2}P(|X|>x)\,dx,&\quad\mbox{
for}\quad\alpha=\frac{1}{2}<\beta\leq 1,\\\\[6.0pt]
\int_{1}^{\infty}\int_{1}^{x}x\frac{1}{y}P(|X|>x)\,dydx\\\\[4.0pt] \hskip
24.0pt=\frac{1}{2}\int_{1}^{\infty}x\log xP(|X|>x)\,dx,&\quad\mbox{
for}\quad\frac{1}{2}<\alpha\leq\beta\leq 1.\end{cases}$ (A.4)
Summarizing this we have shown that, for $0<\alpha\leq\beta<1$,
$\displaystyle\hskip-24.0pt\sum_{m,n}\,\sum_{k,l=1}^{m,n}P(A_{k}^{\alpha-1}A_{l}^{\beta-1}|X|>A_{m}^{\alpha}A_{n}^{\beta})<\infty\quad\mbox{
$\Longleftrightarrow$}\quad$ (A.5) $\displaystyle\hskip
48.0pt\begin{cases}E|X|^{\frac{1}{\alpha}},&\quad\mbox{ for}\quad
0<\alpha<1/2,\,\alpha<\beta\leq 1,\\\\[6.0pt]
E|X|^{\frac{1}{\alpha}}\log^{+}|X|,&\quad\mbox{ for}\quad
0<\alpha=\beta<\frac{1}{2},\\\\[6.0pt] E|X|^{2}(\log^{+}|X|)^{3},&\quad\mbox{
for}\quad\alpha=\beta=\frac{1}{2},\\\\[6.0pt]
E|X|^{2}(\log^{+}|X|)^{2},&\quad\mbox{ for}\quad\alpha=\frac{1}{2}<\beta\leq
1,\\\\[6.0pt] E|X|^{2}\log^{+}|X|,&\quad\mbox{
for}\quad\frac{1}{2}<\alpha\leq\beta\leq 1.\end{cases}$ (A.6)
### Acknowledgement
The work on this paper has been supported by Kungliga Vetenskapssamhället i
Uppsala. Their support is gratefully acknowledged. In addition, the second
author likes to thank his partner Allan Gut for the great hospitality during
two wonderful and stimulating weeks at the University of Uppsala.
## References
* [1] Chow, Y.S. and Lai, T.L. (1973). Limiting behavior of weighted sums of independent random variables. _Ann. Probab._ 1, 810-824.
* [2] Déniel, Y. and Derriennic, Y. (1988). Sur la convergence presque sure, au sens de Cesàro d’ordre $\alpha$, $0<\alpha<1$, de variables aléatoires indépendantes et identiquement distribuées. _Probab. Th. Rel. Fields_ 79, 629-636.
* [3] Feller, W. (1971). _An Introduction to Probability Theory and Its Applications, Vol 2_ , 2nd ed. Wiley, New York.
* [4] Gabriel J.-P. (1977). An inequality for sums of independent random variables indexed by finite dimensional filtering sets and its applications to the convergence of series. _Ann. Probab._ 5, 779-786.
* [5] Gut, A. (1978). Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. _Ann. Probab._ 6, 469-482.
* [6] Gut, A. (1992). Complete convergence for arrays. _Period. Math. Hungar._ 25, 51-75.
* [7] Gut, A. (1993). Complete convergence and Cesàro summation for i.i.d. random variables. _Probab. Th. Rel. Fields_ 97, 169-178.
* [8] Gut, A. (2007). Probability: A Graduate Course, Corr. 2nd printing. Springer-Verlag, New York.
* [9] Gut, A. and Stadtmüller, U. (2008). An asymmetric Marcinkiewicz-Zygmund LLN for random fields. _Report U.U.D.M._ 2008:38, Uppsala University.
* [10] Hardy, G.H. (1949). _Divergent Series._ Oxford University Press.
* [11] Heinkel, B. (1990). An infinite-dimensional law of large numbers in Cesaro’s sense. _J. Theoret. Probab._ 3, 533-546.
* [12] Kolmogorov, A.N. (1928). Über die Summen durch den Zufall bestimmter unabhängiger Größen. _Math. Ann._ 99, 309-319.
* [13] Kolmogorov, A.N. (1930). Bemerkungen zu meiner Arbeit “Über die Summen zufälliger Größen”. _Math. Ann._ 102, 484-488.
* [14] Lorentz G.G. (1955). Borel and Banach properties of methods of summation. _Duke Math. J._ 22, 129-141.
* [15] Marcinkiewicz, J. and Zygmund, A. Sur les fonctions indépendantes. _Fund. Math._ 29, 60-90 (1937).
* [16] Moore, C.N. (1966). _Summable Series and Convergence Factors._ Dover, New York.
* [17] Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. _Z. Wahrsch. verw. Gebiete_ 57, 365-395.
* [18] Smythe, R. (1973). Strong laws of large number for $r$-dimensional arrays of random variables. _Ann. Probab._ 1, 164-170.
* [19] Stadtmüller, U. and Thalmaier, M. (2008). Strong laws for delayed sums of random fields. Preprint, University of Ulm.
* [20] Thalmaier, M. (2008): _Grenzwertsätze für gewichtete Summen von Zufallsvariablen und Zufallsfeldern_. Dissertation, University of Ulm.
* [21] Zygmund, A. (1968). _Trigonometric Series._ Cambridge University Press.
Allan Gut, Department of Mathematics, Uppsala University, Box 480,
SE-751 06 Uppsala, Sweden;
Email: allan.gut@math.uu.se
URL: http://www.math.uu.se/~allan
Ulrich Stadtmüller, Ulm University, Department of Number Theory and
Probability Theory,
D-89069 Ulm, Germany;
Email ulrich.stadtmueller@uni-ulm.de
URL: http://www.mathematik.uni-
ulm.de/matheIII/members/stadtmueller/stadtmueller.html
|
arxiv-papers
| 2009-04-03T09:46:21 |
2024-09-04T02:49:01.651169
|
{
"license": "Public Domain",
"authors": "Allan Gut (Uppsala University), Ulrich Stadtmueller (Ulm University)",
"submitter": "Ulrich Stadtmueller",
"url": "https://arxiv.org/abs/0904.0538"
}
|
0904.0553
|
# Spectral Energy Distributions and Age Estimates of 39 Globular Clusters in
M31
Jun Ma,11affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn Zhou Fan,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn 22affiliation: Graduate University, Chinese Academy
of Sciences, Beijing, 100039, P. R. China Richard de Grijs,33affiliation:
Department of Physics & Astronomy, The University of Sheffield, Hicks
Building, Hounsfield Road, Sheffield S3 7RH, UK 11affiliation: National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, P.
R. China;
majun@vega.bac.pku.edu.cn Zhenyu Wu,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn Xu Zhou,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn Jianghua Wu,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn Zhaoji Jiang,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China;
majun@vega.bac.pku.edu.cn and Jiansheng Chen11affiliation: National
Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, P.
R. China;
majun@vega.bac.pku.edu.cn
###### Abstract
This paper supplements Jiang et al. (2003), who studied 172 M31 globular
clusters (GCs) and globular cluster candidates from Battistini et al. (1987)
on the basis of integrated photometric measurements in the Beijing-Arizona-
Taiwan-Connecticut (BATC) photometric system. Here, we present multicolor
photometric CCD data (in the BATC system) for the remaining 39 M31 GCs and
candidates. In addition, the ages of 35 GCs are constrained by comparing our
accurate photometry with updated theoretical stellar synthesis models. We use
photometric measurements from GALEX in the far- and near-ultraviolet and 2MASS
infrared $JHK_{s}$ data, in combination with optical photometry. Except for
two clusters, the ages of the other sample GCs are all older than 1 Gyr. Their
age distribution shows that most sample clusters are younger than 6 Gyr, with
a peak at $\sim 3$ Gyr, although the ‘usual’ complement of well-known old GCs
(i.e., GCs of similar age as the majority of the Galactic GCs) is present as
well.
###### Subject headings:
galaxies: individual (M31) – galaxies: star clusters – galaxies: stellar
content
††slugcomment: AJ, in press
## 1\. Introduction
The process of galaxy formation and evolution ranks among the most important
outstanding problems in astrophysics (e.g., Perrett et al., 2002). One way to
better understand the underlying questions is by studying globular clusters
(GCs). GCs are often considered the fossils of the galaxy formation process,
since they tend to form in the very early stage of their host galaxy’s
evolution (Barmby et al., 2000). A GC is a densely packed, gravitationally
bound, roughly spherical system of several thousand to about one million
stars. They can be observed out to great distances, which implies that they
can be used to study and probe the properties of extragalactic systems. The
most distant GC systems studied to date are located in the Coma Cluster; their
study has been facilitated by Hubble Space Telescope (HST) Wide Field and
Planetary Camera-2 (WFPC2) observations (Baum et al., 1995; Kavelaars et al.,
2000; Harris et al., 2000; Woodworth & Harris, 2000).
M31 (NGC 224), the Andromeda galaxy, is an early-type spiral galaxy (type Sb),
located at a distance of $\sim 780$ kpc (Stanek & Garnavich, 1998; Macri,
2001). It is the nearest and largest spiral galaxy in the Local Group of
galaxies and has been the subject of many GC studies and surveys. Hubble
(1932) first discovered 140 GC candidates characterized by $m_{\rm pg}\leq 18$
mag. Subsequently, a number of studies (Seyfert & Nassau, 1945; Hiltner, 1958;
Mayall & Eggen, 1953; Kron & Mayall, 1960) identified $\sim 160$ GC candidates
in M31. Vetes̆nik (1962) compiled the first major M31 GC catalog, containing
about 300 GC candidates and including $UBV$ photometric data. The most
extensive GC surveys have since been published by Sargent et al. (1977),
Crampton et al. (1985), and the Bologna group (Battistini et al., 1980, 1987,
1993). In particular, Crampton et al. (1985) and Battistini et al. (1980,
1987, 1993) provided photometric data in either $UBV$ or $UBVR$. These surveys
were mostly based on visual searches of photographic plates and are fairly
complete down to $V=18$ mag ($M_{V}\sim-6.5$ mag) (Fusi Pecci et al., 1993),
although a number of recent studies searched for fainter GCs in M31 (e.g.,
Mochejska et al., 1998; Barmby & Huchra, 2001; Kim et al., 2007). However,
Galleti et al. (2006) showed that a significant number of class-D and E GCs
with $V\gtrsim 17$ are still to be confirmed (and hence the GC luminosity
function is incomplete), and that large surveys are needed to reach a complete
sample of M31 GCs. Following on from the first extensive spectroscopic survey
of M31 GCs by van den Bergh (1969), a significant number of authors (e.g.,
Huchra et al., 1982, 1991; Dubath & Grillmair, 1997; Federici et al., 1993;
Jablonka et al., 1998; Barmby et al., 2000; Perrett et al., 2002; Galleti et
al., 2006; Lee et al., 2008, and references therein) embarked on studies of
their spatial, kinematic, and metal-abundance properties. The first
comprehensive catalog including photometric and spectroscopic data for M31 GCs
was assembled by Barmby et al. (2000). The Revised Bologna Catalog (RBC) of
M31 GCs was recently published by Galleti et al. (2004) and has since been
revised a number of times (Galleti et al., 2005, 2006, 2007). In the primary
catalog (Galleti et al., 2004), all known M31 GCs and candidates were compiled
based on a literature survey, leading to a total of 1164 entries including 337
confirmed GCs, 688 GC candidates, and 10 objects with undetermined
classification. In addition, Galleti et al. (2004) identified 693 known and
candidate GCs in M31 using the 2MASS database and included their 2MASS
$JHK_{s}$ magnitudes. The latest RBC (V3.5) was updated on March 27, 2008, and
includes the newly discovered star clusters from Mackey et al. (2006), Kim et
al. (2007), and Huxor et al. (2008). In total, 1567 GCs and GC candidates (509
confirmed GCs and 1058 GC candidates) are known in M31; 421 former GC
candidates turned out to be stars, asterisms, galaxies, Hii regions, or
extended clusters. In addition, the RBC V3.5 includes photometric data of M31
GCs and GC candidates in the far- and near-ultraviolet (FUV and NUV) from the
Nearby Galaxies Survey (NGS) of the Galaxy Evolution Explorer (GALEX) (Rey et
al., 2007). Very recently, Caldwell et al. (2009) presented a new catalog of
670 likely star clusters in the field of M31, all with updated high-quality
coordinates accurate to $0.2^{\prime\prime}$, based on images from either the
Local Group Survey (Massey, 2006) or the Digital Sky Survey.
An accurate and reliable analysis of integrated stellar populations (such as
star clusters) is key to understanding the formation and evolutionary process
in galaxies. By means of comparisons of integrated populations with models of
homogeneous stellar systems, i.e., simple stellar populations (SSPs), recent
studies have achieved some success in determining ages and masses for
extragalactic star clusters (e.g., de Grijs et al., 2003a, b, c; Bik et al.,
2003; Ma et al., 2006; Fan et al., 2006; Ma et al., 2007).
Ma et al. (2006) and Fan et al. (2006) derived age estimates for M31 GCs by
fitting SSP models (Bruzual & Charlot, 2003, henceforth BC03) to their
photometric measurements in a large number of intermediate- and broad-band
passbands from the optical to the near-infrared (NIR). For instance, Ma et al.
(2007) constrained the age of the M31 GC S312 (B379), using multicolor
photometry from the NUV to the NIR, to $9.5^{+1.15}_{-0.99}$ Gyr. S312 (B379)
is among the first extragalactic GCs for which the age was estimated
accurately using main-sequence photometry, i.e., Brown et al. (2004) estimated
its age at $10^{+2.5}_{-1}$ Gyr. This was based on their analysis of the
cluster’s color-magnitude diagram (CMD) below the main-sequence turn-off
(MSTO) using extremely deep images obtained with the Advanced Camera for
Surveys (ACS) on board the HST. They performed a quantitative comparison of
their resolved stellar photometry with the isochrones of VandenBerg et al.
(2006).
In this paper we first describe our new observations and the relevant data-
processing steps, as well as the complementary data used from the literature
(§2). In §3 we quantitatively compare the spectral energy distributions (SEDs)
of the GCs in our sample with the galev SSP models. Finally, our results and a
summary are presented in §4.
## 2\. Database
### 2.1. The sample
The GC sample used in this paper was taken from the Bologna catalog of
Battistini et al. (1987), which contains 827 M31 GCs and GC candidates. In
addition, our sample also supplements that of Jiang et al. (2003), who studied
172 GC candidates from Battistini et al. (1987) on the basis of integrated
photometric measurements in the Beijing-Arizona-Taiwan-Connecticut (BATC)
photometric system. In Jiang et al. (2003), all GC candidates of classes A and
B (353 objects) in Battistini et al. (1987) (i.e., their Table IV) were
adopted as their original sample. However, of these only 223 objects are in
their observed CCD fields. They also noted that B007 is a galaxy, and B055,
B132 and B147 are virtually stars (Barmby et al., 2000). These four objects
were therefore not included in Jiang et al. (2003)’s final sample. In summary,
219 class-A or B GCs were observed by Jiang et al. (2003), of which 47 were
excluded because of missing photometric measurements in some filters. In this
paper we analyze these 47 GC candidates on the basis of newly observed data in
the BATC photometric system combined with GALEX FUV/NUV photometry, broad-band
$UBVRI$, and NIR $JHK_{s}$ (2MASS) data. However, we did not manage to obtain
accurate photometric measurements for a number of objects because of either
the dominance of a nearby very bright object (B095, B176, and B202), the GC
candidate being very faint and superimposed on a very high background (B119
and B324), or the GC candidate being located near M32 (B124) or NGC 205
(B331), both also resulting in a very high contribution. In addition, object
B330 is faint and located very close to a brighter object, rendering accurate
photometry impossible. Thus, here we analyze the multicolor photometric
properties of 39 GC candidates. Figure 1 shows the spatial distributions of
both our sample GCs (circles) and the Jiang et al. (2003) GCs (plus signs)
across the M31 field.
Figure 1.— Spatial distribution of the GC candidates in M31. Circles and plus
signs represent the samples discussed in this paper and by Jiang et al.
(2003), respectively. The large ellipse is the M31 disk/halo boundary as
defined by Racine (1991); the two small ellipses are the $D_{25}$ isophotes of
NGC 205 (northwest) and M32 (southeast).
### 2.2. Observations and data reduction
To obtain photometric measurements in the BATC photometric system for the 39
GC candidates for which Jiang et al. (2003) did not obtain photometry in a
number of filters, we re-observed the objects. The BATC photometric system
uses a Ford Aerospace 2048$\times$2048 CCD camera with a pixel size of 15
$\mu$m, mounted at the focus of the 0.6/0.9m $f$/3 Schmidt telescope at
Xinglong Station (National Astronomical Observatories of China; NAOC). The CCD
field of view is $58^{\prime}\times 58^{\prime}$, with a pixel size of
$1.7^{\prime\prime}$. The multicolor BATC filter system includes 15
intermediate-band filters covering the wavelength range from 3300Å to 1
$\mu$m. These filters were specifically designed to avoid contamination from
the brightest and most variable night-sky emission lines. The CCD camera is
not sensitive at the shortest wavelengths covered by the BATC filters. For
this reason, neither Jiang et al. (2003) nor we used the two bluest filters
($a$ and $b$) for our observations. Finding charts of the sample GCs and GC
candidates in the BATC $g$ band (centered at 5795Å), obtained with the NAOC
60/90cm Schmidt telescope, are shown in Fig. 2.
Thirteen hours of imaging of the M31 field of Jiang et al. (2003) were
obtained through the usable set of 13 intermediate-band filters from November
15, 2003 to December 13, 2003. Bias subtraction and flat fielding using dome
flats were done with the BATC automatic data-reduction software, pipeline i,
originally developed for the BATC Multicolor Sky Survey (Fan et al., 1996;
Zheng et al., 1999). The dome flat-field images were taken using a diffuser
plate in front of the Schmidt telescope’s corrector plate. This flat-fielding
technique was verified using photometry obtained for other galaxies and
spectrophotometric observations (see, e.g., Fan et al., 1996; Zheng et al.,
1999; Wu et al., 2002; Yan et al., 2000; Zhou et al., 2001, 2004).
Spectrophotometric calibration of the M31 images was done by observations of
four F-type subdwarfs, HD 19445, HD 84937, BD ${+26^{\circ}2606}$, and BD
${+17^{\circ}4708}$ (Oke & Gunn, 1983). Our magnitudes are therefore defined
in the spectrophotometric AB magnitude system (i.e., the Oke & Gunn
$\tilde{f_{\nu}}$ monochromatic system),
$m_{\rm BATC}=-2.5{\rm log}\tilde{F_{\nu}}-48.60,$ (1)
where $\tilde{F_{\nu}}$ is the appropriately averaged monochromatic flux in
units of erg s-1 cm-2 Hz-1 at the effective wavelength of the specific
passband. In the BATC system $\tilde{F_{\nu}}$ is defined as (Yan et al.,
2000)
$\tilde{F_{\nu}}=\frac{\int{\rm d}({\rm log}\nu)f_{\nu}r_{\nu}}{\int{\rm
d}({\rm log}\nu)r_{\nu}},$ (2)
which relates the magnitude to the number of photons detected rather than to
the input flux (Fukugita et al., 1996). In Eq. (2), $r_{\nu}$ is the system’s
response and $f_{\nu}$ the object’s SED. Spectrophotometric calibration of the
M31 images using the Oke-Gunn standard stars was done during photometric
nights (see for details Yan et al., 2000; Zhou et al., 2001). Using these
standard-star images, we iteratively obtained atmospheric extinction curves
and the variation of these extinction coefficients as a function of the time
of night (cf. Yan et al., 2000; Zhou et al., 2001),
$m_{\rm BATC}=m_{\rm inst}+[K+\Delta K({\rm UT})]X+C,$ (3)
where $X$ is the airmass and $[K+\Delta K({\rm UT})]$ the time-dependent
extinction term. The instrumental magnitudes ($m_{\rm inst}$) of selected
bright, isolated, and unsaturated stars on the M31 images observed on
photometric nights can be readily transformed to the BATC system ($m_{\rm
BATC}$). The calibrated magnitudes of these stars were then used as secondary
standards to uniformly combine images from calibrated nights with their
counterparts observed on non-photometric nights. Table 1 lists the parameters
of the BATC filters and the observational statistics; column 6 provides the
scatter, in magnitudes, for the photometric observations of the four primary
standard stars in each filter.
Figure 2.— Finding charts of the sample GCs and GC candidates in the BATC $g$
band, obtained with the NAOC 60/90cm Schmidt telescope. The field of view of
each image is $11^{\prime}\times 11^{\prime}$.
### 2.3. Integrated photometry
For each M31 GC candidate we used the phot routine in DAOPHOT (Stetson, 1987)
to obtain the integrated photometry. To avoid contamination from nearby
objects, we adopted an aperture of $10.2^{\prime\prime}$ diameter,
corresponding to 6 pixels. Inner and outer radii for background determination
were taken at 8 and 13 pixels from the GC center. Given the small aperture
used for the GC observations, aperture corrections were determined as follows.
We used isolated stars to determine the magnitude difference between diameter
of 6 pixels and the fully integrated magnitude of these stars in each of the
13 BATC filters used. The SEDs for our sample of 39 GCs and GC candidates were
then corrected for the filter-specific differences, and these values are given
in Table 2. Columns 2–14 give the magnitudes in the 13 BATC passbands used for
our observations. For each object the second line lists the $1\sigma$
uncertainties in magnitude for the corresponding passband. The errors for each
filter are given by DAOPHOT. The magnitudes of B129 in the $c$ and $d$
filters, and that of B195 in $p$ filter could not be obtained owing to low
signal-to-noise ratios in these filters.
### 2.4. GALEX, broad-band, and 2MASS photometry
To estimate the ages of the M31 sample GCs and GC candidates accurately, we
use as many photometric data points covering as wide a wavelength range as
possible (cf. de Grijs et al., 2003b; Anders et al., 2004). In addition,
Kaviraj et al. (2007) showed that the combination of FUV and NUV photometry
with optical observations in the standard broad bands enables one to
efficiently break the age-metallicity degeneracy. Worthey (1994) showed that
the age-metallicity degeneracy associated with optical broad-band colors is
$\Delta{\rm age}/\Delta Z\sim 3/2$ (see also MacArthur et al., 2004). However,
de Jong (1996) showed that this degeneracy can be partially broken by adding
NIR photometry to optical colors, which was recently confirmed by Wu et al.
(2005). Cardiel et al. (2003) found that the inclusion of an infrared (IR)
passband can improve the predictive power of the stellar population
diagnostics by $\sim 30$ times compared to using optical photometry alone.
Since NIR photometry is less sensitive to interstellar extinction than the
classical optical passbands, Kissler-Patig et al. (2002) and Puzia et al.
(2002) also suggested that it provides useful complementary information that
can help to disentangle the age-metallicity degeneracy (also see Galleti et
al., 2004).
Rey et al. (2007) published GALEX NUV and FUV photometric data for 485 and 273
M31 GCs, respectively. The photometric data for 28 (NUV) and 17 (FUV) of our
M31 sample GCs in common is listed in Table 3\. Again, for each object the
second line lists the photometric uncertainties for the corresponding
passband. The GALEX photometric system is calibrated to match the
spectrophotometric AB system.
To date, the study of M31 GCs has been largely based on the excellent Bologna
catalog (Battistini et al., 1980, 1987, 1993). Updates to the original RBC
were provided by Galleti et al. (2004) who take as their photometric reference
the dataset of Barmby et al. (2000) in order to obtain the most homogeneous
set of photometric measurements available. Barmby et al. (2000) published
optical and IR photometric data for 285 M31 GCs (see their Table 3), obtained
with the 4-Shooter CCD mosaic camera and the SAO IR imager on the 1.2m
telescope at the Fred Lawrence Whipple Observatory. Photometric measurements
in the $UBVRI$ bands were published by Barmby et al. (2000) for most of our
sample objects. Therefore, we preferentially adopt the $UBVRI$ measurements of
Barmby et al. (2000). For the remaining GCs we follow Galleti et al. (2004),
who updated the Bologna catalog with homogenised optical ($UBVRI$) photometry
collected from the most recent photometric references available. Galleti et
al. (2004) did not include the photometric uncertainties. Although we refer to
the original papers, the uncertainties associated with the same object but
based on the use of different photometric systems are often very different. In
addition, Galleti et al. (2004) transformed their $UBVRI$ photometry to the
reference system of Barmby et al. (2000) by applying offsets derived from
objects in common between the relevant catalog and the data set of Barmby et
al. (2000). The measurements are therefore internally consistent, and
referencing the original uncertainties may be irrelevant. Therefore, we only
adopted photometric uncertainties as suggested by Galleti et al. (2004), i.e.,
0.05 mag in $BVRI$ and 0.08 mag in $U$. In fact, these photometric
uncertainties do not affect our results significantly, as we showed in Fan et
al. (2006) (see their §4.3 for details).
Galleti et al. (2004) identified 693 known and candidate GCs in M31 using the
2MASS database, and determined their 2MASS $JHK_{s}$ photometric magnitudes
(transformed to the CIT photometric system) (Elias et al., 1982, 1983).
However, we need the original 2MASS $JHK_{s}$ magnitudes for our sample GCs to
compare our observational SEDs with the SSP models, so we reversed this
transformation using the same procedures. Since Galleti et al. (2004) did not
provide the 2MASS $JHK_{s}$ uncertainties, we obtained these by comparing the
photometric magnitudes with Fig. 2 of Carpenter et al. (2001). They show the
observed photometric rms uncertainties as a function of magnitude for stars
brighter than their observational completeness limits. We include the broad-
band and 2MASS photometry (and the associated uncertainties) of the sample GCs
in Table 3. We also list the new classification flags, following RBC V3.5
notation. From Table 3 we learn that B052 and B062 are classified as galaxies
based on their radial velocities. We will therefore not estimate their ages
below.
### 2.5. Comparison with previously published photometry
The BATC intermediate-band system can easily be transformed to the $UBVRI$
broad-band system. Zhou et al. (2003) derived the relationships between these
two systems using standard stars from the catalogs of Landolt (1983, 1992) and
Galadí-Enríquez et al. (2000):
$m_{B}=m_{d}+0.2201(m_{c}-m_{e})+0.1278\pm 0.076,$ (4)
$m_{V}=m_{g}+0.3292(m_{f}-m_{h})+0.0476\pm 0.027.$ (5)
To check our photometry we derived the magnitudes in $B$ and $V$ based on Eqs.
(4) and (5). We transformed the magnitudes of our 39 GCs and GC candidates in
the BATC $c,d$, $e$ bands to $B$-band photometry, and BATC $f,g$, and $h$-band
measurements into $V$-band data. Fig. 3 shows a comparison of our $V$ and
$(B-V)$ photometry with previously published measurements of Barmby et al.
(2000) and Galleti et al. (2004). The mean $V$ magnitude and $(B-V)$ color
differences (in the sense of this paper minus Barmby et al. (2000) or Galleti
et al. (2004)) are $\langle\Delta V\rangle=-0.066\pm 0.013$ mag and
$\langle\Delta(B-V)\rangle=-0.040\pm 0.017$ mag, respectively, thus showing
excellent agreement.
Figure 3.— Comparison of our newly obtained cluster photometry with previous
measurements by Barmby et al. (2000) (triangles) and Galleti et al. (2004)
(crosses). The dashed lines enclose $\pm 0.2$ mag in $V$ and $\pm 0.3$ mag in
$B-V$.
### 2.6. Metallicities and reddening values
To estimate the ages of our sample GCs accurately we required that our GCs
have both independently determined metallicities and reddening values. We used
three homogeneous sources for spectroscopic metallicities (Huchra et al.,
1991; Barmby et al., 2000; Perrett et al., 2002) and one reference (Fan et
al., 2008).
Huchra et al. (1991) obtained spectroscopy of 150 M31 GCs and candidates with
the Multiple Mirror Telescope (MMT). The system they used has a resolution of
8–9Å and enhanced blue sensitivity. To obtain many of the strongest and most
metallicity-sensitive spectral features of interest in the ultraviolet, they
extended their observations to the atmospheric cut-off at 3200Å (see details
in Brodie & Huchra, 1990). The metallicities of these 150 objects were
determined using six absorption-line indices from integrated cluster spectra
employing the method of Brodie & Huchra (1990).
Barmby et al. (2000) observed 61 M31 GCs and candidates spectroscopically
using the Keck Low Resolution Imaging Spectrometer (LRIS) and the MMT Blue
Channel spectrograph. With Keck LRIS, they used a 600 $\ell$ mm-1 grating with
a 1.2Å pixel-1 dispersion from 3670–6200Å, and a resolution of 4–5Å. With the
MMT Blue Channel, they used a 300 $\ell$ mm-1 grating with a 3.2Å pixel-1
dispersion from 3400–7200 Å, and a resolution of 9–11Å. They obtained the
cluster metallicities on the basis of the Brodie & Huchra (1990) method as
well.
Perrett et al. (2002) determined metallicities for more than 200 M31 GCs and
candidates using the Wide Field Fibre Optic Spectrograph (WYFFOS) at the 4.2m
William Herschel Telescope. Their spectral range covers $\sim$ 3700–5600Å
using two gratings, one of which (H2400B, 2400 $\ell$ mm-1) yields a
dispersion of 0.8Å pixel-1 and a spectral resolution of 2.5Å over the range
3700–4500Å, and the other (R1200R, 1200 $\ell$ mm-1) yields a dispersion of
1.5Å pixel-1 and a spectral resolution of 5.1Å over the range 4400–5600Å.
Perrett et al. (2002) calculated 12 absorption-line indices, again using the
method of Brodie & Huchra (1990). Through a comparison of the line indices
with published M31 GC [Fe/H] values from previous studies (Bònoli et al.,
1987; Brodie & Huchra, 1990; Barmby et al., 2000), they found that the line
indices of the CH (G band), Mg$b$, and Fe53 lines best represented their
observed GCs. Therefore, Perrett et al. (2002) determined the metallicities of
their sample targets using an unweighted mean of these three [Fe/H] values.
Using metallicities from the literature (Huchra et al., 1991; Barmby et al.,
2000; Perrett et al., 2002) combined with the RBC, Fan et al. (2008)
determined 443 reddening values and intrinsic colors, as well as 209
metallicities for individual clusters without spectroscopic observations.
To use all metallicities as coherently as possible we ranked the sources of
M31 GCs metallicities, choosing Perrett et al. (2002) metallicities whenever
available because of the large number of metallicity determinations.
Metallicities from Barmby et al. (2000) and Huchra et al. (1991) were
preferred if Perrett et al. (2002) determinations were not available. If
spectroscopic metallicities were missing, we used Fan et al. (2008).
Metallicities were not available for B089 and B226. As a consequence, we do
not attempt to determine their ages (see details in §4). The final set of
metallicities for the sample clusters is included in Table 4.
For the reddening values of the sample GCs we refer to Barmby et al. (2000)
and Fan et al. (2008). Barmby et al. (2000) determined the reddening for each
cluster using correlations between optical and IR colors and metallicity, and
by defining various ‘reddening-free’ parameters using their large database of
multicolor photometry. Barmby et al. (2000) found that the M31 and Galactic GC
extinction laws, and the M31 and Galactic GC color-metallicity relations are
similar. They estimated the reddening to M31 objects with spectroscopic data
using the relationship between intrinsic optical color and metallicity for
Galactic clusters. For objects without spectroscopic data they used the
relationships between the reddening-free parameters and certain intrinsic
colors based on Galactic GC data. Following the methods in Barmby et al.
(2000), Fan et al. (2008) (re-)determined reddening values for 443 clusters
and cluster candidates. We choose Fan et al. (2008) reddening values whenever
available because their reddening values comprise a homogeneous data set and
they are larger than those of Barmby et al. (2000). The reddening values for
the sample clusters are listed in Table 4. The values of extinction
coefficient $R_{\lambda}$ are obtained by interpolating the interstellar
extinction curve of Cardelli et al. (1989).
## 3\. Age determination
### 3.1. Stellar populations and synthetic photometry
The most direct method to constrain the ages of different stellar populations
involves comparing the observed luminosity levels of the MSTOs. Unfortunately,
this approach is limited to the nearest GCs, where individual stars can be
resolved and measured down to a few magnitudes fainter than the MSTO. Even in
M31, the nearest large spiral galaxy, the MSTO is only reached for one GC
(S312) (also see Brown et al., 2004; Rey et al., 2007; Ma et al., 2007).
However, since the pioneering work of Tinsley (1968, 1972) and Searle et al.
(1973) evolutionary population synthesis modeling has become a powerful tool
for the interpretation of integrated spectrophotometric observations of
galaxies as well as their components (see e.g. Anders et al., 2004).
In evolutionary synthesis models, SSPs are modeled on the basis of a
collection of evolutionary tracks of stars of different initial masses and a
set of stellar spectra at different evolutionary stages. To estimate the ages
of our sample GCs we compare their SEDs with the galev SSP models (e.g., Kurth
et al., 1999; Schulz et al., 2002; Anders & Fritze-v. Alvensleben, 2003). The
galev SSPs are based on the Padova isochrones (with the most recent versions
using the updated Bertelli et al. (1994) isochrones, which include the
thermally-pulsing asymptotic giant-branch [TP-AGB] phase), and a Salpeter
(1955) stellar initial mass function with a lower-mass limit of $0.10~{}{\rm
M}_{\odot}$ and the upper-mass limit between 50 and 70 ${\rm M}_{\odot}$,
depending on metallicity. The full set of models spans the wavelength range
from 91Å to 160 $\mu$m. These models cover ages from $4\times 10^{6}$ to
$1.6\times 10^{10}$ yr, with an age resolution of 4 Myr for ages up to 2.35
Gyr, and 20 Myr for greater ages.
Since our observational data consists of integrated luminosities through a
given set of filters, we convolved the theoretical SSP SEDs with the GALEX FUV
and NUV, broad-band $UBVRI$, BATC, and 2MASS $JHK_{s}$ filter response curves
to obtain synthetic ultraviolet, optical, and NIR photometry for comparison.
The synthetic magnitude in the AB magnitude system for the $i^{\rm th}$ filter
can be computed as
$m_{i}=-2.5\log\frac{\int_{\lambda}F_{\lambda}\varphi_{i}(\lambda){\rm
d}\lambda}{\int_{\lambda}\varphi_{i}(\lambda){\rm d}\lambda}-48.60,$ (6)
where $F_{\lambda}$ is the theoretical SED and $\varphi_{i}$ the response
curve of the $i^{\rm th}$ filter of the GALEX FUV/NUV, $UBVRI$, BATC, and
2MASS $JHK_{s}$ photometric systems. Here, $F_{\lambda}$ changes as a function
of age and metallicity.
### 3.2. Fitting results
We use a $\chi^{2}$ minimization test to examine which galev SSP models are
most compatible with the observed SEDs, following
$\chi^{2}=\sum_{i=1}^{23}{\frac{[m_{\lambda_{i}}^{\rm
intr}-m_{\lambda_{i}}^{\rm mod}(t)]^{2}}{\sigma_{i}^{2}}},$ (7)
where $m_{\lambda_{i}}^{\rm mod}(t)$ is the integrated magnitude in the
$i^{\rm th}$ filter of a theoretical SSP at age $t$, $m_{\lambda_{i}}^{\rm
intr}$ represents the intrinsic integrated magnitude in the same filter, and
$\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm mod},i}^{2}.$ (8)
Here, $\sigma_{{\rm obs},i}^{2}$ is the observational uncertainty, and
$\sigma_{{\rm mod},i}^{2}$ is the uncertainty associated with the model
itself, for the $i^{\rm th}$ filter. Charlot et al. (1996) estimated the
uncertainty associated with the term $\sigma_{{\rm mod},i}^{2}$ by comparing
the colors obtained from different stellar evolutionary tracks and spectral
libraries. Following Wu et al. (2005), Ma et al. (2006), and Fan et al. (2006)
we adopt $\sigma_{{\rm mod},i}^{2}=0.05$. In fact, the values $\sigma_{{\rm
mod},i}^{2}$ adopted do not change the best fits, but only affect the
$\chi^{2}$ values.
The galev SSP models include five initial metallicities,
$Z=0.0004,0.004,0.008,0.02$ (solar metallicity), and 0.05. Spectra for other
metallicities can be obtained by linear interpolation of the appropriate
spectra for any of these metallicities. In addition, if the metallicity of a
cluster is poorer than $Z=0.0004$, we only use the model of $Z=0.0004$. The
best fits to the SEDs of our GCs are presented in Fig. 4.
Figure 4.— Best-fitting integrated SEDs of the galev SSP models shown in
relation to the intrinsic SEDs for our sample GCs. The photometric data points
are represented by the symbols with error bars (vertical error bars for
uncertainties and horizontal ones for the approximate wavelength coverage of
each filter). Open circles represent the calculated magnitude of the model
SEDs for each filter.
Figure 4.— Continued.
## 4\. Results and summary
In the previous Section we determined the ages of 35 M31 GCs and GC
candidates. The results are listed in Table 5. The metallicity of B089 and
B226, and the reddening of B089 had not been determined previously. From Fig.
5, which shows the age distribution of the sample clusters (see also Table 5)
we conclude that, except for two clusters, the ages of the other sample GCs
are all older than 1 Gyr. Most sample GCs are younger than 6 Gyr, with a peak
at $\sim 3$ Gyr. The ‘usual’ complement of well-known old GCs (i.e., GCs of
similar age as the majority of the Galactic GCs) is also present.
Figure 5.— Age distribution of our sample GCs and GC candidates in M31.
As discussed in §2.6, to estimate the ages of our sample GCs accurately we
required that our GC sample have both independently determined metallicities
and reddening values. For metallicity, we used Huchra et al. (1991), Barmby et
al. (2000), and Perrett et al. (2002) as our reference data set. Since all of
these authors determined the M31 GC metallicities using the calibration of
Brodie & Huchra (1990), all three metallicity determinations are on the same
[Fe/H] scale and there are no systematic offsets among any of these data sets
(see details in Perrett et al., 2002). However, individual metallicity
differences exist, which may affect our age estimates. Twelve GCs and GC
candidates have two metallicity determinations, of which B004, B219, and B238
exhibit the largest differences ($>0.2$ dex). The metallicities of B004, B219,
and B238 from Perrett et al. (2002) are $-0.31\pm 0.74$, $-0.01\pm 0.57$, and
$-0.57\pm 0.66$ dex, compared to $-1.26\pm 0.59$, $-0.53\pm 0.53$, and
$-1.22\pm 0.76$ dex from Huchra et al. (1991). Large metallicity differences
lead to large age differences. The ages of B004, B219, and B238 are estimated
at $4.10\pm 0.55$, $2.50\pm 0.15$, and $5.00\pm 0.45$ Gyr (based on the
metallicities of Perrett et al., 2002), compared to $14.40\pm 0.75$, $11.60\pm
1.45$ and $14.40\pm 0.80$ Gyr (metallicities of Huchra et al., 1991). This
implies that the accuracy of the metallicity determinations is very important
for the corresponding age estimates. The age differences for the other 10 GCs
are less than 1 Gyr based on the metallicities of both Perrett et al. (2002)
and Huchra et al. (1991), except for B045: $8.80\pm 1.45$ Gyr versus $6.30\pm
0.45$ Gyr based on Perrett et al. (2002) and Huchra et al. (1991)
metallicities, respectively. In general, the three different sources of
spectroscopic metallicities provide homogeneous age estimates. For B004, B219,
and B238 the signal-to-noise ratios of the observations of Huchra et al.
(1991) and Perrett et al. (2002) are too low. High-quality spectral
observations of these three GCs are needed. In addition, we point out that the
metallicity calibration of Brodie & Huchra (1990) is solely based on old GCs.
However, the sample GCs and GC candidates discussed in this paper are
estimated to be young or of intermediate age, so that the age estimates may be
somewhat biased by the adopted calibration.
Barmby et al. (2000) discovered that M31 contains GCs exhibiting strong Balmer
lines and A-type spectra, from which one infers that these GCs must be very
young. Beasley et al. (2004) and Puzia et al. (2005) confirmed this conclusion
of Barmby et al. (2000). Burstein et al. (2004) and Fusi Pecci et al. (2005)
increased the sample of young M31 GCs to 67. Very recently, Caldwell et al.
(2009) determined the ages and reddening values of 140 young clusters in M31
by comparing the observed spectra with model spectra, and these clusters are
less than 2 Gyr old. Most have ages between $10^{8}$ and $10^{9}$ yr. The ages
of the M31 clusters determined in this paper are in agreement with previous
determinations.
Many M31 GCs are resolved in HST observations. Some authors, including Ajhar
et al. (1996), Fusi Pecci et al. (1996), Rich et al. (1996), Holland et al.
(1997), Jablonka et al. (2000), Williams & Hodge (2001a), Williams & Hodge
(2001b), and Rich et al. (2005), used WFPC2 images to construct CMDs for
determination of the clusters’ metallicities, reddening values, and ages.
However, these CMDs are usually not deep enough to show conspicuous MSTOs and
thus be useful for robust age determinations. In fact, only Williams & Hodge
(2001a) and Williams & Hodge (2001b) managed to estimate the ages for four
blue massive, compact star clusters and 79 candidate young star clusters by
fitting isochrones to the stellar photometry.
Our sample contains four GCs in common with Ajhar et al. (1996) (B006 and
B045), Fusi Pecci et al. (1996) (also B006 and B045), and Rich et al. (2005)
(B012 and B233). However, only B012 (Rich et al., 2005) could be older than
about 8 Gyr (see Gallart et al. 2005), given the presence of a prominent blue
horizontal branch, which compares rather poorly with the age obtained here
($\sim 2$ Gyr). However, even with multi-passband photometry spanning from the
FUV to $K_{s}$, we can only determine cluster ages in a statistical sense
(also see Gallagher & Grebel, 2002; de Grijs et al., 2005).
This discrepancy of a few Gyr highlights the difficulty of obtaining age
estimates of unresolved intermediate-age clusters based on multi-passband
photometry, given that the color evolution of SSPs is only minimally age
dependent once the population has reached an age of $\sim 1-3$ Gyr. In
addition, although the general age distribution of an entire cluster
population (in a given galaxy) can be retrieved fairly self-consistently, the
clusters’ individual age determinations depend rather strongly on the approach
taken to fitting their ages (cf. de Grijs et al., 2005). This, combined with a
strong dependence on the adopted reddening and metallicity, results in
individual age estimates of intermediate-age clusters associated with large
uncertainties.
Cluster ages can also be derived by comparing observed with model spectra.
Cross-identification of Beasley et al. (2004), Puzia et al. (2005), and
Caldwell et al. (2009) with the sample in this paper reveals that only six GCs
(and GC candidates) overlap with Puzia et al. (2005) (B006, B012, B045, and
B232) and Caldwell et al. (2009) (B049 and B195). The age deerminations of two
of these (B012 and B232) are inconsistent between Puzia et al. (2005) and this
paper: $10.2\pm 2.9$ Gyr and $9.0\pm 3.3$ Gyr obtained by Puzia et al. (2005)
compared to $2.0\pm 0.1$ Gyr and $2.0\pm 0.1$ Gyr, respectively, obtained in
this paper. The ages of the other GCs are consistent among the different
determinations.
In fact, the ages of GCs derived by different authors based on a range of
methods are not always consistent. For example, the ages of B292 and B327
derived by Beasley et al. (2004) and Puzia et al. (2005) are $2.748\pm 1.151$
Gyr and $0.080\pm 0.929$ Gyr (Beasley et al., 2004) compared to $9.2\pm 3.3$
Gyr and $5.4\pm 1.4$ Gyr (Puzia et al., 2005), respectively. On the other
hand, Caldwell et al. (2009) estimated the age of B327 at $0.050$ Gyr. In
addition, the ages of clusters derived in the same paper but based on
different line-index measurements are not always consistent either and may
indeed differ significantly (see the upper panels of Fig. 5 in Puzia et al.,
2005).
## Acknowledgments
We are indebted to the referee for thoughtful comments and insightful
suggestions that improved this paper greatly. This study has been supported by
the Chinese National Natural Science Foundation through grants 10873016,
10803007, 10473012, 10573020, 10633020, 10673012, and 10603006, and by
National Basic Research Program of China (973 Program), No. 2007CB815403.
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Table 1BATC filter parameters and observational statistics
Filter | $\lambda_{\rm central}$ (Å) | FWHM (Å) | Na | Exp.b | rmsc
---|---|---|---|---|---
$c$ | 4210 | 320 | 3 | 01:00 | 0.015
$d$ | 4540 | 340 | 3 | 01:00 | 0.009
$e$ | 4925 | 390 | 3 | 01:00 | 0.015
$f$ | 5270 | 340 | 3 | 01:00 | 0.006
$g$ | 5795 | 310 | 3 | 01:00 | 0.003
$h$ | 6075 | 310 | 3 | 01:00 | 0.003
$i$ | 6656 | 480 | 3 | 01:00 | 0.003
$j$ | 7057 | 300 | 3 | 01:00 | 0.008
$k$ | 7546 | 330 | 3 | 01:00 | 0.004
$m$ | 8023 | 260 | 3 | 01:00 | 0.003
$n$ | 8480 | 180 | 6 | 02:00 | 0.004
$o$ | 9182 | 260 | 6 | 02:00 | 0.003
$p$ | 9739 | 270 | 6 | 02:00 | 0.009
a Number of exposures for each BATC filter
b Exposure time (in hr:min)
cZero-point error (in mag)
Table 2BATC intermediate-band photometry of our sample of 39 GCs and GC candidates in M31. Name | $c$ | $d$ | $e$ | $f$ | $g$ | $h$ | $i$ | $j$ | $k$ | $m$ | $n$ | $o$ | $p$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
B004 | 17.71 | 17.49 | 17.22 | 17.06 | 16.72 | 16.61 | 16.44 | 16.34 | 16.21 | 16.03 | 16.09 | 15.91 | 16.01
| 0.130 | 0.012 | 0.008 | 0.009 | 0.009 | 0.008 | 0.008 | 0.009 | 0.010 | 0.009 | 0.021 | 0.010 | 0.024
B006 | 16.68 | 16.17 | 15.84 | 15.67 | 15.29 | 15.25 | 15.09 | 14.98 | 14.82 | 14.67 | 14.70 | 14.58 | 14.53
| 0.009 | 0.005 | 0.004 | 0.004 | 0.004 | 0.004 | 0.004 | 0.003 | 0.004 | 0.003 | 0.008 | 0.005 | 0.008
B008 | 17.83 | 17.33 | 17.04 | 16.88 | 16.51 | 16.43 | 16.26 | 16.14 | 16.01 | 15.87 | 15.73 | 15.80 | 15.73
| 0.016 | 0.010 | 0.007 | 0.008 | 0.008 | 0.007 | 0.008 | 0.008 | 0.009 | 0.007 | 0.015 | 0.011 | 0.019
B010 | 17.48 | 17.21 | 16.94 | 16.79 | 16.50 | 16.44 | 16.25 | 16.17 | 16.10 | 15.95 | 16.00 | 15.83 | 15.82
| 0.014 | 0.010 | 0.007 | 0.008 | 0.008 | 0.007 | 0.008 | 0.008 | 0.011 | 0.008 | 0.019 | 0.012 | 0.023
B012 | 15.90 | 15.58 | 15.34 | 15.20 | 14.91 | 14.85 | 14.71 | 14.63 | 14.55 | 14.40 | 14.45 | 14.36 | 14.35
| 0.006 | 0.004 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.004 | 0.003 | 0.007 | 0.004 | 0.007
B013 | 18.35 | 17.86 | 17.51 | 17.36 | 16.99 | 16.92 | 16.75 | 16.64 | 16.47 | 16.33 | 16.36 | 16.24 | 16.19
| 0.022 | 0.012 | 0.009 | 0.010 | 0.010 | 0.010 | 0.009 | 0.010 | 0.012 | 0.009 | 0.029 | 0.015 | 0.027
B016 | 18.64 | 18.08 | 17.78 | 17.62 | 17.24 | 17.15 | 16.95 | 16.82 | 16.68 | 16.51 | 16.56 | 16.33 | 16.33
| 0.027 | 0.016 | 0.011 | 0.012 | 0.012 | 0.011 | 0.011 | 0.010 | 0.014 | 0.010 | 0.028 | 0.016 | 0.029
B019 | 15.75 | 15.45 | 15.15 | 14.99 | 14.61 | 14.53 | 14.38 | 14.25 | 14.11 | 13.96 | 13.89 | 13.80 | 13.76
| 0.006 | 0.004 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.006 | 0.004 | 0.007
B020 | 15.61 | 15.29 | 15.06 | 14.92 | 14.54 | 14.48 | 14.36 | 14.24 | 14.12 | 13.95 | 13.99 | 13.92 | 13.85
| 0.006 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.006 | 0.004 | 0.007
B022 | 17.96 | 17.74 | 17.53 | 17.40 | 17.09 | 17.05 | 16.96 | 16.89 | 16.80 | 16.65 | 16.65 | 16.59 | 16.61
| 0.022 | 0.016 | 0.013 | 0.014 | 0.014 | 0.013 | 0.015 | 0.016 | 0.022 | 0.017 | 0.042 | 0.028 | 0.047
B026 | 18.58 | 18.09 | 17.70 | 17.55 | 17.15 | 17.06 | 16.88 | 16.72 | 16.54 | 16.37 | 16.27 | 16.18 | 16.19
| 0.031 | 0.023 | 0.016 | 0.018 | 0.020 | 0.015 | 0.017 | 0.015 | 0.018 | 0.018 | 0.033 | 0.022 | 0.035
B035 | 18.45 | 17.98 | 17.68 | 17.55 | 17.17 | 17.06 | 16.92 | 16.81 | 16.62 | 16.49 | 16.52 | 16.39 | 16.33
| 0.026 | 0.015 | 0.011 | 0.011 | 0.012 | 0.011 | 0.011 | 0.011 | 0.013 | 0.011 | 0.030 | 0.018 | 0.028
B045 | 16.88 | 16.41 | 16.08 | 15.94 | 15.55 | 15.48 | 15.30 | 15.19 | 15.06 | 14.92 | 14.93 | 14.77 | 14.75
| 0.010 | 0.006 | 0.004 | 0.005 | 0.005 | 0.004 | 0.004 | 0.004 | 0.005 | 0.004 | 0.009 | 0.007 | 0.010
B047 | 18.36 | 17.95 | 17.68 | 17.59 | 17.23 | 17.15 | 17.02 | 16.96 | 16.86 | 16.69 | 16.77 | 16.70 | 16.73
| 0.025 | 0.015 | 0.011 | 0.011 | 0.013 | 0.011 | 0.012 | 0.012 | 0.016 | 0.015 | 0.038 | 0.023 | 0.035
B049 | 17.93 | 17.76 | 17.70 | 17.61 | 17.43 | 17.42 | 17.30 | 17.25 | 17.16 | 17.12 | 17.28 | 16.92 | 17.92
| 0.025 | 0.022 | 0.020 | 0.021 | 0.024 | 0.019 | 0.021 | 0.024 | 0.026 | 0.021 | 0.061 | 0.040 | 0.412
B050 | 17.68 | 17.32 | 17.03 | 16.87 | 16.49 | 16.42 | 16.27 | 16.14 | 16.01 | 15.84 | 15.78 | 15.70 | 15.73
| 0.020 | 0.014 | 0.011 | 0.010 | 0.012 | 0.010 | 0.011 | 0.011 | 0.013 | 0.012 | 0.026 | 0.017 | 0.028
B052 | 19.02 | 18.24 | 17.64 | 17.27 | 16.92 | 16.76 | 16.53 | 16.38 | 16.23 | 16.04 | 15.89 | 15.73 | 15.65
| 0.046 | 0.020 | 0.012 | 0.011 | 0.012 | 0.010 | 0.009 | 0.009 | 0.012 | 0.009 | 0.020 | 0.013 | 0.021
B062 | 18.86 | 18.17 | 17.66 | 17.24 | 16.96 | 16.74 | 16.52 | 16.44 | 16.31 | 16.13 | 15.98 | 15.83 | 15.79
| 0.042 | 0.021 | 0.014 | 0.010 | 0.013 | 0.010 | 0.011 | 0.011 | 0.014 | 0.011 | 0.026 | 0.016 | 0.024
B074 | 17.55 | 17.16 | 16.91 | 16.80 | 16.43 | 16.35 | 16.22 | 16.14 | 16.03 | 15.89 | 15.92 | 15.84 | 15.80
| 0.014 | 0.010 | 0.007 | 0.007 | 0.008 | 0.007 | 0.008 | 0.008 | 0.009 | 0.008 | 0.027 | 0.013 | 0.020
B081 | 17.05 | 17.01 | 16.87 | 16.81 | 16.59 | 16.42 | 16.44 | 16.18 | 16.08 | 16.01 | 15.95 | 15.53 | 15.86
| 0.016 | 0.015 | 0.013 | 0.013 | 0.015 | 0.014 | 0.024 | 0.034 | 0.032 | 0.029 | 0.040 | 0.049 | 0.107
B089 | 18.24 | 18.22 | 18.16 | 18.16 | 18.11 | 18.16 | 18.16 | 18.16 | 18.07 | 18.01 | 18.49 | 18.17 | 18.34
| 0.027 | 0.026 | 0.026 | 0.024 | 0.039 | 0.034 | 0.041 | 0.045 | 0.050 | 0.047 | 0.126 | 0.090 | 0.163
Table 2Continued. Name | $c$ | $d$ | $e$ | $f$ | $g$ | $h$ | $i$ | $j$ | $k$ | $m$ | $n$ | $o$ | $p$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
B100 | 18.82 | 18.31 | 18.04 | 17.92 | 17.63 | 17.61 | 17.31 | 17.25 | 17.24 | 17.21 | 17.24 | 16.95 | 16.91
| 0.050 | 0.030 | 0.026 | 0.026 | 0.027 | 0.025 | 0.026 | 0.048 | 0.031 | 0.033 | 0.065 | 0.037 | 0.170
B129 | … | … | 18.41 | 17.71 | 16.83 | 16.59 | 16.07 | 15.79 | 15.47 | 15.21 | 14.99 | 14.71 | 14.62
| … | … | 0.090 | 0.059 | 0.034 | 0.030 | 0.022 | 0.019 | 0.016 | 0.014 | 0.014 | 0.012 | 0.014
B156 | 17.67 | 17.32 | 17.09 | 16.82 | 16.53 | 16.64 | 16.44 | 16.35 | 16.17 | 16.20 | 16.18 | 16.13 | 16.40
| 0.018 | 0.011 | 0.008 | 0.008 | 0.008 | 0.010 | 0.012 | 0.014 | 0.012 | 0.031 | 0.021 | 0.035 | 0.008
B168 | 19.50 | 18.76 | 18.23 | 17.96 | 17.29 | 17.13 | 16.78 | 16.60 | 16.28 | 16.07 | 15.95 | 15.80 | 15.69
| 0.083 | 0.053 | 0.030 | 0.026 | 0.025 | 0.017 | 0.015 | 0.015 | 0.014 | 0.010 | 0.027 | 0.013 | 0.020
B170 | 18.27 | 17.90 | 17.61 | 17.46 | 17.11 | 17.06 | 16.84 | 16.76 | 16.66 | 16.52 | 16.46 | 16.38 | 16.44
| 0.025 | 0.020 | 0.012 | 0.013 | 0.014 | 0.012 | 0.012 | 0.014 | 0.015 | 0.014 | 0.033 | 0.018 | 0.141
B195 | 18.97 | 18.78 | 18.61 | 18.57 | 18.38 | 18.36 | 18.35 | 18.11 | 18.20 | 18.04 | 17.96 | 17.96 | …
| 0.046 | 0.035 | 0.028 | 0.035 | 0.044 | 0.033 | 0.044 | 0.044 | 0.062 | 0.059 | 0.109 | 0.074 | …
B199 | 18.27 | 18.00 | 17.80 | 17.62 | 17.44 | 17.41 | 17.22 | 17.10 | 17.06 | 17.00 | 17.05 | 16.87 | 17.06
| 0.024 | 0.018 | 0.012 | 0.013 | 0.016 | 0.013 | 0.014 | 0.016 | 0.018 | 0.017 | 0.039 | 0.022 | 0.060
B207 | 18.04 | 17.74 | 17.53 | 17.36 | 17.14 | 17.10 | 16.93 | 16.84 | 16.81 | 16.73 | 16.71 | 16.59 | 16.64
| 0.020 | 0.014 | 0.011 | 0.012 | 0.014 | 0.013 | 0.014 | 0.015 | 0.020 | 0.015 | 0.039 | 0.023 | 0.053
B212 | 16.17 | 15.91 | 15.69 | 15.51 | 15.31 | 15.28 | 15.10 | 15.01 | 14.96 | 14.91 | 14.84 | 14.76 | 14.80
| 0.007 | 0.005 | 0.004 | 0.004 | 0.004 | 0.004 | 0.004 | 0.004 | 0.005 | 0.004 | 0.011 | 0.006 | 0.012
B219 | 17.25 | 16.90 | 16.63 | 16.44 | 16.15 | 16.05 | 15.88 | 15.76 | 15.62 | 15.46 | 15.46 | 15.32 | 15.33
| 0.013 | 0.008 | 0.006 | 0.006 | 0.007 | 0.006 | 0.006 | 0.006 | 0.007 | 0.006 | 0.014 | 0.008 | 0.018
B226 | 19.13 | 18.51 | 18.10 | 17.77 | 17.55 | 17.43 | 17.15 | 17.05 | 16.97 | 16.82 | 16.63 | 16.56 | 16.49
| 0.044 | 0.029 | 0.015 | 0.015 | 0.018 | 0.015 | 0.016 | 0.016 | 0.022 | 0.016 | 0.041 | 0.021 | 0.045
B230 | 16.44 | 16.33 | 16.13 | 15.98 | 15.75 | 15.68 | 15.56 | 15.47 | 15.41 | 15.21 | 15.21 | 15.20 | 15.20
| 0.009 | 0.006 | 0.005 | 0.005 | 0.006 | 0.005 | 0.005 | 0.006 | 0.007 | 0.005 | 0.015 | 0.008 | 0.020
B232 | 16.15 | 16.02 | 15.80 | 15.65 | 15.34 | 15.28 | 15.17 | 15.06 | 14.98 | 14.80 | 14.74 | 14.76 | 14.71
| 0.008 | 0.005 | 0.004 | 0.005 | 0.005 | 0.005 | 0.005 | 0.005 | 0.006 | 0.006 | 0.011 | 0.008 | 0.016
B233 | 16.41 | 16.19 | 15.94 | 15.81 | 15.44 | 15.38 | 15.26 | 15.15 | 15.01 | 14.85 | 14.85 | 14.73 | 14.90
| 0.010 | 0.008 | 0.007 | 0.006 | 0.007 | 0.006 | 0.007 | 0.006 | 0.008 | 0.007 | 0.010 | 0.015 | 0.019
B236 | 17.86 | 17.69 | 17.52 | 17.38 | 17.08 | 17.03 | 16.88 | 16.75 | 16.47 | 16.40 | 16.45 | 16.76 | 16.22
| 0.021 | 0.017 | 0.012 | 0.012 | 0.015 | 0.012 | 0.012 | 0.018 | 0.023 | 0.040 | 0.021 | 0.039 | 0.164
B237 | 17.92 | 17.70 | 17.45 | 17.31 | 17.01 | 16.94 | 16.79 | 16.70 | 16.63 | 16.48 | 16.50 | 16.42 | 16.47
| 0.020 | 0.015 | 0.012 | 0.012 | 0.015 | 0.013 | 0.015 | 0.015 | 0.016 | 0.018 | 0.033 | 0.024 | 0.145
B238 | 17.39 | 17.02 | 16.73 | 16.58 | 16.23 | 16.13 | 15.97 | 15.88 | 15.74 | 15.66 | 15.58 | 15.58 | 15.45
| 0.014 | 0.007 | 0.006 | 0.006 | 0.007 | 0.012 | 0.007 | 0.016 | 0.016 | 0.014 | 0.019 | 0.011 | 0.055
B239 | 17.88 | 17.79 | 17.51 | 17.43 | 17.01 | 16.90 | 16.77 | 16.66 | 16.52 | 16.43 | 16.33 | 16.32 | 16.15
| 0.156 | 0.062 | 0.040 | 0.037 | 0.033 | 0.025 | 0.017 | 0.032 | 0.032 | 0.026 | 0.037 | 0.063 | 0.103
Table 3GALEX, broad-band, and 2MASS photometry of the 39 M31 GCs and GC candidates. Name | $c$††footnotemark: | FUV | NUV | $U$ | $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{s}$
---|---|---|---|---|---|---|---|---|---|---|---
B004 | 1 | … | 22.25 | 18.29 | 17.87 | 16.95 | 16.36 | 15.73 | 14.91 | 14.36 | 14.19
| | … | 0.07 | 0.03 | 0.01 | 0.01 | 0.02 | 0.01 | 0.02 | 0.07 | 0.05
B006 | 1 | … | 21.41 | 16.94 | 16.49 | 15.53 | 14.97 | 14.31 | 13.48 | 12.75 | 12.61
| | … | 0.04 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.03 | 0.03
B008 | 1 | … | 22.59 | 18.16 | 17.66 | 16.56 | 16.21 | 15.51 | 14.68 | 14.17 | 13.98
| | … | 0.12 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | 0.07 | 0.06
B010 | 1 | 21.93 | 20.87 | 17.65 | 17.50 | 16.66 | 16.12 | 15.48 | 14.76 | 14.41 | 14.07
| | 0.08 | 0.03 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.07 | 0.06
B012 | 1 | 20.10 | 19.02 | 15.99 | 15.86 | 15.13 | 14.62 | 14.08 | 13.36 | 12.79 | 12.72
| | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.03 | 0.03
B013 | 1 | … | … | 18.56 | 18.06 | 17.19 | 16.60 | 15.96 | 15.18 | 14.61 | 14.34
| | … | … | 0.05 | 0.02 | 0.01 | 0.02 | 0.02 | 0.03 | 0.07 | 0.05
B016 | 1 | … | … | 18.86 | 18.58 | 17.58 | 16.85 | 16.15 | 15.18 | 14.18 | 14.05
| | … | … | 0.08 | 0.04 | 0.01 | 0.03 | 0.02 | 0.08 | 0.07 | 0.10
B019 | 1 | 20.81 | 19.97 | 16.36 | 15.94 | 14.93 | 14.31 | 13.74 | 12.86 | 12.10 | 11.96
| | 0.04 | 0.02 | 0.01 | 0.01 | 0.01 | 0.05 | 0.01 | 0.02 | 0.03 | 0.02
B020 | 1 | 20.05 | 19.20 | 15.98 | 15.74 | 14.91 | 14.37 | 13.65 | 12.97 | 12.26 | 12.21
| | 0.02 | 0.01 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.02 | 0.03 | 0.03
B022 | 1 | 22.69 | 21.20 | 18.14 | 18.09 | 17.36 | 16.97 | 16.35 | 15.75 | 15.04 | 15.48
| | 0.16 | 0.04 | 0.08 | 0.02 | 0.01 | 0.02 | 0.02 | 0.08 | 0.10 | 0.12
B026 | 1 | … | … | 19.14 | 18.60 | 17.53 | 16.88 | 16.22 | 15.10 | 14.48 | 14.00
| | … | … | 0.06 | 0.02 | 0.01 | 0.03 | 0.02 | 0.08 | 0.07 | 0.05
B035 | 1 | … | 22.61 | 18.52 | 18.37 | 17.48 | 16.81 | 16.24 | 15.30 | 14.67 | 14.42
| | … | 0.10 | 0.08 | 0.03 | 0.01 | 0.02 | 0.02 | 0.08 | 0.07 | 0.10
B045 | 1 | … | 21.07 | 17.09 | 16.72 | 15.78 | 15.19 | 14.54 | 13.73 | 13.00 | 12.89
| | … | 0.03 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.04 | 0.03
B047 | 1 | 22.68 | 21.30 | 18.32 | 18.23 | 17.51 | 16.88 | 16.30 | 15.86 | 15.24 | 15.47
| | 0.15 | 0.04 | 0.06 | 0.02 | 0.01 | 0.03 | 0.02 | 0.08 | 0.10 | 0.12
B049 | 1 | … | 21.55 | 18.26 | 18.08 | 17.56 | 17.11 | 16.87 | 15.61 | 15.33 | 14.68
| | … | 0.06 | 0.09 | 0.04 | 0.01 | 0.04 | 0.04 | 0.08 | 0.10 | 0.10
B050 | 1 | … | 22.18 | 18.09 | 17.76 | 16.84 | 16.27 | 15.66 | 14.72 | 14.22 | 13.96
| | … | 0.09 | 0.05 | 0.02 | 0.01 | 0.02 | 0.01 | 0.03 | 0.07 | 0.05
B052 | 4 | … | … | 19.80 | 18.62 | 17.21 | 16.54 | 15.77 | 14.70 | 14.01 | 13.40
| | … | … | 0.08 | 0.02 | 0.01 | 0.02 | 0.02 | 0.04 | 0.07 | 0.05
B062 | 4 | … | … | 19.33 | 18.58 | 17.24 | 16.61 | 15.82 | 14.89 | 14.21 | 13.66
| | … | … | 0.08 | 0.02 | 0.01 | 0.02 | 0.01 | 0.04 | 0.07 | 0.05
B074 | 1 | 22.12 | 20.75 | 17.54 | 17.40 | 16.65 | 16.14 | 15.58 | 14.83 | 13.95 | 14.11
| | 0.07 | 0.02 | 0.03 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.04 | 0.04
B081 | 1 | 21.73 | 20.47 | 17.60 | 17.34 | 16.80 | 16.36 | 15.73 | 14.82 | 14.01 | 13.96
| | 0.13 | 0.04 | 0.02 | 0.01 | 0.01 | 0.02 | 0.02 | 0.03 | 0.07 | 0.05
B089 | 2 | 19.89 | 19.62 | 17.96 | 18.28 | 18.18 | 18.22 | 17.70 | … | … | …
| | 0.03 | 0.02 | 0.05 | 0.04 | 0.03 | 0.06 | 0.06 | … | … | …
Table 3Continued.
Name | $c$ | FUV | NUV | $U$ | $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{s}$
---|---|---|---|---|---|---|---|---|---|---|---
B100 | 1 | 22.37 | … | 18.94 | 19.05 | 17.91 | … | 17.77 | 15.85 | 14.70 | 14.67
| | 0.19 | … | 0.08 | 0.07 | 0.05 | … | 0.07 | 0.08 | 0.07 | 0.10
B129 | 1 | … | … | … | 19.56 | 17.40 | … | 14.69 | 13.25 | 12.40 | 12.19
| | … | … | … | 0.05 | 0.05 | … | 0.05 | 0.03 | 0.03 | 0.04
B156 | 1 | … | 20.99 | 17.89 | 17.63 | 16.84 | 16.37 | 15.87 | … | … | …
| | … | 0.09 | 0.02 | 0.02 | 0.01 | 0.02 | 0.05 | … | … | …
B168 | 1 | … | … | 20.82 | 19.23 | 17.63 | 16.69 | 15.72 | 14.52 | 13.43 | 13.37
| | … | … | 0.08 | 0.06 | 0.01 | 0.03 | 0.02 | 0.06 | 0.04 | 0.08
B170 | 1 | … | … | 18.90 | 18.37 | 17.39 | 16.80 | 16.17 | 15.38 | 14.75 | 14.61
| | … | … | 0.06 | 0.02 | 0.01 | 0.02 | 0.02 | 0.08 | 0.07 | 0.10
B195 | 2 | … | … | 19.94 | 18.97 | 18.57 | 18.02 | 17.59 | … | … | …
| | … | … | 0.08 | 0.06 | 0.01 | 0.07 | 0.05 | … | … | …
B199 | 1 | … | 21.53 | 18.45 | 18.37 | 17.60 | 17.03 | 16.57 | 16.06 | 15.42 | 15.39
| | … | 0.10 | 0.08 | 0.03 | 0.01 | 0.03 | 0.02 | 0.10 | 0.10 | 0.12
B207 | 1 | 21.64 | 21.04 | 18.26 | 18.07 | 17.33 | 16.81 | 16.33 | 15.67 | 14.42 | 14.78
| | 0.11 | 0.06 | 0.03 | 0.02 | 0.01 | 0.02 | 0.02 | 0.05 | 0.07 | 0.08
B212 | 1 | 20.27 | 19.16 | 16.23 | 16.22 | 15.48 | 15.00 | 14.48 | 13.82 | 13.17 | 13.11
| | 0.05 | 0.02 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.04 | 0.05
B219 | 1 | … | 21.59 | 17.74 | 17.32 | 16.39 | 15.82 | 15.19 | 14.32 | 13.71 | 13.51
| | … | 0.10 | 0.08 | 0.05 | 0.05 | 0.05 | 0.05 | 0.02 | 0.04 | 0.04
B226 | 2 | 22.09 | 21.61 | 19.08 | 19.04 | 17.65 | … | 16.32 | 15.21 | 14.47 | 14.14
| | 0.14 | 0.09 | 0.08 | 0.05 | 0.05 | … | 0.05 | 0.08 | 0.07 | 0.10
B230 | 1 | 20.69 | 19.46 | 16.78 | 16.77 | 16.05 | 15.61 | 15.13 | 14.43 | 13.92 | 13.85
| | 0.08 | 0.03 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.04 | 0.05
B232 | 1 | 20.67 | 19.49 | 16.53 | 16.38 | 15.70 | 15.20 | 14.65 | 13.94 | 13.36 | 13.25
| | 0.06 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.03 | 0.04 | 0.05
B233 | 1 | 21.27 | 20.04 | 16.82 | 16.61 | 15.80 | 15.27 | 14.76 | 13.90 | 13.32 | 13.21
| | 0.05 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.04 | 0.03
B236 | 1 | … | 21.07 | 18.21 | 18.20 | 17.38 | 16.97 | 16.24 | … | … | …
| | … | 0.07 | 0.08 | 0.02 | 0.01 | 0.02 | 0.02 | … | … | …
B237 | 1 | … | 21.25 | 18.03 | 17.87 | 17.10 | 16.57 | 16.05 | 15.47 | 15.06 | 14.91
| | … | 0.05 | 0.02 | 0.02 | 0.01 | 0.02 | 0.02 | 0.04 | 0.10 | 0.09
B238 | 1 | … | 21.91 | 17.73 | 17.39 | 16.42 | 15.86 | 15.22 | 14.46 | 13.72 | 13.67
| | … | 0.08 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.04 | 0.04 | 0.05
B239 | 1 | … | … | 18.49 | 18.10 | 17.08 | 16.65 | 16.09 | 15.31 | 14.55 | 14.55
| | … | … | 0.04 | 0.02 | 0.01 | 0.02 | 0.02 | 0.04 | 0.07 | 0.08
${\dagger}$ New classification flag, following RBC V3.5 notation. 1 =
confirmed GC, 2 = GC candidate, 4 = confirmed galaxy
Table 4Reddening values and metallicities for our 39 M31 GCs and GC
candidates.
Name | $E(B-V)$ | ref.a | $\rm[Fe/H]$ | ref.b
---|---|---|---|---
B004 | 0.07$\pm$ 0.02 | 1 | $-0.31\pm$ 0.74 | 1
B006 | 0.09$\pm$ 0.02 | 1 | $-0.58\pm$ 0.10 | 1
B008 | 0.21 | 2 | $-0.41\pm$ 0.38 | 1
B010 | 0.22$\pm$ 0.01 | 1 | $-1.77\pm$ 0.14 | 1
B012 | 0.12$\pm$ 0.01 | 1 | $-1.65\pm$ 0.19 | 1
B013 | 0.13$\pm$ 0.02 | 1 | $-1.01\pm$ 0.49 | 1
B016 | 0.30$\pm$ 0.02 | 1 | $-0.78\pm$ 0.19 | 1
B019 | 0.20$\pm$ 0.01 | 1 | $-1.09\pm$ 0.02 | 1
B020 | 0.12$\pm$ 0.01 | 1 | $-1.07\pm$ 0.10 | 3
B022 | 0.04$\pm$ 0.03 | 1 | $-1.64\pm$ 0.07 | 4
B026 | 0.15$\pm$ 0.02 | 1 | $0.01\pm$ 0.38 | 1
B035 | 0.27$\pm$ 0.05 | 2 | $-0.20\pm$ 0.54 | 1
B045 | 0.18$\pm$ 0.01 | 1 | $-1.05\pm$ 0.25 | 1
B047 | 0.09$\pm$ 0.02 | 1 | $-1.62\pm$ 0.41 | 1
B049 | 0.16$\pm$ 0.02 | 1 | $-2.14\pm$ 0.55 | 1
B050 | 0.24$\pm$ 0.01 | 1 | $-1.42\pm$ 0.37 | 1
B052 | 0.23$\pm$ 0.04 | 1 | $0.12\pm$ 0.17 | 4
B062 | 0.26$\pm$ 0.03 | 1 | $-0.47\pm$ 0.11 | 4
B074 | 0.19$\pm$ 0.01 | 1 | $-1.88\pm$ 0.06 | 1
B081 | 0.11$\pm$ 0.02 | 1 | $-1.74\pm$ 0.40 | 1
B089 | … | … | … | …
B100 | 0.48$\pm$ 0.08 | 1 | $-2.21\pm$ 0.10 | 4
B129 | 1.16$\pm$ 0.06 | 1 | $-1.21\pm$ 0.32 | 1
B156 | 0.10$\pm$ 0.02 | 1 | $-1.51\pm$ 0.38 | 1
B168 | 0.54$\pm$ 0.05 | 1 | $-0.12\pm$ 0.21 | 4
B170 | 0.10$\pm$ 0.02 | 1 | $-0.54\pm$ 0.24 | 1
B195 | 0.12$\pm$ 0.00 | 1 | $-1.48\pm$ 0.63 | 4
B199 | 0.10$\pm$ 0.02 | 1 | $-1.59\pm$ 0.11 | 1
B207 | 0.05$\pm$ 0.02 | 2 | $-0.81\pm$ 0.59 | 1
B212 | 0.13$\pm$ 0.01 | 1 | $-1.75\pm$ 0.13 | 3
B219 | 0.05$\pm$ 0.03 | 1 | $-0.01\pm$ 0.57 | 1
B226 | 1.08$\pm$ 0.06 | 1 | … | …
B230 | 0.15$\pm$ 0.01 | 1 | $-2.17\pm$ 0.16 | 1
B232 | 0.14$\pm$ 0.01 | 1 | $-1.83\pm$ 0.14 | 1
B233 | 0.17$\pm$ 0.01 | 1 | $-1.59\pm$ 0.32 | 3
B236 | 0.07$\pm$ 0.05 | 1 | $-1.01\pm$ 0.17 | 4
B237 | 0.14$\pm$ 0.02 | 1 | $-2.09\pm$ 0.28 | 1
B238 | 0.11$\pm$ 0.02 | 1 | $-0.57\pm$ 0.66 | 1
B239 | 0.09$\pm$ 0.01 | 1 | $-1.18\pm$ 0.61 | 2
aThe reddening values are taken from Fan et al. (2008) (ref.=1) and Barmby et
al. (2000) (ref.=2).
bThe metallicities are taken from Perrett et al. (2002) (ref.=1), Barmby et
al. (2000) (ref.=2), Huchra et al. (1991) (ref.=3), and Fan et al. (2008)
(ref.=4).
Table 5Ages estimates for 35 GCs and GC candidates in M31. Name | Age | $\chi_{\rm min}^{2}$ | Name | Age | $\chi_{\rm min}^{2}$
---|---|---|---|---|---
| (Gyr) | (per degree of freedom) | | (Gyr) | (per degree of freedom)
B004 | $4.10\pm 0.55$ | 3.13 | B100 | $0.50\pm 0.10$ | 14.38
B006 | $12.50\pm 0.65$ | 1.39 | B129 | $15.10\pm 0.70$ | 9.00
B008 | $2.00\pm 0.10$ | 6.54 | B156 | $4.90\pm 0.65$ | 2.09
B010 | $1.80\pm 0.10$ | 1.08 | B168 | $12.60\pm 0.20$ | 3.27
B012 | $2.00\pm 0.10$ | 1.54 | B170 | $4.00\pm 0.45$ | 1.39
B013 | $12.00\pm 2.00$ | 0.96 | B195 | $0.70\pm 0.15$ | 0.82
B016 | $2.40\pm 0.30$ | 1.95 | B199 | $3.30\pm 0.55$ | 1.38
B019 | $2.10\pm 0.10$ | 12.01 | B207 | $1.20\pm 0.10$ | 7.99
B020 | $1.80\pm 0.10$ | 7.50 | B212 | $1.80\pm 0.10$ | 0.75
B022 | $3.40\pm 0.15$ | 4.11 | B219 | $2.50\pm 0.15$ | 3.06
B026 | $3.50\pm 0.25$ | 3.28 | B230 | $1.60\pm 0.10$ | 4.59
B035 | $1.00\pm 0.10$ | 3.97 | B232 | $2.00\pm 0.10$ | 2.62
B045 | $8.80\pm 1.45$ | 0.78 | B233 | $2.30\pm 0.10$ | 3.73
B047 | $2.80\pm 0.20$ | 4.22 | B236 | $2.00\pm 0.25$ | 2.63
B049 | $1.60\pm 0.10$ | 7.82 | B237 | $3.50\pm 0.35$ | 1.41
B050 | $16.00\pm 0.30$ | 2.12 | B238 | $5.00\pm 0.45$ | 2.01
B074 | $2.10\pm 0.15$ | 2.12 | B239 | $14.50\pm 2.05$ | 1.70
B081 | $2.10\pm 0.20$ | 7.82 | | |
|
arxiv-papers
| 2009-04-03T12:37:24 |
2024-09-04T02:49:01.657846
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Ma (1), Zhou Fan (1), Richard de Grijs (2), Zhenyu Wu (1), Xu Zhou\n (1), Jianghua Wu (1), et al. ((1)National Astronomical Observatories, Chinese\n Academy of Sciences; (2)Department of Physics & Astronomy, The University of\n Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK)",
"submitter": "Jun Ma",
"url": "https://arxiv.org/abs/0904.0553"
}
|
0904.0633
|
The Dynamic Radio Sky:
An Opportunity for Discovery
J. Lazio1 (NRL), J. S. Bloom (UC Berkeley), G. C. Bower (UC Berkeley), J.
Cordes (Cornell, NAIC), S. Croft (UC Berkeley), S. Hyman (Sweet Briar), C. Law
(UC Berkeley), & M. McLaughlin (WVU)
Submitted to Astro2010: The Astronomy and Astrophysics Decadal Survey
1 Contact information: 202-404-6329, Joseph.Lazio@nrl.navy.mil; Image credit:
Hallinan et al., NRAO/AUI/NSF
###### Executive Summary
The time domain of the sky has been only sparsely explored. Nevertheless,
recent discoveries from limited surveys and serendipitous discoveries indicate
that there is much to be found on timescales from nanoseconds to years and at
wavelengths from meters to millimeters. These observations have revealed
unexpected phenomena such as rotating radio transients and coherent pulses
from brown dwarfs. Additionally, archival studies have found not-yet
identified radio transients without optical or high-energy hosts. In addition
to the known classes of radio transients, possible other classes of objects
include extrapolations from known classes and exotica such as orphan
$\gamma$-ray burst afterglows, radio supernovae, tidally-disrupted stars,
flare stars, magnetars, and transmissions from extraterrestrial civilizations.
Over the next decade, meter- and centimeter-wave radio telescopes with
improved sensitivity, wider fields of view, and flexible digital signal
processing will be able to explore radio transient parameter space more
comprehensively and systematically.
## 1 Frontier Question: What New Sources and Phenomena Populate the Sky?
The available parameter space for transient surveys is extensive: transients
have been detected at, and are predicted for, all radio wavelengths;
timescales range from nanoseconds to the longest timescales probed; and
transients may originate from nearly all astrophysical environments including
the solar system, star-forming regions, the Galactic center, and other
galaxies.
_By observing the sky so as to preserve information about the time domain, the
past decade has illustrated that there is a considerable potential for
discovery. Over the next decade, a combination of increased sensitivity, field
of view, and algorithmic developments likely would yield transformational
discoveries in a wide range of astronomical fields._
## 2 Science Opportunity: The Dynamic Sky
Transient emission—bursts, flares, and pulses on time scales of less than
about 1 month—marks compact sources or the locations of explosive or dynamic
events. Transient sources offer insight into a variety of fundamental
questions including
* •
Mechanisms of particle acceleration;
* •
Possible physics beyond the Standard Model;
* •
The physics of accretion and outflow;
* •
Stellar evolution and death;
* •
The nature of strong field gravity;
* •
The nuclear equation of state;
* •
The cosmological star formation history;
* •
Probing the intervening medium(a); and
* •
The possibility of extraterrestrial (ET) civilizations.
Much of astronomy’s progress over the last half of the $20^{\mathrm{th}}$
Century resulted from opening new spectral windows. With essentially the
entire spectrum having been explored at some level, we must look to other
parts of parameter space—such as increased sensitivity, field of view, or the
time domain—for future transformational discoveries.
The time domain appears ripe for new exploration as observations over the past
decade have emphasized that the sky may be quite dynamic—known sources have
been discovered to behave in new ways and what may be entirely new classes of
sources have been discovered. Radio observations triggered by high-energy
observations (e.g., observations of $\gamma$-ray burst [GRB] afterglows),
monitoring programs of known high-energy transients (e.g., radio monitoring of
X-ray binaries), giant pulses from the Crab pulsar, a small number of
dedicated radio transient surveys, and the serendipitous discovery of
transient radio sources (e.g., near the Galactic center, brown dwarfs) all
suggest that the sky is likely to be quite active on timescales from
nanoseconds to years and at wavelengths from meters to millimeters.
## 3 Scientific Context: The Transient Sky
Classes of transients are diverse, ranging from nearby stars to cosmological
distances (GRBs), and touching upon nearly every aspect of astronomy,
astrophysics, and astrobiology. Table 1 lists a series of known, hypothesized,
and exotic classes of radio transients. In the remainder of this section, we
provide two case studies and brief discussions of other classes of transients.
Table 1: Illustrations of Classes of Transients Known Classes | Extrapolations of Known Physics | Exotica
---|---|---
brown dwarfs, flare stars | extrasolar planets | signals from ET civilizations
pulsar giant pulses, intermittant pulsars, magnetar flares, X-ray binaries | giant pulses, flares from neutron stars in other galaxies | electromagnetic counterparts to gravitational wave events
radio supernovae, GRB afterglows | prompt emission from GRBs, orphan GRB afterglows | annihilating black holes
variability from interstellar propagation | variability from intergalactic propagation |
### 3.1 Case Study: Rotating Radio Transients—A New Population of Neutron
Stars
The first pulsars were discovered through visual inspection of pen chart
recordings, which revealed the presence of individual radio pulses spaced by
the neutron star rotation period. It was soon realized that Fourier methods
were far more sensitive to the periodic emission believed to be characteristic
of all radio pulsars, and periodicity searches have been used in the discovery
of over 1800 radio pulsars.
In 2003, the Parkes Multibeam Survey had covered the entire Galactic plane
visible from Parkes, finding over 700 new pulsars. The data were then re-
analyzed for single, dispersed pulses, revealing a new population of neutron
stars only detectable through their individual radio bursts (McLaughlin et al.
2006). The average pulse rates of these 11 sources were (3 min)-1 to (3 hr)-1.
Periods ranging from 0.7–7 s were eventually inferred from the differences
between the pulse arrival times. These periods are comparable to those of
traditional radio pulsars, and confirmed the neutron star nature of these
sources, dubbed Rotating Radio Transients (RRATs).
Since the discovery of the original 11 RRATs, interest in single radio pulse
searches has increased dramatically. Single pulse searches are incorporated in
the pipeline of current pulsar surveys, and a great deal of archival pulsar
search data has been reanalyzed. Currently, roughly 30 RRATs are known, with
this number increasing steadily.
What makes RRATs so different from normal pulsars, and how might they be
related to other classes of neutron stars? Perhaps fundamental properties such
as magnetic field or age contribute to the radio sporadicity, or their
emission could be due to external influences such as a debris disk (Cordes &
Shannon 2008). Another fundamental issue is the total number of these sources.
Their sporadicity makes them difficult to detect, and it is likely that the
population of RRATs outnumbers that of normal pulsars, leading Keane & Kramer
(2008) to conclude that the neutron star population is _not_ consistent with
the Galactic supernova rate.
In summary, the RRATs are an example of an unexpected source class discovered
through simple but new transient detection algorithms.
### 3.2 Case Study: Unexplained Transient Events
Figure 1: Illustration of the diversity of the light curves for transients
toward the Galactic center (Hyman et al. 2002, 2005, 2009). The transient GCRT
J1745$-$3009 burst several times (duration $\sim 10$ min.) during a 6-hr
observation, with subsequent bursts detected over the next 1.5 yr; GCRT
J1742$-$3001 brightened and faded over several months, preceded 6 months
earlier by intermittent bursts; and GCRT J1746$-$2757 was detected in only a
single epoch. None of these objects has been identified nor has a multi-
wavelength counterpart been found. The background image is the Galactic center
at 330 MHz, and the total time devoted to the monitoring project, in both new
and archival observations, is about 150 hr.
Figures 1 and 2 illustrate the potential diversity of objects to be
discovered. These transients were discovered in a combination of new and
archival observations toward the Galactic center (Figure 1) or in archival
observations of a “blank field” (Figure 2). Archival data have proven
particularly valuable resources for these programs as both span 1–2 decades of
time. Most of the transients shown in these figures have no multi-wavelength
counterparts, nor are they associated with any known transient classes.
Possible explanations for the various transients range from rare, extremely
luminous flares from Galactic M dwarfs and brown dwarfs to GRB afterglows.
Figure 2: Two radio transients found in a survey of 944 epochs of a blank
field from the VLA archives (Bower et al. 2007); there is no clear object
class identification for these or eight other transients. (Top) Contours
indicate the transients’ locations on the deep radio image. (Bottom) The
positions of the radio transients overlaid on deep Keck G and R band images.
RT 19840613 is offset by 3 kpc from the nucleus of a spiral galaxy at
$z=0.04$; RT 19860115 has no radio or optical counterpart.
### 3.3 Diverse Populations: Opportunity for Discovery
Flare Stars, Brown Dwarfs, and Extrasolar Planets:
Active stars and star systems have long been known to produce radio flares
attributed to particle acceleration from magnetic field activity (Güdel 2002).
More recently, flares from late-type stars (dM) and brown dwarfs have been
discovered (Berger et al. 2001; Hallinan et al. 2007), in some cases with
periodicities indicative of rotation. The radio emission from these late-type
stellar objects is far stronger than expected from the Benz-Güdel relation for
X-ray and radio emission from main-sequence stars. Finally, Jupiter is radio
bright below 40 MHz, and many stars with “hot Jupiters” show signatures of
magnetic star-planet interactions (Shkolnik et al. 2005), so extrasolar
planets may also be radio sources (Zarka 2007).
Pulsar Giant Pulses—Relativistic Magnetohydymamics and the Intergalactic
Medium:
While all pulsars show pulse-to-pulse intensity variations, some pulsars emit
so-called “giant” pulses, with strengths 100 or even 1000 times the mean pulse
intensity. The Crab was the first pulsar found to exhibit this phenomenon, and
giant pulses have since been detected from numerous other pulsars (Cognard et
al. 1996; Romani & Johnston 2001; Johnston & Romani 2003). Pulses with flux
densities of order $10^{3}$ Jy at 5 GHz and with durations of only 2 ns have
been detected from the Crab (Hankins et al. 2003). These “nano-giant” pulses
imply brightness temperatures of 1038 K, by far the most luminous emission
from any astronomical object. In addition to being probes of particle
acceleration in the pulsar magnetosphere, giant pulses may serve as probes of
the local intergalactic medium (McLaughlin & Cordes 2003).
Radio Supernovae and GRBs:
Observations of the kind possible with the new radio telescopes (i.e.,
frequent monitoring of large areas of sky) can be used to find those GRBs and
supernovae that emit in the radio, as well as to follow up on such transients
detected at other wavelengths. Multi-wavelength, multi-epoch observations
(e.g., Cenko et al. 2006) can provide information on progenitors, the
surrounding medium, and models of GRB energetics and beaming. Of special
interest is finding so-called “orphan afterglows,” those without $\gamma$-ray
trigger. The demographics of orphan afterglows directly inform the geometry
and hence energetics of the events (e.g., Levinson et al. 2002).
Intraday Variability, AGN Central Engines, and Interstellar & Intergalactic
Media:
Intraday variability (IDV)—interstellar scintillation of extremely compact
components ($\sim 10$ $\mu$as) in AGN—occurs at frequencies near 5 GHz. The
typical modulation amplitude is a few percent, but occasional sources display
much larger modulations (Kedziora-Chudczer et al. 2001; Lovell et al. 2003);
in _extreme scattering events_ , modulations greater than 50% on time scales
of days to months are obtained (Fielder et al. 1987). The existence of compact
components in AGN may prove to be a sensitive probe of their central engines,
innermost regions of the jet, or both, complementing $\gamma$-ray
observations. Finally, in order for AGN to be sufficiently compact to
scintillate, their signals must not have been affected substantially by
propagation through the _intergalactic medium_. Given that the dominant
baryonic component of the Universe is likely to be in a warm-hot intergalactic
medium, the presence of IDV can also constrain the properties of the
intergalactic medium.
Annihilating Black Holes:
Annihilating black holes are predicted to produce radio bursts (Rees 1977).
Advances in $\gamma$-ray detectors has renewed interest in possible high-
energy signatures from primordial black holes (Dingus et al. 2002; Linton et
al. 2006). Observations at the extremes of the electromagnetic spectrum are
complementary as radio observations attempt to detect the pulse from an
individual primordial black hole, while high-energy observations generally
search for the integrated emission.
Gravitational Wave Events:
The progenitors for gravitational wave events may generate associated
electromagnetic signals or pulses. For example, the in-spiral of a binary
neutron star system, one of the key targets for LIGO, may produce
electromagnetic pulses, both at high energies and in the radio due to the
interaction of the magnetospheres of the neutron stars (e.g., Hansen &
Lyutikov 2001). More generally, the combined detection of both electomagnetic
and gravitational wave signals may be required to produce localizations and
understanding of the gravitational wave emitters (Kocsis et al. 2008). See
also the whitepaper on the GW-EM connection (Bloom et al. 2009).
Extraterrestrial transmitters:
While none are known, searches for extraterrestrial intelligence (SETI) have
found non-repeating signals that are otherwise consistent with the expected
signal from an ET transmitter. Cordes et al. (1997) show how ET signals could
appear transient, even if intrinsically steady.
## 4 Advancing the Science: Exploring Phase Space
_Over the next decade, great progress is possible in the study of transients.
Specific steps include (1) Explicit time-domain processing of data coupled
with algorithmic developments, particularly in the area of identification and
classification of transients; and (2) Exploitation of telescopes with higher
sensitivities, wider fields of view, or both._
The transient detection figure of merit at radio wavelengths is
$\mathrm{FoM}_{t}=\Omega\left(\frac{A_{\mathrm{eff}}}{T_{\mathrm{sys}}}\right)^{2}K(\eta
W,\tau W),$ (1)
which is a function of the telescope sensitivity
$A_{\mathrm{eff}}/T_{\mathrm{sys}}$, instantaneous solid angle $\Omega$,
typical time duration of the transient $W$, event rate $\eta$, and the time
per telescope pointing (“dwell time”) $\tau$. The function $K(\eta W,\tau W)$
incorporates the likelihood of detecting a particular kind of transient.
Roughly, one can separate transients surveys into two classes: (1) Burst
searches that probe timescales of less than about 1 s for which $\Omega$ is
large but $A_{\mathrm{eff}}/T_{\mathrm{sys}}$ is small; and (2) Imaging
surveys conducted with interferometers that typically probe timescales of tens
of seconds and longer and for which $\Omega$ is small but
$A_{\mathrm{eff}}/T_{\mathrm{sys}}$ is large.
1. 1.
Explicit time-domain processing of data and algorithmic developments: Since
the discovery of RRATs, interest in single radio pulse searches has increased
dramatically. Searches for single, dispersed pulses now are incorporated in
the software pipeline of current pulsar surveys, such as those at Arecibo, the
GBT, and Parkes, and archival pulsar data have been reanalyzed. While time-
domain processing is not yet standard for many interferometers, the ASKAP,
ATA, EVLA, LOFAR, LWA, MWA, and eventually the SKA offer new possibilities for
expanding time-domain processing to interferometric imaging. Further, the
interferometers offer the possibility of much higher positional information
for transients, which is essential for multi-wavelength study.
A number of algorithmic improvements would yield improved use of the existing
telescopes and likely a higher yield from future telescopes.
* •
The vast storage and computational requirements of transient searches,
particularly in the case of imaging interferometers, requires the development
of near real-time transient analysis pipelines. The ATA, LOFAR, and MWA
projects are all engaged in the development of such first-generation
pipelines.
* •
The identification, avoidance, and excision of radio frequency interference
(RFI) produced by civil or military transmitters operating in the radio
spectrum is required. These transmitters are often orders of magnitude
stronger than the desired astronomical signal.
* •
The identification and classification of transients is a challenge that is
broader than simply radio wavelength transients.
2. 2.
Exploitation of telescopes with higher sensitivities, wider fields of view:
Generally, both $A_{\mathrm{eff}}/T_{\mathrm{sys}}$ and $\Omega$ should be
large, though depending upon the class of transient and its luminosity
function (if known), it may be possible to trade
$A_{\mathrm{eff}}/T_{\mathrm{sys}}$ vs. $\Omega$. For instance, X- and
$\gamma$-ray instruments with large solid angle coverage and high time
resolution have had great success in finding transients, even if the detectors
were not particularly sensitive.
In the last decade, the field of view of the Arecibo telescope around 1 GHz
was expanded by a factor of 7 with a new feed system (ALFA). In the next
decade, additional field of view expansion technologies such as _phased-array
feeds_ offer the potential of expanding the fields of view of single-dish
telescopes such as Arecibo and the GBT by factors of 10 or more.
For imaging surveys, LOFAR, the LWA and the MWA promise much higher
sensitivities at low radio frequencies for which the fields of view are
naturally large ($\sim 10$ deg.2). The ASKAP and ATA both offer the promise of
much larger fields of view ($\sim 10$ deg.2) at frequencies near 1 GHz, while
the EVLA will provide a factor of 10 in sensitivity improvements across its
entire operational range (1–50 GHz). All of these imaging interferometers also
can be _sub-arrayed_ , providing improvements in field of view ($\sim 100$
deg.2), at the cost of sensitivity.
Looking toward the next decade and to the era of the SKA, the above advances
in searches for transient radio sources promise to transform our understanding
of the dynamic Universe.
## References
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|
arxiv-papers
| 2009-04-03T18:35:26 |
2024-09-04T02:49:01.671911
|
{
"license": "Public Domain",
"authors": "J. Lazio (NRL), J. S. Bloom (Berkeley), G. C. Bower (Berkeley), J.\n Cordes (Cornell, NAIC), S. Croft (Berkeley), S. Hyman (Sweet Briar), C. Law\n (Berkeley), M. McLaughlin (WVU)",
"submitter": "Joseph Lazio",
"url": "https://arxiv.org/abs/0904.0633"
}
|
0904.0674
|
Vol.0 (200x) No.0, 000–000
11institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100012, P. R. China
11email: majun@vega.bac.pku.edu.cn 22institutetext: Department of Physics &
Astronomy, The University of Sheffield, Hicks Building, Hounsfield Road,
Sheffield S3 7RH, UK
33institutetext: Graduate University, Chinese Academy of Sciences, Beijing,
100039, P. R. China
44institutetext: Department of Astronomy and Space Science, Chungnam National
University, Daejeon 305-764, Korea
55institutetext: Center for Space Astrophysics, Yonsei University, Seoul
120-749, Korea
66institutetext: California Institute of Technology, MC 405-47, 1200 E.
California Boulevard, Pasadena, CA 91125
# Old stellar population synthesis: New age and mass estimates for Mayall II =
G1
Jun Ma 11 Richard de Grijs 2211 Zhou Fan 1133 Soo-Chang Rey 44 Zhenyu Wu 11 Xu
Zhou 11 Jianghua Wu 11 Zhaoji Jiang 11 Jiansheng Chen 11 Kyungsook Lee 44 S.
T. Sohn 5566
(Received 2001 month day; accepted 2001 month day)
###### Abstract
Mayall II = G1 is one of the most luminous globular clusters (GCs) in M31.
Here, we determine its age and mass by comparing multicolor photometry with
theoretical stellar population synthesis models. Based on far- and near-
ultraviolet GALEX photometry, broad-band $UBVRI$, and infrared $JHK_{\rm s}$
2MASS data, we construct the most extensive spectral energy distribution of G1
to date, spanning the wavelength range from 1538 to 20,000Å. A quantitative
comparison with a variety of simple stellar population (SSP) models yields a
mean age that is consistent with G1 being among the oldest building blocks of
M31 and having formed within $\sim$1.7 Gyr after the Big Bang. Irrespective of
the SSP model or stellar initial mass function adopted, the resulting mass
estimates (of order $10^{7}M_{\odot}$) indicate that G1 is one of the most
massive GCs in the Local Group. However, we speculate that the cluster’s
exceptionally high mass suggests that it may not be a genuine GC. We also
derive that G1 may contain, on average, $(1.65\pm 0.63)\times 10^{2}L_{\odot}$
far-ultraviolet-bright, hot, extreme horizontal-branch stars, depending on the
SSP model adopted. On a generic level, we demonstrate that extensive multi-
passband photometry coupled with SSP analysis enables one to obtain age
estimates for old SSPs to a similar accuracy as from integrated spectroscopy
or resolved stellar photometry, provided that some of the free parameters can
be constrained independently.
###### keywords:
galaxies: individual (M31) – galaxies: star clusters – galaxies: stellar
content
## 1 Introduction
Globular clusters (GCs) are among the oldest bound stellar systems in the
Universe, and they thus provide a fossil record of the earliest stages of
galaxy formation and evolution. GCs are internally homogeneous in age and
metallicity, i.e. they are simple stellar systems composed of coeval stellar
populations. In addition, GCs are the oldest systems in our own and other
galaxies for which we can estimate reasonably reliable ages (and realistic
uncertainties); they can thus independently provide vitally important
information regarding the minimum age of the Universe. In a detailed study of
17 Galactic GCs, Chaboyer et al. (1998) used the improved Hipparcos parallaxes
having just become available at that time to determine updated distances, and
hence improved ages, of their GC sample. They concluded that the mean age of
the oldest GCs is $11.5\pm 1.3$ Gyr, although their age histogram (their fig.
2) shows a tail toward ages as old as $\sim 16$ Gyr. Gratton et al. (2003)
obtain improved ages (and distances) for three Galactic GCs, NGC 6397, NGC
6752, and 47 Tuc, and conclude that the age of the oldest GCs is $13.4\pm 0.8$
(random) $\pm 0.6$ (systematic) Gyr, in good agreement with the 3-year results
from the Wilkinson Microwave Anisotropy Probe (WMAP). This led them to suggest
that the oldest Galactic GCs formed within the first 1.7 Gyr after the Big
Bang, at the $1\sigma$ confidence level. We note that this is still fully
compatible with the 5-year WMAP results constraining the age of the Universe
to $13.73\pm 0.12$ Gyr (Hinshaw et al. 2008). While the ages of the oldest GCs
in the Galaxy are now reasonably well determined, this is certainly not the
case even for our nearest large neighbour, the Andromeda galaxy (M31).
The most direct method for determining the age of a star cluster is by means
of individual stellar photometry, since the main-sequence turn-off location is
mostly affected by age (see, e.g., Puzia et al. 2002b, and references
therein). However, this method has only been applied to Galactic GCs and to
GCs associated with the Milky Way’s satellites (e.g., Riich 2001), in which
individual stars can be resolved and their photometry determined to
satisfactory accuracy, to a few magnitudes fainter than the main-sequence
turn-offs. This is difficult, if not impossible, to achieve even for GCs as
close as those associated with M31 (but see Brown et al. 2004), at a distance
of $D=772\pm 44$ kpc (e.g., Ribas et al. 2005). Fortunately, starting from the
pioneering work of Tinsley (1968, 1972) and Searle et al. (1973), evolutionary
population synthesis modeling has become a powerful tool to address many
outstanding problems in astrophysics, from determining the ages of star
clusters to interpreting integrated spectrophotometric observations of
galaxies. Therefore, extragalactic GC ages can, in general, also be inferred
from composite colors and/or integrated spectroscopy.
The evolution of GCs is usually modeled by means of the simple stellar
population (SSP) approximation. An SSP is a single generation of coeval stars
formed from the same progenitor molecular cloud (thus implying a single
metallicity), and governed by a given stellar initial mass function (IMF). GCs
are ideal templates to test the compatibility between the population synthesis
models and reality. Barmby & Huchra (2000) compared the predicted SSP colors
of three stellar population synthesis models to the intrinsic broad-band
$UBVRIJHK$ colors of Galactic and M31 GCs, and concluded that the best-fitting
models match the cluster colors very well. Subsequently, many authors have
used SSP modeling to determine the parameters of cluster populations. For
instance, de Grijs et al. (2003a) determined the ages and masses of star
clusters in the fossil starburst region B of M82 by comparing the observed
cluster spectral energy distributions (SEDs) with both the Starburst99 SSP
models (Leitherer et al. 1999) and those developed by Bruzual & Charlot
(2000), based on Hubble Space Telescope (HST) observations from the blue
optical to the near-infrared (NIR) (see also de Grijs et al. 2003b, 2003c);
Bik et al. (2003) and Bastian et al. (2005) estimated ages, initial masses and
extinction values for M51 star cluster candidates by comparing the Starburst99
and the Frascati models (Romaniello 1998) for instantaneous star formation
with the observed SEDs based on HST/WFPC2 observations in six broad-band and
two narrow-band filters. Ma et al. (2006) and Fan et al. (2006) obtained age
estimates for M31 GCs by fitting theoretical stellar population synthesis
models (Bruzual & Charlot 2003, henceforth BC03) to their photometric
measurements in a large number of intermediate- and broad-band passbands from
the optical to the NIR. Based on the same method and models, Ma et al. (2007a)
constrained the age of the M31 GC S312, using multicolor photometry from the
near-ultraviolet (NUV) to the NIR, to $9.5^{+1.15}_{-0.99}$ Gyr. S312 is among
the first extragalactic GCs for which the age was estimated accurately using
main-sequence photometry, i.e., Brown et al. (2004) estimated the age of S312
at $10^{+2.5}_{-1}$ Gyr by means of a quantitative comparison with the
isochrones of VandenBerg et al. (2006). This was based on their analysis of
the cluster’s color-magnitude diagram (CMD) below the main-sequence turn-off
using extremely deep images obtained with the HST/Advanced Camera for Surveys
(ACS).
It is a common misconception that spectroscopic age estimates are always much
better than those based on broad-band photometry. Schweizer et al. (2004)
recently showed convincingly that spectroscopic age determinations are not
necessarily better or more accurate than photometrically obtained ages, at
least in the age range from $\sim 100-500$ Myr. Anders et al. (2004) published
a detailed theoretical investigation of the accuracy of retrieved star cluster
properties, including their ages, based on sophisticated fits of SSP models to
observed broad-band SEDs spanning varying wavelength ranges. They concluded
that if one has access to as large a wavelength range as possible, ideally
including both ultraviolet and NIR data points, the resulting age estimates
are reasonably accurate, even for ages as old as $\sim 10$ Gyr – particularly
if one or more of the other free parameters (e.g., metallicity or extinction)
can be constrained reliably and independently (see Anders et al. 2004, e.g.,
their fig. 14). We will use this promising approach as our basic premise in
this paper.
Of the Local Group members, M31 is particularly important as it provides a
direct comparison with our own Galaxy. In addition, it contains a large number
of GCs and GC candidates, including 496 genuine, 367 probable, and 301
possible GCs (Kim et al. 2007). The M31 GC system is among the extragalactic
GC systems studied most often (Harris 1991; Brodie & Strader 2006). As one of
the brightest M31 GCs, Mayall II = G1 has attracted much scientific interest
(see, e.g., Barmby et al. 2007; Ma et al. 2007b, and references therein).
In this paper, we first determine the age and mass of G1 by comparing
observational SEDs (§2) with population synthesis models (§3). We will use the
lessons learned from studies of broad-band photometric SED fits to minimize
the associated uncertainties. We discuss our results along the way, as
appropriate, and provide a summary in §4.
## 2 Ultraviolet, optical, and infrared observations of G1
### 2.1 Historical observations
G1 was first detected by Mayall & Eggen (1953) (their No. 2, and hence
referred to as Mayall II), who searched for nebulous objects associated with
M31 using a $6^{\circ}\times 6^{\circ}$ Palomar 48-inch Schmidt plate taken in
1948 and centered on M31. Subsequently, Sargent et al. (1977) rediscovered the
cluster (their No. 1, i.e., G1) based on their survey of 29 plates associated
with the general field of M31, which had been obtained at the $f/2.7$ prime
focus of the Kitt Peak National Observatory’s (KPNO) 4-m telescope. The
cluster is located in the halo of M31, at a projected distance of about 40 kpc
from the galaxy’s nucleus (see Meylan et al. 2001).
### 2.2 GALEX ultraviolet, optical broad-band, and 2MASS NIR photometry
Although the cluster is generally believed to be among the oldest GCs in M31,
to the best of our knowledge there is no CMD-based or spectroscopic age
estimate available in the literature to date. The lack of a CMD-based age
estimate is due to the challenges associated with probing the cluster’s CMD
down to below the main-sequence turn-off. The current deepest CMD of the
cluster (Meylan et al. 2001) does not reach these faint levels. Although both
integrated and spatially-resolved spectra of the cluster are available (e.g.,
Huchra et al. 1991; Gebhardt et al. 2005; Cohen 2006), they have thus far not
been used to determine an age for G1. This may be partially due to the limited
wavelength range covered by most of these spectra, and the difficulties one
faces when trying to constrain ages in the regime beyond $\sim 10$ Gyr (see
below).
To constrain the age of G1 accurately, with the smallest uncertainty allowed
by the observational data, we use as many photometric data points covering as
large a wavelength range as possible. Kaviraj et al. (2007) showed that the
combination of far (FUV) and near-ultraviolet photometry with optical
observations in the standard broad bands enables one to efficiently break the
age-metallicity degeneracy; Worthey (1994) showed that the age-metallicity
degeneracy associated with optical broad-band colors is $\Delta{\rm
age}/\Delta Z\sim 3/2$ (also see MacArthur et al. 2004). However, de Jong
(1996) showed that this degeneracy can be partially broken by adding NIR
photometry to the optical colors, which was recently supported by Wu et al.
(2005). Cardiel et al. (2003) found that inclusion of an infrared passband can
improve the predictive power of the stellar population diagnostics by $\sim$30
times compared to using optical photometry alone. Since NIR photometry is less
sensitive to interstellar extinction than the classical optical passbands,
Kissler-Patig et al. (2002) and Puzia et al. (2002a) also suggested that it
provides useful complementary information that can help to disentangle the
age-metallicity degeneracy (also see Galleti et al. 2004).
The M31 field was observed as part of the Nearby Galaxies Survey (NGS) by the
Galaxy Evolution Explorer (GALEX) in two ultraviolet bands (see for details
from Rey et al 2005, 2007). Rey et al. (2007) published photometric data for
485 and 273 M31 GCs in the GALEX NUV and FUV bands, respectively. G1 was
detected in these two ultraviolet bands. The GALEX photometric system is
calibrated to match the spectrophotometric AB system.
van den Bergh (1969) determined photo-electric photometry for 45 M31 GCs,
including G1, in the $UBV$ bands. Using CCD imaging from the KPNO 0.9m
telescope, Reed, Harris & Harris (1994) published integrated $BVR$ magnitudes
and color indices for 41 GCs and GC candidates, including G1, in the outer
halo of M31. We compared the photometry of G1 in the $B$ and $V$ bands between
these two studies; the results match closely. In this paper, we adopt the CCD
$BVR$ photometry of Reed, Harris & Harris (1994), and the photographic
$U$-band photometry of van den Bergh (1969), with a photometric uncertainty of
0.08 mag as suggested by Galleti et al. (2004). Based on HST images, Barmby &
Huchra (2001) detected and published photometry for 114 GC candidates
associated with M31, including 32 new objects. Their $V$-band photometry is in
good agreement with that of van den Bergh (1969) and Reed, Harris & Harris
(1994), although they do not provide their photometric uncertainties. However,
Barmby & Huchra (2001) compared their HST photometry with the ground-based
measurements compiled by Barmby et al. (2000), and found that the median
offset in $I$ is $0.06\pm 0.04$ mag. Therefore, we adopt $0.06$ mag as the
photometric uncertainty in the $I$ band. Using the Two Micron All Sky Survey
(2MASS) database, Galleti et al. (2004) identified 693 known and candidate GCs
in M31, and listed their 2MASS $JHK_{\rm s}$ magnitudes. Galleti et al. (2004)
transformed all 2MASS magnitudes to the CIT photometric system (Elias et al.
1982, 1983) using the color transformations in Carpenter (2001). However, we
need the original 2MASS $JHK_{\rm s}$ magnitudes to compare our observational
SEDs with the SSP models, so we reversed this transformation using the same
procedures. Since Galleti et al. (2004) do not provide the photometric
uncertainties in $JHK_{\rm s}$, we obtained these by comparing the magnitudes
with fig. 2 of Carpenter et al. (2001), where the observed photometric rms
uncertainties in the time series are shown as a function of magnitude, for
stars brighter than the observational completeness limits. In fact, the
photometric uncertainties adopted do not affect our results significantly, as
we showed in Fan et al. (2006) (see their section 4.3 for details). The full
set of ultraviolet, optical broad-band, and 2MASS NIR photometry of G1 is
listed in Table 1\. The $UBVRI$ and 2MASS magnitudes are given in the Vega
system Schneider et al. (1977). For convenience, we converted all
observational magnitudes to the AB system, following the procedures
recommended in BC03.
Table 1: Ultraviolet, optical broad-band, and NIR 2MASS photometry of G1. Filter | Magnitude (uncertainty) | Reference
---|---|---
FUV | 18.972 (0.031) | Rey et al. (2007)
NUV | 18.014 (0.012) |
$U$ | 14.85 (0.08) | van den Bergh (1969)
$B$ | 14.584 (0.013) | Reed, Harris & Harris (1994)
$V$ | 13.750 (0.007) |
$R$ | 13.191 (0.010) |
$I$ | 12.684 (0.060) | Barmby & Huchra (2001)
$J$ | 11.858 (0.054) | Galleti et al. (2004)
$H$ | 11.127 (0.054) |
$K_{\rm s}$ | 11.016 (0.054) |
### 2.3 Reddening and metallicity
To obtain the intrinsic SED of G1, the photometric data must first be
dereddened. Since G1 is located in the halo of M31, i.e., far away from the
galaxy’s disk, it is (for all practical purposes) only affected by Galactic
foreground extinction. In fact, some authors have demonstrated that G1 is
affected by a negligible amount of reddening. Meylan et al. (2001) used
HST/WFPC2 observations in the F555W and F814W filters, and applied Sarajedini
(1994)’s method to simultaneously determine the cluster’s reddening and
metallicity; they obtained a reddening of $E(V-I)=0.05\pm 0.02$ mag toward G1,
which is less than the Galactic foreground extinction. van den Bergh (1969)
studied the reddening in the halo of M31 by comparing the colors of clusters
with the same line-strength index in the Galaxy and in M31, and obtained a
mean reddening of $E(B-V)=0.08\pm 0.02$ mag for the clusters in the halo of
M31. Barmby et al. (2000) determined the reddening for each individual cluster
using correlations between optical and infrared colors and metallicity, and by
defining various ‘reddening-free’ parameters based on their large database of
multi-color photometry. For G1, Barmby et al. (2000, also P. Barmby, priv.
comm.) obtained $E(B-V)=0.09\pm 0.02$ mag. In this paper, we adopt the
reddening value from Barmby et al. (2000). The values for the extinction
coefficient, $R_{\lambda}$, were obtained by interpolating the interstellar
extinction curve of Cardelli et al. (1989).
Cluster SEDs are determined by the combination of their ages and
metallicities, which is often referred to as the age-metallicity degeneracy.
Therefore, the age of a cluster can only be constrained accurately if the
metallicity is known with confidence, from independent determinations. There
exist two metallicity determinations for G1. Huchra et al. (1991) derived
metallicities for 150 M31 GCs, including G1, using the strengths of six
absorption features in the clusters’ integrated spectra. The resulting
metallicity of G1 is $\rm[Fe/H]=-1.08\pm 0.09$. Meylan et al. (2001) used
HST/WFPC2 photometry to construct deep CMDs for G1, combined with the shape of
the red-giant branch as calibrated by Sarajedini et al. (2000), to derive the
mean metallicity of G1 on the scale of Zinn & West (1984), $\rm[Fe/H]=-0.95\pm
0.09$. In this paper, we adopt $\rm[Fe/H]=-1.08\pm 0.09$ for G1.
## 3 The stellar population of G1
### 3.1 Stellar populations and synthetic photometry
In this section, we compare the SED of G1 with theoretical stellar population
synthesis models. We start by using the BC03 SSP models, which have been
upgraded from the earlier Bruzual & Charlot (1993, 1996) versions, and now
provide the evolution of the spectra and photometric properties for a wider
range of stellar metallicities. BC03 provide 26 SSP models (both of high and
low spectral resolution) using the Padova-1994 evolutionary tracks, half of
which were computed based on the Salpeter (1955) IMF with lower and upper-mass
cut-offs of $m_{\rm L}=0.1~{}M_{\odot}$ and $m_{\rm U}=100~{}M_{\odot}$,
respectively. The other thirteen were computed using the Chabrier (2003) IMF
with the same mass cut-offs. In addition, BC03 provide 26 SSP models using the
Padova-2000 evolutionary tracks. In this paper, we will use all of these SSP
models to determine the most appropriate age and mass for G1. These SSP models
contain 221 spectra describing the spectral evolution of SSPs from $1.0\times
10^{5}$ yr to 20 Gyr. The evolving spectra include the contribution of the
stellar component at wavelengths from 91Å to $160\mu$m.
Since our observational data are integrated luminosities through a given set
of filters, we convolved the theoretical SSP SEDs of BC03 with the FUV and
NUV, broad-band $UBVRI$, and 2MASS $JHK_{\rm s}$ filter response curves to
obtain synthetic ultraviolet, optical, and NIR photometry for comparison. The
synthetic magnitude in the AB magnitude system for the $i{\rm th}$ filter can
be computed as
$m_{\lambda_{i}}=-2.5\log\frac{\int_{\lambda}F_{\lambda}\varphi_{i}(\lambda){\rm
d}\lambda}{\int_{\lambda}\varphi_{i}(\lambda){\rm d}\lambda}-48.60,$ (1)
where $F_{\lambda}$ is the theoretical SED and $\varphi_{i}$ the response
curve of the $i{\rm th}$ filter. $F_{\lambda}$ varies as a function of age and
metallicity.
### 3.2 Fit results
We use a $\chi^{2}$ minimization approach to examine which SSP models are most
compatible with the observed SEDs, following
$\chi^{2}=\sum_{i=1}^{10}{\frac{[m_{\lambda_{i}}^{\rm
intr}-m_{\lambda_{i}}^{\rm mod}(t)]^{2}}{\sigma_{i}^{2}}},$ (2)
where $m_{\lambda_{i}}^{\rm mod}(t)$ is the integrated magnitude in the $i{\rm
th}$ filter of a theoretical SSP at age $t$, $m_{\lambda_{i}}^{\rm intr}$
represents the intrinsic integrated magnitude in the same filter, and
$\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm mod},i}^{2}.$ (3)
Here, $\sigma_{{\rm obs},i}^{2}$ is the observational uncertainty, and
$\sigma_{{\rm mod},i}^{2}$ is the uncertainty associated with the model
itself, for the $i{\rm th}$ filter. Charlot et al. (1996) estimated the
uncertainty associated with the term $\sigma_{{\rm mod},i}^{2}$ by comparing
the colors obtained from different stellar evolutionary tracks and spectral
libraries. Following Wu et al. (2005), we adopt $\sigma_{{\rm
mod},i}^{2}=0.05$.
The BC03 SSP models based on the Padova-1994 evolutionary tracks include six
initial metallicities, $Z=0.0001,0.0004,0.004,0.008,0.02\,(Z_{\odot})$, and
0.05, corresponding to ${\rm[Fe/H]}=-2.25,-1.65,-0.64,-0.33,+0.09$, and
$+0.56$. However, the BC03 SSP models based on the Padova-2000 evolutionary
tracks include six partially different initial metallicities, $Z=0.0004$,
0.001, 0.004, 0.008, 0.019 $(Z_{\odot})$, and 0.03, i.e.,
${\rm[Fe/H]}=-1.65,-1.25,-0.64,-0.33,+0.07$, and $+0.29$. Spectra for other
metallicities can, in principle, be obtained by interpolation of the
appropriate spectra for any of these metallicities, although this is not
necessarily advisable or straightforward (Frayn & Gilmore 2002). Instead, we
adopt the most appropriate model metallicity for the analysis performed in
this paper. Since we have good estimates of the metallicity and reddening
values of G1 (see §2.3), the cluster age is therefore the sole parameter to be
estimated (for a given IMF and extinction law, both of which we assume to be
universal).
None of the SSP models fit the photometric data point in the GALEX FUV band as
well as the other nine data points. (We checked that the image of G1 in the
FUV band is not affected by instrumental problems.) Given that G1 contains an
old stellar population, the most likely physical explanation for this FUV
excess compared to the ‘standard’ BC03 SSP models is the presence of a
significant number of FUV-bright, hot, extreme horizontal-branch (EHB) stars
giving rise to the well-known ‘ultraviolet upturn’ below $\lambda\simeq 2000$Å
(see, e.g., the review of O’Connell 1999, and references therein; see also
Landsman et al. 1998; Sohn et al. 2006). (Alternative species, such as AGB-
manqué stars or blue stragglers are expected to be fewer in number in any
‘normal’ stellar population.)
Since ‘standard’ SSP models do not contain EHB populations, we are forced to
deselect the photometric FUV data point when applying our fitting routines. In
Fig. 1, we show the intrinsic SED of G1 and the integrated SEDs of the best-
fitting models. The dereddened data are shown as the symbols with error bars
(vertical errors for photometric uncertainties and horizontal error bars for
the approximate wavelength coverage of each filter); open circles represent
the calculated magnitudes of the model SED for each filter, obtained by
convolving the theoretical SSP SEDs with the appropriate filter response
curves. The best reduced-$\chi^{2}$ values and ages are listed in Table 2. The
mass of G1, also listed in Table 2, can be estimated by comparing the measured
luminosity in the $V$ band with the theoretical mass-to-light ($M/L$) ratios.
These $M/L$ ratios are a function of the cluster age and metallicity. The
mass-to-light ratios of G1, calculated based on the metallicity adopted and
the age obtained in this paper, are listed in Table 2 for the BC03 SSP models.
Based on its present luminosity, $V=13.750\pm 0.007$ mag, and extinction,
$E(B-V)=0.09$ mag, the cluster’s visual magnitude corrected for the extinction
is $V_{0}=13.471\pm 0.007$ mag, assuming a Cardelli et al. (1989) Galactic
reddening law with $A_{V}=0.279$ mag.
(We note that the NUV data point is also marginally affected by the onset of
the UV upturn, which causes a slight mismatch between the observations and the
best-fitting theoretical SSP models.)
Figure 1: Best-fitting integrated theoretical BC03 SEDs compared to the
intrinsic SED of G1. The photometric measurements are shown as the symbols
with error bars (vertical for uncertainties and horizontal for the approximate
wavelength coverage of each filter). Open circles represent the calculated
magnitudes of the model SED for each filter. We did not use the FUV
photometric data point for the fits (see text).
### 3.3 Age and mass
In the previous section we determined the best-fitting age and mass of G1
based on different theoretical SSP models. From Table 2 we conclude that,
within the errors, the ages obtained from the different BC03 models are
internally consistent. The mean age of G1 is $18.23\pm{1.76}$ Gyr. This is in
excellent agreement with the only other available (rough) age estimate for the
cluster by Meylan et al. (2001), who estimated its age to be $\sim$15 Gyr.
However, we note that the age of G1 obtained in this paper is older than the
current-best estimate of the age of the Universe, of order 13.7 Gyr, as
discussed in §3.1. We will discuss this problem in §4.
Table 2: Age and mass estimates of G1 based on the BC03 models. Evolutionary Track | IMF | Age | $\chi^{2}/\rm{degree~{}~{}of~{}~{}freedom}$ | $M/L_{V}$ | Mass
---|---|---|---|---|---
| | (Gyr) | | $(M/L_{V})_{\odot}$ | $(10^{7}M_{\odot})$
Padova 1994 | Salpeter (1955) | $19.68\pm{0.75}$ | 3.04 | 5.10 | $1.06\pm{0.07}$
Padova 1994 | Chabrier (2003) | $19.79\pm{0.50}$ | 2.69 | 3.15 | $0.65\pm{0.04}$
Padova 2000 | Salpeter (1955) | $15.44\pm{0.78}$ | 5.36 | 4.14 | $0.86\pm{0.05}$
Padova 2000 | Chabrier (2003) | $18.01\pm{2.00}$ | 5.27 | 2.79 | $0.58\pm{0.04}$
We conclude that the various mass estimates listed in Table 2 place G1 firmly
at the top of the cluster mass function in the Local Group. Meylan et al.
(2001) presented three estimates of the total mass of G1, (i) a King-model
mass (King 1966) of $1.5\times 10^{7}M_{\odot}$, (ii) a virial mass of
$0.75\times 10^{7}~{}M_{\odot}$; and (iii) a mass based on a King-Michie model
(as defined by Gunn & Griffi 1979) fitted simultaneously to the surface
brightness profile and the central velocity dispersion value, estimated
between $1.4\times 10^{7}M_{\odot}$ and $1.7\times 10^{7}M_{\odot}$. Our
results are in reasonable agreement with Meylan et al. (2001), although we are
aware that the King and King-Michie mass estimates of Meylan et al. (2001) are
up to a factor of two greater than our photometric mass estimates. This is not
too surprising in view of the model assumptions made. Cohen (2006) recently
obtained an optical velocity dispersion for the cluster using the Keck/HIRES
spectrograph, and derived an aperture-corrected line-of-sight velocity
dispersion, $\sigma_{\rm los}=25.5\pm 1.5$ km s-1 (where we have averaged the
values she obtained for the two reddest wavelength ranges analyzed; see also
Djorgovski et al. 2002). We recently redetermined a projected half-light
radius for G1 of $r_{\rm h}=6.5\pm 0.3$ pc (Ma et al. 2007b). Thus, based on
these most recent results, the dynamical (virial) mass of G1 is $M_{\rm
vir}=(7.37\pm 2.15)\times 10^{6}M_{\odot}$. This is in excellent agreement
with the photometric mass estimates obtained in this paper. In turn, this
strongly supports the notion that G1 must have had a close-to-‘normal’ stellar
IMF, in order for it to have survived dissolution due to internal two-body
relaxation until the present time (see also Ma et al. 2006). In particular,
this is driven by the observation that if the IMF is too shallow, i.e., if a
cluster is significantly depleted in low-mass stars compared to (for instance)
the solar neighborhood, it will disperse within a few orbital periods around
its host galaxy’s center, and most likely within about a Gyr of its formation
(e.g., Gnedin & Ostriker 1997; Goodwin 1997; Smith & Gallagher 2001; Mengel et
al. 2002; Rose, Kouwenhoven & de Grijs, in prep.).
From the recent work of Ma et al. (2006), the intrinsically most luminous M31
GC, 037-B327, has been suggested to be the most massive GC in the Local Group,
with a total mass of $\sim(3.0\pm 0.5)\times 10^{7}M_{\odot}$, also determined
photometrically and somewhat depending on the SSP models used, the metallicity
and age adopted, and the IMF representation. However, Cohen (2006) pointed out
that the photometric mass of this cluster had likely been overestimated due to
an incorrect extinction correction. Nevertheless, she also confirmed the
nature of 037-B327 as one of the most massive GCs in the Local Group, with a
dynamical mass similar to that of G1. It is intriguing that these two most
massive GC in M31 both are significantly more massive than the most massive
Galactic GC, $\omega$ Cen [$\sim(2.9-5.1)\times 10^{6}M_{\odot}$; Meylan
(2002)].
In fact, the high mass of these clusters raises additional, intriguing
questions regarding the nature of these objects in general, and of G1 in
particular (see also Federici et al. 2007; Ma et al. 2007b, and references
therein). It has been speculated that these objects may be nucleated dwarf
galaxies instead of genuine GCs. In support of this notion, we point out that
Gieles et al. (2006) suggest that there may be a physical upper limit to the
mass of a star cluster that is not merely the result of size-of-sample
effects. This maximum mass depends to some extent on the galactic environment;
for their example galaxies, M51 and the Antennae system, they find a physical
upper limit to the stellar mass of $\sim(10^{5}-10^{6})M_{\odot}$. Our values
derived for both the photometric and the virial mass of G1 are well above
these suggested upper mass limits. This may, therefore, provide an additional
(although circumstantial) proverbial nail in the coffin of G1 as a normal GC.
### 3.4 Luminosity of the hot, extreme horizontal-branch stars
As discussed in §3.3, none of the BC03 SSP models fit the photometric data
point in the GALEX FUV band as well as the other nine data points. EHB stars
may be responsible for this excess in FUV band. In fact, Rich et al. (1996)
found some bluer horizontal-branch stars extending to $(V-I)=0.0$ mag, based
on their HST/WFPC2 observations. In this section, we will calculate the
luminosity of the EHB stars. We assume that the excess in the GALEX FUV band
is solely due to these EHB stars. The magnitude differences between the four
SSP models employed in this paper and the photometric data points are 0.63,
1.00, 0.56, and 0.94 mag, respectively, corresponding $(2.44,1.07,2.11,\mbox{
and }1.00)\times 10^{2}L_{\odot}$, respectively, with a mean number of EHB
stars in G1 of $(1.65\pm 0.63)\times 10^{2}L_{\odot}$.
### 3.5 Comparison between G1 and S312
Brown et al. (2004) showed that a 10 Gyr old population in M31 has a main-
sequence turnoff at about $m_{\rm F814W}=28.8$ mag. Their deep observations
needed exposures of 39.1 hours in F606W and 45.5 hours F814W, spanning 120
orbits of HST/ACS imaging observations (Brown et al. 2003, 2004). In the
future, such deep HST observations will only be obtained for a very small
number of fields (e.g., Rich et al. 2005). As discussed in §1, Ma et al.
(2007a) constrained the age of the M31 GC S312 by comparing multicolor
photometry and theoretical stellar population synthesis models. It is
encouraging that the age obtained by Ma et al. (2007a) is in good agreement
with the previous determination based on main-sequence photometry (Brown et
al. 2004), i.e., $9.5^{+1.15}_{-0.99}$ Gyr versus $10^{+2.5}_{-1}$ Gyr. S312
is one of the few extragalactic GCs for which the age can be determined from
main-sequence photometry. By comparing the ages of S312 and G1 determined
using the same method, we can conclude that S312 is younger than the majority
of the Galactic GCs at the same metallicity, and G1 is as old as the oldest
Galactic GCs. In fact, if we try to fit the intrinsic SEDs of G1 by the
theoretical BC03 SSP SEDs at an age of 10 Gyr, the resulting fit is very poor
indeed, particularly in the ultraviolet. Therefore, we conclude that the
method used in this paper, by which the ages of both S312 and G1 have been
determined, can be used to determine the ages of old stellar populations to
satisfactory precision, and in particular to distinguish between young and old
populations.
## 4 Summary and Conclusions
In this paper, we first determined the age and mass of the M31 GC G1, as well
as the realistic uncertainties associated with these estimates, by comparing
its multicolor photometry with theoretical stellar population synthesis
models. Our multicolor photometric data were obtained from GALEX FUV and NUV,
broad-band optical $UBVRI$, and 2MASS $JHK_{\rm s}$ observations, which form
an SED covering the wavelength range from 2267 to 20,000Å. Our results confirm
that G1 is one of the oldest and most massive GCs in the Local Group – that
is, if it is indeed a genuine GC given that its mass is well in excess of the
physical maximum mass predicted by the models of Gieles et al. (2006).
The age and mass obtained in this paper are somewhat dependent on the SSP
model adopted. It is evident that the age of $18.23\pm{1.76}$ Gyr for G1 based
on the BC03 models is greater than the currently accepted age of the Universe.
However, we must keep in mind that the BC03 SSP models were calculated for
ages up to 20 Gyr. In fact, ages derived for objects such as GCs and galaxies
in excess of that of the Universe only mean that these objects are among the
oldest objects in the Universe.
In the context of the BC03 models and their associated age range up to 20 Gyr,
our derived age for G1 is consistent with the suggestion by the WMAP team that
the oldest GCs may have formed within the first 1.7 Gyr after the Big Bang
(see §1).
The integrated FUV flux depends mainly on the fractional number of horizontal-
branch (HB) stars with temperatures hotter than $T_{e}\sim 10,000$ K, with a
modest dependence on their temperature distribution (see Rey et al. 2007 and
references therein). Older GCs produce more of these hot HB stars and they are
thus more likely to produce stronger FUV fluxes at a given metallicity (see
Rey et al. 2007 and references therein) Lee et al. (2003) showed that the
addition of FUV photometry to optical data can discriminate cleanly among
young ($<$1 Gyr), intermediate-age (3–5 Gyr), and old ($>$ 14 Gyr) GCs. Young
and very old GCs exhibit a significant FUV-to-optical spectral continuum
slope, but intermediate-age clusters are relatively faint in the FUV (see fig.
2 of Lee et al. 2003). Figure 6 of Rey et al. (2007) implies that the age of
G1 may be similar to that of the oldest GCs.
Overall, we therefore conclude that G1 is indeed among the oldest and most
massive building blocks of M31, and provides a key limitation to the age of
the Universe, although we caution that our results also provide circumstantial
support to the suggestion that the cluster may not be a genuine GC.
## Acknowledgments
We are indebted to the referee for thoughtful comments and insightful
suggestions that improved this paper greatly. This work has been supported by
the Chinese National Natural Science Foundation through Grant Nos. 10873016,
10803007, 10473012, 10573020, 10633020, 10673012, and 10603006; and by
National Basic Research Program of China (973 Program) No. 2007CB815403. RdG
acknowledges partial financial support from the Royal Society in the form of a
UK-China International Joint Project. SCR acknowledges partial support from
KOSEF through the Astrophysical Research Center for the Structure and
Evolution of the Cosmos (ARCSEC). This paper makes use of data from the Two
Micron All Sky Survey, which is a joint project of the University of
Massachusetts and the Infrared Processing and Analysis Center, funded by NASA
and the National Science Foundation. This paper is also partially based on
archival observations with the NASA/ESA Hubble Space Telescope, obtained at
the Space Telescope Science Institute (STScI), which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-26555. This research has made use of NASA’s Astrophysics Data
System Abstract Service. This reasearch is partially based on archival data
from the NASA GALEX mission developed in cooperation with the Centre National
d’Etudes Spatiales of France and the Korean Ministry of Science and
Technology.
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|
arxiv-papers
| 2009-04-04T03:02:40 |
2024-09-04T02:49:01.678517
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Ma, Richard de Grijs, Zhou Fan, Soo-Chang Rey, Zhenyu Wu, Xu Zhou,\n et al",
"submitter": "Jun Ma",
"url": "https://arxiv.org/abs/0904.0674"
}
|
0904.0700
|
LPTh-Ji 09/002
Sphalerons on Orbifolds
Amine Ahriche
LPTh, University of Jijel, PB 98, Ouled Aissa, DZ-18000 Jijel, Algeria.
LPMPS, University of Constantine, Ain El-Bey, DZ-25000 Constantine, Algeria.
Faculty of Physics, University of Bielefeld, Postfach 100131, D-33501
Bielefeld, Germany.
Abstract
In this work, we study the electroweak sphalerons in a 5D background, where
the fifth dimension lies on an interval. We consider two specific cases: flat
space-time and the anti-de Sitter space-time compactified on $S^{1}/Z_{2}$. In
our work, we take the $SU(2)$ gauge-Higgs model, where the gauge fields reside
in the 5D bulk; but the Higgs doublet is confined in one brane. We find that
the results in this model are close to those of the 4D Standard Model (SM).
The existence of the warp effect, as well as the heaviness of the gauge
Kaluza-Klein modes make the results extremely close to the SM ones.
Keywords: Sphalerons, Kalauza-Klein modes, Warp Factor.
## 1 Introduction
The Standard Model (SM) of the electroweak and strong interactions has been
very successful in describing nature at energies around the electroweak scale
($\sim 100~{}$GeV). However, it fails in answering many fundamental questions
in particle physics, like, e.g., the hierarchy problem and the neutrino mass
and its smallness, as well other problems related to cosmology like the baryon
asymmetry in the universe and dark matter. Therefore a more fundamental
theory, which describes nature at higher scales, needs to become known to
explain the problems of particle physics and related topics.
It has been realized that the hierarchy problem could be a consequence of the
existence of extra dimensions [1]. A popular realization of this concept is
the so-called Randall-Sundrum model [2]. There are several variants of this
scenario, depending on whether the extra dimension is finite (RS1) or infinite
(RS2), and on which of the fields is confined in a brane or lying on the bulk.
In the RS1 models, the space-time has the 5D anti-de Sitter ($AdS_{5}$)
geometry
$\displaystyle ds^{2}$
$\displaystyle=g_{MN}dx^{M}dx^{N}=a^{2}\left(y\right)\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}$
(1) $\displaystyle=e^{-2ky}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2},$ (2)
where $y$ is the fifth dimension that has the properties $y\equiv y+2\pi R$;
and $y\equiv-y$; it is compactified on a half-circle $S^{1}/Z_{2}$ with two 4D
boundaries ($y=0,\pi R$). The metric $\eta_{\mu\nu}=diag(-1,1,1,1)$ is the
usual 4-dimensional one; and $k$ is the $AdS_{5}$ curvature. In this model,
the relation between the Planck and the TeV scales seems to be natural,
TeV$\sim w^{-1}M_{Pl}=e^{-\pi kR}M_{Pl}$, where the two fixed points of the
fifth dimension $y=0,\pi R$ represent the Planck and the TeV branes,
respectively.
In the first paper [2], only gravity resides in the 5D bulk, while the SM
fields are confined in the TeV brane. But problems with some of the SM fields
are that propagating in the bulk were also considered, like the case of gauge
fields [3, 4], scalars [5], fermions [6], the whole SM content [7]; and even
supersymmetry [8].
As mentioned above, the SM fails to explain the origin of matter in the
universe [9], it does not fulfill the second and the third Sakharov criteria
for baryogenesis [10]. Although, the first criterion, baryon number violation,
is achieved through the B+L anomaly [11], where both of the baryon and lepton
numbers are violated by 3 units due to the possible transition between two
equivalent neighboring vacua of the nontrivial topology of the SU(2) model. It
was shown [11] that this transition probability is extremely suppressed, $\sim
10^{-162}$, but this is not the case at higher temperatures. The rate of $B$
violating processes is proportional to $T^{4}$ at the symmetric phase [12] and
suppressed like $e^{-E_{Sp}/T}$ in the broken phase [13], where $E_{Sp}$ is
the system’s static energy within the so-called sphaleron configuration [14,
15]; a field configuration that corresponds to the top of the barrier between
two neighboring vacua. Due to their relevance to the electroweak baryogenesis
scenario [9], sphalerons were extensively studied in the literature in
extended SM variants as in the SM with a singlet [16, 17], the Minimal
Supersymmetric Standard Model [18]; and in the next-to-Minimal Supersymmetric
Standard Model [19].
In this work, we will study the sphaleron configuration for a SU(2) gauge-
Higgs model in a 5D background, where the gauge fields propagate in the 5D
bulk and the Higgs doublet is confined in a brane. We will focus on the warp
effect, by comparing the $AdS_{5}$ results with the flat geometry case. In the
second section, the model is shown, where the equations of motion (EOM) for
the Higgs field and the Kaluza-Klein (KK) gauge modes are given. The sphaleron
configuration within this model is expressed in section three. In the fourth
section, we show the profile functions of the gauge and Higgs fields, as well
the values of the sphaleron energy in different cases. These results will be
compared by those of the SM. Finally, we give our conclusion.
## 2 SU(2) Gauge Fields in the Bulk
Let us consider a SU(2) Higgs model in the 5D background (2), with a general
warp factor $a(y)$. The warp factor $a\left(y\right)=1$ refers to the 5D flat
geometry; and $a\left(y\right)=e^{-ky}$ refers to the AdS5 one. We have
$\mu=0,3$ and $M=\mu,5$. In our model, only the gauge fields propagate in the
bulk and the Higgs field is confined in one brane. The action that obeys the
symmetry is
$S=\int
d^{4}xdy\sqrt{G}\left\\{\mathcal{L}_{bulk}+\Delta(y)\mathcal{L}_{brane}\right\\},$
(3)
with $G=\det(g_{MN})$, and $\Delta(y)\equiv
2\delta\left(y\right),2\delta\left(y-\pi R\right)$ refers to the Higgs
localization in the Planck or TeV branes respectively. The boundary Lagrangian
is given by
$\mathcal{L}_{brane}=g^{\mu\nu}\left(D_{\mu}H\right)^{{\dagger}}\left(D_{\nu}H\right)-V\left(H^{{\dagger}}H\right),$
(4)
with the covariant derivative
$D_{M}H=\left(\partial_{M}-\frac{i}{2}g_{5}\sigma^{a}A_{M}^{a}\right)H;$ (5)
and $g_{5}=g\sqrt{\pi R}$ is the 5D SU(2) dimensionful gauge coupling, where
$g$ is the 4D one . The bulk Lagrangian is given by
$\mathcal{L}_{bulk}=-\frac{1}{4}g^{MN}g^{QW}F_{MQ}^{a}F_{NW}^{a},$ (6)
where the 5D field strength is given by
$F_{MN}^{a}=\partial_{M}A_{N}^{a}-\partial_{N}A_{M}^{a}+g_{5}\epsilon^{abc}A_{M}^{b}A_{N}^{c}.$
(7)
In what follows, we work in the gauge ($\partial^{\mu}A_{\mu}^{a}=0,$
$A_{5}^{a}=0$). The scalar potential has the usual Mexican hat form
$V\left(HH^{{\dagger}}\right)=\lambda\left(H^{{\dagger}}H-\upsilon^{2}/2\right)^{2},$
(8)
where $\upsilon$ is the Higgs vev. The equations of motion (EOM) can be
obtained by the vanishing of the action variation, $\delta S=0$, and we get
$\displaystyle\Delta(y)a^{4}\left(y\right)\left[g^{\mu\nu}D_{\mu}D_{\nu}H+\frac{\partial}{\partial
H^{{\dagger}}}V\left(H^{{\dagger}}H\right)\right]=0,$ (9)
$\displaystyle\frac{i}{2}g_{5}\Delta(y)a^{4}\left(y\right)\left[H^{{\dagger}}\sigma^{a}D_{\mu}H-\left(D_{\mu}H\right)^{{\dagger}}\sigma^{a}H\right]-\partial_{5}a^{2}\left(y\right)\partial_{5}A_{\mu}^{a}+\eta^{\alpha\beta}\partial_{\beta}F_{\alpha\mu}^{a}=0,$
(10)
with the boundary condition $\partial_{5}A_{\mu}^{a}=0$ at both boundaries,
$y=0,\pi R$. The gauge fields have to be factorized using the KK decomposition
as
$A_{\mu}^{a}\left(x,y\right)=\sum\limits_{n}A_{\mu}^{a(n)}\left(x\right)\chi^{(n)}(y),$
(11)
with
$\int_{0}^{\pi R}\chi^{(n)}(y)\chi^{(m)}(y)dy=\delta_{nm}.$ (12)
Then, the functions $\chi^{(n)}$ should be the eigenstates of the operator
$-\partial_{5}a^{2}\left(y\right)\partial_{5}\chi^{(n)}=M_{n}^{2}\chi^{(n)},$
(13)
with the condition $\partial_{5}\chi^{(n)}=0$ at both boundaries; $M_{n}$ are
the KK masses. The zero mode $\chi^{(0)}\left(y\right)=1/\sqrt{\pi R}$; does
not depend on the space-time geometry. In flat space-time, the heavy modes
(13) are given by
$\chi^{(n)}\left(y\right)=\sqrt{\frac{2}{\pi
R}}\cos\left(\frac{2ny}{R}\right),$ (14)
with the eigenvalues $M_{n}^{2}=4n^{2}/R^{2}$. However, in the $AdS_{5}$
space-time, they have the form111This result is given in many works, like for
e.g. [3, 4] and [20].
$\displaystyle\chi^{(n)}(y)$
$\displaystyle=\frac{e^{ky}}{a_{n}}\left[J_{1}\left(\alpha_{n}e^{ky}\right)-b_{n}Y_{1}\left(\alpha_{n}e^{ky}\right)\right],$
(15) $\displaystyle b_{n}$
$\displaystyle=J_{0}\left(\alpha_{n}\right)/Y_{0}\left(\alpha_{n}\right),$
(16)
with $\alpha_{n}=M_{n}/k$, and $J_{i}$ and $Y_{i}$ are the $i-th$ order Bessel
functions of first and second kind, respectively; and $a_{n}$ is a
normalization factor which is computed using (12):
$a_{n}^{2}=\left.\frac{e^{2ky}}{2k}\left\\{J_{1}\left(\alpha_{n}e^{ky}\right)-b_{n}Y_{1}\left(\alpha_{n}e^{ky}\right)\right\\}^{2}\right|_{y=0}^{y=\pi
R}.$ (17)
The eigenvalues $M_{n}$ are determined by imposing the boundary condition
$\left.\partial_{5}\chi^{(n)}=0\right|_{y=\pi R}$, which are the zeros of the
quantity
$Y_{0}\left(\alpha_{n}e^{\pi
kR}\right)J_{0}\left(\alpha_{n}\right)-J_{0}\left(\alpha_{n}e^{\pi
kR}\right)Y_{0}\left(\alpha_{n}\right).$ (18)
These eigenvalues can be obtained numerically.
When inserting (11) in (3) and integrating over $y$, we get a 4D Lagrangian
$\mathcal{L}_{\mathit{4D}}$ as a function of the Higgs doublet and an infinite
number of gauge KK modes. The Higgs doublet is coupled to the KK modes through
the parameters $\tau_{i}$. In addition to the quartic couplings between the KK
modes, which are characterized by the parameters $\xi_{ijkl}$, there exist
also new cubic couplings characterized by $\gamma_{ijk}$. This feature does
exist only in non-Abelian theories unlike in the Abelian case [3, 4]. The 4D
Lagrangian is given explicitly in the appendix.
There are some geometry-independent properties of these parameters, like the
invariance under the permutation between each two indices. Also we have the
equalities: $\gamma_{ij0}=\xi_{00ij}=\delta_{ij}$. The 4D SM can be recovered
by keeping only zero modes in (32), since all the indices of zeroth order in
(33) are exactly $1$, whatever the nature of space-time.
The physics at the electroweak scale is more sensitive to the first (and maybe
the second) KK mode interactions; therefore, we will give in the appendix only
the numerical values of the coupling of heavy modes with the first and second
KK modes. The existence of the warp factor makes a difference in the masses of
the KK modes ($M_{i}$) and their couplings ($\gamma_{ijk}$ and $\xi_{ijkl}$).
In what follows, we will investigate the behavior of the sphaleron
configuration with respect to these differences.
## 3 Sphaleron Solutions
It was shown that the 5D anomaly is independent of the bulk physics; the
cancelation of the 4D anomaly is sufficient to eliminate the 5D one in
orbifold theories [21]. Then the problem of fermionic current non-conservation
can be treated as in a 4D theory. In the case of a 5D fermion coupled to an
external gauge potential $A_{M}^{a}(x,y)$ on an $S^{1}/Z_{2}$ orbifold, the
divergent current is given by [21]
$\partial_{M}\mathbf{J}^{M}(x,y)=\frac{1}{2}[\delta(y)+\delta(y-\pi
R)]F^{a\mu\nu}\tilde{F}_{\mu\nu}^{a}/16,$ (19)
where $\mathbf{J}^{M}$ is the 5D fermionic current and
$\tilde{F}_{\mu\nu}^{a}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}F^{a\alpha\beta}$
is the dual field strength. The last term in (19) represents the usual 4D
chiral anomaly for a Dirac fermion in an external gauge potential
$A_{M}^{a}(x,y)$. Since the fermions in our model are confined in one brane,
the expression (19) becomes, after the integration over the fifth coordinate
$y$, like the usual 4D formula,
$\partial_{\mu}J^{\mu}(x)=F^{a\mu\nu(0)}\tilde{F}_{\mu\nu}^{a(0)}/32,$ (20)
where the label $(0)$ means that only zero modes are taken into account [21];
and $J^{\mu}$ is the 4D fermionic current. This means that there is no new
contribution to the fermionic currents divergences beside the 4D ones. In our
model, the Higgs doublet potential on the brane admits of a minimum, therefore
the static energy is bounded from below. In this case, NCS=1/2 represents the
so-called sphaleron configuration [14, 15].
Our system has a 5D SU(2) gauge symmetry; it is invariant under the gauge
transformation
$H\rightarrow UH,~{}i\frac{g}{2}\sigma^{a}A_{M}^{a}\rightarrow
i\frac{g}{2}\sigma^{a}A_{M}^{a}+\partial_{M}UU^{{\dagger}},$ (21)
where $U$ is a SU(2) element. In the gauge $A_{5}^{a}=0$, the matrix U should
be independent of the fifth dimension; and only the zero mode will ensure the
SU(2) gauge invariance. This means that the sphaleron configuration can be
defined for the system ($H$,$~{}A^{a(0)}$) using the 4D transformation matrix
$U(\mu,x)$ [15],
$U\left(\mu,x\right)=\left(\begin{array}[c]{cc}e^{i\mu}\left(\cos\mu-i\sin\mu\cos\theta\right)&e^{i\varphi}\sin\mu\sin\theta\\\
-e^{-i\varphi}\sin\mu\sin\theta&e^{-i\mu}\left(\cos\mu+i\sin\mu\cos\theta\right)\end{array}\right);$
(22)
but this system ($H$,$~{}A^{a(0)}$) is coupled to the heavy KK modes
$A^{a(n\neq 0)}$; this effect will be investigated in this work. The sphaleron
configuration can be obtained by making $\mu=\pi/2$.
For reasons of simplicity, we will not use the sphaleron configuration [15],
but another, equivalent, representation [22]:
$H\left(x\right)=\frac{\upsilon}{\sqrt{2}}L\left(r\right)\left(\begin{array}[c]{c}0\\\
1\end{array}\right),~{}A_{0}^{a}=0,~{}A_{k}^{a}\left(x,y\right)=2\frac{\epsilon_{akj}x_{j}}{gr^{2}}\sum_{i}\left[1-f^{(i)}\left(r\right)\right]\chi^{(i)}\left(y\right).$
(23)
Here the heavy modes are represented by a similar form as the zero one in
order to make the generalization of the orthogonal gauge $x_{i}A_{i}^{a}=0$
consistent for all the KK modes.
Then, when inserting (23) in (9) and (10), we get the differential equations
governing the $f^{(i)}(r)$ modes and $L(r)$. The field’s profile functions $L$
and $f^{(i)}$ are given by the solutions of the system
$\displaystyle\frac{\partial}{\partial\zeta}\zeta^{2}\frac{\partial}{\partial\zeta}L=2L\sum\limits_{n}\sum\limits_{m}\tau_{n}\tau_{m}\left(1-f^{(n)}\right)\left(1-f^{(m)}\right)+\frac{\lambda}{2g^{2}}\zeta^{2}L\left(L^{2}-1\right),$
(24)
$\displaystyle\begin{array}[c]{c}\zeta^{2}\frac{\partial^{2}}{\partial\zeta^{2}}f^{(i)}=-\frac{\zeta^{2}}{4}L^{2}\tau_{i}\sum\limits_{m}\tau_{m}\left(1-f^{(m)}\right)-2\left(1-f^{(i)}\right)-\zeta^{2}\frac{M_{i}^{2}}{g^{2}\upsilon^{2}}\left(1-f^{(i)}\right)+6\sum\limits_{m}\sum\limits_{k}\gamma_{imk}\left(1-f^{(m)}\right)\left(1-f^{(k)}\right)\\\
-4\sum\limits_{m}\sum\limits_{k}\sum\limits_{l}\xi_{imkl}\left(1-f^{(m)}\right)\left(1-f^{(k)}\right)\left(1-f^{(l)}\right),\end{array}$
(27)
where $\zeta=g\upsilon r$ is the dimensionless radial coordinate, $M_{i}$ are
the KK modes eigenmasses; and the $\tau_{i}$ parameters, $\gamma_{ijk}$ and
$\xi_{ijkl}$ are given in the appendix. Here, one needs to mention that the
equations (24), (27) and (28) are referring to both cases where the Higgs
doublet is localized on the Planck or TeV branes. Here one needs to mention
that in the TeV brane case, the Higgs doublet as well the 4D brane parameters
needs to be redefined (for e.g. $a(\pi R)H\rightarrow H$) in order to be
canonically normalized.
The static energy of the system is given by
$\begin{array}[c]{c}E=\frac{4\pi\upsilon}{g}\int_{0}^{\infty}d\zeta\left[\frac{\zeta^{2}}{2}\left(\frac{\partial}{\partial\zeta}L\right)^{2}+\frac{\lambda}{g^{2}}\frac{\zeta^{2}}{4}\left(L^{2}-1\right)^{2}+L^{2}\sum\limits_{n}\sum\limits_{m}\tau_{n}\tau_{m}\left(1-f^{(n)}\right)\left(1-f^{(m)}\right)\right.\\\
+4\sum\limits_{n}\left\\{\left(\frac{\partial}{\partial\zeta}f^{(n)}\right)^{2}+\left[\frac{2}{\zeta^{2}}+\frac{M_{n}^{2}}{g^{2}\upsilon^{2}}\right]\left(1-f^{(n)}\right)^{2}\right\\}-\frac{16}{\zeta^{2}}\sum\limits_{n}\sum\limits_{m}\sum\limits_{k}\gamma_{nmk}\gamma_{nmk}\left(1-f^{(m)}\right)\left(1-f^{(k)}\right)\left(1-f^{(n)}\right)\\\
+\left.\frac{8}{\zeta^{2}}\sum\limits_{n}\sum\limits_{m}\sum\limits_{k}\sum\limits_{l}\xi_{nmkl}\left(1-f^{(n)}\right)\left(1-f^{(m)}\right)\left(1-f^{(k)}\right)\left(1-f^{(l)}\right)\right].\end{array}$
(28)
When comparing equations (24), (27) and (28) with their corresponding
equations in [15]; we find that instead of the gauge profile function $f$, we
have a summation over an infinite number of $f^{(i)}$; and also the Higgs-
gauge, cubic and quartic gauge-gauge couplings get modified as
$\begin{array}[c]{c}L^{2}\left(1-f\right)^{2}\rightarrow\sum\limits_{m}\tau_{n}\tau_{m}L^{2}\left(1-f^{(n)}\right)\left(1-f^{(m)}\right),\\\
\left(1-f\right)^{3}\rightarrow\sum\limits_{m}\sum\limits_{k}\gamma_{nmk}\left(1-f^{(n)}\right)\left(1-f^{(m)}\right)\left(1-f^{(k)}\right),\\\
\left(1-f\right)^{4}\rightarrow\sum\limits_{m}\sum\limits_{k}\sum\limits_{l}\xi_{nmkl}\left(1-f^{(n)}\right)\left(1-f^{(m)}\right)\left(1-f^{(k)}\right)\left(1-f^{(l)}\right),\end{array}$
(29)
in addition to the presence of mass terms for non-zero gauge KK modes. Indeed,
when neglecting the massive gauge KK modes, the EOM (24) and (27) tend to
(11); and (28) tends to (10) in [15].
The convergence of the energy functional (28) implies the following boundary
conditions on the profiles functions $L$ and $f^{(i)}$.
$\displaystyle\mathit{For}\mathit{~{}}\zeta$ $\displaystyle\rightarrow
0:~{}L\sim\zeta;~{}~{}f^{(0)}\sim\zeta^{2};~{}~{}f^{(i)}\sim 1,$ (30)
$\displaystyle\mathit{and}\mathit{~{}}\zeta$
$\displaystyle\rightarrow\infty:~{}L\sim 1;~{}f^{(0)}\sim 1;~{}f^{(i)}\sim 1.$
(31)
We use the relaxation method to integrate this system of differential
equations. The infinite summations in (24), (27) and (28) over the gauge KK
modes are practically impossible analytically as well as numerically. We
expect that the contributions of the heavy gauge KK modes ($n\geq 1$) are just
corrections to the energy of the system ($H,$ $A^{a(0)}$); we will consider
only a finite number $N$ of the KK modes and then examine the variation the
energy (28), as well as the profile functions $L$ and $f^{(n)}$ with respect
to this number $N$ for both cases of flat and warped geometries, with
different values of the warp factor and the first KK mass.
## 4 Numerical Results and Discussion
In our computations, we will take the Higgs mass to be around $120$ GeV, i.e.,
$\lambda\simeq 0.12$. For a rigorous comparison between the flat and warped
cases, we fix the mass of the first heavy KK mode, which represents in a way
the scale of the new physics beyond SM, and we will consider the values $600$
GeV, $2$ TeV and $10$ TeV. In general, the warp factor $w=e^{\pi kR_{w}}$
value is chosen in a way as to represent the hierarchy between the Planck and
TeV scales, i.e. $w\sim 10^{16}$. But since we are interested also to
investigate its effect on the sphaleron configuration, we will vary the size
of the extra dimension to give it different values for the warp factor:
$w=10^{4}$, $10^{8}$ and the desired one, $10^{16}$.
Figure 1: The masses of the gauge KK modes for the cases of flat and warped
geometry with different values of the warp factor w.
In Fig. 1, the masses of KK modes are shown for both flat and warped
backgrounds, where the first KK heavy mode mass is chosen to be $1$ TeV. It is
clear that the flat modes are just multipliers of the first heavy one, while
the existence of the warp factor makes the warped mode masses increasing with
respect to the warp factor $w$.
For the Higgs-gauge and gauge-gauge couplings, they are given in unit of the
SU(2) coupling $g$; by the parameters $\tau$, $\gamma$ and $\xi$. All these
parameters are of order $\mathcal{O}(1)$ in the flat geometry. In warped
geometry, the situation is different, the $\tau$ parameters; that represent
the couplings of the Higgs with gauge KK modes, depend on which boundary the
Higgs filed is located in. If the Higgs field is located in the Planck brane,
these parameters are negative and their modulus is less than unity and
decaying with respect to the KK masses, and also with respect to the warp
effect. If the Higgs doublet is located in the TeV brane, the values of the
$\tau$ parameters are of the order $\mathcal{O}(1)$ but positive for odd modes
and negative for the even ones; and their modules are almost stable with
respect to the KK masses. The previous difference between the two cases will
not change significantly the profile functions of $L$ and $f^{(i)}$ or the
sphaleron energy (28). The difference between the sphaleron energy in both
cases is less than $0.004$% for $w=10^{16}$ and $M_{1}=1$ TeV.
The $\gamma$ parameters that describe the cubic couplings between the gauge KK
modes are also small in the $AdS_{5}$ background and decaying with respect to
the KK masses. However, the $\xi$ parameters that represent the quartic
couplings between the gauge KK modes are large (for e.g. $\xi_{1,1,1,1}\sim
46$) and decaying with respect to the KK masses but still remaining large (for
e.g. $\xi_{30,30,30,30}\sim 27$).
Figure 2: The profile functions $L$ (upper curve) and $f^{(0)}$ (lower curve)
for the SM case, flat geometry and the warped geometry as a function of the
dimensionless radial coordinate $\zeta$. Each profile function is almost
identical for the different cases. This plot was performed taking into account
the first 10 heavy KK modes for both flat and warped geometries for $M_{1}=1$
TeV and $w=10^{16}$.
The profile functions $L$ and $f^{(0)}$ are given in Fig. 2. They are very
close to the SM ones to a very high precision for both the cases of flat and
warped geometries. This feature does not depend on $N$, the number of the
heavy modes taken into account to solve (24) and (27). However, the profile
functions of the heavy modes $f^{(i)}$, as shown in Fig. 3, are just
deviations from 1; and these deviations decrease with respect to the KK
masses.
Figure 3: From up to down, here are the profile functions $f^{(i)}$, of the
first five heavy modes for the flat geometry case (up) and warped geometry
(down) for the same values of $M_{1}$ and $w$ taken in Fig. 2, as a function
of the dimensionless radial coordinate $\zeta$.
We remark that the profile functions of the heavy modes $f^{(i)}$, are more
suppressed in the case of warped geometry than in the flat one. However, the
suppression effect decreases if we decrease the warp factor; for, e.g., when
taking the warp factor to be $w=10^{4}$ instead of $10^{16}$, the maximum of
$f^{(1)}$ (the upper curve in the right side of Fig. 3) increases from
$1.00037$ to $1.00075$. This suppression increases also if we increase the
first KK mode mass.
Due to the fact that the profile functions of $L$ and $f^{(0)}$ practically do
not change with respect the SM results, and in addition to the suppression of
the heavy modes profile functions, one expects that the sphaleron energy
should not be very different from the SM value, but this is not guaranteed due
to the infinite number of terms in Eq. (28), as well the increasing KK mass
values, unless confirmed numerically.
To check this, we compute the sphaleron energy (28) taking into account a
finite number $N$ of KK modes for the different values of the first heavy KK
mode mass and the warped factor mentioned above. The sphaleron energy
dependence on the index $N$ is shown in Fig. 4.
Figure 4: The dependence of the sphaleron energy on the number of heavy KK
modes that are taken into account to estimate (28); for different values of
the first KK mode mass.
The first remark on the results in Fig. 4; is that the sphaleron energy does
differ significantly from the SM value; its largest deviation is in the case
of a small mass of the first KK heavy mode with flat geometry (first plot in
Fig. 4), which is $-0.06~{}\%$, i.e. much less than $1~{}\%$. Also, the
existence and largeness of the warp factor makes the sphaleron energy
practically identical to the SM value. However, this feature is due to the
sphaleron configuration itself, i.e. $(H,A_{\mu}^{(0)})$, rather than the
decoupling effect of the heavy KK modes, because if we consider an extreme
case of a flat geometry with a small mass for the first KK mode (for e.g.
$300$ GeV, and then $100$ GeV), the sphaleron energy decreases only by
$-0.9~{}\%$ and $-6~{}\%$, respectively. This can be explained by the fact
that most of the sphaleron energy is coming from the contributions of the
gauge zero mode and the Higgs fields; and the profile functions of these
fields are determined by self-interactions as well as interactions with each
other rather than their interactions with the heavy KK modes. Then one can say
that the heavy KK modes are just compensating fields in the EOM (24) and (27),
as in the case of the singlet in the model of SM+singlet [17]. This could
explain the fact that the contributions of the KK modes to the sphaleron
energy (28) are very small even though their cubic ($\gamma$) and quartic
($\xi$) coupling are (very) large. Indeed, the sphaleron energy (28) is more
sensitive to the first KK eigenmass rather than to the couplings $\tau$,
$\gamma$ and $\xi$.
At finite temperature, we do not expect to have a deviation in sphaleron
field’s profile functions as well as in the values of sphaleron energy from
the results of the SM [13]; and the B+L anomaly is almost the same as in the
standard theory. Then the criterion for a strongly first-order phase
transition remains the known one, $\upsilon_{c}/T_{c}\geq 1$ [23].
## 5 Conclusion
In this work, the sphaleron configuration for a Higgs model in a 5D space-time
is studied, where the Higgs is confined in a brane and the gauge field resides
in the 5D bulk. When we made the KK decomposition of the gauge field, we found
that possible interactions (cubic and quartic) between different KK modes are
possible due to the non-Abelian nature of the symmetry group unlike the
Abelian case [3, 4]. The strength of these interactions depends on the space-
time nature. The strength of the interaction with the Higgs doublet depends on
where it is located in.
We defined the sphaleron configuration in this case, where we got the
equations like the SM case, but corrected by the existence of the KK heavy
modes. Practically the profile functions of the Higgs and zero mode gauge
fields do not change when comparing with the SM results; and the heavy mode
profile functions are just little deviations from 1\. The suppression of this
deviation from unity is proportional to the KK order. Also the existence of a
strong warp factor (like $w=10^{16}$) suppresses these deviations by one order
of magnitude.
We checked also that the sphaleron energy has the same value as the SM one.
The heavy KK modes do not practically contribute to the sphaleron energy; and
their presence decreases the value of sphaleron energy by $-0.25\%$ for a
light mass of the first KK heavy mode (600 GeV) in a flat geometry. The
existence of a warp factor; or the increasing of the mass of the first KK
heavy mode, which represents somehow the new physics scale, suppresses the
deviation from the SM results.
This allows us to suppose that at finite temperature, the previous results
should differ from those of the SM. In addition to the fact that the 5D B+L
anomaly is identical to the 4D one, the criterion of a strong first-order
phase transition, $\upsilon_{c}/T_{c}\geq 1$, is still valid for these models.
Acknowledgements: I want to thank Mikko Laine for his useful comments as well
for the warm hospitality at Bielefeld University. This work was supported by
both the German Academic Exchange Service (DAAD) and the Algerian Ministry of
Higher Education and Scientific Research under the cnepru-project D0092007148.
## Appendix A Explicit 4D Lagrangian
The 4D theory can be obtained by integrating over the fifth dimension. Here we
explicitly give the 4D Lagrangian with its different parameters that describe
the couplings of the gauge KK modes with themselves as well as with the Higgs
doublet. It is given by
$\begin{array}[c]{l}\mathcal{L}_{\mathit{4D}}=\eta^{\mu\nu}\partial_{\mu}H^{{\dagger}}\partial_{\nu}H-V\left(H^{{\dagger}}H\right)-\frac{i}{2}g\eta^{\mu\nu}\left[\partial_{\nu}H^{{\dagger}}\sigma^{a}H-H^{{\dagger}}\sigma^{a}\partial_{\nu}H\right]\sum\limits_{n}\tau_{n}A_{\mu}^{a(n)}+\frac{1}{2}\eta^{\mu\nu}\sum\limits_{n}\sum\limits_{m}(\tau_{n}\tau_{m}\frac{g^{2}}{2}H^{{\dagger}}H\\\
+\delta_{nm}M_{n}^{2})A_{\mu}^{a(n)}A_{\nu}^{a(m)}-\frac{1}{2}\eta^{\mu\nu}\eta^{\alpha\beta}\sum\limits_{n}\left[\partial_{\mu}A_{\alpha}^{a(n)}\partial_{\nu}A_{\beta}^{a(n)}-\partial_{\alpha}A_{\mu}^{a(n)}\partial_{\nu}A_{\beta}^{a(n)}\right]-g\eta^{\mu\nu}\eta^{\alpha\beta}\epsilon^{abc}\sum\limits_{n}\sum\limits_{m}\sum\limits_{k}\gamma_{nmk}\times\\\
A_{\nu}^{b(m)}A_{\beta}^{c(k)}\partial_{\mu}A_{\alpha}^{a(n)}-\frac{g^{2}}{4}\eta^{\mu\nu}\eta^{\alpha\beta}\epsilon^{abc}\epsilon^{ade}\sum\limits_{n}\sum\limits_{m}\sum\limits_{k}\sum\limits_{l}\xi_{nmkl}A_{\mu}^{b(n)}A_{\alpha}^{c(m)}A_{\nu}^{d(k)}A_{\beta}^{e(l)}.\end{array}$
(32)
The parameters $\tau_{n}$, $\gamma_{nmk}$ and $\xi_{nmkl}$ are given by
$\begin{array}[c]{c}\tau_{n}=\sqrt{\pi R}\int\limits_{0}^{\pi
R}\sqrt{G}\mathbf{\Delta}\left(y\right)\chi^{(n)}\left(y\right)dy,~{}\gamma_{nmk}=\sqrt{\pi
R}\int\limits_{0}^{\pi
R}dy\chi^{(n)}\left(y\right)\chi^{(m)}\left(y\right)\chi^{(k)}\left(y\right),\\\
\xi_{nmkl}=\pi R\int\limits_{0}^{\pi
R}dy\chi^{(n)}\left(y\right)\chi^{(m)}\left(y\right)\chi^{(k)}\left(y\right)\chi^{(l)}\left(y\right).\end{array}$
(33)
In a flat space-time, these parameters can be reduced to
$\begin{array}[c]{l}\tau_{n}=1/\sqrt{2},~{}\gamma_{nmk}=\left\\{\delta_{0,m+k-n}+\delta_{0,m-k-n}+\delta_{0,m-k+n}\right\\}/\sqrt{2},\\\
\xi_{nmkl}=\left\\{\delta_{0,n+m-k-l}+\delta_{0,n+m+k-l}+\delta_{0,n+m-k+l}+\delta_{0,n-m+k+l}+\delta_{0,n-m-
k-l}\right.\\\
\left.+\delta_{0,n-m+k-l}+\delta_{0,n-m-k+l}\right\\}/2.\end{array}$ (34)
In the $AdS_{5}$ space-time, the formulae of the $\tau_{i}$ parameters are
given in both the cases where Higgs field is confined in the Planck (Pl) and
TeV branes by
$\tau_{n}^{(Pl)}=\sqrt{\pi
R}\chi^{(n)}(0),~{}\tau_{n}^{(\mathit{TeV})}=\sqrt{\pi R}\chi^{(n)}(\pi R).$
(35)
In the following table, we give the first 10 values of the $\tau_{i}$
parameters for different values of the warp factor.
$\tau_{n}^{(Pl)}$ i $w=10^{4}$ $w=10^{8}$ $w=10^{16}$ 1 -0.1955 -0.1352
-0.0945 2 -0.1453 -0.0950 -0.0645 3 -0.1236 -0.0782 -0.0523 4 -0.1107 -0.0683
-0.0453 5 -0.1018 -0.0617 -0.0405 6 -0.0952 -0.0568 -0.0371 7 -0.0900 -0.0530
-0.0344 8 -0.0858 -0.0500 -0.0322 9 -0.0823 -0.0473 -0.0304 10 -0.07936
-0.0452 -0.0289 $\tau_{n}^{(TeV)}$ $w=10^{4}$ $w=10^{8}$ $w=10^{16}$ 2.1549
3.0379 4.2930 -2.1509 -3.0363 -4.2924 2.1495 3.0359 4.2923 -2.1488 -3.0356
-4.2922 2.1484 3.0355 4.2921 -2.1481 -3.0354 -4.2921 2.1479 3.0353 4.2921
-2.1477 -3.0353 -4.2921 2.1475 3.0352 4.2920 -2.1474 -3.0352 -4.2920
Table 1: Different values of the parameters $\tau_{i}$ for different values of
the warp factor in both the cases where the Higgs doublet is confined in the
Planck brane (left) or TeV brane (right).
For the parameters $\gamma$ and $\xi$, it is easy to check that they depend
only on the warp factor $w$, and not on the first KK mass $M_{1}$. Their
formulae are complicated; and therefore they could be computed numerically.
As stated above in section 2, it is important to estimate the couplings of the
heavy modes with the zero and first one (and maybe the second one). Here we
give the numerical values of$~{}\gamma_{1,1,i}$, which represents the cubic
coupling of two one modes with a heavier one ($i\geq 2$), or equivalently, the
quartic coupling of a zero mode, two one modes and a heavier one. We give also
the value of $\xi_{1,1,1,i}$, which represents the quartic coupling of three
one modes and a heavier one, taking the value of the warp factor to be
$w=10^{4},~{}10^{8},~{}10^{16}$.
$\begin{array}[c]{c}w=10^{4}\\\ \begin{tabular}[c]{|c|c|c||c|c|}\hline\cr
i&$\gamma_{1,1,i}$&$\xi_{1,1,1,i}$&$\gamma_{1,2,i}$&$\xi_{1,1,2,i}$\\\
\hline\cr 1&2.9616&10.9652&-1.0925&-5.4800\\\ \hline\cr
2&-1.0925&-5.4800&2.0279&6.6308\\\ \hline\cr
3&0.0253&1.3800&-1.1505&-4.6203\\\ \hline\cr 4&5.78$\times
10^{-4}$&-0.0676&0.0364&1.4808\\\ \hline\cr 5&9.46$\times 10^{-4}$&3.95$\times
10^{-3}$&7.45$\times 10^{-4}$&-0.0895\\\ \hline\cr 6&1.49$\times
10^{-4}$&-2.51$\times 10^{-3}$&1.83$\times 10^{-3}$&6.39$\times 10^{-3}$\\\
\hline\cr 7&1.48$\times 10^{-4}$&2.96$\times 10^{-4}$&2.73$\times
10^{-4}$&-4.23$\times 10^{-3}$\\\ \hline\cr 8&4.32$\times
10^{-5}$&-3.59$\times 10^{-4}$&3.33$\times 10^{-4}$&6.06$\times 10^{-4}$\\\
\hline\cr 9&4.01$\times 10^{-5}$&4.54$\times 10^{-5}$&9.21$\times
10^{-5}$&-6.90$\times 10^{-4}$\\\ \hline\cr 10&1.60$\times
10^{-5}$&-8.83$\times 10^{-5}$&9.86$\times 10^{-5}$&1.12$\times 10^{-4}$\\\
\hline\cr\end{tabular}\\\ w=10^{8}\\\
\begin{tabular}[c]{|c|c|c||c|c|}\hline\cr
i&$\gamma_{1,1,i}$&$\xi_{1,1,1,i}$&$\gamma_{1,2,i}$&$\xi_{1,1,2,i}$\\\
\hline\cr 1&4.4339&22.7671&-1.4517&-10.8886\\\ \hline\cr
2&-1.4517&-10.8886&3.0403&13.7351\\\ \hline\cr
3&0.0250&2.4249&-1.5438&-9.2595\\\ \hline\cr 4&-4.39$\times
10^{-4}$&-0.0966&0.0365&2.6336\\\ \hline\cr 5&8.48$\times 10^{-4}$&9.39$\times
10^{-3}$&-1.11$\times 10^{-3}$&-0.1304\\\ \hline\cr 6&5.61$\times
10^{-5}$&-3.62$\times 10^{-3}$&1.68$\times 10^{-3}$&0.0152\\\ \hline\cr
7&1.27$\times 10^{-4}$&7.90$\times 10^{-4}$&6.61$\times 10^{-5}$&-6.29$\times
10^{-3}$\\\ \hline\cr 8&2.41$\times 10^{-5}$&-4.99$\times 10^{-4}$&2.91$\times
10^{-4}$&1.57$\times 10^{-3}$\\\ \hline\cr 9&3.35$\times 10^{-5}$&1.43$\times
10^{-4}$&4.43$\times 10^{-5}$&-1.00$\times 10^{-3}$\\\ \hline\cr
10&1.01$\times 10^{-5}$&-1.18$\times 10^{-4}$&8.39$\times 10^{-5}$&3.23$\times
10^{-4}$\\\ \hline\cr\end{tabular}\\\ w=10^{16}\\\
\begin{tabular}[c]{|c|c|c||c|c|}\hline\cr
i&$\gamma_{1,1,i}$&$\xi_{1,1,1,i}$&$\gamma_{1,2,i}$&$\xi_{1,1,2,i}$\\\
\hline\cr 1&6.4532&46.5505&-1.990&-21.6920\\\ \hline\cr
2&-1.9899&-21.6920&4.4233&28.0336\\\ \hline\cr
3&0.0308&4.5373&-2.1258&-18.5267\\\ \hline\cr 4&8.10$\times
10^{-4}$&-0.1620&0.0431&4.9618\\\ \hline\cr 5&2.75$\times
10^{-3}$&0.0193&-1.68$\times 10^{-3}$&-0.2216\\\ \hline\cr 6&1.68$\times
10^{-3}$&-6.44$\times 10^{-3}$&3.00$\times 10^{-3}$&0.0317\\\ \hline\cr
7&1.64$\times 10^{-3}$&1.41$\times 10^{-3}$&1.13$\times 10^{-3}$&-0.0110\\\
\hline\cr 8&1.33$\times 10^{-3}$&-1.09$\times 10^{-3}$&1.36$\times
10^{-3}$&3.11$\times 10^{-3}$\\\ \hline\cr 9&0.0988&2.94$\times
10^{-4}$&-1.82$\times 10^{-5}$&-1.73$\times 10^{-3}$\\\ \hline\cr
10&1.01$\times 10^{-3}$&-3.98$\times 10^{-4}$&5.29$\times 10^{-5}$&7.00$\times
10^{-4}$\\\ \hline\cr\end{tabular}\end{array}$
Table 2: Different values of the cubic ($\gamma_{1,1,i}$ and $\gamma_{1,2,i}$)
and quartic ($\xi_{1,1,1,i}$ and $\xi_{1,1,2,i}$) gauge-gauge couplings for
$w=10^{4}$, $10^{8}$, $10^{16}$.
## References
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* [8] T. Gherghetta and A. Pomarol, Nucl. Phys. B586, 141 (2000).
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* [10] A.D. Sakharov, JETP Lett. 5, 24 (1967).
* [11] G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976); Phys. Rev. D14, 3432 (1976), Erratum-ibid. D18, 2199 (1978).
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* [13] S. Braibant, Y. Brihaye, and J. Kunz, Int. J. Mod. Phys. A8, 5563 (1993).
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* [21] N. Arkani-Hamed, A.G. Cohen and H. Georgi, Phys. Lett. B516, 395 (2001).
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|
arxiv-papers
| 2009-04-04T10:26:19 |
2024-09-04T02:49:01.688358
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Amine Ahriche (U. Jijel, U. Constantine & U. Bielefeld)",
"submitter": "Amine Ahriche",
"url": "https://arxiv.org/abs/0904.0700"
}
|
0904.0707
|
# Optimal Multi-Modes Switching Problem in Infinite Horizon
Brahim EL ASRI Université du Maine, Département de Mathématiques, Equipe
Statistique et Processus, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9,
France. e-mail: brahim.elasri@univ-lemans.fr
###### Abstract
This paper studies the problem of the deterministic version of the
Verification Theorem for the optimal $m$-states switching in infinite horizon
under Markovian framework with arbitrary switching cost functions. The problem
is formulated as an extended impulse control problem and solved by means of
probabilistic tools such as the Snell envelop of processes and reflected
backward stochastic differential equations. A viscosity solutions approach is
employed to carry out a fine analysis on the associated system of $m$
variational inequalities with inter-connected obstacles. We show that the
vector of value functions of the optimal problem is the unique viscosity
solution to the system. This problem is in relation with the valuation of
firms in a financial market.
AMS Classification subjects: 60G40 ; 62P20 ; 91B99 ; 91B28 ; 35B37 ; 49L25.
$\bf Keywords$: Real options; Backward stochastic differential equations;
Snell envelope; Stopping times ; Switching; Viscosity solution of PDEs;
Variational inequalities.
## 1 Introduction
First let us deal with an example in order to introduce the problem we
consider in this paper:
Assume we have a power station/plant which produces electricity and which has
several modes of production, e.g., the lower, the middle and the intensive
modes. The price of electricity in the market, given by an adapted stochastic
process $(X_{t})_{t\geq 0}$, fluctuates in reaction to many factors such as
demand level, weather conditions, unexpected outages and so on. On the other
hand, electricity is non-storable, once produced it should be almost
immediately consumed. Therefore, as a consequence, the station produces
electricity in its instantaneous most profitable mode known that when the
plant is in mode $i\in{\cal I}$, the yield per unit time $dt$ is given by
means of $\psi_{i}(X_{t})dt$ and, on the other hand, switching the plant from
the mode $i$ to the mode $j$ is not free and generates expenditures given by
$g_{ij}(X_{t})$ and possibly by other factors in the energy market. The
switching from one regime to another one is realized sequentially at random
times which are part of the decisions. So the manager of the power plant faces
two main issues:
$(i)$ when should she decide to switch the production from its current mode to
another one?
$(ii)$ to which mode the production has to be switched when the decision of
switching is made?
In other words she faces the issue of finding the optimal strategy of
management of the plant. This issue is in relation with the price of the power
plant in the energy market.
Optimal switching problems for stochastic systems were studied by several
authors (see e.g. [1, 2, 3, 4, 9, 10, 11, 12, 13, 17, 20, 23, 24] and the
references therein). The motivations are mainly related to decision making in
the economic sphere. Several variants of the problem we deal with here,
including finite and infinite horizons, have been considered during the recent
years. In order to tackle those problems, authors use mainly two approaches.
Either a probabilistic one [10, 11, 17] or an approach which uses partial
differential inequalities (PDIs for short) [1, 2, 4, 12, 20, 24, 23].
In the finite horizon framework Djehiche et al. have studied the multi-modes
switching problem in using probabilistic tools. For general stochastic
processes, they have shown that a value of the problem and an optimal strategy
exits. The partial differential equation version of this work has been carried
out by El-Asri and Hamadène [13]. They showed that when the price process
$X_{t}$ is solution of a Markovian standard differential equation, then with
this problem is associated a system of variational inequalities with
interconnected obstacles for which they provide a solution in viscosity sense.
This solution is bind to the value function of the problem. The solution of
the system is unique.
In the case when the horizon is infinite, there still much to do and this is
the novelty of this paper. Actually, authors treat mainly the case when the
price process $X_{t}$ is of Markovian Itô type, the switching costs are
deterministic functions of time $t$ and the profit functions are deterministic
functions of $(t,X_{t})$ and have linear growth at most (see e.g. [1, 2, 12,
20, 24]). Therefore the main objective of this paper is to fill in the gap
between finite and infinite horizon by providing a complete treatment of the
optimal multiple switching problem in infinite horizon when the price is only
a continuous process. This is what we did in the first part of this paper.
Actually inspired by the work of Djehiche et al. [11], using probabilistic
tools such the Snell envelope of processes and BSDEs we provide a verification
theorem which shapes the problem and then we have constructed a solution for
this latter. This solution provides an optimal strategy for the switching
problem. Later on, in the Markovian framework of randomness, i.e. in the case
when $X$ is a solution of a SDE, we show that with the value function of the
problem is associated an uplet of deterministic functions
$(v^{1},\dots,v^{m})$ which is the unique solution of the following system of
partial differential inequalities (PDIs for short):
$\left\\{\begin{array}[]{l}\min\\{v_{i}(x)-\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(x)+v_{j}(x)),rv_{i}(x)-{\cal A}v_{i}(x)-\psi_{i}(x)\\}=0\\\
\forall x\in I\\!\\!R^{k},\,\,i\in{\cal I}=\\{1,...,m\\},\end{array}\right.$
(1.1)
where $\cal A$ an infinitesimal generator associated with a diffusion process
and ${\cal I}^{-i}:=\\{1,...,i-1,i+1,...,m\\}$. This system is the
deterministic version of the Verification Theorem of the optimal multi-modes
switching problem in infinite horizon.
This paper is organized as follows: In Section 2, we formulate the problem and
we give the related definitions. In Section 3, we introduce the optimal
switching problem under consideration and give its probabilistic Verification
Theorem. It is expressed by means of a Snell envelope of processes. Then we
introduce the approximating scheme which enables to construct a solution for
the Verification Theorem. Moreover we give some properties of that solution,
especially the dynamic programming principle. Section 4 is devoted to the
connection between the optimal switching problem, the Verification Theorem and
the associated system of PDIs. This connection is made through backward
stochastic differential equations with one reflecting obstacle in the case
when randomness comes from a solution of a standard stochastic differential
equation. Further we provide some estimate for the optimal strategy of the
switching problem which, in combination with the dynamic programming
principle, plays a crucial role in the proof of existence of a solution for
(1.1) which we address. In Section 5, we show that the solution of PDIs is
unique in the class of continuous functions which satisfy a polynomial growth
condition. In Section 6, we give some numerical examples.$\Box$
## 2 Assumptions and formulation of the problem
Throughout this paper $k$ is a fixed integer positive constant. Let us now
consider the followings assumption:
$\bf H1$: $b:R^{k}\rightarrow I\\!\\!R^{k}$ and
$\sigma:I\\!\\!R^{k}\rightarrow I\\!\\!R^{k\times d}$ are two continuous
functions for which there exists a constant $C\geq 0$ such that for any
$x,x^{\prime}\in I\\!\\!R^{k}$
$|b(x)|+|\sigma(x)|\leq C(1+|x|)\quad\mbox{ and
}\quad|\sigma(x)-\sigma(x^{\prime})|+|b(x)-b(x^{\prime})|\leq C|x-x^{\prime}|$
(2.1)
$\bf H2$: for $i,j\in{\cal I}=\\{1,...,m\\}$, $g_{ij}:I\\!\\!R^{k}\rightarrow
I\\!\\!R$ is a continuous function. Moreover we assume that there exists a
constant $\alpha>0$ such that for any $x\in I\\!\\!R^{k}$,
$\frac{1}{\alpha}\leq g_{ij}(x)\leq\alpha,\quad\forall i,j\in{\cal I},\quad
i\neq j.$ (2.2)
$\bf H3$: for $i\in{\cal I}$ $\psi_{i}:I\\!\\!R^{k}\rightarrow I\\!\\!R$ is a
continuous function of polynomial growth, $i.e.$, there exist a constant $C$
and $\gamma$ such that for each $i\in\cal I$:
$|\psi_{i}(x)|\leq C(1+|x|^{\gamma}),\,\,\forall x\in I\\!\\!R^{k}.$ (2.3)
We now consider the following system of $m$ variational inequalities with
inter-connected obstacles: $\forall\,\,i\in{\cal I}$
$\begin{array}[]{l}\min\\{v_{i}(x)-\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(x)+v_{j}(x)),rv_{i}(x)-{\cal
A}v_{i}(x)-\psi_{i}(x)\\}=0\end{array}.$ (2.4)
where ${\cal I}^{-i}:={\cal I}-\\{i\\}$, $r$ is a positive discount factor and
${\cal A}$ is the following infinitesimal generator:
${\cal
A}=\frac{1}{2}\sum_{i,j=1,k}(\sigma\sigma^{*})_{ij}(x)\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}+\sum_{i=1,k}b_{i}(x)\frac{\partial}{\partial x_{i}}\,;$
(2.5)
hereafter the superscript $(^{*})$ stands for the transpose, $Tr$ is the trace
operator and finally $<x,y>$ is the inner product of $x,y\in I\\!\\!R^{k}$.
The main objective of this paper is to focus on the existence and uniqueness
of the solution in viscosity sense of (2.4) whose definition is:
###### Definition 1
Let $(v_{1},...,v_{m})$ be a $m$-uplet of continuous functions defined on
$I\\!\\!R^{k}$, $I\\!\\!R$-valued. The $m$-uplet $(v_{1},...,v_{m})$ is
called:
* $(i)$
a viscosity supersolution (resp. subsolution) of the system (2.4) if for each
fixed $i\in{\cal I}$, for any $x_{0}\in I\\!\\!R^{k}$ and any function
$\varphi_{i}\in C^{1,2}(I\\!\\!R^{k})$ such that
$\varphi_{i}(x_{0})=v_{i}(x_{0})$ and $x_{0}$ is a local maximum of
$\varphi_{i}-v_{i}$ (resp. minimum), we have:
$\begin{array}[]{l}\min\left\\{v_{i}(x_{0})-\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(x_{0})+v_{j}(x_{0})),\right.\\\
\qquad\qquad\qquad\left.r\varphi_{i}(x_{0})-{\cal
A}\varphi_{i}(x_{0})-\psi_{i}(x_{0})\right\\}\geq 0\quad(\mbox{resp.}\leq
0).\end{array}$ (2.6)
* $(ii)$
a viscosity solution if it is both a viscosity supersolution and subsolution.
$\Box$
There is an equivalent formulation of this definition (see e.g. [6]) which we
give since it will be useful later. So firstly we define the notions of
superjet and subjet of a continuous function $v$.
###### Definition 2
Let $v\in C(I\\!\\!R^{k})$, $x$ an element of $I\\!\\!R^{k}$ and finally
$S_{k}$ the set of $k\times k$ symmetric matrices. We denote by $J^{2,+}v(x)$
(resp. $J^{2,-}v(x)$), the superjets (resp. the subjets) of $v$ at $x$, the
set of pairs $(q,X)\in I\\!\\!R^{k}\times S_{k}$ such that:
$\begin{array}[]{c}v(y)\leq v(x)+\langle q,y-x\rangle+\frac{1}{2}\langle
X(y-x),y-x\rangle+o(|y-x|^{2})\\\ (resp.\quad v(y)\geq v(x)+\langle
q,y-x\rangle+\frac{1}{2}\langle
X(y-x),y-x\rangle+o(|y-x|^{2})).\Box\end{array}$
Note that if $\varphi-v$ has a local maximum (resp. minimum) at $x$, then we
obviously have:
$\left(D_{x}\varphi(x),D^{2}_{xx}\varphi(x)\right)\in
J^{2,-}v(x)\,\,\,(\mbox{resp. }J^{2,+}v(x)).\Box$
We now give an equivalent definition of a viscosity solution of the elliptic
system with inter-connected obstacles (2.4).
###### Definition 3
Let $(v_{1},...,v_{m})$ be a $m$-uplet of continuous functions defined on
$I\\!\\!R^{k}$ and $I\\!\\!R$-valued. The $m$-uplet $(v_{1},...,v_{m})$ is
called a viscosity supersolution (resp. subsolution) of (2.4) if for any
$i\in{\cal I}$, $x\in I\\!\\!R^{k}$ and $(q,X)\in J^{2,-}v_{i}(t,x)$ (resp.
$J^{2,+}v_{i}(x)$),
$min\left\\{v_{i}(x)-\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(x)+v_{j}(x)),rv_{i}(x)-\frac{1}{2}Tr[\sigma^{*}X\sigma]-\langle
b,q\rangle-\psi_{i}(x)\right\\}\geq 0\,\,(resp.\leq 0).$
It is called a viscosity solution if it is both a viscosity subsolution and
supersolution .$\Box$
As pointed out previously we will show that system (2.4) has a unique solution
in viscosity sense. This system is the deterministic version of the
verification theorem of the optimal $m$-states switching problem in infinite
horizon which is well documented in [11] in the case of finite horizon and
which we will describe briefly in the next section.
## 3 The optimal $m$-states switching problem
### 3.1 Setting of the problem
Let $(\Omega,{\cal F},P)$ be a fixed probability space on which is defined a
standard $d$-dimensional Brownian motion $B=(B_{t})_{t\geq 0}$ whose natural
filtration is $({\cal F}_{t}^{0}:=\sigma\\{B_{s},s\leq t\\})_{t\geq 0}$. Let
${\bf F}=({\cal F}_{t})_{t\geq 0}$ be the completed filtration of $({\cal
F}_{t}^{0})_{t\geq 0}$ with the $P$-null sets of ${\cal F}$, hence $({\cal
F}_{t})_{t\geq 0}$ satisfies the usual conditions, $i.e.$, it is right
continuous and complete. Furthermore, let:
\- ${\cal P}$ be the $\sigma$-algebra on $[0,+\infty)\times\Omega$ of ${\bf
F}$-progressively measurable sets;
\- ${\cal M}^{2,k}$ be the set of $\cal P$-measurable and
$I\\!\\!R^{k}$-valued processes $w=(w_{t})_{t\geq 0}$ such that
$E[\int_{0}^{+\infty}|w_{s}|^{2}ds]<\infty$ and ${\cal S}^{2}$ be the set of
$\cal P$-measurable, continuous processes ${w}=({w}_{t})_{t\geq 0}$ such that
$E[\sup_{t\geq 0}|{w}_{t}|^{2}]<\infty$;
\- for any stopping time $\tau\in I\\!\\!R^{+}$, ${\cal T}_{\tau}$ denotes the
set of all stopping times $\theta$ such that $\tau\leq\theta;$
\- for any stopping time $\tau$, ${\cal F}_{\tau}$ is the $\sigma$-algebra on
$\Omega$ which contains the sets $A$ of $\cal{F}$ such that $A\cap\\{\tau\leq
t\\}\in{\cal F}_{t}$ for every $t\geq 0$. $\Box$
A decision (strategy) of the problem of multiple switching, on the one hand,
consists of the choice of a sequence of nondecreasing stopping times
$(\tau_{n})_{n\geq 1}$ $(i.e.\tau_{n}\leq\tau_{n+1}$) where the manager
decides to switch the activity from its current mode to another one. On the
other hand, it consists of the choice of the mode $\xi_{n}$, a r.v. ${\cal
F}_{\tau_{n}}$-measurable with values in ${\cal I}$, to which the production
is switched at $\tau_{n}$. Therefore the admissible management strategies are
the pairs $(\delta,\xi):=((\tau_{n})_{n\geq 1},(\xi_{n})_{n\geq 1})$ and we
denote by $\cal D$ the set of these strategies.
Let now $X:=(X_{t})_{t\geq 0}$ be an $\cal P$-measurable,
$I\\!\\!R^{k}$-valued continuous stochastic process which stands for the
market price of $k$ factors which determine the market price of the commodity.
On the other hand, assuming that the production activity is in mode 1 at the
initial time $t=0$, let $(u_{t})_{t\geq 0}$ denote the indicator of the
production activity’s mode at time $t\in I\\!\\!R^{+}$ :
$u_{t}=1\\!\\!1_{[0,\tau_{1}]}(t)+\sum_{n\geq
1}\xi_{n}1\\!\\!1_{(\tau_{n},\tau_{n+1}]}(t).$ (3.1)
Then for any $t\geq 0$, the state of the whole economic system related to the
project at time $t$ is given by the vector:
$\begin{array}[]{ll}(t,X_{t},u_{t})\in I\\!\\!R^{+}\times
I\\!\\!R^{k}\times{\cal I}.\end{array}$ (3.2)
Finally, let $\psi_{i}(X_{t})$ be the instantaneous profit when the system is
in state $(t,X_{t},i)$, and for $i,j\in{\cal I}\quad i\neq j$, let
$g_{ij}(X_{t})$ denote the switching cost of the production at time $t$ from
the current mode $i$ to another mode $j$. When the plant is run under the
strategy $(\delta,\xi)=((\tau_{n})_{n\geq 1},(\xi_{n})_{n\geq 1})$ the
expected total profit is given by:
$\begin{array}[]{l}J(\delta,\xi)=E[\displaystyle\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X_{s})ds-\sum_{n\geq
1}e^{-r\tau_{n}}g_{u_{\tau_{n-1}}u_{\tau_{n}}}(X_{\tau_{n}})].\end{array}$
Then the problem we are interested in is to find an optimal strategy, $i.e$, a
strategy $(\delta^{*},\xi^{*})$ such that $J(\delta^{*},\xi^{*})\geq
J(\delta,\xi)$ for any $(\delta,\xi)\in\cal D$.
Note that in order that the quantity $J(\delta,\xi)$ makes sense we assume
throughout this paper that for any $i\in{\cal I}$ the processes
$(e^{-rt}\psi_{i}(X_{t}))_{t\geq 0}$ belong to ${\cal M}^{2,1}$. On the other
hand there is a bijective correspondence between the pairs $(\delta,\xi)$ and
the pairs $(\delta,u)$. Then throughout this paper one refers indifferently to
$(\delta,\xi)$ or $(\delta,u)$.
### 3.2 The Verification Theorem
To tackle the problem described above in the finite horizon case, Djehiche et
al. [11] have introduced a Verification Theorem which is expressed by means of
Snell envelope of processes which we describe briefly now. The Snell envelope
of a stochastic process $(\eta_{t})_{t\geq 0}$ of ${\cal S}^{2}$ (with a
possible positive jump at $+\infty$ and
$\lim\limits_{t\rightarrow\infty}\eta_{t}=M\in L^{2}(\Omega,{\cal F},P)$) is
the lowest supermartingale $R(\eta):=(R(\eta)_{t})_{t\geq 0}$ of ${\cal
S}^{2}$ such that for any $t\geq 0$, $R(\eta)_{t}\geq\eta_{t}$. It has the
following expression:
$\forall t\geq 0,R(\eta)_{t}=esssup_{\tau\in{\cal T}_{t}}E[\eta_{\tau}|{\bf
F}_{t}]\quad\mbox{(then it satisfies
}\lim\limits_{t\rightarrow+\infty}R(\eta)_{t}=M.)$
For more details on the Snell envelope notion on can see e.g. [7, 14, 16].
The Verification Theorem for the $m$-states optimal switching problem in
infinite horizon is the following:
###### Theorem 1
. Assume that there exist $m$ processes $(Y^{i}:=(Y^{i}_{t})_{t\geq
0},i=1,...,m)$ of ${\cal S}^{2}$ such that:
$\begin{array}[]{l}\forall t\geq 0,\,\,e^{-rt}Y^{i}_{t}=\mbox{ess
sup}_{\tau\geq
t}E[\int_{t}^{\tau}e^{-rs}\psi_{i}(X_{s})ds+e^{-r\tau}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{\tau})+Y^{j}_{\tau})|{\cal
F}_{t}],\quad\lim\limits_{t\rightarrow+\infty}(e^{-rt}Y^{i}_{t})=0.\end{array}$
(3.3)
Then:
* $(i)$
$Y^{1}_{0}=\sup\limits_{(\delta,\xi)\in{\cal D}}J(\delta,u).$
* $(ii)$
Define the sequence of ${\bf F}$-stopping times
$\delta^{*}=(\tau_{n}^{*})_{n\geq 1}$ as follows :
$\begin{array}[]{lll}\tau^{*}_{1}&=&\inf\\{s\geq 0,\quad
Y1_{s}=\max\limits_{j\in{{\cal I}^{-1}}}(-g_{1j}(X_{s})+Y^{j}_{s})\\},\\\
\tau^{*}_{n}&=&\inf\\{s\geq\tau^{*}_{n-1},\quad
Y^{u_{\tau^{*}_{n-1}}}_{s}=\max\limits_{k\in{\cal
I}\backslash\\{u_{\tau^{*}_{n-1}}\\}}(-g_{u_{\tau^{*}_{n-1}}k}(X_{s})+Y^{k}_{s})\\},\quad\mbox{for}\quad
n\geq 2,\end{array}$
where:
* $\bullet$
$u_{\tau^{*}_{1}}=\sum\limits_{j\in{\cal
I}}j1\\!\\!1_{\\{\max\limits_{k\in{\cal
I}^{-1}}(-g_{1k}(X_{\tau^{*}_{1}})+Y^{k}_{\tau^{*}_{1}})=-g_{1j}(X_{\tau^{*}_{1}})+Y^{j}_{\tau^{*}_{1}}\\}};$
* $\bullet$
for any $n\geq 1$ and $t\geq\tau^{*}_{n},$
$Y^{u_{\tau^{*}_{n}}}_{t}=\sum\limits_{j\in{\cal
I}}1\\!\\!1_{[u_{\tau^{*}_{n}}=j]}Y^{j}_{t}$
* $\bullet$
for any $n\geq 2,\,\,u_{\tau^{*}_{n}}=l$ on the set
$\left\\{\max\limits_{k\in{\cal
I}\backslash\\{{u_{\tau^{*}_{n-1}}}\\}}(-g_{u_{\tau^{*}_{n-1}}k}(X_{\tau^{*}_{n}})+Y^{k}_{\tau^{*}_{n}})=-g_{u_{\tau^{*}_{n-1}l}}(X_{\tau^{*}_{n}})+Y^{l}_{\tau^{*}_{n}}\right\\}$
with $g_{u_{\tau^{*}_{n-1}k}}(X_{\tau^{*}_{n}})=\sum\limits_{j\in{\cal
I}}1\\!\\!1_{[u_{\tau^{*}_{n-1}}=j]}g_{jk}(X_{\tau^{*}_{n}})$ and ${\cal
I}\backslash\\{u_{\tau^{*}_{n-1}}\\}=\sum\limits_{j\in{\cal
I}}1\\!\\!1_{[u_{\tau^{*}_{n-1}}=j]}{\cal I}^{-j}$.
Then the strategy $(\delta^{*},u^{*})$ satisfies $E[\sum_{n\geq
0}e^{-r\tau^{*}_{n}}]<+\infty$ and it is optimal i.e. $J(\delta^{*},u^{*})\geq
J(\delta,u)$ for any $(\delta,u)\in\cal D$. $\Box$
Proof. The arguments of this proof are standard, based on the properties the
Snell envelope. We defer the proof in the Appendix.$\Box$
The issue of existence of the processes $Y^{1},...,Y^{m}$ which satisfy (3.3)
is also addressed in [11]. For $n\geq 0$ let us define the processes
$(Y^{n,1},...,Y^{n,m})$ recursively as follows: for $i\in{\cal I}$ we set,
$e^{-rt}Y^{0,i}_{t}=E[\displaystyle\int_{t}^{+\infty}e^{-rs}\psi_{i}(X_{s})ds|{\cal
F}_{t}],\,\,t\geq 0,$ (3.4)
and for $n\geq 1$,
$e^{-rt}Y^{n,i}_{t}=\mbox{ess sup}_{\tau\geq
t}E[\displaystyle\int_{t}^{\tau}e^{-rs}\psi_{i}(X_{s})ds+e^{-r\tau}\max\limits_{k\in{\cal
I}^{-i}}(-g_{ik}(X_{\tau})+Y^{n-1,k}_{\tau})|{\cal F}_{t}],\,\,t\geq 0.$ (3.5)
Then the sequence of processes $((Y^{n,1},...,Y^{n,m}))_{n\geq 0}$ have the
following properties:
###### Proposition 1
([11], Pro.3 and Th.2)
* $(i)$
for any $i\in{\cal I}$ and $n\geq 0$, the processes $Y^{n,1},...,Y^{n,m}$ are
well-posed, continuous and belong to ${\cal S}^{2}$, and verify
$\forall t\geq 0,\,\,e^{-rt}Y^{n,i}_{t}\leq e^{-rt}Y^{n+1,i}_{t}\leq
E[\int_{t}^{+\infty}e^{-rs}\\{\max_{i=1,m}|\psi_{i}(X_{s})|\\}ds|{\cal
F}_{t}];$ (3.6)
* $(ii)$
there exist $m$ processes $Y^{1},...,Y^{m}$ of ${\cal S}^{2}$ such that for
any $i\in{\cal I}$:
* $(a)$
$\forall t\geq 0$, $Y^{i}_{t}=\lim_{n\rightarrow\infty}\nearrow Y^{n,i}_{t}$
* $(b)$
$\forall t\geq 0$,
$e^{-rt}{Y}^{i}_{t}=\mbox{ess sup}_{\tau\geq
t}E[\displaystyle\int_{t}^{\tau}e^{-rs}\psi_{i}(X_{s})ds+e^{-r\tau}\max\limits_{k\in{\cal
I}^{-i}}(-g_{ik}(X_{\tau})+{Y}^{k}_{\tau})|{\cal F}_{t}]$ (3.7)
i.e. ${Y}^{1},...,{Y}^{m}$ satisfy the Verification Theorem 1 ;
* $(c)$
$\forall t\geq 0$,
$e^{-rt}{Y}^{i}_{t}=esssup_{(\delta,\xi)\in{\cal
D}^{i}_{t}}E[\displaystyle\int_{t}^{+\infty}e^{-rs}\psi_{u_{s}}(X_{s})ds-\sum_{n\geq
1}e^{-r\tau_{n}}g_{u_{\tau_{n-1}}u_{\tau_{n}}}(X_{\tau_{n}})|{\cal F}_{t}]$
(3.8)
where ${\cal D}^{i}_{t}=\\{(\delta,\xi)=((\tau_{n})_{n\geq 1},(\xi_{n})_{n\geq
1})\mbox{ such that }u_{0}=i\mbox{ and }\tau_{1}\geq t\\}$. This
characterization means that if at time $t$ the production activity is in its
regime $i$ then the optimal expected profit is $Y^{i}_{t}$.
* $(d)$
the processes $Y^{1},...,Y^{m}$ verify the dynamical programming principle of
the $m$-states optimal switching problem, $i.e.$, $\forall t\leq T$,
$\\!\\!\\!\\!\\!\begin{array}[]{ll}e^{-rt}Y^{i}_{t}&=\mbox{ess
sup}_{(\delta,u)\in{\cal
D}_{t}^{i}}E[\displaystyle\int_{t}^{\tau_{n}}e^{-rs}\psi_{u_{s}}(X_{s})ds-\sum_{1\leq
k\leq
n}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{{\tau_{k}}}}(X_{{\tau}_{k}})+e^{-r\tau_{n}}Y^{u_{\tau_{n}}}_{\tau_{n}}|{\cal
F}_{t}].\Box\end{array}$ (3.9)
Note that except $(ii-d)$, the proofs of the other points are the same as in
[11] in the framework of finite horizon. The proof of $(ii-d)$ can be easily
deduced in using relation (3.7). Actually from (3.7) for any $i\in{\cal I}$,
$t\geq 0$ and $(\delta,\xi)\in{\cal D}^{i}_{t}$ we have:
$e^{-rt}Y^{i}_{t}\geq
E[\displaystyle\int_{t}^{\tau_{n}}e^{-rs}\psi_{u_{s}}(X_{s})ds-\sum_{1\leq
k\leq
n}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{u_{\tau_{k}}}}(X_{{\tau}_{k}})+e^{-r\tau_{n}}Y^{u_{\tau_{n}}}_{\tau_{n}}|{\cal
F}_{t}].$ (3.10)
Next using the optimal strategy we obtain the equality instead of inequality
in (3.10). Therefore the relation (3.9) holds true. $\Box$
###### Remark 1
The characterization (3.8) implies that the processes $Y^{1},...,Y^{m}$ of
${\cal S}^{2}$ which satisfy the Verification Theorem are unique.
## 4 Existence of a solution for the system of variational inequalities
### 4.1 Connection with BSDEs with one reflecting barrier
Let $x\in I\\!\\!R^{k}$ and let $X^{x}$ be the solution of the following
standard SDE:
$dX_{t}^{x}=b(X^{x}_{t})dt+\sigma(X^{x}_{t})dB_{t},\quad X^{x}_{0}=x$ (4.1)
where the functions $b$ and $\sigma$ are the ones of $\bf H1$. These
properties of $\sigma$ and $b$ imply in particular that $X^{x}$ solution of
the standard SDE (4.1) exists and is unique in $I\\!\\!R^{k}$. The operator
$\cal A$ defined in (2.5) is the infinitesimal generator associated with
$X^{x}$.
In the following result we collect some properties of $X^{x}$.
###### Proposition 2
(see e.g. [22]) The process $X^{x}$ satisfies the following estimates:
* $(i)$
For any $q\geq 2$ there exists $C_{q}$ such that,
$E[|X^{x}_{t}|^{q}]\leq C_{q}e^{C_{q}t}(1+|x|^{q})\quad\forall t\geq 0.$ (4.2)
* $(ii)$
There exists a constant $C$ such that for any $x,x^{\prime}\in I\\!\\!R^{k}$
and $T\geq 0$,
$E[\sup\limits_{0\leq s\leq T}|X^{x}_{s}-X^{x^{\prime}}_{s}|^{2}]\leq
Ce^{CT}|x-x^{\prime}|^{2}.\Box$ (4.3)
In the sequel we consider the following condition:
$\bf H4$: Assume $\gamma\geq 2$ and
$-r+C_{\gamma}<0,$ (4.4)
where $\gamma$ is the growth exponent of the functions $\psi_{i}$ and
$C_{\gamma}$ is the constant in (4.2). $\Box$
###### Remark 2
: If $\gamma<2$, there exists a constant $\gamma_{1}\geq 2$ such that
$\gamma_{1}$ verifies the growth exponent of the functions $\psi_{i}$.
We are going now to introduce the notion of a BSDE with one reflecting barrier
considered in [19]. This notion will allow us to make the connection between
the variational inequalities system (2.4) and the $m$-states optimal switching
problem described in the previous section.
Let us introduce the pair of process $(Y^{x},Z^{x})\in{\cal S}^{2}\times{\cal
M}^{2,d}$ solution of the following BSDE:
$Y^{x}_{s}=Y_{T}^{x}+\int_{s}^{T}F(X_{l}^{x},Y_{l}^{x},Z_{l}^{x})dl-\int_{s}^{T}Z^{x}_{l}dB_{l},\quad\mbox{for
all}\quad T\geq 0\quad\mbox{and}\quad t\leq T,$ (4.5)
where $F:I\\!\\!R^{k}\times I\\!\\!R\times I\\!\\!R^{d}\rightarrow I\\!\\!R$
is continuous and satisfies: there exist a continuous increasing function
$\phi:I\\!\\!R^{+}\rightarrow I\\!\\!R^{+}$ and constant $K$, $K^{\prime}$,
$\mu<0$, $p>0$ such that,
$\begin{array}[]{lll}|F(x,y,z)|\leq K^{\prime}(1+|x|^{p}+\phi(|y|)+|z|),\\\
\langle
y-y^{\prime},F(x,y,z)-F(x,y^{\prime},z)\rangle\leq\mu|y-y^{\prime}|^{2},\\\
|F(x,y,z)-F(x,y,z^{\prime})|\leq K||z-z^{\prime}||.\end{array}$ (4.6)
We assume moreover that for some $\lambda>2\mu+K2,$
$E[\int_{0}^{+\infty}e^{\lambda s}|F(X_{s}^{x},0,0)|^{2}ds]<+\infty,$ (4.7)
which essentially implies that $\lambda+C_{2\gamma}<0$.
Let us consider the following semilinear elliptic PDE in $I\\!\\!R^{k}$:
${\cal A}u(x)+F(x,u(x),\sigma(x)^{*}\nabla u(x))=0,\quad x\in I\\!\\!R^{k}.$
(4.8)
Then we have the following result:
###### Theorem 2
([21], Th. 5.2) Under the above assumptions, $u(x)=Y^{x}_{0}$ is a continuous
function and it is a viscosity solution of (4.8) which satisfies,
$|Y^{x}_{0}|\leq C\sqrt{E[\int_{0}^{+\infty}e^{\lambda
s}|F(X_{s}^{x},0,0)|^{2}ds}],$ (4.9)
for any $\lambda>2\mu+K2.$$\Box$
Let us now introduce the following functions:
* $(i)$
$f:I\\!\\!R^{k}\rightarrow I\\!\\!R$ is continuous and of polynomial growth,
$i.e.$, there exist some positive constants $C$ and $\gamma$ such that:
$|f(x)|\leq C(1+|x|^{\gamma}),\,\,\forall x\in I\\!\\!R^{k}.$ (4.10)
* $(ii)$
$h:I\\!\\!R^{k}\rightarrow I\\!\\!R$ is continuous and bounded.
Then we have the following result related to BSDEs with one reflecting
barrier:
###### Theorem 3
For any $x\in I\\!\\!R^{k}$, there exits a unique triple of processes
$(Y^{x},Z^{x},K^{x})$ such that:
$\left\\{\begin{array}[]{l}Y^{x},K^{x}\in{\cal S}^{2}\mbox{ and }Z^{x}\in{\cal
M}^{2,d};\,K^{x}\mbox{ is non-decreasing and }K^{x}_{0}=0,\\\
e^{-rs}Y^{x}_{s}=\int_{s}^{+\infty}e^{-rl}f(X_{l}^{x})dl-\int_{s}^{+\infty}Z^{x}_{l}dB_{l}+K_{+\infty}^{x}-K^{x}_{s},\\\
e^{-rs}Y^{x}_{s}\geq e^{-rs}h(X^{x}_{s}),\,\forall s\geq 0\mbox{ and
}\int_{0}^{+\infty}(e^{-rl}Y^{x}_{l}-e^{-rl}h(X^{x}_{l}))dK^{x}_{l}=0.\end{array}\right.$
(4.11)
Moreover the following characterization of $Y^{x}$ as a Snell envelope holds
true:
$\forall s\geq 0,\,\,e^{-rs}Y^{x}_{s}=esssup_{\tau\in{\cal
T}_{s}}E[\int_{s}^{\tau}e^{-rl}f(X_{l}^{x})dl+e^{-r\tau}h(X^{x}_{\tau})|{\cal
F}_{s}].$ (4.12)
On the other hand there exists a deterministic continuous with polynomial
growth function $u:I\\!\\!R^{k}\rightarrow I\\!\\!R$ such that:
$\forall x\in I\\!\\!R^{k}\quad Y^{x}_{0}=u(x).$
Moreover the function $u$ is the viscosity solution in the class of continuous
function with polynomial growth of the following PDE with obstacle:
$\begin{array}[]{l}\min\\{u(x)-h(x),ru(x)-{\cal A}u(x)-f(x)\\}=0.\end{array}$
(4.13)
$Proof$: Existence and uniqueness of the triple
$(Y^{x}_{t},Z^{x}_{t},K^{x}_{t})_{t\geq 0}$ of (4.11) follow from Theorem 3.2
in [19]. Now we consider the infinite horizon BSDE:
$^{n}Y^{x}_{s}e^{-rs}=\int_{s}^{+\infty}e^{-rl}f(X_{l}^{x})dl-\int_{s}^{+\infty}Z^{n,x}_{l}dB_{l}+\int_{s}^{+\infty}ne^{-rl}(^{n}Y^{x}_{l}-h(X^{x}_{l}))^{-}dl.$
(4.14)
From Theorem 1 in [5] there exists a unique solution
$(^{n}Y^{x},Z^{n,x})\in{\cal S}^{2}\times{\cal M}^{2,d}$ satisfying the BSDE
(4.14).
Next let us define
$K^{n,x}_{s}=\int_{0}^{s}ne^{-rl}(^{n}Y^{x}_{l}-h(X^{x}_{l}))^{-}dl,$
then
$\begin{array}[]{ll}\int_{0}^{+\infty}e^{-rl}(^{n}Y^{x}_{l}-h(X^{x}_{l})\wedge^{n}Y^{x}_{l})dK_{l}^{n,x}&=n\int_{0}^{+\infty}e^{-rl}(^{n}Y^{x}_{l}-h(X^{x}_{l})\wedge^{n}Y^{x}_{l})e^{-rl}(^{n}Y^{x}_{l}-h(X^{x}_{l}))^{-}dl\\\
&=0.\end{array}$
Since $K^{n,x}$ is non-decreasing and $K^{n,x}_{0}=0$, we rewrite Eq. (4.14)
in RBSDE form
$\left\\{\begin{array}[]{l}{}^{n}Y^{x}_{s}e^{-rs}=\int_{s}^{+\infty}e^{-rl}f(X_{l}^{x})dl-\int_{s}^{+\infty}Z^{n,x}_{l}dB_{l}+K_{\infty}^{n,x}-K^{n,x}_{s},\\\
{}^{n}Y^{x}_{s}e^{-rs}\geq e^{-rs}(h(X^{x}_{s})\wedge^{n}Y^{x}_{s}),\,\forall
s\geq 0\mbox{ and
}\int_{0}^{+\infty}e^{-rl}(^{n}Y^{x}_{l}-h(X^{x}_{l})\wedge^{n}Y^{x}_{l})dK^{x}_{l}=0.\end{array}\right.$
(4.15)
Then from property (4.12) we have:
$^{n}Y^{x}_{s}e^{-rs}=esssup_{\tau\in{\cal
T}_{s}}E[\int_{s}^{\tau}e^{-rl}f(X_{l}^{x})dl+e^{-r\tau}(^{n}Y^{x}_{\tau}\wedge
h(X^{x}_{\tau}))|{\cal F}_{s}].$ (4.16)
Note that if we define
$f_{n}(t,x,y,z)=e^{-rt}f(x,y,z)+ne^{-rt}(y-h(x))^{-}$ $f_{n}(t,x,y,z)\leq
f_{n+1}(t,x,y,z).$
Then it follows from the comparison Theorem 2.2 in [19]
${}^{n}Y_{s}^{x}e^{-rs}\leq^{n+1}Y_{s}^{x}e^{-rs},$ $s\geq 0,$ a.s. and from
(4.12) and (4.16) ${}^{n}Y_{s}^{x}e^{-rs}\leq Y_{s}^{x}e^{-rs}.$ This implies
that there exits a càdlàg process $(\widetilde{Y}^{x}_{s})_{s\geq 0}$ such
that $P-a.s$. for any $s\geq 0$,
${}^{n}Y_{s}^{x}e^{-rs}\uparrow e^{-rs}\widetilde{Y}_{s}^{x},\quad\quad a.s.$
Let us actually show that $\tilde{Y}^{x}$ is càdlàg . By (4.16), for any
$n\geq 1$, the process
$(^{n}Y^{x}_{t}+\int_{0}^{t}e^{-rs}f(X_{s}^{x})ds)_{t\geq 0}$ is an ${\bf
F}$-supermartingale which converges increasingly and pointwisely to
$(\widetilde{Y}^{x}_{t}+\int_{0}^{t}e^{-rs}f(X_{s}^{x})ds)_{t\geq 0}$.
Therefore, the limit is also a càdlàg ${\bf F}$-supermartingale (see e.g.
Dellacherie and Meyer (1980), pp. 86). Hence, the process $\tilde{Y}^{x}$ is
càdlàg .
Then it follows from Proposition 2 in [11], as $n\rightarrow+\infty$,
$\widetilde{Y}^{x}_{s}e^{-rs}=esssup_{\tau\in{\cal
T}_{s}}E[\int_{s}^{\tau}e^{-rl}f(X_{l}^{x})dl+e^{-r\tau}(\widetilde{Y}^{x}_{\tau}\wedge
h(X^{x}_{\tau}))|{\cal F}_{s}].$ (4.17)
From(4.14) we have:
$\begin{array}[]{ll}E[\int_{s}^{+\infty}e^{-rl}(^{n}Y^{x}_{l}-h(X^{x}_{l}))^{-}dl]&=\frac{1}{n}E[^{n}Y^{x}_{s}e^{-rs}+\int_{s}^{+\infty}e^{-rl}f(X_{l}^{x})dl]\\\
&\leq\frac{1}{n}E[|Y^{x}_{s}e^{-rs}|+\int_{s}^{+\infty}|e^{-rl}f(X_{l}^{x})|dl]\\\
&\leq\frac{1}{n}(E[|Y^{x}_{s}e^{-rs}|]+C\int_{s}^{+\infty}e^{-rl}e^{C_{\gamma}l}|x|^{\gamma}dl)\end{array}$
for a constant $C$ independent of $n$ and $\bf H4$. Then
$E[\int_{s}^{+\infty}e^{-rl}(^{n}Y^{x}_{l}-h(X^{x}_{l}))^{-}dl]\leq\frac{C_{x}}{n}.$
Hence as $n\rightarrow+\infty$ we obtain,
$E[\int_{s}^{+\infty}e^{-rl}(\widetilde{Y}^{x}_{l}-h(X^{x}_{l}))^{-}dl]=0$,
and since $(\widetilde{Y}^{x}_{s})_{s\geq 0}$ (resp. $h(x)$) is a càdlàg
process (resp. continuous), we have
$\widetilde{Y}^{x}_{t}\geq h(X^{x}_{t}).$ (4.18)
From (4.12), (4.17) and (4.18) we get:
$\widetilde{Y}^{x}_{t}=Y^{x}_{t}\quad\forall t\geq 0.$
Now rewrite Eq. (4.14) in differential form
$\begin{array}[]{l}d(^{n}Y^{x}_{s}e^{-rs})=-[e^{-rs}f(X_{s}^{x})+ne^{-rs}(^{n}Y^{x}_{s}-h(X^{x}_{s}))^{-}]ds+Z^{n,x}_{s}dB_{s}.\\\
\end{array}$
So for arbitrary $T>0$ and $0\leq s\leq T$, Eq. (4.14) is equivalent to
$\begin{array}[]{l}{}^{n}Y^{x}_{s}=^{n}Y^{x}_{T}+\int_{s}^{T}[(f(X_{l}^{x})+n(^{n}Y^{x}_{l}-h(X^{x}_{l}))^{-})-r^{n}Y^{x}_{l}]dl-\int_{s}^{T}\widetilde{Z}^{n,x}_{l}dB_{l},\\\
\end{array}$ (4.19)
with $\widetilde{Z}^{n,x}_{s}=Z^{n,x}_{s}e^{rs}$. Let us set
$F_{n}(x,y,z)=f(x)+n(y-h(x))^{-})-ry$.
In order that it satisfies the assumptions of Theorem 2, we just need to
verify that $F_{n}$ satisfy condition (4.6) and (4.7). It is obvious that
$F_{n}$ satisfy (4.6) where $\mu>-r$, and we show that $F_{n}$ satisfy (4.7).
From the polynomial growth of $f$ and since $h$ bounded and estimate (4.2), we
deduce
$\begin{array}[]{lll}E[\int_{0}^{+\infty}e^{\lambda
s}|F_{n}(X_{s}^{x},0,0)|^{2}ds]&=E[\int_{0}^{+\infty}e^{\lambda
s}|f(X_{s}^{x})+n(-h(X_{s}^{x}))^{-}|^{2}ds]\\\ &\leq
2E[\int_{0}^{+\infty}e^{\lambda s}((1+|X_{s}^{x}|^{\gamma})2+n^{2}C2)ds]\\\
&\leq C\int_{0}^{+\infty}e^{\lambda
s}e^{C_{2\gamma}s}(|x|^{2\gamma}+n2)ds,\end{array}$
for $\lambda+C_{2\gamma}<0$. This proves assumption (4.7). Then
$u_{n}(x)=^{n}Y^{x}_{0},$
and is a viscosity solution of the elliptic PDE
${\cal A}u_{n}(x)+F_{n}(x,u_{n}(x),\sigma(x)^{*}\nabla u_{n}(x))=0.$
We now define
$u(x)=Y^{x}_{0},\quad\forall x\in I\\!\\!R^{k},$
which is a deterministic quantity. Let us admit for a moment the following
Lemma:
###### Lemma 1
The function $u$ is continuous in $R^{k}$.$\Box$
From the previous results we have, for each $x\in I\\!\\!R^{k},$
$u_{n}(x)\uparrow u(x)\quad\mbox{as}\quad n\rightarrow+\infty.$
Since $u_{n}$ and $u$ are continuous, it follow from Dini’s theorem that the
above convergence is uniform on compacts.
We now show that $u$ is a subsolution of (4.13). Let $x$ be a point at which
$u(x)>h(x),$ and let $(q,X)\in J^{2,+}u(x).$ From Lemma 6.1 in [6], there
exists sequences:
$\begin{array}[]{l}n_{j}\rightarrow+\infty,\quad x_{j}\rightarrow
x,\quad(q_{j},X_{j})\in J^{2,+}u_{n_{j}}(x_{j}),\end{array}$
such that
$(q_{j},X_{j})\rightarrow(q,X).$
But for any $j$,
$\begin{array}[]{ll}-\frac{1}{2}Tr[\sigma^{*}X_{j}\sigma]-\langle
b,q_{j}\rangle-F_{n}(x_{j},u_{n_{j}}(x_{j}),\sigma(x_{j})^{*}\nabla
u_{n_{j}}(x_{j}))\leq 0,\\\ -\frac{1}{2}Tr[\sigma^{*}X_{j}\sigma]-\langle
b,q_{j}\rangle-f(x_{j})-n_{j}(u_{n_{j}}(x_{j})-h(x_{j}))^{-})+ru_{n_{j}}(x_{j})\leq
0.\end{array}$
From the assumption that $u(x)>h(x)$ and the uniform convergence of $u_{n},$
it follows that for $j$ large enough $u_{n_{j}}(x_{j})>h(x_{j})$. Hence,
taking the limit as $j\rightarrow+\infty$ in the above inequality yields:
$-\frac{1}{2}Tr[\sigma^{*}X\sigma]-\langle b,q\rangle-f(x)+ru(x)\leq 0,$
and we have proved that $u$ is a subsolution of (4.13).
We now show that $u$ is a supersolution of (4.13). Let $x$ be arbitrary in
$I\\!\\!R^{k}$, and $(q,X)\in J^{2,-}u(x).$ We already know that $u(x)\geq
h(x).$ By the same argument as above, there exist sequences:
$\begin{array}[]{l}n_{j}\rightarrow+\infty,\quad x_{j}\rightarrow
x,\quad(q_{j},X_{j})\in J^{2,-}u_{n_{j}}(x_{j}),\end{array}$
such that
$(q_{j},X_{j})\rightarrow(q,X).$
But for any $j$,
$\begin{array}[]{ll}-\frac{1}{2}Tr[\sigma^{*}X_{j}\sigma]-\langle
b,q_{j}\rangle-F_{n}(x_{j},u_{n_{j}}(x_{j}),\sigma(x_{j},i)^{*}\nabla
u_{n_{j}}(x_{j})))\geq 0,\\\ -\frac{1}{2}Tr[\sigma^{*}X_{j}\sigma]-\langle
b,q_{j}\rangle-f(x_{j})-n_{j}(u_{n_{j}}(x_{j})-h(x_{j}))^{-})+ru_{n_{j}}(x_{j})\geq
0.\end{array}$
Hence,
$-\frac{1}{2}Tr[\sigma^{*}X_{j}\sigma]-\langle
b,q_{j}\rangle-f(x_{j})+ru_{n_{j}}(x_{j})\geq 0,$
and taking the limit as $j\rightarrow+\infty$, we conclude that:
$-\frac{1}{2}Tr[\sigma^{*}X\sigma]-\langle b,q\rangle-f(x)+ru(x)\geq 0.$
We conclude by showing that $u$ is of polynomial growth. From (4.12) we have,
$\begin{array}[]{ll}|Y^{x}_{0}|&\leq sup_{\tau\geq
0}E[\int_{0}^{\tau}e^{-rs}|f(X_{s}^{x})|ds+|h(X^{x}_{\tau})|1\\!\\!1_{[\tau<+\infty]}]\\\
&\leq sup_{\tau\geq
0}E[\int_{0}^{\tau}e^{-rs}|f(X_{s}^{x})|ds+e^{-r\tau}|h(X^{x}_{\tau})|]\\\
&\leq E[\int_{0}^{+\infty}e^{-rs}|f(X_{s}^{x})|ds]+C_{1}.\end{array}$ (4.20)
From polynomial growth of $f$ and $u(x)=Y^{x}_{0}$, we deduce that $u$ is of
polynomial growth. Now we proceed to the proof of Lemme1.
$Proof$ of Lemma 2. It suffices to show that whenever $x_{n}\rightarrow x$,
$|Y_{0}^{x_{n}}-Y_{0}^{x}|\rightarrow 0$.
From (4.12) we have,
$Y^{x}_{0}=\sup_{\tau\in{\cal
T}_{0}}E[\int_{0}^{\tau}e^{-rl}f(X_{l}^{x})dl+e^{-r\tau}h(X^{x}_{\tau})],$
$Y^{x_{n}}_{0}=\sup_{\tau\in{\cal
T}_{0}}E[\int_{0}^{\tau}e^{-rl}f(X_{l}^{x_{n}})dl+e^{-r\tau}h(X^{x_{n}}_{\tau})]$
then,
$\begin{array}[]{ll}|Y^{x_{n}}_{0}-Y^{x}_{0}|&\leq\sup\limits_{\tau\in{\cal
T}_{0}}E[\int_{0}^{\tau}e^{-rl}|f(X_{l}^{x_{n}})-f(X_{l}^{x})|dl+e^{-r\tau}|h(X^{x_{n}}_{\tau})-h(X^{x}_{\tau})|]\\\
&\leq
E[\int_{0}^{+\infty}e^{-rl}|f(X_{l}^{x_{n}})-f(X_{l}^{x})|dl]+E[\sup\limits_{t\geq
0}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|].\end{array}$ (4.21)
In the right-hand side of (4.21) the first term converges to 0 as
$x_{n}\rightarrow x$. Next let us show that,
$E[\sup\limits_{t\geq 0}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|]\rightarrow
0\quad\mbox{as}\quad x_{n}\rightarrow x.$
For any $T\geq 0$ we have
$E[\sup\limits_{t\geq 0}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|]\leq
E[\sup\limits_{0\leq t\leq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|]+E[\sup\limits_{t\geq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|].$
Since $h$ is bounded there exists $C$ such that,
$E[\sup\limits_{t\geq 0}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|]\leq
E[\sup\limits_{0\leq t\leq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|]+Ce^{-rT}.$
For any $\rho>0$ we have:
$\begin{array}[]{ll}E[\sup\limits_{0\leq t\leq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|]&=E[\sup\limits_{0\leq t\leq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|1\\!\\!1_{[\sup\limits_{t\leq
T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq T}|X_{t}^{x}|\leq\rho]}]\\\
&+E[\sup\limits_{0\leq t\leq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|1\\!\\!1_{[\sup\limits_{t\leq
T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq T}|X_{t}^{x}|>\rho]}].\end{array}$
But since $h$ is continuous then it is uniformly continuous on compact
subsets, then there exists $\pi:R^{k}\rightarrow R$ increasing with
$\pi(0)=0$, such that:
$|h(X^{x_{n}}_{t})-h(X^{x}_{t})|\leq\pi(|X^{x_{n}}_{t}-X^{x}_{t}|),$
we have
$\begin{array}[]{ll}E[\sup\limits_{0\leq t\leq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|1\\!\\!1_{[\sup\limits_{t\leq
T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq T}|X_{t}^{x}|\leq\rho]}]&\leq
E[\sup\limits_{0\leq t\leq
T}\pi(|X^{x_{n}}_{t}-X^{x}_{t}|)1\\!\\!1_{[\sup\limits_{t\leq
T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq T}|X_{t}^{x}|\leq\rho]}]\\\ &\leq
E[\pi(\sup\limits_{0\leq t\leq
T}|X^{x_{n}}_{t}-X^{x}_{t}|)1\\!\\!1_{[\sup\limits_{t\leq
T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq T}|X_{t}^{x}|\leq\rho]}].\end{array}$
Using the continuity proprety (4.3), $\pi(0)=0$ and the Lebesgue dominated
convergence theorem to obtain that
$E[\sup\limits_{0\leq t\leq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|1\\!\\!1_{[\sup\limits_{t\leq
T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq T}|X_{t}^{x}|\leq\rho]}]\rightarrow
0\quad\mbox{as}\quad x_{n}\rightarrow x.$ (4.22)
The second term satisfies:
$\begin{array}[]{ll}E[\sup\limits_{0\leq t\leq
T}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|1\\!\\!1_{[\sup\limits_{t\leq
T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq T}|X_{t}^{x}|>\rho]}]\\\ {}\qquad\leq
E[\sup\limits_{0\leq t\leq
T}e^{-2rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|^{2}]\\}^{\frac{1}{2}}\\{E[1\\!\\!1_{[\sup\limits_{t\leq
T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq
T}|X_{t}^{x}|>>\rho]}]\\}^{\frac{1}{2}}\\\ {}\qquad\leq
E[\\{\sup\limits_{0\leq t\leq
T}e^{-2rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|^{2}]\\}^{\frac{1}{2}}\\{\
\rho^{-1}E[\sup\limits_{t\leq T}|X_{t}^{x_{n}}|+\sup\limits_{t\leq
T}|X_{t}^{x}|]\\}^{\frac{1}{2}}.\end{array}$
Since $h$ is bounded, it follows that, when $x_{n}\rightarrow x$, the right-
hand side of the last inequality is smaller than $\rho^{-\frac{1}{2}}C_{x}$.
However, from previous results we have,
$\limsup\limits_{x_{n}\rightarrow x}E[\sup\limits_{t\geq
0}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|]\leq\rho^{-\frac{1}{2}}C_{x}+Ce^{-rT}.$
As $\rho$ and $T$ are arbitrary then making $\rho\rightarrow+\infty$ and
$T\rightarrow+\infty$ to obtain that,
$\lim\limits_{x_{n}\rightarrow x}E[\sup\limits_{t\geq
0}e^{-rt}|h(X^{x_{n}}_{t})-h(X^{x}_{t})|]=0.$ (4.23)
From (4.21) and (4.23), we deduce
$|Y_{0}^{x_{n}}-Y_{0}^{x}|\rightarrow 0\quad\mbox{as}\quad x_{n}\rightarrow
x.\Box$
### 4.2 Existence of a solution for the system of variational inequalities
Let $(Y^{1,x}_{s},...,Y^{m,x}_{s})_{s\geq 0}$ be the processes which satisfy
the Verification Theorem 1 in the case when the process $X\equiv X^{x}$.
Therefore using the characterization (4.12), there exist processes $K^{i,x}$
and $Z^{i,x}$, such that the triples ($Y^{i,x},Z^{i,x},K^{i,x})$ are unique
solutions (thanks to Remark 2) of the following reflected BSDEs: for any
$i=1,...,m$ we have,
$\left\\{\begin{array}[]{l}Y^{i,x},K^{i,x}\in{\cal S}^{2}\mbox{ and
}Z^{i,x}\in{\cal M}^{2,d};\,K^{i,x}\mbox{ is non-decreasing and
}K^{i,x}_{0}=0,\\\
e^{-rs}Y^{i,x}_{s}=\int_{s}^{+\infty}e^{-rl}\psi_{i}(X_{l}^{x})ds-\int_{s}^{+\infty}Z^{i,x}_{l}dB_{l}+K_{+\infty}^{i,x}-K^{i,x}_{s},\,\,\,s\in
I\\!\\!R^{+},\,\,\lim\limits_{s\rightarrow+\infty}(e^{-rs}Y^{i,x}_{s})=0,\\\
e^{-rs}Y^{i,x}_{s}\geq-e^{-rs}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{s}^{x})+Y^{j,x}_{s}),\,\,s\in I\\!\\!R^{+},\\\
\int_{0}^{+\infty}e^{-rl}(Y^{i,x}_{l}-\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{l}^{x})+Y^{j,x}_{l}))dK^{i,x}_{l}=0.\end{array}\right.$
(4.24)
Moreover we have the following result.
###### Proposition 3
There are deterministic functions $v^{1},...,v^{m}$ $:I\\!\\!R^{k}\rightarrow
I\\!\\!R$ such that:
$\forall x\in I\\!\\!R^{k},Y_{0}^{i,x}=v^{i}(x),\,\,i=1,...,m.$
Moreover the functions $v^{i}$, $i=1,...,m,$ are of polynomial growth.
$Proof$: For $n\geq 0$ let $(Y^{n,1,x}_{s},...,Y^{n,m,x}_{s})_{s\geq 0}$ be
the processes constructed in (3.4)-(3.5). Therefore using an induction
argument and Theorem 2 there exist deterministic continuous with polynomial
growth functions $v^{n,i}$ ($i=1,...,m$) such that for any $x\in
I\\!\\!R^{k}$, $Y^{n,i,x}_{0}=v^{n,i}(x)$. Using now inequality (3.6) we get:
$Y^{n,i,x}_{t}\leq Y^{n+1,i,x}_{t}\leq
CE[\int_{0}^{+\infty}\\{\max_{i=1,m}|e^{-rs}\psi_{i}(X^{x}_{s})|\\}ds]$
since $Y^{n,i,x}_{t}$ is deterministic. Therefore combining the polynomial
growth of $\psi_{i}$ and estimate (4.2) for $X^{x}$ we obtain:
$v^{n,i}(x)\leq v^{n+1,i}(x)\leq C(1+|x|^{\gamma})$
for a constant $C$ independent of $n$. In order to complete the proof it is
enough now to set $v^{i}(x):=\lim_{n\rightarrow\infty}v^{n,i}(x),x\in
I\\!\\!R^{k}$ since $Y^{n,i,x}\nearrow Y^{i,x}$ as $n\rightarrow\infty$.
$\Box$
We are now going to focus on the continuity of the functions
$v^{1},...,v^{m}$. But first let us deal with some properties of the optimal
strategy which exist thanks to Theorem 1.
###### Proposition 4
Let $(\delta,u)=((\tau_{n})_{n\geq 1},(\xi_{n})_{n\geq 1})$ be an optimal
strategy, then there exists a constant $C$ which does not depend on $t$ and
$x$ such that:
$\forall n\geq 1,\,\,E[e^{-r\tau_{n}}]\leq\frac{C(1+|x|^{\gamma})}{n}.$ (4.25)
$Proof$: Recall the characterization of (3.8) that reads as:
$\begin{array}[]{l}Y^{i,x}_{0}=sup_{(\delta,u)\in{\cal
D}}E[\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X_{s}^{x})ds-\sum_{k\geq
1}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X^{x}_{\tau_{k}})].\end{array}$
Now if $(\delta,u)=((\tau_{n})_{n\geq 1},(\xi_{n})_{n\geq 1})$ is the optimal
strategy then we have:
$\begin{array}[]{l}Y^{i,x}_{0}=E[\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X_{s}^{x})ds-\sum_{k\geq
1}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X^{x}_{\tau_{k}})].\end{array}$
Taking into account that $g_{ij}\geq\frac{1}{\alpha}>0$ for any $i\neq j$ we
obtain:
$\begin{array}[]{ll}\frac{1}{\alpha}E[\sum_{k=1,n}e^{-r\tau_{k}}]+Y^{i,x}_{0}&\leq
E[\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X_{s}^{x})ds-\sum_{k\geq
n+1}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X^{x}_{\tau_{k}})].\end{array}$
But for any $k\leq n$, $e^{-r\tau_{n}}\leq e^{-r\tau_{k}}$ then:
$\begin{array}[]{ll}\frac{n}{\alpha}E[e^{-r\tau_{n}}]+Y^{i,x}_{0}&\leq
E[\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X_{s}^{x})ds-\sum_{k\geq
n+1}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X^{x}_{\tau_{k}})]\\\ {}&\leq
E[\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X_{s}^{x})ds].\end{array}$
and then
$\begin{array}[]{ll}\frac{n}{\alpha}E[e^{-r\tau_{n}}]&\leq
E[\int_{0}^{+\infty}e^{-rs}\mid\psi_{u_{s}}(X_{s}^{x})\mid ds]-Y^{i,x}_{0}\\\
{}&\leq E[\int_{0}^{+\infty}e^{-rs}\mid\psi_{u_{s}}(X_{s}^{x})\mid
ds]-Y^{0,i,x}_{0}.\end{array}$
Finally taking into account the facts that $\psi_{i}$ and $Y^{0,i,x}$ are of
polynomial growth, estimate (4.2) for $X^{x}$ and $\bf H4$ to obtain the
desired result. Note that the polynomial growth of $Y^{0,i,x}$ stems from
Proposition 3. $\Box$
###### Remark 3
The estimate (4.25) is also valid for the optimal strategy if at the initial
time the state of the plant is an arbitrary $i\in{\cal I}$. $\Box$
We are now ready to give the main result of this article.
###### Theorem 4
The functions $(v^{1},...,v^{m}):I\\!\\!R^{k}\rightarrow I\\!\\!R$ are
continuous and solution in viscosity sense of the system of variational
inequalities with inter-connected obstacles (2.4).
$Proof$: First let us focus on continuity and let us show that $v1$ is
continuous. The same proof will be valid for the continuity of the other
functions $v^{i}$ ($i=2,...,m$). First the characterization (3.8) implies
that:
$Y^{1,x}_{0}=\sup_{(\delta,\xi)\in{\cal
D}}E[\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X^{x}_{s})ds-\sum_{n\geq
1}e^{-r\tau_{n}}g_{u_{\tau_{n-1}}u_{\tau_{n}}}(X^{x}_{\tau_{n}})]$
On the other hand an optimal strategy $(\delta^{*},\xi^{*})$ exists and
satisfies the estimates (4.25) with the same constant $C$. Next let
$\epsilon>0$ and $x^{\prime}\in B(x,\epsilon)$ and let us consider the
following set of strategies:
$\tilde{D}:=\\{(\delta,\xi)=((\tau_{n})_{n\geq 1},(\xi_{n})_{n\geq 0})\in{\cal
D}\mbox{ such that }\forall n\geq
1,E[e^{-r\tau_{n}}]\leq\frac{C(1+(\epsilon+|x|^{\gamma}))}{n}\\}.$
Therefore the strategy $(\delta^{*},\xi^{*})$ belongs to $\tilde{D}$ and then
we have:
$\begin{array}[]{ll}Y^{1,x}_{0}&=\sup_{(\delta,\xi)\in{\tilde{D}}}E[\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X^{x}_{s})ds-\sum_{n\geq
1}e^{-ru_{\tau_{n}}}g_{u_{\tau_{n-1}}u_{\tau_{n}}}(X^{x}_{\tau_{n}})]\\\
{}&=sup_{(\delta,u)\in{\tilde{D}}}E[\int_{0}^{\tau_{n}}e^{-rs}\psi_{u_{s}}(X^{x}_{s})ds\\\
{}&\qquad\qquad\qquad-\sum_{1\leq k\leq
n}e^{-ru_{\tau_{k}}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X^{x}_{{\tau}_{k}})+e^{-r\tau_{n}}Y^{u_{\tau_{n}},x}_{\tau_{n}}]\end{array}$
and
$\begin{array}[]{ll}Y^{1,x^{\prime}}_{0}&=\sup_{(\delta,\xi)\in{\tilde{D}}}E[\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X^{x^{\prime}}_{s})ds-\sum_{n\geq
1}e^{-ru_{\tau_{n}}}g_{u_{\tau_{n-1}}u_{\tau_{n}}}(X^{x^{\prime}}_{\tau_{n}})]\\\
{}&=sup_{(\delta,u)\in{\tilde{D}}}E[\int_{0}^{\tau_{n}}e^{-rs}\psi_{u_{s}}(X^{x^{\prime}}_{s})ds\\\
{}&\qquad\qquad\qquad-\sum_{1\leq k\leq
n}e^{-ru_{\tau_{k}}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X^{x^{\prime}}_{{\tau}_{k}})+e^{-r\tau_{n}}Y^{u_{\tau_{n}},x^{\prime}}_{\tau_{n}}]\end{array}$
The second equalities it due to the dynamical programming principle. It
follows that:
$\begin{array}[]{lll}|Y^{1,x^{\prime}}_{0}-Y^{1,x}_{0}|&\leq
sup_{(\delta,u)\in{\tilde{D}}}E[\int_{0}^{\tau_{n}}e^{-rs}|\psi_{u_{s}}(X^{x^{\prime}}_{s})-\psi_{u_{s}}(X^{x}_{s})|ds\\\
{}&\qquad+\sum_{1\leq k\leq
n}e^{-ru_{\tau_{k}}}|g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X^{x^{\prime}}_{\tau_{k}})-g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X^{x}_{\tau_{k}})|\\\
{}&\qquad+e^{-r\tau_{n}}|Y^{u_{\tau_{n}},x^{\prime}}_{\tau_{n}}-Y^{u_{\tau_{n}},x}_{\tau_{n}}|]\\\
{}&\leq
E[\int_{0}^{+\infty}\max_{j=1,m}e^{-rs}|\psi_{j}(X^{x^{\prime}}_{s})-\psi_{j}(X^{x}_{s})|ds\\\
{}&\qquad+n\max_{i\neq j\in{\cal I}}\\{\sup_{s\geq
0}e^{-rs}|g_{ij}(X^{x^{\prime}}_{s})-g_{ij}(X^{x}_{s})|\\}]\\\
{}&\qquad+sup_{(\delta,u)\in{\tilde{D}}}(E[e^{-2r\tau_{n}}])^{\frac{1}{2}}(2E[(Y^{u_{\tau_{n}},x^{\prime}}_{\tau_{n}})2+(Y^{u_{\tau_{n}},x}_{\tau_{n}})2])^{\frac{1}{2}}.\end{array}$
(4.26)
In the right-hand side of (4.26) the first and the second term converges to
$0$ as $x^{\prime}\rightarrow x$.
Now let us focus on the last one. Since $(\delta,u)\in\tilde{D}$ then:
$\begin{array}[]{ll}sup_{(\delta,u)\in{\tilde{D}}}(E[e^{-2r\tau_{n}}])^{\frac{1}{2}}(2E[(Y^{u_{\tau_{n}},x^{\prime}}_{\tau_{n}})2+(Y^{u_{\tau_{n}},x}_{\tau_{n}})2])^{\frac{1}{2}}&\leq
sup_{(\delta,u)\in{\tilde{D}}}(E[e^{-r\tau_{n}}])^{\frac{1}{2}}(2E[(Y^{u_{\tau_{n}},x^{\prime}}_{\tau_{n}})2+(Y^{u_{\tau_{n}},x}_{\tau_{n}})2])^{\frac{1}{2}}\\\
&\leq
n^{-\frac{1}{2}}\sup_{(\delta,u)\in{\tilde{D}}}(2E[(Y^{u_{\tau_{n}},x^{\prime}}_{\tau_{n}})2+(Y^{u_{\tau_{n}},x}_{\tau_{n}})2])^{\frac{1}{2}}\\\
{}&\leq Cn^{-\frac{1}{2}}(1+|x|^{\gamma}+|x^{\prime}|^{\gamma})\end{array}$
where $C$ an appropriate constant which comes from the polynomial growth of
$\psi_{i}$, $i\in{\cal I}$, estimate (4.2) for the process $X^{x}$ and
inequality (3.6). Going back now to (4.26), taking the limit as
$x^{\prime}\rightarrow x$ to obtain:
$\lim_{x^{\prime}\rightarrow x}|Y^{1,x^{\prime}}_{0}-Y^{1,x}_{0}|\leq
Cn^{-\frac{1}{2}}(1+2|x|^{\gamma}).$
As $n$ is arbitrary then putting $n\rightarrow+\infty$ to obtain:
$Y^{1,x^{\prime}}_{0}\rightarrow Y^{1,x}_{0}.$
Therefore $v^{1}$ is continuous. In the same way we can show that
$v^{2}$,…,$v^{m}$ are continuous. As they are of polynomial growth then taking
into account Theorem 2 to obtain that $(v^{1},\dots,v^{m})$ is a viscosity
solution for the system of variational inequalities with inter-connected
obstacles (2.4). $\Box$
## 5 Uniqueness of the solution of the system
We are going now to address the question of uniqueness of the viscosity
solution of the system (2.4). We have the following:
###### Theorem 5
The solution in viscosity sense of the system of variational inequalities with
inter-connected obstacles (2.4) is unique in the space of continuous functions
on $R^{k}$ which satisfy a polynomial growth condition, i.e., in the space
$\begin{array}[]{l}{\cal C}:=\\{\varphi:I\\!\\!R^{k}\rightarrow
I\\!\\!R,\mbox{ continuous and for any }\\\ \qquad\qquad\qquad
x,\,|\varphi(x)|\leq C(1+|x|^{\gamma})\mbox{ for some constants
}C\quad\mbox{and}\quad\gamma\\}.\end{array}$
Proof. We will show by contradiction that if $u_{1},...,u_{m}$ and
$w_{1},...,w_{m}$ are a subsolution and a supersolution respectively for (2.4)
then for any $i=1,...,m$, $u_{i}\leq w_{i}$. Therefore if we have two
solutions of (2.4) then they are obviously equal. Actually for some $R>0$
suppose there exists $(x_{0},i_{0})\in B_{R}\times{\cal I}$ $(B_{R}:=\\{x\in
I\\!\\!R^{k};|x|\leq R\\})$ such that:
$\max\limits_{(x,i)}(u_{i}(x)-w_{i}(x))=u_{i_{0}}(x_{0})-w_{i_{0}}(x_{0})=\eta>0.$
(5.1)
Then, for a small $\epsilon>0$, and $\theta,\lambda\in(0,1)$ small enough, let
us define:
$\Phi^{i}_{\epsilon}(x,y)=u_{i}(x)-(1-\lambda)w_{i}(y)-\frac{1}{2\epsilon}|x-y|^{2\gamma}-\theta(|x-x_{0}|^{2\gamma+2}+|y-x_{0}|^{2\gamma+2}).$
(5.2)
By the polynomial growth assumption on $u_{i}$ and $w_{i}$, there exists a
$(x_{\epsilon},y_{\epsilon},i_{\epsilon})\in B_{R}\times B_{R}\times{\cal I}$,
such that:
$\Phi^{i_{\epsilon}}_{\epsilon}(x_{\epsilon},y_{\epsilon})=\max\limits_{(x,y,i)}\Phi^{i}_{\epsilon}(x,y).$
On the other hand, from
$2\Phi^{i_{\epsilon}}_{\epsilon}(x_{\epsilon},y_{\epsilon})\geq\Phi^{i_{\epsilon}}_{\epsilon}(x_{\epsilon},x_{\epsilon})+\Phi^{i_{\epsilon}}_{\epsilon}(y_{\epsilon},y_{\epsilon})$,
we have
$\begin{array}[]{ll}\frac{1}{2\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2\gamma}&\leq(u_{i_{\epsilon}}(x_{\epsilon})-u_{i_{\epsilon}}(y_{\epsilon}))+(1-\lambda)(w_{i_{\epsilon}}(x_{\epsilon})-w_{i_{\epsilon}}(y_{\epsilon}))\\\
&\leq\sum\limits_{i\in{\cal
I}}|u_{i}(x_{\epsilon})-u_{i}(y_{\epsilon})|+(1-\lambda)\sum\limits_{i\in{\cal
I}}|w_{i}(x_{\epsilon})-w_{i}(y_{\epsilon})|\end{array}$ (5.3)
and consequently $\frac{1}{2\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2\gamma}$ is
bounded, and as $\epsilon\rightarrow 0$,
$|x_{\epsilon}-y_{\epsilon}|\rightarrow 0$. Since $u_{i}$ and $w_{i}$ are
uniformly continuous on $B_{R}$, then
$\frac{1}{2\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2\gamma}\rightarrow 0$ as
$\epsilon\rightarrow 0.$
Since
$u_{i_{0}}(x_{0})-(1-\lambda)w_{i_{0}}(x_{0})\leq\Phi^{i_{\epsilon}}_{\epsilon}(x_{\epsilon},y_{\epsilon})\leq
u_{i_{\epsilon}}(x_{\epsilon})-(1-\lambda)w_{i_{\epsilon}}(y_{\epsilon}),$
it follow as $\lambda\rightarrow 0$ and the continuity of $u_{i}$ and $w_{i}$
that, up to a subsequence,
$(x_{\epsilon},y_{\epsilon},i_{\epsilon})\rightarrow(x_{0},x_{0},i_{0}).$
(5.4)
We now claim that:
$u_{i_{\epsilon}}(x_{\epsilon})-\max\limits_{j\in{\cal
I}^{-i_{\epsilon}}}\\{-g_{i_{\epsilon}j}(x_{\epsilon})+u_{j}(x_{\epsilon})\\}>0.$
(5.5)
Indeed if
$u_{i_{\epsilon}}(x_{\epsilon})-\max\limits_{j\in{\cal
I}^{-i_{\epsilon}}}\\{-g_{i_{\epsilon}j}(x_{\epsilon})+u_{j}(x_{\epsilon})\\}\leq
0$
then there exists $k\in{\cal I}^{-i_{\epsilon}}$ such that:
$u_{i_{\epsilon}}(x_{\epsilon})\leq-
g_{i_{\epsilon}k}(x_{\epsilon})+u_{k}(x_{\epsilon}).$
From the supersolution property of $w_{i_{\epsilon}}(y_{\epsilon})$, we have
$w_{i_{\epsilon}}(y_{\epsilon})\geq\max\limits_{j\in{\cal
I}^{-i_{\epsilon}}}(-g_{i_{\epsilon}j}(y_{\epsilon})+w_{j}(y_{\epsilon}))$
then
$w_{i_{\epsilon}}(y_{\epsilon})\geq-
g_{i_{\epsilon}k}(y_{\epsilon})+w_{k}(y_{\epsilon}).$
It follows that:
$u_{i_{\epsilon}}(x_{\epsilon})-(1-\lambda)w_{i_{\epsilon}}(y_{\epsilon})-(u_{k}(x_{\epsilon})-(1-\lambda)w_{k}(y_{\epsilon}))\leq(1-\lambda)g_{i_{\epsilon}k}(y_{\epsilon})-g_{i_{\epsilon}k}(x_{\epsilon}).$
Now since $g_{ij}\geq\alpha>0$, for every $i\neq j$, and taking into account
of (5.2) to obtain:
$\begin{array}[]{ll}\Phi^{i_{\epsilon}}_{\epsilon}(x_{\epsilon},y_{\epsilon})-\Phi^{k}_{\epsilon}(x_{\epsilon},y_{\epsilon})&<-\alpha\lambda+g_{i_{\epsilon}k}(y_{\epsilon})-g_{i_{\epsilon}k}(x_{\epsilon})\\\
\end{array}$
But this contradicts the definition of $i_{\epsilon}$, since
$g_{i_{\epsilon}k}$ is uniformly continuous on $B_{R}$ and the claim (5.5)
holds.
Next let us denote
$\varphi_{\epsilon}(x,y)=\frac{1}{2\epsilon}|x-y|^{2\gamma}+\theta(|x-x_{0}|^{2\gamma+2}+|y-x_{0}|^{2\gamma+2}).$
(5.6)
Then we have:
$\left\\{\begin{array}[]{lll}D_{x}\varphi_{\epsilon}(t,x,y)=\frac{\gamma}{\epsilon}(x-y)|x-y|^{2\gamma-2}+\theta(2\gamma+2)(x-x_{0})|x-x_{0}|^{2\gamma},\\\
D_{y}\varphi_{\epsilon}(t,x,y)=-\frac{\gamma}{\epsilon}(x-y)|x-y|^{2\gamma-2}+\theta(2\gamma+2)(y-y_{0})|y-y_{0}|^{2\gamma},\\\
\\\
B(t,x,y)=D_{x,y}^{2}\varphi_{\epsilon}(t,x,y)=\frac{1}{\epsilon}\begin{pmatrix}a_{1}(x,y)&-a_{1}(x,y)\\\
-a_{1}(x,y)&a_{1}(x,y)\end{pmatrix}+\begin{pmatrix}a_{2}(x)&0\\\
0&a_{2}(y)\end{pmatrix}\\\ \\\ \mbox{ with
}a_{1}(x,y)=\gamma|x-y|^{2\gamma-2}I+\gamma(2\gamma-2)(x-y)(x-y)^{*}|x-y|^{2\gamma-4}\mbox{
and }\\\
a_{2}(x)=\theta(2\gamma+2)|x-x_{0}|^{2\gamma}I+2\theta\gamma(2\gamma+2)(x-x_{0})(x-x_{0})^{*}|x-x_{0}|^{2\gamma-2}.\end{array}\right.$
(5.7)
Taking into account (5.5) then applying the result by Crandall et al. (Theorem
3.2, [6]) to the function
$u_{i}(x)-(1-\lambda)w_{i}(y)-\varphi_{\epsilon}(x,y)$
at the point $(x_{\epsilon},y_{\epsilon})$, for any $\epsilon_{1}>0$, we can
find $X,Y\in S_{k}$, such that:
$\left\\{\begin{array}[]{lllll}(\frac{\gamma}{\epsilon}(x_{\epsilon}-y_{\epsilon})|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2}+\theta(2\gamma+2)(x_{\epsilon}-x_{0})|x_{\epsilon}-x_{0}|^{2\gamma},X)\in
J^{2,+}(u_{i_{\epsilon}}(x_{\epsilon})),\\\
(\frac{\gamma}{\epsilon}(x_{\epsilon}-y_{\epsilon})|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2}-\theta(2\gamma+2)(y_{\epsilon}-y_{0})|y_{\epsilon}-y_{0}|^{2\gamma},Y)\in
J^{2,-}((1-\lambda)w_{i_{\epsilon}}(y_{\epsilon})),\\\
-(\frac{1}{\epsilon_{1}}+||B(x_{\epsilon},y_{\epsilon})||)I\leq\begin{pmatrix}X&0\\\
0&-Y\end{pmatrix}\leq
B(x_{\epsilon},y_{\epsilon})+\epsilon_{1}B(x_{\epsilon},y_{\epsilon})2.\end{array}\right.$
(5.8)
Taking now into account (5.5), and the definition of viscosity solution, we
get:
$\begin{array}[]{l}ru_{i_{\epsilon}}(x_{\epsilon})-\frac{1}{2}Tr[\sigma^{*}(x_{\epsilon})X\sigma(x_{\epsilon})]-\langle\frac{\gamma}{\epsilon}(x_{\epsilon}-y_{\epsilon})|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2}\\\
\qquad\qquad\qquad\qquad\qquad+\theta(2\gamma+2)(x_{\epsilon}-x_{0})|x_{\epsilon}-x_{0}|^{2\gamma},b(x_{\epsilon})\rangle-\psi_{i_{\epsilon}}(x_{\epsilon})\leq
0\mbox{ and }\\\
r(1-\lambda)w_{i_{\epsilon}}(y_{\epsilon})-\frac{1}{2}Tr[\sigma^{*}(y_{\epsilon})Y\sigma(y_{\epsilon})]-\langle\frac{\gamma}{\epsilon}(x_{\epsilon}-y_{\epsilon})|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2}\\\
\qquad\qquad\qquad\qquad\qquad-\theta(2\gamma+2)(y_{\epsilon}-x_{0})|y_{\epsilon}-x_{0}|^{2\gamma},b(y_{\epsilon})\rangle-(1-\lambda)\psi_{i_{\epsilon}}(y_{\epsilon})\geq
0\end{array}$
which implies that:
$\begin{array}[]{llll}&ru_{i_{\epsilon}}(x_{\epsilon})-r(1-\lambda)w_{i_{\epsilon}}(y_{\epsilon})\leq\frac{1}{2}Tr[\sigma^{*}(x_{\epsilon})X\sigma(x_{\epsilon})-\sigma^{*}(y_{\epsilon})Y\sigma(y_{\epsilon})]\\\
&\qquad+\langle\frac{\gamma}{\epsilon}(x_{\epsilon}-y_{\epsilon})|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2},b(x_{\epsilon})-b(y_{\epsilon})\rangle\\\
&\qquad+\langle\theta(2\gamma+2)(x_{\epsilon}-x_{0})|x_{\epsilon}-x_{0}|^{2\gamma},b(x_{\epsilon})\rangle+\langle\theta(2\gamma+2)(y_{\epsilon}-x_{0})|y_{\epsilon}-x_{0}|^{2\gamma},b(y_{\epsilon})\rangle\\\
&\qquad+\psi_{i_{\epsilon}}(x_{\epsilon})-(1-\lambda)\psi_{i_{\epsilon}}(y_{\epsilon}).\end{array}$
(5.9)
But from (5.7) there exist two constants $C$ and $C_{1}$ such that:
$||a_{1}(x_{\epsilon},y_{\epsilon})||\leq
C|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2}\mbox{ and
}(||a_{2}(x_{\epsilon})||\vee||a_{2}(y_{\epsilon})||)\leq C_{1}\theta.$
As
$B=B(x_{\epsilon},y_{\epsilon})=\frac{1}{\epsilon}\begin{pmatrix}a_{1}(x_{\epsilon},y_{\epsilon})&-a_{1}(x_{\epsilon},y_{\epsilon})\\\
-a_{1}(x_{\epsilon},y_{\epsilon})&a_{1}(x_{\epsilon},y_{\epsilon})\end{pmatrix}+\begin{pmatrix}a_{2}(x_{\epsilon})&0\\\
0&a_{2}(y_{\epsilon})\end{pmatrix}$
then
$B\leq\frac{1}{\epsilon}\begin{pmatrix}I&-I\\\ -I&I\end{pmatrix}+C_{1}\theta
I.$
It follows that:
$B+\epsilon_{1}B2\leq
C(\frac{1}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2}+\frac{\epsilon_{1}}{\epsilon
2}|x_{\epsilon}-y_{\epsilon}|^{4\gamma-4})\begin{pmatrix}I&-I\\\
-I&I\end{pmatrix}+C_{1}\theta I$ (5.10)
where $C$ and $C_{1}$ which hereafter may change from line to line. Choosing
now $\epsilon_{1}=\epsilon$, yields the relation
$B+\epsilon_{1}B2\leq\frac{C}{\epsilon}(|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2}+|x_{\epsilon}-y_{\epsilon}|^{4\gamma-4})\begin{pmatrix}I&-I\\\
-I&I\end{pmatrix}+C_{1}\theta I.$ (5.11)
Now, from $\bf H1$, (5.8) and (5.11) we get:
$\frac{1}{2}Tr[\sigma^{*}(x_{\epsilon})X\sigma(x_{\epsilon})-\sigma^{*}(y_{\epsilon})Y\sigma(y_{\epsilon})]\leq\frac{C}{\epsilon}(|x_{\epsilon}-y_{\epsilon}|^{2\gamma}+|x_{\epsilon}-y_{\epsilon}|^{4\gamma-2})+C_{1}\theta(1+|x_{\epsilon}|^{2}+|y_{\epsilon}|^{2}).$
Next
$\langle\frac{\gamma}{\epsilon}(x_{\epsilon}-y_{\epsilon})|x_{\epsilon}-y_{\epsilon}|^{2\gamma-2},b(x_{\epsilon})-b(y_{\epsilon})\rangle\leq\frac{C2}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2\gamma}$
and finally,
$\langle\theta(2\gamma+2)(x_{\epsilon}-x_{0})|x_{\epsilon}-x_{0}|^{2\gamma},b(x_{\epsilon})\rangle\leq\theta
C(1+|x_{\epsilon}|)|x_{\epsilon}-x_{0}|^{2\gamma+1}$
$\langle\theta(2\gamma+2)(y_{\epsilon}-x_{0})|y_{\epsilon}-x_{0}|^{2\gamma},b(y_{\epsilon})\rangle\leq\theta
C(1+|y_{\epsilon}|)|y_{\epsilon}-x_{0}|^{2\gamma+1}.$
So that by plugging into (LABEL:viscder) we obtain:
$\begin{array}[]{l}ru_{i_{\epsilon}}(x_{\epsilon})-r(1-\lambda)w_{i_{\epsilon}}(y_{\epsilon})\leq\frac{C}{\epsilon}(|x_{\epsilon}-y_{\epsilon}|^{2\gamma}+|x_{\epsilon}-y_{\epsilon}|^{4\gamma-2})+C_{1}\theta(1+|x_{\epsilon}|^{2}+|y_{\epsilon}|^{2})+\frac{C2}{\epsilon}|x_{\epsilon}-y_{\epsilon}|^{2\gamma}+\\\
\qquad\qquad\theta C(1+|x_{\epsilon}|)|x_{\epsilon}-x_{0}|^{2\gamma+1}+\theta
C(1+|y_{\epsilon}|)|y_{\epsilon}-x_{0}|^{2\gamma+1}+\psi_{i_{\epsilon}}(x_{\epsilon})-(1-\lambda)\psi_{i_{\epsilon}}(y_{\epsilon}).\end{array}$
By sending $\epsilon\rightarrow 0$, $\lambda\rightarrow 0$, $\theta\rightarrow
0$ and taking into account of the continuity of $\psi_{i_{\epsilon}}$, we
obtain $u_{i_{0}}(x_{0})-w_{i_{0}}(x_{0})<0$ which is a contradiction. The
proof of Theorem 5 is now complete. $\Box$
As a by-product we have the following Corollary:
###### Corollary 1
Let $(v^{1},...,v^{m})$ be a viscosity solution of (2.4) which satisfies a
polynomial growth condition then for $i=1,...,m$ and $(t,x)\in I\\!\\!R^{k}$,
$v^{i}(x)=\sup_{(\delta,\xi)\in{\cal
D}^{i}_{0}}E[\displaystyle\int_{0}^{+\infty}e^{-rs}\psi_{u_{s}}(X^{x}_{s})ds-\sum_{n\geq
1}e^{-r\tau_{n}}g_{u_{\tau_{n-1}}u_{\tau_{n}}}(X^{x}_{\tau_{n}})].$
## 6 Numerical results
We consider now some numerical examples of the optimal switching problem
(2.4).
Example1: In this example we consider an optimal switching problem with two
modes, where
$r=100$, $b=x$, $\sigma=\sqrt{2}x$, $g_{12}(x)={\frac{1}{2}}|x|+0.1$,
$g_{21}(t,x)=|x|+0.48$, $\psi_{1}(x)={\frac{1}{2}}x^{2}-0.3x+1$,
$\psi_{2}(t,x)=x^{2}+1$.
Figure 1: Curves of $v^{2}$ and $v^{1}$.
Example2: We now consider the case of 3 modes where $r=100$, $b=x$,
$\sigma=\sqrt{2}x$, $g_{12}(t,x)=0.5|x|+1$, $g_{13}(t,x)=x^{2}+0.5$,
$g_{21}(t,x)=|x|+4$, $g_{23}(t,x)=|x|+5$, $g_{31}(t,x)=0.001|x|+0.1$,
$g_{32}(t,x)=x^{2}+|x|+0.5$, $\psi_{1}(t,x)=x+1$, $\psi_{2}(t,x)=-x-2$ and
finally $\psi_{3}(t,x)=-x-2$.
Figure 2: Curves of $v^{1}$, $v^{3}$ and $v^{2}$.
Acknowledgement: The author thanks gratefully Prof. S. Hamadène for the
fructuous discussions during the preparation of this paper.$\Box$
## Appendix: proof of Theorem 1
The proof consists in showing that for any $t\leq T,$ $Y^{i}_{t}$, as defined
by (3.3), is nothing but the expected total profit or the value function of
the optimal problem, given that the system is in mode $i$ at time $t$. More
precisely,
$e^{-rt}Y^{i}_{t}=\mbox{ess sup}_{(\delta,u)\in{\cal
D}_{t}}E[\int_{t}^{+\infty}e^{-rs}\psi_{i}(X_{s})ds-\sum_{k\geq
1}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X_{{\tau_{k}}})|{\cal F}_{t}],$
where ${\cal D}_{t}$ is the set of strategies such that $\tau_{1}\geq t$,
P-a.s. if at time $t$ the system is in the mode i.
Let us admit for a moment the following Lemma.
###### Lemma 2
For every $t\geq\tau^{*}_{1}$.
$e^{-rt}Y^{u_{\tau^{*}_{1}}}_{t}=\mbox{ess sup}_{\tau\geq
t}E[\int_{t}^{\tau}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds+e^{-r\tau}\max\limits_{j\in{\cal
I}^{-u_{\tau^{*}_{1}}}}(-g_{ij}(X_{\tau})+Y^{j}_{\tau})|{\cal F}_{t}].\Box$
(6.1)
From properties of the Snell envelope and at time $t=0$ the system is in mode
$1$, we have:
$\begin{array}[]{ll}Y^{1}_{0}&=E[\int_{0}^{\tau^{*}_{1}}e^{-rs}\psi_{1}(X_{s})ds+e^{-r\tau^{*}_{1}}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{\tau^{*}_{1}})+Y^{j}_{\tau^{*}_{1}})]\\\
&=E[\int_{0}^{\tau^{*}_{1}}e^{-rs}\psi_{1}(X_{s})ds+e^{-r\tau^{*}_{1}}(-g_{iu_{\tau^{*}_{1}}}(X_{\tau^{*}_{1}})+Y^{u_{\tau^{*}_{1}}}_{\tau^{*}_{1}})].\end{array}$
Now from Lemma 2 and the definition of $\tau^{*}_{2}$ we have:
$\begin{array}[]{ll}e^{-r\tau^{*}_{1}}Y^{u_{\tau^{*}_{1}}}_{\tau^{*}_{1}}&=E[\int_{\tau^{*}_{1}}^{\tau^{*}_{2}}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds+e^{-r\tau^{*}_{2}}\max\limits_{j\in{\cal
I}^{-u_{\tau^{*}_{1}}}}(-g_{u_{\tau^{*}_{1}}j}(X_{\tau^{*}_{2}})+Y^{j}_{\tau^{*}_{2}})|{\cal
F}_{\tau^{*}_{1}}]\\\
&=E[\int_{\tau^{*}_{1}}^{\tau^{*}_{2}}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds+e^{-r\tau^{*}_{2}}(-g_{u_{\tau^{*}_{1}}u_{\tau^{*}_{2}}}(X_{\tau^{*}_{2}})+Y^{u_{\tau^{*}_{2}}}_{\tau^{*}_{2}})|{\cal
F}_{\tau^{*}_{1}}].\end{array}$
It implies that
$\begin{array}[]{lll}Y^{1}_{0}&=E[\int_{0}^{\tau^{*}_{1}}e^{-rs}\psi_{1}(X_{s})ds-e^{-r\tau^{*}_{1}}g_{iu_{\tau^{*}_{1}}}(X_{\tau^{*}_{1}})\\\
&+E[\int_{\tau^{*}_{1}}^{\tau^{*}_{2}}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds+e^{-r\tau^{*}_{2}}(-g_{u_{\tau^{*}_{1}}u_{\tau^{*}_{2}}}(X_{\tau^{*}_{2}})+Y^{u_{\tau^{*}_{2}}}_{\tau^{*}_{2}})|{\cal
F}_{\tau^{*}_{1}}]]\\\
&=E[\int_{0}^{\tau^{*}_{1}}e^{-rs}\psi_{1}(X_{s})ds+\int_{\tau^{*}_{1}}^{\tau^{*}_{2}}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds-e^{-r\tau^{*}_{1}}g_{iu_{\tau^{*}_{1}}}(X_{\tau^{*}_{1}})-e^{-r\tau^{*}_{2}}g_{u_{\tau^{*}_{1}}u_{\tau^{*}_{2}}}(X_{\tau^{*}_{2}})\\\
&+e^{-r\tau^{*}_{2}}Y^{u_{\tau^{*}_{2}}}_{\tau^{*}_{2}}].\end{array}$
Therefore
$Y^{1}_{0}=E[\int_{0}^{\tau^{*}_{2}}e^{-rs}\psi(X_{s},u_{s})ds-e^{-r\tau^{*}_{1}}g_{iu_{\tau^{*}_{1}}}(X_{\tau^{*}_{1}})-e^{-r\tau^{*}_{2}}g_{u_{\tau^{*}_{1}}u_{\tau^{*}_{2}}}(X_{\tau^{*}_{2}})+e^{-r\tau^{*}_{2}}Y^{u_{\tau^{*}_{2}}}_{\tau^{*}_{2}}],$
since between 0 and $\tau^{*}_{1}$ (resp. $\tau^{*}_{1}$ and $\tau^{*}_{2}$)
the production is in regime $1$ (resp. regime $u_{\tau^{*}_{1}}$) and then
$u_{t}=1$ (resp. $u_{t}=u_{\tau^{*}_{1}}$) which implies that
$\displaystyle\int_{0}^{\tau^{*}_{2}}e^{-rs}\psi(X_{s},u_{s})ds=\displaystyle\int_{0}^{\tau^{*}_{1}}e^{-rs}\psi_{1}(X_{s})ds+\displaystyle\int_{\tau^{*}_{1}}^{\tau^{*}_{2}}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds.$
Now repeating this reasoning as many times as necessary we obtain that for any
$n\geq 0,$
$\begin{array}[]{l}Y^{1}_{0}=E[\displaystyle\int_{0}^{\tau^{*}_{n}}e^{-rs}\psi(X_{s},u_{s})ds-\sum_{1\leq
k\leq
n}e^{-r\tau^{*}_{k}}g_{u_{\tau^{*}_{k-1}}u_{\tau^{*}_{k}}}(X_{{\tau^{*}_{k}}})+e^{-r\tau^{*}_{n}}Y^{u_{\tau^{*}_{n}}}_{\tau^{*}_{n}}].\end{array}$
Then, the strategy $(\delta^{*},u^{*})$ verify $E[\sum_{n\geq
0}e^{-r\tau^{*}_{n}}]<+\infty$, otherwise $Y^{1}_{0}$ would be equal to
$-\infty$ contradicting the assumption that the processes $Y^{i}$ belong to
${\cal S}^{2}$. Therefore, taking the limit as $n\rightarrow+\infty$ we obtain
$Y^{1}_{0}=J(\delta^{*},u^{*})$.
To complete the proof it remains to show that the strategy
$(\delta^{*},u^{*})$ it is optimal i.e. $J(\delta^{*},u^{*})\geq J(\delta,u)$
for any $(\delta,u)\in\cal D$.
The definition of the Snell envelope yields
$\begin{array}[]{ll}Y^{1}_{0}&\geq
E[\int_{0}^{\tau_{1}}e^{-rs}\psi_{1}(X_{s})ds+e^{-r\tau_{1}}\max\limits_{j\in{\cal
I}^{-1}}(-g_{1j}(X_{\tau_{1}})+Y^{j}_{\tau_{1}})]\\\ &\geq
E[\int_{0}^{\tau_{1}}e^{-rs}\psi_{1}(X_{s})ds+e^{-r\tau_{1}}(-g_{1u_{\tau^{*}_{1}}}(X_{\tau_{1}})+Y^{u_{\tau_{1}}}_{\tau_{1}})].\end{array}$
But, once more using a similar characterization as (6.1), we get
$\begin{array}[]{ll}e^{-r\tau_{1}}Y^{u_{\tau_{1}}}_{\tau_{1}}&\geq
E[\int_{\tau_{1}}^{\tau_{2}}e^{-rs}\psi_{u_{\tau_{1}}}(X_{s})ds+e^{-r\tau_{2}}\max\limits_{j\in{\cal
I}^{-u_{\tau_{1}}}}(-g_{u_{\tau_{1}}j}(X_{\tau_{2}})+Y^{j}_{\tau_{2}})|{\cal
F}_{\tau_{1}}]\\\ &\geq
E[\int_{\tau_{1}}^{\tau_{2}}e^{-rs}\psi_{u_{\tau_{1}}}(X_{s})ds+e^{-r\tau_{2}}(-g_{u_{\tau_{1}}u_{\tau_{2}}}(X_{\tau_{2}})+Y^{u_{\tau_{2}}}_{\tau_{2}})|{\cal
F}_{\tau_{1}}].\end{array}$
Therefore,
$\begin{array}[]{lll}Y^{1}_{0}&\geq
E[\int_{0}^{\tau_{1}}e^{-rs}\psi_{1}(X_{s})ds-e^{-r\tau_{1}}g_{1u_{\tau_{1}}}(X_{\tau_{1}})]\\\
&+E[\int_{\tau_{1}}^{\tau_{2}}e^{-rs}\psi_{u_{\tau_{1}}}(X_{s})ds+e^{-r\tau_{2}}(-g_{u_{\tau_{1}}u_{\tau_{2}}}(X_{\tau_{2}})+Y^{u_{\tau_{2}}}_{\tau_{2}})]\\\
&=E[\int_{0}^{\tau_{2}}e^{-rs}\psi(X_{s},u_{s})ds-e^{-r\tau_{1}}g_{1u_{\tau_{1}}}(X_{\tau_{1}})-e^{-r\tau_{2}}g_{u_{\tau_{1}}u_{\tau_{2}}}(X_{\tau_{2}})+e^{-r\tau_{2}}Y^{u_{\tau_{2}}}_{\tau_{2}}].\end{array}$
Repeat this argument $n$ times to obtain
$\begin{array}[]{l}Y^{1}_{0}\geq
E[\displaystyle\int_{0}^{\tau_{n}}e^{-rs}\psi(X_{s},u_{s})ds-\sum_{1\leq k\leq
n}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X_{{\tau_{k}}})+e^{-r\tau_{n}}Y^{u_{\tau_{n}}}_{\tau_{n}}].\end{array}$
Finally, taking the limit as $n\rightarrow+\infty$ yields
$\begin{array}[]{l}Y1_{0}\geq
E[\displaystyle\int_{0}^{+\infty}e^{-rs}\psi(X_{s},u_{s})ds-\sum_{k\geq
1}e^{-r\tau_{k}}g_{u_{\tau_{k-1}}u_{\tau_{k}}}(X_{{\tau_{k}}})].\end{array}$
Hence, the strategy $(\delta^{*},u^{*})$ is optimal. We proceed to the proof
of Lemma 2.
$Proof$ of Lemma 2. From (3.3) we have for any $i\in{\cal I}$ and $t\geq 0$
$\begin{array}[]{l}e^{-rt}Y^{i}_{t}=\mbox{ess sup}_{\tau\geq
t}E[\int_{t}^{\tau}e^{-rs}\psi_{i}(X_{s})ds+e^{-r\tau}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{\tau})+Y^{j}_{\tau})|{\cal F}_{t}].\end{array}$ (6.2)
This also means that the process
$(e^{-rt}Y^{i}_{t}+\int_{0}^{t}e^{-rs}\psi_{i}(X_{s})ds)_{t\geq 0}$ is a
supermartingale which dominates
$(\int_{0}^{t}e^{-rs}\psi_{i}(X_{s})ds+e^{-rt}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{t})+Y^{j}_{t}))_{t\geq 0}.$
This implies that the process
$(1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}(e^{-rt}Y^{i}_{t}+\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{i}(X_{s})ds))_{t\geq\tau^{*}_{1}}$
is a supermartingale which dominates
$(1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}(\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{i}(X_{s})ds+e^{-rt}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{t})+Y^{j}_{t}))_{t\geq\tau^{*}_{1}}.$
Since ${\cal I}$ is finite, the process $(\sum_{i\in{\cal
I}}1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}(e^{-rt}Y^{i}_{t}+\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{i}(X_{s})ds))_{t\geq\tau^{*}_{1}}$
is also a supermartingale which dominates $(\sum_{i\in{\cal
I}}1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}(\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{i}(X_{s})ds+e^{-rt}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{t})+Y^{j}_{t}))_{t\geq\tau^{*}_{1}}.$
Thus, the process
$(e^{-rt}Y^{u_{\tau^{*}_{1}}}_{t}+\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds)_{t\geq\tau^{*}_{1}}$
is a supermartingale which is greater than
$(\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds+e^{-rt}\max\limits_{j\in{\cal
I}^{-u_{\tau^{*}_{1}}}}(-g_{u_{\tau^{*}_{1}}j}(X_{t})+Y^{j}_{t}))_{t\geq\tau^{*}_{1}}.$
To complete the proof it remains to show that it is the smallest one which has
this property and use the characterization of the Snell envelope see e.g. [7,
14, 16].
Indeed, let $(Z_{t})_{t\geq\tau^{*}_{1}}$ be a supermartingale of class $[D]$
such that, for any $t\geq\tau^{*}_{1}$,
$Z_{t}\geq\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds+e^{-rt}\max\limits_{j\in{\cal
I}^{-u_{\tau^{*}_{1}}}}(-g_{u_{\tau^{*}_{1}}j}(X_{t})+Y^{j}_{t}).$
It follows that for every $t\geq\tau^{*}_{1}$,
$1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}Z_{t}\geq
1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}(\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{i}(X_{s})ds+e^{-rt}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{t})+Y^{j}_{t})).$
But, the process $(1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}Z_{t})_{t\geq\tau^{*}_{1}}$
is a supermartingale and for every $t\geq\tau^{*}_{1}$,
$1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}e^{-rt}Y^{i}_{t}=\mbox{ess sup}_{\tau\geq
t}E[1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}(\int_{t}^{\tau}e^{-rs}\psi_{i}(X_{s})ds+e^{-r\tau}\max\limits_{j\in{\cal
I}^{-i}}(-g_{ij}(X_{\tau})+Y^{j}_{\tau}))|{\cal F}_{t}].$
It follows that, for every $t\geq\tau^{*}_{1}$,
$1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}Z_{t}\geq
1\\!\\!1_{[u_{\tau^{*}_{1}}=i]}(e^{-rt}Y^{i}_{t}+\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{i}(X_{s})ds).$
Summing over $i$, we get, for every $t\geq\tau^{*}_{1}$,
$Z_{t}\geq
e^{-rt}Y^{u_{\tau^{*}_{1}}}_{t}+\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds.$
Hence, the process
$(e^{-rt}Y^{u_{\tau^{*}_{1}}}_{t}+\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds)_{t\geq\tau^{*}_{1}}$
is the Snell envelope of
$(\int_{\tau^{*}_{1}}^{t}e^{-rs}\psi_{u_{\tau^{*}_{1}}}(X_{s})ds+e^{-rt}\max\limits_{j\in{\cal
I}^{-u_{\tau^{*}_{1}}}}(-g_{u_{\tau^{*}_{1}}j}(X_{t})+Y^{j}_{t}))_{t\geq\tau^{*}_{1}},$
whence Lemma 2.$\Box$
## References
* [1] Bayraktar, E. and Egami, M. (2007): On the One-Dimensional Optimal Switching Problem. Preprint.
* [2] Brekke, K. A. and Øksendal, B. (1994): Optimal switching in an economic activity under uncertainty. SIAM J. Control Optim. (32), pp. 1021-1036.
* [3] Brennan, M. J. and Schwartz, E. S. (1985): Evaluating natural resource investments. J.Business 58, pp. 135-137.
* [4] Carmona, R. and Ludkovski, M. (2005): Optimal Switching with Applications to Energy Tolling Agreements. Preprint.
* [5] Chen, Z. (1998): Existence and uniqueness for BSDE’s with stopping time, Chinese Science Bulletin, 43, p.96-99.
* [6] Crandall, M., Ishii, H. and P.L. Lions (1992) : User s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27, 1-67.
* [7] Cvitanic, J. and Karatzas, I (1996): Backward SDEs with reflection and Dynkin games. Annals of Probability 24 (4), pp. 2024-2056.
* [8] Dellacherie, C. and Meyer, P.A. (1980). Probabilités et Potentiel V-VIII, Hermann, Paris.
* [9] Dixit, A. and Pindyck, R. S. (1994): Investment under uncertainty. Princeton Univ. Press.
* [10] Djehiche, B. and Hamadène, S (2009): On a finite horizon Starting and Stopping Problem with Default risk. to appear in the International J. of Theoretical and Applied Finance (IJTAF).
* [11] Djehiche, B. Hamadène, S. and Popier, A. (2007): A finite horizon optimal multiple switching problem. Preprint, Université du Maine, F.
* [12] Duckworth, K. and Zervos, M. (2001): A model for investment decisions with switching costs. Annals of Applied probability 11 (1), pp. 239-260.
* [13] El Asri, B. and Hamadène, S. (2009): The Finite Horizon Optimal Multi-Modes Switching Problem: the Viscosity Solution Approach, Applied Mathematics and Optimization, DOI 10.1007/s00245-009-9071-3.
* [14] El Karoui, N. (1980): Les aspects probabilistes du contrôle stochastique. Ecole d’été de probabilités de Saint-Flour, Lect. Notes in Math. No 876, Springer Verlag.
* [15] El-Karoui, N. Kapoudjian, C. Pardoux, E. Peng, S. and Quenez, M. C. (1997): Reflected solutions of backward SDEs and related obstacle problems for PDEs. Annals of Probability 25 (2), pp. 702-737.
* [16] Hamadène, S. (2002): Reflected BSDEs with discontinuous barriers. Stochastics and Stochastic Reports 74 (3-4), pp. 571-596.
* [17] Hamadène, S. and Jeanblanc, M (2007): On the Starting and Stopping Problem: Application in reversible investments, Math. of Operation Research, vol.32, No.1, pp.182-192.
* [18] Hamadène, S. and Hdhiri, I. (2006): On the starting and stopping problem in the model with jumps. Preprint , Université du Maine, Le Mans, F.
* [19] Hamadène, S. Lepeltier, J-P and Wu, Z. (1999): nfinite Horizon Reflected BSDE’s and Applications in Mixed Control and Game Problems. Probability and Mathematical Statistics International Journal vol.19, pp.211-234
* [20] Ly Vath, V. Pham, H and Zhou, X. (2007): Optimal switching over multiple regimes. Preprint..
* [21] Pardoux, E. (1999): Weak convergence and homogenization of semilinear PDEs. Nonlin. Anal, Dif. Equa. and Control, pp. 503-549.
* [22] Revuz, D and Yor, M. (1991): Continuous Martingales and Brownian Motion. Springer Verlag, Berlin.
* [23] Tang, S. and Yong, J. (1993): Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach. Stoch. and Stoch. Reports, 45, 145-176.
* [24] Zervos, M. (2003): A Problem of Sequential Enty and Exit Decisions Combined with Discretionary Stopping. SIAM J. Control Optim. 42 (2), pp. 397-421.
|
arxiv-papers
| 2009-04-04T12:53:19 |
2024-09-04T02:49:01.697365
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Brahim El Asri",
"submitter": "Brahim El Asri",
"url": "https://arxiv.org/abs/0904.0707"
}
|
0904.0817
|
# On CON(${\mathfrak{d}}_{\lambda}>$ covλ(meagre))
Saharon Shelah Einstein Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 91904, Israel
and
Department of Mathematics
Hill Center - Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Road
Piscataway, NJ 08854-8019 USA shelah@math.huji.ac.il http://shelah.logic.at
###### Abstract.
We prove the consistency of: for suitable strongly inaccessible cardinal
$\lambda$ the dominating number, i.e., the cofinality of
${}^{\lambda}\lambda$, is strictly bigger than covλ(meagre), i.e. the minimal
number of nowhere dense subsets of ${}^{\lambda}2$ needed to cover it. This
answers a question of Matet.
The author thanks Alice Leonhardt for the beautiful typing. I thank Shimoni
Garti for some corrections. This research was supported by the United States-
Israel Binational Science Foundation. Publication 945.
## 0\. Introduction
Cardinal invariants on the continuum have a long tradition of research. For a
topologist, it can be viewed as investigating the space $\beta(\omega)$, the
Stone Čzech compactification of $\omega$. This point of view is taken, for
example, in the celebrated paper of Van Douwen [vD84].
For set theorists, it is interesting to check the relationship between the
relevant cardinal invariants. In this context, it is natural to generalize the
problems to higher cardinals, above $\aleph_{0}$. One finds out, very soon,
that for the class of (strongly) inaccessible cardinals, the generalizations
are more reasonable and have more affinity to the $\aleph_{0}$ case.
We shall define three cardinal invariants (but the paper deals, actually, just
with two of them):
###### Definition 0.1.
The bounding and dominating numbers.
Let $\lambda$ be an inaccessible cardinal.
Let $f,g\in{}^{\lambda}\lambda$
1. $(a)$
$f\leq^{*}g$ if $|\\{\alpha<\lambda:f(\alpha)>g(\alpha)\\}|<\lambda$
2. $(b)$
$A\subseteq{}^{\lambda}\lambda$ is unbounded if there is no
$h\in{}^{\lambda}\lambda$ so that $f\in A\Rightarrow f\leq^{*}h$
3. $(c)$
$A\subseteq{}^{\lambda}\lambda$ is dominating when for every
$f\in{}^{\lambda}\lambda$ there exists $g\in A$ so that $f\leq^{*}g$
4. $(d)$
the bounding number for $\lambda$, denoted by ${\mathfrak{b}}_{\lambda}$, is
min$\\{|A|:A$ is unbounded in ${}^{\lambda}\lambda\\}$
5. $(e)$
the dominating number for $\lambda$, denoted by ${\mathfrak{d}}_{\lambda}$, is
min$\\{|A|:A$ is dominating in ${}^{\lambda}\lambda\\}$.
Notice that the usual definitions of ${\mathfrak{b}}$ and ${\mathfrak{d}}$ are
${\mathfrak{b}}_{\aleph_{0}}$ and ${\mathfrak{d}}_{\aleph_{0}}$ according to
Definition 0.1. The definition of covλ(meagre) involves some topology.
###### Definition 0.2.
The meagre covering number.
Let $\lambda$ be a regular cardinal
1. $(a)$
${}^{\lambda}2$ is the space of functions from $\lambda$ into 2
2. $(b)$
$({}^{\lambda}2)^{[\nu]}=\\{\eta\in{}^{\lambda}2:\nu\triangleleft\eta\\}$, for
$\nu\in\bigcup\limits_{\alpha<\lambda}{}^{\alpha}2$
3. $(c)$
${\mathcal{U}}\subseteq{}^{\lambda}2$ is open in the topology
$({}^{\lambda}2)_{<\lambda}$, iff for every $\eta\in{\mathcal{U}}$ there
exists $i<\lambda$ so that
$({}^{\lambda}2)^{[\eta{\restriction}i]}\subseteq{\mathcal{U}}$
4. $(d)$
covλ(meagre) is the minimal cardinality of a family of meagre subsets of
$({}^{\lambda}2)_{<\lambda}$, which covers this space.
The paper deals with the relationship between ${\mathfrak{d}}_{\lambda}$ and
covλ(meagre). Matet asked (a personal communication) whether
${\mathfrak{d}}_{\lambda}\leq\text{ cov}_{\lambda}$(meagre) is provable in
ZFC. We give here a negative answer.
For $\lambda$ a supercompact cardinal and $\lambda<\kappa=\text{
cf}(\kappa)<\mu=\mu^{\lambda}$, we force large ${\mathfrak{d}}_{\lambda}$
i.e., ${\mathfrak{d}}_{\lambda}=\mu$ and small covering number (i.e.,
covλ(meagre) $=\kappa$). A similar result should hold also for a wider class
of cardinals and we intend to return to this subject.
We try to use standard notation. We use $\theta,\kappa,\lambda,\mu,\chi$ for
cardinals $\alpha,\beta,\gamma,\delta,\varepsilon,\zeta$ for ordinals. We use
also $i$ and $j$ as ordinals. We adopt the Cohen convention that $p\leq q$
means that $q$ gives more information, in forcing notions. The symbol
$\triangleleft$ is preserved for “being an initial segment”. Also recall
${}^{B}A=\\{f:f$ a function from $B$ to $A\\}$ and let
${}^{\alpha>}A=\cup\\{{}^{\beta}A:\beta<\alpha\\}$, some prefer
${}^{<\alpha}A$, but ${}^{\alpha>}A$ is used systematically in the author’s
papers. At last, $J^{\text{\rm bd}}_{\lambda}$ denotes the ideal of the
bounded subsets of $\lambda$.
The picture of cardinal invariants related to uncountable $\lambda$ is related
but usually quite different than the one for $\aleph_{0}$, they are more
similar if $\kappa$ is “large” enough, mainly strongly inaccessible.
Let us sketch some known results. These results are related to the unequality
number and the covering number for category. Recall:
###### Definition 0.3.
The unequality number.
Let $\kappa$ be an infinite cardinal. The unequality number of
$\kappa,{\mathfrak{e}}_{\kappa}$, is the minimal cardinal $\lambda$ such that
there is a set ${\mathcal{F}}\subseteq{}^{\lambda}\lambda$ of cardinality
$\lambda$ such that there is no $g\in{}^{\lambda}\lambda$ satisfying $(\forall
f\in{\mathcal{F}})(\exists^{\kappa}\alpha<\lambda)(f(\alpha)=g(\alpha))$.
For $\kappa=\aleph_{0},{\mathfrak{e}}_{\kappa}=\text{
cov}_{\aleph_{0}}(\text{meagre})$; see Bartosynski (in [Bar87]) and Miller (in
[Mil82]).
Now
1. $(a)$
the statement ${\mathfrak{e}}_{\kappa}=\text{ cov}_{\kappa}(\text{meagre})$ is
valid for $\kappa>\aleph_{0}$, in the case that $\kappa$ is strongly
inaccessible, by [Lan92]. But if $\kappa$ is a successor cardinal, it may fail
2. $(b)$
if ${\mathfrak{d}}_{\kappa}$ is only finitely many cardinals away from
$\kappa$, then ${\mathfrak{e}}_{\kappa}={\mathfrak{d}}_{\kappa}$. This can be
found in Matet-Shelah [MtSh:804]
3. $(c)$
if $\kappa<\kappa^{<\kappa}$, then cov${}_{\kappa}({\mathcal{M}})=\kappa^{+}$.
This is due to Landver (in [Lan92])
4. $(d)$
it is consistent to get (a) and (b) together, so that
cov${}_{\kappa}(\text{meagre})<{\mathfrak{e}}_{\kappa}$. This follows from
Cummings-Shelah (in [CuSh:541]).
## 1\. The forcing
###### Theorem 1.1.
Assume
1. $(a)$
$\lambda$ is supercompact
2. $(b)$
$\lambda<\kappa=\text{\rm cf}(\kappa)=\kappa^{<\kappa}<\mu=\text{\rm
cf}(\mu)=\mu^{\lambda}$
3. $(c)$
$\kappa>\lambda^{+}$ and111The assumption $\kappa>\lambda^{+}$ is technical,
to allow $\kappa=\lambda^{+}$ we should just use $\delta(*)\kappa$ instead of
$\kappa$. $\delta(*)=(\lambda^{+})^{\lambda^{+}}$ ordinal exponentiation and
${\mathcal{U}}_{*}=\\{\delta(*)(\alpha+1):\alpha<\kappa\\}$ not used till
$(*)_{8}$ in the proof of 1.3.
Then for some forcing notion ${\mathbb{P}}$ not collapsing cardinals
$\geq\lambda,\lambda$ is still supercompact in $\mathbb{V}^{{\mathbb{P}}}$ and
covλ(meagre) $=\kappa,{\mathfrak{d}}_{\lambda}=\mu$.
###### Proof..
By 1.3 below. ∎
Recall
###### Definition 1.2.
1) We say that a forcing notion $\mathbb{P}$ is $\alpha$-strategically
complete when for each $p\in\mathbb{P}$ in the following game
$\Game_{\alpha}(p,\mathbb{P})$ between the players COM and INC, the player COM
has a winning strategy.
A play lasts $\alpha$ moves; in the $\beta$-th move, first the player COM
chooses $p_{\beta}\in\mathbb{P}$ such that $p\leq_{\mathbb{P}}p_{\beta}$ and
$\gamma<\beta\Rightarrow q_{\gamma}\leq_{\mathbb{P}}p_{\beta}$ and second the
player INC chooses $q_{\beta}\in\mathbb{P}$ such that
$p_{\beta}\leq_{\mathbb{P}}q_{\beta}$.
The player COM wins a play if he has a legal move for every $\beta<\alpha$.
2) We say that a forcing notion $\mathbb{P}$ is $(<\lambda)$-strategically
complete when it is $\alpha$-strategically complete for every
$\alpha<\lambda$.
###### Lemma 1.3.
1) If $\lambda$ is supercompact then after some preliminary forcing of
cardinality $\lambda,\lambda$ is still supercompact and $\boxdot_{\lambda}$
below holds.
2) If $\lambda$ is strongly inaccessible and $\boxdot_{\lambda}$ below holds
and $\lambda^{+}<\kappa=\,\text{\rm cf}(\kappa)<\mu=\mu^{\lambda}$, then for
some $\lambda^{+}$-c.c., $(<\lambda)$-strategically complete forcing notion
$\mathbb{P}$ we have $\Vdash_{\mathbb{P}}``{\mathfrak{d}}_{\lambda}=\mu$,
cov${}_{\lambda}(\text{\rm meagre})=\kappa"$
where
1. $\boxdot_{\lambda}$
for any regular cardinal $\chi>\lambda$ and forcing notion
${\mathbb{P}}\in{\mathcal{H}}(\chi)$ which is $(<\lambda)$-strategically
complete (see Definition 1.2(2)) the following set
${{\mathcal{S}}}={{\mathcal{S}}}_{{\mathbb{P}}}$ is a stationary subset of
$[{{\mathcal{H}}}(\chi)]^{<\lambda}$:
${{\mathcal{S}}}$ is the set of $N$’s such that for some
$\lambda_{N},\chi_{N},\mathbb{j}=\mathbb{j}_{N},N^{\prime}=N^{\prime}_{N},M=M_{N},\mathbb{G}=\mathbb{G}_{N}$
we have:
1. $(a)$
$N\prec({{\mathcal{H}}}(\chi)^{\mathbb{V}},\in)$ and ${\mathbb{P}}\in N$
2. $(b)$
the Mostowski collapse $N^{\prime}$ of $N$ is
$\subseteq{{\mathcal{H}}}(\chi_{N})$, and let $\mathbb{j}_{N}:N\rightarrow
N^{\prime}$ be the unique isomorphism
3. $(c)$
$N\cap\lambda=\lambda_{N}$ and ${}^{(\lambda_{N})>}N\subseteq N$ and
$\lambda_{N}$ is strongly inaccessible
4. $(d)$
$N^{\prime}\subseteq M:=({{\mathcal{H}}}(\chi_{N}),\in)$ so both $N^{\prime}$
and $M$ are transitive
5. $(e)$
$\mathbb{G}\subseteq\mathbb{j}_{N}({\mathbb{P}})$ is generic over $N^{\prime}$
for the forcing notion $\mathbb{j}({\mathbb{P}})$
6. $(f)$
$M=N^{\prime}[\mathbb{G}]$.
###### Remark 1.4.
1) Recall that:
1. $(a)$
$\bar{{\mathbb{Q}}}=\langle{\mathbb{P}}_{\alpha},\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\beta}:\alpha\leq\delta,\beta<\delta\rangle$ be a
$(<\lambda)$-support iteration of $(<\lambda)$-strategically complete forcing
notions, then ${\mathbb{P}}_{\delta}$ is also $\lambda$-strategically
complete.
2. $(b)$
If ${\mathbb{P}}$ is $(<\lambda)$-strategically complete forcing notion then
$({}^{\lambda>}\text{Ord})^{\mathbb{V}}=({}^{\lambda>}\text{Ord})^{\mathbb{V}^{{\mathbb{P}}}}$,
and consequently $\lambda$ is strongly inaccessible in
$\mathbb{V}^{{\mathbb{P}}}$.
2) In part (1) the “$\lambda^{+}<\kappa$” rather than “$\lambda<\kappa$” is
not essential, see in the proof.
3) Is the use of $\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\restriction}{\mathcal{U}}_{*}$ rather than
$\textstyle\bar{g}$ $\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$ in the
proof necessary? See on this [Sh:F979].
###### Definition 1.5.
We may say
$(N,\lambda_{N},\chi_{N},\mathbb{j}_{N},N^{\prime}_{N},M_{N},\mathbb{G}_{N})$
is a witness for $(N,{\mathbb{P}})$ when clauses (a)-(f) from 1.3 hold.
###### Proof..
Proof of Claim 1.3 1) This is essentially by Laver [Lav78] using Laver’s
diamond.
2) We use a $(<\lambda)$-support iteration
$\bar{{\mathbb{Q}}}=\langle\mathbb{P}_{\alpha},\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\beta}:\alpha\leq\mu+\kappa,\beta<\mu+\kappa\rangle$ such
that
1. $(A)$
if $\alpha<\mu$ then
$\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}$ is the $({\mathbb{P}}_{\alpha}$-name of the)
dominating forcing, ${\mathbb{Q}}^{\text{dom}}_{\lambda}$, i.e.
$({\mathbb{Q}}^{\text{dom}}_{\lambda})^{\mathbb{V}[{\mathbb{P}}_{\alpha}]}$
where in the universe $\mathbb{V}^{{\mathbb{P}}_{\alpha}}$ the forcing
${\mathbb{Q}}={\mathbb{Q}}^{\text{dom}}_{\lambda}$ is
1. $(\alpha)$
$p\in{\mathbb{Q}}$ iff
1. $(a)$
$p=(\eta,f)=(\eta^{p},f^{p})$
2. $(b)$
$\eta\in{}^{\varepsilon}\lambda$ for some $\varepsilon<\lambda$, ($\eta$ is
called the trunk of $p$)
3. $(c)$
$f\in{}^{\lambda}\lambda$
4. $(d)$
$\eta\triangleleft f$
2. $(\beta)$
$p\leq_{\mathbb{Q}}q$ iff
1. $(a)$
$\eta^{p}\trianglelefteq\eta^{q}$
2. $(b)$
$f^{p}\leq f^{q}$, i.e. $(\forall\varepsilon<\lambda)f^{p}(\varepsilon)\leq
f^{q}(\varepsilon)$
3. $(c)$
if $\ell g(\eta^{p})\leq\varepsilon<\ell g(\eta^{q})$ then
$\eta^{q}(\varepsilon)\in[f^{p}(\varepsilon),\lambda)$; this follows
2. $(B)$
fix $\bar{\theta}=\langle\theta_{\alpha}:\alpha<\lambda\rangle$ with
$\theta_{\alpha}=(2^{|\alpha|+\aleph_{0}})^{+}$, or any sequence of cardinals
$\in\text{ Reg }\cap\lambda$, increasing fast enough
3. $(C)$
if $\alpha\in[\mu,\mu+\kappa]$ then
$\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}$ is the $\bar{\theta}$-dominating forcing, i.e.,
$({\mathbb{Q}}_{\bar{\theta}})^{\mathbb{V}[{\mathbb{P}}_{\alpha}]}$ where in
the universe $\mathbb{V}^{{\mathbb{P}}_{\alpha}}$ the forcing notion
${\mathbb{Q}}={\mathbb{Q}}_{\bar{\theta}}$ is defined as follows:
1. $(\alpha)$
$p\in{\mathbb{Q}}$ iff
1. $(a)$
$p=(\eta,f)=(\eta^{p},f^{p})$
2. $(b)$
$\eta\in\prod_{\zeta<\ell g(\eta)}\theta_{\zeta}$ and $\ell g(\eta)$ is an
ordinal $<\lambda$
3. $(c)$
$f\in\prod_{\zeta<\lambda}\theta_{\zeta}$
4. $(d)$
$\eta\triangleleft f$
2. $(\beta)$
order: as in $(A)(\beta)$.
Let $\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}$ be the generic object for
$\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}$ for $\alpha<\mu$ and
$\mathchoice{\oalign{$\displaystyle g$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{i}$ be the generic object for
$\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\mu+i}$ for $i<\kappa$.
Now:
1. $(*)_{1}$
for $\alpha\leq\mu+\kappa$ the forcing notion ${\mathbb{P}}_{\alpha}$ is
$(<\lambda)$-strategically complete and, when
$\alpha<\mu+\kappa,\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}$ is $(<\lambda)$-strategically complete222for this,
$\theta_{\alpha}>\alpha$ is enough, in fact
1. $(\alpha)$
for $\alpha\in[\mu,\mu+\kappa)$ it is not $(<\lambda)$-complete but it is
$(<\lambda)$-strategically complete, and even $\lambda$-strategically
complete; simply, in a play, COM can keep having the trunk being of length
$\geq$ length of the play so far
2. $(\alpha)^{+}$
moreover, COM can guarantee that in limit stage $\beta$ of the game, $\langle
p_{\alpha}:\alpha<\beta\rangle$ has a lub
3. $(\beta)$
for
$\alpha\in[0,\mu),\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}$ is $(<\lambda)$-complete even for directed systems
(hence ${\mathbb{P}}_{\beta}$ for $\beta\leq\mu$ is)
4. $(\beta)^{+}$
moreover, for such systems there is a lub.
[Why? We prove this by induction on $\alpha$ for ${\mathbb{P}}_{\alpha}$,
using 1.4.]
1. $(*)_{2}$
for each $\alpha\leq\mu+\kappa,{\mathbb{P}}_{\alpha}$ and for
$\alpha<\mu+\kappa$, the forcing notions
$\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}$ satisfy a strong form of the $\lambda^{+}$-c.c.,
(see [Sh:80] for definition, preservation and history; or pedantically
[Sh:546, §1])
hence
1. $(*)_{3}$
$(a)\quad$ forcing with ${\mathbb{P}}_{\mu+\kappa}$ collapses no cardinal,
changes no cofinality, and adds no sequence to ${}^{\lambda>}\mathbb{V}$;
2. $(b)\quad({}^{\lambda}\lambda)^{\mathbb{V}[{\mathbb{P}}_{\mu+\kappa}]}=\cup\\{({}^{\lambda}\lambda)^{\mathbb{V}[{\mathbb{P}}_{\mu+i}]}:i<\kappa\\}$
3. $(c)\quad({}^{\lambda}\lambda)^{\mathbb{V}[{\mathbb{P}}_{\mu}]}=\cup\\{({}^{\lambda}\lambda)^{\mathbb{V}[{\mathbb{P}}_{\alpha}]}:\alpha<\mu\\}$.
[Why? By $(*)_{2}+(*)_{1}$ clause (a) holds, for clauses (b),(c) use also the
support in the iteration being $<\lambda$ recalling that $\mu,\kappa$ are
regular $>\lambda$.]
1. $(*)_{4}$
in
$\mathbb{V}^{{\mathbb{P}}_{\mu}},{\mathfrak{b}}_{\lambda}={\mathfrak{d}}_{\lambda}=\mu$
as witnessed by $\mathchoice{\oalign{$\displaystyle\bar{f}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{f}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{f}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{f}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}=\langle\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}:\alpha<\mu\rangle$, in fact
$\Vdash_{{\mathbb{P}}_{\alpha+1}}``\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}\in{}^{\lambda}\lambda$ dominates
$({}^{\lambda}\lambda)^{\mathbb{V}[{\mathbb{P}}_{\alpha}]}$ modulo
$J^{\text{bd}}_{\lambda}"$.
[Why? Easy using $(*)_{3}(c)$.]
1. $(*)_{5}$
$\Vdash_{{\mathbb{P}}_{\mu+i+1}}``\mathchoice{\oalign{$\displaystyle
g$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{i}\in\prod_{\varepsilon<\lambda}\theta_{\varepsilon}$
dominates
$(\prod_{\varepsilon<\lambda}\theta_{\varepsilon})^{\mathbb{V}[{\mathbb{P}}_{\mu+i}]}"$,
the order being modulo $J^{\text{bd}}_{\lambda}$.
[Why? As in $\mathbb{V}^{{\mathbb{P}}_{\mu+i}}$ for each
$g\in\prod_{\varepsilon<\lambda}\theta_{\varepsilon}$ the set
$\\{(\eta,f)\in{\mathbb{Q}}_{\bar{\theta}}$ : for every $\varepsilon\in[\ell
g(\eta),\lambda)$ we have $g(\varepsilon)\leq f(\varepsilon)\\}$ is a dense
open subset of $\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\mu+i}$.]
1. $(*)_{6}$
$\Vdash_{{\mathbb{P}}_{\mu+\kappa}}``\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}=\langle\mathchoice{\oalign{$\displaystyle g$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{i}:i<\kappa\rangle$ is
$<_{J^{\text{bd}}_{\lambda}}$-increasing and cofinal in
$(\prod_{\varepsilon<\lambda}\theta_{\varepsilon},<_{J^{\text{bd}}_{\lambda}})"$.
[Why? By $(*)_{5}$ noting that
$(\prod_{\varepsilon<\lambda}\theta_{\varepsilon})^{\mathbb{V}[{\mathbb{P}}_{\mu+\kappa}]}=\cup\\{(\prod_{\varepsilon<\lambda}\theta_{\varepsilon})^{\mathbb{V}[{\mathbb{P}}_{\mu+i}]}:i<\kappa\\}$
which holds by $(*)_{3}(b)$.]
Now
1. $(*)_{7}$
$\Vdash_{{\mathbb{P}}_{\mu+\kappa}}$ “covλ(meagre) $\leq\kappa$”.
[Why? As we can look at
$\prod\limits_{\varepsilon<\lambda}\theta_{\varepsilon}$ instead333E.g. let
$F:{}^{\lambda}2\rightarrow\prod\limits_{\varepsilon<\lambda}\theta_{\varepsilon}$
be $F(\eta)=\rho$ iff $\eta\in{}^{\lambda}2$ and for every
$\varepsilon<\lambda,\rho(\varepsilon)=0$ iff $(\forall
i<\theta_{\varepsilon})(\eta\sum\limits_{\zeta<\varepsilon}\theta_{\zeta}+i)=0)$
and $\rho(\varepsilon)=1+i$ iff
$\eta(\sum\limits_{\zeta<\varepsilon}\theta_{\zeta}+i)=1\wedge(\forall
j<i)(\eta(\sum\limits_{\zeta<\varepsilon}\theta_{\zeta}+j)=0)$. Now if
$\prod\limits_{\varepsilon}\theta_{\varepsilon}=\cup\\{{\mathcal{U}}_{i}:i<\kappa\\}$,
each ${\mathcal{U}}_{i}$ closed nowhere dense then $\langle
F^{-1}({\mathcal{U}}_{i}):i<\kappa\rangle$ witnesses covλ(meagre)
$\leq\kappa$. of ${}^{\lambda}2$ and for each $\varepsilon<\lambda,i<\kappa$
the set
$B_{\varepsilon,i}=\\{\eta\in\prod_{\varepsilon<\lambda}\theta_{\varepsilon}$:
for every $\zeta\in[\varepsilon,\lambda)$ we have
$\eta(\zeta)\leq\mathchoice{\oalign{$\displaystyle g$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{i}(\zeta)<\theta_{\zeta}\\}$ is closed nowhere dense, and
by $(*)_{6}$
$\mathbb{V}^{{\mathbb{P}}_{\mu+\kappa}}\models``\prod_{\zeta<\lambda}\theta_{\zeta}=\cup\\{B_{\varepsilon,i}:\varepsilon<\lambda,i<\kappa\\}"$.]
Now we come to the main and last point
1. $(*)_{8}$
letting ${\mathcal{U}}_{*}=\\{\lambda^{+}(\gamma+1):\gamma<\kappa\\}$, it is
forced, i.e. $\Vdash_{\mathbb{P}_{\mu+\kappa}}$, that
$\mathbb{V}^{\prime}:=\mathbb{V}[\mathchoice{\oalign{$\displaystyle\bar{f}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{f}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{f}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{f}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}},\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\restriction}{\mathcal{U}}_{*}]$ satisfies:
1. $(a)$
$\mathbb{V}^{\prime}$ has the same cardinals as $\mathbb{V}$
2. $(b)$
the cofinality of a cardinal is the same in $\mathbb{V}^{\prime}$ and
$\mathbb{V}$
3. $(c)$
$({}^{\lambda>}\text{Ord})^{\mathbb{V}^{\prime}}=({}^{\lambda>}\text{Ord})^{\mathbb{V}}$
4. $(d)$
if $\theta\geq\mu$ then
$(2^{\theta})^{\mathbb{V}^{\prime}}=(2^{\theta})^{\mathbb{V}}$
5. $(e)$
$(2^{\lambda})^{\mathbb{V}^{\prime}}=\mu$
6. $(f)$
$\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\restriction}{\mathcal{U}}_{*}$ is
$<_{J^{\text{bd}}_{\lambda}}$-increasing cofinal in
$(\prod\limits_{i<\lambda}\theta_{i})^{\mathbb{V}^{\prime}}$.
[Why? Straight forward.]
1. $(*)_{9}$
it is forced, i.e. $\Vdash_{{\mathbb{P}}_{\mu+\kappa}}$ that no
$\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\in({}^{\lambda}\lambda)^{\mathbb{V}^{\prime}}$ dominate
$\\{\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}:\alpha<\mu\\}$.
We shall note that it suffices to prove $(*)_{9}$ for proving 1.3, and that
$(*)_{9}$ holds, thus finishing.
Why it suffices? As $\langle\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}:\alpha<\mu\rangle$ is
$<_{J^{\text{bd}}_{\lambda}}$-increasing and cf$(\mu)=\mu>\lambda$, this
implies ${\mathfrak{d}}_{\lambda}\geq\mu$, and this is the last piece missing.
The rest of the proof is dedicated to proving that $(*)_{9}$ holds.
Let $\mathbb{G}_{\mu}\subseteq{\mathbb{P}}_{\mu}$ be generic over $\mathbb{V}$
and so $\langle
f_{\alpha}:\alpha<\mu\rangle=\langle\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}[\mathbb{G}_{\mu}]:\alpha<\mu\rangle$ is well
defined. Now ${\mathbb{P}}_{\mu+\kappa}/\mathbb{G}_{\mu}$ is just the limit of
the $(<\lambda)$-support iteration of
$\langle{\mathbb{P}}_{\mu+i}/\mathbb{G}_{\mu},\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\mu+j}:i\leq\kappa,j<\kappa\rangle$. Let
$p\in{\mathbb{P}}_{\mu+\kappa}/\mathbb{G}_{\mu}$. For $i\leq\kappa$ let
${\mathbb{P}}_{0,i}={\mathbb{P}}_{\mu+i}/\mathbb{G},{\mathbb{Q}}_{0,i}$ be the
${\mathbb{P}}_{0,i}$-name of
$\mathchoice{\oalign{$\displaystyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathbb{Q}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\mu+i}$, i.e., of ${\mathbb{Q}}_{\bar{\theta}}$ in the
universe $\mathbb{V}[\mathbb{G}_{\mu}]^{{\mathbb{P}}_{0,i}}$.
We shall apply $\boxdot_{\lambda}$. Let $\gamma(*)=\kappa$ (but we shall use
$\gamma(*)$ since the proof applies to any $\gamma(*)$ of cofinality
$>\lambda$). The condition $\boxdot_{\lambda}$ is preserved by forcing by
${\mathbb{P}}_{\mu}$ recalling $(*)_{1}(\beta)$ so
$\mathbb{V}[\mathbb{G}_{\mu}]=\mathbb{V}^{{\mathbb{P}}_{\mu}}$ satisfies
$\boxdot_{\lambda}$. So it suffices to prove:
1. $(*)^{\prime}_{9}$
if $\mathbb{V}$ satisfies $\boxdot_{\lambda}$ and
$\mathbb{q}=\langle{\mathbb{P}}_{0,i},\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{0,j}:i\leq\gamma(*),j<\gamma(*)\rangle$ is a
$(<\lambda)$-support iteration, such that for every $j<\gamma(*)$ the forcing
notion ${\mathbb{Q}}_{0,j}$ is
$({\mathbb{Q}}_{\bar{\theta}})^{\mathbb{V}[{\mathbb{P}}_{0,j}]}$ and
$\gamma(*)$ is a regular cardinal $>\delta(*),\lambda^{+}$ or just
$\lambda^{+}\cdot\gamma(*)=\gamma(*)\wedge\text{
cf}(\gamma(*))\geq\lambda^{+}$ then it is forced, i.e.
$\Vdash_{{\mathbb{P}}_{0,\gamma(*)}}$, that no
$f\in({}^{\lambda}\lambda)^{\mathbb{V}[\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{0}{\restriction}{\mathcal{U}}_{*}]}$ dominate
$({}^{\lambda}\lambda)^{\mathbb{V}}$ letting
$\bar{g}_{0}=\langle\mathchoice{\oalign{$\displaystyle g$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{0,i}:i<\gamma(*)\rangle$ where
$\mathchoice{\oalign{$\displaystyle g$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{0,i}\in\prod\limits_{\varepsilon<\lambda}\theta_{\varepsilon}$
is the name of the generic for ${\mathbb{Q}}_{0,i}$ so
$\mathchoice{\oalign{$\displaystyle g$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{0,i}=\mathchoice{\oalign{$\displaystyle g$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\mu+i}$.
[Why does $(*)^{\prime}_{9}$ suffice? We apply it with $\mathbb{V},\gamma(*)$
in $(*)^{\prime}_{9}$ standing for
$\mathbb{V}^{{\mathbb{P}}_{\mu}}=\mathbb{V}[\mathbb{G}_{\mu}],\kappa$ here. So
in
$\mathbb{V}[\mathbb{G}_{\mu}]^{{\mathbb{P}}_{0,\gamma(*)}}=\mathbb{V}^{{\mathbb{P}}_{\mu+\kappa}}$,
letting $f_{\alpha}=\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\alpha}[\mathbb{G}_{\mu}]$ for $\alpha<\mu$ we have:
1. $(a)$
$f_{\alpha}\in{}^{\lambda}\lambda$, for every $\alpha<\mu$
2. $(b)$
$\bar{f}=\langle f_{\alpha}:\alpha<\mu\rangle$ is
$<_{J^{\text{bd}}_{\lambda}}$-increasing cofinal in $\mathbb{V}$
3. $(c)$
$\\{f_{\alpha}:\alpha<\mu\\}$ has no common
$\leq_{J^{\text{bd}}_{\lambda}}$-upper bound in
$\mathbb{V}[\bar{f},\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{0}{\restriction}{\mathcal{U}}_{*}]$.
This implies that
$\Vdash_{{\mathbb{P}}_{\mu+\kappa}}``\mathbb{V}[\bar{f},\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\restriction}{\mathcal{U}}_{*}]$ satisfies
${{\mathfrak{d}}}_{\lambda}\geq\mu"$ as required.]
For $i\leq\gamma(*)$ let ${\mathbb{P}}_{1,i}$ be the completion of
${\mathbb{P}}_{0,i}$ and let ${\mathbb{P}}^{\prime}_{i}={\mathbb{P}}_{2,i}$ be
the complete subforcing of ${\mathbb{P}}_{1,\delta(*)(i+1)}$ generated by
$g^{\prime}_{j}=\langle
g^{\prime}_{j}:j<i\rangle=\langle\mathchoice{\oalign{$\displaystyle
g$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{0,\delta(*)(j+1)}:j<i\rangle$.
We shall use the nice properties of
${\mathbb{P}}^{\prime}_{i},\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{\prime}_{i}$.
Note that
1. $\boxplus_{1}$
$(a)\quad\langle\mathchoice{\oalign{$\displaystyle g$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{\prime}_{\gamma}:\gamma<\gamma(*)\rangle$ is generic for
${\mathbb{P}}^{\prime}_{\gamma(*)}$, i.e., if $\mathbb{G}$ is a subset of
${\mathbb{P}}^{\prime}_{\gamma(*)}$ generic
over $\mathbb{V}$ and $g^{\prime}_{\gamma}=\mathchoice{\oalign{$\displaystyle
g$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{\prime}_{\gamma}[\mathbb{G}]$ then
$\mathbb{V}[\mathbb{G}]=\mathbb{V}[\langle
g^{\prime}_{\gamma}:\gamma<\gamma(*)\rangle]$
2. $(b)\quad$ if $g^{\prime\prime}_{\gamma}\in\prod\limits_{\zeta<\lambda}\theta_{\zeta}$ for $\gamma<\gamma(*)$ and the set $\\{(\gamma,\zeta):\gamma<\gamma(*)$ and $\zeta<\lambda$
and $g^{\prime\prime}_{\gamma}(\zeta)\neq g^{\prime}_{\gamma}(\zeta)\\}$ has
cardinality $<\lambda$ then $\langle
g^{\prime\prime}_{\gamma}:\gamma<\gamma(*)\rangle$
is generic for ${\mathbb{P}}^{\prime}_{\gamma(*)}$
3. $(c)\quad{\mathbb{P}}^{\prime}_{\gamma+1}/{\mathbb{P}}^{\prime}_{\gamma}$ is equivalent to ${\mathbb{Q}}_{\bar{\theta}}^{\mathbb{V}[{\mathbb{P}}^{\prime}_{\gamma}]}$
4. $(d)\quad$ if $\langle\zeta(\gamma):\gamma<\gamma(*)\rangle$ is an increasing sequence of ordinals $<\gamma(*)$, then
$\langle g^{\prime}_{\zeta(\gamma)}:\gamma<\gamma(*)\rangle$ is generic for
${\mathbb{P}}^{\prime}_{\gamma(*)}$
5. $(e)\quad$ if $\bar{\zeta}=\langle\zeta(\gamma):\gamma<\gamma(*)\rangle$ is an increasing sequence of ordinals $<\gamma(*)$,
then the sequence $\langle
g_{\mathbb{h}(\gamma,\bar{\zeta})}:\gamma<\gamma(*)\rangle$ is generic for
${\mathbb{P}}_{0,\gamma(*)}$ where
we define $\mathbb{h}(\gamma,\bar{\zeta})<\gamma(*)$ for $\gamma<\gamma(*)$ by
induction on $\gamma$ as:
$\cup\\{\mathbb{h}(\beta,\bar{\zeta})+1:\beta<\gamma\\}$ if
$\beta\notin{\mathcal{U}}_{*}$ and $\delta(*)(\zeta(\gamma)+1)$ if
$\beta\in{\mathcal{U}}$.
[Why? The serious point is clause (d) and (e) which is done similarly. For
this it suffices to show that: if $\langle g_{\gamma}:\gamma<\gamma(*)\rangle$
is generic for ${\mathbb{P}}_{0,\gamma(*)}$ and
$\langle\zeta(\gamma):\gamma<\gamma(*)\rangle$ is as there then not only
$\langle g_{\zeta(\gamma)}:\gamma<\gamma(*)\rangle$ is generic for
${\mathbb{P}}^{\prime}_{\gamma(*)}$ but also $\langle
g_{\delta(*)(\zeta(\gamma)+1)}:\gamma<\gamma(*)\rangle$ is. This holds and it
straightforward translates to saying that the sequence
$\langle\delta(*)(\gamma+1):\gamma<\gamma(*)\rangle$ and
$\langle\delta(*)(\zeta(\gamma)+1):\gamma<\gamma(*)\rangle$ realizes the same
${\mathbb{L}}_{\lambda^{+},\lambda}$-type in the structure $(\gamma(*),<)$,
which holds by Kino [Kin66]. See more in [Sh:F976].
We shall use $\boxplus_{1}$ freely.]
To prove $(*)^{\prime}_{9}$ assume toward contradiction that this fails, so
${\mathbb{P}}^{\prime}_{\gamma(*)}$ satisfies the $\lambda^{+}$-c.c. and for
some ${\mathbb{P}}^{\prime}_{\gamma(*)}$-name $\textstyle f$
$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$ and $\lambda$-Borel function
$\mathbb{B}$ and $\rho\in{}^{\lambda}\gamma(*)$, moreover
$\rho\in{}^{\lambda}({\mathcal{U}}_{*})$ we have (noting: the “moreover” holds
as
$f\in({}^{\lambda}\lambda)^{\mathbb{V}[\mathchoice{\oalign{$\displaystyle\bar{g}$\crcr\vbox
to0.60275pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\bar{g}$\crcr\vbox
to0.60275pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\bar{g}$\crcr\vbox
to0.60275pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\bar{g}$\crcr\vbox
to0.60275pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{0}{\restriction}{\mathcal{U}}_{*}]})$
1. $\circledast_{0}$
$p^{*}\Vdash_{{\mathbb{P}}^{\prime}_{\gamma(*)}}``\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\in{}^{\lambda}\lambda$ and dominates
$({}^{\lambda}\lambda)^{\mathbb{V}}"$ and $\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}=\mathbb{B}(\langle g_{\rho(i)}:i<\lambda\rangle)$.
Now we choose $\bar{N}=\langle N_{\varepsilon}:\varepsilon<\lambda\rangle$
such that
1. $\circledast_{1}$
$(a)\quad N_{\varepsilon}$ is as in $\boxdot_{\lambda}$ for the forcing notion
${\mathbb{P}}^{\prime}_{\gamma(*)}$
2. $(b)\quad\bar{N}\restriction\varepsilon\in N_{\varepsilon}$ hence $\bigcup\limits_{\zeta<\varepsilon}N_{\zeta}\subseteq N_{\varepsilon}$ and $\lambda_{\varepsilon}:=N_{\varepsilon}\cap\lambda>\lambda^{-}_{\varepsilon}:=$
$\Sigma\\{\lambda_{\zeta}:\zeta<\varepsilon\\}$
3. $(c)\quad\bar{\theta},\mathbb{q},p^{*},\mathchoice{\oalign{$\displaystyle f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}},\mathbb{B},\rho$ belong to $N_{\varepsilon}$
4. $(d)\quad$ let $\delta(\varepsilon)=\text{ otp}(\delta(*)\cap N_{\varepsilon})$
5. $(e)\quad\kappa_{\varepsilon}=\kappa^{<\kappa_{\varepsilon}}_{\varepsilon}$ where $\kappa_{\varepsilon}=\text{ otp}(\kappa_{\varepsilon}\cap N_{\varepsilon})$.
We can find $f^{*}\in{}^{\lambda}\lambda$, i.e.
$\in({}^{\lambda}\lambda)^{\mathbb{V}}$, such that
1. $\circledast_{2}$
for arbitrarily large $\varepsilon<\lambda$ for some
$\zeta\in[\lambda^{-}_{\varepsilon},\lambda_{\varepsilon})$ we have
$f^{*}(\zeta)>\lambda_{\varepsilon}$.
For $\varepsilon<\lambda$ let
$(\lambda_{\varepsilon},\chi_{\varepsilon},\mathbb{j}_{\varepsilon},M_{\varepsilon},N^{\prime}_{\varepsilon},\mathbb{G}_{\varepsilon})$
be a witness for $(N_{\varepsilon},{\mathbb{P}}^{\prime}_{\gamma(*)})$
recalling Definition 1.5 so $\lambda_{\varepsilon}\in(\varepsilon,\lambda)$ is
strongly inaccessible and
$\varepsilon<\zeta<\lambda\Rightarrow\lambda_{\varepsilon}<\lambda_{\zeta}$,
recalling $\circledast_{1}$ and
$\delta(\varepsilon)=\mathbb{j}_{\varepsilon}(\delta(*))$, etc.
Let
1. $\circledast_{3}$
$u_{\varepsilon}=N_{\varepsilon}\cap\gamma(*),\bar{\gamma}^{\varepsilon}=\langle\gamma_{i}(\varepsilon):i<i(\varepsilon)\rangle$
list $u_{\varepsilon}$ in increasing order and for $i<\text{
otp}(u_{\varepsilon})$, equivalently $i<\mathbb{j}_{\varepsilon}(\gamma(*))$
let
$\eta^{\varepsilon}_{i}=(\mathbb{j}_{\varepsilon}(\mathchoice{\oalign{$\displaystyle
g$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{\prime}_{i}))^{N^{\prime}_{\varepsilon}}[\mathbb{G}_{\varepsilon}]\in\prod\limits_{\zeta<\lambda_{\varepsilon}}\theta_{\zeta}$
and let $\bar{\eta}^{\varepsilon}=\langle\eta^{\varepsilon}_{i}:i<\text{
otp}(u_{\varepsilon})\rangle$.
Note
1. $\circledast_{4}$
$(a)\quad\bar{\eta}^{\varepsilon}$ is generic for
$(N^{\prime}_{\varepsilon},\mathbb{j}_{\varepsilon}({\mathbb{P}}^{\prime}_{\gamma(*)}))$,
moreover
2. $(b)\quad$ for each $\varepsilon<\lambda$, if we change $\eta^{\varepsilon}_{i}(\zeta)$ (legally, i.e. $<\theta_{\zeta}$) for $<\lambda_{\varepsilon}$ pairs
$(i,\zeta)\in\text{ otp}(u_{\varepsilon})\times\lambda_{\varepsilon}$ and get
$\bar{\eta}^{\prime}$, then also $\bar{\eta}^{\prime}$ is generic for
$(N^{\prime}_{\varepsilon},\mathbb{j}_{\varepsilon}({\mathbb{P}}^{\prime}_{\gamma(*)}))$
and $N^{\prime}_{\varepsilon}[\bar{\eta}^{\prime}]=M_{\varepsilon}$
3. $(c)\quad$ like $\boxplus_{1}$ with $\mathbb{V},{\mathbb{P}}^{\prime}_{\gamma(*)},\lambda$ there standing for $N_{\varepsilon},\mathbb{j}_{\varepsilon}({\mathbb{P}}^{\prime}_{\gamma(*)}),\lambda_{\varepsilon}$ here.
Hence
1. $\circledast^{\prime}_{4}$
for $\varepsilon<\lambda$, if
$\bar{\eta}^{\prime}=\langle\nu_{i}:i<i(\varepsilon)\rangle$ where
$i(\varepsilon)=\text{ otp}(u_{\varepsilon})$ is as in $\circledast_{4}(b)$,
and $q\in{\mathbb{P}}^{\prime}_{\gamma(*)}$ satisfies
$i<i(\varepsilon)\Rightarrow
q\Vdash_{{\mathbb{P}}^{\prime}_{\gamma(*)}}``\mathchoice{\oalign{$\displaystyle
g$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{\prime}_{\gamma_{i}(\varepsilon)}{\restriction}\lambda_{\varepsilon}=\nu_{i}"$
then $q$ is $(N_{\varepsilon},{\mathbb{P}}^{\prime}_{\gamma(*)})$-generic
naturally and
$q\Vdash_{{\mathbb{P}}^{\prime}_{\gamma(*)}}``\mathbb{j}_{\varepsilon}$ can be
extended naturally to an isomorphism from
$N_{\varepsilon}[\mathchoice{\oalign{$\displaystyle G$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle G$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle G$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle G$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{{\mathbb{P}}^{\prime}_{\gamma(*)}}]=N_{\varepsilon}[\langle\mathchoice{\oalign{$\displaystyle
g$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\gamma}:\gamma\in u_{\varepsilon}\rangle]$ onto
$N^{\prime}_{\varepsilon}[\bar{\eta}^{\prime}]"$.
[Why? Should be clear, see $\boxplus_{1}+\circledast_{4}(c)$.]
By the assumption toward contradiction, $\circledast_{0}$, and
${\mathbb{P}}^{\prime}_{\gamma(*)}$ being $(<\lambda)$-strategically closed
recalling $(*)_{1}(\beta)^{+}$, there are $\zeta(*),p^{**}$ and $p^{+}$ such
that (recall
$p^{*}\in{\mathbb{P}}^{\prime}_{\gamma(*)}=\lessdot{\mathbb{P}}_{0,\gamma(*)}$):
1. $\circledast_{5}$
$(a)\quad p^{*}\leq p^{**}\in{\mathbb{P}}^{\prime}_{\gamma(*)}$ and
$p^{**}\leq p^{+}\in{\mathbb{P}}_{0,\gamma(*)}$
2. $(b)\quad\zeta(*)<\lambda$
3. $(c)\quad p^{**}\Vdash_{{\mathbb{P}}^{\prime}_{\gamma(*)}}``f^{*}(\zeta)<\mathchoice{\oalign{$\displaystyle f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}(\zeta)$ whenever $\zeta(*)\leq\zeta<\lambda"$
4. $(d)\quad$ if $\gamma\in\text{ Dom}(p^{+})$ then $\eta^{p^{+}(\gamma)}$ is an object (not just a ${\mathbb{P}}^{\prime}_{\gamma}$-name)
of length $\geq\zeta(*)$ (recall that $\eta^{p^{+}(\gamma)}$ is the trunk of
the condition,
see clause $(\alpha)(b)$ above).
Note that possibly
Dom$(p^{+})\nsubseteq\cup\\{u_{\varepsilon}:\varepsilon<\lambda\\}$. Choose
$\varepsilon(*)<\lambda$ such that
$\lambda_{\varepsilon(*)}>\zeta(*)+|\text{Dom}(p^{+})|$ and $\gamma\in\text{
Dom}(p^{+})\Rightarrow\varepsilon(*)>\ell g(\eta^{p^{+}(\gamma)})$ recalling
clause (d) of $\circledast_{5}$ and $|\text{Dom}(p^{+})|<\lambda$ as
$p^{+}\in{\mathbb{P}}_{0,\gamma(*)}$ and ${\mathbb{P}}_{0,\gamma(*)}$ is the
limit of a $(<\lambda)$-support iteration.
By $\circledast_{2}$ we can add
$(\exists\zeta)[\lambda^{-}_{\varepsilon(*)}\leq\zeta<\lambda_{\varepsilon(*)}<f^{*}(\zeta)]$.
Our intention is to find $q\in{\mathbb{P}}_{0,\gamma(*)}$ above $p^{+}$ which
is above some $q^{\prime}\in{\mathbb{P}}^{\prime}_{\gamma(*)}$ which is
$(N_{\varepsilon(*)},{\mathbb{P}}^{\prime}_{\gamma(*)})$-generic and forces it
to include a generic subset of
$({\mathbb{P}}^{\prime}_{\gamma(*)})^{N_{\varepsilon(*)}}$ which is induced by
some $\bar{\eta}^{\prime}$ as in $\circledast_{4}(b)$. Toward this in
$\circledast_{6}$ below the intention is that $p^{+}_{i(*)}$ will serve as
$q$.
Let $i(*)=i(\varepsilon(*))$ and
$\gamma_{i}=\gamma_{2,i}=\gamma_{\delta(*)(i+1)}(\varepsilon(*))$ for $i<i(*)$
so $\langle\gamma_{i}:i<i(*)\rangle$ list
$u_{\varepsilon(*)}\cap{\mathcal{U}}_{*}$ in increasing order and let
$\gamma_{i(*)}=\gamma(*)$ so $\\{\mathbb{j}_{\varepsilon(*)}(\gamma):\gamma\in
u_{\varepsilon(*)}\\}=\mathbb{j}_{\varepsilon(*)}(\gamma(*))$ and
$N_{\varepsilon(*)}\models``i(*)$ is a regular cardinal
$>\lambda_{\varepsilon}$” hence $i(*)$ is really a regular cardinal so call it
$\sigma$. Now we define a game $\Game$ as follows444The idea is to scatter the
$\eta^{\varepsilon(*)}_{\gamma_{i}}$’s. Why not use the original places? as
then we have a problem in $\circledast_{10}$.:
1. $\boxplus_{2}$
$(A)\quad$ each play lasts $i(*)+1$ moves and in the $i$-th move,
1. $(a)\quad$ if $i=j+1$ the antagonist player chooses $\xi(j)<\sigma$ such that $j_{1}<j\Rightarrow\zeta(j_{1})<\xi(j)$
2. $(b)\quad$ then, if $i=j+1$ the protagonist chooses $\zeta(j)\in(\xi(j),\sigma)\cap{\mathcal{U}}_{*}$, but there are more restrictions implicit in $\boxplus_{3}$
3. $(c)\quad$ in any case the protagnoist chooses $p^{+}_{i},\bar{\nu}^{i}$ such that $\boxplus_{3}$ below holds;
2. $(B)\quad$ in the end of the play the protagonist wins the play iff he always has a legal move and in the end $\\{\zeta(i):i<i(*)\\}\in N^{\prime}_{\varepsilon(*)}$; where
3. $\boxplus_{3}$
$(a)\quad p^{+}_{i}\in{\mathbb{P}}_{0,\gamma_{i}}$
4. $(b)\quad$ if $j<i$ then ${\mathbb{P}}_{0,\gamma_{i}}\models``p^{+}_{j}\leq p^{+}_{i}"$
5. $(c)\quad$ if $\gamma\in\cup\\{\text{Dom}(p^{+}_{j}):j<i\\}$ then
$p^{+}_{i}\restriction\gamma\Vdash_{{\mathbb{P}}_{0,\gamma_{i}}}``\mathchoice{\oalign{$\displaystyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{p^{+}_{i}(\gamma)}$ has length $\geq i(*)$ and
$\geq\lambda_{\varepsilon(*)}"$
moreover $\mathchoice{\oalign{$\displaystyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{p^{+}_{i}(\gamma_{j})}$ is an object,
$\eta^{p^{+}_{i}(\gamma_{j})}$ for $j<i$
6. $(d)\quad{\mathbb{P}}_{0,\gamma_{i}}\models``p^{+}\restriction\gamma_{i}\leq p^{+}_{i}"$
7. $(e)\quad\bar{\nu}^{i}=\langle\nu_{\gamma_{j}}:j<i\rangle$ and $\nu_{\gamma_{j}}\in\prod\limits_{\iota<\lambda_{\varepsilon(*)}}\theta_{\iota}$
8. $(f)\quad$ for $j<i$ we have $\nu_{\gamma_{j}}\trianglelefteq\eta^{p^{+}_{i}(\gamma_{j})}$ so $p^{+}_{i}\restriction\gamma_{j}\Vdash``\nu_{\gamma_{j}}\triangleleft\mathchoice{\oalign{$\displaystyle g$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}^{\prime}_{\gamma_{j}}"$ recalling $\boxplus_{1}$
9. $(g)\quad$ for $j<i$ we have (recall $\bar{\eta}^{\varepsilon}$ from $\circledast_{3}$)
1. $(\alpha)\quad\nu_{\gamma_{j}}=\eta^{\varepsilon(*)}_{\gamma_{\zeta(j)}}$ recalling $\eta^{\varepsilon(*)}_{\gamma_{j}}$ is from $\circledast_{3}$ or
2. $(\beta)\quad\gamma_{j}\in\text{ Dom}(p^{+})$ and $\\{\iota<\lambda_{\varepsilon(*)}:\eta^{\varepsilon(*)}_{\zeta(j)}(\iota)\neq\nu_{\gamma_{j}}(\iota)\\}$ is a bounded
subset of $\lambda_{\varepsilon(*)}$.
We shall prove
1. $\circledast_{6}$
in the game $\Game$
1. $(a)$
the antagonist has no winning strategy
2. $(b)$
in any move the protagonist has a legal move, moreover for any
$\zeta(i)\in(\xi(i),\sigma)$ large enough the protagonist can choose it.
Why $\circledast_{6}$ suffice:
By clause (a) of $\circledast_{6}$ we can choose a play
$\langle(\xi(i),\zeta(i),p^{+}_{i},\bar{\nu}^{i}):i\leq i(*)\rangle$ in which
the protagonist wins. Recalling
${\mathbb{P}}^{\prime}_{\gamma(*)}\lessdot{\mathbb{P}}_{1,\gamma(*)}$ and
${\mathbb{P}}_{0,\gamma(*)}$ is a dense subforcing of
${\mathbb{P}}_{1,\gamma(*)}$, clearly
1. $\circledast_{7}$
there is $p$ such that
1. $(a)$
$p\in{\mathbb{P}}^{\prime}_{\gamma(*)}$
2. $(b)$
if ${\mathbb{P}}^{\prime}_{\gamma(*)}\models``p\leq p^{\prime}"$ then
$p^{\prime},p^{+}$ are compatible in ${\mathbb{P}}_{0,\gamma(*)}$
3. $(c)$
$p$ is above $p^{**}$ and it forces $\mathchoice{\oalign{$\displaystyle
g$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle g$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle g$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{\prime}_{i}{\restriction}\lambda_{\varepsilon(*)}=\nu_{\gamma_{i}}$
for $i<\gamma(*)$.
Then on the one hand
1. $\circledast^{\prime}_{7}$
$p\in{\mathbb{P}}^{\prime}_{\gamma(*)}$ being above $p^{**}$ forces
$f^{\gamma}\restriction[\zeta(*),\lambda)<\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\restriction[\zeta(*),\lambda)$ hence
$f^{*}\restriction[\zeta(*),\lambda_{\varepsilon(*)})<\mathchoice{\oalign{$\displaystyle
f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\restriction[\zeta(*),\lambda_{\varepsilon(*)})$ recalling
that $\zeta(*)<\lambda_{\varepsilon(*)}$.
On the other hand,
1. $\circledast^{\prime\prime}_{7}$
$p$ is $(N_{\varepsilon(*)},{\mathbb{P}}^{\prime}_{\gamma(*)})$-generic.
[Why? As it forces $\mathchoice{\oalign{$\displaystyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\eta$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\gamma_{1,i}}\restriction\lambda_{\varepsilon(*)}=\nu_{\gamma_{i}}$
for $i<i(*)$ and $\langle\nu_{\gamma_{i}}:i<i(*)\rangle$ is (see
$\circledast_{4}$) “almost equal” to
$\langle\eta^{\varepsilon(*)}_{\zeta(i)}:i<i(*)\rangle$ which is a subsequence
of the sequence from $\circledast_{3}$ and recalling clause (g) of
$\boxplus_{3}$. That is
$\\{(i,\iota):\iota<\lambda_{\varepsilon(*)},i<i(*)=\sigma$ and
$\nu_{\gamma_{i}}(\iota)\neq\eta^{\varepsilon(*)}_{\zeta(i)}(\iota)\\}\subseteq\cup\\{\\{(i,\iota):\iota<\lambda_{\varepsilon(*)}$
and
$\nu_{\gamma_{i}}(\iota)\neq\eta^{\varepsilon(*)}_{\zeta(i)}(\iota)\\}:\gamma\in
u_{\varepsilon(*)}\cap\text{ Dom}(p^{+})\\}$ so is the union of
$\leq|\text{Dom}(p^{+})|<\lambda_{\varepsilon(*)}$ sets each of cardinality
$<\lambda_{\varepsilon(*)}$ hence is of cardinality
$<\lambda_{\varepsilon(*)}$. Hence by $\circledast_{4}(c)+\boxplus_{1}(d)$ the
sequence $\bar{\nu}^{i(*)}$ is generic for
$(N_{\varepsilon(*)},{\mathbb{P}}^{\prime}_{\gamma(*)})$.]
As $\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\in N_{\varepsilon(*)}$ it follows from
$\circledast^{\prime\prime}_{7}$ that
1. $\circledast^{\prime\prime\prime}_{7}$
$p\Vdash``\mathchoice{\oalign{$\displaystyle f$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\restriction\lambda_{\varepsilon(*)}$ is a function from
$\lambda_{\varepsilon(*)}$ to $\lambda_{\varepsilon(*)}"$.
Together $\circledast^{\prime}_{7}+\circledast^{\prime\prime\prime}_{7}$ gives
a contradiction by the choice of $f^{*}$ in $\circledast_{2}$ and of
$\varepsilon(*)$ above, hence it is enough to use $\circledast_{6}$.
Why $\circledast_{6}$ is true:
Let us prove $\circledast_{6}$; first for clause (a) choose any strategy st
for the antagonist and fix a partial strategy st′ for the protagonist choosing
$(p^{+}_{i},\bar{\nu}^{i})$ from the previous choices and $\zeta(i)$ if
relevant and possible. So the only freedom left is for the protagonist to
choose the $\zeta(i)$. So we have in $\mathbb{V}$ a function
$F:{}^{\sigma>}(i(*))\rightarrow\sigma$ such that:
1. $(*)_{F}$
playing the game such that the antagonist uses st and the protagonist uses
st′, arriving to the $i$-th move, $\bar{\zeta}=\langle\zeta(j):j<i\rangle$ is
well defined and for the protagonist any choice
$\zeta_{i}\in(F(\bar{\zeta}),\sigma)$ is legal.
Now we have to find an increasing sequence
$\bar{\zeta}=\langle\zeta(i):i<i(*)\rangle$ such that
$F(\bar{\zeta}{\restriction}i)<\zeta(i)\in{\mathcal{U}}_{*}$ and
$\bar{\zeta}\in N^{\prime}_{\varepsilon(*)}$. As
$F\in{\mathcal{H}}(\chi_{\varepsilon})$ and
${\mathcal{H}}(\chi_{\varepsilon})=N^{\prime}_{\varepsilon}[\mathbb{G}_{\varepsilon}]$
where $\mathbb{G}_{\varepsilon}$ is a subset of
$\mathbb{j}_{\varepsilon}({\mathbb{P}}^{\prime}_{\gamma(*)})\in
N^{\prime}_{\varepsilon}$ and
$\mathbb{j}_{\varepsilon}({\mathbb{P}}_{0,\gamma(*)})$ satisfies the
$\lambda^{+}_{\varepsilon}$-c.c. and $\sigma=\text{
cf}(\sigma)>\lambda_{\varepsilon}$ this is possible. We are left with proving
$\circledast_{6}(b)$.
Case 1: $i=0$.
Let $p^{+}_{0}=p^{+}\restriction\gamma_{0}$.
Case 2: $i$ limit.
By clauses (a) and (b), there is $p^{+}_{i}\in{\mathbb{P}}_{0,\gamma_{i}}$
which is an upper bound (even l.u.b.) of $\\{p^{+}_{j}:j<i\\}$ and it is
easily as required. Also $\bar{\nu}^{i}$ is well defined and as required.
Case 3: $i=j+1$ and $\gamma_{j}\notin\text{ Dom}(p^{+})$.
Clearly $\gamma_{i}=\gamma_{j}+\delta(*)$ and $\gamma_{j}\in
u_{\varepsilon(*)}$. As in case 4 below but easier by the properties of the
iteration.
Case 4: $i=j+1$ and $\gamma_{j}\in\text{ Dom}(p^{+})$
Again $\gamma_{i}=\gamma_{j}+\delta(*)$ and $\gamma_{j}\in
u_{\varepsilon(*)}$. First we find $p^{\prime}_{j}$ such that:
1. $\circledast_{8}$
$(a)\quad p^{+}_{j}\leq p^{\prime}_{j}\in{\mathbb{P}}_{0,\gamma_{j}}$
2. $(b)\quad$ if $\gamma\in\text{ Dom}(p^{+}_{j})$ then $p^{\prime}_{j}\restriction\gamma\Vdash``\ell g(\eta^{p^{\prime}_{j}(\gamma)})>i"$
3. $(c)\quad p^{\prime}_{j}$ forces 555recall that $\eta^{p^{*}(\gamma)}$ is an object, not a name and $p^{+}_{j}$ is $(N_{\varepsilon(*)},{\mathbb{P}}^{\prime}_{\gamma_{j}})$-generic a value to the pair $(\eta^{p^{+}(\gamma_{i})},\mathchoice{\oalign{$\displaystyle f$\crcr\vbox to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\textstyle f$\crcr\vbox to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle f$\crcr\vbox to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle f$\crcr\vbox to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern 3.0mu}{}$}\vss}}}^{p^{+}(\gamma_{j})}\restriction\lambda_{\varepsilon(*)})$; we call this
pair $q_{j}$.
This should be clear.
Second
1. $\circledast_{9}$
$p^{+}_{j}$ hence $p^{\prime}_{j}$ is
$(N_{\varepsilon(*)},{\mathbb{P}}^{\prime}_{\gamma_{j}})$-generic and
$\langle\nu_{\gamma_{j(1)}}:j(1)<j\rangle$ induces the generic.
[Why? As in the proof of $\circledast^{\prime\prime}_{7}$ above when we assume
that we have carried the induction, by $\boxplus_{2}$, clause (g) and
$\circledast_{4}$.]
Now
1. $\circledast_{10}$
$(a)\quad
f^{q_{j}}\in(\prod_{\zeta<\lambda_{\varepsilon(*)}}\theta_{\zeta})^{N^{\prime}_{\varepsilon(*)}[\bar{\nu}^{j}]}$
2. $(b)\quad$ for some $\zeta\in(\xi(i),\sigma)$ we have
1. $\bullet\quad f^{q_{j}}\leq\eta^{\varepsilon(*)}_{\zeta}$
2. $\bullet\quad f^{q_{j}}\in N^{\prime}_{\varepsilon(*)}[\bar{\eta}^{\varepsilon(*)}{\restriction}\zeta]$
3. $\bullet\quad\langle\zeta(j_{1}):j_{1}\langle j\rangle\in N^{\prime}_{\varepsilon(*)}[\bar{\eta}^{\varepsilon(*)}{\restriction}\zeta]$.
3. $(c)\quad\eta^{q_{j}}\triangleleft f^{q_{j}}$.
[Why? Clause (a) follows from clause (b) and clause (b) should be clear by
$\circledast_{9}$ as we can choose $\zeta(i)$ large enough recalling
$\circledast_{6}$. Also clause (c) follows from (b).]
Now we choose $\zeta(j)$ as in clause (b) of $\circledast_{10}$ and
$\nu_{j}\in\prod\limits_{\varepsilon<\lambda_{\varepsilon(*)}}\theta_{\varepsilon}$
such that $\eta^{p^{+}(j)}\triangleleft\nu_{j},f^{q_{j}}\leq\nu_{j}$ and
$\\{\iota<\lambda_{\varepsilon(*)}:\nu_{j}(\iota)\neq\eta^{\varepsilon(*)}_{\zeta(j)}\\}$
is a bounded subset of $\lambda_{\varepsilon(*)}$. Next choose
$p^{+}_{i}\in{\mathbb{P}}^{\prime}_{\gamma(*)}$ such that
$p^{+}_{i}{\restriction}\gamma_{j}=p^{\prime}_{j},\eta^{p^{+}_{i}(\gamma_{i})}=\nu_{j}$
and
$f^{p^{+}_{i}(\gamma_{i})}{\restriction}[\lambda_{\varepsilon},\lambda)=f^{p^{+}(\gamma)}{\restriction}[\lambda_{\varepsilon},\lambda)$.
So we have carried the induction hence proved $\circledast_{6}$ so we are
done. ∎
## References
* [Bar87] Tomek Bartoszyński, _Combinatorial aspects of measure and category_ , Fundamenta Mathematicae 127 (1987), 225–239.
* [Kin66] Akiko Kino, _On definability of ordinals in logic with infinitely long expressions_ , Journal of Symbolic Logic 31 (1966), 365–375.
* [Lan92] A. Landver, _Baire numbers, uncountable cohen sets and perfect-set forcing_ , Journal of Symbolic Logic 57 (1992), 1086–1107.
* [Lav78] Richard Laver, _Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing_, Israel J. of Math. 29 (1978), 385–388.
* [Mil82] Arnold W. Miller, _A characterization of the least cardinal for which the baire category theorem fails_ , Proceedings of the American Mathematical Society 86 (1982), 498–502.
* [vD84] Eric K. van Douwen, _The integers and topology_ , Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), Elsevier Science Publishers, 1984, pp. 111–167.
* [Sh:80] Saharon Shelah, _A weak generalization of MA to higher cardinals_ , Israel Journal of Mathematics 30 (1978), 297–306.
* [CuSh:541] James Cummings and Saharon Shelah, _Cardinal invariants above the continuum_ , Annals of Pure and Applied Logic 75 (1995), 251–268, math.LO/9509228.
* [Sh:546] Saharon Shelah, _Was Sierpiński right? IV_ , Journal of Symbolic Logic 65 (2000), 1031–1054, math.LO/9712282.
* [MtSh:804] Pierre Matet and Saharon Shelah, _Positive partition relations for $P_{\kappa}(\lambda)$_, Preprint, math.LO/0407440.
* [Sh:F976] Saharon Shelah, _Nice logics_.
* [Sh:F979] by same author, _Iterating reasonable $\lambda$-complete definable forcing_.
|
arxiv-papers
| 2009-04-05T20:36:54 |
2024-09-04T02:49:01.710537
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Saharon Shelah",
"submitter": "shlhetal",
"url": "https://arxiv.org/abs/0904.0817"
}
|
0904.0833
|
# On the Rate of Convergence for the Pseudospectral Optimal Control of
Feedback Linearizable Systems ††thanks: The research was supported in part by
AFOSR and AFRL
Wei Kang
Department of Applied Mathematics
Naval Postgraduate School
Monterey, CA 93943
wkang@nps.edu
###### Abstract
Over the last decade, pseudospectral (PS) computational methods for nonlinear
constrained optimal control have been applied to many industrial-strength
problems, notably the recent zero-propellant-maneuvering of the International
Space Station performed by NASA. In this paper, we prove a theorem on the rate
of convergence for the optimal cost computed using PS methods. It is a first
proved convergence rate in the literature of PS optimal control. In addition
to the high-order convergence rate, two theorems are proved for the existence
and convergence of the approximate solutions. This paper contains several
essential differences from existing papers on PS optimal control as well as
some other direct computational methods. The proofs do not use necessary
conditions of optimal control. Furthermore, we do not make coercivity type of
assumptions. As a result, the theory does not require the local uniqueness of
optimal solutions. In addition, a restrictive assumption on the cluster points
of discrete solutions made in existing convergence theorems are removed.
## 1 Introduction
Despite the fact that optimal control is one of the oldest problems in the
history of control theory, practical tools of solving nonlinear optimal
control problems are limited. Preferably, a feedback control law is derived
from a solution to the famously difficult Hamilton-Jacobi-Bellman (HJB)
equation. However, analytic solutions of this partial differential equation
can rarely be found for systems with nonlinear dynamics. Numerical
approximation of such solutions suffers from the well-known curse of
dimensionality and it is still an open problem for systems with moderately
high dimension. A practical alternative is to compute one optimal trajectory
at a time so that the difficulty of solving HJB equations is circumvented.
Then this open-loop optimal control can be combined with an inner-loop
tracking controller; or it can be utilized as a core instrument in a real-time
feedback control architecture such as a moving horizon feedback. A critical
challenge in this approach is to develop reliable and efficient computational
methods that generate the required optimal trajectories. In this paper, we
focus on some fundamental issues of pseudospectral computational optimal
control methods.
As a result of significant progress in large-scale computational algorithms
and nonlinear programming, the so-called direct computational methods have
become popular for solving nonlinear optimal control problems [1, 2, 17],
particularly in aerospace applications [16, 18]. In simple terms, in a direct
method, the continuous-time problem of optimal control is discretized, and the
resulting discretized optimization problem is solved by nonlinear programming
algorithms. Over the last decade, pseudospectral (PS) methods have emerged as
a popular direct methods for optimal control. They have been applied to many
industrial-strength problems, notably the recent attitude maneuvers of the
International Space Station performed by NASA. By following an attitude
trajectory developed using PS optimal control, the International Space Station
(ISS) was maneuvered 180 degrees on March 3, 2007, by using the gyroscopes
equipped on the ISS without propellant consumption. This single maneuver have
saved NASA about one million dollars’ worth of fuel [12]. The Legendre PS
optimal control method has already been developed into software named DIDO, a
MATLAB based package commercially available [19]. In addition, the next
generation of the OTIS software package [15] will have the Legendre PS method
as a problem solving option.
PS methods have been widely applied in scientific computation for models
governed by partial differential equations. The method is well known for being
very efficient in approximating solutions of differential equations. However,
despite its success and several decades of development, the intersection
between PS methods and nonlinear optimal control becomes an active research
area only after the mid-1990’s ([5, 6]). As yet, many fundamental theoretical
issues are still widely open. For the last decade, active research has been
carried out in the effort of developing a theoretical foundation for PS
optimal control methods. Among the research focuses, there are three
fundamental issues, namely the state and costate approximation, the existence
and convergence of approximate solutions, and the convergence rate. The
general importance of these issues is not limited to PS methods. They are
essential to other computational methods suchlike those based on Euler [10]
and Runge-Kutta [9] discretization. Similar to other direct computational
optimal control methods, PS method are based upon the Karush-Kuhn-Tucker (KKT)
conditions rather than the Pontryagin’s Minimum Principle (MPM). In [6] and
[8], a covector mapping was derived between the costate from KKT condition and
the costate from PMP. The covector mapping facilitates a verification and
validation of the computed solution. For the problem of convergence, some
theorems were published in [7]; and then the results were generalized in [13]
to problems with non-smooth control.
Among the three fundamental issues mentioned above, the most belated activity
of research is on the rate of convergence. In fact, there have been no results
published on the convergence rate for PS optimal control methods. Although
some results on the issue of convergence were proved in [7] and [13], a main
drawback of these results is the strong assumption in which the derivatives of
the discrete approximate solutions are required to converge uniformly. In this
paper, we prove a rate of convergence for the approximate optimal cost
computed using PS methods. Then, we prove theorems on existence and
convergence without the restrictive assumption made in [7] and [13]. In
addition to the high-order convergence rate addressed in Section 3, which is
the first proved convergence rate in the literature of PS optimal control,
this paper contains several essential differences from existing papers on PS
optimal control as well as some other direct computational methods. First of
all, the proof is not based on necessary conditions of optimal control.
Furthermore, we do not make coercivity type of assumptions. As a result, the
theory does not require the local uniqueness of optimal solutions. Therefore,
it is applicable to problems with multiple optimal solutions. Secondly, the
proof is not build on the bases of consistent approximation theory [17]. Thus,
we can remove the assumption in [7] and [13] on the existence of cluster
points for the derivatives of discrete solutions. The key that makes these
differences possible is that we introduce a set of sophisticated
regularization conditions in the discretization so that the computational
algorithm has a greater control of the boundedness of the approximate
solutions and their derivatives. Different from the existing results in the
literature of direct methods for optimal control, the desired boundedness is
achieved not by making assumptions on the original system, but by implementing
specially designed search region for the discrete problem of nonlinear
programming. This new boundary of search region automatically excludes
possible bad solutions that are numerically unstable.
The paper is organized as follows. In Section 2, the formulations of the
optimal control problem and its PS discretization are introduced. In 3, we
prove two theorems on the rate of convergence. In Section 4, two theorems on
the existence and convergence are proved.
## 2 Problem Formulation
For the rate of convergence, we focus on the following Bolza problem of
control systems in the feedback linearizable normal form. A more complicated
problem with constraints is studied in Section 4 for the existence and
convergence of approximate solutions.
Problem B: Determine the state-control function pair $(x(t),u(t))$,
$x\in\Re^{r}$ and $u\in\Re$, that minimizes the cost function
$\displaystyle J(x(\cdot),u(\cdot))$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}F(x(t),u(t))\ dt+E(x(-1),x(1))$ (2.1)
subject to the following differential equations and initial condition
$\displaystyle\left\\{\begin{array}[]{lll}\dot{x}_{1}=x_{2}\\\ \;\;\;\vdots\\\
\dot{x}_{r-1}=x_{r}\\\ \dot{x}_{r}=f(x)+g(x)u\end{array}\right.$ (2.6)
$\displaystyle x(-1)=x_{0}$ (2.7)
where $x\in\Re^{r}$, $u\in\Re$, and $F:\Re^{r}\times\Re\to\Re$,
$E:\Re^{r}\times\Re^{r}\to\Re$, $f:\Re^{r}\to\Re$, and $g:\Re^{r}\to\Re$ are
all Lipschitz continuous functions with respect to their arguments. In
addition, we assume $g(x)\neq 0$ for all $x$.
Throughout the paper we make extensive use of Sobolev spaces, $W^{m,p}$, that
consists of functions, $\xi:[-1,1]\to\mathbb{R}$ whose $j$-th order weak
derivative, $\xi^{(j)}$, lies in $L^{p}$ for all $0\leq j\leq m$ with the
norm,
$\parallel\xi\parallel_{W^{m,p}}\quad=\sum_{j=0}^{m}\parallel\xi^{(j)}\parallel_{L^{p}}$
In this paper, we only consider the problems that have at least one optimal
solution in which $x_{r}^{\ast}(t)$ has bounded $m$-th order weak derivative,
i.e. $x_{r}^{\ast}(t)$ is in $W^{m,\infty}$. For some results, we assume
$m\geq 3$. For others, $m$ is smaller. Unless the term ‘strong derivative’ is
emphasized, all derivatives in the paper are in the weak sense.
The PS optimal control method addressed in this paper is an efficient direct
method. In typical direct methods, the original optimal control problem, not
the associated necessary conditions, is discretized to formulate a nonlinear
programming problem. The accuracy of the discretization is largely determined
by the accuracy of the underlying approximation method. Given any function
$f(t):[a,b]\rightarrow\Re$, a conventional method of approximation is to
interpolate at uniformly spaced nodes: $t_{0}=a$, $t_{1}=(b-a)/N$, $\cdots$,
$t_{N}=b$. However, it is known that uniform spacing is not efficient. More
sophisticated node selection methods are able to achieve significantly
improved accuracy with fewer nodes. It is important to emphasize that, for
optimal control problems, the rate of convergence is not merely an issue of
efficiency; more importantly it is about feasibility. An increased number of
nodes in discretization results in a higher dimension in the nonlinear
programming problem. A computational method becomes practically infeasible
when the dimension and complexity of the nonlinear programming exceed the
available computational power. In a PS approximation based on Legendre-Gauss-
Lobatto (LGL) quadrature nodes, a function $f(t)$ is approximated by $N$-th
order Lagrange polynomials using the interpolation at these nodes. The LGL
nodes, $t_{0}=-1<t_{1}<\cdots<t_{N}=1$, are defined by
$\begin{array}[]{llll}t_{0}=-1,\;\;t_{N}=1,\mbox{ and }\\\ \mbox{for
}k=1,2,\ldots,N-1,t_{k}\mbox{ are the roots of }\dot{L}_{N}(t)\end{array}$
where $\dot{L}_{N}(t)$ is the derivative of the $N$-th order Legendre
polynomial $L_{N}(t)$. The discretization works in the interval of $[-1,1]$.
An example of LGL nodes with $N=16$ is shown in Figure 1.
Figure 1: LGL nodes $N=16$
It was proved in approximation theory that the polynomial interpolation at the
LGL nodes converges to $f(t)$ under $L^{2}$ norm at the rate of $1/N^{m}$,
where $m$ is the smoothness of $f(t)$ (see for instance [4] Section 5.4). If
$f(t)$ is $C^{\infty}$, then the polynomial interpolation at the LGL nodes
converges at a spectral rate, i.e. it is faster than any given polynomial
rate. This is a very impressive convergence rate.
PS methods have been widely applied in scientific computation for models
governed by partial differential equations, such as complex fluid dynamics.
However, PS optimal control has several fundamental differences from the
computation of PDEs. Solving optimal control problems asks for the
approximation of several objects collectively, including the differential
equation that defines the control system, the integration in the cost
function, and the state and control trajectories. In addition to the various
types of approximations, a nonlinear programming must be applied to the
overall discretized optimization problem to find an approximate optimal
control. All these factors may deteriorate the final approximate solution. The
existing theory of PS approximation of differential equations is not
applicable. New theory needs to be developed for the existence, convergence,
and the rate of convergence for optimal control problems.
In the following, we introduce the notations used in this paper. Then, the
discretized nonlinear programming problem is formulated. In a PS optimal
control method, the state and control functions, $x(t)$ and $u(t)$, are
approximated by $N$-th order Lagrange polynomials based on the interpolation
at the LGL quadrature nodes. In the discretization, the state variables are
approximated by the vectors $\bar{x}^{Nk}\in\Re^{r}$, i.e.
$\bar{x}^{Nk}=\left[\begin{array}[]{cccccccccccccc}\bar{x}_{1}^{Nk}\\\
\bar{x}_{2}^{Nk}\\\ \vdots\\\ \bar{x}_{r}^{Nk}\end{array}\right]$
is an approximation of $x(t_{k})$. Similarly, $\bar{u}^{Nk}$ is the
approximation of $u(t_{k})$. Thus, a discrete approximation of the function
$x_{i}(t)$ is the vector
$\bar{x}_{i}^{N}=\left[\begin{array}[]{cccccccccccccc}\bar{x}_{i}^{N1}&\bar{x}_{i}^{N2}&\cdots&\bar{x}_{i}^{NN}\end{array}\right]$
A continuous approximation is defined by its polynomial interpolation, denoted
by $x_{i}^{N}(t)$, i.e.
$\displaystyle x_{i}(t)$ $\displaystyle\approx$ $\displaystyle
x_{i}^{N}(t)=\sum_{k=0}^{N}\bar{x}_{i}^{Nk}\phi_{k}(t),$ (2.8)
where $\phi_{k}(t)$ is the Lagrange interpolating polynomial [4]. Instead of
polynomial interpolation, the control input is approximated by the following
non-polynomial interpolation
$\displaystyle
u^{N}(t)=\displaystyle\frac{\dot{x}_{r}^{N}(t)-f(x^{N}(t))}{g(x^{N}(t))}$
(2.9)
In the notations, the discrete variables are denoted by letters with an upper
bar, such as $\bar{x}^{Nk}_{i}$ and $\bar{u}^{Nk}$. If $k$ in the superscript
and/or $i$ in the subscript are missing, it represents the corresponding
vector or matrix in which the indices run from minimum to maximum. For
example,
$\displaystyle\bar{x}^{N}_{i}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccccccccccc}\bar{x}_{i}^{N0}&\bar{x}_{i}^{N1}&\cdots&\bar{x}_{i}^{NN}\end{array}\right]$
$\displaystyle\bar{x}^{Nk}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccccccccccc}\bar{x}_{1}^{Nk}\\\
\bar{x}_{2}^{Nk}\\\ \vdots\\\ \bar{x}_{r}^{Nk}\end{array}\right]$
$\displaystyle\bar{x}^{N}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccccccccccc}\bar{x}_{1}^{N0}&\bar{x}_{1}^{N1}&\cdots&\bar{x}_{1}^{NN}\\\
\bar{x}_{2}^{N0}&\bar{x}_{2}^{N1}&\cdots&\bar{x}_{2}^{NN}\\\
\vdots&\vdots&\vdots&\vdots\\\
\bar{x}_{r}^{N0}&\bar{x}_{r}^{N1}&\cdots&\bar{x}_{r}^{NN}\end{array}\right]$
Similarly,
$\bar{u}^{N}=\left[\begin{array}[]{cccccccccccccc}\bar{u}^{N0}&\bar{u}^{N1}&\cdots&\bar{u}^{NN}\end{array}\right]$
Given a discrete approximation of a continuous function, the interpolation is
denoted by the same notation without the upper bar. For example,
$x_{i}^{N}(t)$ in (2.8), $u^{N}(t)$ in (2.9). The superscript $N$ represents
the number of LGL nodes used in the approximation. Throughout this paper, the
interpolation of $(\bar{x}^{N},\bar{u}^{N})$ is defined by (2.8)-(2.9), in
which $u^{N}(t)$ is not necessarily a polynomial. It is proved in Lemma 5 that
(2.9) is indeed an interpolation.
Existing results in the analysis of spectral methods show that PS method is an
approach that is easy and accurate in the approximation of smooth functions,
integrations, and differentiations, all critical to optimal control problems.
For differentiation, the derivative of $x^{N}_{i}(t)$ at the LGL node $t_{k}$
is easily computed by the following matrix multiplication [4]
$\displaystyle\left[\begin{array}[]{cccccccccccccc}\dot{x}_{i}^{N}(t_{0})&\dot{x}_{i}^{N}(t_{1})&\cdots&\dot{x}_{i}^{N}(t_{N})\end{array}\right]^{T}=D(\bar{x}^{N}_{i})^{T}$
(2.14)
where the $(N+1)\times(N+1)$ differentiation matrix $D$ is defined by
$\displaystyle D_{ik}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{L_{N}(t_{i})}{L_{N}(t_{k})}\frac{1}{t_{i}-t_{k}},&\mbox{if}\
\ i\neq k;\\\ \\\ -\frac{N(N+1)}{4},&\mbox{if}\ \ i=k=0;\\\ \\\
\frac{N(N+1)}{4},&\mbox{if}\ \ i=k=N;\\\ \\\
0,&\mbox{otherwise}\end{array}\right.$
The cost functional $J[x(\cdot),u(\cdot)]$ is approximated by the Gauss-
Lobatto integration rule,
$\displaystyle J[x(\cdot),u(\cdot)]\ \approx\
\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N}F(\bar{x}^{Nk},\bar{u}^{Nk})w_{k}+E(\bar{x}^{N0},\bar{x}^{NN})$
where $w_{k}$ are the LGL weights defined by
$\displaystyle w_{k}$ $\displaystyle=$
$\displaystyle\frac{2}{N(N+1)}\frac{1}{[L_{N}(t_{k})]^{2}},$
The approximation is so accurate that it has zero error if the integrand
function is a polynomial of degree less than or equal to $2N-1$, a degree that
is almost a double of the number of nodes [4]. Now, we are ready to define
Problem ${\rm B}^{\rm N}$, a PS discretization of Problem B.
For any integer $m_{1}>0$, let
$\\{a_{0}^{N}(m_{1}),a_{1}^{N}(m_{1}),\cdots,a_{N-r-m_{1}+1}^{N}(m_{1})\\}$
denote the coefficients in the Legendre polynomial expansion for the
interpolation polynomial of the vector $\bar{x}_{r}^{N}(D^{T})^{m_{1}}$. Note
that the interpolation of $\bar{x}_{r}^{N}(D^{T})^{m_{1}}$ equals the
polynomial of $\frac{d^{m_{1}}x_{r}^{N}(t)}{dt^{m_{1}}}$. Thus, there are only
$N-r-m_{1}+2$ nonzero spectral coefficients because it is proved in Section 3
that the order of $\frac{d^{m_{1}}x^{N}_{r}(t)}{dt^{m_{1}}}$ is at most degree
of $N-r-m_{1}+1$. These coefficients depend linearly on $\bar{x}_{r}^{N}$ [3],
$\begin{array}[]{llllllllll}{\tiny\left[\begin{array}[]{cccccccccccccc}a^{N}_{0}(m_{1})\\\
\vdots\\\ a^{N}_{N-r-m_{1}+1}(m_{1})\end{array}\right]=}\\\
{\tiny\left[\begin{array}[]{cccccccccccccc}\frac{1}{2}&&\\\ &\ddots&\\\ &&N-r-
m_{1}+1+\frac{1}{2}\end{array}\right]\left[\begin{array}[]{cccccccccccccc}L_{0}(t_{0})&\cdots&L_{0}(t_{N})\\\
&\vdots&\\\ L_{N-r-m_{1}+1}(t_{0})&\cdots&L_{N-r-
m_{1}+1}(t_{N})\end{array}\right]\left[\begin{array}[]{cccccccccccccc}w_{0}&&\\\
&\ddots&\\\ &&w_{N}\\\
\end{array}\right]D^{m_{1}}\left[\begin{array}[]{cccccccccccccc}\bar{x}^{N0}_{r}\\\
\vdots\\\ \bar{x}^{NN}_{r}\\\ \end{array}\right]}\end{array}$ (2.16)
The PS discretization of Problem ${\rm B}^{\rm N}$is defined as follows.
Problem ${\bf B}^{\bf N}$: Find $\bar{x}^{Nk}\in\Re^{r}$ and
$\bar{u}^{Nk}\in\Re$, $k\ =\ 0,1,\ldots,N$, that minimize
$\displaystyle\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N}F(\bar{x}^{Nk},\bar{u}^{Nk})w_{k}+E(\bar{x}^{N0},\bar{x}^{NN})$
(2.17)
subject to
$\displaystyle\left\\{\begin{array}[]{rcl}D(\bar{x}_{1}^{N})^{T}&=&(\bar{x}_{2}^{N})^{T}\\\
D(\bar{x}_{2}^{N})^{T}&=&(\bar{x}_{3}^{N})^{T}\\\ &\vdots&\\\
D(\bar{x}_{r-1}^{N})^{T}&=&(\bar{x}_{r}^{N})^{T}\\\
D(\bar{x}_{r}^{N})^{T}&=&\left[\begin{array}[]{cccccccccccccc}f(\bar{x}^{N0})+g(\bar{x}^{N0})\bar{u}^{N0}\\\
\vdots\\\ f(\bar{x}^{NN})+g(\bar{x}^{NN})\bar{u}^{NN}\end{array}\right]\\\
\end{array}\right.$ (2.26) $\displaystyle\bar{x}^{N0}=x_{0}$ (2.27)
$\displaystyle\underline{{\boldsymbol{b}}}\leq\left[\begin{array}[]{cccccccccccccc}\bar{x}^{Nk}\\\
\bar{u}^{Nk}\end{array}\right]\ \leq\ \bar{\boldsymbol{b}},\;\;\;\;\mbox{ for
all }0\leq k\leq N$ (2.30)
$\displaystyle\underline{{\boldsymbol{b}}}_{j}\leq\left[\begin{array}[]{cccccccccccccc}1&0&\cdots&0\end{array}\right]D^{j}(\bar{x}_{r}^{N})^{T}\
\leq\bar{\boldsymbol{b}}_{j},\mbox{ if }1\leq j\leq m_{1}-1\mbox{ and
}m_{1}\geq 2$ (2.32) $\displaystyle\displaystyle\sum_{n=0}^{N-r-
m_{1}+1}|a^{N}_{n}(m_{1})|\leq{\boldsymbol{d}}$ (2.33)
Comparing to Problem B, (2.26) is the discretization of the control system
defined by the differential equation. The regularization condition (2.33)
assures that the derivative of the interpolation up to the order of $m_{1}$ is
bounded. It is proved in the following sections that the integer $m_{1}$ is
closely related to the convergence rate. The inequalities (2.30), (2.32) and
(2.33) are regularization conditions that do not exist in Problem B. It is
proved in the next few sections that these additional constraints do not
affect the feasibility of Problem ${\rm B}^{\rm N}$. Therefore, it does not
put an extra limit to the family of problems to be solved.
In searching for a discrete optimal solution, it is standard for software
packages of nonlinear programming to require a search region. Typically, the
search region is defined by a constraint (2.30). However, this box-shaped
region may contain solutions that are not good approximations of the
continuous-time solution. To guarantee the rate of convergence, the search
region is refined to a smaller one by imposing constraints (2.32) and (2.33).
It is proved in this paper that there always exist feasible solutions that
satisfy all the constraints and the optimal cost converges, provided the upper
and lower bounds are large enough. In (2.32),
$\underline{{\boldsymbol{b}}}_{j}$ and $\bar{\boldsymbol{b}}_{j}$ represent
the bounds of initial derivatives. In (2.33), ${\boldsymbol{d}}$ is the bound
determined by $x_{r}^{(m_{1}+1)}$ satisfying the inequality (3.6). Without
known the optimal solution, these bounds of search region have to be estimated
before computation or they are determined by numerical experimentations. The
constraints (2.32) and (2.33) are necessary to avoid the restrictive
consistent approximation assumption made in [7]. At a more fundamental level,
the order of derivatives, $m_{1}$ in (2.33), determines the convergence rate
of the approximate optimal control. Another interesting fact that amply
justify these additional constraints is that Problem ${\rm B}^{\rm N}$ may not
even have an optimal solution if we do not enforce (2.30). This is shown by
the following counter example.
###### Example 1
Consider the following problem of optimal control.
$\displaystyle\min_{(x(\cdot),u(\cdot))}\int_{-1}^{1}\displaystyle\frac{(x(t)-u(t))^{2}}{u(t)^{4}}dt$
$\displaystyle\dot{x}=u$ (2.34) $\displaystyle x(-1)=e^{-1}$
It is easy to check that the optimal solution is
$\displaystyle u=e^{t},$ $\displaystyle x(t)=e^{t}$
and the optimal cost value is zero. Although the solution to the problem
(2.34) is simple and analytic, the PS discretization of (2.34) does not have
an optimal solution if the constraint (2.30) is not enforced. To prove this
claim, consider the PS discretization,
$\displaystyle\min_{(\bar{x}^{N},\bar{u}^{N})}\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})=\displaystyle\sum_{k=0}^{N}\displaystyle\frac{(\bar{x}^{Nk}-D_{k}(\bar{x}^{N})^{T})^{2}}{\left(D_{k}(\bar{x}^{N})^{T}\right)^{4}}w_{k}$
$\displaystyle D(\bar{x}^{N})^{T}=(\bar{u}^{N})^{T}$ (2.35)
$\displaystyle\bar{x}^{N0}=e^{-1}$
where $D_{k}$ is the $k$th row of the differentiation matrix $D$. Let
$x^{N}(t)$ be the interpolation polynomial of $\bar{x}^{N}$, then it is
obvious that
$x^{N}(t)-\dot{x}^{N}(t)\not\equiv 0$
Thus, there exists $k$ so that
$\bar{x}^{Nk}-D_{k}(\bar{x}^{N})^{T}\neq 0$
So,
$\begin{array}[]{llllllllll}\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})>0\end{array}$
(2.36)
for all feasible pairs $(\bar{x}^{N},\bar{u}^{N})$. For any $\alpha>0$, define
$\bar{x}^{Nk}=e^{-1}+\alpha(t_{k}+1)$
The interpolation of $\bar{x}^{N}$ is the linear polynomial
$x^{N}(t)=e^{-1}+\alpha(t+1)$
Then,
$D_{k}(\bar{x}^{N})^{T}=\dot{x}^{N}(t_{k})=\alpha$
The cost function is
$\displaystyle\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{k=0}^{N}\displaystyle\frac{(e^{-1}+\alpha(t_{k}+1)-\alpha)^{2}}{\alpha^{4}}w_{k}$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{k=0}^{N}\displaystyle\frac{(e^{-1}+\alpha
t_{k})^{2}}{\alpha^{4}}w_{k}$ $\displaystyle\leq$
$\displaystyle\displaystyle\sum_{k=0}^{N}\displaystyle\frac{(e^{-1}+\alpha)^{2}}{\alpha^{4}}w_{k}$
$\displaystyle=$ $\displaystyle
2\displaystyle\frac{(e^{-1}+\alpha)^{2}}{\alpha^{4}}$
Therefore, $\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})$ can be arbitrarily small as
$\alpha$ approaches $\infty$. However, $\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})$
is always positive as shown by (2.36). We conclude that the discretization
(2.35) has no minimum value for $\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})$.
## 3 Convergence Rate
Given a solution to Problem ${\rm B}^{\rm N}$, we use (2.9) to approximate the
optimal control. In this section we prove that, under this approximate optimal
control, the value of the cost function converges to the optimal cost of
Problem B as the number of nodes is increased. More importantly, we can prove
a high-order rate of convergence. In the literature, it has been proved that
PS methods have a spectral rate when approximating $C^{\infty}$ functions,
i.e. the rate is faster than any polynomial rate. However, there are no
results in the literature thus far on the convergence rate of PS optimal
control. Meanwhile, in many problems solved by PS optimal control we clearly
observed a rate of high-order in the convergence. In this section, we prove a
convergence rate that depends on the smoothness of the optimal control. More
specifically, the rate is about $\frac{1}{N^{2m/3-1}}$, where $m$ is defined
by the smoothness of the optimal trajectory. If the cost function can be
accurately computed, then the convergence rate is improved to
$\frac{1}{N^{2m-1}}$. In the special case of $C^{\infty}$, it is proved that
PS method is able to converge faster than any given polynomial rate. Before we
introduce the main theorems of this section, the following example in [7] is
briefly presented to show the rapid convergence of the PS optimal control
method.
###### Example 2
Consider the following nonlinear optimal control problem:
$\displaystyle\left\\{\begin{array}[]{lrl}{\rm
Minimize}&J[x(\cdot),u(\cdot)]=&4x_{1}(2)+x_{2}(2)+4\displaystyle\int_{0}^{2}u^{2}(t)\
dt\\\ {\rm Subject\ to}&\dot{x}_{1}(t)=&x_{2}^{3}(t)\\\
&\dot{x}_{2}(t)=&u(t)\\\ &(x_{1}(0),x_{2}(0))=&(0,1)\end{array}\right.$ (3.5)
It can be shown that the exact optimal control is defined by
$u^{\ast}(t)=-\frac{8}{(2+t)^{3}}$. For this problem, the PS method achieves
the accuracy in the magnitude of $10^{-8}$ with only 18 nodes. A detail
comparison of the PS method with some other discretization methods are
addressed in [7]. From Figure 2 in logarithmically scaled coordinates, it is
obvious that the computation using the PS method converges exponentially.
Figure 2: Error vs number of the nodes for the pseudospectral method
Problem ${\rm B}^{\rm N}$ has several bounds in its definition,
$\underline{{\boldsymbol{b}}}$, $\bar{\boldsymbol{b}}$,
$\underline{{\boldsymbol{b}}}_{j}$, $\bar{\boldsymbol{b}}_{j}$, and
${\boldsymbol{d}}$. These bounds can be selected from a range determined by
Problem B. The constraints $\underline{{\boldsymbol{b}}}$ and
$\bar{\boldsymbol{b}}$ are lower and upper bounds so that the optimal
trajectory of Problem B is contained in the interior of the region. Suppose
Problem B has an optimal solution $(x^{\ast}(t),u^{\ast}(t))$ in which
$(x_{r}^{\ast}(t))^{(m)}$ has bounded variation for some $m\geq 3$, where
$x_{r}^{\ast}(t)$ is the $r$th component of the optimal trajectory. Suppose
$m_{1}$ in Problem ${\rm B}^{\rm N}$ satisfies $2\leq m_{1}\leq m-1$. Then, we
can select the bounds $\underline{{\boldsymbol{b}}}_{j}$ and
$\bar{\boldsymbol{b}}_{j}$ so that $(x_{r}^{\ast}(t))^{(j)}$ is contained in
the interior of the region. For ${\boldsymbol{d}}$, we assume
$\displaystyle{\boldsymbol{d}}$ $\displaystyle>$
$\displaystyle\displaystyle\frac{6}{\sqrt{\pi}}(U(\left.x_{r}^{\ast}\right.^{(m_{1}+1)})+V(\left.x_{r}^{\ast}\right.^{(m_{1}+1)}))\zeta(3/2)$
(3.6)
where $U(\left.x_{r}^{\ast}\right.^{(m_{1}+1)})$ is the upper bound and
$V(\left.x_{r}^{\ast}\right.^{(m_{1}+1)})$ is the total variation of
$\left.x_{r}^{\ast}\right.^{(m_{1}+1)}(t)$; and $\zeta(s)$ is the $\zeta$
function defined by
$\displaystyle\zeta(s)=\sum_{k=1}^{\infty}\displaystyle\frac{1}{k^{s}}$ (3.7)
If all the bounds are selected as above, then it is proved in Section 4 that
Problem ${\rm B}^{\rm N}$ is always feasible provided $m\geq 2$. Note that in
practical computation, $\underline{{\boldsymbol{b}}}$, $\bar{\boldsymbol{b}}$,
$\underline{{\boldsymbol{b}}}_{j}$, $\bar{\boldsymbol{b}}_{j}$, and
${\boldsymbol{d}}$ are unknown. They must be estimated based upon experience
or other information about the system.
###### Theorem 1
Suppose Problem B has an optimal solution $(x^{\ast}(t),u^{\ast}(t))$ in which
the $m$-th order derivative $(x_{r}^{\ast}(t))^{(m)}$ has a bounded variation
for some $m\geq 3$. In Problem ${\rm B}^{\rm N}$, select $m_{1}$ and $\alpha$
so that $1\leq m_{1}\leq m-1$ and $0<\alpha<m_{1}-1$. Suppose $f(\cdot)$,
$g(\cdot)$, $F(\cdot)$, and $E(x_{0},\cdot)$ are $C^{m}$ and globally
Lipschitz. Suppose all other bounds in Problem ${\rm B}^{\rm N}$ are large
enough. Given any sequence
$\begin{array}[]{llllllllll}\\{(\bar{x}^{\ast N},\bar{u}^{\ast N})\\}_{N\geq
N_{1}}\end{array}$ (3.8)
of optimal solutions of Problem ${\rm B}^{\rm N}$. Then the approximate cost
converge to the optimal value at the following rate
$\displaystyle\left|J(x^{\ast}(\cdot),u^{\ast}(\cdot))-J(x^{\ast
N}(\cdot),u^{\ast N}(\cdot))\right|$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{M_{1}}{(N-r-
m_{1}-1)^{2m-2m_{1}-1}}+\displaystyle\frac{M_{2}}{N^{\alpha}}$ (3.9)
$\displaystyle\left|J(x^{\ast}(\cdot),u^{\ast}(\cdot))-\bar{J}^{N}(\bar{x}^{\ast
N},\bar{u}^{\ast N})\right|$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{M_{1}}{(N-r-
m_{1}-1)^{2m-2m_{1}-1}}+\displaystyle\frac{M_{2}}{N^{\alpha}}$ (3.10)
where $M_{1}$ and $M_{2}$ are some constants independent of $N$. In (3.9),
$(x^{\ast N}(t),u^{\ast N}(t))$ is the interpolation of (3.8) defined by
(2.8)-(2.9). In fact, $x^{\ast N}(t)$ is the trajectory of (2.6) under the
control input $u^{\ast N}(t)$.
Theorem 1 implies that the costs of any sequence of discrete optimal solutions
must converge to the optimal cost of Problem B, no matter the sequence of the
discrete state and control trajectories converge or not. In other words, it is
possible that the sequence of discrete optimal controls does not converge to a
unique continuous-time control; meanwhile the costs using these approximate
optimal controls converge to the true optimal cost of Problem B. Therefore,
this theorem does not require the local uniqueness of solutions for Problem B.
This is different from many existing convergence theorems of computational
optimal control, in which a unique optimal solution and coercivity are
assumed. This is made possible because the proofs in this paper do not rely on
the necessary conditions of optimal control. The key idea in the proof is to
shape the search region in Problem ${\rm B}^{\rm N}$by regulating the discrete
solutions using (2.30)-(2.32)-(2.33). We would like to emphasize that the
regulation constraints are added to the discretized problem, not the original
Problem B. So, the constraints do not restrict the problem to be solved, and
they do not need to be verified before computation. In addition, increasing
the number of constraints results in smaller search region for an optimal
solution.
###### Remark 3.1
If $f(\cdot)$, $g(\cdot)$, $F(\cdot)$ and $x_{r}^{\ast}(t)$ are $C^{\infty}$,
then we can select $m$ and $m_{1}$ arbitrarily large. In this case, we can
make the optimal cost of Problem ${\rm B}^{\rm N}$converge faster than any
given polynomial rate.
###### Remark 3.2
From (3.9) and (3.10), the convergence rate is determined by $m$, the
smoothness of the optimal trajectory, and $m_{1}$, the order in the regulation
of discrete solutions. While $m$ is a property of Problem B that cannot be
changed, $m_{1}$ in Problem ${\rm B}^{\rm N}$ can be selected within a range.
However, the errors in (3.9) and (3.10) have two parts, one is an increasing
function of $m_{1}$ and the other is a decreasing function of $m_{1}$. In
Corollary 1, we show an optimal selection of $m_{1}$ to maximize the combined
convergence rate.
The proof is convoluted involving results from several different areas,
including nonlinear functional analysis, orthogonal polynomials, and
approximation theory. First, we introduce the concept of Fréchet derivative.
Let us consider the continuous cost function, $J(x(\cdot),u(\cdot))$, subject
to (2.6)-(2.7) as a nonlinear functional of $u(\cdot)$, denoted by ${\cal
J}(u)$. For any $u$ in the Banach space $W^{m-1,\infty}$, suppose there exists
a linear bounded operator ${\cal L}$: $W^{m-1,\infty}\rightarrow\Re$ such that
$|{\cal J}(u+\Delta u)-{\cal J}(u)-L\Delta u|=o(||\Delta
u||_{W^{m-1,\infty}})$
for all $u+\Delta u$ in an open neighborhood of $u$ in $W^{m-1,\infty}$. Then,
${\cal L}$ is called the Fréchet derivative of ${\cal J}(u)$ at $u$, denoted
by ${\cal J}^{\prime}(u)=\cal L$. If ${\cal J}^{\prime}(u)$ exists at all
points in an open subset of $W^{m-1,\infty}$, then ${\cal J}^{\prime}(u)$ is a
functional from this open set to the Banach space $L(W^{m-1,\infty},\Re)$ of
all bounded linear operators. If this new functional has a Fréchet derivative,
then it is called the second order Fréchet derivative, denoted by ${\cal
J}^{\prime\prime}(u)$. The following lemma is standard in nonlinear functional
analysis [21].
###### Lemma 1
Suppose ${\cal J}$ takes a local minimum value at $u^{\ast}$. Suppose ${\cal
J}$ has second order Fréchet derivative at $u^{\ast}$. Then,
${\cal J}(u^{\ast}+\Delta u)=({\cal J}^{\prime\prime}(u^{\ast})\Delta u)\Delta
u+o(||\Delta u||^{2})$
The rate of convergence for the spectral coefficients can be estimated by the
following Jackson’s Theorem.
###### Lemma 2
(Jackson’s Theorem [20]) Let $h(t)$ be of bounded variation in $[-1,1]$.
Define
$H(t)=H(-1)+\displaystyle{\int_{-1}^{t}}h(s)ds$
then $\\{a_{n}\\}_{n=0}^{\infty}$, the sequence of spectral coefficients of
$H(t)$, satisfies the following inequality
$a_{n}<\displaystyle\frac{6}{\sqrt{\pi}}(U(h(t))+V(h(t)))\displaystyle\frac{1}{n^{3/2}}$
for $n\geq 1$.
Given a continuous function $h(t)$ defined on $[-1,1]$. Let $\hat{p}^{N}(t)$
be the best polynomial of degree $N$, i.e. the $N$th order polynomial with the
smallest distance to $h(t)$ under $||\cdot||_{\infty}$ norm. Let $I_{N}h(t)$
be the polynomial interpolation using the value of $h(t)$ at the LGL nodes.
Then, we have the following inequality from the theory of approximation and
orthogonal polynomials [4], [11].
###### Lemma 3
$||h(t)-I_{N}h||_{\infty}\leq(1+\Lambda_{N})||h(t)-\hat{p}^{N}(t)||_{\infty}$
where $\Lambda_{N}$ is called Lebesgue constant. It satisfies
$\Lambda_{N}\leq\displaystyle\frac{2}{\pi}log(N+1)+0.685\cdots$
The best polynomial approximation represents the closest polynomial to a
function under $||\cdot||_{\infty}$. The error can be estimated by the
following Lemma [4].
###### Lemma 4
(1) Suppose $h(t)\in W^{m,\infty}$. Let $\hat{p}^{N}(t)$ be the best
polynomial approximation. Then
$||\hat{p}^{N}(t)-h(t)||_{\infty}\leq\displaystyle\frac{C}{N^{m}}||h(t)||_{W^{m,\infty}}$
for some constant $C$ independent of $h(t)$, $m$ and $N$.
(2) If $h(t)\in W^{m,2}$, then
$||h(t)-P_{N}h(t)||_{\infty}\leq\displaystyle\frac{C||h(t)||_{W^{m,2}}}{N^{m-3/4}}$
where $P_{N}h$ is the N-th order truncation of the Legendre series of $h(t)$.
(3) If $h(t)$ has the $m$-th order strong derivative with a bounded variation,
then
$||h(t)-P_{N}h(t)||_{\infty}\leq\displaystyle\frac{CV(h^{(m)}(t))}{N^{m-1/2}}$
The following lemmas are proved specifically for PS optimal control methods.
Similar results can be found in [14] except that some assumptions on $m_{1}$
are relaxed.
###### Lemma 5
([14]) (i) For any trajectory, $(\bar{x}^{N},\bar{u}^{N})$, of the dynamics
(2.26), the pair $(x^{N}(t),u^{N}(t))$ defined by (2.8)-(2.9) satisfies the
differential equations defined in (2.6). Furthermore,
$\displaystyle\bar{x}^{Nk}=x^{N}(t_{k}),\;\bar{u}^{Nk}=u^{N}(t_{k}),\;\mbox{for
}k=0,1,\cdots,N$ (3.11)
(ii) For any pair $(x^{N}(t),u^{N}(t))$ in which $x^{N}(t)$ consists of
polynomials of degree less than or equal to $N$ and $u^{N}(t)$ is a function,
if $(x^{N}(t),u^{N}(t))$ satisfies the differential equations in (2.6), then
$(\bar{x}^{N},\bar{u}^{N})$ defined by (3.11) satisfies (2.26).
(iii) If $(\bar{x}^{N},\bar{u}^{N})$ satisfies (2.26), then the degree of
$x_{i}^{N}(t)$ is less than or equal to $N-i+1$.
Proof. (i) Suppose $(\bar{x}^{N},\bar{u}^{N})$ satisfies the equations in
(2.26). Because $x^{N}(t)$ is the polynomial interpolation of $\bar{x}^{N}$,
and because of equations (2.14), we have
$\begin{array}[]{llllllllll}\left[\begin{array}[]{cccccccccccccc}\dot{x}_{i}^{N}(t_{0})&\dot{x}_{i}^{N}(t_{1})&\cdots&\dot{x}_{i}^{N}(t_{N})\end{array}\right]\\\
=\bar{x}_{i}^{N}D^{T}\\\ =\bar{x}_{i+1}^{N}\\\
=\left[\begin{array}[]{cccccccccccccc}x_{i+1}^{N}(t_{0})&x_{i+1}^{N}(t_{1})&\cdots&x_{i+1}^{N}(t_{N})\end{array}\right]\end{array}$
Therefore, the polynomials $\dot{x}^{N}_{i}(t)$ and $x^{N}_{i+1}(t)$ must
equal each other because they coincide at $N+1$ points and because the degrees
of $x_{i}^{N}(t)$ and $x_{i+1}^{N}(t)$ are less than or equal to $N$. In
addition, (2.9), the definition of $u^{N}(t)$, implies the last equation in
(2.6). So, the pair $(x^{N}(t),u^{N}(t))$ satisfies all equations in (2.6).
Now, we prove (3.11). Because $x^{N}(t)$ is an interpolation of $\bar{x}^{N}$,
we know $\bar{x}^{Nk}=x^{N}(t_{k})$ for $0\leq k\leq N$. From (2.9),
$\displaystyle u^{N}(t_{k})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\dot{x}_{r}^{N}(t_{k})-f(x^{N}(t_{k}))}{g(x^{N}(t_{k}))}$
(3.12) $\displaystyle=$
$\displaystyle\displaystyle\frac{\dot{x}^{N}_{r}(t_{k})-f(\bar{x}^{Nk})}{g(\bar{x}^{Nk})}$
Because of (2.14), we have
$\left[\begin{array}[]{cccccccccccccc}\dot{x}_{r}^{N}(t_{0})&\dot{x}_{r}^{N}(t_{1})&\cdots&\dot{x}_{r}^{N}(t_{N})\end{array}\right]^{T}=D(\bar{x}_{r}^{N})^{T}$
Therefore, (3.12) is equivalent to
$\displaystyle\left[\begin{array}[]{cccccccccccccc}u^{N}(t_{0})&u^{N}(t_{1})&\cdots&u^{N}(t_{N})\end{array}\right]^{T}$
$\displaystyle=$
$\displaystyle\mbox{diag}\left(\displaystyle\frac{1}{g(\bar{x}^{N0})},\cdots,\displaystyle\frac{1}{g(\bar{x}^{NN})}\right)\left(D(\bar{x}_{r}^{N})^{T}-\left[\begin{array}[]{cccccccccccccc}f(\bar{x}^{N0})\\\
\vdots\\\ f(\bar{x}^{NN})\end{array}\right]\right)$
Comparing to the last equation in (2.26), it is obvious that
$u^{N}(t_{k})=\bar{u}^{Nk}$. So, (3.11) holds true. Part (i) is proved.
(ii) Assume $(x^{N}(t),u^{N}(t))$ satisfies the differential equations in
(2.6). Because $x^{N}(t)$ are polynomials, (2.14) implies
$\displaystyle\bar{x}_{i}^{N}D^{T}$
$\displaystyle=\left[\begin{array}[]{cccccccccccccc}\dot{x}_{i}^{N}(t_{0})&\dot{x}_{i}^{N}(t_{1})&\cdots&\dot{x}_{i}^{N}(t_{N})\end{array}\right]$
(3.15)
$\displaystyle=\left[\begin{array}[]{cccccccccccccc}x_{i+1}^{N}(t_{0})&x_{i+1}^{N}(t_{1})&\cdots&x_{i+1}^{N}(t_{N})\end{array}\right]$
(3.17) $\displaystyle=\bar{x}_{i+1}^{N}$
Furthermore,
$\displaystyle\bar{x}^{N}_{r}D^{T}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccccccccccc}\dot{x}_{r}^{N}(t_{0})&\dot{x}_{r}^{N}(t_{1})&\cdots&\dot{x}_{r}^{N}(t_{N})\end{array}\right]$
(3.19) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccccccccccc}f(x^{N}(t_{0}))+g(x^{N}(t_{0}))u^{N}(t_{0})&\cdots&f(x^{N}(t_{N}))+g(x^{N}(t_{N}))u^{N}(t_{N})\end{array}\right]$
(3.21)
Equations (3.15) and (3.19) imply that $(\bar{x}^{N},\bar{u}^{N})$ satisfies
(2.26). Part (ii) is proved.
(iii) We know that the degree of $x_{1}^{N}(t)$, the interpolation polynomial,
is less than or equal to $N$. From (i), we know
$x_{2}^{N}(t)=\dot{x}_{1}^{N}(t)$. Therefore, the degree of $x_{2}^{N}(t)$
must be less than or equal to $N-1$. In general, the degree of $x_{i}^{N}(t)$
is less than or equal to $N-i+1$. $\Box$
###### Lemma 6
Suppose $\\{(\bar{x}^{N},\bar{u}^{N})\\}_{N=N_{1}}^{\infty}$ is a sequence
satisfying (2.26), (2.30), (2.32) and (2.33), where $m_{1}\geq 1$. Then,
$\displaystyle\left\\{\left.||(x^{N}(t))^{(l)}||_{\infty}\right|N\geq
N_{1},\,l=0,1,\cdots,m_{1}\right\\}$
is bounded. If $f(x)$ and $g(x)$ are $C^{m_{1}-1}$, then
$\displaystyle\left\\{\left.||(u^{N}(t))^{(l)}||_{\infty}\right|N\geq
N_{1},\,l=0,1,\cdots,m_{1}-1\right\\}$
is bounded.
Proof. Consider $(x_{r}^{N}(t))^{(m_{1})}$. From Lemma 5, it is a polynomial
of degree less than or equal to $N-r-m_{1}+1$. Therefore,
$\displaystyle(x_{r}^{N}(t))^{(m_{1})}=\displaystyle\sum_{n=0}^{N-r-
m_{1}+1}a_{n}^{N}(m_{1})L_{n}(t)$
where $L_{n}(t)$ is the Legendre polynomial of degree $n$. It is known that
$|L_{n}(t)|\leq 1$. Therefore, (2.33) implies that
$||(x_{r}^{N}(t))^{(m_{1})}||_{\infty}$ is bounded by ${\boldsymbol{d}}$ for
all $N\geq N_{1}$. Now, let us consider $(x_{r}^{N}(t))^{(m_{1}-1)}$. From
(2.14) we have,
$\displaystyle(x_{r}^{N}(t))^{(m_{1}-1)}$ $\displaystyle=$
$\displaystyle(x_{r}^{N}(t))^{(m_{1}-1)}|_{t=-1}+\int_{0}^{t}(x_{r}^{N}(s))^{(m_{1})}ds$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccccccccccc}1&0&\cdots&0\end{array}\right]D^{m_{1}-1}(\bar{x}_{r}^{N})^{T}+\int_{0}^{t}(x_{r}^{N}(s))^{(m_{1})}ds$
So, $||(x_{r}^{N}(t))^{(m_{1}-1)}||_{\infty}$, $N\geq N_{1}$, is bounded
because of (2.32). Similarly, we can prove all derivatives of $x_{r}^{N}(t)$
of order less than $m_{1}$ are bounded. The same approach can also be applied
to prove the bound
$u^{N}(t)=\displaystyle\frac{\dot{x}_{r}^{N}(t)-f(x^{N}(t))}{g(x^{N}(t))}$
Because $f(x)$ and $g(x)$ have continuous derivatives of order less than or
equal to $m_{1}-1$, the boundedness of
$\left\\{\left.||(u^{N}(t))^{(l)}||_{\infty}\right|N\geq
N_{1},\,j=0,1,\cdots,m_{1}-1\right\\}$
follows the boundedness of $(x_{r}^{N}(t))^{(l)}$ proved above. $\Box$
Given any function $h(t)$ defined on $[-1,1]$. In the following, $U(h)$
represents an upper bound of $h(t)$ and $V(h)$ represents the total variation.
###### Lemma 7
Let $(x(t),u(t))$ be a solution of the differential equation (2.6). Suppose
$x_{r}^{(m)}(t)$ has bounded variation for some $m\geq 2$. Let $m_{1}$ be an
integer satisfying $1\leq m_{1}\leq m-1$. Then, there exist constants $M>0$
and $N_{1}>0$ so that for each integer $N\geq N_{1}$ the differential equation
(2.6) has a solution $(x^{N}(t),u^{N}(t))$ in which $x^{N}(t)$ consists of
polynomials of degree less than or equal to $N$. Furthermore, the pair
$(x^{N}(t),u^{N}(t))$ satisfies
$\displaystyle||x_{i}^{N}(t)-x_{i}(t)||_{\infty}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{M||x_{r}||_{W^{m,2}}}{(N-r-
m_{1}+1)^{(m-m_{1})-3/4}},\;\;\;i=1,2,\cdots,r$ (3.23)
$\displaystyle||(x_{r}^{N}(t))^{(l)}-(x_{r}(t))^{(l)}||_{\infty}$
$\displaystyle\leq$
$\displaystyle\displaystyle\frac{M||x_{r}||_{W^{m,2}}}{(N-r-
m_{1}+1)^{(m-m_{1})-3/4}},\;\;\;\;l=1,2,\cdots,m_{1}$ (3.24)
$\displaystyle||u^{N}(t)-u(t)||_{\infty}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{M||x_{r}||_{W^{m,2}}}{(N-r-
m_{1}+1)^{(m-m_{1})-3/4}}$ (3.25)
Furthermore, the spectral coefficients of $(x^{N}_{r})^{(m_{1})}(t)$ satisfy
$\displaystyle|a^{N}_{n}(m_{1})|\leq\displaystyle\frac{6(U(x^{(m_{1}+1)}_{r})+V(x^{(m_{1}+1)}_{r}))}{\sqrt{\pi}n^{3/2}},\;\;n=1,2,\cdots,N-r-1$
(3.26)
If $f(x)$ and $g(x)$ have Lipschitz continuous $L$th order partial derivatives
for some $L\leq m_{1}-1$, then
$\displaystyle||(u^{N}(t))^{(l)}-(u(t))^{(l)}||_{\infty}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{M||x_{r}||_{W^{m,2}}}{(N-r-
m_{1}+1)^{(m-m_{1})-3/4}},\;\;\;l=1,\cdots,L$ (3.27)
Furthermore,
$\displaystyle\begin{array}[]{lll}x^{N}(-1)=x(-1)\\\ u^{N}(-1)=u(-1),&\mbox{
If }m_{1}\geq 2\end{array}$ (3.30)
###### Remark 3.3
In this lemma, if $x_{r}(t)$ has the $m$-th order strong derivative and if
$x_{r}^{(m)}(t)$ has bounded variation for some $m\geq 2$, then the
inequalities (3.23), (3.24), and (3.25) are slightly tighter.
$\displaystyle||x_{i}^{N}(t)-x_{i}(t)||_{\infty}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{M||x_{r}||_{W^{m,2}}}{(N-r-
m_{1}+1)^{(m-m_{1})-1/2}},\;\;\;i=1,2,\cdots,r$ (3.31)
$\displaystyle||(x_{r}^{N}(t))^{(l)}-(x_{r}(t))^{(l)}||_{\infty}$
$\displaystyle\leq$
$\displaystyle\displaystyle\frac{M||x_{r}||_{W^{m,2}}}{(N-r-
m_{1}+1)^{(m-m_{1})-1/2}},\;\;\;\;l=1,2,\cdots,m_{1}$ (3.32)
$\displaystyle||u^{N}(t)-u(t)||_{\infty}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{M||x_{r}||_{W^{m,2}}}{(N-r-
m_{1}+1)^{(m-m_{1})-1/2}}$ (3.33)
The proof is identical as that of Lemma 7 except that the error estimation in
(3) of Lemma 4 is used.
Proof. Consider the Legendre series
$(x_{r})^{(m_{1})}(t)\sim\displaystyle\sum_{n=0}^{N-r-
m_{1}+1}a_{n}^{N}(m_{1})L_{n}(t)$
A sequence of polynomials $x_{1}^{N}(t),\cdots,x_{r+m_{1}}^{N}(t)$ is defined
as follows,
$\displaystyle x_{r+m_{1}}^{N}(t)$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{n=0}^{N-r-m_{1}+1}a_{n}^{N}(m_{1})L_{n}(t)$
$\displaystyle x_{r+m_{1}-1}^{N}(t)$ $\displaystyle=$
$\displaystyle(x_{r})^{(m_{1}-1)}(-1)+\displaystyle{\int_{-1}^{t}}x_{r+m_{1}}^{N}(s)ds$
$\displaystyle\vdots$ $\displaystyle x_{r+1}^{N}(t)$ $\displaystyle=$
$\displaystyle\dot{x}_{r}(-1)+\displaystyle{\int_{-1}^{t}}x_{r+2}^{N}(s)ds$
and
$\displaystyle x_{i}^{N}(t)$ $\displaystyle=$ $\displaystyle
x_{i}(-1)+\displaystyle{\int_{-1}^{t}}x_{i+1}^{N}(s)ds,\;\;\mbox{ for }1\leq
i\leq r$
Define
$x^{N}(t)=\left[\begin{array}[]{cccccccccccccc}x_{1}^{N}(t)&\cdots&x^{N}_{r}(t)\end{array}\right]^{T}$
and define
$u^{N}(t)=\displaystyle\frac{x_{r+1}^{N}(t)-f(x^{N}(t))}{g(x^{N}(t))}$
From the definition of $x^{N}(t)$, we have $x^{N}(-1)=x(-1)$. If $m_{1}\geq
2$, then $x_{r+1}(-1)=\dot{x}_{r}(-1)$. From the definition of $u^{N}(t)$, we
know $u^{N}(-1)=u(-1)$ provided $m_{1}\geq 2$. Therefore,
$(x^{N}(t),u^{N}(t))$ satisfies (3.30). It is obvious that $x_{i}^{N}(t)$ is a
polynomial of degree less than or equal to $N$; and $(x^{N}(t),u^{N}(t))$
satisfies the differential equation (2.6). Because we assume
$V(x_{r}^{(m)})<\infty$, we have $x_{r}^{(m)}\in L^{2}$. From Lemma 4
$\displaystyle||x_{r+m_{1}}^{N}(t)-x_{r}^{(m_{1})}(t)||_{\infty}$
$\displaystyle=$
$\displaystyle||x_{r}^{(m_{1})}(t)-\displaystyle\sum_{n=0}^{N-r-
m_{1}+1}a_{n}^{N}(m_{1})L_{n}(t)||_{\infty}$ $\displaystyle\leq$
$\displaystyle C_{1}||x_{r}||_{W^{m,2}}(N-r-m_{1}+1)^{-(m-m_{1})+3/4}$
for some constant $C_{1}>0$. Therefore,
$\displaystyle|x^{N}_{r+m_{1}-1}(t)-(x_{r})^{(m_{1}-1)}(t)|$
$\displaystyle\leq$
$\displaystyle\displaystyle{\int_{-1}^{t}}|x^{N}_{r+m_{1}}(s)-(x_{r})^{(m_{1})}(s)|ds$
$\displaystyle\leq$ $\displaystyle 2C_{1}||x_{r}||_{W^{m,2}}(N-r-
m_{1}+1)^{-(m-m_{1})+3/4}$
Similarly, we can prove (3.23) and (3.24).
To prove (3.26), note that the spectral coefficient $a^{N}_{n}(m_{1})$ of
$(x_{r}^{N})^{(m_{1})}(t)$ is the same as the spectral coefficients of
$(x_{r})^{(m_{1})}(t)$. From Jackson’s Theorem (Lemma 2), we have
$|a^{N}_{n}(m_{1})|<\displaystyle\frac{6}{\sqrt{\pi}}(U(x^{(m_{1}+1)}_{r})+V(x^{(m_{1}+1)}_{r}))\displaystyle\frac{1}{n^{3/2}}$
In a bounded set around $x(t)$, we have $g(x)>\alpha>0$ for some $\alpha>0$
because $f$ and $g$ are Lipschitz continuous (Definition of Problem B).
Therefore, the function
$\displaystyle\frac{s-f(x)}{g(x)}$
is Lipschitz in a neighborhood of $(x,s)$, i.e. there exists a constant
$C_{2}$ independent of $N$ such that
$\displaystyle|u^{N}(t)-u(t)|$ $\displaystyle=$
$\displaystyle\left|\frac{x_{r+1}^{N}(t)-f(x^{N}(t))}{g(x^{N}(t))}-\frac{\dot{x}_{r}(t)-f(x(t))}{g(x(t))}\right|$
(3.34) $\displaystyle\leq$ $\displaystyle
C_{2}(|x_{r+1}^{N}(t)-\dot{x}_{r}(t)|+|x_{1}^{N}(t)-x_{1}(t)|+\cdots+|x_{r}^{N}(t)-x_{r}(t)|)$
Hence, (3.25) follows (3.23), (3.24) and (3.34) when $l=0$. Similarly, we can
prove (3.27) for $l\leq L$. $\Box$
Now, only after this lengthy work of preparation, we are ready to prove
Theorem 1.
Proof of Theorem 1: Let $(x^{\ast}(t),u^{\ast}(t))$ be an optimal solution to
Problem B. According to Lemma 7 and Remark 3.3, for any positive integer $N$
that is large enough, there exists a pair of functions
$(\hat{x}^{N}(t),\hat{u}^{N}(t))$ in which $\hat{x}^{N}(t)$ consists of
polynomials of degree less than or equal to $N$. Furthermore, the pair
satisfies the differential equation with initial conditions in Problem B and
$\displaystyle||\hat{x}^{N}(t)-x^{\ast}(t)||_{\infty}$ $\displaystyle<$
$\displaystyle\displaystyle\frac{L}{(N-r-m_{1}+1)^{m-m_{1}-1/2}}$ (3.35)
$\displaystyle||\hat{u}^{N}(t)-u^{\ast}(t)||_{\infty}$ $\displaystyle<$
$\displaystyle\displaystyle\frac{L}{(N-r-m_{1}+1)^{m-m_{1}-1/2}}$ (3.36)
$\displaystyle||(\hat{x}_{r}^{N}(t))^{(l)}-(x^{\ast}_{r}(t))^{(l)}||_{\infty}$
$\displaystyle<$ $\displaystyle\displaystyle\frac{L}{(N-r-
m_{1}+1)^{m-m_{1}-1/2}},\;\;\;1\leq l\leq m_{1}$ (3.37)
If we define
$\hat{\bar{u}}^{Nk}=\hat{u}^{N}(t_{k}),\hat{\bar{x}}^{Nk}=\hat{x}^{N}(t_{k})$
Then $\\{(\hat{\bar{x}}^{N},\hat{\bar{u}}^{N})\\}$ satisfies (2.26) and (2.27)
(Lemma 5 and 7). Because $\hat{x}_{r}^{N}(t)$ is a polynomial of degree less
than or equal to $N$ and because of (2.14), we know
$(\hat{x}^{N}_{r}(t))^{(j)}$ equals the interpolation polynomial of
$\hat{\bar{x}}^{N}_{r}(D^{T})^{j}$. So,
$\left[\begin{array}[]{cccccccccccccc}1&0&\cdots&0\end{array}\right]D^{j}(\hat{\bar{x}}_{r}^{N})^{T}=(\hat{x}^{N}_{r}(t))^{(j)}|_{t=-1}$
Therefore, (3.37) implies (2.32) if the bounds
$\underline{{\boldsymbol{b}}}_{j}$ and ${\boldsymbol{b}}_{j}$ are large
enough. In addition, the spectral coefficients of
$\hat{\bar{x}}^{N}_{r}(D^{T})^{m_{1}}$ is the same as the spectral
coefficients of $(\hat{x}^{N}_{r}(t))^{(m_{1})}$. From (3.26), (3.7) and
(3.6), we have
$\displaystyle\sum_{n=0}^{N-r-m_{1}+1}|a^{N}_{n}(m_{1})|\leq{\boldsymbol{d}}$
So, the spectral coefficients of $(\hat{x}_{r}^{N})^{(m_{1})}$ satisfies
(2.33). Because we select $\underline{{\boldsymbol{b}}}$ and
$\bar{\boldsymbol{b}}$ large enough so that the optimal trajectory of the
original continuous-time problem is contained in the interior of the region,
then (3.35) and (3.36) imply (2.30) for $N$ large enough. In summary, we have
proved that $(\hat{\bar{x}}^{N},\hat{\bar{u}}^{N})$ is a discrete feasible
trajectory satisfying all constraints, (2.26)-(2.33), in Problem ${\rm B}^{\rm
N}$.
Given any bounded control input $u(\cdot)$, because the system is globally
Lipschitz, it uniquely determines the trajectory $x(\cdot)$ if the initial
state is fixed. Therefore, the cost $J(x^{\ast}(\cdot),u^{\ast}(\cdot))$ can
be considered as a functional, denoted by ${\cal J}(u)$. Because all the
functions in Problem B are $C^{m}$ with $m\geq 2$, we know that ${\cal J}(u)$
has second order Fréchet derivative. By Lemma 1
$\displaystyle|J(x^{\ast}(\cdot),u^{\ast}(\cdot))-J(\hat{x}^{N}(\cdot),\hat{u}^{N}(\cdot))|$
(3.38) $\displaystyle=$ $\displaystyle|{\cal J}(u^{\ast})-{\cal
J}(\hat{u}^{N})|$ $\displaystyle\leq$ $\displaystyle
C_{1}(||u^{\ast}-\hat{u}^{N}||_{W^{m_{1}-1,\infty}}^{2})$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{C_{2}}{(N-r-m_{1}+1)^{2m-2m_{1}-1}}$
for some constant numbers $C_{1}$ and $C_{2}$. The last inequality is from
(3.36).
Now, consider $F(\hat{x}^{N}(t),\hat{u}^{N}(t))$ as a function of $t$. Let
$F^{N}(t)$ represent the polynomial interpolation of this function at
$t=t_{0},t_{1},\cdots,t_{N}$. Let $\hat{p}(t)$ be the best polynomial
approximation of $F(\hat{x}^{N}(t),\hat{u}^{N}(t))$ under the norm of
$L^{\infty}[-1,1]$. Then we have
$\displaystyle|J(\hat{x}^{N}(\cdot),\hat{u}^{N}(\cdot))-\bar{J}^{N}(\hat{\bar{x}}^{N},\hat{\bar{u}}^{N})|$
(3.39) $\displaystyle=$
$\displaystyle|J(\hat{x}^{N}(\cdot),\hat{u}^{N}(\cdot))-\displaystyle\sum_{k=0}^{N}F(\hat{\bar{x}}^{Nk},\hat{\bar{u}}^{Nk})w_{k}-E(\hat{\bar{x}}^{N0},\hat{\bar{x}}^{NN})|$
$\displaystyle=$
$\displaystyle\left|\int_{-1}^{1}F(\hat{x}^{N}(t),\hat{u}^{N}(t))dt-\int_{-1}^{1}F^{N}(t)dt\right|$
$\displaystyle\leq$
$\displaystyle\int_{-1}^{1}|F(\hat{x}^{N}(t),\hat{u}^{N}(t))-F^{N}(t)|dt$
$\displaystyle\leq$ $\displaystyle
2(1+\Lambda_{N})||\hat{p}(t)-F(\hat{x}^{N}(t),\hat{u}^{N}(t))||_{\infty}$
where
$\begin{array}[]{llllllllll}\Lambda_{N}\leq\displaystyle\frac{2}{\pi}log(N+1)+0.685\cdots\end{array}$
(3.40)
is the Lebesgue constant. The inequality (3.39) is a corollary of Lemma 3.
Because $f(\cdot)$, $g(\cdot)$, and $F(\cdot)$ are $C^{m}$, it is known (Lemma
4) that the best polynomial approximation satisfies
$||\hat{p}(t)-F(\hat{x}^{N}(t),\hat{u}^{N}(t))||_{\infty}\leq\displaystyle\frac{C_{3}}{N^{m_{1}-1}}||F(\hat{x}^{N}(t),\hat{u}^{N}(t))||_{W^{m_{1}-1,\infty}}$
Because of Lemma 6,
$\\{||F(\hat{x}^{N}(t),\hat{u}^{N}(t))||_{W^{m_{1}-1,\infty}}|N\geq N_{1}\\}$
is bounded. Therefore,
$\displaystyle|J(\hat{x}^{N}(\cdot),\hat{u}^{N}(\cdot))-\bar{J}^{N}(\hat{\bar{x}}^{N},\hat{\bar{u}}^{N})|\leq\displaystyle\frac{(1+\Lambda_{N})C_{4}}{N^{m_{1}-1}}\leq\displaystyle\frac{C_{5}}{N^{\alpha}}$
(3.41)
for some constant numbers $C_{4}$ and $C_{5}$ independent of $N$ and any
$\alpha<m_{1}-1$. Let
$\begin{array}[]{llllllllll}\\{(\bar{x}^{\ast N},\bar{u}^{\ast
N})\\}_{N=N_{0}}^{\infty}\end{array}$ (3.42)
be a sequence of optimal discrete solutions. Its interpolation is denoted by
$(x^{\ast N}(t),u^{\ast N}(t))$. Then, similar to the derivation above, we can
prove
$\displaystyle|J(x^{\ast N}(\cdot),u^{\ast
N}(\cdot))-\bar{J}^{N}(\bar{x}^{\ast N},\bar{u}^{\ast N})|$
$\displaystyle\leq$ $\displaystyle 2(1+\Lambda_{N})||p^{N}(t)-F(x^{\ast
N}(t),u^{\ast N}(t))||_{\infty}$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{C_{6}(1+\Lambda_{N})}{N^{m_{1}-1}}||F(x^{\ast
N}(t),u^{\ast N}(t))||_{W^{m_{1}-1,\infty}}$
where $p^{N}(t)$ is the best polynomial approximation of $F(x^{\ast
N}(t),u^{\ast N}(t))$ with degree less than or equal to $N$. Because of Lemma
6, $||F(x^{\ast N}(t),u^{\ast N}(t))||_{W^{m_{1}-1,\infty}}|N\geq N_{1}\\}$ is
bounded. So
$\displaystyle|J(x^{\ast N}(\cdot),u^{\ast
N}(\cdot))-\bar{J}^{N}(\bar{x}^{\ast N},\bar{u}^{\ast
N})|\leq\displaystyle\frac{C_{7}}{N^{\alpha}}$ (3.44)
for some constant $C_{7}>0$. Now, we are ready to piece together the puzzle of
inequalities and finalize the proof.
$\begin{array}[]{rcllll}&&J(x^{\ast}(\cdot),u^{\ast}(\cdot))\\\
&\leq&J(x^{\ast N}(\cdot),u^{\ast N}(\cdot))&\left(\begin{array}[]{ll}(x^{\ast
N}(t),u^{\ast N}(t))\mbox{ is a feasible }\\\ \mbox{trajectory (Lemma
\ref{lemma1})}\end{array}\right)\\\ &\leq&\bar{J}^{N}(\bar{x}^{\ast
N},\bar{u}^{\ast N})+\displaystyle\frac{C_{7}}{N^{\alpha}}&\left(\mbox{
inequality }(\ref{eq3_11})\right)\\\
&\leq&\bar{J}^{N}(\hat{x}^{N},\hat{u}^{N})+\displaystyle\frac{C_{7}}{N^{\alpha}}&\left(\begin{array}[]{ll}(\hat{x}^{N},\hat{u}^{N})\mbox{
is a feasible discrete }\\\ \mbox{trajectory and }(\bar{x}^{\ast
N},\bar{u}^{\ast N})\mbox{ is optimal}\end{array}\right)\\\
&\leq&J(\hat{x}^{N}(\cdot),\hat{u}^{N}(\cdot))+\displaystyle\frac{C_{5}}{N^{\alpha}}+\displaystyle\frac{C_{7}}{N^{\alpha}}&\left(\mbox{
inequality }(\ref{eq3_8})\right)\\\
&\leq&J(x^{\ast}(\cdot),u^{\ast}(\cdot))+\displaystyle\frac{C_{2}}{(N-r-
m_{1}-1)^{2m-2m_{1}-1}}\\\
&&+\displaystyle\frac{C_{5}}{N^{\alpha}}+\displaystyle\frac{C_{7}}{N^{\alpha}}&\left(\mbox{
inequality }(\ref{eq3_6})\right)\end{array}$
Therefore,
$0\leq J(x^{\ast N}(\cdot),u^{\ast
N}(\cdot))-J(x^{\ast}(\cdot),u^{\ast}(\cdot))\leq\displaystyle\frac{C_{2}}{(N-r-
m_{1}-1)^{2m-2m_{1}-1}}+\displaystyle\frac{C_{5}}{N^{\alpha}}+\displaystyle\frac{C_{7}}{N^{\alpha}}$
This inequality implies (3.9). Furthermore, (3.9) and (3.44) imply (3.10).
$\Box$
According to Theorem 1, the convergence rate of the approximate cost is
determined by two terms with the rates
$\displaystyle\displaystyle\frac{1}{(N-r-
m_{1}-1)^{2m-2m_{1}-1}}\sim\displaystyle\frac{1}{N^{2m-2m_{1}-1}}$ (3.45)
and
$\displaystyle\displaystyle\frac{1}{N^{\alpha}}\sim\displaystyle\frac{1}{N^{m_{1}-1}}$
(3.46)
where $m$, the smoothness of $x^{\ast}(t)$, is fixed. However, $m_{1}$ can be
selected provided $f(\cdot)$, $g(\cdot)$, and $F(\cdot)$ are smooth enough.
Note that increasing $m_{1}$ will increase the rate defined by (3.46), but
decrease the rate defined by (3.45). There is a value of $m_{1}$ that
determines the maximum rate. Given any real number $a\in\Re$, let $[a]$ be the
greatest integer less than or equal to $a$.
###### Corollary 1
Under the same assumption as Theorem 1, the convergence rate of $J(x^{\ast
N}(\cdot),u^{\ast N}(\cdot))$ and $\bar{J}^{N}(\bar{x}^{\ast N},\bar{u}^{\ast
N})$ is
$O\left(\frac{1}{N^{[\frac{2m}{3}]-\delta}}\right)$
in which
$\delta=\left\\{\begin{array}[]{ll}1&0<\gamma<\frac{2}{3}\\\
3\left(1-\gamma\right)&\gamma\geq\frac{2}{3}\\\ 1-\mbox{any positive
number},&\gamma=0\end{array}\right.$
where $\gamma=\frac{2m}{3}-[\frac{2m}{3}]$. To achieve this rate,
$m_{1}=\left\\{\begin{array}[]{lll}\left[\displaystyle\frac{2m}{3}\right],&0\leq\gamma<\frac{2}{3}\\\
\\\
\left[\displaystyle\frac{2m}{3}\right]+1,&\gamma\geq\frac{2}{3}\end{array}\right.$
Proof. The optimal convergence rate is determined by
$\max_{2\leq m_{1}\leq m-1}\min\\{2m-2m_{1}-1,\,m_{1}-1\\}$
The maxmin is achieved at
$m_{1}=\frac{2m}{3}$
However, it may not be an integer. If $m_{1}$ is not an integer, we have two
options,
$m_{1}=[\frac{2m}{3}]\mbox{ or }[\frac{2m}{3}]+1$
If we define
$\gamma=\frac{2m}{3}-[\frac{2m}{3}]$
then either $m_{1}=\frac{2m}{3}-\gamma$ or $m_{1}=\frac{2m}{3}-\gamma+1$. It
is straightforward to verify that
$\min\\{2m-2m_{1}-1,\,m_{1}-1\\}=\left\\{\begin{array}[]{lll}\frac{2m}{3}-\gamma-1,&m_{1}=\frac{2m}{3}-\gamma\\\
\frac{2m}{3}-\gamma-3(1-\gamma),&m_{1}=\frac{2m}{3}-\gamma+1\end{array}\right.$
Therefore, $\frac{2m}{3}-1-\gamma$ is larger when $\gamma<\frac{2}{3}$, and
$\frac{2m}{3}-1-2(1-\gamma)$ is larger if $\gamma\geq\frac{2}{3}$. The special
case at $\gamma=0$ is because of (3.46) when $\frac{2m}{3}$ equals an integer.
$\Box$
Different from numerical computations of differential equations, solving an
optimal control problem requires the approximation, (2.1), of the integration
as an addition to the approximation, (2.26), of the differential equation. The
contributions of these approximations to the overall approximation error are
different; and the errors are inversely related to each other. The following
theorem indicates that the rate (3.45) is due to the approximation error of
the differential equation and the rate (3.46) is due to the approximation
error of the quadrature integration rule (4.9). To verify this fact, we define
the following discretization problem with exact integration.
Problem ${\rm B}^{\rm N}$(J) Find $\bar{x}^{Nk}\in\Re^{r}$ and
$\bar{u}^{Nk}\in\Re$, $k\ =\ 0,1,\ldots,N$, that minimize
$\displaystyle J(x^{N}(\cdot),u^{N}(\cdot))$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}F(x^{N}(t),u^{N}(t))\ dt+E(x^{N}(-1),x^{N}(1))$
(3.47)
subject to
$\displaystyle\left\\{\begin{array}[]{rcl}D(\bar{x}_{1}^{N})^{T}&=&(\bar{x}_{2}^{N})^{T}\\\
D(\bar{x}_{2}^{N})^{T}&=&(\bar{x}_{3}^{N})^{T}\\\ &\vdots&\\\
D(\bar{x}_{r-1}^{N})^{T}&=&(\bar{x}_{r}^{N})^{T}\\\
D(\bar{x}_{r}^{N})^{T}&=&\left[\begin{array}[]{cccccccccccccc}f(\bar{x}^{N0})+g(\bar{x}^{N0})\bar{u}^{N0}\\\
\vdots\\\ f(\bar{x}^{NN})+g(\bar{x}^{NN})\bar{u}^{NN}\end{array}\right]\\\
\end{array}\right.$ (3.56) $\displaystyle\bar{x}^{N0}=x_{0}$ (3.57)
$\displaystyle\underline{{\boldsymbol{b}}}\leq\left[\begin{array}[]{cccccccccccccc}\bar{x}^{Nk}\\\
\bar{u}^{Nk}\end{array}\right]\ \leq\ \bar{\boldsymbol{b}},\;\;\;\;\mbox{ for
all }0\leq k\leq N$ (3.60)
$\displaystyle\underline{{\boldsymbol{b}}}_{j}\leq\left[\begin{array}[]{cccccccccccccc}1&0&\cdots&0\end{array}\right]D^{j}(\bar{x}_{r}^{N})^{T}\
\leq\bar{\boldsymbol{b}}_{j},\;\;1\leq j\leq m_{1}-1$ (3.62)
$\displaystyle\displaystyle\sum_{n=0}^{N-r-
m_{1}+1}|a^{N}_{n}(m_{1})|\leq{\boldsymbol{d}}$ (3.63)
In Problem ${\rm B}^{\rm N}$(J), $(x^{N}(t),u^{N}(t))$ is the interpolation of
$(\bar{x}^{N},\bar{u}^{N})$. In this discretization, we approximate the
differential equation by the PS method. However, the integration in the cost
function is exact. In this case, the overall error is controlled by the single
rate (3.45) rather than the two-rate convergence of Problem ${\rm B}^{\rm N}$.
Without the integration error of the cost function, the convergence rate is
improved to $\displaystyle\frac{1}{N^{2m-3}}$; and the smoothness requirement
can be reduced to $m\geq 2$.
###### Theorem 2
Suppose Problem B has an optimal solution $(x^{\ast}(t),u^{\ast}(t))$ in which
the strong derivative $(x_{r}^{\ast}(t))^{(m)}$ has bounded variation for some
$m\geq 2$. In Problem ${\rm B}^{\rm N}$(J), select $m_{1}$ so that $1\leq
m_{1}\leq m-1$. Suppose $f(\cdot)$, $g(\cdot)$, and $F(\cdot)$ are $C^{m}$.
Suppose all other bounds in Problem ${\rm B}^{\rm N}$ are large enough. Given
any sequence
$\begin{array}[]{llllllllll}\\{(\bar{x}^{\ast N},\bar{u}^{\ast N})\\}_{N\geq
N_{1}}\end{array}$ (3.64)
of optimal solutions of Problem ${\rm B}^{\rm N}$(J). Then the cost of (3.64)
converges to the optimal cost at the following rate
$\displaystyle\left|J(x^{\ast}(\cdot),u^{\ast}(\cdot))-J(x^{\ast
N}(\cdot),u^{\ast N}(\cdot))\right|\leq\displaystyle\frac{M_{1}}{(N-r-
m_{1}-1)^{2m-2m_{1}-1}}$ (3.65)
for some constants $M_{1}$ independent of $N$.
Proof. Let
$\begin{array}[]{llllllllll}\\{(\bar{x}^{\ast N},\bar{u}^{\ast
N})\\}_{N=N_{0}}^{\infty}\end{array}$ (3.66)
be a sequence of optimal solutions of Problem ${\rm B}^{\rm N}$(J). According
to Lemma 7 and Remark 3.3, for any positive integer $N$ that is large enough,
there exists a pair of functions $(\hat{x}^{N}(t),\hat{u}^{N}(t))$ in which
$\hat{x}^{N}(t)$ consists of polynomials of degree less than or equal to $N$.
Furthermore, the pair satisfies the differential equation in Problem B and the
inequalities (3.35), (3.36), and (3.37). If we define
$\hat{\bar{u}}^{Nk}=\hat{u}^{N}(t_{k}),\hat{\bar{x}}^{Nk}=\hat{x}^{N}(t_{k})$
Then, from the first part in the proof of Theorem 1,
$(\hat{\bar{x}}^{N},\hat{\bar{u}}^{N})$ is a discrete feasible solution
satisfying all constraints in Problem ${\rm B}^{\rm N}$(J). By Lemma 1
$\displaystyle|J(x^{\ast}(\cdot),u^{\ast}(\cdot))-J(\hat{x}^{N}(\cdot),\hat{u}^{N}(\cdot))|$
(3.67) $\displaystyle=$ $\displaystyle|{\cal J}(u^{\ast})-{\cal
J}(\hat{u}^{N})|$ $\displaystyle\leq$ $\displaystyle
C_{1}(||u^{\ast}-\hat{u}^{N}||_{W^{m_{1}-1,\infty}}^{2})$ $\displaystyle\leq$
$\displaystyle\displaystyle\frac{C_{2}}{(N-r-m_{1}+1)^{2m-2m_{1}-1}}$
for some constant numbers $C_{1}$ and $C_{2}$. The last inequality is from
(3.25). The interpolation $(x^{\ast N}(t),u^{\ast N}(t))$ of (3.66) is a
feasible trajectory of Problem B (Lemma 5). Thus,
$\begin{array}[]{rcllll}&&J(x^{\ast}(\cdot),u^{\ast}(\cdot))\\\
&\leq&J(x^{\ast N}(\cdot),u^{\ast N}(\cdot))&\left(\begin{array}[]{ll}(x^{\ast
N}(t),u^{\ast N}(t))\mbox{ is a feasible }\\\ \mbox{trajectory (Lemma
\ref{lemma1})}\end{array}\right)\\\
&\leq&J(\hat{x}^{N}(\cdot),\hat{u}^{N}(\cdot))&\left(\begin{array}[]{ll}(\hat{\bar{x}}^{N},\hat{\bar{u}}^{N})\mbox{
is a feasible discrete }\\\ \mbox{trajectory and }(\bar{x}^{\ast
N},\bar{u}^{\ast N})\mbox{ is optimal}\end{array}\right)\\\
&\leq&J(x^{\ast}(\cdot),u^{\ast}(\cdot))+\displaystyle\frac{C_{2}}{(N-r-
m_{1}-1)^{2m-2m_{1}-1}}&\left(\mbox{ inequality
}(\ref{eq3_6a})\right)\end{array}$
Therefore,
$0\leq J(x^{\ast N}(\cdot),u^{\ast
N}(\cdot))-J(x^{\ast}(\cdot),u^{\ast}(\cdot))\leq\displaystyle\frac{C_{2}}{(N-r-
m_{1}-1)^{2m-2m_{1}-1}}$
$\Box$
## 4 Existence and Convergence of Approximate Optimal Solutions
In Section 3, the rate of convergence for the cost function is proved.
However, the results do not guarantee the convergence of the approximate
optimal trajectory $\\{(x^{N}(t),u^{N}(t))\\}$. In this section, we prove the
existence of feasible trajectories for Problem ${\rm B}^{\rm N}$ and the
existence of a convergent subsequence in any set of approximate optimal
solutions. In addition, we consider a larger family of problems. Different
from Section 2 where Problem B does not contain constraints other than the
control system, in this section the problem of optimal control may contain
nonlinear path constraints. Furthermore, general endpoint conditions are
allowed, rather than being limited to the initial value problem as in the
previous sections.
Problem B: Determine the state-control function pair $(x(t),u(t))$,
$x\in\Re^{r}$ and $u\in\Re$, that minimizes the cost function
$\displaystyle J(x(\cdot),u(\cdot))$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}F(x(t),u(t))\ dt+E(x(-1),x(1))$ (4.1)
subject to the state equation
$\displaystyle\left\\{\begin{array}[]{lll}\dot{x}_{1}=x_{2}\\\ \;\;\;\vdots\\\
\dot{x}_{r-1}=x_{r}\\\ \dot{x}_{r}=f(x)+g(x)u\end{array}\right.$ (4.6)
end-point conditions
$\displaystyle e(x(-1),x(1))$ $\displaystyle=$ $\displaystyle 0$ (4.7)
and state-control constraints
$\displaystyle h(x(t),u(t))$ $\displaystyle\leq$ $\displaystyle 0$ (4.8)
where $x\in\Re^{r}$, $u\in\Re$, and $F:\Re^{r}\times\Re\to\Re$,
$E:\Re^{r}\times\Re^{r}\to\Re$, $f:\Re^{r}\to\Re$, $g:\Re^{r}\to\Re$
$e:\Re^{r}\times\Re^{r}\to\Re^{N_{e}}$ and $h:\Re^{r}\times\Re\to\Re^{N_{h}}$
are all Lipschitz continuous functions with respect to their arguments. In
addition, we assume $g(x)\neq 0$ for all $x$. The corresponding discretization
is defined as follows.
Problem ${\bf B}^{\bf N}$: Find $\bar{x}^{Nk}\in\Re^{r}$ and
$\bar{u}^{Nk}\in\Re$, $k\ =\ 0,1,\ldots,N$, that minimize
$\displaystyle\bar{J}^{N}(\bar{x}^{N},\bar{u}^{N})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N}F(\bar{x}^{Nk},\bar{u}^{Nk})w_{k}+E(\bar{x}^{N0},\bar{x}^{NN})$
(4.9)
subject to
$\displaystyle\left\\{\begin{array}[]{rcl}D(\bar{x}_{1}^{N})^{T}&=&(\bar{x}_{2}^{N})^{T}\\\
D(\bar{x}_{2}^{N})^{T}&=&(\bar{x}_{3}^{N})^{T}\\\ &\vdots&\\\
D(\bar{x}_{r-1}^{N})^{T}&=&(\bar{x}_{r}^{N})^{T}\\\
D(\bar{x}_{r}^{N})^{T}&=&\left[\begin{array}[]{cccccccccccccc}f(\bar{x}^{N0})+g(\bar{x}^{N0})\bar{u}^{N0}\\\
\vdots\\\ f(\bar{x}^{NN})+g(\bar{x}^{NN})\bar{u}^{NN}\end{array}\right]\\\
\end{array}\right.$ (4.18)
$\displaystyle\|e(\bar{x}^{N0},\bar{x}^{NN})\|_{\infty}\leq(N-r-1)^{-\beta}$
(4.19) $\displaystyle
h(\bar{x}^{Nk},\bar{u}^{Nk})\leq(N-r-1)^{-\beta}\cdot\mathbf{1},\qquad\ \ \ \
\mbox{ for all }0\leq k\leq N$ (4.20)
$\displaystyle\underline{{\boldsymbol{b}}}\leq\left[\begin{array}[]{cccccccccccccc}\bar{x}^{Nk}\\\
\bar{u}^{Nk}\end{array}\right]\ \leq\ \bar{\boldsymbol{b}},\;\;\;\;\mbox{ for
all }0\leq k\leq N$ (4.23)
$\displaystyle\underline{{\boldsymbol{b}}}_{j}\leq\left[\begin{array}[]{cccccccccccccc}1&0&\cdots&0\end{array}\right]D^{j}(\bar{x}_{r}^{N})^{T}\
\leq\bar{\boldsymbol{b}}_{j},\mbox{ if }1\leq j\leq m_{1}-1\mbox{ and
}m_{1}\geq 2$ (4.25) $\displaystyle\displaystyle\sum_{n=0}^{N-r-
m_{1}+1}|a^{N}_{n}(m_{1})|\leq{\boldsymbol{d}}$ (4.26)
The discretization is almost identical to the one used in the previous
sections except for the path constraints and the endpoint conditions, which
must be treated with care. Note that in Problem ${\rm B}^{\rm N}$the right
sides of (4.19) and (4.20) are not zero. It is necessary to relax (4.7) and
(4.8) by a small margin for the reason of feasibility. The margin approaches
zero as $N$ is increased. Without this relaxation, it is shown by a counter
example in [7] that Problem ${\rm B}^{\rm N}$ may have no feasible
trajectories.
Some feasibility and convergence results were proved in [7], which take the
form of consistent approximation theory based on the convergence assumption
about $\\{\dot{x}_{r}^{N}(t)\\}$ and $\\{\bar{x}^{N0}\\}$. The goal of this
section is to remove this bothersome assumption by using a fundamentally
different approach. In addition, the proofs in this section are not based on
necessary conditions of optimal control and any coercivity assumption, which
are widely used in existing work on the convergence of direct optimal control
methods. Before we introduce main results in this section, some useful results
from [7] are summarized in the following Lemma.
###### Lemma 8
([7]) Suppose Problem B has an optimal solution $(x^{\ast}(t),u^{\ast}(t))$
satisfying $x^{\ast}_{r}(t)\in W^{m,\infty}$, $m\geq 2$. Let
$\\{(\bar{x}^{N},\bar{u}^{N})\\}_{N=N_{1}}^{\infty}$ be a sequence of feasible
solutions to (4.18)-(4.23). Suppose there is a subsequence
$\\{N_{j}\\}_{j=1}^{\infty}$ of $\\{N\\}_{N=1}^{\infty}$ such that the
sequence $\left\\{\bar{x}^{N_{j}0}\right\\}_{j=1}^{\infty}$ converges as
$N_{j}\rightarrow\infty$. Suppose there exists a continuous function $q(t)$
such that $\dot{x}_{r}^{N_{j}}(t)$ converges to $q(t)$ uniformly in $[-1,1]$.
Then, there exists $(x^{\infty}(t),u^{\infty}(t))$ satisfying (4.6)-(4.8) such
that the following limits converge uniformly in $[-1,1]$.
$\displaystyle\lim_{N_{j}\rightarrow\infty}(x^{N_{j}}(t)-x^{\infty}(t))=0$
(4.27)
$\displaystyle\lim_{N_{j}\rightarrow\infty}(u^{N_{j}}(t)-u^{\infty}(t))=0$
(4.28)
$\displaystyle\lim_{N_{j}\rightarrow\infty}\bar{J}^{N_{j}}(\bar{x}^{N_{j}},\bar{u}^{N_{j}})=J(x(\cdot),u(\cdot))$
(4.29)
$\displaystyle\lim_{N_{j}\rightarrow\infty}J(x^{N_{j}},u^{N_{j}})=J(x(\cdot),u(\cdot))$
(4.30)
In addition to the above assumptions, if
$\\{(\bar{x}^{N},\bar{u}^{N})\\}_{N=N_{1}}^{\infty}$ is a sequence of optimal
solutions subject to the constraints (4.18)-(4.23), then
$(x^{\infty}(t),u^{\infty}(t))$ must be an optimal solution to Problem B.
The following are the two main theorems of this section. Relative to [14],
these results has a tightened bounds for $m_{1}$ and $\beta$.
###### Theorem 3
(Existence of solutions) Consider Problem B and Problem ${\rm B}^{\rm N}$
defined in Section 2. Suppose Problem B has a feasible trajectory
$(x(t),u(t))$ in which $(x_{r}(t))^{(m)}$ has bounded variation for some
$m\geq 2$. In Problem ${\rm B}^{\rm N}$, let $m_{1}$ be any integer and
$\beta$ be any real number satisfying $1\leq m_{1}\leq m-1$ and
$0<\beta<(m-m_{1})-\frac{3}{4}$. Then, there exists $N_{1}>0$ so that, for all
$N\geq N_{1}$, Problem ${\rm B}^{\rm N}$ has a feasible trajectory satisfying
(4.18)-(4.26). Furthermore, $(x(t),u(t))$ and the interpolation
$(x^{N}(t),u^{N}(t))$ satisfy (3.23)-(3.26).
###### Theorem 4
(Convergence) Consider Problem B and Problem ${\rm B}^{\rm N}$ defined in
Section 2. Suppose Problem B has an optimal solution
$(x^{\ast}(t),u^{\ast}(t))$ in which $(x^{\ast}_{r}(t))^{(m)}$ has bounded
variation for some $m\geq 3$. In Problem ${\rm B}^{\rm N}$, let $m_{1}$ be any
integer and $\beta$ be any real number satisfying $2\leq m_{1}\leq m-1$ and
$0<\beta<(m-m_{1})-\frac{3}{4}$. Then for any sequence $\\{(\bar{x}^{\ast
N},\bar{u}^{\ast N})\\}_{N=N_{1}}^{\infty}$ of optimal solutions of Problem
${\rm B}^{\rm N}$, there exists a subsequence, $\\{(\bar{x}^{\ast
N_{j}},\bar{u}^{\ast N_{j}})\\}_{j\geq 1}^{\infty}$, and an optimal solution,
$(x^{\ast}(t),u^{\ast}(t))$, of Problem B so that the following limits
converge uniformly in $[-1,1]$
$\displaystyle\lim_{N_{j}\rightarrow\infty}(x^{\ast N_{j}}(t)-x^{\ast}(t))$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\lim_{N_{j}\rightarrow\infty}(u^{\ast N_{j}}(t)-u^{\ast}(t))$
$\displaystyle=$ $\displaystyle 0$ (4.31)
$\displaystyle\lim_{N_{j}\rightarrow\infty}\bar{J}^{N_{j}}(\bar{x}^{\ast
N_{j}},\bar{u}^{\ast N_{j}})$ $\displaystyle=$ $\displaystyle
J(x^{\ast}(\cdot),u^{\ast}(\cdot))$
$\displaystyle\lim_{N_{j}\rightarrow\infty}J(x^{\ast N_{j}}(\cdot),u^{\ast
N_{j}}(\cdot))$ $\displaystyle=$ $\displaystyle
J(x^{\ast}(\cdot),u^{\ast}(\cdot))$
where $(x^{\ast N_{j}}(t),u^{\ast N_{j}}(t))$ is the interpolation of
$(\bar{x}^{\ast N},\bar{u}^{\ast N})$.
###### Remark 4.1
The integers $m$ and $m_{1}$ in Theorem 3 are smaller than those in Theorem 4,
i.e. the existence theorem is proved under a weaker smoothness assumption than
the convergence theorem.
To prove these theorems, we first briefly review some results on real analysis
and then prove a lemma. Given a sequence of functions
$\\{f_{k}(t)\\}_{k=1}^{\infty}$ defined on $[a,b]$. It is said to be uniformly
equicontinuous if for every $\epsilon>0$, there exists a $\delta>0$ such that
for all $t$, $t^{\prime}$ in $[a,b]$ with $|t^{\prime}-t|<\delta$, we have
$|f_{k}(t)-f_{k}(t^{\prime})|<\epsilon$
for all $k\geq 1$. The following Proposition and Theorem are standard in real
analysis [20].
###### Proposition 1
If $f_{k}(t)$ is differentiable for all $k$, and if
$\\{\dot{f}_{k}(t)\\}_{k=1}^{\infty}$ is bounded. Then,
$\\{f_{k}(t)\\}_{k=1}^{\infty}$ is uniformly equicontinuous.
###### Theorem 5
(Arzelà-Ascoli Theorem) Consider a sequence of continuous functions
$\\{h_{n}(t)\\}_{n=1}^{\infty}$ defined on a closed interval $[a,b]$ of the
real line with real values. If this sequence is uniformly bounded and
uniformly equicontinuous, then it admits a subsequence which converges
uniformly.
###### Lemma 9
([14])Let $\\{(\bar{x}^{N},\bar{u}^{N})\\}_{N=N_{1}}^{\infty}$ be a sequence
satisfying (4.18)-(4.23). Assume the set
$\displaystyle\left\\{\left.||\ddot{x}^{N}_{r}(t)||_{\infty}\right|N\geq
N_{1}\right\\}$ (4.32)
is bounded. Then, there exists $(x^{\infty}(t),u^{\infty}(t))$ satisfying
(4.6)-(4.8) and a subsequence
$\\{(\bar{x}^{N_{j}},\bar{u}^{N_{j}})\\}_{N_{j}\geq N_{1}}^{\infty}$ such that
(4.27), (4.28), (4.29) and (4.30) hold. Furthermore, if
$\\{(\bar{x}^{N},\bar{u}^{N})\\}_{N=N_{1}}^{\infty}$ is a sequence of optimal
solutions to Problem ${\rm B}^{\rm N}$, then $(x^{\infty}(t),u^{\infty}(t))$
must be an optimal solution to Problem B.
Proof. Let $x_{r}^{N}(t)$ be the interpolation polynomial of
$\bar{x}_{r}^{N}$. Because (4.32) is a bounded set, we know that the sequence
of functions $\\{\dot{x}_{r}^{N}(t)|N\geq N_{1}\\}$ is uniformly
equicontinuous (Proposition 1). By the Arzelà-Ascoli Theorem, a subsequence
$\\{\dot{x}_{r}^{N_{j}}(t)\\}$ converges uniformly to a continuous function
$q(t)$. In addition, because of (4.23), we can select the subsequence so that
$\\{\bar{x}^{N_{j}0}\\}_{N_{j}\geq N_{1}}^{\infty}$ is convergent. Therefore,
all conclusions in Lemma 8 hold true. $\Box$
Now, we are ready to prove the theorems.
Proof of Theorem 3: For the feasible trajectory $(x(t),u(t))$, consider the
pair $(x^{N}(t),u^{N}(t))$ in Lemma 7 that satisfies the differential equation
(4.6). Define
$\displaystyle\begin{array}[]{rcl}\bar{x}^{Nk}&=&x^{N}(t_{k})\\\
\bar{u}^{Nk}&=&u^{N}(t_{k})\end{array}$ (4.35)
for $0\leq k\leq N$. From Lemma 5, we know that
$\\{(\bar{x}^{N},\bar{u}^{N})\\}$ satisfies the discrete equations in (4.18).
In the next we prove that the mixed state-control constraint (4.20) is
satisfied. Because $h$ is Lipschitz continuous and because of (3.23) and
(3.25), there exists a constant $C$ independent of $N$ so that
$\displaystyle\|h(x(t),u(t))-h(x^{N}(t),u^{N}(t))\|$ $\displaystyle\leq$
$\displaystyle
C(|x_{1}(t)-x^{N}_{1}(t)|+\cdots+|x_{r}(t)-x^{N}_{r}(t)|+|u(t)-u^{N}(t)|)$
$\displaystyle\leq$ $\displaystyle CMV||x_{r}||_{W^{m,2}}(r+1)(N-r-
m_{1}+1)^{-(m-m_{1})+3/4}$
Hence
$\displaystyle h(x^{N}(t),u^{N}(t))$ $\displaystyle\leq$ $\displaystyle
h(x(t),u(t))+CM||x_{r}||_{W^{m,2}}(r+1)(N-r-
m_{1}+1)^{-(m-m_{1})+3/4}\cdot\mathbf{1}$ $\displaystyle\leq$ $\displaystyle
CM||x_{r}||_{W^{m,2}}(r+1)(N-r-m_{1}+1)^{-(m-m_{1})+3/4}$
Because $\beta<m-m_{1}-\frac{3}{4}$, there exists a positive integer $N_{1}$
such that, for all $N>N_{1}$,
$\displaystyle CM||x_{r}||_{W^{m,2}}(r+1)(N-r-m_{1}+1)^{-(m-m_{1})+3/4}$
$\displaystyle\leq$ $\displaystyle(N-r-1)^{-\beta}$
Therefore $x^{N}_{1}(t_{k})$, $\ldots$, $x^{N}_{r}(t_{k})$, $u^{N}(t_{k})$,
$k=0,1,\ldots,N$, satisfy the mixed state and control constraint (4.20) for
all $N>N_{1}$.
By a similar procedure, we can prove that the endpoint condition (4.19) is
satisfied. Because $x_{r}^{N}(t)$ is a polynomial of degree less than or equal
to $N$, and because of (2.14) and (4.35), we know $(x^{N}_{r}(t))^{(j)}$
equals the interpolation polynomial of $\bar{x}^{N}_{r}(D^{T})^{j}$. So,
$\left[\begin{array}[]{cccccccccccccc}1&0&\cdots&0\end{array}\right]D^{j}(\bar{x}_{r}^{N})^{T}=(x^{N}_{r}(t))^{(j)}|_{t=-1}$
Therefore, (3.24) implies (4.25) if the interval between
$\underline{{\boldsymbol{b}}}_{j}$ and ${\boldsymbol{b}}_{j}$ is large enough.
In addition, the spectral coefficients of $\bar{x}^{N}_{r}(D^{T})^{m_{1}}$ is
the same as the spectral coefficients of $(x^{N}_{r}(t))^{(m_{1})}$. From
(3.26) and (3.6), we have
$\displaystyle\sum_{n=0}^{N-r-m_{1}+1}|a^{N}_{n}(m_{1})|\leq{\boldsymbol{d}}$
So, $\\{(\bar{x}^{N},\bar{u}^{N})\\}$ satisfies (4.26). Because we select
$\underline{{\boldsymbol{b}}}$ and $\bar{\boldsymbol{b}}$ large enough so that
the optimal trajectory of the original continuous-time problem is contained in
the interior of the region, we can assume that $(x(t),u(t))$ is also bounded
by $\underline{{\boldsymbol{b}}}$ and $\bar{\boldsymbol{b}}$. Then, (3.23) and
(3.25) imply (4.23) for $N$ large enough. To summarize,
$(\bar{x}^{N},\bar{u}^{N})$ satisfies (4.18)-(4.26). Therefore, it is a
feasible trajectory of (4.18)-(4.23). $\Box$
Proof of Theorem 4: Consider $\\{(\bar{x}^{\ast N},\bar{u}^{\ast
N})\\}_{N=N_{1}}^{\infty}$, a sequence of optimal solutions of Problem ${\rm
B}^{\rm N}$. From Lemma 6,
$\\{||\ddot{x}_{r}^{\ast N}(t)||_{\infty}|N\geq N_{1}\\}$
is bounded. Now, we can apply Lemma 9 to conclude that there exists a
subsequence of $\\{(x^{\ast N}(t),u^{\ast N}(t))\\}_{N=N_{1}}^{\infty}$ and an
optimal solution of Problem B so that the limits in (4.31) converge uniformly.
$\Box$
## 5 Simulation results
The rate of convergence for the optimal cost is illustrated in the following
example
$\displaystyle\min_{u}\int_{0}^{\pi}(1-x_{1}+x_{1}x_{2}+x_{1}u)^{2}dt$ subject
to $\displaystyle\dot{x}_{1}=-x_{1}^{2}x_{2}$
$\displaystyle\dot{x}_{2}=-1+\frac{1}{x_{1}}+x_{2}+\sin t+u$ $\displaystyle
x(0)=\left[\begin{array}[]{cccccccccccccc}1\\\
0\end{array}\right],\;\;x(\pi)=\left[\begin{array}[]{cccccccccccccc}\frac{1}{\pi+1}\\\
2\end{array}\right]$
The analytic solution of this problem is known so that the approximation error
can be computed
$\displaystyle x_{1}(t)=\frac{1}{1-\sin t+t}$ $\displaystyle x_{2}(t)=1-\cos
t$ $\displaystyle u=-(t+1)+\sin t+\cos t$ $\displaystyle\mbox{optimal cost}=0$
The problem is solved using PS optimal control method. The approximated
optimal cost is compared to the true value. The number of nodes, N, ranges
from $4$ to $16$. The error decreases rapidly as shown in Table 1.
N | 4 | 6 | 8 | 10 | 12 | 14 | 16
---|---|---|---|---|---|---|---
Error | $7.5\times 10^{-2}$ | $1.1\times 10^{-3}$ | $2.1\times 10^{-4}$ | $7.1\times 10^{-5}$ | $6.7\times 10^{-6}$ | $1.0\times 10^{-6}$ | $5.8\times 10^{-7}$
Table 1: The error of optimal cost
The rate of convergence is illustrated in the following Figure 3.
Figure 3: Log-scale plot of the error of optimal cost (the solid curve)
Because the analytic solution is $C^{\infty}$, the rate of convergence of the
PS method is faster than any polynomial rate. As a result, it converges
exponentially. Of course, in practical computations the accuracy is limited by
the machine precision of the computers. Therefore, the accuracy cannot be
improved after $N$ is sufficiently large.
## 6 Conclusions
It is proved that the PS optimal control has a high-order rate of convergence.
According to the theorems in Section 3, the approximate cost computed using
the Legendre PS method converges at an order determined by the smoothness of
the original problem. More specifically, the rate is about
$\frac{1}{N^{2m/3-1}}$, where $m$ is defined by the smoothness of the optimal
trajectory. If the cost function can be accurately computed, then the
convergence rate is improved to $\frac{1}{N^{2m-1}}$. If the optimal control
is $C^{\infty}$, then the convergence rate can be made faster than any given
polynomial rate. The results in Section 4 imply that the discretization using
the Legendre PS method is feasible; and there always exists a convergent
subsequence from the approximate discrete optimal solutions, provided some
smoothness assumptions are satisfied.
## References
* [1] J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, Philadelphia, PA, 2001.
* [2] J. T. Betts, “Survey of Numerical Methods for Trajectory Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1998, pp. 193-207.
* [3] J. P. Boyd, Chebyshev and Fourier Spectral Methods, second edition, Dover, 2001,
* [4] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Method in Fluid Dynamics. New York: Springer-Verlag, 1988.
* [5] G. Elnagar, and M. A. Kazemi, Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems, Computational Optimization and Applications, 11, 1998, pp. 195-217.
* [6] Fahroo, F., Ross, I. M., ”Costate Estimation by a Legendre Pseudospectral Method,” Proceedings of the AIAA Guidance, Navigation and Control Conference, 10-12 August 1998, Boston, MA.
* [7] Q. Gong, W. Kang, and I. M. Ross, A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems, IEEE Trans. Automat. Contr., Vol. 51, No. 7, pp. 1115-1129, 2006.
* [8] Q. Gong, M. Ross, W. Kang, F. Fahroo, Connections Between the Covector Mapping Theorem and Convergence of Pseudospectral Methods for Optimal Control, Computational Optimization and Applications, to appear.
* [9] W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Numerische Mathematik, Vol. 87, pp. 247-282, 2000.
* [10] A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control, Mathematics of Computation, Vol. 70, pp. 173-203, 2000.
* [11] J. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, 2007.
* [12] W. Kang, N. Bedrossian, Pseudospectral Optimal Control Theory Makes Debut Flight - Saves NASA $1M in under 3 hrs, SIAM News, September, 2007.
* [13] W. Kang, Q. Gong, and I. M. Ross, On the Convergence of Nonlinear Optimal Control using Pseudospectral Methods for Feedback Linearizable Systems, International Journal of Robust and Nonlinear Control, Vol. 17, 1251-1277, online publication, 3 January, 2007.
* [14] W. Kang, I. M. Ross, Q. Gong, Pseudospectral Optimal Control and Its Convergence Theorems, Analysis and Design of Nonlinear Control Systems - In Honor of Alberto Isidori, A. Astolfi and L. Marconi eds., Springer, 2008.
* [15] S. W. Paris and C. R. Hargraves, OTIS 3.0 Manual, Boeing Space and Defense Group, Seattle, WA, 1996.
* [16] S. W. Paris, J. P. Riehl, and W. K. Sjauw, “Enhanced Procedures for Direct Trajectory Optimization Using Nonlinear Programming and Implicit Integration,” Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 21-24 August 2006, Keystone, CO. AIAA Paper No. 2006-6309.
* [17] E. Polak, Optimization: Algorithms and Consistent Approximations, Springer-Verlag, Heidelberg, 1997.
* [18] J. P. Riehl, S. W. Paris, and W. K. Sjauw, “Comparision of Implicit Integration Methods for Solving Aerospace Trajectory Optimization Problems,” Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 21-24 August 2006, Keystone, CO. AIAA Paper No. 2006-6033.
* [19] Ross, I. M., A Beginner s Guide to DIDO: A MATLAB Application Package for Solving Optimal Control Problems, Elissar Inc., Monterey, CA, October 2007.
* [20] G. Sansone, A. H. Diamond, and E. Hille, Orthogonal Functions, Robert E. Krieger Publishing Co., Huntington, New York, 1977.
* [21] A. Wouk, A Course of Applied Functional Analysis, John Wiley & Sons, New York, 1979.
|
arxiv-papers
| 2009-04-06T02:08:15 |
2024-09-04T02:49:01.721441
|
{
"license": "Public Domain",
"authors": "Wei Kang",
"submitter": "Wei Kang",
"url": "https://arxiv.org/abs/0904.0833"
}
|
0904.0917
|
# Probing Quantum Hall Pseudospin Ferromagnet by Resistively Detected NMR
G. P. Guo(1) Y. J. Zhao(1) T. Tu(1) tutao@ustc.edu.cn X. J. Hao(1) G. C.
Guo(1) H. W. Jiang(2) jiangh@physics.ucla.edu (1) Key Laboratory of Quantum
Information, University of Science and Technology of China, Chinese Academy of
Sciences, Hefei 230026, P. R. China
(2) Department of Physics and Astronomy, University of California at Los
Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095, USA
###### Abstract
Resistively Detected Nuclear Magnetic Resonance (RD-NMR) has been used to
investigate a two-subband electron system in a regime where quantum Hall
pseudo-spin ferromagnetic (QHPF) states are prominently developed. It reveals
that the easy-axis QHPF state around the total filling factor $\nu=4$ can be
detected by the RD-NMR measurement. Approaching one of the Landau level (LL)
crossing points, the RD-NMR signal strength and the nuclear spin relaxation
rate $1/T_{1}$ enhance significantly, a signature of low energy spin
excitations. However, the RD-NMR signal at another identical LL crossing point
is surprisingly missing which presents a puzzle.
###### pacs:
73.43.Nq, 71.30.+h, 72.20.My
The multi-component electron systems have been continuously drawing intensive
research interest because of its novel ground states and excitations DasSarma
. In experimental systems, different Landau levels (LLs) can be tuned to cross
by varying gate voltage, charge density, magnetic field or the magnetic field
tilted angle to the sample. Electron-electron correlations become particularly
prominent when two or more sets of LLs with different layer, subband, valley,
spin, or Landau level indices are brought into degeneracy DasSarma ;
Wescheider1999 ; Shayegan2000 ; Hirayama2001 ; Jiang2005 ; Jiang2006 ;
Tsui2006 ; Shayegan2006 . Recent experiments in single quantum well with two
subbands occupied systems Jiang2005 ; Jiang2006 , showed evidence of the
formation of quantum Hall pseudospin ferromagnets (QHPFs) due to the
interactions of the two subbands (termed as pseudospins) around the LLs
crossing point. The QHPFs taking place at total filling factor $\nu=3,5$ and
$\nu=4$ are easy-plane or easy-axis QHPFs respectively, depending on the
details of the two subbands configurations. In spite of various theoretical
models MacDonald2000 ; DasSarma2003 ; Hao2008 motivated by these findings, a
comprehensive understanding is not yet achieved. Thus far, experimental and
theoretical studies all focused on the pseudospin freedom. However, in this
work we would address the unique spin excitations in the QHPF states.
To address the question whether spin states in two-subband systems in nature,
measurements other than the conventional transport and optical means are
needed. Since the Zeeman energy of nuclear spin is about $3$ orders of
magnitude smaller than that of electron spin, exchange of spin angular
momentum between the electron and nuclear spin is allowed only when the
electron system supports spin excitations with low energy. The nuclear spin
relaxation rate $1/T_{1}$ thus probes the density of states at low energy of
the electron spin system that cannot be accessed by other means. The
resistively detected NMR technique has recently emerged as an effective method
to probe collective spin states in the fractional quantum Hall regime
KlitzingNMR1 ; KlitzingNMR2 , the Skyrmion spin texture close to the filling
factor $1$ PortalNMR ; TsuiNMR , the role of electron spin polarization in the
phase transition of a bilayer system EisensteinNMR ; HirayamaNMR1 , and the
ferromagnetic state accompanied by collective spin excitations of a two-
subband system JiangNMR . Here we use this technique to study spin freedom and
its relation with pseudospin in the vicinity of the QHPF states at filling
factor $\nu=3,4,5$. It reveals that the easy-axis QHPF state at $\nu=4$ is
sensitive to the RD-NMR measurement. As approaching to one LL crossing point
at $\nu=4$ where the easy-axis QHPF phase is well developed, the RD-NMR signal
strength and the nuclear spin relaxation rate $1/T_{1}$ enhance quickly which
may be due to the low energy spin excitations there. Furthermore, the RD-NMR
signal can be suppressed anomaly at another identical LL crossing point of
$\nu=4$.
The sample was grown by molecular-beam epitaxy and consists of a symmetrical
modulation-doped $24$ nm wide single GaAs quantum well bounded on each side by
Si $\delta$-doped layers of AlGaAs with doping level $n_{d}=10^{12}$ cm-2.
Heavy doping creates a very dense 2DEG, resulting in the filling of two
subbands in the well. As determined from the Hall resistance data and
Shubnikov-de Haas oscillations in the longitudinal resistance, the total
density is $n=8.0\times 10^{11}$ cm-2, where the first and the second subband
have a density of $n_{1}=6.1\times 10^{11}$ cm-2 and $n_{2}=1.9\times 10^{11}$
cm-2. The sample has a low-temperature mobility $\mu=4.1\times 10^{5}$ cm2/V
s, which is extremely high for a 2DEG with two filled subbands. A $100$ $\mu$m
wide Hall bar with $270$ $\mu$m between voltage probes was patterned by
standard lithography techniques. A NiCr top gate was evaporated on the top of
the sample, approximately $350$ nm away from the center of the quantum well.
By applying a negative gate voltage on the NiCr top gate, the electron density
can be varied continuously. Several turns of NMR coil were wound around the
sample, which was placed in a Top-Loading Dilution Refrigerator with a base
temperature of $15$ mK. A small radio frequency (rf) magnetic field generated
by the coil with a matching frequency $f=\gamma H_{0}$ will cause NMR for 75As
nuclei, where the gyromagnetic ratio $\gamma=7.29$ MHz/T. The resistance was
measured using quasi-dc lock-in technique with $11.3$ Hz.
Figure 1: (a) The longitudinal resistance $R_{xx}$ in the density ($n$) -
magnetic field ($B_{\bot}$) phase diagram at filling factor $\nu=3,4,5$, which
are measured at the base temperature. (b) Schematic drawing of the crossing
between different indices Landau levels and resulting easy-plane or easy-axis
pseudo-spin states at points B, D and A, C, as correspondingly marked in Fig.
2a.
Figure 2: (a) The NMR signals phase diagram of the sample at $\nu=3,4,5$. The
cross and circle symbols in the map denote the places where the NMR signals
are measured. The ’$\times$’ mean places where there are no NMR signals, while
the ’$\circ$’ show the places where the NMR signals are observed. And the size
of ’$\circ$’ symbols give a schematic illustration of the strength of NMR
signals. The dashed line L1 is the trace along which we measured NMR signal as
shown in Fig. 5. (b) Typical resistively detected NMR spectrum measured around
point C and A, B, D.
In the present work, we refer the first and second subbands, to as symmetric
and antisymmetric states. In the pseudo-spin language, one of them can be
labeled as pseudo-spin up ($\Uparrow$) and the other as pseudo-spin down
($\Downarrow$). When a magnetic field $B_{\bot}$ is applied, the energy
spectrum of the quantum well discretizes into a sequence of Landau levels. We
label the single-particle levels ($i,N,\sigma$), which $i$
($=\Uparrow,\Downarrow$), $N$, and $\sigma$ ($=\uparrow,\downarrow$) are the
pseudo-spin, orbital and spin quantum numbers. In the present work we have
concentrated our study around the filling factor $\nu=3,4,5$, where the
filling factor $\nu$ denotes the number of filled Landau levels. The
longitudinal resistance $R_{xx}$ in the density ($n$) - perpendicular magnetic
field ($B_{\bot}$) plane exhibits a square-like structure around $\nu=3,4,5$,
as shown in Fig. 1a. The most noticeable feature of the square-like structure
is the disappearance of the extended states (i.e., bright lines) on its four
boundaries, marked by A, B, C, D in Fig. 1a. Here point A corresponds to the
degeneracy point of $\left|(\Uparrow,1,\downarrow)\right\rangle$ and
$\left|(\Downarrow,0,\uparrow)\right\rangle$, point B corresponds to that of
$\left|(\Uparrow,1,\uparrow)\right\rangle$ and
$\left|(\Downarrow,0,\uparrow)\right\rangle$, point C corresponds to that of
$\left|(\Uparrow,1,\uparrow)\right\rangle$ and
$\left|(\Downarrow,0,\downarrow)\right\rangle$, point D corresponds to that of
$\left|(\Uparrow,1,\downarrow)\right\rangle$ and
$\left|(\Downarrow,0,\downarrow)\right\rangle$, as illustrated schematically
in the Landau level fan diagram Fig. 1b. The disappearance and result square
structure represents a pseudo-spin ferromagnet, which is due to the opening
pseudo-spin gaps of easy-plane or easy-axis pseudo-spin ferromagnetic states,
respectively at the level crossing points of B, D and A, C, as depicted in
Fig. 1b Jiang2005 ; Jiang2006 ; Hirayama2001 ; MacDonald2000 .
RD-NMR, performed in the proximity of the square structure, reveals prominent
(absent) NMR signal at different regions. In order to get a clear signal and
minimize heat effect, most of experiments were carried out with a rf power of
$0$ dBm. The ac current $I_{ac}$ was $50$ nA, and a large dc current
$I_{dc}=250$ nA were applied to enhance the NMR signal. All the measurements
were carried out at temperature below $120$ mK. The measurement result under
the same condition are shown in Fig. 2a, the cross and circle symbols in the
map denote the places where the NMR signals are measured. The cross ’$\times$’
means the places where there are no NMR signals, while the circle ’$\circ$’
shows the places where the NMR signals are observed. And the size of ’$\circ$’
symbols give a schematic illustration of the strength of NMR signals. From
this map we found that the NMR signals only occur at the upper arm of the
square structure around crossing point C, while we didn’t find any signal at
the lower arm of this square structure around another crossing point A and its
two sides around crossing point B and D.
Now we focused on the region around the LL crossing point C, where pronounced
NMR signals were observed. Typical NMR lines around point C are shown in Fig.
2b. The relative change of $R_{xx}$ is typically about 1% at resonance. Upon
resonance, $R_{xx}$ in all NMR lines shows a sharp decrease followed by a much
slower relaxation process back to its original value, which is characterized
by the nuclear spin relaxation time owing to the interaction with the electron
spin system, $T_{1}$, as will be discussed below. In these experiments, we
have changed the rf amplitude from $-15$ dBm to $2$ dBm. Even very weak, the
NMR signal can be recognized at $-15$ dBm.
We believe the RD-NMR described here is due to the electron and nuclear spin
flip-flop effect JiangNMR . For the two dimensional electron system in GaAs,
the contact hyperfine interaction with the polarized nuclei acts as an
effective magnetic field $B_{N}$ for the electron spin. The effective electron
spin-flip energy is then reduced, $E_{z}=g^{\ast}\mu_{B}BS_{z}+A\left\langle
I_{z}\right\rangle S_{z}=g^{\ast}\mu_{B}(B+B_{N})S_{z}$ as $g^{\ast}<0$. When
the NMR resonance condition is matched, the nuclear spins are depolarized and
the electron Zeeman energy increases consequently. Since $R_{xx}$ is dependent
on the thermally activated energy gap $E_{a}$,
$R_{xx}\propto\exp(-E_{a}/2k_{B}T)$, the NMR is manifested by a drop in
$R_{xx}$, as shown by all the NMR lines in Fig. 2b. This allows the nuclear
spin polarization to be sensitively detected by a change in the transport
coefficient of the electron system $R_{xx}$.
The above observations reveals the spin excitation in the square structure is
of intrinsic interest and is well correlated with the spin excitations of the
easy-axis QHPF states. At point C, when the two competing pseudospin (up and
down) states acquire the same energy and leads to easy-axis anisotropy, they
separate into domains with opposite pseudospin states Hirayama2001 ; Jiang2006
; MacDonald2000 ; MacDonald2001 . On the other hand, the pseudospin up and
down states have opposite spins. As a result, magnetic domains form and the
electronic state within each domain is described as an Ising-like QH
ferromagnet with either one of two possible spin orientations. As the applied
current forces electrons to scatter between adjacent domains with different
spin but almost degenerate energy, the nuclei in the neighborhood can become
polarized and probed by the RD-NMR measurement. However at other crossing
point B and D, the QHPF states are easy-plane, which means that the two
degenerate Landau levels are mixing and no spin magnetization formation. Since
easy-plane QHPF state can not spontaneously separate into magnetic domains,
there is no nuclear polarization and the NMR signals are destroyed.
To support the mechanism of the polarized nuclear spins, current dependence of
the NMR signal was studied. In this measurement, the sample resistance was
measured with a low ac current of $20$ nA, while ramping the dc current in a
wide range to bias the sample. The result indicates that the NMR signal is
enhanced by a factor of $8$ in the low current range from $100$ nA up to $250$
nA. The data thus consist with the picture of current induced dynamic
polarization.
Figure 3: Measuring nuclear spin relaxation time $T_{1}$ around point C by
recording time evolution of $R_{xx}$ irradiated by rf, initially off
resonance, on resonance and finally off resonance. $T_{1}$ is determined by an
exponential fit to the experiment data.
Figure 4: (a) Plot of the resistively detected NMR signal ratio $\Delta
R_{xx}/R_{xx}$ (black square), nuclear spin relaxation rate $1/T_{1}$ (blue
circle) against gate voltage $V_{g}$ along the line L1 (in Fig. 2a). (b) Plot
of electron activation energy gap $E_{a}$ against gate voltage $V_{g}$ along
the same line.
To gain more support of our observation of the nature of the spin in the easy-
axis QHPF states, we studied the coupling between the nuclei and the electrons
by measuring the nuclear spin relaxation time $T_{1}$, at various positions
near the crossing point C. First, rf was tuned into resonance, and $R_{xx}$
shows a sharp decrease due to the nuclear depolarization. Then, the frequency
was switched back to off resonance. Nuclear spins that have once flopped
hardly relax back because of their longer relaxation time $T_{1}$, which is on
the order of minutes, relative to that of the electrons. Hence, $R_{xx}$
slowly relaxes back to its original value, and $T_{1}$ can be derived by
fitting $R_{xx}$ to the relation $R_{xx}=\alpha+\beta\exp(-t/T_{1})$. Fig. 3
shows the data around point C to determine $T_{1}$.
Further insight is gained by investigating the NMR signals along the line L1
(please see Fig. 2a). As depicted in Fig. 4a, our measurement shows a clear
peak of NMR ratio $\Delta R_{xx}/R_{xx}$ at the crossing point C where the
easy-axis pseudo-spin ferromagnetic states is well developed. The obtained
values of nuclear spin relaxation rate $1/T_{1}$ along line L1 are also
plotted in Fig. 4a. $1/T_{1}$ rapidly increases from nearly zero to $8\times
10^{-3}$ (1/s) toward to the crossing point C, as electron becomes the pseudo-
spin ferromagnetic states. For comparison, in Fig. 4b we also show the
electron activation energy gap $E_{a}$ along the line L1. The single particle
energy difference $E_{z}$ acts as effective Zeeman energy, and $E_{a}$ shows a
slope of $5$ times greater than the single particle Zeeman gap $E_{z}$. This
unusual behavior is likely to be caused by the easy-axis ferromagnetism
Jiang2006 ; Hirayama2001 . These quantities all show an obvious change as
approaching to the crossing point and demonstrate that $1/T_{1}$ is a
sensitive indicator of the pseudo-spin ferromagnetic formation. The similarity
between these phenomenon strongly suggest that an intimate link between the
spin and pseudo-spin in the easy-axis pseudo-spin ferromagnetic states.
Interestingly, the data shown in Fig. 4b shows that the slop of activation
energy gap $E_{a}$ to single particle Zeeman gap is as large as $5$, which
implies many spin flips within the magnetic domain walls and support low
energy mode of spin excitations Eisenstein1995 ; MacDonald2001 . As
approaching to the crossing point C, there are low energy spin excitations
which give new channel to relax the nuclear spin through the electron and
nuclear spin flip-flop process. Thus the NMR signal ratio $\Delta
R_{xx}/R_{xx}$ and the nuclear spin relaxation rate $1/T_{1}$ enhanced.
Despite the fact that the bulk of the results can be understood within the
framework of pseudo-spin quantum Hall ferromagnetism, there is still an
apparent puzzle. While we can find very strong NMR signals at the upper arm of
the square structure around point C, there is no detectible signal at the
lower arm of this square structure around point A. Since the two points have
equivalent LLs crossing configurations, one would expect that they are the
same easy-axis QHPF states and should produce similar NMR responses. In
principle, the NMR signal can be suppressed by spin-orbital coupling
HirayamaNMR2 or mobility of domains HirayamaNMR3 . However, in our case,
point A and C have identical strength in spin-orbital coupling and disorder.
Therefore, the anomalous suppression of NMR signal at point A may suggest that
there could be some additional physics which has not yet been recognized in
the theory of pseudo-spin quantum Hall ferromagnetism.
In summary, RD-NMR has been measured in a two-subband electron system around
the LLs crossing points at total filling factor $\nu=3,5$ and $4$ where easy-
plane or easy-axis QHPF states are well developed. It reveals that the easy-
axis quantum Hall pseudospin state of $\nu=4$ is sensitive to the RD-NMR
measurement. As approaching to one LL crossing point at $\nu=4$, the RD-NMR
signal strength and the nuclear spin relaxation rate $1/T_{1}$ enhance quickly
which may be due to the low energy spin excitations. At another identical LL
crossing point of $\nu=4$, the RD-NMR signal is found to be suppressed and
remains as a puzzle to be understood. Of course further study is necessary to
access the detailed mechanism.
This work at USTC was funded by National Basic Research Programme of China
(Grants No. 2006CB921900 and No. 2009CB929600), the Innovation funds from
Chinese Academy of Sciences, and National Natural Science Foundation of China
(Grants No. 10604052 and No. 10874163 and No.10804104). The work at UCLA was
supported by the NSF under Grant No. DMR-0804794.
## References
* (1) Chap 2 and 5 in Perspectives on Quantum Hall Effects, S. Das Sarma and A. Pinczuk eds., (Wiley, New York, 1997).
* (2) V. Piazza, V. Pellegrini, F. Beltram, W. Wegscheider, T. Jungwirth, and A. H. MacDonald, Nature 402, 638 (1999).
* (3) E. P. De Portere, E. Tutuc, S. J. Papadakis, M. Shayegan, Science 290, 1546 (2000).
* (4) K. Muraki, T. Saku, and Y. Hirayama, Phys. Rev. Lett. 87, 196801 (2001).
* (5) X. C. Zhang, D. R. Faulhaber and H. W. Jiang, Phys. Rev. Lett. 95, 216801 (2005).
* (6) X. C. Zhang, I. Martin and H. W. Jiang, Phys. Rev. B 74, 073301 (2006).
* (7) K. Lai, W. Pan, D.C. Tsui, S. Lyon, M. Muhlberger and F. Schaffler, Phys. Rev. Lett. 96, 076805 (2006).
* (8) K. Vakili, T. Gokmen, O. Gunawan, Y. P. Shkolnikov, E. P. De Poortere and M. Shayegan, Phys. Rev. Lett. 97. 116803 (2006).
* (9) T. Jungwirth and A. H. MacDonald, Phys. Rev. B 63, 035305 (2000).
* (10) D. W. Wang, E. Demler, and S. Das Sarma, Phys. Rev. B 68, 165303 (2003).
* (11) X. J. Hao et al., arXiv:0807.0297.
* (12) J. H. Smet et al., Nature (London) 415, 281 (2002).
* (13) O. Stern et al., Phys. Rev. B 70, 075318 (2004).
* (14) W. Desrat et al., Phys. Rev. Lett. 88, 256807 (2002).
* (15) G. Gervais et al., Phys. Rev. Lett. 94, 196803 (2005).
* (16) I. B. Spielman et al., Phys. Rev. Lett. 94, 076803 (2005).
* (17) N. Kumada et al., Phys. Rev. Lett. 94, 096802 (2005).
* (18) X. C. Zhang, G. D. Scott and H. W. Jiang, Phys. Rev. Lett. 98, 246802 (2007).
* (19) T. Jungwirth and A. H. MacDonald, Phys. Rev. Lett. 87, 216801 (2001).
* (20) A. Schmeller, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 75, 4290 (1995).
* (21) K. Hashimoto et al., Phys. Rev. Lett. 94, 146601 (2005).
* (22) Y. Hirayama et al., Physica E. 20, 133 (2003).
|
arxiv-papers
| 2009-04-06T13:33:03 |
2024-09-04T02:49:01.729469
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. P. Guo, Y. J. Zhao, T. Tu, X. J. Hao, G. C. Guo, H. W. Jiang",
"submitter": "Tao Tu",
"url": "https://arxiv.org/abs/0904.0917"
}
|
0904.0977
|
# Bayesian MAP Model Selection of Chain Event Graphs
G. Freeman g.freeman@warwick.ac.uk J.Q. Smith j.q.smith@warwick.ac.uk
Department of Statistics, University of Warwick, Coventry, CV4 7AL
###### Abstract
The class of chain event graph models is a generalisation of the class of
discrete Bayesian networks, retaining most of the structural advantages of the
Bayesian network for model interrogation, propagation and learning, while more
naturally encoding asymmetric state spaces and the order in which events
happen. In this paper we demonstrate how with complete sampling, conjugate
closed form model selection based on product Dirichlet priors is possible, and
prove that suitable homogeneity assumptions characterise the product Dirichlet
prior on this class of models. We demonstrate our techniques using two
educational examples.
###### keywords:
chain event graphs , Bayesian model selection , Dirichlet distribution
††journal: Journal of Multivariate Analysis
## 1 Introduction
Bayesian networks (BNs) are currently one of the most widely used graphical
models for representing and analysing finite discrete graphical multivariate
distributions with their explicit coding of conditional independence
relationships between a system’s variables [1, 2]. However, despite their
power and usefulness, it has long been known that BNs cannot fully or
efficiently represent certain common scenarios. These include situations where
the state space of a variable is known to depend on other variables, or where
the conditional independence between variables is itself dependent on the
values of other variables. Some examples of such latter scenarios are given by
Poole and Zhang [3]. In order to overcome such deficiencies, enhancements have
been proposed to the basic Bayesian network in order to create so-called
“context-specific” Bayesian networks [3]. These have their own problems,
however: either they represent too much of the information about a model in a
non-graphical way, thus undermining the rationale for using a graphical model
in the first place, or they struggle to represent a general class of models
efficiently. Other graphical approaches that seek to account for “context-
specific” beliefs suffer from similar problems.
This has led to the proposal of a new graphical model — the chain event graph
(CEGs) — which first propounded in [4]. As well as solving the aforementioned
problems associated with Bayesian networks and related graphical models, CEGs
are able, not unrelatedly, to encode far more efficiently the common structure
in which models are elicited — as asymmetric processes — in a single graph. To
this end, CEGs are based not on Bayesian networks, but on event trees (ETs)
[5]. Event trees are trees where nodes represent situations — i.e. scenarios
in which a unit might find itself — and each node’s extending edges represent
possible future situations that can develop from the current one. It follows
that every atom of the event space is encoded by exactly one root-to-leaf
path, and each root-to-leaf path corresponds to exactly one atomic event. It
has been argued that ETs are expressive frameworks to directly and accurately
represent beliefs about a process, particularly when the model is described
most naturally, as in the example below, through how situations might unfold
[5]. However, as explained in [4], ETs can contain excessive redundancy in
their structure, with subtrees describing probabilistically isomorphic
unfoldings of situations being represented separately. They are also unable to
explicitly express a model’s non-trivial conditional independences. The CEG
deals with these shortcomings by combining the subtrees that describe
identical subprocesses (see [4] for further details), so that the CEG derived
from a particular ET has a simpler topology while in turn expressing more
conditional independence statements than is possible through an ET.
We illustrate the construction and the types of symmetries it is possible to
code using a CEG with the following running example.
###### Example 1
Successful students on a one year programme study components $A$ and $B$, but
not everyone will study the components in the same order: each student will be
allocated to study either module $A$ or $B$ for the first 6 months and then
the other component for the final 6 months. After the first 6 months each
student will be examined on their allocated module and be awarded a
distinction (denoted with $D$), a pass ($P$) or a fail ($F$), with an
automatic opportunity to resit the module in the last case. If they resit then
they can pass and be allowed to proceed to the other component of their
course, or fail again and be permanently withdrawn from the programme.
Students who have succeeded in proceeding to the second module can again
either fail, pass or be awarded a distinction. On this second round, however,
there is no possibility of resitting if the component is failed. With an
obvious extension of the labelling, this system can be depicted by the event
tree given in Figure 1.
[treemode=R,nodesep=1pt]$V_{0}$$A$ $F_{1,A}$$F_{R,A}$ $P_{R,A}$$F_{2,R,B}$
$P_{2,R,B}$$D_{2,R,B}$ $P_{1,A}$$F_{2,B}$ $P_{2,B}$ $D_{2,B}$
$D_{1,A}$$F_{2,B}$ $P_{2,B}$ $D_{2,B}$ $B$$F_{1,B}$ $F_{R,B}$
$P_{R,B}$$F_{2,R,A}$ $P_{2,R,A}$ $D_{2,R,A}$ $P_{1,B}$$F_{2,A}$ $P_{2,A}$
$D_{2,A}$ $D_{1,B}$$F_{2,A}$ $P_{2,A}$ $D_{2,A}$
Figure 1: Event tree of a student’s potential progress through a hypothetical
course described in Example 1. Each non-leaf node represents a juncture at
which a random event will take place, with the selection of possible outcomes
represented by the edges emanating from that node. Each edge distribution is
defined conditional on the path passed through earlier in the tree to reach
the specific node.
To specify a full probability distribution for this model it is sufficient to
only specify the distributions associated with the unfolding of each situation
a student might reach. However, in many applications it is often natural to
hypothesise a model where the distribution associated with the unfolding from
one situation is assumed identical to another. Situations that are thus
hypothesised to have the same transition probabilities to their children are
said to be in the same _stage_. Thus in Example 1 suppose that as well as
subscribing to the ET of Figure 1 we want to consider a model also embodying
the following three hypotheses:
1. 1.
The chances of doing well in the second component are the same whether the
student passed first time or after a resit.
2. 2.
The components $A$ and $B$ are equally hard.
3. 3.
The distribution of marks for the second component is unaffected by whether
students passed or got a distinction for the first component.
These hypotheses can be identified with a partitioning of the non-leaf nodes
(situations). In Figure 1 the set of situations is
$\mathcal{S}=\\{V_{0},A,B,P_{1,A},P_{1,B},D_{1,A},D_{1,B},F_{1,A},F_{1,B},P_{R,A},P_{R,B}\\}.$
The partition $C$ of $\mathcal{S}$ that encodes exactly the above three
hypotheses consists of the stages $u_{1}=\left\\{A,B\right\\}$,
$u_{2}=\left\\{F_{1,A},F_{1,B}\right\\}$, and
$u_{3}=\left\\{P_{1,A},P_{1,B},P_{R,A},P_{R,B},D_{1,A},D_{1,B}\right\\}$
together with the singleton $u_{0}=\left\\{V_{0}\right\\}$. Thus the second
stage $u_{2}$, for example, implies that the probabilities on the edges
$\left(F_{1,B},F_{R,B}\right)$ and $\left(F_{1,A},F_{R,A}\right)$ are equal,
as are the probabilities on $\left(F_{1,B},P_{R,B}\right)$ and
$\left(F_{1,A},P_{R,A}\right)$. Clearly the joint probability distribution of
the model – whose atoms are the root to leaf paths of the tree – is determined
by the conditional probabilities associated with the stages. A CEG is the
graph that is constructed to encode a model that can be specified through an
event tree combined with a partitioning of its situations into stages.
In this paper we suppose that we are in a context similar to that of Example
1, where, for any possible model, the sample space of the problem must be
consistent with a single event tree, but where on the basis of a sample of
students’ records we want to select one of a number of different possible CEG
models, i.e. we want to find the “best” partitioning of the situations into
stages. We take a Bayesian approach to this problem and choose the model with
the highest posterior probability — the Maximum A Posteriori (MAP) model. This
is the simplest and possibly most common Bayesian model selection method,
advocated by, for example, Dennison et al [6], Castelo [7], and Heckerman [8],
the latter two specifically for Bayesian network selection.
The paper is structured as follows. In the next section we review the
definitions of event trees and CEGs. In Section 3 we develop the theory of how
conjugate learning of CEGs is performed. In Section 4 we apply this theory by
using the posterior probability of a CEG as its score in a model search
algorithm that is derived using an analogous procedure to the model selection
of BNs. We characterise the product Dirichlet distribution as a prior
distribution for the CEGs’ parameters under particular homogeneity conditions.
In Section 5 the algorithm is used to discover a good explanatory model for
real students’ exam results. We finish with a discussion.
## 2 Definitions of event trees and chain event graphs
In this section we briefly define the event tree and chain event graph. We
refer the interested reader to [4] for further discussion and more detail
concerning their construction. Bayesian networks, which will be referenced
throughout the paper, have been defined many times before. See [8] for an
overview.
### 2.1 Event Trees
Let $T=(V(T),E(T))$ be a directed tree where $V(T)$ is its node set and $E(T)$
its edge set. Let $S(T)=\\{v:v\in V(T)-L(T)\\}$ be the set of situations of
$T$, where $L(T)$ is the set of leaf (or terminal) nodes. Furthermore, define
$\mathbb{X}=\\{\lambda(v_{0},v):v\in V(T)\backslash S(T)\\}$, where
$\lambda(a,b)$ is the path from node $a$ to node $b$, and $v_{0}$ is the root
node, so that $\mathbb{X}$ is the set of root-to-leaf paths of $T$. Each
element of $\mathbb{X}$ is called an atomic event, each one corresponding to a
possible unfolding of events through time by using the partial ordering
induced by the paths. Let $\mathbb{X}(v)$ denote the set of children of $v\in
V(T)$. In an event tree, each situation $v\in S(T)$ has an associated random
variable $X(v)$ with sample space $\mathbb{X}(v)$, defined conditional on
having reached $v$. The distribution of $X(v)$ is determined by the primitive
probabilities
$\\{\pi(v^{\prime}|v)=p(X(v)=v^{\prime}):v^{\prime}\in\mathbb{X}(v)\\}$. With
random variables on the same path being mutually independent, the joint
probability of events on a path can be calculated by multiplying the
appropriate primitive probabilities together. Each primitive probability
$\pi(v^{\prime}|v)$ is a colour for the directed edge $e=(v,v^{\prime})$, so
that we can have $\pi(e)=\pi(v^{\prime}|v)$.
###### Example 2
Figure 2 shows a tree for two Bernoulli random variables, $X$ and $Y$, with
$X$ occurring before $Y$. In an educational example $X$ could be the indicator
variable of a student passing one module, and $Y$ the indicator variable for a
subsequent module.
[treemode=R,nodesep=1pt]$v_{0}$$v_{1}$ $v_{3}$ $v_{4}$ $v_{2}$$v_{5}$ $v_{6}$
Figure 2: Simple event tree. The non-zero-probability events in the joint
probability distribution of two Bernoulli random variables, $X$ and $Y$, with
$X$ observed before $Y$, can be represented by this tree. Here, all four joint
states are possible, because there are four root-to-leaf paths through the
nodes.
Here we have random variables $X(v_{0})=X$, $X(v_{1})=Y|(X=0)$ and
$X(v_{2})=Y|(X=1)$, and primitive probabilities $\pi(v_{1}|v_{0})=p(X=0)$,
$\pi(v_{3}|v_{1})=p(Y=0|X=0)$ and so on for every other edge. Joint
probabilities can be found by multiplying primitive probabilities along a
path, e.g. $p(X=0,Y=0)=p(X=0)p(Y=0|X=0)=\pi(v_{1}|v_{0})\pi(v_{3}|v_{1})$ as
$v_{0}$ and $v_{1}$ are on a path.
### 2.2 Chain Event Graphs
Starting with an event tree $T$, define a floret of $v\in S(T)$ as
$\mathcal{F}(v,T)=\left(V\left(\mathcal{F}(v,T)\right),E\left(\mathcal{F}(v,T)\right)\right)$
where $V(\mathcal{F}(v,T))=\\{v\\}\cup\\{v^{\prime}\in V(T):(v,v^{\prime})\in
E(T)\\}$ and $E(\mathcal{F}(v,T))=\\{e\in E(T):e=(v,v^{\prime})\\}$. The
floret of a vertex $v$ is thus a sub-tree consisting of $v$, its children, and
the edges connecting $v$ and its children, as shown in Figure 3. This
represents, as defined in section 2.1, the random variable $X(v)$ and its
sample space $\mathbb{X}(v)$.
[treemode=R,nodesep=1pt]$v$$v_{1}$ $v_{2}$ … $v_{k-1}$ $v_{k}$
Figure 3: Floret of $v$. This subtree represents both the random variable
$X(v)$ and its state space $\mathbb{X}(v)$.
One of the redundancies that can be eliminated from an ET is that of the
florets’ edges of two situations, $v$ and $v^{\prime}$ say, which have
identical associated edge probabilities despite being defined by different
conditioning paths. We say these two situations are at the same stage. This
concept is formally defined as follows.
###### Definition 3
Two situations $v,v^{\prime}\in S(T)$ are in the same stage $u$ if and only if
$X(v)$ and $X(v^{\prime})$ have the same distribution under a bijection
$\psi_{u}(v,v^{\prime}):E(\mathcal{F}(v,T))\rightarrow
E(\mathcal{F}(v^{\prime},T))$
i.e.
$\psi_{u}(v,v^{\prime}):\mathbb{X}(v)\rightarrow\mathbb{X}(v^{\prime})$
The set of stages of an ET $T$ is written $J(T)$. This set partitions the set
of situations $S(T)$.
We can construct a staged tree $\mathcal{G}(T,L(T))$ with
$V(\mathcal{G})=V(T)$, $E(\mathcal{G})=E(T)$, and colour its edges such that:
* 1.
If $v\in u$ and $u$ contains no other vertices, then all $(v,v^{*})\in
E(\mathcal{G})$ are left uncoloured;
* 2.
If $v\in u$ and $u$ contains other vertices, then all $(v,v^{*})\in
E(\mathcal{G})$ are coloured; and
* 3.
Whenever $e(v,v^{*})\mapsto e(v^{\prime},v^{\prime*})$ under
$\psi_{u}(v,v^{\prime})$, then the two edges must have the same colour.
There is another type of situation that is of further interest. When the whole
development from two situations $v$ and $v^{\prime}$ have identical
distributions, i.e. there exists a bijection between their respective subtrees
similar to that between stages as defined in Definition 3, then the situations
are said to be in the same position. This is defined formally as follows.
###### Definition 4
Two situations $v,v^{\prime}\in S(T)$ are in the same position $w$ if and only
if there exists a bijection
$\phi_{w}(v,v^{\prime}):\Lambda(v,T)\rightarrow\Lambda(v^{\prime},T)$
where $\Lambda(v,T)$ is the set of paths in $T$ from $v$ to a leaf node of
$T$, such that
* 1.
all edges in all of the paths in $\Lambda(v,T)$ and $\Lambda(v^{\prime},T)$
are coloured in $\mathcal{G}(T,L(T))$; and
* 2.
for every path $\lambda(v)\in\Lambda(v,T)$, the ordered sequence of colours in
$\lambda(v)$ equals the ordered sequence of colours in
$\lambda(v^{\prime}):=\phi_{w}(v,T)(\lambda(v))\in\Lambda(v^{\prime},T)$
This ensures that when $v$ and $v^{\prime}$ are in the same position, then
under the map $\phi_{w}(v,v^{\prime})$ future development from either node
follows identical probability distributions.
We denote the set of positions as $K(T)$. Positions are an obvious way of
equating situations, because the different conditioning variables of different
nodes in the same position have no effect on any subsequent development. It is
clear that $K(T)$ is a finer partition of $V(T)$ than $J(T)$, and indeed that
$J(T)$ partitions $K(T)$, as situations in the same position will also be in
the same stage.
We now use stages and positions to compress the event tree into a chain event
graph. First, the probability graph of the event tree
$\mathcal{H}(\mathcal{G}(T))=\mathcal{H}(T)=(V(\mathcal{H}),E(\mathcal{H}))$
is drawn, where $V(\mathcal{H})=K(T)\cup\\{w_{\infty}\\}$ and $E(\mathcal{H})$
is constructed as follows.
* 1.
For each pair of positions $w,w^{\prime}\in K(T)$, if there exists
$v,v^{\prime}\in S(T)$ such that $v\in w$,$v^{\prime}\in w^{\prime}$ and
$e(v,v^{\prime})\in E(T)$, then an associated edge $e(w,w^{\prime})\in
E(\mathcal{H})$ is drawn. Furthermore, if for a position $w$ there exists
$v\in S(T)$, $v^{\prime}\in L(T)$ and $e(v,v^{\prime})\in E(T)$ such that
$v\in w$, then an associated edge $e(w,w_{\infty})\in E(\mathcal{H})$ is
drawn.
* 2.
The colour of this edge, $e(w,w^{\prime})$, is the same as the colour of the
associated edge $e(v,v^{\prime})$.
Now the CEG can finally be constructed by taking the probability graph
$\mathcal{H}(T)$ and connecting the positions that are in the same stage using
undirected edges: Let $\mathcal{C}(T)$ be a mixed graph with vertex set
$V(\mathcal{C})=V(\mathcal{H})$, directed edge set
$E_{d}(\mathcal{C})=E(\mathcal{H})$, and undirected edge set
$E_{u}(\mathcal{C})=\\{(w,w^{\prime}):u(w)=u(w^{\prime}),\,w,w^{\prime}\in
V(\mathcal{C})\\}$.
An example of a CEG that could be constructed from the event tree in Figure 1
is shown in Figure 5.
## 3 Conjugate learning of CEGs
One convenient property of CEGs is that conjugate updating of the model
parameters proceeds in a closely analogous fashion to that on a BN. Conjugacy
is a crucial part of the model selection algorithm that will be described in
Section 4, because it leads to closed form expressions for the posterior
probabilities of candidate CEGs. This in turn makes it possible to search the
often very large model space quickly to find optimal models. We demonstrate
here how a conjugate analysis on a CEG proceeds.
Let a CEG $C$ have set of stages $J(C)=\\{u_{1},\dots,u_{k}\\}$, and let each
stage $u_{i}$ have $k_{i}$ emanating edges (labelled $e_{1},\dots,e_{k_{i}}$)
with associated probability vector
$\boldsymbol{\pi}_{i}=(\pi_{i1},\pi_{i2},\ldots,\pi_{ik_{i}})^{\prime}$ (where
$\sum_{j=1}^{k_{i}}\pi_{ij}=1$ and $\pi_{ij}>0$ for $j\in\\{1,\dots,k\\}$).
Then, under random sampling, the likelihood of the CEG can be decomposed into
a product of the likelihood of each probability vector, i.e.
$p(\boldsymbol{x}|\boldsymbol{\pi},C)=\prod\limits_{i=1}^{k}p_{i}(\boldsymbol{x}_{i}|\boldsymbol{\pi}_{i},C)$
where
$\boldsymbol{\pi}=\left\\{\boldsymbol{\pi}_{1},\boldsymbol{\pi}_{2},\ldots,\boldsymbol{\pi}_{k}\right\\}$,
and
$\boldsymbol{x}=\left\\{\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{k}\right\\}$
is the complete sample data such that each
$\boldsymbol{x}_{i}=(x_{i1},\dots,x_{ik_{i}})^{\prime}$ is the vector of the
number of units in the sample (for example, the students in Example 1) that
start in stage $u_{i}$ and move to the stage at the end of edge $e_{ij}$ for
$j\in\\{1,\dots,k_{i}\\}$.
If it is further assumed that
$\boldsymbol{x}_{i}\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}\boldsymbol{x}_{j}|\boldsymbol{\pi},\forall
i\neq j$ then
$p_{i}(\boldsymbol{x}_{i}|\boldsymbol{\pi}_{i},C)=\prod\limits_{j=1}^{k_{i}}\pi_{ij}^{x_{ij}}$
(1)
Thus, just as for the analogous situation with BNs, the likelihood of a random
sample also separates over the components of $\boldsymbol{\pi}$. With BNs, a
common modelling assumption is of local and global independence of the
probability parameters [9]; the corresponding assumption here is that the
parameters
$\boldsymbol{\pi}_{1}$,$\boldsymbol{\pi}_{2}$,$\ldots$,$\boldsymbol{\pi}_{k}$
of $\boldsymbol{\pi}$ are all mutually independent a priori. It will then
follow, with the separable likelihood, that they will also be independent a
posteriori.
If the probabilities $\boldsymbol{\pi}_{i}$ are assigned a Dirichlet
distribution, $\operatorname{Dir}(\boldsymbol{\alpha}_{i})$, a priori, where
$\boldsymbol{\alpha}_{i}=(\alpha_{i1},\alpha_{i2},\ldots,\alpha_{ik_{i}})^{\prime}$,
so that for values of $\pi_{ij}$ such that $\sum_{j=1}^{k_{i}}\pi_{ij}=1$ and
$\pi_{ij}>0$ for $1\leq j\leq k_{i}$, the density of $\boldsymbol{\pi}_{i}$,
$q_{i}(\boldsymbol{\pi}_{i}|C)$, can be written
$q_{i}(\boldsymbol{\pi}_{i}|C)=\frac{\Gamma(\alpha_{i1}+\ldots+\alpha_{ik_{i}})}{\Gamma(\alpha_{i1})\ldots\Gamma(\alpha_{ik_{i}})}\prod\limits_{j=1}^{k_{i}}\pi_{ij}^{\alpha_{ij}-1}$
where $\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt$ is the Gamma function. It
then follows that $\boldsymbol{\pi}_{i}|\boldsymbol{x}$
$(=\boldsymbol{\pi}_{i}|\boldsymbol{x}_{i})$ also has a Dirichlet
distribution, $\operatorname{Dir}(\boldsymbol{\alpha}_{i}^{\ast})$, a
posteriori, where
$\boldsymbol{\alpha}^{*}_{i}=(\alpha_{i1}^{*},\dots,\alpha_{ik_{i}}^{*})^{\prime}$,
$\alpha_{ij}^{\ast}=\alpha_{ij}+x_{ij}$ for $1\leq j\leq k_{i},1\leq i\leq k$.
The marginal likelihood of this model can be written down explicitly as the
function of the prior and posterior Dirichlet parameters:
$p(\boldsymbol{x}|C)=\prod_{i=1}^{k}\left[\frac{\Gamma(\sum_{j}\alpha_{ij})}{\Gamma(\sum_{j}\alpha_{ij}^{*})}\prod_{j=1}^{k_{i}}\frac{\Gamma(\alpha_{ij}^{*})}{\Gamma(\alpha_{ij})}\right].$
The computationally more useful logarithm of the marginal likelihood is
therefore a linear combination of functions of $\alpha_{ij}$ and
$\alpha_{ij}^{*}$. Explicitly,
$\log
p(\boldsymbol{x}|C)=\sum_{i=1}^{k}{\left[s(\boldsymbol{\alpha}_{i})-s(\boldsymbol{\alpha}_{i}^{\ast})\right]}+\sum_{i=1}^{k}{\left[t(\boldsymbol{\alpha}_{i}^{\ast})-t(\boldsymbol{\alpha}_{i})\right]}$
(2)
where for any vector $\mathbf{c}=(c_{1},c_{2},\dots,c_{n})^{\prime}$,
$s(\mathbf{c})=\log\Gamma(\sum_{v=1}^{n}{c_{v}})\mbox{ and
}t(\mathbf{c})=\sum_{v=1}^{n}{\log\Gamma(c_{v})}$ (3)
So the posterior probability of a CEG $C$ after observing $\boldsymbol{x}$,
$q(C|\boldsymbol{x})$, can be calculated using Bayes’ Theorem, given a prior
probability $q(C)$:
$\log q(C|\boldsymbol{x})=\log p(\boldsymbol{x}|C)+\log q(C)+K$ (4)
for some value $K$ which does not depend on $C$. This is the score that will
be used when searching over the candidate set of CEGs for the model that best
describes the data.
## 4 A Local Search Algorithm for Chain Event Graphs
### 4.1 Preliminaries
With the log marginal posterior probability of a CEG model, $\log
q(C|\boldsymbol{x})$, as its score, searching for the highest-scoring CEG in
the set of all candidate models is equivalent to trying to find the Maximum A
Posteriori (MAP) model [10]. The intuitive approach for searching
$\boldsymbol{C}$, the candidate set of CEGs — calculating
$q(C|\boldsymbol{x})$ (or $\log q(C|\boldsymbol{x})$) for every
$C\in\boldsymbol{C}$ and choosing
$C^{*}:=\max_{C}q(C|\boldsymbol{x})=\max_{C}\log q(C|\boldsymbol{x})$ — is
infeasible for any but the most trivial problems. We describe in this section
an algorithm for efficiently searching the model space by reformulating the
model search problem as a clustering problem.
As mentioned in Section 2.2, every CEG that can be formed from a given event
tree can be identified exactly with a partition of the event tree’s nodes into
stages. The coarsest partition $C_{\infty}$ has all nodes with $k$ outgoing
edges in the same stage, $u_{k}$; the finest partition $C_{0}$ has each
situation in its own stage, except for the trivial cases of those nodes with
only one outgoing edge. Defined this way, the search for the highest-scoring
CEG is equivalent to searching for the highest-scoring clustering of stages.
Various Bayesian clustering algorithm exist [11], including many involving
MCMC [12]. We show here how to implement an Bayesian agglomerative
hierarchical clustering (AHC) exact algorithm related to that of Heard et al
[13]. The AHC algorithm here is a local search algorithm that begins with the
finest partition of the nodes of the underlying ET model (called $C_{0}$ above
and henceforth) and seeks at each step to find the two nodes that will yield
the highest-scoring CEG if combined.
Some optional steps can be taken to simplify the search, which we will
implement here. The first of these involves the calculation of the scores of
the proposed models in the algorithm. By assuming that the probability
distributions of stages that are formed from the same nodes of the underlying
ET are equal in all CEGs, i.e.
$p(\boldsymbol{x_{i}}|\boldsymbol{\pi_{i}},C_{1})=p(\boldsymbol{x_{i}}|\boldsymbol{\pi_{i}},C_{2}),\forall
C_{1},C_{2}\in\boldsymbol{C}$, it becomes more efficient to calculate the
differences of model scores, i.e. the logarithms of the relevant Bayes
factors, than to calculate the two individual model scores absolutely. This is
because, if for two CEGs their stage sets $J(C_{1})$ and $J(C_{2})$ differ
only in that stages $u_{1a},u_{1b}\in C_{1}$ are combined into $u_{2c}\in
C_{2}$, with all other stages unchanged, then the calculation of the logarithm
of their posterior Bayes factor depends only on the stages involved; using the
notation of Equation (3),
$\displaystyle\log{\frac{q(C_{1}|\boldsymbol{x})}{q(C_{2}|\boldsymbol{x})}}$
$\displaystyle=\log{q(C_{1}|\boldsymbol{x})}-\log{q(C_{2}|\boldsymbol{x})}$
(5)
$\displaystyle=\log{q(C_{1})}-\log{q(C_{2})}+\log{q(\boldsymbol{x}|C_{1})}-\log{q(\boldsymbol{x}|C_{2})}$
(6)
$\displaystyle\begin{split}&=\log{q(C_{1})}-\log{q(C_{2})}+\sum_{i}{\left[s(\boldsymbol{\alpha}_{1i})-s(\boldsymbol{\alpha}_{1i}^{\ast})\right]}+\sum_{i}{\left[t(\boldsymbol{\alpha}_{1i}^{\ast})-t(\boldsymbol{\alpha}_{1i})\right]}\\\
&\qquad{}-\sum_{i}{\left[s(\boldsymbol{\alpha}_{2i})-s(\boldsymbol{\alpha}_{2i}^{\ast})\right]}-\sum_{i}{\left[t(\boldsymbol{\alpha}_{2i}^{\ast})-t(\boldsymbol{\alpha}_{2i})\right]}\end{split}$
(7)
$\displaystyle\begin{split}&=\log{q(C_{1})}-\log{q(C_{2})}+s(\boldsymbol{\alpha}_{1a})-s(\boldsymbol{\alpha}_{1a}^{*})+t(\boldsymbol{\alpha}_{1a}^{*})-t(\boldsymbol{\alpha}_{1a})\\\
&\qquad{}+s(\boldsymbol{\alpha}_{1b})-s(\boldsymbol{\alpha}^{*}_{1b})+t(\boldsymbol{\alpha}^{*}_{1b})-t(\boldsymbol{\alpha}_{1b})\\\
&\qquad\qquad{}-s(\boldsymbol{\alpha}_{2c})+s(\boldsymbol{\alpha}_{2c}^{*})-t(\boldsymbol{\alpha}_{2c}^{*})+t(\boldsymbol{\alpha}_{2c})\end{split}$
(8)
Using the trivial result that for any three CEGs
$\log q(C_{3}|\boldsymbol{x})-\log q(C_{2}|\boldsymbol{x})=\left[\log
q(C_{3}|\boldsymbol{x})-\log q(C_{1}|\boldsymbol{x})\right]-\left[\log
q(C_{2}|\boldsymbol{x})-\log q(C_{1}|\boldsymbol{x})\right],$
it can be seen that in the course of the AHC algorithm, comparing two proposal
CEGs from the current CEG can be done equivalently by comparing their log
Bayes factors with the current CEG, which as shown above requires fewer
calculations.
The calculation of the score for each CEG $C$, as shown by Equation (4), shows
that it is formed of two components: the prior probability of the CEG being
the true model and the marginal likelihood of the data. These must therefore
be set before the algorithm can be run, and it is here that the other
simplifications are made.
### 4.2 The prior over the CEG space
For any practical problem $\boldsymbol{C}$, the set of all possible CEGs for a
given ET, is likely to be a very large set, making setting a value for
$q(C),\forall C\in\boldsymbol{C}$ a non-trivial task. An obvious way to set a
non-informative or exploratory prior is to choose the uniform prior, so that
$q(C)=\frac{1}{\left|\boldsymbol{C}\right|}$. This has the advantages of being
simple to set and of eliminating the $\log{q(C_{1})}-\log{q(C_{2})}$ term in
Equation (8).
A more sophisticated approach is to consider which potential clusters are more
or less likely a priori, according to structural or causal beliefs, and to
exploit the modular nature of CEGs by stating that the prior log Bayes factor
of a CEG relative to $C_{0}$ is the sum of the prior log Bayes factors of the
individual clusters relative to their components completely unclustered, and
that these priors are modular across CEGs. This approach makes it simple to
elicit priors over $\boldsymbol{C}$ from a lay expert, by requiring the
elicitation only of the prior probability of each possible stage.
A particular computational benefit of this approach is when the prior Bayes
factor of any CEG $C$ with $C_{0}$ is believed to be zero, because one or more
of its clusters is considered to be impossible. This is equivalent in the
algorithm to not including the CEG in its search at all, as though it was
never in $\boldsymbol{C}$ in the first place, with the obvious simplification
of the search following.
### 4.3 The prior over the parameter space
Just as when attempting to set $q(C)$, the size of most CEGs in practise leads
to intractability of setting $p(\boldsymbol{x}|C)$ for each CEG $C$
individually. However, the task is again made possible by exploiting the
structure of a CEG with judicious modelling assumptions.
Assuming independence between the likelihoods of the stages for every CEG, so
that $p(\boldsymbol{x}|\boldsymbol{\pi},C)$ is as determined by Equation (1),
and the fact that $p(\boldsymbol{x}|C)=\int
p(\boldsymbol{x}|\boldsymbol{\pi},C)p(\boldsymbol{\pi}|C)d\boldsymbol{\pi}$,
it is clear that to set the marginal likelihood for each CEG is equivalent to
setting the prior over the CEG’s parameters, i.e. setting
$p(\boldsymbol{\pi}|C)$ for each $C$. With the two further structural
assumptions that the stage priors are independent for all CEGs (so that
$p(\boldsymbol{\pi}|C)=\prod_{i=1}^{k}p(\boldsymbol{\pi}_{i}|C)$) and that
equivalent stages in different CEGs have the same prior distributions on their
probability vectors, (i.e.
$p(\boldsymbol{\pi}_{i}|C_{1})=p(\boldsymbol{\pi}_{i}|C_{2})$), it can be seen
that the problem of setting $p(\boldsymbol{x}|\boldsymbol{\pi},C)$ is reduced
to setting the parameter priors of each non-trivial floret in $C_{0}$
($p(\boldsymbol{\pi}_{i}|C_{0}),i=1,\dots,k$) and the parameter priors of
stages that are clusters of stages of $C_{0}$.
The usual prior put on the probability parameters of finite discrete BNs is
the product Dirichlet distribution. In Geiger and Heckerman [14] the
surprising result was shown that a product Dirichlet prior is inevitable if
local and global independence are assumed to hold over all Markov equivalent
graphs on at least two variables. In this paper we show that a similar
characterisation can be made for CEGs given the assumptions in the previous
paragraph. We will first show that the floret parameters in $C_{0}$ must have
Dirichlet priors, and second that all CEGs formed by clustering the florets in
$C_{0}$ have Dirichlet priors on the stage parameters. One characterisation of
$C_{0}$ is given by Theorem 5.
###### Theorem 5
If it is assumed a priori that the rates at which units take the root-to-leaf
paths in $C_{0}$ are independent (“path independence”) and that the
probability of which edge units take after arriving at a situation $v$ is
independent of the rate at which units arrive at $v$ (“floret independence”),
then the non-trivial florets of $C_{0}$ have independent Dirichlet priors on
their probability vectors.
###### Proof 1
The proof is in the Appendix.
Thus $p(\boldsymbol{\pi}_{i}|C_{0})$ is entirely determined by the stated
rates $\gamma(\lambda)$ on the root-to-leaf paths $\lambda\in\Lambda(C_{0})$
of $C_{0}$. This is similar to the “equivalent sample sizes” method of
assessing prior uncertainty of Dirichlet hyperparameters in BNs as discussed
in Section 2 of Heckerman [8].
Another way to show that all non-trivial situations in $C_{0}$ have Dirichlet
priors on their parameter spaces is to use the characterisation of the
Dirichlet distribution first proven by Geiger and Heckerman [14], repeated
here as Theorem 6.
###### Theorem 6
Let $\\{\theta_{ij}\\},1\leq i\leq k,1\leq j\leq n,\sum_{ij}{\theta_{ij}}=1$,
where $k$ and $n$ are integers greater than 1, be positive random variables
having a strictly positive pdf $f_{U}(\\{\theta_{ij}\\})$. Define
$\theta_{i.}=\sum_{j=1}^{n}{\theta_{ij}}$,
$\theta_{I.}=\\{\theta_{i.}\\}_{i=1}^{k-1}$,
$\theta_{j|i}=\theta_{ij}/{\sum_{j}{\theta_{ij}}}$, and
$\theta_{J|i}=\\{\theta_{j|i}\\}_{j=1}^{n-1}$.
Then if $\\{\theta_{I.},\theta_{J|1},\dots,\theta_{J|k}\\}$ are mutually
independent, $f_{U}(\\{\theta_{ij}\\})$ is Dirichlet.
###### Proof 2
Theorem 2 of Geiger and Heckerman [14].
###### Corollary 7
If $C_{0}$ has a composite number $m$ of root-to-leaf paths and all Markov
equivalent CEGs have independent floret distributions then the vector of
probabilities on the root-to-leaf paths of $C_{0}$ must have a Dirichlet
prior. This means in particular that, from the properties of the Dirichlet
distribution, the floret of each situation with at least two outgoing edges
has a Dirichlet prior on its edges.
###### Proof 3
Construct an event tree $C_{0}^{\prime}$ with $m$ root-to-leaf paths, where
the floret of the root node $v_{0}^{\prime}$ has $k$ edges and each of the
florets extending from the children of $v_{0}^{\prime}$ have $n$ edges
terminating in leaf nodes, where $m=kn,k\geq 2,n\geq 2$. This will always be
possible with a composite $m$. $C_{0}^{\prime}$ describes the same atomic
events as $C_{0}$ with a different decomposition.
Let the random variable associated with the root floret of $C_{0}^{\prime}$ be
$X$, and let the random variable associated with each of the other florets be
$Y|X=i,i=1,\dots,k$. Let $\theta_{ij}=P(X=i,Y=j)$. Then by the definition of
event trees, $P(\theta_{ij}>0)>0,1\leq i\leq k,1\leq j\leq n$ and
$\sum\theta_{ij}=1$. By the notation of Theorem 6, $\theta_{i.}=P(X=i)$ and
$\theta_{j|i}=P(Y=j|X=i)$.
By hypothesis the floret distributions of $C_{0}^{\prime}$ are independent.
Therefore the condition of Theorem 6 holds and hence $f_{U}(\theta_{ij})$ is
Dirichlet. From the equivalence of the atomic events, the probability
distribution over the root-to-leaf path probabilities of $C_{0}$ is also
Dirichlet, and so by Lemma 16, all non-trivial florets of $C_{0}$ therefore
have Dirichlet priors on their probability vectors.
To show that the stage parameters of all the other CEGs in $\boldsymbol{C}$
have independent Dirichlet priors, an inductive approach will be taken.
Because of the assumption of consistency – that two identically composed
stages in different CEGs have identical priors on their parameter space – for
any given CEG $C$ whose stages all have independent Dirichlet priors on their
parameters spaces, it is known that another CEG $C^{*}$ formed by clustering
two stages $u_{1c},u_{2c}$ from $C$ into one stage $u_{c^{*}}$ will have
independent Dirichlet priors on all its stages apart from $u_{c^{*}}$. It is
thus only required to show that $\boldsymbol{\pi}_{c^{*}}$ has a Dirichlet
prior. We prove this result for a class of CEGs called regular CEGs.
###### Definition 8
A stage $u$ is regular if and only if every path $\lambda\in\Lambda(C)$
contains either one situation in $u$ or none of the situations in $u$.
###### Definition 9
A CEG is regular if and only if every situation $u\in\boldsymbol{u}(C)$ is
regular.
###### Theorem 10
Let $C$ be a regular CEG, and let $C^{*}$ be the CEG that is formed from $C$
by setting two of its stages, $u_{1c}$ and $u_{2c}$, as being in the same
stage $u_{c^{*}}$, where $u_{c^{*}}$ is a regular stage, with all other
attributes of the CEG unchanged from $C$.
If all stages in $C$ have Dirichlet priors, then assuming that equivalent
stages in different CEGs have equivalent priors, all stages in $C^{*}$ have
Dirichlet priors.
###### Proof 4
Without loss of generality, let all situations in $u_{1c}$ and $u_{2c}$ have
$s$ children each, and let the total number of situations in $u_{1c}$ and
$u_{2c}$ be $r$. Thus there are $r$ situations in $u_{c^{*}}$, each with $s$
children. By the assumption of prior consistency across stages, all stages in
$C^{*}$ have Dirichlet priors on their parameter spaces, so it is only
required to prove that $u_{c^{*}}$ has a Dirichlet prior.
Consider the CEG $C^{\prime}$ formed as follows: Let the root node of
$C^{\prime}$, $v_{0}$, have 2 children, $v_{1}$ and $v^{\prime}$. Let
$v^{\prime}$ be a terminal node, and let $v_{1}$ have $r$ children,
$\\{v_{1}(1),\dots,v_{1}(r)\\}$, which are equivalent to the situations in
$u_{c^{*}}$, including the property that they are in the same stage
$u_{c^{\prime}}$. Lastly, let the children of $\\{v_{1}(1),\dots,v_{1}(r)\\}$,
$\\{v_{1}(1,1),\dots,v_{1}(1,s),\dots,v_{1}(r,1),\dots,v_{1}(r,s)\\}$, be leaf
nodes in $C^{\prime}$.
By construction, the prior for $u_{c^{\prime}}$ is the same as that for
$u_{c^{*}}$.
Now construct another CEG $C^{*\prime}$ from $C^{\prime}$ by reversing the
order of the stages $v_{1}$ and $u_{c^{\prime}}$. The new CEG has root node
$v_{0}$ with the same distribution as $v_{0}\in C^{\prime}$. $v_{0}$ now has
two children $v^{\prime}$ – the same as before – and $v_{2}$, which has $s$
children $\\{v_{2}(1),\dots,v_{2}(s)\\}$ in the same stage. Each node
$v_{2}(i),i=1,\dots,s$ has $r$ children $v_{2}(i,1),\dots,v_{2}(i,r)$, all of
which are leaf nodes.
The two CEGs $C^{*\prime}$ and $C^{\prime}$ are Markov equivalent, as it is
clear that $P(v_{1}(i,j))=P(v_{2}(j,i)),i=1,\dots,r,j=1,\dots,s$. The
probabilities on the floret of $v_{2}$ are thus equal to the probabilities of
the situations in the stage of $u_{c^{\prime}}$, and hence $u_{c^{*}}$.
Because $v_{2}$ is a stage with only one situation, Theorem 5 implies that it
has a Dirichlet prior. Therefore $u_{c^{*}}$ has a Dirichlet prior.
An alternative justification for assigning a Dirichlet prior to any stage that
is formed by clustering situations with Dirichlet priors on their state spaces
can be obtained which does not depend on assuming Markov equivalency between
CEGs derived from different event trees by assuming a property analogous to
that of “parameter modularity” for BNs [15]. This property states that the
distribution over structures common to two CEGs should be identical.
###### Definition 11
Let $u$ be a stage in a CEG $C$ composed of the situations $v_{1},\dots,v_{n}$
from $C_{0}$, each of which has $m$ children $v_{i1},\dots,v_{im},i=1,\dots,n$
such that $v_{ij}$ are the same colour for all $i$ for each $j$. Then $u$ has
the property of margin equivalency if
$\displaystyle\pi_{uj}$ $\displaystyle=P(v_{1j}\mbox{ or }v_{2j}\mbox{ or
}\dots\mbox{ or }v_{nj}|v_{1}\mbox{ or }v_{2}\mbox{ or }\dots\mbox{ or
}v_{n})$ (9)
$\displaystyle=\frac{\sum_{i=1}^{n}{P(v_{ij})}}{\sum_{i=1}^{n}{P(v_{i})}}$
(10)
is the same for both $C$ and $C_{0}$ for $j=1,\dots,m$.
###### Definition 12
$C$ has margin equivalency if all of its stages have margin equivalency.
###### Theorem 13
Let $u_{c}$ be a stage as defined in Definition 11 with $m\geq 2$. Then
assuming independent priors between the situations for the associated finest-
partition CEG $C_{0}$ of $C$,
$\boldsymbol{\pi}_{v_{i}}\thicksim\operatorname{Dir}(\boldsymbol{\alpha}_{i})$
where $\boldsymbol{\alpha}_{i}=\left(\alpha_{i1},\dots,\alpha_{im}\right)$ for
each $v_{i}$, $i=1,\dots,n$. Furthermore, for both $C$ and $C_{0}$,
$\boldsymbol{\pi}_{u}\thicksim\operatorname{Dir}(\boldsymbol{\alpha}_{u})$,
where
$\boldsymbol{\alpha}_{u}=\left(\sum_{i}\alpha_{i1},\dots,\sum_{i}\alpha_{im}\right)$.
###### Proof 5
From Theorem [5] or Corollary [7], every non-trivial floret in $C_{0}$ has a
Dirichlet prior on its edges, which includes in this case the situations
$v_{1},\dots,v_{n}$.
Let $\gamma_{ij}=\gamma\pi_{ij}$ for $i=1,\dots,n,\>j=1,\dots,m$ for some
$\gamma\in\mathbb{R^{+}}$. Then it is a well-known fact that
$\gamma_{ij}\thicksim\operatorname{Gamma}(\alpha_{ij},\beta)$ for all $1\leq
i\leq n,1\leq j\leq m$ for some $\beta>0$, and that
$\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}_{j}\gamma_{ij}$. As
$\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}_{i}\boldsymbol{\pi}_{v_{i}}$,
$\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}_{ij}\gamma_{ij}$. Then by
Lemma 15, letting $I[j]$ be the set of edges
$\left\\{e_{ij}=e(v_{i},v_{ij}),i=1,\dots,n\right\\}$ for $j=1,\dots,m$,
$\boldsymbol{\pi}_{u}\thicksim\operatorname{Dir}(\sum_{i}\alpha_{i1},\dots,\sum_{i}\alpha_{im})$
By margin equivalency, $\boldsymbol{\pi}_{u}$ must be set the same way for
$C$.
Note that the posterior of $\boldsymbol{\pi}_{u}$ for a stage $u$ that is
composed of the $C_{0}$ situations $v_{1},\dots,v_{n}$ is thus
$\boldsymbol{\pi}_{u}|\boldsymbol{x}\sim\operatorname{Dir}(\boldsymbol{\alpha}_{u}^{*})$
where
$\boldsymbol{\alpha}_{u}^{*}=\boldsymbol{\alpha}_{u}+\boldsymbol{x}_{u}=\sum_{i=1}^{n}{\boldsymbol{\alpha}_{v_{n}}}+\sum_{i=1}^{n}{\boldsymbol{x}_{v_{n}}}$.
Equation (8), therefore, becomes
$\log{\frac{q(C_{1}|\boldsymbol{x})}{q(C_{2}|\boldsymbol{x})}}=\log{q(C_{1})}-\log{q(C_{2})}+s(\boldsymbol{\alpha}_{1a})-s(\boldsymbol{\alpha}_{1a}^{*})+t(\boldsymbol{\alpha}_{1a}^{*})-t(\boldsymbol{\alpha}_{1a})\\\
{}+s(\boldsymbol{\alpha}_{1b})-s(\boldsymbol{\alpha}^{*}_{1b})+t(\boldsymbol{\alpha}^{*}_{1b})-t(\boldsymbol{\alpha}_{1b})-s(\boldsymbol{\alpha}_{1a}+\boldsymbol{\alpha}_{1b})\\\
{}+s(\boldsymbol{\alpha}_{1a}^{*}+\boldsymbol{\alpha}^{*}_{1b})-t(\boldsymbol{\alpha}_{1a}^{*}+\boldsymbol{\alpha}^{*}_{1b})+t(\boldsymbol{\alpha}_{1a}+\boldsymbol{\alpha}_{1b})$
(11)
### 4.4 The algorithm
The algorithm thus proceeds as follows:
1. 1.
Starting with the initial ET model, form the CEG $C_{0}$ with the finest
possible partition, where all leaf nodes are placed in the terminal stage
$u_{\infty}$ and all nodes with only one emanating edge are placed in the same
stage. Calculate $\log q(C_{0}|\boldsymbol{x})$ using (4).
2. 2.
For each pair of situations $v_{i},v_{j}\in C_{0}$ with the same number of
edges, calculate
$\log{\frac{q(C_{1}^{*}|\boldsymbol{x})}{q(C_{0}|\boldsymbol{x})}}$ where
$C_{1}^{*}$ is the CEG formed by having $v_{i},v_{j}$ in the same stage and
keeping all others in their own stage; do not calculate if $q(C_{1}^{*})=0$.
3. 3.
Let
$C_{1}=\max_{C_{1}^{*}}(\log{\frac{q(C_{1}^{*}|\boldsymbol{x})}{q(C_{0}|\boldsymbol{x})}})$.
4. 4.
Now calculate $C_{2}^{*}$ for each pair of stages in $C_{1}$ except where
$q(C_{2}^{*})=0$, and record $C_{2}=\max(q(C_{2}^{*}|\boldsymbol{x}))$.
5. 5.
Continue for $C_{3}$, $C_{4}$ and so on until the coarsest partition
$C_{\infty}$ has been reached.
6. 6.
Find $C=\max(C_{0},C_{1},\dots,C_{\infty})$, and select this as the MAP model.
We note that the algorithm can also be run backwards, starting from
$C_{\infty}$ and splitting one cluster in two at each step. This has the
advantage of making the identification of positions in the MAP model easier.
## 5 Examples
### 5.1 Simulated data
To first demonstrate the efficacy of the algorithm described above we
implement the algorithm using simulated data for Example 1, where the CEG
generating the data was as known and described in Section 1. Figure 4 shows
the number of students in the sample who reached each situation in the tree.
[treemode=R,nodesep=1pt]$V_{0}$$A$500 $F_{1,A}$108$F_{R,A}$41
$P_{R,A}$67$F_{2,R,B}$25 $P_{2,R,B}$35 $D_{2,R,B}$7 $P_{1,A}$261$F_{2,B}$21
$P_{2,B}$182 $D_{2,B}$58 $D_{1,A}$131$F_{2,B}$2 $P_{2,B}$30 $D_{2,B}$99
$B$500$F_{1,B}$100 $F_{R,B}$40 $P_{R,B}$60 $F_{2,R,A}$23 $P_{2,R,A}$33
$D_{2,R,A}$4 $P_{1,B}$251$F_{2,A}$26 $P_{2,A}$175 $D_{2,A}$50
$D_{1,B}$159$F_{2,A}$3 $P_{2,A}$48 $D_{2,A}$108
Figure 4: The event tree from Example 1 with the numbers representing the
number of students in a simulated sample who reached each situation.
In this complete dataset the progress of 1000 students has been tracked
through the event tree. Half are assigned to take module $A$ first and the
other half $B$. By finding the MAP CEG model in the light of this data we may
find out whether the three hypotheses posed in the introduction are valid. We
repeat them here for convenience:
1. 1.
The chances of doing well in the second component are the same whether the
student passed first time or after a resit.
2. 2.
The components $A$ and $B$ are equally hard.
3. 3.
The distribution of marks for the second component is unaffected by whether
students passed or got a distinction for the first component.
We set a uniform prior on the CEG priors and on the root-to-leaf paths of
$C_{0}$, the finest partition of the tree, for illustration purposes. The
algorithm is then implemented as follows.
There are only two florets with two edges; with Beta(1,3) priors on each and a
Beta(2,6) prior on the combined stage, the log Bayes factor is -1.85. Carrying
out similar calculations for all the pairs of nodes with three edges, it is
first decided to merge the nodes $P_{1,A}$ and $P_{1,B}$, which has a log
Bayes factor of -3.76 against leaving them apart. Applying the algorithm to
the updated set of nodes and iterating, the CEG in Figure 5 is found to be the
MAP one.
$\psmatrix[mnode=circle]&w_{2}\\\
w_{0}w_{1}w_{3}w_{\infty}\endpsmatrix\psset{shortput=tablr,arrows=->,nodesep=4.0pt}{\psset{arcangle=35.0}\ncarc{->}{2,1}{2,2}}^{A}{\psset{arcangle=-35.0}\ncarc{->}{2,1}{2,2}}^{B}{\psset{arcangle=35.0}\ncarc{->}{2,2}{1,3}}^{F_{1}}{\psset{arcangle=35.0}\ncarc{->}{2,2}{2,3}}^{P_{1}}{\psset{arcangle=-35.0}\ncarc{->}{2,2}{2,3}}^{D_{1}}{\ncline{->}{1,3}{2,6}}>{F_{R}}{\ncline{->}{1,3}{2,3}}>{P_{R}}{\psset{arcangle=10.0}\ncarc{->}{2,3}{2,6}}{\ncline{->}{2,3}{2,6}}{\psset{arcangle=-10.0}\ncarc{->}{2,3}{2,6}}$
Figure 5: The MAP CEG for that event tree in Figure 4
Under this model, it can be seen that all three hypotheses above are satisfied
and that the MAP model is the correct one.
### 5.2 Student test data
In our second example we apply the learning algorithm to a real dataset in
order to test the algorithm’s efficacy in a real-life situation and to
identify remaining issues with its usage. The dataset we used was an
appropriately disguised set of marks taken over a 10-year period from four
core modules of the MORSE degree course taught at the University of Warwick. A
part of the event tree used as the underlying model for the first two modules
is shown in Figure 6, along with a few illustrative data points. This is a
simplification of a much larger study that we are currently investigating but
large enough to illustrate the richness of inference possible with our model
search.
* [tnpos=l]1036 * [tnpos=l]936^$A$ * [tnpos=a]601^$1$ * [tnpos=a]601 * 288 * 272 * 41 * * [tnpos=a]257^$2$ * * * * * * [tnpos=a]78^$3$ * * * * * * [tnpos=l]100^$NA$ * * * * *
Figure 6: Sub-tree of the event tree of possible grades for the MORSE degree
course at the University of Warwick. Each floret of two edges describes
whether a student’s marks are available for a particular module (denoted by
the edge labelled $A$ for the first module) or whether they are missing
($NA$). If they are available, then they are counted as grade 1 if are 70% or
higher, grade 2 if they are between 50% and 69% inclusive, and grade 3 if they
are below 50%. Some illustrative count data are shown on corresponding nodes.
For simplicity, the prior distributions on the candidate models and on the
root-to-leaf paths for $C_{0}$ were both chosen to be uniform distributions.
The MAP CEG model was not $C_{0}$, so that there were some non-trivial stages.
In total, 170 situations were clustered into 32 stages. Some of the more
interesting stages of this model are described in Table 1.
Stage | Probability vector | Students | Situations | Locations | Comments
---|---|---|---|---|---
7 | (0.47, 0.44, 0.08) | 685 | 2 | 1; 1,1,1 | High achievers
11 | (0.22, 0.43, 0.35) | 412 | 6 | 3; 1,2; 3,1; 1,1,3 | Middling students
13 | (0.33, 0.33, 0.33) | 16 | 18 | 4; 4,2; 4,3 | No students appeared in 17 of these situations
17 | (0.07, 0.27, 0.66) | 86 | 4 | 1,3; 3,2; 3,2,4 | Struggling students
27 | (0.19, 0.56, 0.25) | 464 | 7 | 1,1,4; 1,2,2; 1,3,2; 1,4,2 | More likely to get grade 2 than stage 11
28 | (0.11, 0.51, 0.38) | 436 | 6 | 1,2,3; 3,1,3; 1,2,4 | More likely to get grade 3 than stage 27
Table 1: Selected stages of MAP CEG model formed from data described in
Section 5.2. The columns respectively detail the stage number, posterior
expectation of the probability vector of that stage (rounded to two decimal
places), number of students passing through that stage in the dataset, number
of situations from the original ET in that stage, examples of situations in
that stage (shown as sequence of grades, where “4” means that grade is
missing), and any comments or observations related to that stage.
From inspecting the membership of stages it was possible to identify various
situations which were discovered to share distributions. From example,
students who reach one of the two situations in stage 7 have an expected
probability of 0.47 in getting a high mark, an expected probability of 0.44 of
getting a middling grade, and only an expected probability of 0.08 of
achieving the lowest grade. From being in a stage of their own, it can be
deduced that students in these situations have qualitatively different
prospects from students in any other situations. In contrast, students who
reach one of the four situations in stage 17 have an expected probability of
0.66 of getting the lowest grade.
## 6 Discussion
In this paper we have shown that chain event graphs are not just an efficient
way of storing the information contained in an event tree, but also a natural
way to represent the information that is most easily elicited from a domain
expert: the order in which events happen, the distributions of variables
conditional on the process up to the point they are reached, and prior beliefs
about the relative homogeneity of different situations. This strength is
exploited when the MAP CEG is discovered, as this can be used in a qualitative
fashion to detect homogeneity between seemingly disparate situations.
There are a number extensions to the theory in this paper that are currently
being pursued. These fall mostly into the two categories: creating even richer
model classes than those considered here; and developing even more efficient
algorithms for selecting the MAP model in these model classes.
The first category includes dynamic chain event graphs. This framework can
supply a number of different model classes. The simplest case involves
selecting a CEG structure that is constant across time, but with a time series
on its parameters. A bigger class would allow the MAP CEG structure to change
over time. These larger model classes would clearly be useful in the
educational setting considered in this paper, as they would allow for
background changes in the students’ abilities, for example.
Another important model class is that which arises from uncertainty about the
underlying event tree. A similar model search algorithm to the one described
in this paper is possible in this case after setting a prior distribution on
the candidate event trees.
In order to search any of these model classes more effectively, the problem of
finding the MAP model can be reformulated as a weighted MAX-SAT problem, for
which algorithms have been developed. This approach was used to great effect
for finding a MAP BN by Cussens [16].
## Appendix
Theorem 5 is based on three well-known results concerning properties of the
Dirichlet distribution, which we review below.
###### Lemma 14
Let $\gamma_{j}\thicksim\operatorname{Gamma}(\alpha_{j},\beta),j=1,\dots,n$
where $\alpha_{j}>0$ for $j\in\\{1,\dots,n\\}$, $\beta>0$ and
$\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}\limits_{i\in\\{1\dots
n\\}}\gamma_{i}$. Furthermore, let $\theta_{j}=\frac{\gamma_{j}}{\gamma}$ for
$j\in\\{1,\dots,n\\}$, where $\gamma=\sum_{i=1}^{n}{\gamma_{i}}$.
Then
$\boldsymbol{\theta}=\operatorname*{\left(\theta_{i}\right)}_{i=\\{1,\dots,n\\}}\thicksim\operatorname{Dir}\left(\alpha_{1},\dots,\alpha_{n}\right)$.
###### Proof 6
Kotz et al [17].
###### Lemma 15
Let $I[j]\subseteq\\{1,\dots,n\\}$, $\gamma(I[j])=\sum_{i\in I[j]}\gamma_{i}$
and $\theta(I[j])=\sum_{i\in I[j]}\theta_{i}$.
Then for any partition $I=\\{I[1],\dots,I[k]\\}$ of $\\{1,\dots,n\\}$,
$\theta(I)=(\theta(I[1]),\theta(I[2]),\dots,\theta(I[k]))\thicksim\operatorname{Dir}\left(\alpha(I[1]),\dots,\alpha(I[k])\right)$
where $\alpha(I[j])=\sum_{i\in I[j]}{\alpha_{i}}$.
###### Proof 7
For any $I[j]\subseteq\\{1,\dots,n\\}$,
$\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}\limits_{i\in
I[j]}\gamma_{i}$,
$\gamma(I[j])\thicksim\operatorname{Gamma}{\left(\alpha(I[j]),\beta\right)}$
(a well-known result; see, for example, Weatherburn [18]), and for any
partition $I=\\{I[1],\dots,I[k]\\}$ of $\\{1,\dots,n\\}$,
$\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}\limits_{i\in\\{1,\dots,k\\}}\gamma(I[j])$.
Therefore, as
$\theta(I[j])=\sum_{i\in I[j]}{\theta_{i}}=\sum_{i\in
I[j]}{\frac{\gamma_{i}}{\gamma}}=\frac{\gamma(I[j])}{\gamma},\quad
j=1,\dots,k$
and $\gamma=\sum_{i=1}^{k}{\gamma(I[i])}$, the result follows from Lemma 14.
###### Lemma 16
For any $I[j]\subseteq\\{1,\dots,n\\}$ where $\left|I[j]\right|\geq 2$,
$\theta_{I[j]}=\left(\frac{\theta_{i}}{\theta(I[j])}\right)_{i\in
I[j]}\thicksim\operatorname{Dir}\left((\alpha_{i})_{i\in I[j]}\right)$
###### Proof 8
Wilks [19].
###### Theorem 17
Let the rates of units along the root-to-leaf paths
$\lambda_{i}\in\Lambda,i\in\\{1,\dots,\left|\Lambda\right|\\}$ of an event
tree $T$ have independent Gamma distributions with the same scale parameter,
i.e.
$\gamma_{i}=\gamma(\lambda_{i})\thicksim\operatorname{Gamma}(\alpha_{i},\beta),i\in\\{1,\dots,\left|\Lambda\right|\\}$
and
$\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}\limits_{i\in\\{1,\dots,\left|\Lambda\right|\\}}\gamma_{i}$.
Then the distribution on each floret in the tree will be Dirichlet.
###### Proof 9
Consider a floret $\mathcal{F}$ with root node $v$ and edge set
$\\{e_{1},\dots,e_{l}\\}$. The rate for each edge $e_{i}$, $\gamma(e_{i})$, is
equal to $\gamma(\lambda_{e_{i}})$, where $\lambda_{e_{i}}$ is the root-to-
leaf path that intersects with $e_{i}$, so that
$\gamma(e_{i})\thicksim\operatorname{Gamma}(\alpha_{e_{i}},\beta)$ and
$\operatorname*{{\;\bot\\!\\!\\!\\!\\!\\!\bot\;}}\limits_{i\in\\{1,\dots,l\\}}\gamma(e_{i})$.
Let $I=\\{I[\mathcal{F}],I[\mathcal{\overline{F}}]\\}$ partition $\Lambda$,
where $I[\mathcal{F}]=\\{\lambda_{e_{1}},\dots,\lambda_{e_{l}}\\}$ and
$I[\mathcal{\overline{F}}]=I-I[\mathcal{F}]$. Then by Lemma 16, the
probability vector on $\mathcal{F}$ is Dirichlet, where
$\theta_{I[\mathcal{F}]}\thicksim\operatorname{Dir}\left((\alpha_{e_{i}})_{i\in\\{1,\dots,l\\}}\right)$
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* [13] N. A. Heard, C. C. Holmes, D. A. Stephens, A quantitative study of gene regulation involved in the immune response of anopheline mosquitoes: An application of bayesian hierarchical clustering of curves, Journal of the American Statistical Association 101 (473) (2006) 18–29.
* [14] D. Geiger, D. Heckerman, A characterization of the dirichlet distribution through global and local parameter independence, The Annals of Statistics 25 (3) (1997) 1344–1369.
* [15] D. Heckerman, M. P. Wellman, Bayesian networks, Communications of the ACM 38 (3) (1995) 27–30.
* [16] J. Cussens, Bayesian network learning by compiling to weighted MAX-SAT, in: D. A. McAllester, P. Myllymäki (Eds.), Proceedings of the 24th Conference in Uncertainty in Artificial Intelligence, AUAI Press, Helsinki, Finland, 2008, pp. 105–112.
* [17] S. Kotz, N. Balakrishnan, N. L. Johnson, Continuous Multivariate Distributions, 2nd Edition, Wiley series in probability and statistics. Applied probability and statistics, Wiley, New York, 2000.
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|
arxiv-papers
| 2009-04-06T17:51:33 |
2024-09-04T02:49:01.734931
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guy Freeman and Jim Q. Smith",
"submitter": "Guy Freeman",
"url": "https://arxiv.org/abs/0904.0977"
}
|
0904.0981
|
1em1em
# Dependency Pairs and Polynomial Path Orders††thanks: This research is
partially supported by FWF (Austrian Science Fund) projects P20133.
Martin Avanzini and Georg Moser
{martin.avanzini georg.moser}@uibk.ac.at
(March 2009)
###### Abstract
We show how polynomial path orders can be employed efficiently in conjunction
with weak innermost dependency pairs to automatically certify polynomial
runtime complexity of term rewrite systems and the polytime computability of
the functions computed. The established techniques have been implemented and
we provide ample experimental data to assess the new method.
###### Contents
1. 1 Introduction
2. 2 The Polynomial Path Order on Sequences
3. 3 Complexity Analysis Based on the Dependency Pair Method
4. 4 The Polynomial Path Order over Quasi-Precedences
5. 5 Dependency Pairs and Polynomial Path Orders
6. 6 Experimental Results
7. 7 Conclusion
8. A Appendix
1. A.1 Proof of Theorem 5.11
2. A.2 Proof of Theorem 5.14
## 1 Introduction
In order to measure the complexity of a (terminating) term rewrite system (TRS
for short) it is natural to look at the maximal length of derivation
sequences—the _derivation length_ —as suggested by Hofbauer and Lautemann in
Hofbauer and Lautemann (1989). More precisely, the _runtime complexity
function_ with respect to a (finite and terminating) TRS $\mathcal{R}$ relates
the maximal derivation length to the size of the initial term, whenever the
set of initial terms is restricted to constructor based terms, also called
_basic_ terms. The restriction to basic terms allows us to accurately express
the complexity of a program through the runtime complexity of TRSs. In this
paper we study and combine recent efforts for the _automatic_ analysis of
runtime complexities of TRSs. In Avanzini and Moser (2008) we introduced a
restriction of the multiset path order, called _polynomial path order_
(_$\textsc{POP}^{\ast}$_ for short) that induces polynomial runtime complexity
if restricted to innermost rewriting. The definition of $\textsc{POP}^{\ast}$
employs the idea of _tiered recursion_ Simmons (1988). Syntactically this
amounts to a separation of arguments into _normal_ and _safe_ arguments, cf.
Bellantoni and Cook (1992). Furthermore, Hirokawa and the second author
introduced a variant of dependency pairs, dubbed _weak dependency pairs_ ,
that makes the dependency pair method applicable in the context of complexity
analysis, cf. Hirokawa and Moser (2008b, a).
We show how weak innermost dependency pairs can be successfully applied in
conjunction with $\textsc{POP}^{\ast}$. The following example (see Fuhs et al.
(2007)) motivates this study. Consider the TRS $\mathcal{R}_{\textsf{bin}}$
encoding the function $\lambda x.\lceil{\log(x+1)}\rceil$ for natural numbers
given as tally sequences:
$\displaystyle 1\colon$ $\displaystyle\mathsf{half}(\mathsf{0})$
$\displaystyle\to\mathsf{0}$ $\displaystyle 4\colon$
$\displaystyle\mathsf{bits}(\mathsf{0})$ $\displaystyle\to\mathsf{0}$
$\displaystyle 2\colon$ $\displaystyle\mathsf{half}(\mathsf{s}(\mathsf{0}))$
$\displaystyle\to\mathsf{0}$ $\displaystyle 5\colon$
$\displaystyle\mathsf{bits}(\mathsf{s}(\mathsf{0}))$
$\displaystyle\to\mathsf{s}(\mathsf{0})$ $\displaystyle 3\colon$
$\displaystyle\mathsf{half}(\mathsf{s}(\mathsf{s}(x)))$
$\displaystyle\to\mathsf{s}(\mathsf{half}(x))$ $\displaystyle\hskip
8.61108pt6\colon$ $\displaystyle\mathsf{bits}(\mathsf{s}(\mathsf{s}(x)))$
$\displaystyle\to\mathsf{s}(\mathsf{bits}(\mathsf{s}(\mathsf{half}(x))))$
It is easy to see that the TRS $\mathcal{R}_{\textsf{bin}}$ is not compatible
with $\textsc{POP}^{\ast}$, even if we allow quasi-precedences, see Section 4.
On the other hand, employing (weak innermost) dependency pairs, argument
filtering, and the usable rules criteria in conjunction with
$\textsc{POP}^{\ast}$, polynomial innermost runtime complexity of
$\mathcal{R}_{\textsf{bin}}$ can be shown fully automatically.
The combination of dependency pairs and polynomial path orders, while
conceptually quite clear, turns out to be technical involved. One of the first
obstacles one encounters is that the pair
$(\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}},\mathrel{{>}_{\mathsf{pop*}}})$
cannot be used as a reduction pair in the spirit of Hirokawa and Moser
(2008b), as
$\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}$
fails to be closed under contexts. Conclusively, we start from scratch and
study polynomial path orders in the context of _relative rewriting_ Geser
(1990). Based on this study an incorporation of argument filterings becomes
possible so that we can employ the pair
$(\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}},\mathrel{{>}^{\pi}_{\mathsf{pop*}}})$
in conjunction with dependency pairs successfully. Here,
$\mathrel{{>}^{\pi}_{\mathsf{pop*}}}$ refers to the order obtained by
combining $\mathrel{{>}_{\mathsf{pop*}}}$ with the argument filtering $\pi$ as
expected, and
$\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}$
denotes the extension of $\mathrel{{>}^{\pi}_{\mathsf{pop*}}}$ by term
equivalence, preserving the separation of safe and normal argument positions.
Note that for polynomial path orders, the integration of argument filterings
is not only non-trivial, but indeed a challenging task. This is mainly due to
the embodiment of tiered recursion in $\textsc{POP}^{\ast}$. Thus we establish
a combination of two syntactic techniques in complexity analysis. The
experimental evidence given below indicates the power and in particular the
efficiency of the provided results.
Our next contribution is concerned with _implicit complexity theory_ , see for
example Bonfante et al. (2009). A careful analyis of our main result shows
that polynomial path orders in conjunction with (weak innermost) dependency
pairs even induce polytime computability of the functions defined by the TRS
studied. This result fits well with recent results by Marion and Péchoux on
the use of restricted forms of the dependency pair method to charcterise
complexity classes like PTIME or PSPACE, cf. Marion and Péchoux (2008). Note
that both results allow to conclude, based on different restrictions, polytime
computability of the functions defined by constructor TRSs, whose termination
can be shown by the dependency pair method. Note that the results in Marion
and Péchoux (2008) also capture programs admitting infeasible runtime
complexities but define functions that are computable in polytime if suitable
(and non-trivial) program transformations are used. Such programs are outside
the scope of our results. Thus it seems that our results more directly assess
the complexity of the given programs. Note that our tool provides (for the
first time) a fully automatic application of the dependency pair method in the
context of implicit complexity theory.111In this context it is perhaps
interesting to note that for a variant of the TRS
$\mathcal{R}_{\textsf{bin}}$, studied in Marion and Péchoux (2008), our tool
verifies polytime computability fully automatically. See also Avanzini et al.
(2008) for the description of a small tool that implements related
characterisations of of the class of polynomial time computable functions.
The rest of the paper is organised as follows. In Section 2 we present basic
notions and recall (briefly) the _path order for FP_ from Arai and Moser
(2005). We then briefly recall dependency pairs in the context of complexity
analysis from Hirokawa and Moser (2008b, a), cf. Section 3. In Section 4 we
present polynomial path orders over quasi-precedences. Our main results are
presented in Section 5. We continue with experimental results in Section 6,
and conclude in Section 7.
## 2 The Polynomial Path Order on Sequences
We assume familiarity with the basics of term rewriting, see Baader and Nipkow
(1998); Terese (2003). Let $\mathcal{V}$ denote a countably infinite set of
variables and $\mathcal{F}$ a signature, containing at least one constant. The
set of terms over $\mathcal{F}$ and $\mathcal{V}$ is denoted as
$\mathcal{T}(\mathcal{F},\mathcal{V})$ and the set of ground terms as
$\mathcal{T}(\mathcal{F})$. We write $\operatorname{\mathsf{Fun}}(t)$ and
$\operatorname{\mathsf{Var}}(t)$ for the set of function symbols and variables
appearing in $t$, respectively. The root symbol
$\operatorname{\mathsf{rt}}(t)$ of a term $t$ is defined as usual and the
(proper) subterm relation is denoted as $\mathrel{\unlhd}$ ($\mathrel{\lhd}$).
We write $s|_{p}$ for the _subterm_ of $s$ at position $p$. The _size_
$\lvert{t}\rvert$ of a term $t$ is defined as usual and the _width_ of $t$ is
defined as
$\operatorname{\mathsf{width}}(t)\mathrel{:=}\max\\{{n,{\operatorname{\mathsf{width}}}({t}_{1}),\ldots,{\operatorname{\mathsf{width}}}({t}_{n})}\\}$
if $t=f({t}_{1},\ldots,{t}_{n})$ and $n>0$ or
$\operatorname{\mathsf{width}}(t)=1$ else. Let $\succsim$ be a preorder on the
signature $\mathcal{F}$, called _quasi-precedence_ or simply _precedence_.
Based on $\succsim$ we define an equivalence $\approx$ on terms: $s\approx t$
if either (i) $s=t$ or (ii) $s=f({s}_{1},\ldots,{s}_{n})$,
$t=g({t}_{1},\ldots,{t}_{n})$, $f\approx g$ and there exists a permutation
$\pi$ such that $s_{i}\approx t_{\pi(i)}$. For a preorder $\succsim$, we use
$\mathrel{\succsim}^{\mathsf{mul}}$ for the multiset extension of $\succsim$,
which is again a preorder. The proper order (equivalence) induced by
$\mathrel{\succsim}^{\mathsf{mul}}$ is written as
$\mathrel{\succ}^{\mathsf{mul}}$ ($\mathrel{\approx}^{\mathsf{mul}}$).
A _term rewrite system_ (_TRS_ for short) $\mathcal{R}$ over
$\mathcal{T}(\mathcal{F},\mathcal{V})$ is a _finite_ set of rewrite rules
$l\to r$, such that $l\notin\mathcal{V}$ and
$\operatorname{\mathsf{Var}}(l)\supseteq\operatorname{\mathsf{Var}}(r)$. We
write
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$$}}}_{\mathcal{R}}}$
($\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}$)
for the induced (innermost) rewrite relation. The set of defined function
symbols is denoted as $\mathcal{D}$, while the constructor symbols are
collected in $\mathcal{C}$, clearly $\mathcal{F}=\mathcal{D}\cup\mathcal{C}$.
We use $\operatorname{\mathsf{NF}}(\mathcal{R})$ to denote the set of normal
forms of $\mathcal{R}$ and set
$\mathsf{Val}\mathrel{:=}\mathcal{T}(\mathcal{C},\mathcal{V})$, the elements
of $\mathsf{Val}$ are called _values_. A TRS is called _completely defined_ if
normal forms coincide with values. We define
$\mathcal{T}_{\mathsf{b}}\mathrel{:=}\\{f({v}_{1},\ldots,{v}_{n})\mid
f\in\mathcal{D}\text{ and }v_{i}\in\mathsf{Val}\\}$ as the set of _basic
terms_. A TRS $\mathcal{R}$ is a _constructor TRS_ if
$l\in\mathcal{T}_{\mathsf{b}}$ for all ${l\to r}\in\mathcal{R}$. Let
$\mathcal{Q}$ denote a TRS. The _generalised restricted rewrite relation
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{Q}$}}}_{\mathcal{R}}}$_
is the restriction of
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$$}}}_{\mathcal{R}}}$
where all arguments of the redex are in normal form with respect to the TRS
$\mathcal{Q}$ (see Thiemann (2007)). We define the (innermost) relative
rewriting relation (denoted as
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}/\mathcal{S}}}$)
as follows:
${\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}/\mathcal{S}}}}\mathrel{:=}{{\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{R}\cup\mathcal{S}$}}}^{\ast}_{\mathcal{S}}}}\cdot{\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{R}\cup\mathcal{S}$}}}_{\mathcal{R}}}}\cdot{\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{R}\cup\mathcal{S}$}}}^{\ast}_{\mathcal{S}}}}}\hbox
to0.0pt{$\;$.\hss}$
Similarly, we set
${\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}}}\mathrel{:=}{{\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{R}\cup\mathcal{S}$}}}^{\ast}_{\mathcal{S}}}}\cdot{\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{R}\cup\mathcal{S}$}}}^{\varepsilon}_{\mathcal{R}}}}\cdot{\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{R}\cup\mathcal{S}$}}}^{\ast}_{\mathcal{S}}}}}$,
to define an _(innermost) relative root-step_.
A _polynomial interpretation_ is a well-founded and monotone algebra
$(\mathcal{A},>)$ with carrier $\mathbb{N}$ such that $>$ is the usual order
on natural numbers and all interpretation functions $f_{\mathcal{A}}$ are
polynomials. Let $\alpha\colon\mathcal{V}\to\mathcal{A}$ denote an
_assignment_ , then we write $[\alpha]_{\mathcal{A}}(t)$ for the evaluation of
term $t$ with respect to $\mathcal{A}$ and $\alpha$. A polynomial
interpretation is called a _strongly linear interpretation_ (_SLI_ for short)
if all function symbols are interpreted by _weight functions_
$f_{\mathcal{A}}({x}_{1},\ldots,{x}_{n})=\sum_{i=1}^{n}x_{i}+c$ with
$c\in\mathbb{N}$. The _derivation length_ of a terminating term $s$ with
respect to $\to$ is defined as
$\operatorname{dl}(s,\to)\mathrel{:=}\max\\{{n\mid\exists t.\;s\to^{n}t}\\}$,
where $\to^{n}$ denotes the $n$-fold application of $\to$. The _innermost
runtime complexity function_
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}$
with respect to a TRS $\mathcal{R}$ is defined as
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}(n)\mathrel{:=}\max\\{\operatorname{dl}(t,\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}})\mid\text{$t\in\mathcal{T}_{\mathsf{b}}$
and $\lvert{t}\rvert\leqslant n$}\\}$. If no confusion can arise
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}$
is simply called _runtime complexity function_.
Below we recall the bare essentials of the polynomial path order
$\mathrel{\blacktriangleright}$ on sequences (POP for short) as put forward in
Arai and Moser (2005). We kindly refer the reader to Arai and Moser (2005);
Avanzini and Moser (2008) for motivation and examples. We recall the
definition of _finite approximations $\mathrel{\blacktriangleright}_{k}^{l}$_
of $\mathrel{\blacktriangleright}$. The latter is conceived as the _limit_ of
these approximations. The domain of this order are so called _sequences_
$\operatorname{\mathcal{S}eq}(\mathcal{F},\mathcal{V})\mathrel{:=}\mathcal{T}(\mathcal{F}\cup\\{\circ\\},\mathcal{V})$.
Here $\mathcal{F}$ is a finite signature and $\circ\not\in\mathcal{F}$ a fresh
variadic function symbol, used to form sequences. We denote sequences
$\circ({s}_{1},\ldots,{s}_{n})$ by $[{s}_{1}\cdots{s}_{n}]$ and write
$a\mathrel{::}[{b}_{1}\cdots{b}_{n}]$ for the sequence $[a\leavevmode\nobreak\
{b}_{1}\cdots{b}_{n}]$.
Let $\succsim$ denote a precedence. The order
$\mathrel{\blacktriangleright}_{k}^{l}$ is based on an auxiliary order
$\mathrel{\gtrdot}_{k}^{l}$ (and the equivalence $\approx$ on terms defined
above). Below we set
${\mathrel{\not{\gtrsim}}_{k}^{l}}\mathrel{:=}{\mathrel{\gtrdot}_{k}^{l}}\cup{\approx}$.
We write ${\not{\\{}}t_{1},\dots,t_{n}{\not{\\}}}$ to denote multisets and
$\uplus$ for the multiset sum.
###### Definition 2.1.
Let $k,l\geqslant 1$. The order $\mathrel{\gtrdot}_{k}^{l}$ induced by
$\succsim$ is inductively defined as follows: $s\mathrel{\gtrdot}_{k}^{l}t$
for $s=f({s}_{1},\ldots,{s}_{n})$ or $s=[{s}_{1}\cdots{s}_{n}]$ if either
1. (i)
$s_{i}\leavevmode\nobreak\ \mathrel{\not{\gtrsim}}_{k}^{l}\leavevmode\nobreak\
t$ for some $i\in\\{{1,\dots,n}\\}$, or
2. (ii)
$s=f({s}_{1},\ldots,{s}_{n})$, $t=g({t}_{1},\ldots,{t}_{m})$ with $f\succ g$
or $t=[{t}_{1}\cdots{t}_{m}]$, $s\mathrel{\gtrdot}_{k}^{l-1}t_{j}$ for all
$j\in\\{{1,\dots,m}\\}$, and $m<k+\operatorname{\mathsf{width}}(s)$,
3. (iii)
$s=[{s}_{1}\cdots{s}_{n}]$, $t=[{t}_{1}\cdots{t}_{m}]$ and the following
properties hold:
* –
${\not{\\{}}{t}_{1},\ldots,{t}_{m}{\not{\\}}}=N_{1}\uplus\cdots\uplus N_{n}$
for some multisets $N_{1},\dots,N_{n}$, and
* –
there exists $i\in\\{{1,\dots,n}\\}$ such that
${\not{\\{}}s_{i}{\not{\\}}}\not\approx^{\mathsf{mul}}N_{i}$, and
* –
for all $1\leqslant i\leqslant n$ such that
${\not{\\{}}s_{i}{\not{\\}}}\not\approx^{\mathsf{mul}}N_{i}$ we have
$s_{i}\mathrel{\gtrdot}_{k}^{l}r$ for all $r\in N_{i}$, and
$m<k+\operatorname{\mathsf{width}}(s)$.
###### Definition 2.2.
Let $k,l\geqslant 1$. The _approximation
$\mathrel{\blacktriangleright}_{k}^{l}$ of the polynomial path order on
sequences_ induced by $\succsim$ is inductively defined as follows:
$s\mathrel{\blacktriangleright}_{k}^{l}t$ for $s=f({s}_{1},\ldots,{s}_{n})$ or
$s=[{s}_{1}\cdots{s}_{n}]$ if either $s\mathrel{\gtrdot}_{k}^{l}t$ or
1. (i)
$s_{i}\mathrel{\not{\gtrsim}}_{k}^{l}t$ for some $i\in\\{{1,\dots,n}\\}$,
2. (ii)
$s=f({s}_{1},\ldots,{s}_{n})$, $t=[{t}_{1}\cdots{t}_{m}]$, and the following
properties hold:
* –
$s\mathrel{\blacktriangleright}_{k}^{l-1}t_{j_{0}}$ for some
$j_{0}\in\\{{1,\dots,m}\\}$,
* –
$s\mathrel{\gtrdot}_{k}^{l-1}t_{j}$ for all $j\neq j_{0}$, and
$m<k+\operatorname{\mathsf{width}}(s)$,
3. (iii)
$s=f({s}_{1},\ldots,{s}_{n})$, $t=g({t}_{1},\ldots,{t}_{m})$, $f\sim g$ and
$[{s}_{1}\cdots{s}_{n}]\mathrel{\blacktriangleright}_{k}^{l}[{t}_{1}\cdots{t}_{m}]$,
or
4. (iv)
$s=[{s}_{1}\cdots{s}_{n}]$, $t=[{t}_{1}\cdots{t}_{m}]$ and the following
properties hold:
* –
${\not{\\{}}{t}_{1},\ldots,{t}_{m}{\not{\\}}}=N_{1}\uplus\cdots\uplus N_{n}$
for some multisets $N_{1},\dots,N_{n}$, and
* –
there exists $i\in\\{{1,\dots,n}\\}$ such that
${\not{\\{}}s_{i}{\not{\\}}}\not\approx^{\mathsf{mul}}N_{i}$, and
* –
for all $1\leqslant i\leqslant n$ such that
${\not{\\{}}s_{i}{\not{\\}}}\not\approx^{\mathsf{mul}}N_{i}$ we have
$s_{i}\mathrel{\blacktriangleright}_{k}^{l}r$ for all $r\in N_{i}$, and
$m<k+\operatorname{\mathsf{width}}(s)$.
Above we set
${\mathrel{\not{\gtrsim}}_{k}^{l}}\mathrel{:=}{\mathrel{\blacktriangleright}_{k}^{l}}\cup{\approx}$
and abbreviate $\mathrel{\blacktriangleright}_{k}^{k}$ as
$\mathrel{\blacktriangleright}_{k}$ in the following. Note that the empty
sequence is minimal with respect to both orders. It is easy to see that for
$k\leqslant l$, we have
${\mathrel{\gtrdot}_{k}}\subseteq{\mathrel{\gtrdot}_{l}}$ and
${\mathrel{\blacktriangleright}_{k}}\subseteq{\mathrel{\blacktriangleright}_{l}}$.
Note that $s\mathrel{\blacktriangleright}_{k}t$ implies that
$\operatorname{\mathsf{width}}(t)<\operatorname{\mathsf{width}}(s)+k$. For a
fixed approximation $\mathrel{\blacktriangleright}_{k}$, we define the length
of its longest decent as follows: $\mathsf{G}_{k}(t)\mathrel{:=}\max\\{{n\mid
t=t_{0}\mathrel{\blacktriangleright}_{k}t_{1}\mathrel{\blacktriangleright}_{k}\dots\mathrel{\blacktriangleright}_{k}t_{n}}\\}$.
The following proposition is a reformulation of (Arai and Moser, 2005, Lemma
6).
###### Proposition 2.3.
Let $k\in\mathbb{N}$. There exists a polynomial interpretation $\mathcal{A}$
such that $\mathsf{G}_{k}(t)\leqslant[\alpha]_{\mathcal{A}}(t)$ for all
assignments $\alpha\,\colon\,\mathcal{V}\to\mathbb{N}$. As a consequence, for
all terms $f({t}_{1},\ldots,{t}_{n})$ with
$[\alpha]_{\mathcal{A}}(t_{i})=\operatorname{\mathsf{O}}(\lvert{t_{i}}\rvert)$,
$\mathsf{G}_{k}(f({t}_{1},\ldots,{t}_{n}))$ is bounded by a polynomial $p$ in
the size of $t$, where $p$ depends on $k$ only.
Observe that the polynomial interpretation $\mathcal{A}$ as employed in the
proposition fulfils:
$\circ_{\mathcal{A}}({m}_{1},\ldots,{m}_{n})=\sum_{i=1}^{n}m_{i}+n$. In
particular, we have $[\alpha]_{\mathcal{A}}([])=0$.
## 3 Complexity Analysis Based on the Dependency Pair Method
In this section, we briefly recall the central definitions and results
established in Hirokawa and Moser (2008b, a). We kindly refer the reader to
Hirokawa and Moser (2008b, a) for further examples and underlying intuitions.
Let $\mathcal{X}$ be a set of symbols. We write
$C\langle{t}_{1},\ldots,{t}_{n}\rangle_{\mathcal{X}}$ to denote
$C[{t}_{1},\ldots,{t}_{n}]$, whenever
$\operatorname{\mathsf{rt}}(t_{i})\in\mathcal{X}$ for all
$i\in\\{{1,\dots,n}\\}$ and $C$ is a $n$-hole context containing no symbols
from $\mathcal{X}$. We set
$\mathcal{D}^{\sharp}\mathrel{:=}\mathcal{D}\cup\\{{f^{\sharp}\mid
f\in\mathcal{D}}\\}$ with each $f^{\sharp}$ a fresh function symbol. Further,
for $t=f({t}_{1},\ldots,{t}_{n})$ with $f\in\mathcal{D}$, we set
$t^{\sharp}\mathrel{:=}f^{\sharp}({t}_{1},\ldots,{t}_{n})$.
###### Definition 3.1.
Let $\mathcal{R}$ be a TRS. If $l\to r\in\mathcal{R}$ and
$r=C\langle{u}_{1},\ldots,{u}_{n}\rangle_{\mathcal{D}}$ then
$l^{\sharp}\to\operatorname{COM}(u_{1}^{\sharp},\ldots,u_{n}^{\sharp})$ is
called a _weak innermost dependency pair_ of $\mathcal{R}$. Here
$\operatorname{COM}(t)=t$ and
$\operatorname{COM}({t}_{1},\ldots,{t}_{n})=\mathsf{c}(t_{1},\ldots,t_{n})$,
$n\not=1$, for a fresh constructor symbol $\mathsf{c}$, the _compound symbol_.
The set of all weak innermost dependency pairs is denoted by
$\mathsf{WIDP}(\mathcal{R})$.
###### Example 3.2.
Reconsider the example $\mathcal{R}_{\textsf{bits}}$ from the introduction.
The set of weak innermost dependency pairs
$\mathsf{WIDP}(\mathcal{R}_{\textsf{bits}})$ is given by
$\displaystyle 7\colon$ $\displaystyle\mathsf{half}^{\sharp}(\mathsf{0})$
$\displaystyle\to\mathsf{c_{1}}$ $\displaystyle 10\colon$
$\displaystyle\mathsf{bits}^{\sharp}(\mathsf{0})$
$\displaystyle\to\mathsf{c_{3}}$ $\displaystyle 8\colon$
$\displaystyle\mathsf{half}^{\sharp}(\mathsf{s}(\mathsf{0}))$
$\displaystyle\to\mathsf{c_{2}}$ $\displaystyle 11\colon$
$\displaystyle\mathsf{bits}^{\sharp}(\mathsf{s}(\mathsf{0}))$
$\displaystyle\to\mathsf{c_{4}}$ $\displaystyle 9\colon$
$\displaystyle\mathsf{half}^{\sharp}(\mathsf{s}(\mathsf{s}(x)))$
$\displaystyle\to\mathsf{half}^{\sharp}(x)$ $\displaystyle\hskip
12.91663pt12\colon$
$\displaystyle\mathsf{bits}^{\sharp}(\mathsf{s}(\mathsf{s}(x)))$
$\displaystyle\to\mathsf{bits}^{\sharp}(\mathsf{s}(\mathsf{half}(x)))$
We write $f\rhd_{\mathrm{d}}g$ if there exists a rewrite rule $l\to
r\in\mathcal{R}$ such that $f=\operatorname{\mathsf{rt}}(l)$ and $g$ is a
defined symbol in $\operatorname{\mathsf{Fun}}(r)$. For a set $\mathcal{G}$ of
defined symbols we denote by $\mathcal{R}{\restriction}\mathcal{G}$ the set of
rewrite rules $l\to r\in\mathcal{R}$ with
$\operatorname{\mathsf{rt}}(l)\in\mathcal{G}$. The set $\mathcal{U}(t)$ of
usable rules of a term $t$ is defined as
$\mathcal{R}{\restriction}\\{{g\mid\text{$f\rhd_{\mathrm{d}}^{*}g$ for some
$f\in\operatorname{\mathsf{Fun}}(t)$}}\\}$. Finally, we define
$\mathcal{U}(\mathcal{P})=\bigcup_{l\to r\in\mathcal{P}}\mathcal{U}(r)$.
###### Example 3.3 (Example 3.2 continued).
The usable rules of $\mathsf{WIDP}(\mathcal{R}_{\textsf{bits}})$ consist of
the following rules: $1\colon\mathsf{half}(\mathsf{0})\to\mathsf{0}$,
$2\colon\mathsf{half}(\mathsf{s}(\mathsf{0}))\to\mathsf{0}$, and
$3\colon\mathsf{half}(\mathsf{s}(\mathsf{s}(x)))\to\mathsf{half}(x)$.
The following proposition allows the analysis of the (innermost) runtime
complexity through the study of (innermost) relative rewriting, see Hirokawa
and Moser (2008b) for the proof.
###### Proposition 3.4.
Let $\mathcal{R}$ be a TRS, let $t$ be a basic terminating term, and let
$\mathcal{P}=\mathsf{WIDP}(\mathcal{R})$. Then
$\operatorname{dl}(t,\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}})\leqslant\operatorname{dl}(t^{\sharp},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{U}(\mathcal{P})\,\cup\,\mathcal{P}}})$.
Moreover, if $\mathcal{P}$ is non-duplicating and
${\mathcal{U}(\mathcal{P})}\subseteq{>_{\mathcal{A}}}$ for some SLI
$\mathcal{A}$. Then there exist constants $K,L\geqslant 0$ (depending on
$\mathcal{P}$ and $\mathcal{A}$ only) such that
$\operatorname{dl}(t,\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}})\leqslant
K\cdot\operatorname{dl}(t^{\sharp},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{P}/\mathcal{U}(\mathcal{P})}})+L\cdot\lvert{t^{\sharp}}\rvert$.
This approach admits also an integration of _dependency graphs_ Arts and Giesl
(2000) in the context of complexity analysis. The nodes of the _weak innermost
dependency graph_ $\mathsf{WIDG}(\mathcal{R})$ are the elements of
$\mathcal{P}$ and there is an arrow from $s\to t$ to $u\to v$ if there exist a
context $C$ and substitutions $\sigma$, $\tau$ such that
$t\sigma\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{R}}}C[u\tau]$.
Let $\mathcal{G}=\mathsf{WIDG}(\mathcal{R})$; a _strongly connected component_
(_SCC_ for short) in $\mathcal{G}$ is a maximal _strongly connected subgraph_.
We write ${\mathcal{G}}/_{\\!\equiv}$ for the _congruence graph_ , where
$\equiv$ is the equivalence relation induced by SCCs.
###### Example 3.5 (Example 3.2 continued).
$\mathcal{G}=\mathsf{WIDG}(\mathcal{R}_{\textsf{bits}})$ consists of the nodes
(7)–(12) as mentioned in Example 3.2 and has the following shape:
798101211
The only non-trivial SCCs in $\mathcal{G}$ are $\\{9\\}$ and $\\{12\\}$. Hence
${\mathcal{G}}/_{\\!\equiv}$ consists of the nodes
$[7]_{\equiv}$–$[12]_{\equiv}$, and edges $([a]_{\equiv},[b]_{\equiv})$ for
edges $(a,b)$ in $\mathcal{G}$. Here $[a]_{\equiv}$ denotes the equivalence
class of $a$.
We set
$\operatorname{\mathsf{L}}(t)\mathrel{:=}\max\\{{\operatorname{dl}(t,\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{P}_{m}/\mathcal{S}}})\mid\text{$(\mathcal{P}_{1},\ldots,\mathcal{P}_{m})$
a path in ${\mathcal{G}}/_{\\!\equiv}$,
$\mathcal{P}_{1}\in\mathsf{Src}$}}\\}$, where $\mathsf{Src}$ denote the set o
f source nodes from ${\mathcal{G}}/_{\\!\equiv}$ and
$\mathcal{S}=\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{m-1}\cup\mathcal{U}(\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{m})$.
The proposition allows the use of different techniques to analyse polynomial
runtime complexity on separate paths, cf. Hirokawa and Moser (2008a).
###### Proposition 3.6.
Let $\mathcal{R}$, $\mathcal{P}$, and $t$ be as above. Then there exists a
polynomial $p$ (depending only on $\mathcal{R}$) such that
$\operatorname{dl}(t^{\sharp},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{P}/\mathcal{U}(\mathcal{P})}})\leqslant
p(\operatorname{\mathsf{L}}(t^{\sharp}))$.
## 4 The Polynomial Path Order over Quasi-Precedences
In this section, we briefly recall the central definitions and results
established in Avanzini and Moser (2008); Avanzini et al. (2008) on the
_polynomial path order_. We employ the variant of $\textsc{POP}^{\ast}$ based
on quasi-precendences, cf. Avanzini et al. (2008).
As mentioned in the introduction, $\textsc{POP}^{\ast}$ relies on tiered
recursion, which is captured by the notion of _safe mapping_. A _safe mapping_
$\operatorname{\mathsf{safe}}$ is a function that associates with every
$n$-ary function symbol $f$ the set of _safe argument positions_. If
$f\in\mathcal{D}$ then
$\operatorname{\mathsf{safe}}(f)\subseteq\\{{1,\dots,n}\\}$, for
$f\in\mathcal{C}$ we fix $\operatorname{\mathsf{safe}}(f)=\\{{1,\dots,n}\\}$.
The argument positions not included in $\operatorname{\mathsf{safe}}(f)$ are
called _normal_ and denoted by $\operatorname{\mathsf{nrm}}(f)$. We extend
$\operatorname{\mathsf{safe}}$ to terms $t\not\in\mathcal{V}$ as follows: we
define
$\operatorname{\mathsf{safe}}(f({t}_{1},\ldots,{t}_{n}))\mathrel{:=}\\{{t_{i_{1}},\dots,t_{i_{p}}}\\}$
where $\operatorname{\mathsf{safe}}(f)=\\{{{i}_{1},\ldots,{i}_{p}}\\}$,
likewise we define
$\operatorname{\mathsf{nrm}}(f({t}_{1},\ldots,{t}_{n}))\mathrel{:=}\\{{t_{j_{1}},\dots,t_{j_{q}}}\\}$
where $\operatorname{\mathsf{nrm}}(f)=\\{{{j}_{1},\ldots,{j}_{q}}\\}$. Not
every precedence is suitable for $\mathrel{{>}_{\mathsf{pop*}}}$, in
particular we need to assert that constructors are minimal.
We say that a precedence $\succsim$ is _admissible_ for $\textsc{POP}^{\ast}$
if the following is satisfied: (i) $f\succ g$ with $g\in\mathcal{D}$ implies
$f\in\mathcal{D}$, and (ii) if $f\approx g$ then $f\in\mathcal{D}$ if and only
if $g\in\mathcal{D}$. In the sequel we assume any precedence is admissible. We
extend the equivalence $\approx$ to the context of safe mapping:
$s\mathrel{\text{\raisebox{-1.00006pt}{$\stackrel{{\scriptstyle\text{{\raisebox{-0.70004pt}{\tiny{$\operatorname{\mathsf{safe}}$}}}}}}{{\approx}}$}}}t$,
if (i) $s=t$, or (ii) $s=f({s}_{1},\ldots,{s}_{n})$,
$t=g({t}_{1},\ldots,{t}_{n})$, $f\approx g$ and there exists a permutation
$\pi$ so that
$s_{i}\mathrel{\text{\raisebox{-1.00006pt}{$\stackrel{{\scriptstyle\text{{\raisebox{-0.70004pt}{\tiny{$\operatorname{\mathsf{safe}}$}}}}}}{{\approx}}$}}}t_{\pi(i)}$,
where $i\in\operatorname{\mathsf{safe}}(f)$ if and only if
$\pi(i)\in\operatorname{\mathsf{safe}}(g)$ for all $i\in\\{{1,\dots,n}\\}$.
Similar to POP, the definition of the polynomial path order
$\mathrel{{>}_{\mathsf{pop*}}}$ makes use of an auxiliary order
$\mathrel{{>}_{\mathsf{pop}}}$.
###### Definition 4.1.
The auxiliary order $\mathrel{{>}_{\mathsf{pop}}}$ induced by $\succsim$ and
$\operatorname{\mathsf{safe}}$ is inductively defined as follows:
$s=f({s}_{1},\ldots,{s}_{n})\mathrel{{>}_{\mathsf{pop}}}t$ if either
1. (i)
$s_{i}\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop}$}}}$}}}t$
for some $i\in\\{{1,\dots,n}\\}$, and if $f\in\mathcal{D}$ then
$i\in\operatorname{\mathsf{nrm}}(f)$, or
2. (ii)
$t=g({t}_{1},\ldots,{t}_{m})$, $f\succ g$, $f\in\mathcal{D}$ and
$s\mathrel{{>}_{\mathsf{pop}}}t_{j}$ for all $j\in\\{{1,\dots,m}\\}$.
###### Definition 4.2.
The _polynomial path order_ $\mathrel{{>}_{\mathsf{pop*}}}$ induced by
$\succsim$ and $\operatorname{\mathsf{safe}}$ is inductively defined as
follows: $s=f({s}_{1},\ldots,{s}_{n})\mathrel{{>}_{\mathsf{pop*}}}t$ if either
$s\mathrel{{>}_{\mathsf{pop}}}t$ or
1. (i)
$s_{i}\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}t$
for some $i\in\\{{1,\dots,n}\\}$, or
2. (ii)
$t=g({t}_{1},\ldots,{t}_{m})$, $f\succ g$, $f\in\mathcal{D}$, and
* –
$s\mathrel{{>}_{\mathsf{pop*}}}t_{j_{0}}$ for some
$j_{0}\in\operatorname{\mathsf{safe}}(g)$, and
* –
for all $j\neq j_{0}$ either $s\mathrel{{>}_{\mathsf{pop}}}t_{j}$, or $s\rhd
t_{j}$ and $j\in\operatorname{\mathsf{safe}}(g)$, or
3. (iii)
$t=g({t}_{1},\ldots,{t}_{m})$, $f\approx g$,
$\operatorname{\mathsf{nrm}}(s)\mathrel{>_{\mathsf{pop*}}^{\mathsf{mul}}}\operatorname{\mathsf{nrm}}(t)$
and
$\operatorname{\mathsf{safe}}(s)\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\mathsf{mul}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}\operatorname{\mathsf{safe}}(t)$.
Above we set
${\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop}$}}}$}}}}\mathrel{:=}{\mathrel{{>}_{\mathsf{pop}}}}\cup{\mathrel{\text{\raisebox{-1.00006pt}{$\stackrel{{\scriptstyle\text{{\raisebox{-0.70004pt}{\tiny{$\operatorname{\mathsf{safe}}$}}}}}}{{\approx}}$}}}}$
and
${\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}\mathrel{:=}{\mathrel{{>}_{\mathsf{pop*}}}}\cup{\mathrel{\text{\raisebox{-1.00006pt}{$\stackrel{{\scriptstyle\text{{\raisebox{-0.70004pt}{\tiny{$\operatorname{\mathsf{safe}}$}}}}}}{{\approx}}$}}}}$
below. Here $\mathrel{>_{\mathsf{pop*}}^{\mathsf{mul}}}$ and
$\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\mathsf{mul}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}$
refer to the strict and weak multiset extension of
$\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}$
respectively.
The intuition of $\mathrel{{>}_{\mathsf{pop}}}$ is to deny any recursive call,
whereas $\mathrel{{>}_{\mathsf{pop*}}}$ allows predicative recursion: by the
restrictions imposed by $\operatorname{\mathsf{safe}}$, recursion needs to be
performed on normal arguments, while a recursively computed result must only
be used in a safe argument position, compare Bellantoni and Cook (1992). Note
that the alternative $s\rhd t_{j}$ for $j\in\operatorname{\mathsf{safe}}(g)$
in Definition 4.2(ii) guarantees that $\textsc{POP}^{\ast}$ characterises the
class of polytime computable functions, cf. Avanzini and Moser (2008). The
proof of the next theorem follows the pattern of the proof of main theorem in
Avanzini and Moser (2008), but the result is stronger due to the extension to
quasi-precedences.
###### Theorem 4.3.
Let $\mathcal{R}$ be a constructor TRS. If $\mathcal{R}$ is compatible with
$\mathrel{{>}_{\mathsf{pop*}}}$, i.e.,
${\mathcal{R}}\subseteq{\mathrel{{>}_{\mathsf{pop*}}}}$, then the innermost
runtime complexity
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}$
induced is polynomially bounded.
Note that Theorem 4.3 is too weak to handle the TRS
$\mathcal{R}_{\textsf{bits}}$ as the (necessary) restriction to an admissible
precedence is too strong. To rectify this, we suit $\textsc{POP}^{\ast}$ so
that it can be used in conjunction with weak (innermost) dependency pairs.
An argument filtering (for a signature $\mathcal{F}$) is a mapping $\pi$ that
assigns to every $n$-ary function symbol $f\in\mathcal{F}$ an argument
position $i\in\\{1,\dots,n\\}$ or a (possibly empty) list
$\\{{i}_{1},\ldots,{i}_{m}\\}$ of argument positions with $1\leqslant
i_{1}<\cdots<i_{m}\leqslant n$. The signature $\mathcal{F}_{\pi}$ consists of
all function symbols $f$ such that $\pi(f)$ is some list
$\\{{i}_{1},\ldots,{i}_{m}\\}$, where in $\mathcal{F}_{\pi}$ the arity of $f$
is $m$. Every argument filtering $\pi$ induces a mapping from
$\mathcal{T}(\mathcal{F},\mathcal{V})$ to
$\mathcal{T}(\mathcal{F}_{\pi},\mathcal{V})$, also denoted by $\pi$:
$\pi(t)=\begin{cases}t&\text{if $t$ is a variable}\\\ \pi(t_{i})&\text{if
$t=f({t}_{1},\ldots,{t}_{n})$ and $\pi(f)=i$}\\\
f(\pi(t_{k_{1}}),\dots,\pi(t_{k_{m}}))&\text{if $t=f({t}_{1},\ldots,{t}_{n})$
and $\pi(f)=\\{{i}_{1},\ldots,{i}_{m}\\}$}\hbox to0.0pt{$\;$.\hss}\end{cases}$
###### Definition 4.4.
Let $\pi$ denote an argument filtering, and $\mathrel{{>}_{\mathsf{pop*}}}$ a
polynomial path order. We define $s\mathrel{{>}^{\pi}_{\mathsf{pop*}}}t$ if
and only if $\pi(s)\mathrel{{>}_{\mathsf{pop*}}}\pi(t)$, and likewise
$s\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}t$
if and only if
$\pi(s)\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}\pi(t)$.
###### Example 4.5 (Example 3.2 continued).
Let $\pi$ be defined as follows: $\pi(\mathsf{half})=1$ and
$\pi(f)=\\{1,\dots,n\\}$ for each $n$-ary function symbol other than
$\mathsf{half}$. Compatibility of $\mathsf{WIDP}(\mathcal{R}_{\textsf{bits}})$
with $\mathrel{{>}^{\pi}_{\mathsf{pop*}}}$ amounts to the following set of
order constraints:
$\displaystyle\mathsf{half}^{\sharp}(0)$
$\displaystyle\mathrel{{>}_{\mathsf{pop*}}}\mathsf{c_{1}}$
$\displaystyle\mathsf{bits}^{\sharp}(0)$
$\displaystyle\mathrel{{>}_{\mathsf{pop*}}}\mathsf{c_{3}}$
$\displaystyle\mathsf{half}^{\sharp}(\mathsf{s}(\mathsf{s}(x)))$
$\displaystyle\mathrel{{>}_{\mathsf{pop*}}}\mathsf{half}^{\sharp}(x)$
$\displaystyle\mathsf{half}^{\sharp}(\mathsf{s}(0))$
$\displaystyle\mathrel{{>}_{\mathsf{pop*}}}\mathsf{c_{2}}$
$\displaystyle\mathsf{bits}^{\sharp}(\mathsf{s}(0))$
$\displaystyle\mathrel{{>}_{\mathsf{pop*}}}\mathsf{c_{4}}$
$\displaystyle\mathsf{bits}^{\sharp}(\mathsf{s}(\mathsf{s}(x)))$
$\displaystyle\mathrel{{>}_{\mathsf{pop*}}}\mathsf{bits}^{\sharp}(\mathsf{s}(x))$
In order to define a $\textsc{POP}^{\ast}$ instance
$\mathrel{{>}_{\mathsf{pop*}}}$, we set
$\operatorname{\mathsf{safe}}(\mathsf{bits}^{\sharp})=\operatorname{\mathsf{safe}}(\mathsf{half})=\operatorname{\mathsf{safe}}(\mathsf{half}^{\sharp})=\varnothing$
and $\operatorname{\mathsf{safe}}(\mathsf{s})=\\{{1}\\}$. Furthermore, we
define an (admissible) precedence:
$0\approx\mathsf{c_{1}}\approx\mathsf{c_{2}}\approx\mathsf{c_{3}}\approx\mathsf{c_{4}}$.
The easy verification of
$\mathsf{WIDP}(\mathcal{R}_{\textsf{bits}})\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$
is left to the reader.
## 5 Dependency Pairs and Polynomial Path Orders
Motivated by Example 4.5, we show in this section that the pair
($\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}},\mathrel{{>}^{\pi}_{\mathsf{pop*}}}$)
can play the role of a _safe_ reduction pair, cf. Hirokawa and Moser (2008b,
a). Let $\mathcal{R}$ be a TRS over a signature $\mathcal{F}$ that is
innermost terminating. In the sequel $\mathcal{R}$ is kept fixed. Moreover, we
fix some safe mapping $\operatorname{\mathsf{safe}}$, an admissible precedence
$\succsim$, and an argument filtering $\pi$. We refer to the induced
$\textsc{POP}^{\ast}$ instance by $\mathrel{{>}^{\pi}_{\mathsf{pop*}}}$.
We adapt $\operatorname{\mathsf{safe}}$ to $\mathcal{F}_{\pi}$ in the obvious
way: for each $f_{\pi}\in\mathcal{F}_{\pi}$ with corresponding
$f\in\mathcal{F}$, we define
$\operatorname{\mathsf{safe}}(f_{\pi})\mathrel{:=}\operatorname{\mathsf{safe}}(f)\cap\pi(f)$,
and likewise
$\operatorname{\mathsf{nrm}}(f_{\pi})\mathrel{:=}\operatorname{\mathsf{nrm}}(f)\cap\pi(f)$.
Set
${\mathsf{Val}}_{\pi}\mathrel{:=}\mathcal{T}({\mathcal{C}}_{\pi},\mathcal{V})$.
Based on $\mathcal{F}_{\pi}$ we define the _normalised signature_
$\mathcal{F}^{\operatorname{\mathsf{n}}}_{\pi}\mathrel{:=}\\{{f^{\operatorname{\mathsf{n}}}\mid
f\in\mathcal{F}_{\pi}}\\}$ where the arity of $f^{\operatorname{\mathsf{n}}}$
is $\lvert{\operatorname{\mathsf{nrm}}(f)}\rvert$. We extend $\succsim$ to
$\mathcal{F}^{\operatorname{\mathsf{n}}}_{\pi}$ by
$f^{\operatorname{\mathsf{n}}}\succsim g^{\operatorname{\mathsf{n}}}$ if and
only if $f\succsim g$. Let $\mathsf{s}$ be a fresh constant that is minimal
with respect to $\succsim$. We introduce the _Buchholz norm_ of $t$ (denoted
as $\lVert{t}\rVert$) a term complexity measure that fits well with the
definition of $\textsc{POP}^{\ast}$. Set
$\lVert{t}\rVert\mathrel{:=}1+\max\\{{n,\lVert{t_{1}}\rVert,\dots,\lVert{t_{n}}\rVert}\\}$
for $t=f({t}_{1},\ldots,{t}_{n})$ and $\lVert{t}\rVert\mathrel{:=}1$,
otherwise. In the following we define an embedding from the relative rewriting
relation
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}}$
into $\mathrel{\blacktriangleright}_{k}$, such that $k$ depends only on TRSs
$\mathcal{R}$ and $\mathcal{S}$. This embedding provides the technical tool to
measure the number of root steps in a given derivation through the number of
descent in $\mathrel{\blacktriangleright}_{k}$. Hence Proposition 2.3 becomes
applicable to establishing our main result. This intuition is cast into the
next definition.
###### Definition 5.1.
A _predicative interpretation_ is a pair of mappings
$(\mathsf{S}_{\pi},\mathsf{N}_{\pi})$ from terms to sequences
$\operatorname{\mathcal{S}eq}(\mathcal{F}^{\operatorname{\mathsf{n}}}_{\pi}\cup\\{{\mathsf{s}}\\},\mathcal{V})$
defined as follows. We assume for $\pi(t)=f(\pi(t_{1}),\dots,\pi(t_{n}))$ that
$\operatorname{\mathsf{safe}}(f)=\\{{i_{1},\dots,i_{p}}\\}$ and
$\operatorname{\mathsf{nrm}}(f)=\\{{j_{1},\dots,j_{q}}\\}$.
$\displaystyle\mathsf{S}_{\pi}(t)$
$\displaystyle\mathrel{:=}\begin{cases}[\,]&\text{if
$\pi(t)\in{\mathsf{Val}}_{\pi}$},\\\
[f^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(t_{j_{1}}),\dots,\mathsf{N}_{\pi}(t_{j_{q}}))\leavevmode\nobreak\
\mathsf{S}_{\pi}(t_{i_{1}})\leavevmode\nobreak\ \cdots\leavevmode\nobreak\
\mathsf{S}_{\pi}(t_{i_{p}})]&\text{if
$\pi(t)\not\in{\mathsf{Val}}_{\pi}$.}\end{cases}$
$\displaystyle\mathsf{N}_{\pi}(t)$
$\displaystyle\mathrel{:=}\mathsf{S}_{\pi}(t)\mathrel{::}\operatorname{\mathsf{BN_{\pi}}}(t)$
Here the function $\operatorname{\mathsf{BN_{\pi}}}$ maps a term $t$ to the
sequence $[\mathsf{s}\cdots\mathsf{s}]$ with $\lVert{\pi(t)}\rVert$
occurrences of the constant $\mathsf{s}$.
Note that as a direct consequence of the definitions we obtain
$\operatorname{\mathsf{width}}(\mathsf{N}_{\pi}(t))=\lVert{\pi(t)}\rVert+1$
for all terms $t$.
###### Lemma 5.2.
There exists a polynomial $p$ such that
$\mathsf{G}_{k}(\mathsf{N}_{\pi}(t))\leqslant p(\lvert{t}\rvert)$ for every
basic term $t$. The polynomial $p$ depends only on $k$.
###### Proof.
Suppose $t=f({v}_{1},\ldots,{v}_{n})$ is a basic term with
$\operatorname{\mathsf{safe}}(f)=\\{{{i}_{1},\ldots,{i}_{p}}\\}$ and
$\operatorname{\mathsf{nrm}}(f)=\\{{{j}_{1},\ldots,{j}_{q}}\\}$. The only non-
trivial case is when $\pi(t)\not\in{\mathsf{Val}}_{\pi}$. Then
$\mathsf{N}_{\pi}(t)=[u\leavevmode\nobreak\
\mathsf{S}_{\pi}(v_{i_{1}})\cdots\mathsf{S}_{\pi}(v_{i_{p}})]\mathrel{::}\operatorname{\mathsf{BN_{\pi}}}(t)$
where
$u=f^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(v_{j_{1}}),\dots,\mathsf{N}_{\pi}(v_{j_{q}}))$.
Note that $\mathsf{S}_{\pi}(v_{i})=[\,]$ for
$i\in\\{{{i}_{1},\ldots,{i}_{q}}\\}$. Let $\mathcal{A}$ denote a polynomial
interpretation fulfilling Proposition 2.3. Using the assumption
$\circ_{\mathcal{A}}({m}_{1},\ldots,{m}_{n})=\sum_{i=1}^{n}m_{i}+n$, it is
easy to see that $\mathsf{G}_{k}(\mathsf{N}_{\pi}(t))$ is bounded linear in
$\lVert{\pi(t)}\rVert\leqslant\lvert{t}\rvert$ and
$[\alpha]_{\mathcal{A}}(u)$. As
$\mathsf{N}_{\pi}(v_{j})=[[]\leavevmode\nobreak\ \mathsf{s}\cdots\mathsf{s}]$
with $\lVert{\pi(v_{j})}\rVert\leqslant\lvert{t}\rvert$ occurrences of
$\mathsf{s}$, $\mathsf{G}_{k}(\mathsf{N}_{\pi}(v_{j}))$ is linear in
$\lvert{t}\rvert$. Hence from Proposition 2.3 we conclude that
$\mathsf{G}_{k}(\mathsf{N}_{\pi}(t))$ is polynomially bounded in
$\lvert{t}\rvert$. ∎
The next sequence of lemmas shows that the relative rewriting relation
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}}$
is embeddable into $\mathrel{\blacktriangleright}_{k}$.
###### Lemma 5.3.
Suppose $s\mathrel{{>}^{\pi}_{\mathsf{pop*}}}t$ such that
$\pi(s\sigma)\in{\mathsf{Val}}_{\pi}$. Then
$\mathsf{S}_{\pi}(s\sigma)=[\,]=\mathsf{S}_{\pi}(t\sigma)$ and
$\mathsf{N}_{\pi}(s\sigma)\mathrel{\blacktriangleright}_{1}\mathsf{N}_{\pi}(t\sigma)$.
###### Proof.
Let $\pi(s\sigma)\in{\mathsf{Val}}_{\pi}$, and suppose
$s\mathrel{{>}^{\pi}_{\mathsf{pop*}}}t$, i.e.,
$\pi(s)\mathrel{{>}_{\mathsf{pop*}}}\pi(t)$ holds. Observe that since
$\pi(s)\in{\mathsf{Val}}_{\pi}$ and due to our assumptions on safe mappings,
only clause $(1)$ from the definition of $\mathrel{{>}_{\mathsf{pop*}}}$ (or
respectively $\mathrel{{>}_{\mathsf{pop}}}$) is applicable. And thus $\pi(t)$
is a subterm of $\pi(s)$ modulo the equivalence $\approx$. We conclude
$\pi(t\sigma)\in{\mathsf{Val}}_{\pi}$, and hence
$\mathsf{S}_{\pi}(s\sigma)=[\,]=\mathsf{S}_{\pi}(t\sigma)$. Finally, notice
that $\lVert{\pi(s\sigma)}\rVert>\lVert{\pi(t\sigma)}\rVert$ as $\pi(t\sigma)$
is a subterm of $\pi(s\sigma)$. Thus
$\mathsf{N}_{\pi}(s\sigma)\mathrel{\blacktriangleright}_{1}\mathsf{N}_{\pi}(t\sigma)$
follows as well. ∎
To improve the clarity of the exposition, we concentrate on the curcial cases
in the proofs of the following lemma. The interested reader is kindly referred
to Avanzini (2009) for the full proof.
###### Lemma 5.4.
Suppose $s\mathrel{{>}^{\pi}_{\mathsf{pop}}}t$ such that
$\pi(s\sigma)=f(\pi(s_{1}\sigma),\dots,\pi(s_{n}\sigma))$ with
$\pi(s_{i}\sigma)\in{\mathsf{Val}}_{\pi}$ for $i\in\\{{1,\dots,n}\\}$.
Moreover suppose $\operatorname{\mathsf{nrm}}(f)=\\{{j_{1},\dots,j_{q}}\\}$.
Then
$f^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(s_{j_{1}}\sigma),\dots,\mathsf{N}_{\pi}(s_{j_{q}}\sigma))\mathrel{\gtrdot}_{3\cdot\lVert{\pi(t)}\rVert}\mathsf{N}_{\pi}(t\sigma)$
holds.
###### Proof.
Note that the assumption implies that the argument filtering $\pi$ does not
collapse $f$. We show the lemma by induction on
$\mathrel{{>}^{\pi}_{\mathsf{pop}}}$. We consider the subcase that
$s\mathrel{{>}^{\pi}_{\mathsf{pop}}}t$ follows as
$t=g({t}_{1},\ldots,{t}_{m})$, $\pi$ does not collapse on $g$, $f\succ g$, and
$s\mathrel{{>}^{\pi}_{\mathsf{pop}}}t_{j}$ for all $j\in\pi(g)$, cf.
Definition 4.1(ii). We set
$u\mathrel{:=}f^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(s_{j_{1}}\sigma),\dots,\mathsf{N}_{\pi}(s_{j_{q}}\sigma))$
and $k\mathrel{:=}3\cdot\lVert{\pi(t)}\rVert$ and first prove
$u\mathrel{\gtrdot}_{k-1}\mathsf{S}_{\pi}(t\sigma)$.
If $\pi(t\sigma)\in{\mathsf{Val}}_{\pi}$, then
$\mathsf{S}_{\pi}(t\sigma)=[\,]$ is minimal with respect to
$\mathrel{\gtrdot}_{k-1}$. Thus we are done. Hence suppose
$\operatorname{\mathsf{nrm}}(g)=\\{{j^{\prime}_{1},\dots,j^{\prime}_{q}}\\}$,
$\operatorname{\mathsf{safe}}(g)=\\{{i^{\prime}_{1},\dots,i^{\prime}_{p}}\\}$
and let
$\mathsf{S}_{\pi}(t\sigma)=[g^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(t_{j^{\prime}_{1}}\sigma),\dots,\mathsf{N}_{\pi}(t_{j^{\prime}_{q}}\sigma))\leavevmode\nobreak\
\mathsf{S}_{\pi}(t_{i^{\prime}_{1}}\sigma)\cdots\mathsf{S}_{\pi}(t_{i^{\prime}_{p}}\sigma)]\hbox
to0.0pt{$\;$.\hss}$
We set
$v\mathrel{:=}g^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(t_{j^{\prime}_{1}}\sigma),\dots,\mathsf{N}_{\pi}(t_{j^{\prime}_{q}}\sigma))$.
It suffices to show $u\mathrel{\gtrdot}_{k-2}v$ and
$u\mathrel{\gtrdot}_{k-2}\mathsf{S}_{\pi}(t_{j}\sigma)$ for
$j\in\operatorname{\mathsf{safe}}(g)$. Both assertions follow from the
induction hypothesis.
Now consider
$\mathsf{N}_{\pi}(t\sigma)=[\mathsf{S}_{\pi}(t\sigma)\leavevmode\nobreak\
\mathsf{s}\cdots\mathsf{s}]$ with $\lVert{\pi(t\sigma)}\rVert$ occurrences of
the constant $\mathsf{s}$. Recall that
$\operatorname{\mathsf{width}}(\mathsf{N}_{\pi}(t\sigma))=\lVert{\pi(t\sigma)}\rVert+1$.
Observe that $f^{\operatorname{\mathsf{n}}}\succ\mathsf{s}$. Hence to prove
$u\mathrel{\gtrdot}_{k}\mathsf{S}_{\pi}(t\sigma)$ it suffices to observe that
$\operatorname{\mathsf{width}}(u)+k>\lVert{\pi(t\sigma)}\rVert+1$ holds. For
that note that $\lVert{\pi(t\sigma)}\rVert$ is either
$\lVert{\pi(t_{j}\sigma)}\rVert+1$ for some $j\in\pi(g)$ or less than $k$. In
the latter case, we are done. Otherwise
$\lVert{\pi(t\sigma)}\rVert=\lVert{\pi(t_{j}\sigma)}\rVert+1$. Then from the
definition of $\mathrel{\gtrdot}_{k}$ and the induction hypothesis
$u\mathrel{\gtrdot}_{3\cdot\lVert{\pi(t_{j})}\rVert}\mathsf{N}_{\pi}(t_{j}\sigma)$
we can conclude
$\operatorname{\mathsf{width}}(u)+3\cdot\lVert{\pi(t_{j})}\rVert>\operatorname{\mathsf{width}}(\mathsf{N}_{\pi}(t_{j}\sigma))=\lVert{\pi(t_{j}\sigma)}\rVert+1$.
Since $k\geqslant 3\cdot(\lVert{\pi(t_{j})}\rVert+1)$,
$\operatorname{\mathsf{width}}(u)+k>\lVert{\pi(t\sigma)}\rVert+1$ follows. ∎
###### Lemma 5.5.
Suppose $s\mathrel{{>}^{\pi}_{\mathsf{pop*}}}t$ such that
$\pi(s\sigma)=f(\pi(s_{1}\sigma),\dots,\pi(s_{n}\sigma))$ with
$\pi(s_{i}\sigma)\in{\mathsf{Val}}_{\pi}$ for $i\in\\{{1,\dots,n}\\}$. Then
for $\operatorname{\mathsf{nrm}}(f)=\\{{j_{1},\dots,j_{q}}\\}$,
1. (i)
$f^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(s_{j_{1}}\sigma),\dots,\mathsf{N}_{\pi}(s_{j_{q}}\sigma))\mathrel{\blacktriangleright}_{3\cdot\lVert{\pi(t)}\rVert}\mathsf{S}_{\pi}(t\sigma)$,
and
2. (ii)
$f^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(s_{j_{1}}\sigma),\dots,\mathsf{N}_{\pi}(s_{j_{q}}\sigma))\mathrel{::}\operatorname{\mathsf{BN_{\pi}}}(s\sigma)\mathrel{\blacktriangleright}_{3\cdot\lVert{\pi(t)}\rVert}\mathsf{N}_{\pi}(t\sigma)$.
###### Proof.
The lemma is shown by induction on the definition of
$\mathrel{{>}^{\pi}_{\mathsf{pop*}}}$. For the following, we set
$u=f^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(s_{j_{1}}\sigma),\dots,\mathsf{N}_{\pi}(s_{j_{q}}\sigma))$.
Suppose $s\mathrel{{>}^{\pi}_{\mathsf{pop*}}}t$ follows due to Definition
4.2(ii). We set $k\mathrel{:=}3\cdot\lVert{\pi(t)}\rVert$. Let
$\operatorname{\mathsf{nrm}}(g)=\\{{j^{\prime}_{1},\dots,j^{\prime}_{q}}\\}$
and let
$\operatorname{\mathsf{safe}}(g)=\\{{i^{\prime}_{1},\dots,i^{\prime}_{p}}\\}$.
Property $(\ref{en:embed:hlp:a})$ is immediate for
$\pi(t\sigma)\in{\mathsf{Val}}_{\pi}$, so assume otherwise. We see that
$s\mathrel{{>}^{\pi}_{\mathsf{pop}}}t_{j}$ for all
$j\in\operatorname{\mathsf{nrm}}(g)$ and obtain
$u\mathrel{\gtrdot}_{k-1}g^{\operatorname{\mathsf{n}}}(\mathsf{N}_{\pi}(t_{j^{\prime}_{1}}\sigma),\dots,\mathsf{N}_{\pi}(t_{j^{\prime}_{q}}\sigma))$
as in Lemma 5.4. Furthermore, $s\mathrel{{>}^{\pi}_{\mathsf{pop*}}}t_{j_{0}}$
for some $j_{0}\in\operatorname{\mathsf{safe}}(g)$ and by induction
hypothesis:
$u\mathrel{\blacktriangleright}_{k-1}\mathsf{S}_{\pi}(t_{j_{0}}\sigma)$. To
conclude property $(\ref{en:embed:hlp:a})$, it remains to verify
$u\mathrel{\gtrdot}_{k-1}\mathsf{S}_{\pi}(t_{j}\sigma)$ for the remaining
$j\in\operatorname{\mathsf{safe}}(g)$. We either have
$s\mathrel{{>}^{\pi}_{\mathsf{pop}}}t_{j}$ or
$\pi(s_{i})\mathrel{\unrhd}\pi(t_{j})$ (for some $i$). In the former subcase
we proceed as in the claim, and for the latter we observe
$\pi(t_{j}\sigma)\in{\mathsf{Val}}_{\pi}$, and thus
$\mathsf{S}_{\pi}(t_{j}\sigma)=[\,]$ follows. This establishes property
$(\ref{en:embed:hlp:a})$.
To conclude property $(\ref{en:embed:hlp:b})$, it suffices to show
$\operatorname{\mathsf{width}}(u\mathrel{::}\operatorname{\mathsf{BN_{\pi}}}(s\sigma))+k>\operatorname{\mathsf{width}}(\mathsf{N}_{\pi}(t\sigma))$,
or equivalently $\lVert{\pi(s\sigma)}\rVert+1+k>\lVert{\pi(t\sigma)}\rVert$.
The latter can be shown, if we proceed similar as in the claim. ∎
Recall the definition of
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{Q}$}}}_{\mathcal{R}}}$
from Section 2 and define
$\mathcal{Q}\mathrel{:=}\\{f({x}_{1},\ldots,{x}_{n})\to\bot\mid
f\in\mathcal{D}\\}$, and set
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{R}}}\mathrel{:=}{\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{Q}$}}}_{\mathcal{R}}}}$.
As the normal forms of $\mathcal{Q}$ coincide with $\mathsf{Val}$,
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{R}}}$
is the restriction of
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}$,
where arguments need to be values instead of normal forms of $\mathcal{R}$.
From Lemma 5.3 and 5.5 we derive an embedding of root steps
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\varepsilon}_{\mathcal{R}}}$.
Suppose the step
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{R}}}t$
takes place below the root. Observe that $\pi(s)\not=\pi(t)$ need not hold in
general. Thus we cannot hope to prove
$\mathsf{N}_{\pi}(s)\mathrel{\blacktriangleright}_{k}\mathsf{N}_{\pi}(t)$.
However, we have the following stronger result.
###### Lemma 5.6.
There exists a uniform $k\in\mathbb{N}$ (depending only on $\mathcal{R}$) such
that if $\mathcal{R}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ holds then
${s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\varepsilon}_{\mathcal{R}}}t}$
implies
${\mathsf{N}_{\pi}(s)\mathrel{\blacktriangleright}_{k}\mathsf{N}_{\pi}(t)}$.
Moreover, if
$\mathcal{R}\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
holds then
${s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{R}}}t}$
implies ${\mathsf{N}_{\pi}(s)\mathrel{\not{\gtrsim}}_{k}\mathsf{N}_{\pi}(t)}$.
###### Proof.
We consider the first half of the assertion. Suppose
$\mathcal{R}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ and
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\varepsilon}_{\mathcal{R}}}t$,
that is for some rule ${f({l}_{1},\ldots,{l}_{n})\to r}\in\mathcal{R}$ and
substitution $\sigma\,\colon\,\mathcal{V}\to\mathsf{Val}$ we have
$s=f(l_{1}\sigma,\dots,l_{n}\sigma)$ and $t=r\sigma$. Depending on whether
$\pi$ collapses $f$, the property either directly follows from Lemma 5.3 or is
a consequence of Lemma $\ref{l:embed:hlp}(\ref{en:embed:hlp:b})$.
In order to conclude the second half of the assertion, one performs induction
on the rewrite context. In addition, one shows that for the special case
$\mathsf{S}_{\pi}(s)\approx\mathsf{S}_{\pi}(t)$, still
$\lVert{\pi(s)}\rVert\geqslant\lVert{\pi(t)}\rVert$ holds. From this the lemma
follows. ∎
For constructor TRSs, we can simulate
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}$
using
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{R}}}$.
We extend $\mathcal{R}$ with suitable rules $\Phi(\mathcal{R})$, which replace
normal forms that are not values by some constructor symbol. To simplfy the
argument we re-use the symbol $\bot$ from above. We define the TRS
$\Phi(\mathcal{R})$ as
$\Phi(\mathcal{R})\mathrel{:=}\\{{f({t}_{1},\ldots,{t}_{n})\to\bot\mid
f({t}_{1},\ldots,{t}_{n})\in{\operatorname{\mathsf{NF}}(\mathcal{R})\cap\mathcal{T}(\mathcal{F})}\text{
and }f\in\mathcal{D}}\\}\hbox to0.0pt{$\;$.\hss}$
Moreover, we define
$\phi_{\mathcal{R}}(t)\mathrel{:=}t{\downarrow}_{\Phi(\mathcal{R})}$. Observe
that $\phi_{\mathcal{R}}(\cdot)$ is well-defined since $\Phi(\mathcal{R})$ is
confluent and terminating.
###### Lemma 5.7.
Let $\mathcal{R}\cup\mathcal{S}$ be a constructor TRS. Define
$\mathcal{S}^{\prime}\mathrel{:=}\mathcal{S}\cup\Phi(\mathcal{R}\cup\mathcal{S})$.
For $s\in\mathcal{T}(\mathcal{F})$,
${s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}}t}\quad\text{implies}\quad{\phi_{\mathcal{R}\cup\mathcal{S}}(s)\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}^{\prime}}}\phi_{\mathcal{R}\cup\mathcal{S}}(t)}\hbox
to0.0pt{$\;$,\hss}$
where
${\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{R}/\mathcal{S}^{\prime}}}}$
abbreviates
${\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{S}^{\prime}}}\cdot\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{R}}}\cdot\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{S}^{\prime}}}}$.
###### Proof.
It is easy to see that
${s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}t}$
implies
$\phi_{\mathcal{R}}(s)\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{R}}}\cdot\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{!}_{\Phi(\mathcal{R})}}\phi_{\mathcal{R}}(t)$.
Suppose
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}}t$,
then there exist ground terms $u$ and $v$ such that
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{S}}}u\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}}}v\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{S}}}t$.
Let $\phi(t)\mathrel{:=}\phi_{\mathcal{R}\cup\mathcal{S}}(t)$. From the above,
$\phi(s)\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{S}^{\prime}}}\phi(u)\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\varepsilon}_{\mathcal{R}}}\cdot\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{S}^{\prime}}}\phi(v)\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{S}^{\prime}}}\phi(t)$
follows as desired. ∎
Suppose $\mathcal{R}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ and
$\mathcal{S}\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
holds. Together with Lemma 5.6, the above simulation establishes the promised
embedding of
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}}$
into $\mathrel{\blacktriangleright}_{k}$.
###### Lemma 5.8.
Let $\mathcal{R}\cup\mathcal{S}$ be a constructor TRS, and suppose
$\mathcal{R}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ and
${\mathcal{S}}\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
hold. Then for $k$ depending only on $\mathcal{R}$ and $\mathcal{S}$ and
$s\in\mathcal{T}(\mathcal{F})$, we have
${s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}}t}\quad\text{implies}\quad{\mathsf{N}_{\pi}(\phi(s))\mathrel{\blacktriangleright}_{k}^{+}\mathsf{N}_{\pi}(\phi(t))}\hbox
to0.0pt{$\;$.\hss}$
###### Proof.
Consider a step
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}}t$
and set $\phi(t)\mathrel{:=}\phi_{\mathcal{R}\cup\mathcal{S}}(t)$. By Lemma
5.7 there exist terms $u$ and $v$ such that
$\phi(s)\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{S}\cup\Phi(\mathcal{R}\cup\mathcal{S})}}u\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\varepsilon}_{\mathcal{R}}}v\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{S}\cup\Phi(\mathcal{R}\cup\mathcal{S})}}\phi(t)$.
Since $\mathcal{R}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ holds, by
Lemma 5.6
$\mathsf{N}_{\pi}(u)\mathrel{\blacktriangleright}_{k_{1}}\mathsf{N}_{\pi}(v)$
follows. Moreover from
$\mathcal{S}\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
together with Lemma 5.6 we conclude that
$r_{1}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{S}\cup\Phi(\mathcal{R}\cup\mathcal{S})}}r_{2}$
implies
$\mathsf{N}_{\pi}(r_{1})\mathrel{\not{\gtrsim}}_{k_{2}}\mathsf{N}_{\pi}(r_{2})$.
Here it suffices to see that steps from
$\mathcal{V}(\mathcal{R}\cup\mathcal{S})$ are easy to embed into
$\mathrel{\not{\gtrsim}}_{k_{2}}$ using the predicative interpretation
$\mathsf{N}_{\pi}$ independent of $k_{2}$. In both cases $k_{1}$ and $k_{2}$
depend only on $\mathcal{R}$ and $\mathcal{S}$ respectively; set
$k\mathrel{:=}\max\\{{k_{1},k_{2}}\\}$. In sum we have
$\mathsf{N}_{\pi}(\phi(s))\mathrel{\not{\gtrsim}}_{k}^{*}\mathsf{N}_{\pi}(u)\mathrel{\blacktriangleright}_{k}\mathsf{N}_{\pi}(v)\mathrel{\not{\gtrsim}}_{k}^{*}\mathsf{N}_{\pi}(\phi(t))$,
employing
${\mathrel{\blacktriangleright}_{l_{1}}}\subseteq{\mathrel{\blacktriangleright}_{l_{2}}}$
for $l_{1}\leqslant l_{2}$. It is an easy exercise to show that
${{\mathrel{\blacktriangleright}_{k}}\cdot{\approx}}\subseteq{\mathrel{\blacktriangleright}_{k}}$
and likewise
${{\approx}\cdot{\mathrel{\blacktriangleright}_{k}}}\subseteq{\mathrel{\blacktriangleright}_{k}}$
holds. Hence the lemma follows. ∎
###### Theorem 5.9.
Let $\mathcal{R}\cup\mathcal{S}$ be a constructor TRS, and suppose
$\mathcal{R}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ and
${\mathcal{S}}\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
holds. Then there exists a polynomial $p$ depending only on
$\mathcal{R}\cup\mathcal{S}$ such that for any basic and ground term $t$,
$\operatorname{dl}(t,\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}})\leqslant
p(\lvert{t}\rvert)$.
###### Proof.
Assume $t\not\in\operatorname{\mathsf{NF}}(\mathcal{R}\cup\mathcal{S})$,
otherwise
$\operatorname{dl}(t,\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}})$
is trivially bounded. Moreover $t$ is a basic term, hence
$\phi_{\mathcal{R}\cup\mathcal{S}}(t)=t$. From Lemma 5.8 we infer that
$\operatorname{dl}(t,\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}/\mathcal{S}}})\leqslant\mathsf{G}_{k}(\mathsf{N}_{\pi}(\phi_{\mathcal{R}\cup\mathcal{S}}(t)))=\mathsf{G}_{k}(\mathsf{N}_{\pi}({t}))$
for some $k$, where the latter is polynomially bounded in $\lvert{t}\rvert$
and the polynomial only depends on $k$, cf. Lemma 5.2. Finally $k$ depends
only on $\mathcal{R}\cup\mathcal{S}$. ∎
Suppose $\mathcal{R}$ is a constructor TRS, and let $\mathcal{P}$ denote the
set of weak innermost dependency pairs. For the moment, suppose that all
compound symbols of $\mathcal{P}$ are nullary. Provided that $\mathcal{P}$ is
non-duplicating and compatible with some SLI, as a consequence of the above
theorem paired with Proposition 3.4, the inclusions
$\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ and
$\mathcal{U}(\mathcal{P})\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
certify that
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}$
is polynomially bounded. Observe that for the application of
$\mathrel{{>}^{\pi}_{\mathsf{pop*}}}$ and
$\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}$
in the context of $\mathcal{P}$ and $\mathcal{U}(\mathcal{P})$, we alter
Definitions 4.1 and 4.2 such that $f\in\mathcal{D}^{\sharp}$ is demanded.
###### Example 5.10 (Example 4.5 continued).
Reconsider the TRS $\mathcal{R}_{\textsf{bits}}$, and let $\mathcal{P}$ denote
$\mathsf{WIDP}(\mathcal{R}_{\textsf{bits}})$ as drawn in Example 3.2. By
taking the SLI $\mathcal{A}$ with $0_{\mathcal{A}}=0$,
$\mathsf{s}_{\mathcal{A}}(x)=x+1$ and $\mathsf{half}_{\mathcal{A}}(x)=x+1$ we
obtain $\mathcal{U}(\mathcal{P})\subseteq{\mathrel{>_{\mathcal{A}}}}$ and
moreover, observe that $\mathcal{P}$ is both non-duplicating and contains only
nullary compound symbols. In Example 4.5 we have seen that
$\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ holds. Similar,
$\mathcal{U}(\mathsf{WIDP}(\mathcal{R}_{\textsf{bits}}))\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
can easily be shown. From the above observation we thus conclude a polynomial
runtime-complexity of $\mathcal{R}_{\textsf{bits}}$.
The assumption that all compound symbols from $\mathcal{P}$ need to be nullary
is straightforward to lift, but technical. Hence, we do not provide a complete
proof here, but only indicate the necessary changes. The formal construction
can be found in the Appendix.
Note that in the general case, it does not suffice to embed root steps of
$\mathcal{P}$ into $\mathrel{\blacktriangleright}_{k}$, rather we have to
embed steps of form
$C[s_{1}^{\sharp},\dots,s_{i}^{\sharp},\dots,s_{n}^{\sharp}]\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{P}}}C[s_{1}^{\sharp},\dots,t_{i}^{\sharp},\dots,s_{n}^{\sharp}]$
with $C$ being a context built from compound symbols. As first measure we
require that the argument filtering $\pi$ is _safe_ Hirokawa and Moser
(2008b), that is $\pi(c)=[1,\dots,n]$ for each compound symbol $\mathsf{c}$ of
arity $n$. Secondly, we adapt the predicative interpretation
$\mathsf{N}_{\pi}$ in such a way that compound symbols are interpreted as
sequences, and their arguments by the interpretation $\mathsf{N}_{\pi}$. This
way, a proper embedding using $\mathsf{N}_{\pi}$ requires
$\mathsf{N}_{\pi}(s_{i}^{\sharp})\mathrel{\blacktriangleright}_{k}\mathsf{N}_{\pi}(t_{i}^{\sharp})$
instead of
$\mathsf{S}_{\pi}(s_{i}^{\sharp})\mathrel{\blacktriangleright}_{k}\mathsf{S}_{\pi}(t_{i}^{\sharp})$.
###### Theorem 5.11.
Let $\mathcal{R}$ be a constructor TRS, and let $\mathcal{P}$ denote the set
of weak innermost dependency pairs. Assume $\mathcal{P}$ is non-duplicating,
and suppose ${\mathcal{U}(\mathcal{P})}\subseteq{>_{\mathcal{A}}}$ for some
SLI $\mathcal{A}$. Let $\pi$ be a safe argument filtering. If
$\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ and
$\mathcal{U}(\mathcal{P})\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
then
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}$
is polynomially bounded.
Above it is essential that $\mathcal{R}$ is a constructor TRS. This even holds
when $\textsc{POP}^{\ast}$ is applied directly.
###### Example 5.12.
Consider the TRS $\mathcal{R}_{\operatorname{\mathsf{exp}}}$ below:
$\begin{array}[]{c}\operatorname{\mathsf{exp}}(x)\to\mathsf{e}(\mathsf{g}(x))\qquad\mathsf{e}(\mathsf{g}(\mathsf{s}(x)))\to\operatorname{\mathsf{dp}}_{1}(\mathsf{g}(x))\qquad\mathsf{g}(\mathsf{0})\to\mathsf{0}\rule[-7.74998pt]{0.0pt}{0.0pt}\\\
\operatorname{\mathsf{dp}}_{1}(x)\to\operatorname{\mathsf{dp}}_{2}(\mathsf{e}(x),x)\qquad\qquad\operatorname{\mathsf{dp}}_{2}(x,y)\to\operatorname{\mathsf{pr}}(x,\mathsf{e}(y))\end{array}$
The above rules are oriented (directly) by $\mathrel{{>}_{\mathsf{pop*}}}$
induced by $\operatorname{\mathsf{safe}}$ and $\succsim$ such that: (i) the
argument position of $\mathsf{g}$ and $\operatorname{\mathsf{exp}}$ are
normal, the remaining argument positions are safe, and (ii)
$\operatorname{\mathsf{exp}}\succ\mathsf{g}\succ\operatorname{\mathsf{dp}}_{1}\succ\operatorname{\mathsf{dp}}_{2}\succ\mathsf{e}\succ\operatorname{\mathsf{pr}}\succ\mathsf{0}$.
On the other hand, $\mathcal{R}_{\operatorname{\mathsf{exp}}}$ admits at least
exponential innermost runtime-complexity, as for instance
$\operatorname{\mathsf{exp}}(s^{n}(\mathsf{0}))$ normalizes in exponentially
(in $n$) many innermost rewrite steps.
To overcome this obstacle, we adapt the definition of
$\mathrel{{>}_{\mathsf{pop*}}}$ in the sense that we refine the notion of
defined function symbols as follows. Let $\mathcal{G}_{\mathcal{C}}$ denote
the least set containing $\mathcal{C}$ and all symbols appearing in arguments
to left-hand sides in $\mathcal{R}$. Moreover, set
$\mathcal{G}_{\mathcal{D}}\mathrel{:=}\mathcal{F}\setminus\mathcal{G}_{\mathcal{C}}$
and set
$\mathsf{Val}\mathrel{:=}\mathcal{T}(\mathcal{G}_{\mathcal{C}},\mathcal{V})$.
Then in order to extend Theorem 5.11 to non-constructor TRS it suffices to
replace $\mathcal{D}$ by $\mathcal{G}_{\mathcal{D}}$ and $\mathcal{C}$ by
$\mathcal{G}_{\mathcal{C}}$ in all above given definitions and arguments (see
Avanzini (2009) for the formal construction). Thus the next theorem follows
easily from combining Proposition 3.6 and Theorem 5.11. Note that this theorem
can be easily extended so that in each path different termination techniques
(inducing polynomial runtime complexity) are employed, see Hirokawa and Moser
(2008a) and Section 6.
###### Theorem 5.13.
Let $\mathcal{R}$ be a TRS. Let $\mathcal{G}$ denote the weak innermost
dependency graph, and let
$\mathcal{F}=\mathcal{G}_{\mathcal{D}}\uplus\mathcal{G}_{\mathcal{C}}$ be
separated as above. Suppose for every path
$(\mathcal{P}_{1},\ldots,\mathcal{P}_{n})$ in ${\mathcal{G}}/_{\\!\equiv}$
there exists an SLI $\mathcal{A}$ and a pair
$(\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}},\mathrel{{>}^{\pi}_{\mathsf{pop*}}})$
based on a safe argument filtering $\pi$ such that (i)
$\mathcal{U}(\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{n})\subseteq{>_{\mathcal{A}}}$
(ii)
$\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{n-1}\cup\mathcal{U}(\mathcal{P}_{1}\cup\cdots\cup\mathcal{P}_{n})\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$,
and (iii) $\mathcal{P}_{n}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$
holds. Then
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}$
is polynomially bounded.
The next theorem establishes that $\textsc{POP}^{\ast}$ in conjunction with
(weak innermost) dependency pairs induces polytime computability of the
function described through the analysed TRS. We kindly refer the reader to the
Appendix for the proof.
###### Theorem 5.14.
Let $\mathcal{R}$ be an orthogonal, $S$-sorted and completely defined
constructor TRS such that the underlying signature is simple. Let
$\mathcal{P}$ denote the set of weak innermost dependency pairs. Assume
$\mathcal{P}$ is non-duplicating, and suppose
${\mathcal{U}(\mathcal{P})}\subseteq{>_{\mathcal{A}}}$ for some SLI
$\mathcal{A}$. If $\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$
and
$\mathcal{U}(\mathcal{P})\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
then the functions computed by $\mathcal{R}$ are computable in polynomial
time.
Here _simple_ signature Marion (2003) essentially means that the size of any
constructor term depends polynomially on its depth. Such a restriction is
always necessary in this context. A detailed account is given in the Appendix
(see alo Marion (2003)). This restriction is also responsible for the
introduction of sorts.
## 6 Experimental Results
All described techniques have been incorporated into the _Tyrolean Complexity
Tool_ TCT, an open source complexity analyser222Available at http://cl-
informatik.uibk.ac.at/software/tct.. We performed tests on two testbeds: T
constitutes of the $1394$ examples from the Termination Problem Database
Version 5.0.2 that were used in the runtime-complexity category of the
termination competition 2008333See http://termcomp.uibk.ac.at.. Moreover,
testbed C is the restriction of testbed T to constructor TRSs ($638$ in
total). All experiments were conducted on a machine that is identical to the
official competition server ($8$ AMD Opteron${}^{\text{\textregistered}}$ 885
dual-core processors with 2.8GHz, $8\text{x}8$ GB memory). As timeout we use 5
seconds. We orient TRSs using $\mathrel{{>}^{\pi}_{\mathsf{pop*}}}$ by
encoding the constraints on precedence and so forth in _propositional logic_
(cf. Avanzini (2009) for details), employing $\mathsf{MiniSat}$ Eén and
Sörensson (2003) for finding satisfying assignments. In a similar spirit, we
check compatibility with SLIs via translations to SAT. In order to derive an
estimated dependency graph, we use the function $\mathsf{ICAP}$ (cf. Giesl et
al. (2005)).
Experimental findings are summarised in Table 1.444See http://cl-
informatik.uibk.ac.at/~zini/rta09 for extended results. In each column, we
highlight the total on yes-, maybe- and timeout-instances. Furthermore, we
annotate average times in seconds. In the first three columns we contrast
$\textsc{POP}^{\ast}$ as direct technique to $\textsc{POP}^{\ast}$ as base to
(weak innermost) dependency pairs. I.e., the columns WIDP and WIDG show
results concerning Proposition 3.4 together with Theorem 5.11 or Theorem 5.13
respectively.
In the remaining four columns we assess the power of Proposition 3.4 and 3.6
in conjunction with different base orders, thus verifying that the use of
$\textsc{POP}^{\ast}$ in this context is independent to existing techniques.
Column P asserts that the different paths are handled by _linear and quadratic
restricted interpretations_ Hirokawa and Moser (2008b). In column PP, in
addition $\textsc{POP}^{\ast}$ is employed. Similar, in column M _restricted
matrix interpretations_ (that is matrix interpretations Endrullis et al.
(2008), where constructors are interpreted by triangular matrices) are used to
handle different paths. Again column MP extends column M with
$\textsc{POP}^{\ast}$. Note that all methods induce polynomial innermost
runtime complexity.
| polynomial path orders | dependency graphs mixed
---|---|---
| DIRECT | WIDP | WIDG | P | PP | M | MP
T | Yes | 46 | /0.03 | 69 | /0.09 | 80 | /0.07 | 198 | /0.54 | 198 | /0.51 | 200 | /0.63 | 207 | /0.48
| Maybe | 1348 | /0.04 | 1322 | /0.10 | 1302 | /0.14 | 167 | /0.77 | 170 | /0.82 | 142 | /0.61 | 142 | /0.63
| Timeout | 0 | | 3 | | 12 | | 1029 | | 1026 | | 1052 | | 1045 |
C | Yes | 40 | /0.03 | 48 | /0.08 | 55 | /0.05 | 99 | /0.40 | 100 | /0.38 | 98 | /0.26 | 105 | /0.23
| Maybe | 598 | /0.05 | 587 | /0.10 | 576 | /0.13 | 143 | /0.72 | 146 | /0.77 | 119 | /0.51 | 119 | /0.54
| Timeout | 0 | | 3 | | 7 | | 396 | | 392 | | 421 | | 414 |
Table 1: Experimental Results
Table 1 reflects that the integration of $\textsc{POP}^{\ast}$ in the context
of (weak) dependency pairs, significantly extends the direct approach. Worthy
of note, the extension of Avanzini and Moser (2008) with quasi-precedences
alone gives 5 additional examples. As advertised, $\textsc{POP}^{\ast}$ is
incredibly fast in all settings. Consequently, as evident from the table,
polynomial path orders team well with existing techniques, without affecting
overall performance: notice that due to the additional of
$\textsc{POP}^{\ast}$ the number of timeouts is reduced.
## 7 Conclusion
In this paper we study the runtime complexity of rewrite systems. We combine
two recently developed techniques in the context of complexity analysis: weak
innermost dependency pairs and polynomial path orders. If the conditions of
our main result are met, we can conclude the innermost polynomial runtime
complexity of the studied term rewrite system. And we obtain that the function
defined are _polytime computable_. We have implemented the technique and
experimental evidence clearly indicates the power and in particular the
efficiency of the new method.
## Appendix A Appendix
Below we present the missing proofs of Theorem 5.11 and Theorem 5.14
respectively.
As mentioned in Section 5, we now introduce an _extended predicative
interpretation_ whose purpose is to interpret compound symbols as sequences,
and their arguments via the interpretation $\mathsf{N}_{\pi}$.
###### Definition A.1.
The _extended predicative interpretation_ $\mathsf{N}_{\pi}^{\mathsf{s}}$ from
terms $\mathcal{T}(\mathcal{F},\mathcal{V})$ to sequences
$\operatorname{\mathcal{S}eq}(\mathcal{F}^{\operatorname{\mathsf{n}}}_{\pi}\cup\\{{\mathsf{s}}\\},\mathcal{V})$
is defined as follows: if $t=\mathsf{c}({t}_{1},\ldots,{t}_{n})$ and
$\mathsf{c}\in{\mathcal{C}}_{\text{\tiny{$com$}}}$ then
$\mathsf{N}_{\pi}^{\mathsf{s}}(t)\mathrel{:=}[\mathsf{N}_{\pi}^{\mathsf{s}}(t_{1})\leavevmode\nobreak\
\cdots\leavevmode\nobreak\ \mathsf{N}_{\pi}^{\mathsf{s}}(t_{n})]$, and
otherwise $\mathsf{N}_{\pi}^{\mathsf{s}}(t)\mathrel{:=}[\mathsf{N}_{\pi}(t)]$.
Following (Terese, 2003, Section 6.5), we briefly recall _typed rewriting_.
Let $S$ be a finite set representing the set of _types_ or _sorts_. An _$S$
-sorted set $A$_ is a family of sets $\\{{A_{s}\mid s\in S}\\}$ such that all
sets $A_{s}$ are pairwise disjoint. In the following, we suppose that
$\mathcal{V}$ denotes an $S$-sorted set of variables. An _$S$ -sorted
signature $\mathcal{F}$_ is like a signature, but the _arity_ of
$f\in\mathcal{F}$ is defined by
$\operatorname{\mathsf{ar}}(f)=(s_{1},\dots,s_{n})$ for $s_{1},\dots,s_{n}\in
S$. Additionally, each symbol $f\in\mathcal{F}$ is associated with a sort
$s\in S$, called the _type of $f$_ and denoted by
$\operatorname{\mathsf{st}}(f)$. We adopt the usual notion and write
$f\,\colon\,(s_{1},\dots,s_{n})\to s$ when
$\operatorname{\mathsf{ar}}(f)=(s_{1},\dots,s_{n})$ and
$\operatorname{\mathsf{st}}(f)=s$. The _$S$ -sorted set of terms
$\mathcal{T}(\mathcal{F},\mathcal{V})_{S}$_ consists of the sets
$\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$ for $s\in S$, where
$\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$ is inductively defined by (i)
$\mathcal{V}_{s}\subseteq\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$, and (ii)
$f({t}_{1},\ldots,{t}_{n})\in\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$ for all
function symbols $f\in\mathcal{F}$, $f\,\colon\,(s_{1},\dots,s_{n})\to s$ and
terms $t_{i}\in\mathcal{T}(\mathcal{F},\mathcal{V})_{s_{i}}$ for
$i\in\\{{1,\dots,n}\\}$. We say that a term $t$ is _well-typed_ if
$t\in\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$ for some sort $s$. An
$S$-sorted term rewrite system $\mathcal{R}$ is a TRS such that for ${l\to
r}\in\mathcal{R}$, it holds that
$l,r\in\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$ for some sort $s\in S$. As a
consequence, for $s\in\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$ and
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$$}}}_{\mathcal{R}}}t$,
we have that $t\in\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$.
###### Example A.2.
Let $S=\\{{\mathsf{Bool},\mathsf{List},\mathsf{Nat},\mathsf{Pair}}\\}$. The
$S$-sorted rewrite system $\mathcal{R}_{\mathsf{Lst}}$ is given by the
following rules:
$\displaystyle\mathsf{f}(\mathsf{s}(x))$
$\displaystyle\to\mathsf{cons}(\mathsf{pair}(x,\mathsf{g}(x)),\mathsf{f}(x))$
$\displaystyle\hskip 12.91663pt\mathsf{g}(\mathsf{s}(x))$
$\displaystyle\to\mathsf{g}(x)$ $\displaystyle\mathsf{f}(0)$
$\displaystyle\to\mathsf{nil}$ $\displaystyle\mathsf{g}(0)$
$\displaystyle\to\mathsf{tt}$
Here we assign arities and sorts as follows: for the constructors we set
$0\,\colon\,\mathsf{Nat}$, $\mathsf{s}\,\colon\,\mathsf{Nat}\to\mathsf{Nat}$,
$\mathsf{pair}\,\colon\,(\mathsf{Nat},\mathsf{Bool})\to\mathsf{Pair}$,
$\mathsf{tt}\,\colon\,\mathsf{Bool}$, $\mathsf{nil}\,\colon\,\mathsf{List}$,
$\mathsf{cons}\,\colon\,(\mathsf{Pair},\mathsf{List})\to\mathsf{List}$; for
the defined symbols we set $\mathsf{f}\,\colon\,\mathsf{Nat}\to\mathsf{List}$
and $\mathsf{g}\,\colon\,\mathsf{Nat}\to\mathsf{Bool}$.
A _simple_ signature Marion (2003) is a sorted signature such that each sort
has a finite _rank_ $r$ in the following sense: the sort $s$ has rank $r$ if
for every constructor $c\,\colon\,({s}_{1},\ldots,{s}_{n})\to s$, the rank of
each sort $s_{i}$ is less than the rank of $s$, except for at most one sort
which can be of rank $r$. Simple signatures allow the definition of enumerated
datatypes and inductive datatypes like words and lists but prohibit for
instance the definition of tree structures. Observe that the signature
underlying $\mathcal{R}_{\mathsf{Lst}}$ from Example A.2 is simple. A crucial
insight is that sizes of values formed from a simple signature can be
estimated polynomially in their depth. The easy proof of the following
proposition can be found in (Marion, 2003, Proposition 17).
###### Proposition A.3.
Let $\mathcal{C}$ be a set of constructors from a simple signature
$\mathcal{F}$. There exists a constant $d\in\mathbb{N}$ such that for each
term $t\in\mathcal{T}(\mathcal{C},\mathcal{V})_{S}$ whose rank is $r$,
$\lvert{t}\rvert\leqslant d^{r}\cdot\operatorname{\mathsf{dp}}(t)^{r+1}$.
In order to give a polytime algorithm for the functions computed by a TRS, it
is essential that sizes of reducts do not exceed a polynomial bound with
respect to the size of the start term. Recall that approximations
$\mathrel{\blacktriangleright}_{k}$ tightly control the size growth of terms.
For simple signatures, we can exploit this property for a space-complexity
analysis. Although predicative interpretations remove values, by the above
proposition sizes of those can be estimated based on the Buchholz-norm record
in $\mathsf{N}_{\pi}$. And so we derive the following Lemma, essential for the
proof of Theorem 5.14.
###### Lemma A.4.
Let $\mathcal{F}$ be a simple signature. There exists a (monotone) polynomial
$p$ depending only on $\mathcal{F}$ such that for each well-typed term
$t\in\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$, $\lvert{t}\rvert\leqslant
p(\mathsf{G}_{k}(\mathsf{N}^{\mathsf{s}}(t)))$.
###### Proof.
The Lemma follows as: (i) for all sequences
$s\in\operatorname{\mathcal{S}eq}$,
$\lvert{s}\rvert\leqslant\mathsf{G}_{k}(s)+1$, and (ii) for all terms
$t\in\mathcal{T}(\mathcal{F},\mathcal{V})_{s}$, $\lvert{t}\rvert\leqslant
c\cdot\lvert{\mathsf{N}^{\mathsf{s}}(t)}\rvert^{d}$ for some uniform constants
$0<c,d\in\mathbb{N}$. These properties are simple to verify: property (i)
follows from induction on $s$ where we employ for the inductive step that
$f({s}_{1},\ldots,{s}_{n})\mathrel{\blacktriangleright}_{k}[{s}_{1}\cdots{s}_{n}]$
and
$\mathsf{G}_{k}([{s}_{1}\cdots{s}_{n}])=\sum_{i=1}^{n}\mathsf{G}_{k}(s_{i})+n$.
For property (ii), set $d=r+2$ where $r$ is the maximal rank of a symbol in
$\mathcal{C}$, and set $c=e^{r}$ where $e$ is as given from Proposition A.3.
First one shows by a straight forward induction on $t$ that
$\lvert{t}\rvert\leqslant
c\cdot(\lvert{\mathsf{S}(t)}\rvert\cdot\lVert{t}\rVert^{r+1})$ (employing
Proposition A.3 and $\operatorname{\mathsf{dp}}(t)\leqslant\lVert{t}\rVert$).
As $\lvert{\mathsf{S}(t)}\rvert<\lvert{\mathsf{N}(t)}\rvert$ and
$\lVert{t}\rVert<\lvert{\mathsf{N}(t)}\rvert$, we derive
$\lvert{t}\rvert<c\cdot\lvert{\mathsf{N}(t)}\rvert^{d}$. By induction on the
definition of $\mathsf{N}^{\mathsf{s}}$ we finally obtain property (ii). ∎
Let $\mathcal{R}$ be a (not necessarily $S$-sorted) TRS that is innermost
terminating. In the sequel, we keep $\mathcal{R}$ fixed. In order to exploit
Lemma A.4 for an analysis by means of weak innermost dependency pairs, we
introduce the notion of _type preserving weak innermost dependency pairs_.
###### Definition A.5.
If $l\to r\in\mathcal{R}$ and
$r=C\langle{u}_{1},\ldots,{u}_{n}\rangle_{\mathcal{D}}$ then
$l^{\sharp}\to\mathsf{c}(u_{1}^{\sharp},\ldots,u_{n}^{\sharp})$ is called a
_type preserving weak innermost dependency pair_ of $\mathcal{R}$. Here, the
_compound symbol_ $\mathsf{c}$ is supposed to be fresh. We set
$\mathsf{repr}(\mathsf{c})\mathrel{:=}C$ and say that $\mathsf{c}$
_represents_ the context $C$. The set of all type preserving weak innermost
dependency pairs is denoted by $\mathsf{WIDP}(\mathcal{R})$.
We collect all compound symbols appearing in $\mathsf{TPWIDP}(\mathcal{R})$ in
the set ${\mathcal{C}}_{\text{\tiny{$com$}}}$.
###### Example A.6 (Example A.2 continued).
Reconsider the rewrite system $\mathcal{R}_{\mathsf{Lst}}$ given in Example
A.2. The set $\mathsf{TPWIDP}(\mathcal{R}_{\mathsf{Lst}})$ is given by
$\displaystyle\mathsf{f}^{\sharp}(\mathsf{s}(x))$
$\displaystyle\to\mathsf{c}_{1}(\mathsf{g}^{\sharp}(x),\mathsf{f}^{\sharp}(x))$
$\displaystyle\hskip 12.91663pt\mathsf{g}^{\sharp}(\mathsf{s}(x))$
$\displaystyle\to\mathsf{c}_{3}(\mathsf{g}^{\sharp}(x))$
$\displaystyle\mathsf{f}^{\sharp}(0)$ $\displaystyle\to\mathsf{c}_{2}$
$\displaystyle\mathsf{g}^{\sharp}(0)$ $\displaystyle\to\mathsf{c}_{4}$
The constant $\mathsf{c}_{3}$ represents for instance the empty context, and
the constant $\mathsf{c}_{1}$ represents the context
$\mathsf{repr}(\mathsf{c}_{1})=\mathsf{cons}(\mathsf{pair}(x,\Box),\Box)$.
###### Lemma A.7.
Let $\mathcal{R}$ be an $S$-sorted TRS such that the underlying signature
$\mathcal{F}$ is simple. Then
$\mathsf{TPWIDP}(\mathcal{R})\cup\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))$ is
an $S$-sorted TRS, and the underlying signature
$\mathcal{F}^{\sharp}\cup{\mathcal{C}}_{\text{\tiny{$com$}}}$ a simple
signature.
###### Proof.
To conclude the claim, it suffices to type the marked and compound symbols
appropriately. For each rule
${f^{\sharp}({l}_{1},\ldots,{l}_{n})\to\mathsf{c}(r_{1}^{\sharp},\dots,r_{n}^{\sharp})}\in\mathsf{TPWIDP}(\mathcal{R})$
we proceed as follows: we set
$\operatorname{\mathsf{ar}}(f^{\sharp})\mathrel{:=}\operatorname{\mathsf{ar}}(f)$
and
$\operatorname{\mathsf{st}}(f^{\sharp})\mathrel{:=}\operatorname{\mathsf{st}}(f)$.
Moreover, we set
$\operatorname{\mathsf{ar}}(\mathsf{c})\mathrel{:=}({\operatorname{\mathsf{st}}}({r}_{1}),\ldots,{\operatorname{\mathsf{st}}}({r}_{m}))$
and
$\operatorname{\mathsf{st}}(\mathsf{c})\mathrel{:=}\operatorname{\mathsf{st}}(f)$.
It is easy to see that since $\mathcal{R}$ is $S$-sorted,
$\mathsf{TPWIDP}(\mathcal{R})\cup\mathcal{U}(\mathsf{TPWIDP}(\mathcal{R}))$ is
$S$-sorted too. ∎
Note that the above lemma fails for weak innermost dependency pairs: consider
the rule $\mathsf{f}(x)\to\mathsf{d}(\mathsf{g}(x))$, where $\mathsf{f}$ and
$\mathsf{g}$ are defined symbols and $\mathsf{d}$ is a constructor. Moreover,
suppose $\mathsf{f}\,\colon\,\mathsf{s_{2}}\to\mathsf{s_{1}}$,
$\mathsf{g}\,\colon\,\mathsf{s_{2}}\to\mathsf{s_{3}}$ and
$\mathsf{d}\,\colon\,\mathsf{s_{3}}\to\mathsf{s_{1}}$. Then we cannot type the
corresponding weak innermost dependency pair
$\mathsf{f}^{\sharp}(x)\to\mathsf{g}^{\sharp}(x)$ as above because
(return-)types of $\mathsf{f}^{\sharp}$ and $\mathsf{g}^{\sharp}$ differ.
As for practical all termination techniques, compatibility of weak innermost
dependency pairs with polynomial path orders also yield compatibility of type
preserving weak innermost dependency pairs. Moreover, from the definition we
immediately see that
$\operatorname{dl}(t^{\sharp},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathsf{TPWIDP}(\mathcal{R})/\mathcal{U}}})=\operatorname{dl}(t^{\sharp},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathsf{WIDP}(\mathcal{R})/\mathcal{U}}})$
with $\mathcal{U}=\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))$ and basic term $t$.
And so it is clear that in order to proof Theorem 5.11 and Theorem 5.14,
$\mathsf{WIDP}(\mathcal{R})$ can safely be replaced by
$\mathsf{TPWIDP}(\mathcal{R})$. We continue with the proof of Theorem 5.11.
### A.1 Proof of Theorem 5.11
Let $\mathsf{ComCtx}$ abbreviate the set of contexts
$\mathcal{T}({\mathcal{C}}_{\text{\tiny{$com$}}}\cup\\{{\Box}\\},\mathcal{V})$
build from compound symbols. Set $\mathcal{P}=\mathsf{TPWIDP}(\mathcal{R})$
and $\mathcal{U}=\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))$. In order to
highlight the correspondence between
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}$
and
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{P}/\mathcal{U}}}$,
we extend the notion of _representatives_.
###### Definition A.8.
Let $C\in\mathsf{ComCtx}$. We define $\mathsf{reprs}(C)$ as the least set of
(ground) contexts such that (i) if $C=\Box$ then $\Box\in\mathsf{reprs}(C)$,
and (ii) if $C=\mathsf{c}({C}_{1},\ldots,{C}_{n})$,
$C^{\prime}_{i}\in\mathsf{reprs}(C_{i})$ and $\sigma$ is a substitution from
all variables in $\mathsf{repr}(\mathsf{c})$ to ground normal forms of
$\mathcal{R}$ then
$(\mathsf{repr}(\mathsf{c})\sigma)[C^{\prime}_{1},\dots,C^{\prime}_{n}]\in\mathsf{reprs}(C)$.
###### Example A.9 (Example A.6 continued).
Reconsider the TRS $\mathcal{R}_{\mathsf{Lst}}$ from Example A.2, together
with $\mathsf{TPWIDP}(\mathcal{R}_{\mathsf{Lst}})$ as given in Example A.6.
Consider the step
$\mathsf{f}(\mathsf{s}(0))\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$$}}}_{\mathcal{R}_{\mathsf{Lst}}}}\mathsf{cons}(\mathsf{pair}(0,\mathsf{g}(0)),\mathsf{f}(0))$
and the corresponding dependency pair step
$\mathsf{f}^{\sharp}(\mathsf{s}(0))\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$$}}}_{\mathsf{TPWIDP}(\mathcal{R}_{\mathsf{Lst}})}}\mathsf{c}_{1}(\mathsf{g}^{\sharp}(0),\mathsf{f}^{\sharp}(0))\hbox
to0.0pt{$\;$.\hss}$
Let $C=\mathsf{c}_{1}(\Box,\Box)$, remember that
$\mathsf{repr}(\mathsf{c}_{1})=\mathsf{cons}(\mathsf{pair}(x,\Box),\Box)$,
$\mathsf{reprs}(\Box)=\Box$ and observe that
$C^{\prime}=\mathsf{cons}(\mathsf{pair}(0,\Box),\Box)\in\mathsf{reprs}(C)$ by
taking the substitution $\sigma=\\{{x\mapsto 0}\\}$. And hence we can
reformulate the above two steps as
$\mathsf{f}(\mathsf{s}(0))\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$$}}}_{\mathcal{R}_{\mathsf{Lst}}}}C^{\prime}[\mathsf{g}(0),\mathsf{f}(0)]$
and likewise
$\mathsf{f}^{\sharp}(\mathsf{s}(0))\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$$}}}_{\mathsf{TPWIDP}(\mathcal{R}_{\mathsf{Lst}})}}C[\mathsf{g}^{\sharp}(0),\mathsf{f}^{\sharp}(0)]$.
We manifest the above observation in the following lemma.
###### Lemma A.10.
Let $s\in\mathcal{T}_{\mathsf{b}}$ be a ground and basic term. Suppose
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{R}}}t$.
Let $\mathcal{P}=\mathsf{TPWIDP}(\mathcal{R})$ and let
$\mathcal{U}=\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))$. Then there exists
contexts $C^{\prime}\in\mathsf{ComCtx}$, $C\in\mathsf{reprs}(C^{\prime})$ and
terms ${t}_{1},\ldots,{t}_{n}$ such that $t=C[{t}_{1},\ldots,{t}_{n}]$ and
moreover,
$s^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{P}\cup\mathcal{U}}}C^{\prime}[t_{1}^{\sharp},\dots,t_{n}^{\sharp}]$.
###### Proof.
We proof the lemma by induction on the length of the rewrite sequence
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{n}_{\mathcal{R}}}t$.
The base case $n=0$ is trivial, we set $C=C^{\prime}=\Box$. So suppose
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{n}_{\mathcal{R}}}t\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}u$
and the property holds for $n$. And thus we can find contexts
$C^{\prime}_{t}\in\mathsf{ComCtx}$, $C_{t}\in\mathsf{reprs}(C^{\prime}_{t})$
and terms ${t}_{1},\ldots,{t}_{n}$ such that $t=C_{t}[{t}_{1},\ldots,{t}_{n}]$
and moreover,
$s^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{P}\cup\mathcal{U}}}C^{\prime}_{t}[t_{1}^{\sharp},\dots,t_{n}^{\sharp}]$.
Without loss of generality we can assume
$u=C_{t}[t_{1},\dots,u_{i},\dots,t_{n}]$ with
$t_{i}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}u_{i}$,
as the context $C_{t}$ is solely build from constructors and normal forms of
$\mathcal{R}$.
First, suppose
$t_{i}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\varepsilon}_{\mathcal{R}}}u_{i}$,
and hence $t_{i}={l\sigma}$ for ${l\to r}\in\mathcal{R}$ and substitution
$\sigma\,\colon\,\mathcal{V}\to\operatorname{\mathsf{NF}}(\mathcal{R})\cap\mathcal{T}(\mathcal{F})$.
Moreover
${l^{\sharp}\to\mathsf{c}(r_{1}^{\sharp},\dots,r_{m}^{\sharp})}\in\mathcal{P}$
such that
$u_{i}=(\mathsf{repr}(\mathsf{c})\sigma)[r_{1}\sigma,\dots,r_{m}\sigma]$. We
set $C^{\prime}$ as the context obtained from replacing the $i$-th hole of
$C^{\prime}_{t}$ by $\mathsf{c}(\Box,\dots,\Box)$, likewise we set $C$ as the
context obtained from replacing the $i$-th hole of $C_{t}$ by
$\mathsf{repr}(\mathsf{c})\sigma$. Note that $C\in\mathsf{reprs}(C^{\prime})$.
We conclude
$s^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{P}\cup\mathcal{U}}}C^{\prime}[t_{1}^{\sharp},\dots,{r_{1}^{\sharp}\sigma,\dots,r_{m}^{\sharp}\sigma},\dots,t_{n}^{\sharp}]$
and $u=C[t_{1},\dots,r_{1}\sigma,\dots,r_{m}\sigma,\dots,t_{n}]$ which
establishes the lemma for this case.
Now suppose
$t_{i}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}u_{i}$
is a step below the root. Thus we have also
$t_{i}^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}u_{i}^{\sharp}$.
As shown in (Hirokawa and Moser, 2008b, Lemma 16), the latter can be
strengthened to
$t_{i}^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\mathcal{P}\cup\mathcal{U}$}}}_{\mathcal{U}}}u_{i}^{\sharp}$.
We conclude
$s^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{P}\cup\mathcal{U}}}C^{\prime}_{t}[t_{1}^{\sharp},\dots,u_{i}^{\sharp},\dots,t_{n}^{\sharp}]$,
and the lemma follows by setting $C^{\prime}=C^{\prime}_{t}$ and $C=C_{t}$. ∎
Suppose $\mathsf{WIDP}(\mathcal{R})$ contains non-nullary compound symbols. In
order to establish an embedding in the sense of Lemma 5.6 for that case, by
the above lemma we see that it suffices to consider only terms of shape
$s=C[{s_{1}^{\sharp},\dots,s_{n}^{\sharp}}]$ with $C\in\mathsf{ComCtx}$. With
this insight, we adjust Lemma 5.6 as below. Observe that due to the definition
of $\mathsf{N}_{\pi}^{\mathsf{s}}$, we cannot simply apply Lemma 5.6 together
with closure under context of $\mathrel{\blacktriangleright}_{k}$ here.
###### Lemma A.11.
Let $s=C[s_{1}^{\sharp},\dots,s_{n}^{\sharp}]$ for $C\in\mathsf{ComCtx}$ and
${s}_{1},\ldots,{s}_{n}\in\mathcal{T}(\mathcal{F},\mathcal{V})$. Let
$\mathcal{P}=\mathsf{TPWIDP}(\mathcal{R})$ and
$\mathcal{U}=\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))$. There exists a uniform
constant $k\in\mathbb{N}$ depending only on $\mathcal{R}$ such that if
$\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ holds then
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{P}}}t$
implies
$\mathsf{N}_{\pi}^{\mathsf{s}}(s)\mathrel{\blacktriangleright}_{k}\mathsf{N}_{\pi}^{\mathsf{s}}(t)$.
Moreover, if
${\mathcal{U}}\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
holds then
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{U}}}t$
implies
$\mathsf{N}_{\pi}^{\mathsf{s}}(s)\mathrel{\not{\gtrsim}}_{k}\mathsf{N}_{\pi}^{\mathsf{s}}(t)$.
###### Proof.
We proof the lemma for $k\mathrel{:=}\max\\{{3\cdot\lVert{r}\rVert\mid{l\to
r}\in\mathcal{P}\cup\mathcal{U}}\\}$. Suppose
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{P}}}t$
or
$s\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{U}}}t$
respectively, and thus $t=C[s_{1}^{\sharp},\dots,t_{i},\dots,s_{n}^{\sharp}]$
for some term $t_{i}$. There exists a context $C^{\prime}$ (over sequences)
such that
$\mathsf{N}_{\pi}^{\mathsf{s}}(s)=C^{\prime}[\mathsf{N}_{\pi}^{\mathsf{s}}(s_{i}^{\sharp})]$
and
$\mathsf{N}_{\pi}^{\mathsf{s}}(t)=C^{\prime}[\mathsf{N}_{\pi}^{\mathsf{s}}(t_{i}^{\sharp})]$.
First assume
$s_{i}^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{P}}}t_{i}$,
and thus
$\mathsf{N}_{\pi}^{\mathsf{s}}(s_{i}^{\sharp})=[\mathsf{N}_{\pi}(l^{\sharp}\sigma)]$
and
$\mathsf{N}_{\pi}^{\mathsf{s}}(t_{i})=[[\mathsf{N}_{\pi}(r_{1}^{\sharp}\sigma)],\dots,[\mathsf{N}_{\pi}(r_{m}^{\sharp}\sigma)]]$
for ${l\to\mathsf{c}(r_{1}^{\sharp},\dots,r_{m}^{\sharp})}\in\mathcal{P}$. To
verify
$\mathsf{N}_{\pi}^{\mathsf{s}}(s)\mathrel{\blacktriangleright}_{k}\mathsf{N}_{\pi}^{\mathsf{s}}(t)$,
by Definition 2.2(ii) and Definition 2.2(iv), it suffices to verify
$\mathsf{N}_{\pi}(l^{\sharp}\sigma)\mathrel{\blacktriangleright}_{k-1}\mathsf{N}_{\pi}(r_{j}^{\sharp}\sigma)$
for all $j\in\\{{1,\dots,m}\\}$. The latter is an easy consequence of Lemma
5.6, where we employ that (i)
$l^{\sharp}\mathrel{{>}^{\pi}_{\mathsf{pop*}}}r_{j}^{\sharp}$ follows from the
assumption $\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$, and
(ii) $\lVert{\pi(r)}\rVert>\lVert{\pi(r_{j})}\rVert$. Both properties are
straight forward to verify since $\pi$ is safe. For
$s_{i}^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{U}}}t$
we have
$\mathsf{N}_{\pi}^{\mathsf{s}}(s_{i}^{\sharp})=[\mathsf{N}_{\pi}(s_{i}^{\sharp})]$
and
$\mathsf{N}_{\pi}^{\mathsf{s}}(t_{i}^{\sharp})=[\mathsf{N}_{\pi}(t_{i}^{\sharp})]$
for ${l\to r}\in\mathcal{U}$. From Lemma 5.6 we obtain
$\mathsf{N}_{\pi}(s_{i}^{\sharp})\mathrel{\not{\gtrsim}}_{k}\mathsf{N}_{\pi}(t_{i}^{\sharp})$
which establishes the lemma. ∎
The proof of Theorem 5.11 is now easily obtained by incorporating the above
lemma into Theorem 5.9.
###### Theorem.
Let $\mathcal{R}$ be a constructor TRS, and let $\mathcal{P}$ denote the set
of weak innermost dependency pairs. Assume $\mathcal{P}$ is non-duplicating,
and suppose ${\mathcal{U}(\mathcal{P})}\subseteq{>_{\mathcal{A}}}$ for some
SLI $\mathcal{A}$. Let $\pi$ be a safe argument filtering. If
$\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ and
$\mathcal{U}(\mathcal{P})\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
then
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}$
is polynomially bounded.
###### Proof.
According to Proposition 3.4 we need to find a polynomial $p$ such that
$\operatorname{dl}(t^{\sharp},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathsf{WIDP}(\mathcal{R})/\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))}})\leqslant
p(\lvert{t^{\sharp}}\rvert)\hbox to0.0pt{$\;$.\hss}$
We set $\mathcal{P}=\mathsf{TPWIDP}(\mathcal{R})$ and likewise
$\mathcal{U}=\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))$. Clearly, it suffices to
show
$\operatorname{dl}(t^{\sharp},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathsf{TPWIDP}(\mathcal{R})/\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))}})\leqslant
p(\lvert{t^{\sharp}}\rvert)$ for that. Consider a sequence
$t^{\sharp}=t_{0}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{P}/\mathcal{U}}}t_{1}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{P}/\mathcal{U}}}\dots\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{P}/\mathcal{U}}}t_{\ell}\hbox
to0.0pt{$\;$,\hss}$
and pick a relative step
$t_{i}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{P}/\mathcal{U}}}t_{i+1}$.
Define
$\mathcal{U}^{\prime}=\mathcal{U}\cup\mathcal{V}(\mathcal{P}\cup\mathcal{U})$
and $\phi(t)=\phi_{\mathcal{P}\cup\mathcal{U}}(t)$. Clearly Lemma 5.7 can be
extended to account for steps of $\mathcal{P}$ below the root, and thus
$\phi(t_{i})\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{P}/\mathcal{U}^{\prime}}}\phi(t_{i+1})$
follows. Hence for some terms $u$ and $v$,
$\phi(t_{i})\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{U}^{\prime}}}u\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}_{\mathcal{P}}}v\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{U}^{\prime}}}\phi(t_{i+1})$.
As shown in Lemma A.10, all involved terms in the above sequence have the
shape $C[s_{1}^{\sharp},\dots,s_{n}^{\sharp}]$, $C\in\mathsf{ComCtx}$. As
$\mathsf{WIDP}(\mathcal{R})\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$,
and since $\pi$ is safe, it is easy to infer that
$\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$ holds (we just set
every compound symbol from $\mathcal{P}$ minimal in the precedence). And hence
Lemma A.11 translates the above relative step to
$\mathsf{N}_{\pi}^{\mathsf{s}}(\phi(s))\mathrel{\blacktriangleright}^{+}_{k}\mathsf{N}_{\pi}^{\mathsf{s}}(\phi(t))$
for some uniform constant $k$. As a consequence,
$\operatorname{dl}(t,\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathsf{WIDP}(\mathcal{R})/\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))}})\leqslant\mathsf{G}_{k}(\mathsf{N}_{\pi}^{\mathsf{s}}(\phi(t)))$
for all terms $t$. Fix some reducible and basic term
$t\in\mathcal{T}_{\mathsf{b}}$. Observe
$\mathsf{N}_{\pi}^{\mathsf{s}}(\phi(t^{\sharp}))=[\mathsf{N}_{\pi}(t^{\sharp})]$
and so from Lemma 5.1 we see that
$\mathsf{G}_{k}(\mathsf{N}_{\pi}^{\mathsf{s}}(\phi(t^{\sharp})))$ is bounded
polynomially in the size of $t$. The polynomial depends only on $k$. We
conclude the theorem. ∎
### A.2 Proof of Theorem 5.14
We now proceed with the proof Theorem 5.14, which is essentially an extension
to Theorem 5.11.
We first precisely state what it means that a TRS _computes_ some function.
For this, let
$\ulcorner{\cdot}\urcorner\,\colon\,\Sigma^{\ast}\to\mathcal{T}(\mathcal{C})$
denote an _encoding function_ that represents words over the alphabet $\Sigma$
as ground values. We call an encoding $\ulcorner{\cdot}\urcorner$ _reasonable_
if it is bijective and there exists a constant $c$ such that
$\lvert{u}\rvert\leqslant\lvert{\ulcorner{u}\urcorner}\rvert\leqslant
c\cdot\lvert{u}\rvert$ for every $u\in\Sigma^{*}$. Let
$\ulcorner{\cdot}\urcorner$ denote a reasonable encoding function, and let
$\mathcal{R}$ be a completely defined, orthogonal and terminating TRS. We say
that an $n$-ary function $f\colon(\Sigma^{\ast})^{n}\to\Sigma^{*}$ is
_computable_ by $\mathcal{R}$ if there exists a defined function symbol
$\mathsf{f}$ such that for all $w_{1},\dots,w_{n},v\in\Sigma^{\ast}$
$\mathsf{f}(\ulcorner{w_{1}}\urcorner,\dots,\ulcorner{w_{n}}\urcorner)\to^{!}\ulcorner{v}\urcorner\Longleftrightarrow
f(w_{1},\dots,w_{n})=v$. On the other hand the TRS $\mathcal{R}$ _computes_
$f$, if the function $f\colon(\Sigma^{\ast})^{n}\to\Sigma^{*}$ is defined by
the above equation.
Below we abbreviate $\mathsf{Q}_{\pi}$ as $\mathsf{Q}$ for predicative
interpretation
$\mathsf{Q}\in\\{{\mathsf{S},\mathsf{N},\mathsf{N}^{\mathsf{s}}}\\}$ and the
particular argument filtering $\pi$ that induces the identity function on
terms. Consider the following lemma.
###### Lemma A.12.
Let $\mathcal{R}$ be an $S$-sorted and completely defined constructor TRS such
that the underlying signature is simple. If
${\mathsf{TPWIDP}(\mathcal{R})\cup\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))}\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
then there exists a polynomial $p$ such that for all ground and well-typed
basic terms $t\in\mathcal{T}_{\mathsf{b}}$,
$t^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathsf{TPWIDP}(\mathcal{R})\cup\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))}}s$
implies $\lvert{s}\rvert\leqslant p(\lvert{t}\rvert)$.
###### Proof.
Let
$\mathcal{S}=\mathsf{TPWIDP}(\mathcal{R})\cup\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))$.
Suppose
$t^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{S}}}s$,
or equivalently
$t^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{v}}$}}}^{\ast}_{\mathcal{S}}}s$
since $\mathcal{R}$ is completely defined. By Lemma A.11 we derive
$\mathsf{N}^{\mathsf{s}}(t^{\sharp})\mathrel{\not{\gtrsim}}_{k}^{*}\mathsf{N}^{\mathsf{s}}(s)$
for some uniform $k\in\mathbb{N}$. And thus
$\mathsf{G}_{k}(\mathsf{N}^{\mathsf{s}}(s))\leqslant\mathsf{G}_{k}(\mathsf{N}^{\mathsf{s}}(t^{\sharp}))$.
As
$\mathsf{G}_{k}(\mathsf{N}^{\mathsf{s}}(t^{\sharp}))=\mathsf{G}_{k}([\mathsf{N}(t^{\sharp})])$
is bounded polynomially in the size of $t$ according to Lemma 5.2, we see that
there exists a polynomial $p$ such that
$\mathsf{G}_{k}(\mathsf{N}^{\mathsf{s}}(s))\leqslant\mathsf{G}_{k}(\mathsf{N}^{\mathsf{s}}(t^{\sharp}))\leqslant
p(\lvert{t}\rvert)$. Since $\mathcal{R}$ is and $S$-sorted TRS over a simple
signature, the same holds for $\mathcal{S}$ due to Lemma A.7. And thus since
$t^{\sharp}$ is well-typed and
$t^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{S}}}s$
holds, also $s$ is well-typed. Let $q$ be the polynomial as given from Lemma
A.4 with $\lvert{s}\rvert\leqslant
q(\mathsf{G}_{k}(\mathsf{N}^{\mathsf{s}}(s)))$. Summing up, we derive
$\lvert{s}\rvert\leqslant
q(\mathsf{G}_{k}(\mathsf{N}^{\mathsf{s}}(s)))\leqslant q(p(\lvert{t}\rvert))$
as desired. ∎
The above lemma has established that sizes of reducts with respect to the
relation
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathsf{TPWIDP}(\mathcal{R})\cup\mathcal{U}(\mathsf{WIDP}(\mathcal{R}))}}$
are bounded polynomially in the size of the start term, provided we can orient
dependency pairs and usable rules. It remains to verify that this is indeed
sufficient to appropriately estimate sizes of reducts with respect to
$\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}$.
The fact is established in the final Theorem.
###### Theorem.
Let $\mathcal{R}$ be an orthogonal $S$-sorted and completely defined
constructor TRS such that the underlying signature is simple. Let
$\mathcal{P}$ denote the set of weak innermost dependency pairs. Assume
$\mathcal{P}$ is non-duplicating, and suppose
${\mathcal{U}(\mathcal{P})}\subseteq{>_{\mathcal{A}}}$ for some SLI
$\mathcal{A}$. If $\mathcal{P}\subseteq{\mathrel{{>}^{\pi}_{\mathsf{pop*}}}}$
and
$\mathcal{U}(\mathcal{P})\subseteq{\mathrel{\text{\raisebox{0.0pt}{${\not{>}}^{\pi}_{\text{\raisebox{2.0pt}{$\mathsf{pop*}$}}}$}}}}$
then the functions computed by $\mathcal{R}$ are computable in polynomial-
time.
###### Proof.
We single out one of the defined symbols $\mathsf{f}\in\mathcal{D}$ and
consider the corresponding function
$f\colon(\Sigma^{\ast})^{n}\to\Sigma^{\ast}$ computed by $\mathcal{R}$. Under
the assumptions, $\mathcal{R}$ is terminating, but moreover
$\operatorname{rc}^{\text{\scriptsize$\operatorname{\mathsf{i}}$}}_{\mathcal{R}}$
is polynomially bounded according to Theorem 5.11. Additionally, from
orthogonality (and hence confluence) of $\mathcal{R}$, normal forms are unique
and so the function $f$ is well-defined. Suppose
$\mathsf{f}(\ulcorner{w_{1}}\urcorner,\dots,\ulcorner{w_{n}}\urcorner)\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$$}}}^{!}_{\mathcal{R}}}\ulcorner{v}\urcorner$
for words $w_{1},\dots,w_{n},v$. In particular, from confluence we see that
$\mathsf{f}(\ulcorner{w_{1}}\urcorner,\dots,\ulcorner{w_{n}}\urcorner)\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}t_{1}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}\cdots\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}t_{\ell}=\ulcorner{v}\urcorner\hbox
to0.0pt{$\;$.\hss}$
It is folklore that there exists a polytime algorithm performing one rewrite
step. Hence to conclude the existence of a polytime algorithm for $f$ it
suffices to bound the size of terms $t_{i}$ for $1\leqslant i\leqslant\ell$
polynomially in $\sum_{i}\lvert{w_{i}}\rvert$. And as we suppose that the
encoding $\ulcorner{\cdot}\urcorner$ is reasonable, it thus suffice to bound
the sizes of $t_{i}$ for $i\in\\{{1,\dots,\ell}\\}$ polynomially in the size
of
$t_{0}=\mathsf{f}(\ulcorner{w_{1}}\urcorner,\dots,\ulcorner{w_{n}}\urcorner)$.
Consider a term $t_{i}$. Without loss of generality, we can assume $t_{i}$ is
ground. According to Lemma A.10 there exists contexts
$C^{\prime}_{i}\in\mathsf{ComCtx}$, $C_{i}\in\mathsf{reprs}(C^{\prime}_{i})$
and terms ${u}_{1},\ldots,{u}_{n}$ such that
$t_{i}=C_{i}[{u}_{1},\ldots,{u}_{n}]$ and moreover,
$t_{0}^{\sharp}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{P}\cup\mathcal{U}}}C^{\prime}_{i}[u_{1}^{\sharp},\dots,u_{n}^{\sharp}]$
for all $i\in\\{{1,\dots,\ell}\\}$. From the assumption
$\mathsf{WIDP}(\mathcal{R})\subseteq{\mathrel{{>}_{\mathsf{pop*}}}}$ we see
$\mathsf{TPWIDP}(\mathcal{R})\subseteq{\mathrel{{>}_{\mathsf{pop*}}}}$. Thus
by Lemma A.12 there exists a polynomial $p$ such that
$\lvert{C^{\prime}_{i}[u_{1}^{\sharp},\dots,u_{n}^{\sharp}]}\rvert\leqslant
p(\lvert{t_{0}}\rvert)$. And so, clearly
$\sum_{j=0}^{n}\lvert{u_{j}}\rvert\leqslant p(\lvert{t_{0}}\rvert)$. It
remains to bound the sizes of contexts $C_{i}$ polynomially in
$\lvert{t_{0}}\rvert$.
Recall Definition A.8, and recall that
$C_{i}\in\mathsf{reprs}(C^{\prime}_{i})$. Thus $C_{i}$ is a context build from
constructors and variables, where the latter are replaced by normal forms of
$\mathcal{R}$. Since $\mathcal{R}$ is completely defined,
$\operatorname{\mathsf{NF}}(\mathcal{R})$ coincides with values. We conclude
that $C_{i}\in\mathcal{T}(\mathcal{C}\cup\\{{\Box_{s}\mid s\in S}\\})$. Here
$\Box_{s}$ denotes the hole of sort $s$. Moreover since $\mathcal{R}$ is
$S$-sorted, and
$t_{0}\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}^{\ast}_{\mathcal{R}}}C_{i}[{u}_{1},\ldots,{u}_{n}]$,
we see that $C_{i}$ is well-typed. We define
$\triangle_{\mathcal{R}}=\max\\{{\operatorname{\mathsf{dp}}(r)\mid{l\to
r}\in\mathcal{R}}\\}$. By a straight forward induction it follows that
$\operatorname{\mathsf{dp}}(t_{i})\leqslant\operatorname{\mathsf{dp}}(t_{0})+\triangle_{\mathcal{R}}\cdot
i\leqslant\lvert{t_{0}}\rvert+\triangle_{\mathcal{R}}\cdot\operatorname{dl}(t_{0},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}})$.
As a consequence,
$\operatorname{\mathsf{dp}}({C_{i}})\leqslant\lvert{t_{0}}\rvert+\triangle_{\mathcal{R}}\cdot\operatorname{dl}(t_{0},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}})$,
and thus by Proposition A.3 there exists constants $c,d\in\mathbb{N}$ such
that $\lvert{C_{i}}\rvert\leqslant
c\cdot\operatorname{\mathsf{dp}}(C_{i})^{d}\leqslant
c\cdot(\lvert{t_{0}}\rvert+\triangle_{\mathcal{R}}\cdot\operatorname{dl}(t_{0},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}}))^{d}$.
As we have that
$\operatorname{dl}(t_{0},\mathrel{\xrightarrow{\raisebox{-2.0pt}[0.0pt][0.0pt]{\text{\scriptsize$\operatorname{\mathsf{i}}$}}}_{\mathcal{R}}})$
is polynomially bounded in the size of $t_{0}$, it follows that
$\lvert{C_{i}}\rvert\leqslant q(\lvert{t_{0}}\rvert)$ for some polynomial $q$.
Summing up, we conclude that for all $i\in\\{{1,\dots,\ell}\\}$,
$\lvert{t_{i}}\rvert\leqslant p(\lvert{t_{0}}\rvert)+q(\lvert{t_{0}}\rvert)$
for the polynomials $p$ and $q$ from above. This concludes the theorem. ∎
## References
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|
arxiv-papers
| 2009-04-06T18:10:53 |
2024-09-04T02:49:01.745525
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Martin Avanzini and Georg Moser",
"submitter": "Martin Avanzini",
"url": "https://arxiv.org/abs/0904.0981"
}
|
0904.1040
|
# Locating critical point of QCD phase transition basing on finite-size
scaling
Chen Lizhu Institute of Particle Physics, Hua-Zhong Normal University, Wuhan
430079, China X.S. Chen Institute of Theoretial Physics, Chinese Academy of
Sciences, Beijing 100190, China Wu Yuanfang Institute of Particle Physics,
Hua-Zhong Normal University, Wuhan 430079, China Key Laboratory of Quark $\&$
Lepton Physics (Huazhong Normal University), Ministry of Education, China
###### Abstract
It is argued that in relativistic heavy ion collisions, due to limited size of
the formed matter, the reliable criterion of critical point is finite-size
scaling, rather than non-monotonous behavior of observable. How to locate
critical point by finite-size scaling is proposed. The data of $p_{\rm t}$
correlation from RHIC/STAR are analyzed. Critical points are likely observed
around $\sqrt{s}=62$ and $200$ GeV. They could be, respectively, the
transition of deconfinement and chiral symmetry restoration predicted by
lattice-QCD. Further confirmation with other observable and energies is
suggested.
###### pacs:
12.38.Mh, 25.75.Nq, 25.75.Gz
Lattice-QCD simulations have shown that the transition of deconfinement in
quantum chromodynamics (QCD) at vanishing baryon chemical potential $\mu_{\rm
B}$ is crossover lattice-1 . There has been much speculation that the
crossover becomes a true first-order phase transition for larger values of
$\mu_{\rm B}$. This suggests that the QCD phase diagram can exhibit a critical
endpoint where the line of first order transition matches that of second order
or analytical crossover 1st .
Chiral symmetry restoration is another QCD originated phase transition. It has
been shown that the transition for $\mu_{\rm B}=0$ is crossover c-crossover .
So there could also be a chiral critical endpoint in phase diagram. But it is
unclear if the critical temperature of chiral symmetry restoration is above
karsch-h , or equal to karsch-s , or below quarkyonic that of the
deconfinement.
Locating the critical endpoints of QCD phase transitions by lattice
calculation is still a formidable challenge. But if the critical endpoint is
in the region accessible to current relativistic heavy ion collisions, it
should be discovered experimentally.
Most of the current signatures for finding the critical point are focused on
the anomalous, or non-monotonous, behavior of the observable at various
incident energies ebye-t . The argument is that in infinite system, the
correlation length $\xi$ diverges when approaching the critical point. The
contribution of this singularity to the observable is supposed to be
proportional to $\xi^{2}$. However, the data from RHIC and SPS in more than a
decade accumulation show no sign of anomalous behavior as a function of
$\sqrt{s}$ ebye-e .
In relativistic heavy ion collisions, two nuclei move with relativistic
velocity and collide as two contracted pancakes. More central collision makes
overlapped area larger. It is just because the large number of strongly
interacting nucleons in more central nuclear collisions make the transition
between hadron and quark-gluon plasma possible. The centrality (or the system
size) dependence of the observable is noticeable starprc .
Due to the finite size of system, no divergence can be practically observed at
critical point. The physical quantities, which are divergent in infinite
system, become finite and have a maximum, i.e., so called non-monotonous
behavior. However, the position of the maximum changes with system size and
deviates from the true critical point.
The appearance of non-monotonous behavior is not always associated with
critical point. Taking one-dimensional Ising model as an example, there is no
critical point in this model, but its specific heat in a finite system has
non-monotonous behavior.
Moreover, the absence of non-monotonous behavior does not mean no critical
point. The physical quantities like order parameter, which are finite in
infinite system, have a monotonous behavior near critical point in a finite
system chen1996 . Therefore, non-monotonous behavior is not a reliable
criterion for the critical point of finite system.
An effective identification of critical point of finite system is the finite-
size scaling, which was proposed from phenomenological fss-1 and
renormalization-group fss-RG theories, and was approved by the Monte Carlo
results of finite systems in different universal classes fss-2 .
In this letter, we first propose how to locate critical point by finite-size
scaling. Then the data of $p_{\rm t}$ correlation at 6 centralities and 4
incident energies from RHIC/STAR are analyzed. The behavior of critical point
is likely observed around $\sqrt{s}=62$ and $200$ GeV. Finally, we suggest how
to confirm the findings and precisely locate the critical point in coming
experimental study.
The main points of finite-size scaling can be described as the following. An
observable $Q$ of finite system is a function of temperature $T$ and system
size $L$. When $L$ is much larger than the microscopic length scale and $T$ is
in the vicinity of critical point $T_{c}$, the observable $Q(T,L)$ can be
written in a finite-size scaling form fss-1 ; fss-RG ; fss-2 ,
$Q(T,L)=L^{\lambda/\nu}F_{Q}(tL^{1/\nu}).$ (1)
$t=(T-T_{c})/T_{c}$ is the reduced temperature and $\lambda$ is the critical
exponent of the observable. $\nu$ is the critical exponent of the correlation
length $\xi=\xi_{0}t^{-\nu}$.
Finite-size scaling not only characterizes the scaling behavior of
thermodynamic quantities of finite system near critical point, but also
provides criterion for locating the critical point. At critical point
$T=T_{c}$, the finite-size scaling function $F_{Q}$ in Eq. (1) becomes
$F_{Q}(0)=Q(T_{c},L)L^{-\lambda/\nu},$ (2)
which is constant and independent of system size $L$. In the plot of
$Q(T,L)L^{-\lambda/\nu}$ vs $T$, the critical point [$T_{c}$, $F_{Q}(0)$] is a
fixed point, where all curves of different system sizes converges to.
Reversely, the appearance of fixed point indicates the existence of a critical
point.
If the critical exponent $\lambda=0$, like Binder cumulant ratio binder1981 ,
the fixed point can be obtained directly from the temperature dependence of
this observable at different system sizes. This is why Binder cumulant ratio
has been used very widely in determining critical point of finite-size system.
If the critical exponent $\lambda\neq 0$ and is unknown, the fixed point can
be found by investigating the temperature dependence of $Q(T,L)L^{-a}$ at
different system sizes. When a fixed point is observed at a certain parameter
$a_{0}$, it indicates the existence of a critical point and the parameter
$a_{0}$ is related to the ratio of critical exponents, i.e.,
$\lambda/\nu=a_{0}$.
The critical point can also be found directly from the system size dependence
of the observable. Taking logarithm in the both sides of Eq. (1), it becomes
$\ln Q(T,L)=\lambda/\nu\ln L+\ln F_{Q}(tL^{1/\nu}).$ (3)
At critical point $t=0$, the second term of Eq. (3) becomes a constant and
$\ln Q(T_{c},L)$ becomes a straight line with respect to $\ln L$. If system is
away from the critical point, the second term of Eq. (3) is no longer a
constant. It gives an additional size dependent contribution to the observable
and makes $\ln Q(T,L)$ deviate from the straight line with respect to $\ln L$.
It is found recently that the finite-size scaling holds not only for
thermodynamic quantities like order-parameter, susceptibility, and so on, but
also for various cluster sizes liangsheng and their fluctuations lizhu-ising
. Therefore, the finite-size scaling of various critical related observable
could be used to identify critical point and its critical exponents.
In relativistic heavy ion collision, correlation and fluctuation of final
state particles is regarded as critical related observable lattice-corr .
Although much attention have been drawn in measuring them, but influence of
system size has been neglected. The available data for system size study is
very few. The $p_{\rm t}$ correlation at Au+ Au collisions from RHIC/STAR
starprc is the only data which can be used for the analysis, where the
centrality dependence of $p_{\rm t}$ correlation at 4 incident energies are
well presented starprc . But the errors of the data at $\sqrt{s}=20$ GeV are
much larger than that at other collision energies. The $p_{\rm t}$ correlation
is defined as
$\displaystyle P(\sqrt{s},L)=\frac{1}{N_{\rm e}}\sum\limits_{k=1}^{N_{\rm
e}}\frac{\sum\limits_{i=1}^{N_{k}}\sum\limits_{j=1,i\not=j}^{N_{k}}(p_{{\rm
t},i}-\langle p_{\rm t}\rangle)(p_{{\rm t},j}-\langle p_{\rm
t}\rangle)}{N_{k}(N_{k}-1)}.$ (4)
$N_{\rm e}$ is the number of event, $p_{{\rm t},i}$ is the transverse-momentum
of the $i$th particle in each event, and $N_{k}$ is the number of particles in
the $k$th event. $\langle\ldots\rangle$ is the average over event sample.
Collision energy is the controllable condition. Here we let it play the role
of temperature in the analysis of finite-size scaling. The size of the formed
matter is mainly limited by the size of overlapping transverse region, which
is proportional to the number of participant nucleons and is quantified as
centrality. So the initial mean size of the formed matter can be approximately
estimated by the square root of participants, $\sqrt{N_{\rm part}}$. We choose
dimensionless (or relative) size,
$\displaystyle L=\sqrt{N_{\rm part}}/\sqrt{2N_{\rm A}},$ (5)
as scaled mean size of initial system, where $N_{\rm A}$ is the number of
nucleons of incident nucleus. The system size at transition should be a
monotonically increasing function of $L$. The position of critical point is
insensitive to the concrete form of this function, but only the critical
exponents changes with it. As the first step, the system size at transition is
assumed to be proportional to $L$.
In the case of a few critical points, the finite-size scaling of $p_{\rm t}$
correlation in the vicinity of each critical collision energy
$\sqrt{s_{c,i}}(i=1,2,...)$ can be written as
$\displaystyle
P(\sqrt{s},L)=L^{\lambda_{i}/\nu_{i}}F_{P,i}[e_{i}L^{1/\nu_{i}}].$ (6)
$e_{i}=(\sqrt{s}-\sqrt{s_{c,i}})/\sqrt{s_{c,i}}$ is the reduced collision
energy at $i$th critical point, which is unknown in priori. $\lambda_{i}$ is
the $i$th critical exponent of $p_{\rm t}$ correlation. In the following, we
demonstrate how to locate the critical point by the data of $p_{\rm t}$
correlation from RHIC/STAR.
Figure 1: (a) The energy dependence of $p_{\rm t}$ correlations at different
sizes $L$ (or centralities). Data come from RHIC/STAR starprc . (b), (c) and
(d) are $p_{\rm t}$ correlation multiplied by the factor, $L^{-a}$, with
$-a=1.0$, 2.09 and 4, respectively
.
Firstly, we change the centrality dependence of $p_{\rm t}$ correlation at
different collision energies in Ref. starprc to the collision energy
dependence at different sizes (or centralities). The results are shown in Fig.
1(a). Since in the most peripheral collisions, the size of the formed matter
is too small to be inside the asymptotic region of finite-size scaling, we
choose six centralities at mid-central and central collisions to do the
analysis. The sizes corresponding to the 6 centralities are indicated in the
legend of Fig. 1(a). It is clear that at a given collision energy, the
correlation strength increases with the decrease of system size. The influence
of finite size is obvious.
If critical collision energy of QCD phase transition is in the range of
incident energy at RHIC, the behavior of fixed point should be observable. So
we multiply $P(\sqrt{s},L)$ by a size factor $L^{-a}$ with different $a$ to
see how it changes with the system size $L$. Varying $-a$ from small to large,
it is interesting to see that at collision energy $\sqrt{s}=62$ GeV, all
points of different sizes move firstly toward each other, then well converge
at $-a_{0,1}=2.09$, and finally move again apart from each other. The
corresponding steps and typical $a$ values are presented in Fig. 1(b), (c),
and (d) respectively, where the errors in each sub-figures come from the
measure of $P(\sqrt{s},L)$ only, and the errors of $N_{\rm part}$ are not
included.
At $\sqrt{s}=200$ GeV, the points of different sizes show the same behavior
and best converge at $-a_{0,2}=2.08$. While in the whole process, the points
of different sizes at energies $\sqrt{s}=20$ (or 130) GeV never move close to
each other as those at $\sqrt{s}=62$ (or 200) GeV do. So there are likely two
fixed points around $\sqrt{s}=62$ and 200 GeV.
In order to confirm the position of fixed points, we study the $\ln L$
dependence of $\ln P(\sqrt{s},L)$ for four incident energies, respectively. A
parabola fit, $c_{2}(\ln L)^{2}+c_{1}\ln L+c_{0}$, is used at each collision
energy. The better straight-line behavior results in smaller $|c_{2}|$ and
larger ratio of $|c_{1}/c_{2}|$. The fit parameters, $c_{2}$ and $c_{1}$, for
4 collision energies are listed in Tab. 1. It shows that the better straight-
line behavior happen to be at $\sqrt{s}=62$ and 200 GeV, which are the same
collision energies of fixed points found above. The data at these two energies
can be well fitted, respectively, by the straight lines with slopes $a_{0,1}$
and $a_{0,2}$ obtained above by the fixed points. The results are shown in
Fig. 2(a). While, the data at $\sqrt{s}=20$ and 130 GeV are better fitted by
parabola as shown in Fig. 2(b).
Table 1: Parameters of parabola fits. $\sqrt{s}$(GeV) | | 20 | | 62 | | 130 | | 200
---|---|---|---|---|---|---|---|---
$|c_{2}|$ | | 1.86$\pm$ 0.93 | | 0.6 $\pm$ 0.09 | | 1.56$\pm$ 0.41 | | 0.77$\pm$ 0.1
$|c_{1}|$ | | 3.9$\pm$0.89 | | 2.59$\pm$ 0.09 | | 3.43$\pm$ 0.41 | | 2.74$\pm$0.1
Figure 2: Double-log plots of $p_{\rm t}$ correlation with respect to size,
(a): straight-line fits with slopes $a_{0,1}$ and $a_{0,2}$ obtained by fixed
points, and (b): parabola fits.
The same analysis has also been applied to the $p_{\rm t}$ correlation
normalized by the average $p_{\rm t}$ over the whole sample starprc . The
analysis for normalized $p_{\rm t}$ correlation at $\sqrt{s}=62$ and $200$ GeV
show exactly the same behavior of fixed points and straight lines as what
$p_{\rm t}$ correlation demonstrates above. The critical exponents of
normalized $p_{\rm t}$ correlation (about 1.1) are smaller than that of
$p_{\rm t}$ correlation.
So the critical collision energies are most probably around $\sqrt{s}=62$ and
$200$ GeV, rather than near $\sqrt{s}=20$ and 130 GeV. The same analysis for
other critical related observable, such as the fluctuations of mean $p_{\rm
t}$ per event, the moments of multiplicity, the ratio of $K$ to $\pi$, and so
on, will be greatly helpful in confirming the observed results. Therefore, the
incident energy and centrality dependence of those observable are called for.
If there were additional collisions around $\sqrt{s}=62$ and $200$ GeV, we
could determine the finite-size scaling function defined in Eq. (6). This is
impossible at present since there are only two collision energies in addition
to the critical ones, and they could be outside of the asymptotic region where
finite-size scaling holds.
The findings of the two critical points may imply that deconfinement and
chiral symmetry restoration occur at different temperatures. Which one is at
the lower or higher temperature (energy) has to be confirmed finally from
theoretical calculation. Two critical collision energies, $\sqrt{s}=62$ and
$200$ GeV, are both within the range estimated by lattice calculation hTc .
The similar ratios of critical exponents at two critical points is consistent
with current theoretical estimation, which shows that all critical exponents
of the deconfinement transition, in the same university as the $3$-dimensional
Ising model 3d-ising , are very close to that of chiral symmetry restoration,
in the same university as the $3$-dimensional O$(4)$ model with spin symmetry
3d-o4 .
To the summary, we argue in this letter that finite-size effects of the formed
matter in relativistic heavy ion collisions is not negligible. The finite-size
scaling, rather than non-monotonous behavior of observable is a reliable
criterion of the existence of critical point. Then we propose how to locate
critical point by finite-size scaling. As an application, we analyze the data
of $p_{\rm t}$ correlation and its normalized one at 6 centralities and 4
incident energies from RHIC/STAR. Two fixed points, and therefore two critical
points, are likely observed around $\sqrt{s}=62$ and 200GeV. They could be,
respectively, related to the transition of deconfinement and chiral symmetry
restoration predicted by lattice-QCD. The ratios of critical exponents at
these two critical points are similar, in consistence with current theoretical
estimation.
The confirmation of this observation requires the efforts from both
theoretical and experimental sides. From experimental side, it is proposed to
get more and better data on other critical related observable at current
collision energies, and a few additional collisions around $\sqrt{s}=62$ and
200 GeV. Then we can more precisely determine the critical endpoints and
critical exponents.
The authors are grateful to Dr. Li Liangsheng, Prof. Liu Lianshou and Prof.
Dr. Hou Defu for very helpful discussions. This work is supported in part by
the NSFC of China with project No. 10835005 and MOE of China with project No.
IRT0624 and No. B08033.
## References
* (1) Y. Aoki, G. Endrodi, Z. Fodor, S.D. Katz, and K.K. Szabo, Nature 443, 675(2006).
* (2) Z. Fodor and S. D. Katz, J. High Energy Phys., 050(2004); Z. Fodor, S.D. Katz, and K.K. Szabo, Phys. Lett. B 568. 73(2003).
* (3) Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, K.K. Szabo, Nature 443, 675(2006); Y. Aoki, Z. Fodor, S.D. Katz, K.K. Szabo, Phys. Lett. B 643, 46(2006).
* (4) Karsch F. And Lutgemeier M., Nucl. Phys. B550, 449(1999). Ágnes Mócsy, Francesco Sannino and Kimmo Tuominen, J. Phys. G 30, S1255(2004).
* (5) Karsch F., Lecture Notes Phys. 583, 209(2002), hep-lat/0106019; Y. Aoki, Z. Fodor, S.D. Katza, and K.K. Szabo, Phys. Lett. B 643, 46(2006).
* (6) L. McLerran, R. D. Pisarski, Nucl. Phys. A 796, 83-100(2007); L. McLerran, K. Redlich, C. Sasaki, arXiv:0812.3585;
* (7) M. A. Stephanov, Phys. Rev. Lett. 102, 032301(2009); M. A. Stephanov, K. Rajagopal, and E. Shuyak, Phys. Rev. Lett. 81, 4816(1998); ibid, Phys. Rev. D 60, 114028(1999).
* (8) Stanisław Mrówczyński, arXiv:0902.0825; T. Nayak, Int. J. Mod. Phys. E 16, 3303(2008); J. T. Mitchell, PoS CFRNC2006, 015(2006); D. Adamová, et. al., (CERES Collaboration.), Nucl. Phys A 727, 97(2003).
* (9) J. Adams, et. al.(STAR collaboration), Phys. Rev. C 72, 044902(2005).
* (10) A. Esser, V. Dohm, and X.S. Chen, Physica A 222, 355 (1995).
* (11) M. E. Fisher, in Critical Phenomena, Proceedings of the International School of Physics Enrico Fermi, Course 51, edited by M. S. Green (Academic, New York, 1971).
* (12) E. Brézin, J. Phys. (Paris) 43, 15 (1982).
* (13) X. S. Chen, V. Dohm, and A. L. Talapov, Physica A 232, 375 (1996); X. S. Chen, V. Dohm, and N. Schultka, Phys. Rev. Lett.77, 3641(1996).
* (14) K. Binder, Z. Phys. B43, 119 (1981).
* (15) Li Liangsheng and X.S. Chen (to be published).
* (16) Chen Lizhu, Li Liangsheng, X.S. Chen and Wu Yuanfang (to be published).
* (17) H. Heiselberg, Phys. Rept. 351, 161(2001); M. Stephanov, J. of Phys. 27, 144(2005).
* (18) Y. Aoki, Z. Fodor, S.D. Katza, and K.K. Szabo, Phys. Lett. B643, 46(2006); F. Karsch, PoS CFRNC2007, arXiv:0711.0661; M. Stephanov, arXiv:hep-lat:0701002.
* (19) Jorge Garcá, Julio A. Gonzalo, Physica A 326, 464(2003).
* (20) Jens Braun1 and Bertram Klein, Phys. Rev.D77, 096008(2008).
|
arxiv-papers
| 2009-04-07T01:21:15 |
2024-09-04T02:49:01.761007
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chen Lizhu, X.S. Chen, Wu Yuanfang",
"submitter": "Yuanfang Wu",
"url": "https://arxiv.org/abs/0904.1040"
}
|
0904.1147
|
# On the Construction for Quantum Code $((n,K,d))_{p}$
via Logic Function over ${\rm{\mathbb{F}}}_{p}$
Shuqin Zhong, Zhi Ma, Yajie Xu and Xin L$\ddot{u}$ Zhengzhou Information
Science and Technology Institute
Zhengzhou, 450002, China
Email: lavenderzhong@live.cn
###### Abstract
This paper studies the construction for quantum codes with parameters
$((n,K,d))_{p}$ by use of an n-variable logic function with APC distance
$d^{\prime}\geq 2$ over ${\rm{\mathbb{F}}}_{p}$, where $d$ is related to
$d^{\prime}$. We obtain $d\leq d^{\prime}$ and the maximal $K$ for all
$d=d^{\prime}-k$, $0\leq k\leq d^{\prime}-2$. We also discuss the basic states
and the equivalent conditions of saturating quantum Singleton bound.
## I Introduction
Quantum error correcting code [1], [2], [3], [4] has become an indispensable
element in many quantum information tasks such as the fault-tolerant quantum
computation [5] the quantum key distribution [6] and the entanglement
purification [7], [8], to fight the noises.
Early in 1998, Calderbank [9] presented systematic mathematical methods to
construct binary quantum codes (stabilizer codes) from classical error
correcting codes over ${\rm{\mathbb{F}}}_{2}$ or ${\rm{\mathbb{F}}}_{4}$. A
series of good binary quantum codes were constructed by using classical codes
(BCH codes, Reed-Muller codes, AG codes, etc.). Schlingemann and Werner [10]
proposed a new way to construct quantum stabilizer codes by finding certain
graphs (or matrices) with special properties. Using this method they
constructed several new non-binary quantum codes. In particular, they gave a
new proof on the existence of quantum code $[[5,1,3]]_{p}$ for all odd primes
$p$ (the first proof was given by Rain [11]). It seems that this method can be
used to obtain many quantum codes saturating quantum Singleton bound (For any
code $[[n,k,d]]_{p}$ , the quantum Singleton bound says that $n\geq k+2d-2$,
see [3] for $p=2$ and [11] for $p\geq 3$). We call this kind of quantum codes
quantum MDS codes. At the same time, Feng Keqin [12] showed there existed
quantum codes $[[6,2,3]]_{p}$ and $[[7,3,3]]_{p}$ for any prime number $p$.
Liu Tailin [13] proved the existence of quantum codes $[[8,2,4]]_{p}$ and
$[[n,n-2,2]]_{p}$ for all odd prime numbers $p$.
In the correspondence, researchers made use of Boolean functions and
projection operators [14] to find quantum error correcting codes. In Ref [15],
the author constructed quantum code with parameters $[[n,0,d]]_{p}$, where $d$
is the APC distance of a Boolean function. Xu [16] generalized the definition
of APC distance for Boolean functions to logic functions over
${\rm{\mathbb{F}}}_{p}$, then constructed quantum code $((n,K,d))_{p}$, where
$d$ is related to APC distance of an n-variable function over
${\rm{\mathbb{F}}}_{p}$. Before talking further more about the ideas and
results of this paper, we need to introduce the logic construction of Ref [16]
which will be used in this paper.
For $d^{\prime}\geq 2$, let $f(x)$ be a function with $n$ variables and APC
distance $d^{\prime}$ over ${\rm{\mathbb{F}}}_{p}$.
$\beta_{i}=\left(\beta_{i1},\cdots,\beta_{in}\right)\in\rm{\mathbb{F}}_{p}^{n}$
for all $1\leq i\leq K$.
###### Lemma 1
[16] The space spanned by
$\\{|\psi_{i}\rangle=p^{-\frac{n}{2}}\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{f(x)+\beta_{i}x}|x\rangle|1\leq
i\leq K\\}$ is a quantum code with parameters $((n,K,d))_{p}$ satisfying:
$d=min\\{W_{s}(u,v)|\exists 1\leq i\leq j\leq
K,W_{s}(u,v-\beta_{i}+\beta_{j})\geq d^{\prime}\\},$
where $\zeta$ is a primitive element in $\mathbb{F}_{p}$.
This result was proved by Xu in [16]. Following the work of Xu, we discussed
the parameters and basic states of the constructed quantum code. The main
results proved in this paper are:
###### Theorem 1
Quantum code $((n,K,d))_{p}$ spanned by
$\\{|\psi_{i}\rangle=p^{-\frac{n}{2}}\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{f(x)+\beta_{i}x}|x\rangle|1\leq
i\leq K\\}$
is with following properties:
1. 1.
$d\leq d^{\prime}$,
2. 2.
$\beta_{1}=\cdots=\beta_{K}=0\;$ for $d=d^{\prime}$,
3. 3.
$W_{H}\left(\beta_{i},\beta_{j}\right)\leq k$ for all $d^{\prime}=d-k$ if
$0<k\leq d^{\prime}-2$.
###### Theorem 2
If quantum code $((n,K,d))_{p}$ is spanned by
$\\{|\psi_{i}\rangle=p^{-\frac{n}{2}}\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{f(x)+\beta_{i}x}|x\rangle|1\leq
i\leq K\\}.$
Then,
$K=\left\\{\begin{array}[]{l}{1\;,\;\;\;\;\;\;d=d^{\prime}}\\\ {\leq
p,\;\;\;d=d^{\prime}-1}\\\ {\leq\max
p^{k-2}(1+n(p-1),p^{2})\;\;,\;d=d^{\prime}-k}\end{array}\right.,$
where $2\leq k\leq d^{\prime}-2$.
We state the logic description of quantum codes in Section II and the proof of
our main results in Section III . Section IV is largely devoted to the basic
states and equivalent conditions of constructing quantum codes saturating
quantum Singleton Bound. Conclusions are drawn in Section V.
## II A Logic Description of Quantum Codes
The logic description of quantum codes given by [16] can be stated in
following element way.
Let $f(x)$ be a function of $n$ variables over ${\rm{\mathbb{F}}}_{p}$, the
quantum state
$|\psi_{f}\rangle=p^{-\frac{n}{2}}\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{f(x)}|x\rangle$
is called logic state corresponding to $f(x)$, where $\zeta$ is a primitive
element in ${\rm{\mathbb{F}}}_{p}$. Specially, $|\psi_{f}\rangle$ is called
Boolean state corresponding to Boolean function $f(x)$ if $p=2$.
Denote quantum error as $E_{\left(a,b\right)}=X\left(a\right)Z\left(b\right)$.
Then,
$E(a,b)\left|{\psi_{f}}\right\rangle=p^{-\frac{n}{2}}\sum\limits_{x\in\mathbb{F}_{p}^{n}}{\xi^{f(x-a)+b(x-a)}}$
(1)
where $\xi$ is a primitive element in ${\rm{\mathbb{F}}}_{p}$,
$a=(a_{1},\cdots,a_{n})\in\mathbb{F}_{p}^{n}$ and
$b=(b_{1},\cdots,b_{n})\in\mathbb{F}_{p}^{n}$, namely,
$\left|{\psi_{f}}\right\rangle\to
E(a,b)\left|{\psi_{f}}\right\rangle\Leftrightarrow f(x)\to f(x-a)+b(x-a)$ (2)
Let ${\rm{\mathbb{F}}}_{p}^{n}$ be the vector space of dimension $n$ over
${\rm{\mathbb{F}}}_{p}$ with the following inner product ( , ) defined by
$\left(a,b\right)=\sum_{i=1}^{n}a_{i}b_{i}$ (3)
for any $a=\left(a_{1},\cdots,a_{n}\right)$,
$b=\left(b_{1},\cdots,b_{n}\right)$$\in{\rm{\mathbb{F}}}_{p}^{n}$. For
convenience, denote $\left(a,b\right)$ as $a\cdot b$ .
For $K$ different vectors $\beta_{1},\cdots,\beta_{K}$ and an n-variable
function $f(x)$, $g_{i}(x)=f(x)+\beta_{i}\cdot x$, $1\leq i\leq K$ are $K$
different functions. Further more,
$|\psi_{i}\rangle=p^{-\frac{n}{2}}\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{g_{i}(x)}|x\rangle,1\leq
i\leq K$ (4)
are $K$ different logical states. Since,
$\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{f(x)-f(x)+(\beta_{i}-\beta_{j})\cdot
x}=0,$ (5)
we have $\langle\psi_{i}|\psi_{j}\rangle=0$, namely, $|\psi_{i}\rangle,1\leq
i\leq K$ are co-orthonogal.
###### Definition 1
The symmetrical distance between $a$ and $b$ is defined by
$W_{s}(a,b)=\\#\\{i|1\leq i\leq n,(a_{i},b_{i})\neq(0,0)\\},$ (6)
where
$a=\left(a_{1},\cdots,a_{n}\right),b=\left(b_{1},\cdots,b_{n}\right)\in{\rm{\mathbb{F}}}_{p}^{n}$.
###### Definition 2
[15] Let $f(x)$ be an n-variable Boolean function. The APC distance of $f(x)$
is the minimum $W_{s}(a,b)$, where
$a=\left(a_{1},\cdots,a_{n}\right),b=\left(b_{1},\cdots,b_{n}\right)\in{\rm{\mathbb{F}}}_{2}^{n}$
satisfying:
$\sum_{x\in{\rm{\mathbb{F}}}_{2}^{n}}\left(-1\right)^{f(x)-f(x-a)-b\cdot
x}\neq 0.$ (7)
Xu [16] generalized the definition of APC distance for a Boolean function to
logic function over ${\rm{\mathbb{F}}}_{p}$ as following.
###### Definition 3
[16] Let $f(x)$ be an n-variable function over ${\rm{\mathbb{F}}}_{p}$. The
APC distance of $f(x)$ is defined by the minimum $W_{s}(a,b)$, where
$a=\left(a_{1},\cdots,a_{n}\right),b=\left(b_{1},\cdots,b_{n}\right)\in
F_{p}^{n}$ satisfying:
$\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{f(x-a)+b\cdot
x-f\left(x\right)}\neq 0,$ (8)
where $\zeta$ is a primitive element in ${\rm{\mathbb{F}}}_{p}$.
###### Definition 4
The Hamming distance between $a$ and $b$ is defined by
$W_{H}(a,b)=\\#\\{i|1\leq i\leq n,a_{i}\neq b_{i}\\}$ (9)
with
$a=\left(a_{1},\cdots,a_{n}\right),b=\left(b_{1},\cdots,b_{n}\right)\in{\rm{\mathbb{F}}}_{p}^{n}$.
## III Proof of Main Results
In this section, let $f(x)$ be an n-variable function with APC distance
$d^{\prime}\geq 2$ over ${\rm{\mathbb{F}}}_{p}$ and
$\beta_{i}=\left({\beta_{i1},\cdots,\beta_{in}}\right)\in{\mathbb{F}}_{p}^{n}$
for all $1\leq i\leq K$.
For function $f(x)$ over ${\rm{\mathbb{F}}}_{p}$, constructing quantum code
$((n,K,d))_{p}$ by Lemma 1 is to find a group of vectors,
$\beta_{1},\cdots,\beta_{K}$, with special properties.The following theorem
tells the properties of $\beta_{1},\cdots,\beta_{K}$.
###### Theorem 1
Quantum code $((n,K,d))_{p}$ spanned by
$\\{|\psi_{i}\rangle=p^{-\frac{n}{2}}\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{f(x)+\beta_{i}x}|x\rangle|1\leq
i\leq K\\}$
is with following properties:
1. 1.
$d\leq d^{\prime}$,
2. 2.
$\beta_{1}=\cdots=\beta_{K}=0\;$ for $d=d^{\prime}$,
3. 3.
$W_{H}\left(\beta_{i},\beta_{j}\right)\leq k$ for all $d^{\prime}=d-k$ if
$0<k\leq d^{\prime}-2$.
###### Proof:
We prove $d\leq d^{\prime}$ in two separate way firstly.
Case 1: $\exists 1\leq i_{0}<j_{0}\leq K$ satisfying
$W_{H}\left(\beta_{i_{0}},\beta_{j_{0}}\right)=t>0$. Then it is reasonable to
suppose $\beta_{2i}-\beta_{1i}\neq 0$ for all $1\leq i\leq t$ and
$\beta_{2i}=\beta_{1i}$ for all $t+1\leq i\leq n$.
If $t\geq d^{\prime}$, set $u_{0}=(1,\underbrace{0,\cdots,0}_{n-1}),v_{0}=0$.
Thus,
$W_{s}\left(u_{0},v_{0}-\beta_{i_{0}}+\beta_{j_{0}}\right)=t\geq d^{\prime}.$
$d=\min\left\\{W_{s}\left(u,v\right)|\exists 1\leq i\leq j\leq
K,W_{s}\left(u,v-\beta_{i}+\beta_{j}\right)\geq d^{\prime}\right\\}$
$\leq W_{s}\left(u_{0},v_{0}\right)<d^{\prime}.$
If $t<d^{\prime}$, set
$u_{0}=(\underbrace{0,\cdots,0}_{t},\underbrace{1,\cdots,1}_{d^{\prime}-t},0,\cdots,0)$,
$v_{0}=0$. Then,
$W_{s}\left(u_{0},v_{0}-\beta_{1}+\beta_{2}\right)=d^{\prime}$
$d\leq W_{s}\left(u_{0},v_{0}\right)=d^{\prime}-t<d^{\prime}$
Therefore,
$d\leq d^{\prime}$
if $\exists 1\leq i_{0}<j_{0}\leq K$ satisfying
$W_{H}\left(\beta_{i_{0}},\beta_{j_{0}}\right)=t>0$.
Case 2: $\beta_{i}=\beta_{j}$ for all $1\leq i<j\leq K$. Suppose
$W_{H}\left(\beta_{i}\right)=t$.
If $t\geq d^{\prime}$, set $u_{0}=(1,\underbrace{0,\cdots,0}_{n-1}),v_{0}=0$.
Accordingly,
$W_{s}\left(u_{0},v_{0}-\beta_{1}+\beta_{2}\right)=t\geq d^{\prime},$
$d\leq W_{s}\left(u_{0},v_{0}\right)<d^{\prime}.$
If $t<d^{\prime}$, set
$u_{0}=(\underbrace{0,\cdots,0}_{t},\underbrace{1,\cdots,1}_{d^{\prime}-t},0,\cdots,0)$,
$v_{0}=0$. As a result,
$W_{s}\left(u_{0},v_{0}-\beta_{1}\right)=d^{\prime},$
$d\leq W_{s}\left(u_{0},v_{0}\right)=d^{\prime}-t\leq d^{\prime}.$
Therefore,
$d\leq d^{\prime}$
if $\beta_{i}=\beta_{j}$ for all $1\leq i<j\leq K$.
We now prove $\beta_{1}=\cdots=\beta_{K}=0$ if $d=d^{\prime}$.
First, we prove $\beta_{1}=\cdots=\beta_{K}$. Suppose $\exists 1\leq
i_{0}<j_{0}\leq K$ satisfying
$W_{H}\left(\beta_{i_{0}},\beta_{j_{0}}\right)=t>0$. Hence, it is reasonable
to suppose $i_{0}=1,j_{0}=2$ and $\beta_{2i}-\beta_{1i}\neq 0$ for all $1\leq
i\leq t$, $\beta_{2i}-\beta_{1i}=0$ for all $t+1\leq i\leq n$.
If $t\geq d^{\prime}$, set $u_{0}=(1,\underbrace{0,\cdots,0}_{n-1}),v_{0}=0$.
Consequently,
$W_{s}\left(u_{0},v_{0}-\beta_{1}+\beta_{2}\right)=t>d^{\prime},$
$d\leq W_{s}\left(u_{0},v_{0}\right)=1<d^{\prime}.$
If $t<d^{\prime}$, set
$u_{0}=(\underbrace{0,\cdots,0}_{t},\underbrace{1,\cdots,1}_{d^{\prime}-t},0,\cdots,0)$,
$v_{0}=0$. Hence,
$W_{s}\left(u_{0},v_{0}-\beta_{1}+\beta_{2}\right)=d^{\prime},$
$d\leq W_{s}\left(u_{0},v_{0}\right)=d^{\prime}-t<d^{\prime}.$
A contradiction, therefore $W_{H}\left(\beta_{i},\beta_{j}\right)=0$ for all
$1\leq i<j\leq n$.
Hence, $\beta_{1}=\cdots=\beta_{K}$. Denote $\beta_{1},\cdots,\beta_{K}$ as
$\beta_{1}$.
Second, we prove $\beta_{1}=0$. Suppose $W_{H}\left(\beta_{1}\right)=t>0$,
thus, it is reasonable to suppose $\beta_{1i}\neq 0$ for all $1\leq i\leq t$
and $\beta_{2i}-\beta_{1i}=0$ for all $t+1\leq i\leq n$.
If $t\geq d^{\prime}$, set $u_{0}=(1,\underbrace{0,\cdots,0}_{n-1}),v_{0}=0$.
As a result,
$W_{s}\left(u_{0},v_{0}-\beta_{1}\right)=t,$
$d=\min\left\\{W_{s}\left(u,v\right)|W_{s}\left(u,v-\beta_{1}\right)\geq
d^{\prime}\right\\}$ $\leq W_{s}\left(u_{0},v_{0}\right)<d^{\prime}.$
If $t<d^{\prime}$, set
$u_{0}=(\underbrace{0,\cdots,0}_{t},\underbrace{1,\cdots,1}_{d^{\prime}-t},0,\cdots,0)$,
$v_{0}=0$. Consequently,
$W_{s}\left(u_{0},v_{0}-\beta_{1}\right)=d^{\prime},$
$d\leq W_{s}\left(u_{0},v_{0}\right)=d^{\prime}-t<d^{\prime}.$
A contradiction, therefore, $W_{H}\left(\beta_{1}\right)=0$.
This completes the proof of property $2)$.
We now prove property $3)$. Suppose $\exists 1\leq i_{0}<j_{0}\leq K$
satisfying $W_{H}\left(\beta_{i_{0}},\beta_{j_{0}}\right)\geq k+1$. Then it is
reasonable to suppose $i_{0}=1,j_{0}=2$. Denote
$W_{H}\left(\beta_{1},\beta_{2}\right)=t$, where $t\geq k+1$. Thus it is
reasonable to suppose $\beta_{1i}\neq\beta_{2i}$ for all $1\leq i\leq t$ and
$\beta_{2i}-\beta_{1i}=0$ for all $t+1\leq i\leq n$.
If $t\geq d^{\prime}$, set $u_{0}=(1,\underbrace{0,\cdots,0}_{n-1}),v_{0}=0$.
Hence,
$W_{s}\left({u_{0},v_{0}-\beta_{1}+\beta_{2}}\right)=t\geq d^{\prime}.$ $d\leq
W_{s}(u_{0},v_{0})<d^{\prime}-k.$
If $t<d^{\prime}$, set
$u_{0}=(\underbrace{0,\cdots,0}_{t},\underbrace{1,\cdots,1}_{d^{\prime}-t},0,\cdots,0),v_{0}=0$.
Accordingly,
$W_{s}\left(u_{0},v_{0}-\beta_{1}+\beta_{2}\right)=t\geq d^{\prime},$
$d\leq W_{s}\left(u_{0},v_{0}\right)=d^{\prime}-t\leq d^{\prime}-k-1.$
A contradiction, therefore $W_{H}\left(\beta_{i},\beta_{j}\right)\leq k$ for
all $1\leq i<j\leq K$ if $0<k\leq d^{\prime}-2$.
This completes the proof of Theorem $1$. ∎
###### Remark 1
It can be easily seem from Theorem $1$ that if the following conditions
satisfy:
1. 1.
There exists an n-variable function with APC distance $d^{\prime}\geq 2$ over
${\rm{\mathbb{F}}}_{p}$,
2. 2.
A group of vectors $\beta_{1},\cdots,\beta_{K}$ over
${\rm{\mathbb{F}}}_{p}^{n}$ satisfy $W_{H}\left(\beta_{i},\beta_{j}\right)\leq
k$ for all $1\leq i<j\leq K$.
Quantum code $((n,K,d^{\prime}-k))_{p}$ can be constructed by Lemma $1$.
In the following theorem, we are going to deal with the parameter $K$.
###### Theorem 2
If quantum code $((n,K,d))_{p}$ is spanned by
$\\{|\psi_{i}\rangle=p^{-\frac{n}{2}}\sum_{x\in{\rm{\mathbb{F}}}_{p}^{n}}\zeta^{f(x)+\beta_{i}x}|x\rangle|1\leq
i\leq K\\}$. Then
$K=\left\\{\begin{array}[]{l}{1\;,\;\;\;\;\;\;d=d^{\prime}}\\\ {\leq
p,\;\;\;d=d^{\prime}-1}\\\ {\leq\max
p^{k-2}(1+n(p-1),p^{2})\;\;,\;d=d^{\prime}-k}\end{array}\right.,$
where $2\leq k\leq d^{\prime}-2$.
###### Proof:
1. 1.
For $d=d^{\prime}$, it can be deduced from Theorem 1 that
$\beta_{1}=\cdots=\beta_{K}=0.$
Thus,
$\textit{K}=1.$
2. 2.
For $d=d^{\prime}-1$, let $W_{ij}=W_{H}(\beta_{i},\beta_{j})$ for all $1\leq
i<j\leq n$.
Suppose $K>p$. Then there exists $1\leq i_{0}<j_{0}\leq K$ satisfying
$W_{i_{0}j_{0}}\geq 2$, a contradiction, thus
$K\leq p.$
3. 3.
Denote $C_{n}^{t}$ as the number of vectors where the Hamming distance between
each other is no more than $t$.
For $k=2$, since $W_{H}(\beta_{i},\beta_{j})\leq 2$ for all $1\leq i<j\leq K$
by Theorem $2$.
Case 1: If $\beta_{1},\cdots,\beta_{K}$ are the same in $n-2$ bits. It can be
deduced that $\beta_{1},\cdots,\beta_{K}$ are different in at most 2 bits,
hence,
$K\leq p^{2}.$
Case 2: If that $\beta_{1},\cdots,\beta_{K}$ are the same in $n-2$ bits
doesn’t satisfy, then, K is the maximal when the different bits are all n
bits. Thus,
$K\leq(p-1)n+1$
Therefore, $K\leq\max\\{p^{2},(p-1)n+1\\}$ for $d=d^{\prime}-2$.
For $3\leq k\leq d^{\prime}-2$, since $W_{H}(\beta_{i},\beta_{j})\leq k$ by
Theorem $1$ for all $1\leq i<j\leq K$. Thus,
$K=C_{n}^{k}\leq pC_{n-1}^{k-1}\leq\cdots\leq p^{k-2}C_{n-k+2}^{2}$
$\leq\max p^{k-2}\\{1+(n-k+2)(p-1),p^{2}\\}$
This completes the proof of Theorem $2$.
∎
###### Remark 2
It can be inferred from Theorem $1$ and Theorem $2$ that for an n-variable
function with APC distance $d^{\prime}\geq 2$ over ${\rm{\mathbb{F}}}_{p}$,
quantum code with parameters $((n,K,d))_{p}$ can be constructed by Lemma $1$
where $d\leq d^{\prime}$. Furthermore, if $d=d^{\prime}-k,0\leq k\leq
d^{\prime}-2$, then $\beta_{1},\cdots,\beta_{K}$ should satisfy
$W_{H}(\beta_{i},\beta_{j})\leq t$ for all $1\leq i<j\leq K$. At the same
time, we obtain the maximal $K$.
## IV Basic States and Equivalent Conditions of Constructing Quantum MDS
Codes
### IV-A The basic states of the constructed quantum code
In this subsection, denote $\beta_{i}$ as
$\beta_{i}=\left({\beta_{i1},\cdots,\beta_{in}}\right)$.
For an n-variable function with APC distance $d^{\prime}$ over
${\rm{\mathbb{F}}}_{p}$ and $\beta_{1},\cdots,\beta_{K}$, quantum code
$((n,K,d))_{p}$ can be constructed by Lemma $1$. The basic states of the
constructed quantum code can be stated as following:
If $p\geq n-k+1$, then
$p^{k}\geq p^{k-2}+p^{k-2}(p-1)(n-k+2).$
Let
$K=p^{k}.$
At this time, we set $\beta_{1},\cdots,\beta_{K}$ be vectors that the first
$k$ bits run all over ${\rm{\mathbb{F}}}_{p}^{k}$ and the last $n-k$ bits are
zeros. Namely,
$\beta_{ij}\in\mathbb{F}_{p}~{}for~{}1\leq j\leq k$ (10)
$\beta_{ij}=0~{}for~{}k+1\leq j\leq n$ (11)
where $1\leq i\leq p^{k}$. It can be checked that
$W_{H}(\beta_{i},\beta_{j})\leq k$ for all $1\leq i<j\leq p^{k}$, thus, the
space spanned by formula $(4)$ corresponding to $\beta_{1},\cdots,\beta_{K}$
satisfying formula (10) and (11) is a quantum code with parameters
${\rm((}n,K,d^{\prime}-k{\rm))}_{p}$.
If $p<n-k+1$, then $p^{k-2}+p^{k-2}\left(n-k+2\right)(p-1)+1>p^{k}$. Let
$K=p^{k-2}+p^{k-2}\left(n-k+2\right)(p-1).$
At this time, we set $\beta_{1},\cdots,\beta_{K}$ be vectors that the first
$k-2$ bits run all over ${\rm{\mathbb{F}}}_{p}^{k-2}$ , the $k+l-2$ -th bit
run all over ${\rm{\mathbb{F}}}_{p}\backslash\left\\{0\right\\}$, $1\leq l\leq
n-k+2$. Namely,
$\beta_{ij}\in\mathbb{F}_{p}~{}for~{}1\leq j\leq k-2$ (12)
$\beta_{i~{}k+l-2}\in\mathbb{F}_{p}\backslash\\{0\\}~{}for~{}1\leq l\leq
n-k+2$ (13)
and the rest bits are all zeros. It can be easily checked that
$W_{H}(\beta_{i},\beta_{j})\leq k-2+2=k$
for all $1\leq i<j\leq K$, thus, the space spanned by formula $(4)$
corresponding to $\beta_{1},\cdots,\beta_{K}$ satisfying formula (12) and
formula (13) is a quantum code with parameters
$((n,p^{k-2}+p^{k-2}(p-1)(n-k+2),d^{\prime}-k))_{p}.$
### IV-B The equivalent conditions of constructing quantum MDS codes
Theory of quantum code has quantum singleton bound as classical code. Quantum
codes saturating quantum Singleton Bound are quantum MDS codes. The following
theorem presents the equivalent conditions of quantum MDS codes constructed by
Lemma 1.
###### Theorem 3
Quantum code ${\rm((}n,K,d^{\prime}-k{\rm))}_{p}$ is constructed by Lemma 1,
where $d^{\prime}-k\leq\frac{n}{2}+1$. Then it saturates quantum Singleton
Bound if and only if the following conditions satisfy:
1. 1.
If $k=0$, then there exists an n-variable function over
${\rm{\mathbb{F}}}_{p}$ with APC distance $d^{\prime}$ over
${\rm{\mathbb{F}}}_{p}$, where $d^{\prime}=\frac{n}{2}+1$ and $n$ is even,
2. 2.
If $k=1$, then there exists an n-variable function with APC distance
$d^{\prime}$ over ${\rm{\mathbb{F}}}_{p}$, where $d^{\prime}=\frac{n}{2}+1$,
3. 3.
If $2\leq k\leq d^{\prime}$ and $p\geq n-k+1$, then there exists an n-variable
function with APC distance $d^{\prime}$ over ${\rm{\mathbb{F}}}_{p}$, where
$2d^{\prime}=n+k+2$,
4. 4.
If $2\leq k\leq d^{\prime}$and$p<n-k+1$, then there exists an n-variable
function with APC distance $d^{\prime}$ over ${\rm{\mathbb{F}}}_{p}$, where
$p^{k-2}+p^{k-2}\left(n-k+2\right)(p-1)=p^{n-2(d^{\prime}-k)+2}$.
###### Proof:
Let quantum code $((n,K,d^{\prime}-k))_{p}$ be constructed by Lemma $1$.
1. 1.
If $k=0$, then
$K=1$
by Theorem 2. Thus, the quantum code saturates Quantum Singleton Bound if and
only if
$n-2d^{\prime}+2=0.$
2. 2.
If $k=1$, we get
$K\leq n(p-1)+1$
by Theorem 2. Thus, the quantum code saturates Quantum Singleton Bound if and
only if
$n(p-1)+1=p^{n-2d^{\prime}+4}.$
3. 3.
If $2\leq k\leq d^{\prime}$ and $p\geq n-k+1$,
$K\leq p^{k}$
by Theorem 2. Thus, the quantum code saturates Quantum Singleton Bound if and
only if
$k=n-2\left(d^{\prime}-k\right)+2\Leftrightarrow 2d^{\prime}=n+k+2.$
4. 4.
If $2\leq k\leq d^{\prime}$ and $p<n-k+1$,
$K<p^{k-2}+p^{k-2}\left(n-k+2\right)(p-1)$
by Theorem 2. Thus, the quantum code saturates Quantum Singleton Bound if and
only if
$p^{k-2}+p^{k-2}\left(n-k+2\right)(p-1)=p^{n-2(d^{\prime}-k)+2}.$
This completes the proof of this Theorem . ∎
## V Conclusion
Ref. [16] presented a new way to construct quantum error correcting codes.
Quantum error correcting codes can be constructed by use of logic functions
with n variables and APC distance $d^{\prime}\geq 2$ over
${\rm{\mathbb{F}}}_{p}$. The minimum distance of the constructed quantum code
is $d=d^{\prime}-t(0\leq t\leq d^{\prime}-2)$. We can also get the maximal
dimension of the corresponding space. In this paper, we also give the basic
states and the equivalent conditions for existence of quantum MDS codes.
It can be seem that logic functions with favorable APC distance play a key
role in logic construction for quantum codes. The presented paper is to re-
cast the construction of QECCs as a problem of construction logic function
with favorable APC distance. Ref [17] proposed a quadratic residue
construction for Boolean function with favorable APC distance. For an
n-variable function over ${\rm{\mathbb{F}}}_{p}$, how to compute the APC
distance fast is still a problem to be researched.
## Acknowledgment
This work is supported by the NFS of China under Grant number 60403004 and the
Outstanding Youth Foundation of Henan Province under Grant No.0612000500.
## References
* [1] P. W. Shor, “ Scheme for Reducing Decoherence in Quantum Computer Memory,” _Phys. Rev. A._ 54 (2), pp. 1098–1105, 1995.
* [2] C. H. Bennettt, D. P. DiVincenco, J. A. Smolin and W. K. Wootters, “ Mixed state entanglement and quantum error correction,” _Phys. Rev._ 54 (5), pp. 3824–3851, 1996.
* [3] E. Knill and R. Laflamme, “ A Theory of quantum error-correcting code saturating quantum Hamming Bound,” _Phys. Rev. A._ 55, pp. 900–911, 1997.
* [4] A. M. Steane, “ Simple quantum error correcting codes,” _Phys. Rev. Lett._ 77, pp. 793–797, 1996.
* [5] D. Gottesman, “ Theory of fault-tolerant quantum computation,” _Phys. Rev. A._ 57, pp. 127–137, 1998.
* [6] C. H. Bennett and G. Brassard, “ Quantum cryptography: public key distribution and coin tossing,” _Proceedings of IEEE International Conference on Computers, Systems, and Sig-nal Processing,_ pp. 175–179, 1984.
* [7] S. Glancy, E. Knill and H. M. Vasconcelos,“ Entanglement purification of any stabilizer state,” _Phys. Rev. A._ 74, no. 032319, 2006.
* [8] A. Ambainis and D. Gottesman, “The minimum distance problem for two-way entanglement purification,” _IEEE Trans. Inform. Theory._ 52, pp. 748–753, 2006.
* [9] A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, “ Quantum error correction via codes over $\mathbb{F}_{4}$,” _IEEE Trans. Inform Theory_ 44, pp. 1369–1387, 1998.
* [10] D. Schlingemann and R. F. Werner,“Quantum error correcting codes associated with graphs,” _Phys. Rev. A._ 65, 012308, 2002.
* [11] E. M. Rain, “ Nonbinary quantum code,” _IEEE Trans. Inform Theory_ 45, pp. 1827–1832, 1999.
* [12] K. Q. Feng, “ Quantum codes $[[6,2,3]]_{p}$ and $[[7,3,3]]_{p}$ ($p\geq 3$) exist,” _IEEE Trans. Inform Theory_ 48 (8), pp. 2384–2391, 2002.
* [13] T. L. Liu, “ On construction for nonbinary cyclic quantum code via graph,” _China Science Inform Theory. E._ 35 (6), pp. 588–596, 2005.
* [14] V. Aggarwal and R. Calderbank, “ Boolean functions, projection operators and quantum error correction codes,” _IEEE Trans. Inform Theory._ , 54 (4) PP. 1700–1707, 2008\.
* [15] L. E. Danielsen, “ On self-dual quantum codes, graphs, and Boolean functions,” http://arxiv.org/abs/quant-ph/0503236, 2005.12.
* [16] Y. J. Xu, “Logic function and quantum code,” http://arxiv.org/abs/quant-ph/0712.3605v4, 2008.01.
* [17] L. E. Danielsen, “ Aperiodic Propagation Criteria for Boolean Functions,” _In Information and Computation_ 204 (5), pp. 741–770, 2006.
|
arxiv-papers
| 2009-04-07T14:14:55 |
2024-09-04T02:49:01.767945
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shuqin Zhong, Zhi Ma, Yajie Xu and Xing Lv",
"submitter": "Xin L\\\"u",
"url": "https://arxiv.org/abs/0904.1147"
}
|
0904.1301
|
# Differential graded Lie algebras controlling infinitesimal deformations of
coherent sheaves
Domenico Fiorenza
Dipartimento di Matematica “Guido Castelnuovo”,
Sapienza Università di Roma,
P.le Aldo Moro 5, I-00185 Roma Italy. fiorenza@mat.uniroma1.it
www.mat.uniroma1.it/~fiorenza/ , Donatella Iacono
Max-Planck Institut für Mathematik,
Vivatsgasse 7, D 53111 Bonn Germany iacono@mpim-bonn.mpg.de and Elena
Martinengo
Dipartimento di Matematica “Guido Castelnuovo”,
Sapienza Università di Roma,
P.le Aldo Moro 5, I-00185 Roma Italy. martinengo@mat.uniroma1.it
www.mat.uniroma1.it/dottorato/
###### Abstract.
We use the Thom-Whitney construction to show that infinitesimal deformations
of a coherent sheaf ${\mathcal{F}}$ are controlled by the differential graded
Lie algebra of global sections of an acyclic resolution of the sheaf
$\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})$, where $\mathcal{E}^{\cdot}$ is any
locally free resolution of ${\mathcal{F}}$. In particular, one recovers the
well known fact that the tangent space to $\operatorname{Def}_{\mathcal{F}}$
is $\operatorname{Ext}^{1}({\mathcal{F}},{\mathcal{F}})$, and obstructions are
contained in $\operatorname{Ext}^{2}({\mathcal{F}},{\mathcal{F}})$.
The main tool is the identification of the deformation functor associated with
the Thom-Whitney DGLA of a semicosimplicial DGLA ${\mathfrak{g}}^{\Delta}$,
whose cohomology is concentrated in nonnegative degrees, with a noncommutative
Čech cohomology-type functor $H^{1}_{\rm sc}(\exp{\mathfrak{g}}^{\Delta})$.
###### Key words and phrases:
Differential graded Lie algebras, functors of Artin rings
###### 1991 Mathematics Subject Classification:
18G30, 18G50, 18G55, 13D10, 17B70
## Introduction
The classical approach to deformation theory, starting with Kodaira and
Spencer’s studies on deformations of complex manifolds, consists in deforming
the objects locally and then glue back together these local deformations.
During the last thirty years, another approach to deformation problems has
been developed. The philosophy underlying it, essentially due to Quillen,
Deligne, Drinfeld and Kontsevich, is that, in characteristic zero, every
deformation problem is controlled by a differential graded Lie algebra, via
solutions of Maurer-Cartan equation modulo gauge equivalence. The aim of this
paper is to exhibit an explicit equivalence between the two approaches for the
problem of infinitesimal deformations of coherent sheaves.
In the particular case of a locally free sheaf $\mathcal{E}$ of
${\mathcal{O}}_{X}$-modules on a complex manifold $X$, the Kodaira-Spencer’s
description of deformations of $\mathcal{E}$ is given in terms of the Čech
functor $H^{1}(X;\exp\mathcal{E}nd(\mathcal{E}))$, where
$\mathcal{E}nd(\mathcal{E})$ is the sheaf of endomorphism of $\mathcal{E}$.
Indeed, a locally free sheaf has only trivial local deformations and so a
deformation of $\mathcal{E}$ is reduced to a deformation of the gluing data of
its local charts, and the compatibility conditions these gluing data have to
satisfy is precisely expressed by the cocycle condition in the Čech functor.
On the other hand, it is well known that deformations of $\mathcal{E}$ are
controlled by the DGLA of global sections of an acyclic resolution of
$\mathcal{E}nd(\mathcal{E})$, e.g., by the DGLA
$A^{0,*}_{X}(\mathcal{E}nd(\mathcal{E}))$ of $(0,*)$-forms on $X$ with values
in the sheaf of endomorphisms of the sheaf $\mathcal{E}$.
The equivalence between these two descriptions is best understood by moving
from set-valued to groupoid-valued deformation functors; see, e.g., [9, 20].
Associating with any open set $U$ in $X$ the groupoid
$\operatorname{Def}_{\mathcal{E}|_{U}}$ of infinitesimal deformations of
$\mathcal{E}$ over $U$ (over a fixed base $\operatorname{Spec}A$, for some
local Artin ring $A$) defines a stack over ${\bf{Top}}_{X}$; this is just a
one-word way of saying that global deformations of $\mathcal{E}$ are the same
thing as the descent data for its local deformations:
$\operatorname{Def}_{\mathcal{E}}\simeq\displaystyle\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}\operatorname{Def}_{\mathcal{E}|_{U}},$
where $\Delta_{\mathcal{U}}$ is the semisimplicial object in ${\bf{Top}}_{X}$
associated with an open cover $\mathcal{U}$ of $X$. Next, one sees that
locally the groupoid of deformations of $\mathcal{E}|_{U}$ is equivalent to
the Deligne groupoid of $\mathcal{E}nd(\mathcal{E})(U)$; since these
equivalences are compatible with restriction maps, one has an equivalence of
semicosimplicial groupoids. Finally, Deligne groupoid commutes with homotopy
limits of DGLA concentrated in positive degree (see [9]), so that
$\operatorname{Def}_{\mathcal{E}}\simeq\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}\operatorname{Del}_{\mathcal{E}nd(\mathcal{E})(U)}\simeq\operatorname{Del}_{\begin{subarray}{c}{\mathop{\rm
holim}\mathcal{E}nd(\mathcal{E})(U)}\\\
{\scriptscriptstyle{U\in\Delta_{\mathcal{U}}\phantom{mmmmmi}}}\end{subarray}}.$
This shows that the problem of infinitesimal deformations of $\mathcal{E}$ is
controlled by the DGLA $\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}\mathcal{E}nd(\mathcal{E})(U)$. It is now a
simple exercise in homological algebra showing that there is a quasi-
isomorphism of DGLAs
$\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}\mathcal{E}nd(\mathcal{E})(U)\simeq
A^{0,*}_{X}(\mathcal{E}nd(\mathcal{E})).$
The reader who prefers to not leave the peaceful realm of set-valued
deformation functors can found a direct (but less enlightening) proof of the
equivalence between the Kodaira-Spencer’s and the DGLA approach to
infinitesimal deformation of locally free sheaves in [7], where the explicit
Thom-Whitney model for $\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}\mathcal{E}nd(\mathcal{E})(U)$ is used.
We now turn our attention to deformations of a coherent sheaf $\mathcal{F}$ of
$\mathcal{O}_{X}$-modules on a complex manifold or an algebraic variety $X$.
The classical approach to this deformation problem is based on a locally free
resolution $\mathcal{E}^{\cdot}\to\mathcal{F}$ of $\mathcal{F}$; then, the
data of a deformation of $\mathcal{F}$ are the data of local deformations of
$\mathcal{E}^{\cdot}$ with appropriate gluing conditions. More precisely, the
sheaf of differential graded Lie algebras
$\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})$ of the endomorphisms of the
resolution $\mathcal{E}^{\cdot}$ controls infinitesimal deformations of
$\mathcal{F}$ via the Čech-type functor $H^{1}_{\rm
Ho}(X;\exp\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))$; the subscript ${\rm Ho}$
refers to the fact that cocycle conditions hold only up to homotopy. The
functor $H^{1}_{\rm Ho}(X;\exp\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))$ is
actually independent of the particular resolution chosen. And again, on the
DGLA side, one proves that infinitesimal deformations of ${\mathcal{F}}$ are
controlled by the DGLA of global sections of an acyclic resolution of
$\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})$; in particular, one recovers the well
known fact that the tangent space to $\operatorname{Def}_{\mathcal{F}}$ is
$\operatorname{Ext}^{1}({\mathcal{F}},{\mathcal{F}})$, and obstructions are
contained in $\operatorname{Ext}^{2}({\mathcal{F}},{\mathcal{F}})$.
To see why such a result should hold, one has to make a further step and go
from groupoid-valued to $\infty$-groupoid-valued deformation functors, and to
think the whole problem in terms of $\infty$-stacks [10, 16, 24]. Indeed, due
to the presence of negative degree components in
$\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})$, the groupoids
$\operatorname{Def}_{\mathcal{F}|_{U}}$ are no more equivalent to the Deligne
groupoids $\operatorname{Del}_{\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})(U)}$;
yet from the $\infty$-groupoid point of view it is natural to expect that the
stack $\operatorname{Def}_{\mathcal{F}}$ is locally homotopy equivalent to the
$\infty$-stack
$\operatorname{MC}_{\bullet}(\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))$. Then
one reasons as in the locally free sheaf case, using the fact that the Kan
complexes-valued functor $\operatorname{MC}_{\bullet}$ commutes with homotopy
limits of DGLAs whose cohomology is concentrated in positive degree [8]:
$\operatorname{Def}_{\mathcal{F}}\simeq\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}\operatorname{Def}_{\mathcal{F}|_{U}}\simeq\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}{\operatorname{MC}_{\bullet}(\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})(U))}\simeq{\operatorname{MC}_{\bullet}}(\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})(U)).$
As above, the homotopy limit $\mathop{\rm
holim}_{U\in\Delta_{\mathcal{U}}}\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})(U)$ is
quasiisomorphic to the DGLA of global sections of an acyclic resolution of
$\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})$, which therefore controls the
infinitesimal deformations of ${\mathcal{F}}$.
The aim of this paper is to give a direct proof of this fact at the level of
set-valued deformation functors. The proof closely follows the argument in [7]
and does not rely on the conjectural homotopy equivalence between
$\operatorname{Def}_{\mathcal{F}|_{U}}$ and
$\operatorname{MC}_{\bullet}(\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})(U))$. More
precisely, we associate with any semicosimplicial DGLA
${\mathfrak{g}}^{\Delta}$ a set-valued functor of Artin rings $Z^{1}_{\rm
sc}(\exp{\mathfrak{g}}^{\Delta})$ together with an equivalence relation $\sim$
on it, such that the quotient functor $H^{1}_{\rm
sc}(\exp{\mathfrak{g}}^{\Delta})=Z^{1}_{\rm
sc}(\exp{\mathfrak{g}}^{\Delta})/\sim$ is an abstract version of $H^{1}_{\rm
Ho}(X;\exp\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))$. The latter is obtained, as
a particular case, by considering the Čech semicosimplicial Lie algebra
${\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}})(\mathcal{U})$
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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Namely,
$H^{1}_{\rm
Ho}(X;\exp\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))=\lim_{\begin{subarray}{c}\longrightarrow\\\
{\mathcal{U}}\end{subarray}}H^{1}_{\rm
sc}(\exp{\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}})(\mathcal{U}))$
and both sides coincide with $H^{1}_{\rm
sc}(\exp\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})(\mathcal{U}))$, for an
$\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})$-acyclic cover of $X$. Next, we
consider the Thom-Whitney model
$\operatorname{Tot}_{TW}{\mathfrak{g}}^{\Delta}$ for $\mathop{\rm
holim}{\mathfrak{g}}^{\Delta}$ and show that there exists a commutative
diagram of functors
$\textstyle{{\rm DGLA}^{\Delta_{\rm mon}}_{H^{\geq
0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tot}_{TW}}$$\scriptstyle{H^{1}_{\rm
sc}(\exp-)}$$\textstyle{{\rm
DGLA}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phantom{mi}\text{Maurer-
Cartan}/\text{gauge}}$$\textstyle{\mathbf{Set}^{{\mathbf{Art}}_{\mathbb{K}}},}$
where ${\rm DGLA}^{\Delta_{\rm mon}}_{H^{\geq 0}}$ is the category of
semicosimplicial DGLAs with no negative cohomology. From the point of view of
$\infty$-groupoids, this can be seen as an explicit description of the set
$\pi_{\leq 0}(\operatorname{MC}_{\bullet}(\mathop{\rm
holim}{\mathfrak{g}}^{\Delta}))$.
The paper is organized as follows: in Section 1 we dicuss deformations of
coherent sheaves from a classical perspective and show how deformation data
can be conveniently encoded into a Čech cohomology group with coefficient in a
sheaf of DGLAs. In Section 2, the functors $H^{1}_{\rm
sc}(\exp{\mathfrak{g}}^{\Delta})$ and $H^{1}_{\rm Ho}(X;\exp{\mathcal{L}})$
are defined; next, in Sections 3 and 4, we recall the definition of the Thom-
Whitney DGLA associated with $\mathfrak{g}^{\Delta}$ and with its truncations
$\mathfrak{g}^{\Delta_{[m,n]}}$. Sections 5 and 6 are rather technical; namely
Section 5 is devoted to a technical lemma on Maurer-Cartan elements in the
Thom-Whitney DGLAs $\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})$
and $\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})$ and Section 6 to
the proof of the isomorphism $H^{1}_{\rm
sc}(\exp{\mathfrak{g}}^{\Delta_{[0,1]}})$ and
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}$.
Finally, in Section 7, we are able to prove our main result (Theorem 7.6):
under the cohomological hypotesis $H^{-1}(\mathfrak{g}_{2})=0$ there is a
natural isomorphism of funtors
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}\cong
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$; moreover, if
$H^{j}(\mathfrak{g}_{i})=0$ for all $i\geq 0$ and $j<0$, then there is a
natural isomorphism of functors
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta})}\cong
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$. In the concluding Section 8, we
use this isomorphism to prove that infinitesimal deformations of a coherent
sheaf ${\mathcal{F}}$ are controlled by the DGLA of global sections of an
acyclic resolution of $\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})$, where
$\mathcal{E}^{\cdot}$ is a locally free resolution of $\mathcal{F}$.
While revising this paper, we became aware of [25] where a similar
construction is developed and investigated.
Throughout this paper we work on a fixed algebraically closed field
$\mathbb{K}$ of characteristic zero; the symbol $\bf{Art}_{\mathbb{K}}$
denotes the category of local Artinian $\mathbb{K}$-algebras
$(A,{\mathfrak{m}}_{A})$, with residue field $\mathbb{K}$.
* Acknowledgement.
We thank Marco Manetti for stimulating discussions on the subject and for
useful comments and suggestions on the first version of this paper; d.i.
thanks the Mathematical Department “Guido Castelnuovo”, Sapienza Università di
Roma for the hospitality.
## 1\. Infinitesimal deformations and sheaves of DGLAs
In this section, we study infinitesimal deformations of a coherent sheaf
$\mathcal{F}$ of $\mathcal{O}_{X}$-modules on a smooth projective variety $X$
and explain how these deformations can be naturally described in terms of a
sheaf of differential graded Lie algebras on $X$.
An infinitesimal deformation of the coherent sheaf of
$\mathcal{O}_{X}$-modules $\mathcal{F}$ over $A\in\bf{Art}_{\mathbb{K}}$ is
given by a coherent sheaf $\mathcal{F}_{A}$ of $\mathcal{O}_{X}\otimes
A$-modules on $X\times\operatorname{Spec}A$, flat over $A$, with a morphism of
sheaves $\pi:\mathcal{F}_{A}\to\mathcal{F}$ inducing an isomorphism
$\mathcal{F}_{A}\otimes_{A}\mathbb{K}\cong\mathcal{F}$.
Two deformations $\mathcal{F}_{A},\mathcal{F^{\prime}}_{A}$ of the coherent
sheaf $\mathcal{F}$ over $A$ are isomorphic if there exists an isomorphism of
sheaves $f:\mathcal{F}_{A}\to\mathcal{F^{\prime}}_{A}$, that commutes with the
morphisms to $\mathcal{F}$. We denote by
$\operatorname{Def}_{\mathcal{F}}:\bf{Art}_{\mathbb{K}}\to\bf{Set}$ the
functor of infinitesimal deformations of the sheaf $\mathcal{F}$.
We start by studying infinitesimal deformations of a coherent sheaf
${\mathcal{F}}$ of $\mathcal{O}_{X}$-modules on an affine variety $X$. Let
$X=\operatorname{Spec}R$, where $R$ is a Noetherian $\mathbb{K}$-algebra and
let $\mathcal{F}$ be the coherent sheaf associated with a finitely generated
$R$-module $M$; in this simple case, deformations of the sheaf $\mathcal{F}$
reduce to deformations of the $R$-module $M$.
An infinitesimal deformation of the $R$-module $M$ over
$A\in\bf{Art}_{\mathbb{K}}$ is given by a $R\otimes A$-module $M_{A}$, flat
over $A$, with a morphism $\pi:M_{A}\to M$ inducing an isomorphism
$M_{A}\otimes_{A}\mathbb{K}\cong M$. Two deformations $M_{A}$ and
$M^{\prime}_{A}$ of the module $M$ over $A$ are isomorphic if there exists an
isomorphism of $R\otimes A$-modules $f:M_{A}\to M^{\prime}_{A}$, that commutes
with the morphisms to $M$.
Next, let
(1) $\cdots\stackrel{{\scriptstyle
d}}{{\longrightarrow}}R^{n_{1}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}R^{n_{0}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}M\longrightarrow 0$
be a presentation of $M$ as $R$-module. If $M_{A}$ is a deformation of $M$
over $A$, then it is an $A$-flat $R\otimes A$-module; therefore, flatness
allows to lift relations between generators and to construct the exact
sequence
$\cdots\stackrel{{\scriptstyle d_{A}}}{{\longrightarrow}}R^{n_{1}}\otimes
A\stackrel{{\scriptstyle d_{A}}}{{\longrightarrow}}R^{n_{0}}\otimes
A\stackrel{{\scriptstyle d_{A}}}{{\longrightarrow}}M_{A}\longrightarrow 0,$
that reduces to (1) when tensored by $\mathbb{K}$ over $A$. On the other hand,
the datum of such an exact sequence assures flatness of the $R\otimes
A$-module $M_{A}$ and so it defines a deformation of $M$ over $A$ (see [1,
par. 3], or [23, Theorem A.31] for details of these correspondences).
Moreover, if $M_{A}$ and $M^{\prime}_{A}$ are isomorphic deformations of $M$
over $A$, the isomorphism between them lifts to an isomorphism between the
correspondent deformed complexes and viceversa.
Next, we return to the global case of a coherent sheaf $\mathcal{F}$ of
$\mathcal{O}_{X}$-modules on a smooth projective variety $X$. Let
$0\longrightarrow\mathcal{E}^{-m}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{E}^{-1}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{E}^{0}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{F}\longrightarrow 0$
be a global syzygy for ${\mathcal{F}}$, and denote by $\mathcal{E}^{\cdot}$
the complex of locally free sheaves
$(\mathcal{E}^{\cdot},d):\qquad\qquad
0\longrightarrow\mathcal{E}^{-m}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{E}^{-1}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{E}^{0}\longrightarrow 0.$
Let $\mathcal{U}=\\{U_{i}\\}_{i\in I}$ be an affine111or Stein, if we work in
the complex analytic category. open cover of $X$, such that every sheaf of
$\mathcal{E}^{\cdot}$ is free on each $U_{i}$.
The Kodaira-Spencer approach to infinitesimal deformations of $\mathcal{F}$
consists in deforming the sheaf $\mathcal{F}$ locally in such a way that local
deformations glue together to a global sheaf, or equivalently, in view of the
above discussion of the affine case, in deforming the complex
$(\mathcal{E}^{\cdot},d)$ on every open set $U_{i}$ in such a way that these
data glue together in cohomology.
Following this approach, let us make explicit the deformation data: the first
datum is an element
$l=\\{l_{i}\\}_{i}\in\prod_{i}{\mathcal{E}nd}^{1}(\mathcal{E}^{\cdot})(U_{i})\otimes\mathfrak{m}_{A}$
defining, on every open set $U_{i}$, a complex
$(\mathcal{E}^{\cdot}|_{U_{i}}\otimes A,d+l_{i})$ which is a deformation of
the complex $(\mathcal{E}^{\cdot}|_{U_{i}},d)$. Note that the condition for
$(\mathcal{E}^{\cdot}|_{U_{i}}\otimes A,d+l_{i})$ to be a complex is the
Maurer-Cartan equation:
$dl_{i}+\frac{1}{2}[l_{i},l_{i}]=0,\quad\mbox{for all }i\in I.$
Also note that, by upper semicontinuity of cohomology, the complex
$(\mathcal{E}^{\cdot}|_{U_{i}}\otimes A,d+l_{i})$ is exact except possibly at
zero level. To glue together the deformed local complexes
$(\mathcal{E}^{\cdot}|_{U_{i}}\otimes A,d+l_{i})$, we need to specify
isomorphisms between the deformed complexes on the double intersections of
open sets of the cover ${\mathcal{U}}$. Since these isomorphisms will have to
be deformations of the identity, they will be of the form
$e^{m_{ij}}:(\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A,d+l_{j})\to(\mathcal{E}^{\cdot}|_{U_{ij}}\otimes A,d+l_{i}),$
with
$m=\\{m_{ij}\\}_{i<j}\in\prod_{i<j}{\mathcal{E}nd}^{0}(\mathcal{E}^{\cdot})(U_{ij})\otimes\mathfrak{m}_{A}$.
The compatibiliy with the differentials, i.e., the commutativity of the
diagrams
$\textstyle{\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{m_{ij}}}$$\scriptstyle{d+l_{j}|_{U_{ij}}}$$\textstyle{\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d+l_{i}|_{U_{ij}}}$$\textstyle{\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{m_{ij}}}$$\textstyle{\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A}$
can be written as
$d+l_{i}|_{U_{ij}}=e^{m_{ij}}(d+l_{j}|_{U_{ij}})e^{-m_{ij}}$, i.e., as
$l_{i}|_{U_{ij}}=e^{m_{ij}}*l_{j}|_{U_{ij}},\quad\mbox{for all }i<j.$
Finally, the above isomorphisms have to satisfy the cocycle condition up to
homotopy. Indeed, in order to obtain a deformation of ${\mathcal{F}}$, we
actually do not want to glue together the complexes
$(\mathcal{E}^{\cdot}|_{U_{i}}\otimes A,d+l_{i})$, but rather their cohomology
sheaves. In other words, we require $e^{m_{jk}}e^{-m_{ik}}e^{m_{ij}}$ to be
homotopic to the identity on triple intersections. Taking logarithm, what we
require is that $m_{jk}\bullet-m_{ik}\bullet m_{ij}$ is homotopy equivalent to
zero, i.e.,
$m_{jk}|_{U_{ijk}}\bullet-m_{ik}|_{U_{ijk}}\bullet
m_{ij}|_{U_{ijk}}=[d+l_{j}|_{U_{ijk}},n_{ijk}],$
for some
$n=\\{n_{ijk}\\}_{ijk}\in\prod_{i<j<k}{\mathcal{E}nd}^{-1}(\mathcal{E}^{\cdot})(U_{ijk})$.
This homotopy cocycle equation is conveniently rewritten as
$m_{jk}|_{U_{ijk}}\bullet-m_{ik}|_{U_{ijk}}\bullet
m_{ij}|_{U_{ijk}}=d_{\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})}n_{ijk}+[l_{j}|_{U_{ijk}},n_{ijk}].$
Next, let explain how the data introduced above are concretely linked with
deformations of the coherent sheaf $\mathcal{F}$ over $A$. As the homotopy
cocycle equation is satisfied, the local $A$-flat sheaves of
$\mathcal{O}_{X}|_{U_{i}}\otimes A$-modules
${\mathcal{F}}_{A,U_{i}}:={\mathcal{H}}^{*}(\mathcal{E}^{\cdot}|_{U_{i}}\otimes
A,d+l_{i})$ glue together to give a global coherent sheaf ${\mathcal{F}}_{A}$
which is a deformation of ${\mathcal{F}}$. On the other hand, every
deformation ${\mathcal{F}}_{A}$ of the sheaf $\mathcal{F}$ can be obtained in
this way. Indeed, the resolution $(\mathcal{E}^{\cdot},d)$ locally extends to
projective resolutions $(\mathcal{E}^{\cdot}|_{U_{i}}\otimes A,d+l_{i})$ of
${\mathcal{F}}_{A}|_{U_{i}}$; these deformed local resolutions are linked each
other on double intersections by isomorphisms of complexes lifting the
identity of ${\mathcal{F}}_{A}$ and the compositions of these isomorphisms on
triple intersections are homotopy to the identity, since they lift the
identity of ${\mathcal{F}}_{A}$ and liftings are unique up to homotopy.
Let now ${\mathcal{F}}_{A}$ and ${\mathcal{F}^{\prime}}_{A}$ be isomorphic
deformations of the sheaf ${\mathcal{F}}$, associated with deformation data
$(l,m)$ and $(l^{\prime},m^{\prime})$, respectively. The restriction to every
open set $U_{i}$ of the isomorphism between ${\mathcal{F}}_{A}$ and
${\mathcal{F}^{\prime}}_{A}$ lifts to local isomorphisms between the
correspondent deformed complexes. Since these isomorphisms specialize to
identities of $(\mathcal{E}^{\cdot}|_{U_{i}},d)$, they are of the form
$e^{a_{i}}:(\mathcal{E}^{\cdot}|_{U_{i}}\otimes
A,d+l_{i})\to(\mathcal{E}^{\cdot}|_{U_{i}}\otimes A,d+l^{\prime}_{i})$, where
$a=\\{a_{i}\\}_{i}\in\prod_{i}{\mathcal{E}nd}^{0}(\mathcal{E}^{\cdot})(U_{i})\otimes\mathfrak{m}_{A}$.
As above, compatibility with the differentials translates into the equations
$e^{a_{i}}*l_{i}=l^{\prime}_{i},\quad\mbox{for all }i\in I.$
Finally, since the local isomorphisms $e^{a_{i}}$ lift a global isomorphism in
cohomology, the diagrams
$\textstyle{(\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A,d+l_{j}|_{U_{ij}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{m_{ij}}}$$\scriptstyle{e^{a_{j}}|_{U_{ij}}}$$\textstyle{(\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A,d+l_{i}|_{U_{ij}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{a_{i}}|_{U_{ij}}}$$\textstyle{(\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A,d+l^{\prime}_{j}|_{U_{ij}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e^{m^{\prime}_{ij}}}$$\textstyle{(\mathcal{E}^{\cdot}|_{U_{ij}}\otimes
A,d+l^{\prime}_{i}|_{U_{ij}}),}$
expressing compatibility with the gluing morphisms, commute in cohomology.
Moreover, since the compositions
$e^{-m_{ij}}e^{-a_{i}}e^{m^{\prime}_{ij}}e^{a_{j}}$ lift the identity of
${\mathcal{F}}_{A}$ on double intersections and liftings are unique up to
homotopy, these compositions are homotopy to identity and, reasoning as above,
we find
$-m_{ij}\bullet-a_{i}|_{U_{ij}}\bullet m^{\prime}_{ij}\bullet
a_{j}|_{U_{ij}}=d_{\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})}b_{ij}+[l_{j}|_{U_{ij}},b_{ij}],$
for some
$b=\\{b_{ij}\\}_{i<j}\in\prod_{i<j}{\mathcal{E}nd}^{-1}(\mathcal{E}^{\cdot})(U_{ij})\otimes\mathfrak{m}_{A}$.
Viceversa, if for the deformation data $(l,m)$ and $(l^{\prime},m^{\prime})$
there exist
$a=\\{a_{i}\\}_{i}\in\prod_{i}{\mathcal{E}nd}^{0}(\mathcal{E}^{\cdot})(U_{i})\otimes\mathfrak{m}_{A}$
and
$b=\\{b_{ij}\\}_{i<j}\in\prod_{i<j}{\mathcal{E}nd}^{-1}(\mathcal{E}^{\cdot})(U_{ij})\otimes\mathfrak{m}_{A}$
that satisfy equations above, the local isomorphisms $e^{a_{i}}$ glue together
in cohomology to give a global isomorphism of the correspondent deformed
sheaves ${\mathcal{F}}_{A}$ and ${\mathcal{F}^{\prime}}_{A}$.
Summing up, we have shown that in the Kodaira-Spencer approach, infinitesimal
deformations of the coherent sheaf ${\mathcal{F}}$ are controlled by the sheaf
of DGLAs ${\mathcal{E}nd}^{*}(\mathcal{E}^{\cdot})$, via the equations above.
At the end of Section 7, we will apply techniques of semicosimplicial DGLAs
developed in this paper to recover the classical well known fact that the
functor of infinitesimal deformations of $\mathcal{F}$ has
$\operatorname{Ext}^{1}({\mathcal{F}},{\mathcal{F}})$ as tangent space and its
obstructions are contained in
$\operatorname{Ext}^{2}({\mathcal{F}},{\mathcal{F}})$.
###### Remark 1.1.
The above description of the functor of infinitesimal deformations of
${\mathcal{F}}$ is actually independent of the resolution chosen. Indeed, the
DGLAs of the endomorphisms of any two locally free resolutions of
${\mathcal{F}}$ are quasi-isomorphic (see,e.g., [22, Lemma 4.4]).
###### Remark 1.2.
If the sheaf $\mathcal{F}$ is locally free, then we can take its trivial
resolution $0\to\mathcal{F}\to\mathcal{F}\to 0$; thus, we recover the well
known fact that the infinitesimal deformations of $\mathcal{F}$ are controlled
by the sheaf $\mathcal{E}nd(\mathcal{F})$ of the endomorphism of $\mathcal{F}$
, via the Čech functor $H^{1}(X,\mathcal{E}nd(\mathcal{F}))$.
###### Remark 1.3.
Note that the results of this section actually hold under the hypotesis that
${\mathcal{F}}$ admists a global syzygy. This hypothesis is always satisfied,
but in the general case the resolution is less obvious. Indeed, following
Illusie [12, Section 1.5], for any sheaf $\mathcal{F}$ of
${\mathcal{O}}_{X}$-modules on a topological space $X$, one can construct the
_standard free resolution_ of $\mathcal{F}$:
$\ldots\longrightarrow{\mathcal{R}}({\mathcal{F}})^{2}\stackrel{{\scriptstyle
D^{2}}}{{\longrightarrow}}{\mathcal{R}}({\mathcal{F}})^{1}\stackrel{{\scriptstyle
D^{1}}}{{\longrightarrow}}{\mathcal{R}}({\mathcal{F}})^{0}\longrightarrow\mathcal{F}\longrightarrow
0.$
Its terms are defined by recurrence: ${\mathcal{R}}({\mathcal{F}})^{0}$ is the
free sheaf of $\mathcal{O}_{X}$-modules associated with the presheaf
$U\mapsto\mathcal{O}_{X}(U)^{{\mathcal{F}}(U)}$, given on every open set
$U\subset X$ by the free $\mathcal{O}_{X}(U)$-module generated by
${\mathcal{F}}(U)$; ${\mathcal{R}}({\mathcal{F}})^{j}$ is the free sheaf of
$\mathcal{O}_{X}$-modules associated with the presheaf
$U\mapsto\mathcal{O}_{X}(U)^{{\mathcal{R}}({\mathcal{F}})^{j-1}(U)}$, given on
every open set $U\subset X$ by the free $\mathcal{O}_{X}(U)$-module generated
by ${\mathcal{R}}({\mathcal{F}})^{j-1}(U)$.
To define morphisms $D^{j}$, let’s write explicitly elements in
${\mathcal{R}}({\mathcal{F}})^{j}(U)$. An element in
${\mathcal{R}}({\mathcal{F}})^{0}(U)$ is of the form $a^{i_{0}}\odot
f_{i_{0}}$, where $a^{i_{0}}\in\mathcal{O}_{X}(U)$,
$f_{i_{0}}\in\mathcal{F}(U)$, and we used the $\odot$ to denote the action of
${\mathcal{O}}_{X}(U)$ on the free $\mathcal{O}_{X}(U)$-module generated by
$\mathcal{F}(U)$, in order to distinguish it from the action of
${\mathcal{O}}_{X}(U)$ on the $\mathcal{O}_{X}(U)$-module $\mathcal{F}(U)$.
Recursively, an element in ${\mathcal{R}}({\mathcal{F}})^{j}(U)$ is of the
form
$a^{i_{j}}\odot a_{i_{j}}^{i_{j-1}}\odot\cdots\odot a^{i_{0}}_{i_{1}}\odot
f_{i_{0}}$
where $a_{i_{k}}^{i_{k-1}}\in{\mathcal{O}}_{X}(U)$,
$f_{i_{0}}\in\mathcal{F}(U)$. The differential of the resolution is defined as
$D^{j}=\sum_{k=0}^{j}(-1)^{i}d^{j}_{k}$, where
$d^{j}_{k}:{\mathcal{R}}({\mathcal{F}})^{j}\longrightarrow{\mathcal{R}}({\mathcal{F}})^{j-1}$
is defined by
$a^{i_{j}}\odot\cdots\odot a_{i_{k+1}}^{i_{k}}\odot
a_{i_{k}}^{i_{k-1}}\odot\cdots\odot a^{i_{0}}_{i_{1}}\odot f_{i_{0}}\mapsto
a^{i_{j}}\odot\cdots\odot
a_{i_{k+1}}^{i_{k}}a_{i_{k}}^{i_{k-1}}\odot\cdots\odot a^{i_{0}}_{i_{1}}\odot
f_{i_{0}}$
The relevant fact is that the sequence of free sheaves of
$\mathcal{O}_{X}$-modules
$({\mathcal{R}}({\mathcal{F}})^{\cdot},D^{\cdot})\to\mathcal{F}$ is a
resolution of $\mathcal{F}$ [12, Theorem 1.5.3]. This construction can be done
for every sheaf $\mathcal{F}$ of $\mathcal{O}_{X}$-modules on a topological
space $X$; Illusie obtains it as an example of the even more general
construction of the standard simplicial resolution of a pair of adjont
functors [12, Section 1.5].
## 2\. Semicosimplicial DGLAs and the functor $H^{1}_{\rm
sc}(\exp{\mathfrak{g}}^{\Delta})$
A _semicosimplicial differential graded Lie algebra_ is a covariant functor
$\mathbf{\Delta}_{\operatorname{mon}}\to\mathbf{DGLA}$, from the category
$\mathbf{\Delta}_{\operatorname{mon}}$, whose objects are finite ordinal sets
and whose morphisms are order-preserving injective maps between them, to the
category of DGLAs. Equivalently, a semicosimplicial DGLA
${\mathfrak{g}}^{\Delta}$ is a diagram
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.90001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathfrak{g}}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.90001pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.90001pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathfrak{g}}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.70003pt\raise 4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.70003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.70003pt\raise-4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.70003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathfrak{g}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50005pt\raise 6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50005pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50005pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50005pt\raise-6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
106.50005pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
where each ${\mathfrak{g}}_{i}$ is a DGLA, and for each $i>0$ there are $i+1$
morphisms of DGLAs
$\partial_{k,i}\colon{\mathfrak{g}}_{i-1}\to{\mathfrak{g}}_{i},\qquad
k=0,\dots,i,$
such that $\partial_{k+1,i+1}\partial_{l,i}=\partial_{l,i+1}\partial_{k,i}$,
for any $k\geq l$.
A classical example is the following: given a sheaf ${\mathcal{L}}$ of DGLAs
on a topological space $X$, and an open cover ${\mathcal{U}}$ of $X$, one has
the Čech cosimplicial DGLA ${\mathcal{L}}({\mathcal{U}})$,
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.62444pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-20.62444pt\raise
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0.0pt\hbox{$\textstyle{{\prod_{i}\mathcal{L}(U_{i})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
44.62444pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
44.62444pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
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44.62444pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{{\prod_{i<j}\mathcal{L}(U_{ij})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.4822pt\raise 4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.4822pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.4822pt\raise-4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
119.4822pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\prod_{i<j<k}\mathcal{L}(U_{ijk})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
204.87886pt\raise 6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
204.87886pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
204.87886pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
204.87886pt\raise-6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
204.87886pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
where the morphisms $\partial_{k,i}$ are the restriction maps.
###### Definition 2.1.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA. The functor
$Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta}):\mathbf{Art}_{\mathbb{K}}\to\mathbf{Set}$
is defined, for all $A\in\mathbf{Art}_{\mathbb{K}}$, by
$Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)=\left\\{(l,m)\in(\mathfrak{g}_{0}^{1}\oplus\mathfrak{g}^{0}_{1})\otimes\mathfrak{m}_{A}\left|\begin{array}[]{l}dl+\frac{1}{2}[l,l]=0,\\\
\partial_{1,1}l=e^{m}*\partial_{0,1}l,\\\
{\partial_{0,2}m}\bullet{-\partial_{1,2}m}\bullet{\partial_{2,2}m}=dn+[\partial_{2,2}\partial_{0,1}l,n]\\\
\qquad\qquad\qquad\qquad\text{for some
$n\in{\mathfrak{g}}_{2}^{-1}\otimes{\mathfrak{m}}_{A}$}\end{array}\right.\right\\}.$
###### Remark 2.2.
In DGLA theory, given a DGLA $L$ and a Maurer-Cartan element $x$ in
$\operatorname{MC}_{L}(A)$, the set
${\rm Stab}(x)=\\{dh+[x,h]\mid h\in L^{-1}\otimes{\mathfrak{m}}_{A}\\}$
is called the _irrelevant stabilizer_ of $x$. Note that ${\rm
Stab}(x)\subseteq{\rm stab}(x)$, where ${\rm stab}(x)=\\{a\in
L^{0}\otimes\mathfrak{m}_{A}\mid e^{a}*x=x\\}$ is the stabilizer of $x$ under
the gauge action of $L^{0}\otimes\mathfrak{m}_{A}$ on
$\operatorname{MC}_{L}(A)$. Also note that, for any $a\in
L^{0}\otimes\mathfrak{m}_{A}$, $e^{a}e^{{\rm Stab}(x)}e^{-a}=e^{{\rm
Stab}(y)}$, with $y=e^{a}*x.$
We now introduce an equivalence relation on the set $Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)$ as follows: we say that two elements
$(l_{0},m_{0})$ and $(l_{1},m_{1})\in Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)$ are equivalent under the relation $\sim$ if
and only if there exist elements
$a\in\mathfrak{g}^{0}_{0}\otimes\mathfrak{m}_{A}$ and
$b\in{\mathfrak{g}}_{1}^{-1}\otimes{\mathfrak{m}}_{A}$ such that
$\begin{cases}e^{a}*l_{0}=l_{1}\\\ -m_{0}\bullet-\partial_{1,1}a\bullet
m_{1}\bullet\partial_{0,1}a=db+[\partial_{0,1}l_{0},b].\end{cases}$
###### Remark 2.3.
The relation $\sim$ is actually an equivalence relation on $Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)$. First note that the set $Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)$ is closed under $\sim$. Indeed, let
$(l_{0},m_{0})$ and
$(l_{1},m_{1})\in(\mathfrak{g}_{0}^{1}\oplus\mathfrak{g}_{1}^{0})\otimes\mathfrak{m}_{A}$
be equivalent under $\sim$ via elements
$a\in\mathfrak{g}^{0}_{0}\otimes\mathfrak{m}_{A}$ and
$b\in{\mathfrak{g}}_{1}^{-1}\otimes{\mathfrak{m}}_{A}$, and suppose that
$(l_{0},m_{0})\in Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(A)$. Then
$l_{1}=e^{a}*l_{0}$ satisfies the Maurer Cartan equation and
$e^{m_{1}}*\partial_{0,1}l_{1}=e^{\partial_{1,1}a\bullet
m_{0}\bullet(db+[\partial_{0,1}l_{0},b])\bullet-\partial_{0,1}a}e^{\partial_{0,1}a}*\partial_{0,1}l_{0}=e^{\partial_{1,1}a}*\partial_{1,1}l_{0}=\partial_{1,1}l_{1}.$
Moreover, an easy calculation, using relations between maps $\partial_{j,k}$
and Remark 2.2, shows that
${\partial_{0,2}m_{1}}\bullet{-\partial_{1,2}m_{1}}\bullet{\partial_{2,2}m_{1}}$
is an element of the irrelevant stabilizer of
$\partial_{2,2}\partial_{0,1}l_{1}$.
Secondly $\sim$ is an equivalent relation. Reflexivity is trivial; for
simmetry, let $(l_{0},m_{0})$ and $(l_{1},m_{1})$ be equivalent via elements
$\ a\in\mathfrak{g}^{0}_{0}\otimes\mathfrak{m}_{A}$ and
$b\in{\mathfrak{g}}_{1}^{-1}\otimes{\mathfrak{m}}_{A}$, then
$e^{-a}*l_{1}=l_{0}$ and $-m_{1}\bullet\partial_{1,1}(a)\bullet
m_{0}\bullet-\partial_{0,1}(a)=\partial_{0,1}(a)\bullet-(db+[\partial_{0,1}l_{0},b])\bullet-\partial_{0,1}(a)$
is an element of the irrelevant stabilizer of $\partial_{0,1}l_{1}$, by Remark
2.2. Next, let $(l_{0},m_{0})\sim(l_{1},m_{1})$ via
$a\in\mathfrak{g}^{0}_{0}\otimes\mathfrak{m}_{A}$ and
$b\in{\mathfrak{g}}_{1}^{-1}\otimes{\mathfrak{m}}_{A}$, and
$(l_{1},m_{1})\sim(l_{2},m_{2})$ via
$\alpha\in\mathfrak{g}^{0}_{0}\otimes\mathfrak{m}_{A}$ and
$\beta\in{\mathfrak{g}}_{1}^{-1}\otimes{\mathfrak{m}}_{A}$; then,
$e^{\alpha\bullet a}*l_{0}=l_{2}$ and
$-m_{0}\bullet\partial_{1,1}(-(b\bullet a))\bullet
m_{2}\bullet\partial_{0,1}(b\bullet
a)=-m_{0}\bullet-\partial_{1,1}(a)\bullet-\partial_{1,1}(b)\bullet
m_{2}\bullet\partial_{0,1}(b)\bullet\partial_{0,1}(a)=$
$-m_{0}\bullet-\partial_{1,1}(a)\bullet
m_{1}\bullet(db+[\partial_{0,1}l_{0},b])\bullet\partial_{0,1}(a),$
by Remark 2.2, it is an element of the irrelevant stabilizer of
$\partial_{0,1}l_{0}$, therefore $\sim$ is transitive.
###### Definition 2.4.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA, the functor
$H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta}):\mathbf{Art}_{\mathbb{K}}\to\mathbf{Set}$
is defined, for all $A\in\mathbf{Art}_{\mathbb{K}}$, by
$H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(A)=\frac{Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)}{\sim}.$
###### Remark 2.5.
Note that, if $\mathfrak{g}^{\Delta}$ is a semicosimplicial Lie algebra, i.e.,
if all the DGLAs ${\mathfrak{g}}_{i}$ are concentrated in degree zero, then
the functor $H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$ reduces to the one
defined in [7].
###### Lemma 2.6.
The projection $\pi:Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})\longrightarrow
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$ is a smooth morphism of functors.
###### Proof.
Let $\beta:B\longrightarrow A$ be a surjection in $\mathbf{Art}_{\mathbb{K}}$,
we prove that the map
$Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(B)\longrightarrow H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(B)\times_{H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)}Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A),$
induced by
$\textstyle{Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\pi}$$\textstyle{Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A),}$
is surjective. Let $([(l,m)],(l_{0},m_{0}))\in H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(B)\times_{H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)}Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)$, then $(\beta l,\beta m)$ and
$(l_{0},m_{0})$ are gauge equivalent in $Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)$, i.e., there exist
$a\in\mathfrak{g}^{0}_{0}\otimes\mathfrak{m}_{A}$ such that $e^{a}*\beta
l=l_{0}$ and $-\beta m\bullet-\partial_{1,1}a\bullet
m_{0}\bullet\partial_{0,1}a=db+[\partial_{0,1}\beta l,b]$, for some
$b\in{\mathfrak{g}}_{1}^{-1}\otimes\mathfrak{m}_{A}$. Let
$\tilde{a}\in\mathfrak{g}^{0}_{0}\otimes m_{B}$ and
$\tilde{b}\in{\mathfrak{g}}_{1}^{-1}\otimes\mathfrak{m}_{B}$ be liftings of
$a$ and $b$, respectively. The element
$(e^{\tilde{a}}*l,\partial_{1,1}\tilde{a}\bullet
m\bullet(d\tilde{b}+[\partial_{0,1}l,\tilde{b}])\bullet-\partial_{0,1}\tilde{a})\in
Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(B)$ is a pre-image of
$([(l,m)],(l_{0},m_{0}))$. ∎
Next, let $\mathcal{L}$ be a sheaf of DGLAs on a topological space $X$ and
$\mathcal{U}=\\{U_{i}\\}_{i\in I}$ an open cover. Considering the Čech
cosimplicial DGLA ${\mathcal{L}}({\mathcal{U}})$, we can define the functor
$H^{1}_{\rm sc}(\exp{\mathcal{L}}(\mathcal{U}))$. This functor depends on the
cover $\mathcal{U}$, but as shown in the following Lemma, the limit over open
covers is a well defined functor:
$H_{\rm Ho}^{1}(X;\exp\mathcal{L})=\lim_{\begin{subarray}{c}\longrightarrow\\\
\mathcal{U}\end{subarray}}H^{1}_{\rm
sc}(\exp\mathcal{L}(\mathcal{U})):\mathbf{Art}\to\mathbf{Set}.$
###### Lemma 2.7.
Let $\mathcal{U}=\\{U_{\alpha}\\}_{\alpha\in I}$ and
$\mathcal{U}^{\prime}=\\{U^{\prime}_{\alpha}\\}_{\alpha\in I^{\prime}}$ be
open covers of $X$ with $\mathcal{U}^{\prime}$ refinement of $\mathcal{U}$ and
let $\phi,\psi:I^{\prime}\to I$ two refinement maps. Then, the induced
morphisms $\rho_{\phi},\rho_{\psi}:H^{1}_{\rm
sc}(\exp\mathcal{L}(\mathcal{U}))\to H^{1}_{\rm
sc}(\exp\mathcal{L}(\mathcal{U}^{\prime}))$ coincide.
###### Proof.
Both $\phi$ and $\psi$ induce, for all $A\in\mathbf{Art}_{\mathbb{K}}$, a
morphism $Z^{1}_{\rm sc}(\exp\mathcal{L}(\mathcal{U}))(A)\to Z^{1}_{\rm
sc}(\exp\mathcal{L}(\mathcal{U}^{\prime}))(A)$, defined sending
$(l_{i},m_{ij})$ to
$\rho_{\phi}(l_{i},m_{ij})=({l_{\phi\alpha}}|_{U^{\prime}_{\alpha}},{m_{\phi\alpha,\phi\beta}}|_{U^{\prime}_{\alpha\beta}})$
and
$\rho_{\psi}(l_{i},m_{ij})=({l_{\psi\alpha}}|_{U^{\prime}_{\alpha}},{m_{\psi\alpha,\psi\beta}}|_{U^{\prime}_{\alpha\beta}})$,
respectively. Therefore, it remains to prove that
$\rho_{\phi}(l_{i},m_{ij})\sim\rho_{\psi}(l_{i},m_{ij})$, for all
$(l_{i},m_{ij})\in Z^{1}_{\rm sc}(\exp\mathcal{L}(\mathcal{U}))(A)$, i.e., for
all $\alpha\in I^{\prime}$, there exists
$a_{\alpha}\in\mathcal{L}^{0}(U^{\prime}_{\alpha})\otimes m_{A}$ such that
$\begin{cases}e^{a_{\alpha}}*{l_{\phi\alpha}}|_{U^{\prime}_{\alpha}}={l_{\psi\alpha}}|_{U^{\prime}_{\alpha}}\\\
-{m_{\phi\alpha,\phi\beta}}|_{U^{\prime}_{\alpha\beta}}\bullet-{a_{\alpha}}|_{U^{\prime}_{\alpha\beta}}\bullet{m_{\psi\alpha,\psi\beta}}|_{U^{\prime}_{\alpha\beta}}\bullet{a_{\beta}}|_{U^{\prime}_{\alpha\beta}}\in{\rm
Stab}(l_{\phi\beta}|_{U^{\prime}_{\alpha\beta}}).\end{cases}$
A simple computation shows that it is enough to choose
${a_{\alpha}}:={m_{\psi\alpha,\phi\alpha}}|_{U^{\prime}_{\alpha}}$, for all
$\alpha$ in $I^{\prime}$. ∎
###### Remark 2.8.
Having introduced the limit $H^{1}_{\rm Ho}(\exp\mathcal{L})$, for a sheaf of
DGLAs $\mathcal{L}$ on a topological space $X$, the results of Section 1 can
be restated as follows: the functor of infinitesimal deformations of a
coherent sheaf ${\mathcal{F}}$ on a projective manifold $X$ is
$\operatorname{Def}_{\mathcal{F}}\cong H^{1}_{\rm
Ho}(X;\exp{\mathcal{E}}nd^{*}(\mathcal{E}^{\cdot})),$
where $\mathcal{E}^{\cdot}$ is a locally free resolution of ${\mathcal{F}}$.
The example of coherent sheaves on projective manifolds together with the DGLA
approach to deformation theory suggests that the functors of Artin rings
$H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$ could actually be isomorphic to
functors $\operatorname{Def}_{L(\mathfrak{g}^{\Delta})}$ for some DGLA
$L(\mathfrak{g}^{\Delta})$ canonically associated with
$\mathfrak{g}^{\Delta}$. We are going to show that, under the cohomological
hypothesis $H^{-1}(\mathfrak{g}_{2})=0$, it is indeed so. More precisely, we
are going to prove that, if $H^{-1}(\mathfrak{g}_{2})=0$, then the functor of
Artin rings $H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$ is isomorphic to the
deformation functor associated with the Thom-Whitney DGLA of the truncation
${\mathfrak{g}}^{\Delta_{[0,2]}}$.
## 3\. The Thom-Whitney DGLA
$\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})$
Let ${\mathfrak{g}}^{\Delta}$ be a semicosimplicial DGLA. The maps
$\partial_{i}=\partial_{0,i}-\partial_{1,i}+\cdots+(-1)^{i}\partial_{i,i}$
endow the vector space $\bigoplus_{i}{\mathfrak{g}}_{i}$ with the structure of
a differential complex. Moreover, being a DGLA, each ${\mathfrak{g}}_{i}$ is
in particular a differential complex
${\mathfrak{g}}_{i}=\bigoplus_{j}{\mathfrak{g}}_{i}^{j};\qquad
d_{i}\colon{\mathfrak{g}}_{i}^{j}\to{\mathfrak{g}}_{i}^{j+1}$
and since the maps $\partial_{k,i}$ are morphisms of DGLAs, the space
${\mathfrak{g}}^{\bullet}_{\bullet}=\bigoplus_{i,j}{\mathfrak{g}}_{i}^{j}$
has a natural bicomplex structure. The associated total complex
$({\rm
Tot}({\mathfrak{g}}^{\Delta}),d_{\operatorname{Tot}})\quad\text{where}\quad{\rm
Tot}({\mathfrak{g}}^{\Delta})=\bigoplus_{i}{\mathfrak{g}}_{i}[-i],\quad
d_{\operatorname{Tot}}=\sum_{i,j}\partial_{i}+(-1)^{j}d_{j}$
has no natural DGLA structure. Yet there is an other bicomplex naturally
associated with a semicosimplicial DGLA, whose total complex is naturally a
DGLA.
For every $n\geq 0$, denote by $\Omega_{n}$ the differential graded
commutative algebra of polynomial differential forms on the standard
$n$-simplex $\Delta^{n}$:
$\Omega_{n}=\frac{{\mathbb{K}}[t_{0},\ldots,t_{n},dt_{0},\ldots,dt_{n}]}{(\sum
t_{i}-1,\sum dt_{i})}.$
Denote by $\delta^{k,n}\colon\Omega_{n}\to\Omega_{n-1}$, $k=0,\ldots,n$, the
face maps; then, one has natural morphisms of bigraded DGLAs
$\delta^{k,n}\colon\Omega_{n}\otimes\mathfrak{g}_{n}\to\Omega_{n-1}\otimes\mathfrak{g}_{n},\qquad\partial_{k,n}\colon\Omega_{n-1}\otimes\mathfrak{g}_{n-1}\to\Omega_{n-1}\otimes\mathfrak{g}_{n},$
for every $0\leq k\leq n$.
The Thom-Whitney bicomplex is defined as
$C^{i,j}_{TW}(\mathfrak{g}^{\Delta})=\\{(x_{n})_{n\in{\mathbb{N}}}\in\bigoplus_{n}\Omega_{n}^{i}\otimes{\mathfrak{g}}_{n}^{j}\mid\delta^{k,n}x_{n}=\partial_{k,n}x_{n-1}\quad\forall\;0\leq
k\leq n\\},$
where $\Omega_{n}^{i}$ denotes the degree $i$ component of $\Omega_{n}$. Its
total complex is a DGLA, called the _Thom-Whitney DGLA_ , and it is denoted by
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta})$; denote by $d_{TW}$ the
differential of the Thom-Whitney DGLA. It is a remarkable fact that the
integration maps
$\int_{\Delta^{n}}\otimes\operatorname{Id}\colon\Omega_{n}\otimes\mathfrak{g}_{n}\to\mathbb{K}[n]\otimes\mathfrak{g}_{n}=\mathfrak{g}_{n}[n]$
give a quasi-isomorphism of differential complexes
$I\colon(\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta}),d_{TW})\to(\operatorname{Tot}({\mathfrak{g}}^{\Delta}),d_{\rm
Tot}).$
Moreover, Dupont has described in [3, 4] an explicit morphism of differential
complexes
$E\colon\operatorname{Tot}({\mathfrak{g}}^{\Delta})\to\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})$
and an explicit homotopy
$h\colon\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})\to\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})[-1]$
such that
$IE={\rm Id}_{{\rm Tot}({\mathfrak{g}}^{\Delta})};\qquad EI-{\rm Id}_{{\rm
Tot}_{TW}({\mathfrak{g}}^{\Delta})}=[h,d_{TW}].$
We also refer to the papers [2, 8, 19] for the explicit description of $E,h$
and for the proof of the above identities. Here, we point out that $E$ and $h$
are defined in terms of integration over standard simplexes and multiplication
with canonical differential forms: in particular, the construction of
$\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})$,
$\operatorname{Tot}({\mathfrak{g}}^{\Delta})$, $I$, $E$ and $h$ is functorial
in the category $\mathbf{DGLA}^{\Delta_{\operatorname{mon}}}$ of
semicosimplicial DGLAs.
Recall that with a DGLA $L$ there is a canonically associated deformation
functor $\operatorname{Def}_{L}$, defined as the solutions of Maurer-Cartan
equation modulo gauge action (or, equivalently, modulo homotopy equivalence).
Moreover, the tangent space to $\operatorname{Def}_{L}$ is $H^{1}(L)$ and
obstructions live in $H^{2}(L)$. Thus, with a semicosimplicial DGLA
$\mathfrak{g}^{\Delta}$ is also associated the deformation functor
$\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta})}$; its
tangent space is
$T\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta})}\cong
H^{1}({\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta}))\cong
H^{1}({\operatorname{Tot}}(\mathfrak{g}^{\Delta}))$
and obstructions live in
$H^{2}({\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta}))\cong
H^{2}({\operatorname{Tot}}(\mathfrak{g}^{\Delta})).$
Let $\mathbf{\Delta}^{+}_{\operatorname{mon}}$ the category obtained by adding
the empty set $\emptyset$ to the category
$\mathbf{\Delta}_{\operatorname{mon}}$. An _augmented semicosimplicial
differential graded Lie algebra_ is a covariant functor
$\mathbf{\Delta}^{+}_{\operatorname{mon}}\to\mathbf{DGLA}$, from the category
$\mathbf{\Delta}^{+}_{\operatorname{mon}}$ to the category of DGLAs.
Equivalently, an augmented semicosimplicial DGLA ${\mathfrak{g}}^{\Delta^{+}}$
is a diagram
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.83333pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-7.83333pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathfrak{g}}_{-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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31.83333pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathfrak{g}}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
69.63335pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
69.63335pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
69.63335pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathfrak{g}}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
107.43336pt\raise 4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
107.43336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
107.43336pt\raise-4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
107.43336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{{\mathfrak{g}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
145.23338pt\raise 6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
145.23338pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
145.23338pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
145.23338pt\raise-6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
145.23338pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
where the truncated diagram ${\mathfrak{g}}^{\Delta}$
$\textstyle{{{\mathfrak{g}}_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathfrak{g}}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathfrak{g}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
is a semicosimplicial DGLA and
$\partial_{0,0}\colon\mathfrak{g}_{-1}\to\mathfrak{g}_{0}$
is a DGLA morphism such that
$\partial_{0,1}\partial_{0,0}=\partial_{1,1}\partial_{0,0}$.
###### Remark 3.1.
There is a morphism of DGLAs
$\displaystyle{\mathfrak{g}}_{-1}$
$\displaystyle\to\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})$
$\displaystyle x$ $\displaystyle\mapsto(\partial_{0,0}x,\
\partial_{1,1}\partial_{0,0}x,\
\partial_{2,2}\partial_{1,1}\partial_{0,0}x,\dots);$
the image of $x$ is an element in
$\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})$ because of equations
$\partial_{1,1}\partial_{0,0}=\partial_{0,1}\partial_{0,0}$ and
$\partial_{k+1,i+1}\partial_{l,i}=\partial_{l,i+1}\partial_{k,i}$, for any
$k\geq l$. This morphism is obtained as the composition of the natural
inclusion
${\mathfrak{g}}_{-1}\hookrightarrow\operatorname{Tot}({\mathfrak{g}}^{\Delta})$
with the morphism
$E:\operatorname{Tot}({\mathfrak{g}}^{\Delta})\to\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})$.
The existence of the DGLA morphism
${\mathfrak{g}}_{-1}\to\operatorname{Tot}_{TW}({\mathfrak{g}}^{\Delta})$ is
not surprising; indeed, it is induced by the natural morphism
$\lim{\mathfrak{g}}^{\Delta}\to\mathop{\rm holim}{\mathfrak{g}}^{\Delta}$.
We use augmentation to link the Thom-Whitney DGLA of the Čech semicosimplicial
DGLA of a sheaf of DGLAs with the DGLA of global sections of an acyclic
resolution of the sheaf. This result is a translation of Theorem 7.2 in [7] in
terms of the Thom-Whitney DGLA.
We recall that if ${\mathcal{L}}$ is a sheaf of DGLAs on a topological space
$X$ and ${\mathcal{U}}$ is an open cover of $X$, the associated Čech
semicosimplicial differential graded Lie algebra is:
$\mathcal{L}(\mathcal{U}):\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.62444pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-20.62444pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\prod_{i}\mathcal{L}(U_{i})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
44.62444pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
44.62444pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
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44.62444pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\prod_{i<j}\mathcal{L}(U_{ij})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.4822pt\raise 4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.4822pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.4822pt\raise-4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
119.4822pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\prod_{i<j<k}\mathcal{L}(U_{ijk})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
204.87886pt\raise 6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
204.87886pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
204.87886pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
204.87886pt\raise-6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
204.87886pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
A morphism $\varphi\colon{\mathcal{L}}\to{\mathcal{A}}$ of sheaves of DGLAs is
a quasi-isomorphism if it is a quasi-isomorphism of sheaves of differential
complexes, i.e., if it induces linear isomorphisms between the cohomology
sheaves,
${\mathcal{H}}^{*}(\varphi)\colon{\mathcal{H}}^{*}({\mathcal{L}})\xrightarrow{\sim}{\mathcal{H}}^{*}({\mathcal{A}}).$
Moreover, if ${\mathcal{A}}^{k}$ is an acyclic sheaf for any $k$, then
$\varphi\colon{\mathcal{L}}\to{\mathcal{A}}$ is called an acyclic resolution
of ${\mathcal{L}}$.
###### Theorem 3.2.
Let $X$ be a paracompact Hausdorff topological space, $\mathcal{L}$ a sheaf of
differential graded Lie algebras on $X$, and
$\varphi\colon{\mathcal{L}}\to{\mathcal{A}}$ an acyclic resolution. Also let
$A={\mathcal{A}}(X)$ be the DGLA of global sections of ${\mathcal{A}}$. Then,
if $\mathcal{U}$ is an open cover of $X$ which is acyclic with respect to both
${\mathcal{L}}$ and ${\mathcal{A}}$, the DGLA
${\operatorname{Tot}_{TW}}({\mathcal{L}}(\mathcal{U}))$ is naturally quasi-
isomorphic to the DGLA $A$.
###### Proof.
The natural inclusion $A\to\mathcal{A}(\mathcal{U})$ gives an augmented
semicosimplicial DGLA, and so it induces a morphism of DGLAs
$A\to\operatorname{Tot}_{TW}(\mathcal{A}(\mathcal{U}))$, that is the
composition of the natural inclusion
$A\to\operatorname{Tot}({\mathcal{A}}({\mathcal{U}}))$ with the quasi-
isomorphism
$E:\operatorname{Tot}({\mathcal{A}}({\mathcal{U}}))\to\operatorname{Tot}_{TW}(\mathcal{A}(\mathcal{U}))$,
by Remark 3.1. Since the sheaves ${\mathcal{A}}^{k}$ are acyclic and
${\mathcal{U}}$-acyclic, and $A^{k}=H^{0}(X;\mathcal{A}^{k})$, the inclusion
$A\to\operatorname{Tot}(\mathcal{A}(\mathcal{U}))$ is a quasiisomorphism.
Indeed, we have a natural identification
$H^{*}({\operatorname{Tot}}(\mathcal{A}(\mathcal{U})))={\mathbb{H}}^{*}(X;{\mathcal{A}})$,
and the spectral sequence abutting to the hypercohomology of $X$ with
coefficients in ${\mathcal{A}}$ degenerates at $E_{2}$, giving
${\mathbb{H}}^{k}(X;{\mathcal{A}})=\bigoplus_{p+q=k}E_{2}^{p,q}=E_{2}^{k,0}=H^{k}(A).$
Then, $A\to\operatorname{Tot}_{TW}(\mathcal{A}(\mathcal{U}))$ is a quasi-
isomorphism of DGLAs.
The morphism $\varphi\colon{\mathcal{L}}\to{\mathcal{A}}$ induces a morphism
of semicosimplicial DGLAs
$\varphi\colon\mathcal{L}(\mathcal{U})\to\mathcal{A}(\mathcal{U}),$
and a morphism of complexes
$\varphi\colon\operatorname{Tot}_{TW}(\mathcal{L}(\mathcal{U}))\to\operatorname{Tot}_{TW}(\mathcal{A}(\mathcal{U})).$
Since the open cover ${\mathcal{U}}$ is ${\mathcal{L}}$-acyclic, the
cohomology of the total complex $\operatorname{Tot}(\mathcal{L}(\mathcal{U}))$
is naturally identified with the hypercohomology of $X$ with coefficients in
${\mathcal{L}}$,
$H^{*}(\operatorname{Tot}(\mathcal{L}(\mathcal{U})))\cong{\mathbb{H}}^{*}(X;{\mathcal{L}}),$
and the induced linear map
$H^{*}(\varphi)\colon H^{*}(\operatorname{Tot}(\mathcal{L}(\mathcal{U})))\to
H^{*}(\operatorname{Tot}(\mathcal{A}(\mathcal{U})))$
is identified with the linear map
${\mathbb{H}}^{*}(\varphi)\colon{\mathbb{H}}^{*}(X;{\mathcal{L}})\rightarrow{\mathbb{H}}^{*}(X;{\mathcal{A}})$
induced in hypercohomology. Since, by hypothesis, $\varphi$ is a quasi-
isomorphism of sheaves of DGLAs, the induced map in hypercohomology is an
isomorphism, and so the morphism
$\varphi\colon\operatorname{Tot}(\mathcal{L}(\mathcal{U}))\to\operatorname{Tot}(\mathcal{A}(\mathcal{U}))$
is a quasi-isomorphism of complexes.
Via the composition with quasi-isomorphisms $E$ and $I$ between the total
complex and the Thom-Whitney total complex of a semicosimplicial DGLA, the
morphism $\varphi$ induces a quasi-isomorphism of DGLAs
$\operatorname{Tot}_{TW}(\mathcal{L}(\mathcal{U}))\to\operatorname{Tot}_{TW}(\mathcal{A}(\mathcal{U})).$
Therefore, we have the chain of quasi-isomorphisms of DGLAs
$\operatorname{Tot}_{TW}(\mathcal{L}(\mathcal{U}))\xrightarrow{\sim}\operatorname{Tot}_{TW}(\mathcal{A}(\mathcal{U}))\xleftarrow{\sim}A.$
∎
## 4\. Truncations
Let
$\mathfrak{g}^{\Delta}:\ \ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.90001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.90001pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.90001pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.70003pt\raise 4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.70003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.70003pt\raise-4.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.70003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50005pt\raise 6.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50005pt\raise 2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50005pt\raise-2.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.50005pt\raise-6.0pt\hbox{\hbox{\kern 0.0pt\raise
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be a semicosimplicial DGLA. Let $m_{1}\in\mathbb{N}$ and
$m_{2}\in\mathbb{N}\cup\\{\infty\\}$ with $m_{1}\leq m_{2}$, we denote by
$\mathfrak{g}^{\Delta_{[m_{1},m_{2}]}}$ the _truncated between levels $m_{1}$
and $m_{2}$_ semicosimplicial DGLA defined by
$(\mathfrak{g}^{\Delta_{[m_{1},m_{2}]}})_{n}=\begin{cases}\mathfrak{g}_{n}&\text{for
}m_{1}\leq n\leq m_{2}\\\ 0&\text{otherwise},\end{cases}$
with the obvious maps $\partial_{k,i}^{[m_{1},m_{2}]}=\partial_{k,i}$, for
$m_{1}<i\leq m_{2}$, and $\partial_{k,i}^{[m_{1},m_{2}]}=0$, otherwise. For
any positive integers $m_{1},m_{2},r_{1},r_{2}$, such that $r_{i}\leq m_{i}$,
the map
$\operatorname{Id}_{[m_{1},r_{2}]}\colon\mathfrak{g}^{\Delta_{[m_{1},m_{2}]}}\to\mathfrak{g}^{\Delta_{[r_{1},r_{2}]}}$
given by
$\operatorname{Id}_{[m_{1},r_{2}]}\biggr{|}_{(\mathfrak{g}^{\Delta_{[m_{1},m_{2}]}})_{n}}=\begin{cases}\operatorname{Id}_{\mathfrak{g}_{n}}&\text{if
}m_{1}\leq n\leq r_{2}\\\ 0&\text{otherwise}.\end{cases}$
is a morphism of semicosimplicial DGLAs; it induces the natural morphism of
complexes
$\phi:\operatorname{Tot}(\mathfrak{g}^{\Delta_{[m_{1},m_{2}]}})\to\operatorname{Tot}(\mathfrak{g}^{\Delta_{[r_{1},r_{2}]}})$
and the natural morphism of DGLAs
$\psi:\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[m_{1},m_{2}]}})\to\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[r_{1},r_{2}]}})$.
Note that we have an homotopy commutative diagram of complexes
$\textstyle{\operatorname{Tot}(\mathfrak{g}^{\Delta_{[m_{1},m_{2}]}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\scriptstyle{\phi}$$\textstyle{\operatorname{Tot}(\mathfrak{g}^{\Delta_{[r_{1},r_{2}]}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\textstyle{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[m_{1},m_{2}]}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{I}$$\textstyle{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[r_{1},r_{2}]}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{I}$
###### Proposition 4.1.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA such that
$H^{j}(\mathfrak{g}_{i})=0$, for all $i\geq 0$ and $j<0$. Then, the morphism
$\operatorname{Id}_{[0,2]}$ induces a natural isomorphism of functors:
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta})}\xrightarrow{\sim}\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}.$
###### Proof.
It is a well known fact (see, e.g., [17] for a proof), that a DGLA morphism
which is surjective on $H^{0}$, bijective on $H^{1}$ and injective on $H^{2}$
induces an isomorphism between the associated deformation functors. Since the
above homotopy commutative diagram identifies $H^{*}(\psi)$ with
$H^{*}(\phi)$, it is enough to prove that $H^{0}(\phi)$ is surjective,
$H^{1}(\phi)$ is bijective and $H^{2}(\phi)$ is injective. This is easily
checked by looking at the spectral sequences associated with double complexes
of ${\mathfrak{g}}^{\Delta}$ and ${\mathfrak{g}}^{\Delta_{[0,2]}}$. ∎
###### Remark 4.2.
Observe that, for any semicosimplicial DGLA $\mathfrak{g}^{\Delta}$, we have
$Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})=Z_{\rm
sc}^{1}(\exp\mathfrak{g}^{\Delta_{[0,2]}})$ and $H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})=H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,2]}})$. Moreover, the inclusion $Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})\hookrightarrow Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})$ induces an injective map $H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})\hookrightarrow H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})$.
###### Remark 4.3.
For later use, we point out that, if $\mathfrak{g}^{\Delta}$ is a
semicosimplicial DGLA with $H^{-1}(\mathfrak{g}_{2})=0$, then
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[2,2]}})}$
is trivial. Indeed,
$H^{1}(\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[2,2]}}))=H^{-1}(\mathfrak{g}_{2})=0$.
###### Remark 4.4.
Note that, by the definition of $H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$ it
follows that, if $H^{-1}(\mathfrak{g}_{2})=0$, then
$TH^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})=H^{1}({\operatorname{Tot}}(\mathfrak{g}^{\Delta_{[0,2]}})).$
Hence, the two functors of Artin rings $H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})$ and
$\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}$
have naturally isomorphic tangent spaces when $H^{-1}(\mathfrak{g}_{2})=0$. We
will show in Section 7 that in this case these two functors are actually
isomorphic.
## 5\. A lemma on Maurer-Cartan elements
We will now give an explicit description of the solutions of Maurer-Cartan
equation for the DGLAs
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})$ and
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})$. Our main tool will
be the following general result [6, Proposition 7.2]:
###### Lemma 5.1.
Let $(L,d,[~{},~{}])$ be a differential graded Lie algebra such that:
1. (1)
$L=M\oplus C\oplus D$ as graded vector spaces.
2. (2)
$M$ is a differential graded subalgebra of $L$.
3. (3)
$d\colon C\to D[1]$ is an isomorphism of graded vector spaces.
Then, for every $A\in\mathbf{Art}_{\mathbb{K}}$ there exists a bijection
$\alpha\colon\operatorname{MC}_{M}(A)\times(C^{0}\otimes\mathfrak{m}_{A}){\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}}\operatorname{MC}_{L}(A),\qquad(x,c)\mapsto
e^{c}\ast x.$
As almost immediate corollaries we obtain:
###### Proposition 5.2.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA. Then, for every
$A\in\mathbf{Art}_{\mathbb{K}}$, the solutions of the Maurer-Cartan equation
for the Thom-Whitney DGLA
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})\otimes\mathfrak{m}_{A}$
are of the form $(x,e^{p(t)}*\partial_{0,1}x)$, where
$x\in\operatorname{MC}_{\mathfrak{g}_{0}}(A)$ and
$p(t)\in(\mathfrak{g}_{1}^{0}[t]\cdot t)\otimes\mathfrak{m}_{A}$. The elements
$x,p$ are uniquely determined, and they satisfy
(2) $\partial_{1,1}x=e^{p(1)}*\partial_{0,1}x.$
###### Proof.
Notice that $\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})$ is a sub-
DGLA of $\mathfrak{g}_{0}\oplus\Omega_{1}\otimes\mathfrak{g}_{1}$. Then, Lemma
5.1 with the decomposition of $\Omega_{1}\otimes\mathfrak{g}_{1}$ given by
$M=\mathfrak{g}_{1},\qquad C=\mathfrak{g}_{1}[t]\cdot t,\qquad D=dC$
tells us that every solution of the Maurer-Cartan equation for
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})\otimes\mathfrak{m}_{A}$
is of the form specified above. ∎
###### Proposition 5.3.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA. Then, for every
$A\in\mathbf{Art}_{\mathbb{K}}$, the solutions of the Maurer-Cartan equation
for the Thom-Whitney DGLA
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})\otimes\mathfrak{m}_{A}$
are of the form
$(x,e^{p(t)}*\partial_{0,1}x,e^{q(s_{0},s_{1})+r(s_{0},s_{1},ds_{0},ds_{1})}*\partial_{0,2}\partial_{0,1}x),$
where $x\in\operatorname{MC}_{\mathfrak{g}_{0}}(A)$,
$p(t)\in(\mathfrak{g}_{1}^{0}[t]\cdot t)\otimes\mathfrak{m}_{A}$,
$q(s_{0},s_{1})\in(\mathfrak{g}_{2}^{0}[s_{0},s_{1}]\cdot
s_{0}+\mathfrak{g}_{2}^{0}[s_{0},s_{1}]\cdot s_{1})\otimes\mathfrak{m}_{A}$
and $r(s_{0},s_{1},ds_{0},ds_{1})\in(\mathfrak{g}_{2}^{-1}[s_{0},s_{1}]\cdot
s_{0}ds_{1})\otimes\mathfrak{m}_{A}$. The elements $x,p,q,r$ are uniquely
determined, and they satisfy
(3) $\begin{cases}\partial_{1,1}x=e^{p(1)}*\partial_{0,1}x,\\\
\partial_{0,2}p(t)=q(0,t),\\\ \partial_{1,2}p(t)=q(t,0),\\\
e^{(-\partial_{2,2}p(t))\bullet(q(t,1-t)+r(t,1-t,dt))\bullet(-q(0,1))}*\partial_{2,2}\partial_{0,1}x=\partial_{2,2}\partial_{0,1}x.\end{cases}$
###### Proof.
Since $\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})$ is a sub-DGLA
of
$\mathfrak{g}_{0}\oplus\Omega_{1}\otimes\mathfrak{g}_{1}\oplus\Omega_{2}\otimes\mathfrak{g}_{2}$,
applying Lemma 5.1 with the decomposition of
$\Omega_{2}\otimes\mathfrak{g}_{2}$ given by
$M=\mathfrak{g}_{2},\qquad C=\mathfrak{g}_{2}[s_{0},s_{1}]\cdot
s_{0}+\mathfrak{g}_{2}[s_{0},s_{1}]\cdot
s_{1}+\mathfrak{g}_{2}[s_{0},s_{1}]\cdot s_{0}ds_{1},\qquad D=dC$
we obtain that every solution of the Maurer-Cartan equation for
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})\otimes\mathfrak{m}_{A}$
is of the form
$(x,e^{p(t)}*y,e^{q(s_{0},s_{1})+r(s_{0},s_{1},ds_{0},ds_{1})}*z),$
with the face conditions
$y=\partial_{0,1}x;\qquad z=\partial_{0,2}\partial_{0,1}x.$
The first relations in (3) are a direct consequence of face conditions and
uniqueness. The last one is obtained as follows. The last face condition is
$\partial_{2,2}(e^{p(t)}*\partial_{0,1}x)=e^{q(t,1-t)+r(t,1-t,dt)}*\partial_{0,2}\partial_{0,1}x;$
using the other face conditions and relations between maps $\partial_{k,i}$,
we obtain that
$\partial_{2,2}\partial_{0,1}x=\partial_{0,2}\partial_{1,1}x=\partial_{0,2}(e^{p(1)}*\partial_{0,1}x)=e^{q(0,1)}*\partial_{0,2}\partial_{0,1}x.$
Then, the above equation becomes
$e^{\partial_{2,2}p(t)}*\partial_{2,2}\partial_{0,1}x=e^{(q(t,1-t)+r(t,1-t,dt))\bullet(-q(0,1))}*\partial_{2,2}\partial_{0,1}x.$
∎
## 6\. The isomorphism $H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})\cong\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}$
###### Proposition 6.1.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA. The map
$\Phi_{[0,1]}:\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)\to(\mathfrak{g}_{0}^{1}\oplus\mathfrak{g}_{1}^{0})\otimes\mathfrak{m}_{A},$
given by
$(x,e^{p(t)}*\partial_{0,1}x)\mapsto(x,p(1)),$
induces a natural transformation of functors of Artin rings
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}\to
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}}).$
###### Proof.
Clearly, if
$(x,e^{p(t)}*\partial_{0,1}x)\in\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)$,
then $(x,p(1))\in Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})$. We have
to show that if two elements
$\eta_{0}=(x_{0},e^{p_{0}(t)}*\partial_{0,1}x_{0})$ and
$\eta_{1}=(x_{1},e^{p_{1}(t)}*\partial_{0,1}x_{1})$ in
$\operatorname{MC}_{{\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)$
are homotopy equivalent, then
$\Phi_{[0,1]}(\eta_{0})\sim\Phi_{[0,1]}(\eta_{1})$ in $Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})$. Let $z(\xi,d\xi)$ be an homotopy
between $\eta_{0}$ and $\eta_{1}$. Therefore, $z(\xi,d\xi)$ is a Maurer-Cartan
element for
${\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})[\xi,d\xi]$ and so,
reasoning as in the proof of Proposition 5.2, we find
$z(\xi,d\xi)=(e^{T(\xi)}*u,e^{U(t,dt;\xi)}*v),$
with $T(0)=U(t,dt;0)=0$. Since $z(0)=\eta_{0}$, we get
$z(\xi,d\xi)=(e^{T(\xi)}*x_{0},e^{U(t,dt;\xi)}*e^{p_{0}(t)}*\partial_{0,1}x_{0}).$
The face conditions for $z(\xi,d\xi)$ and uniqueness imply
$U(0;\xi)=\partial_{0,1}T(\xi)\quad\mbox{ }\quad
U(1;\xi)=\partial_{1,1}T(\xi).$
Moreover, $z(1)=\eta_{1}$, and so
$(e^{T(1)}*x_{0},e^{U(t,dt;1)}*e^{p_{0}(t)}*\partial_{0,1}x_{0})=(x_{1},e^{p_{1}(t)}*\partial_{0,1}x_{1});$
by uniqueness again, we have
$e^{T(1)}*x_{0}=x_{1}.$
Furthermore,
$e^{U(t,dt;1)}*e^{p_{0}(t)}*\partial_{0,1}x_{0}=e^{p_{1}(t)}*\partial_{0,1}x_{1}$,
so, using the face conditions for $\eta_{0}$ and $\eta_{1}$, we obtain
$\partial_{0,1}x_{0}=e^{-p_{0}(t)\bullet-U(t,dt;1)\bullet
p_{1}(t)\bullet\partial_{0,1}T(1)}*\partial_{0,1}x_{0}$
Next, we recall [11, Lemma 6.15] that if $L$ is a DGLA, $x(t,dt)$ is a Maurer-
Cartan element for $L[t,dt]$ and $\mu(t,dt)\in L[t,dt]^{0}$ is such that
$e^{\mu(t,dt)}*x(t,dt)=x(t,dt)$, then $\mu(1)$ is an element of the irrelevant
stabilizer of $x(1)$. Therefore, in our case we get
$-p_{0}(1)\bullet-\partial_{1,1}T(1)\bullet
p_{1}(1)\bullet\partial_{0,1}T(1)\in{\rm Stab}(\partial_{0,1}x_{0}).$
∎
###### Proposition 6.2.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA. The map
$\Phi_{[0,1]}:\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}\to
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})$
is an isomorphism of functors of Artin rings. In particular, $H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})$ is a deformation functor.
###### Proof.
Let $\Psi_{[0,1]}:Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})(A)\to\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})\otimes{\mathfrak{m}}_{A}$
be the map given by $(l,m)\mapsto(l,e^{tm}*\partial_{0,1}l)$; it is immediate
to check that $\Phi_{[0,1]}$ actually takes its values in
$\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)$.
Moreover, $\Psi_{[0,1]}$ induces a map
$H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})(A)\to\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A),$
which is the inverse of $\Phi_{[0,1]}$. Indeed, if
$(l_{0},m_{0})\sim(l_{1},m_{1})$ in $Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})(A)$, then there exist elements
$a\in\mathfrak{g}^{0}_{0}\otimes\mathfrak{m}_{A}$ and
$b\in{\mathfrak{g}}_{1}^{-1}\otimes{\mathfrak{m}}_{A}$ such that
$\begin{cases}e^{a}*l_{0}=l_{1}\\\ -m_{0}\bullet-\partial_{1,1}a\bullet
m_{1}\bullet\partial_{0,1}a=db+[\partial_{0,1}l_{0},b].\end{cases}$
Therefore, the images $(l_{0},e^{tm_{0}}*\partial_{0,1}l_{0})$ and
$(l_{1},e^{tm_{1}}*\partial_{0,1}l_{1})$ are homotopic via the element
$z(\xi,d\xi)=(e^{\xi a}*l_{0},e^{t\bigl{(}\partial_{1,1}(\xi a)\bullet
m_{0}\bullet(d(\xi b)+[\partial_{0,1}l_{0},\xi b])\bullet-\partial_{0,1}(\xi
a)\bigr{)}\bullet\partial_{0,1}(\xi a)}*\partial_{0,1}l_{0}).$
The composition $\Phi_{[0,1]}\circ\Psi_{[0,1]}\colon Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})(A)\to Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})(A)$ is clearly the identity, whereas
the composition
$\Psi_{[0,1]}\circ\Phi_{[0,1]}:\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)\to\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)$
is homotopic to the identity. Indeed, $(x,e^{p(t)}*\partial_{0,1}x)$ and
$(x,e^{tp(1)}*\partial_{0,1}x)$ are homotopic in
$\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)$
via the element $z(\xi,d\xi)=(x,e^{\xi tp(1)+(1-\xi)p(t)}*\partial_{0,1}x)$. ∎
###### Remark 6.3.
A particular case of Proposition 6.2, with an almost identical proof, has been
considered by one of the authors in [11]. Namely, given three DGLAs $L,M$ and
$N$ and two DGLA morphisms $h\colon L\to M$ and $g\colon N\to M$, one can
consider the semicosimplicial DGLA
$\textstyle{L\oplus
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,g)}$$\scriptstyle{(h,0)}$$\textstyle{{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
to reobtain [11, Theorem 6.17].
## 7\. Proof of the main theorem
In this section, we prove the existence of a natural isomorphism of functors
of Artin rings $H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})\cong\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}$,
for any semicosimplicial DGLA $\mathfrak{g}^{\Delta}$ such that
$H^{-1}(\mathfrak{g}_{2})=0$. As an immediate consequence we obtain a natural
isomorphism of deformation functors $H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})\cong\operatorname{Def}_{{\operatorname{Tot}_{TW}}(\mathfrak{g}^{\Delta})}$,
for any semicosimplicial DGLA $\mathfrak{g}^{\Delta}$, such that
$H^{j}(\mathfrak{g}_{i})=0$ for $i\geq 0$ and $j<0$.
The proof is considerably harder than in the case
$\mathfrak{g}^{\Delta_{[0,1]}}$ considered in the previous section. Indeed, we
are still able to define a map
$\Phi\colon\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}\to
Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$ inducing a natural transformation
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}\to
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$, but we will not be able to
explicitly define an homotopy inverse to $\Phi$, so we will have to directly
check that the map
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}\to
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$ is an isomorphism.
###### Proposition 7.1.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA. The map
$\Phi:\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)\to(\mathfrak{g}_{0}^{1}\oplus\mathfrak{g}_{1}^{0})\otimes\mathfrak{m}_{A},$
given by
$(x,e^{p(t)}*\partial_{0,1}x,e^{q(s_{0},s_{1})+r(s_{0},s_{1},ds_{1},ds_{1})}*\partial_{0,2}\partial_{0,1}x)\mapsto(x,p(1)),$
induces a natural transformation of functors of Artin rings
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}\to
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta}).$
###### Proof.
First we check that $\Phi$ takes its values in $Z^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)$. The only nontrivial point consists in
showing that
$-\partial_{2,2}p(1)\bullet\partial_{1,2}p(1)\bullet-\partial_{0,2}p(1)$ is an
element of the irrelevant stabilizer of $\partial_{2,2}\partial_{0,1}x$. This
follows by the face condition
$e^{(-\partial_{2,2}p(t))\bullet(q(t,1-t)+r(t,1-t,dt))\bullet(-q(0,1))}*\partial_{2,2}\partial_{0,1}x=\partial_{2,2}\partial_{0,1}x,$
applying [11, Lemma 6.15] once again. Next, we notice that the equivalence
relation $\sim$ on $Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(A)$ only
involves the DGLAs ${\mathfrak{g}}_{0}$ and ${\mathfrak{g}}_{1}$; hence, we
can conclude verbatim following the proof of Proposition 6.1.
∎
###### Proposition 7.2.
The map
$\Phi:\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)\to
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(A)$ is surjective.
###### Proof.
Let $(l,m)\in Z^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(A)$ and
$n\in\mathfrak{g}_{2}^{-1}\otimes\mathfrak{m}_{A}$, such that
$\partial_{0,2}m\bullet-\partial_{1,2}m\bullet\partial_{2,2}m=dn+\frac{1}{2}[\partial_{2,2}\partial_{0,1}l,n]$.
Consider the element
$w(t)=d(tn)+\frac{1}{2}[\partial_{2,2}\partial_{0,1}l,tn]$ in the irrelevant
stabilizer of $\partial_{2,2}\partial_{0,1}l$ and
$R(s_{0},s_{1})=s_{0}s_{1}\frac{s_{0}\partial_{2,2}m\bullet-w(s_{0})\bullet
s_{0}\partial_{0,2}m\bullet-s_{0}\partial_{1,2}m}{s_{0}(1-s_{0})}\bullet
s_{0}\partial_{1,2}m\bullet s_{1}\partial_{0,2}m.$
Then,
$(l,e^{tm}*\partial_{0,1}l,e^{R(s_{0},s_{1})}*\partial_{0,2}\partial_{0,1}l)$
is an element in
$\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)$
in the fiber of $\Phi$ over $(l,m)$. Indeed, clearly it satisfies the Maurer-
Cartan equation in
$\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}\otimes\Omega_{1}\oplus\mathfrak{g}_{2}\otimes\Omega_{2}$;
the first face conditions follow easly noticing that $R(0,t)=t\partial_{0,2}m$
and $R(t,0)=t\partial_{1,2}m$; for the last one, we have:
$e^{R(t,1-t)}*\partial_{0,2}\partial_{0,1}l=e^{t\partial_{2,2}m\bullet-w(t)\bullet\partial_{0,2}m}*\partial_{0,2}\partial_{0,1}l=$
$=e^{t\partial_{2,2}m\bullet-w(t)}*\partial_{0,2}\partial_{1,1}l=e^{t\partial_{2,2}m\bullet-w(t)}*\partial_{2,2}\partial_{0,1}l=e^{t\partial_{2,2}m}*\partial_{2,2}\partial_{0,1}l.$
∎
We will prove that the map
$\Phi:\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)\to
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(A)$ is injective, under the
hypothesis $H^{-1}(\mathfrak{g}_{2})=0$. For this we need two remarks.
###### Remark 7.3.
Let $(L,d,[\ ,\ ])$ be a DGLA, $A\in\bf{Art}_{\mathbb{K}}$ and $x\in
L^{1}\otimes\mathfrak{m}_{A}$. The linear endomorphism $d_{x}=d+[x,\ ]$ of
$L\otimes\mathfrak{m}_{A}$ is a differential if and only if
$x\in\operatorname{MC}_{L}(A)$, and in this case
$(L\otimes\mathfrak{m}_{A},d_{x},[\ ,\ ])$ is a DGLA. So, we can define the
set of the Maurer-Cartan elements $\operatorname{MC}_{L}^{x}(A)$ and the gauge
action of $(L^{0}\otimes{\mathfrak{m}}_{A},d_{x},[\ ,\ ])$ on it. We denote by
$\operatorname{Def}_{L}^{x}(A)$ the quotient of $\operatorname{MC}_{L}^{x}(A)$
with respect to the gauge action. The affine map
$\begin{array}[]{rll}L\otimes\mathfrak{m}_{A}&\to&L\otimes\mathfrak{m}_{A}\\\
v&\mapsto&v-x.\end{array}$
induces an isomorphism
$\operatorname{Def}_{L}(A)\cong\operatorname{Def}_{L}^{x}(A)$ with obvious
inverse $v\mapsto v+x$.
Next, let $M\subseteq L$ be a sub-DGLA and let $x\in\operatorname{MC}_{L}(A)$.
If $M\otimes\mathfrak{m}_{A}$ is closed under the differential $d_{x}$, then
we can consider the set of Maurer-Cartan elements
$\operatorname{MC}_{M}^{x}(A)$, and its quotient
$\operatorname{Def}_{M}^{x}(A)$. The tangent space to
$\operatorname{Def}_{M}^{x}(A)$ is $H^{1}(M\otimes{\mathfrak{m}}_{A},d_{x})$;
so, by upper semicontinuity of cohomology, $H^{1}(M,d)=0$ implies that
$\operatorname{Def}^{x}_{M}(A)$ is trivial, for all
$x\in\operatorname{MC}_{L}(A)$ such that
$d_{x}(M\otimes\mathfrak{m}_{A})\subseteq M\otimes\mathfrak{m}_{A}$.
###### Remark 7.4.
For any semicosimplicial DGLA $\mathfrak{g}^{\Delta}$, the truncation morphism
$\operatorname{Tot}^{0}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})\to\operatorname{Tot}^{0}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})$
is surjective, i.e., for any
$(a_{0},a_{1})\in\operatorname{Tot}^{0}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})$
there exist $a_{2}\in(\mathfrak{g}_{2}\otimes\Omega_{2})^{0}$ such that
$(a_{0},a_{1},a_{2})\in\operatorname{Tot}^{0}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})$.
To see this, write $a_{1}(t,dt)=a_{1}^{0}(t)+a_{1}^{-1}(t)dt$; then a possible
choice for $a_{2}$ is
$a_{2}(s_{0},s_{1},ds_{0},ds_{1})=a^{0}_{2}(s_{0},s_{1})+a^{-1}_{2,0}(s_{0},s_{1})ds_{0}+a^{-1}_{2,1}(s_{0},s_{1})ds_{1}+a^{-2}_{2}(s_{0},s_{1})ds_{0}ds_{1},$
with
$\displaystyle a^{0}_{2}(s_{0},s_{1})$
$\displaystyle=\partial_{1,2}a^{0}_{1}(s_{0})+\partial_{0,2}a^{0}_{1}(s_{1})-\partial_{1,2}a^{0}_{1}(0)$
$\displaystyle\qquad\qquad+s_{1}\frac{\partial_{2,2}a^{0}_{1}(s_{0})-\partial_{1,2}a^{0}_{1}(s_{0})-\partial_{0,2}a^{0}_{1}(1-s_{0})+\partial_{0,2}a^{0}_{1}(0)}{1-s_{0}};$
$\displaystyle a^{-1}_{2,0}(s_{0},s_{1})$
$\displaystyle=\partial_{1,2}a_{1}^{-1}(s_{0})+\frac{s_{1}}{1-s_{0}}\biggl{(}\partial_{2,2}a_{1}^{-1}(s_{0})-\partial_{1,2}a_{1}^{-1}(s_{0})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\partial_{0,2}a_{1}^{-1}(1-s_{0})-s_{0}\partial_{0,2}a_{1}^{-1}(0)\biggr{)};$
$\displaystyle a^{-1}_{2,1}(s_{0},s_{1})$
$\displaystyle=\partial_{0,2}a_{1}^{-1}(s_{1})ds_{1}-s_{0}\partial_{0,2}a_{1}^{-1}(0);$
$\displaystyle a^{-2}_{2}(s_{0},s_{1})$ $\displaystyle=0.$
It is an easy computation to verify that the element $(a_{0},a_{1},a_{2})$
actually satisfies the face conditions.
###### Proposition 7.5.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA, such that
$H^{-1}(\mathfrak{g}_{2})=0$. The map
$\Phi:\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)\to
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})(A)$ is injective.
###### Proof.
Consider the commutative diagram
$\textstyle{\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}_{[0,1]}}$$\scriptstyle{\Phi}$$\textstyle{\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\Phi_{[0,1]}}$$\textstyle{H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta})(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{H^{1}_{\rm
sc}(\exp\mathfrak{g}^{\Delta_{[0,1]}})(A);}$
since the map $\Phi_{[0,1]}$ is an isomorphism by Proposition 6.2, it is
sufficient to prove that $\operatorname{Id}_{[0,1]}$ is injective. Let
$(x_{0},x_{1},x_{2})$ and $(x^{\prime}_{0},x^{\prime}_{1},x^{\prime}_{2})$ be
two Maurer-Cartan elements for
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})$, such that
$(x_{0},x_{1})$ and $(x^{\prime}_{0},x^{\prime}_{1})$ are gauge equivalent
elements in
$\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})}(A)$.
Let
$(a_{0},a_{1})\in\operatorname{Tot}^{0}_{TW}(\mathfrak{g}^{\Delta_{[0,1]}})\otimes\mathfrak{m}_{A}$
be an element realizing the gauge equivalence between
$(x^{\prime}_{0},x^{\prime}_{1})$ and $(x_{0},x_{1})$, and let
$(a_{0},a_{1},a_{2})$ be a lift of $(a_{0},a_{1})$ in
$\operatorname{Tot}^{0}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})\otimes\mathfrak{m}_{A}$
(see Remark 7.4). Then $(x^{\prime}_{0},x^{\prime}_{1},x^{\prime}_{2})$ is
gauge equivalent via $(a_{0},a_{1},a_{2})$ to the Maurer-Cartan element
$(x_{0},x_{1},e^{a_{2}}*x^{\prime}_{2})$ and we are left to prove that
$(x_{0},x_{1},e^{a_{2}}*x^{\prime}_{2})$ is gauge equivalent to
$(x_{0},x_{1},x_{2})$.
To see this, consider the DGLA
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})\otimes\mathfrak{m}_{A}$
and modify its differential with the Maurer-Cartan element
$(x_{0},x_{1},x_{2})$, as in Remark 7.3. Translation by $(x_{0},x_{1},x_{2})$
gives an isomorphism
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)\cong\operatorname{Def}^{(x_{0},x_{1},x_{2})}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A);$
hence $(x_{0},x_{1},x_{2})$ and $(x_{0},x_{1},e^{a_{2}}*x^{\prime}_{2})$ will
be gauge equivalent in
$\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)$
if and only if $(0,0,0)$ and $(0,0,e^{a_{2}}*x^{\prime}_{2}-x_{2})$ are gauge-
equivalent in
$\operatorname{MC}^{(x_{0},x_{1},x_{2})}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)$.
Next, observe that the sub-DGLA
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[2,2]}})\otimes\mathfrak{m}_{A}$
of
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})\otimes\mathfrak{m}_{A}$
is closed under the modified differential $d_{(x_{0},x_{1},x_{2})}$, so we can
consider the deformation functor
$\operatorname{Def}^{(x_{0},x_{1},x_{2})}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[2,2]}})}(A)$.
Since
$H^{1}(\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[2,2]}}),d_{TW})=H^{1}(\operatorname{Tot}(\mathfrak{g}^{\Delta_{[2,2]}}),d_{\rm
Tot})=H^{-1}(\mathfrak{g}_{2})=0$, this deformation functor is trivial (see
Remark 7.3). Therefore $(0,0,e^{a_{2}}*x^{\prime}_{2}-x_{2})$ is gauge
equivalent to $(0,0,0)$ as an element of
$\operatorname{MC}^{(x_{0},x_{1},x_{2})}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[2,2]}})}(A)$,
and so, a fortiori, as an element of
$\operatorname{MC}^{(x_{0},x_{1},x_{2})}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}(A)$.
∎
Summing up, and recalling Proposition 4.1, we have proved:
###### Theorem 7.6.
Let $\mathfrak{g}^{\Delta}$ be a semicosimplicial DGLA, and let
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta})$ and
$\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})$ be the Thom-Whitney
DGLAs associated with $\mathfrak{g}^{\Delta}$ and
$\mathfrak{g}^{\Delta_{[0,2]}}$, respectively. Assume that
$H^{-1}(\mathfrak{g}_{2})=0$; then, there is a natural isomorphism of funtors
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta_{[0,2]}})}\cong
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$. If moreover
$H^{j}(\mathfrak{g}_{i})=0$ for all $i\geq 0$ and $j<0$, then there is a
natural isomorphism of funtors
$\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak{g}^{\Delta})}\cong
H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$. In particular, in this case, the
tangent space to $H^{1}_{\rm sc}(\exp\mathfrak{g}^{\Delta})$ is $H^{1}({\rm
Tot}(\mathfrak{g}^{\Delta}))$ and obstructions are contained in $H^{2}({\rm
Tot}(\mathfrak{g}^{\Delta}))$.
###### Theorem 7.7.
Let $X$ be a paracompact Hausdorff topological space, and let $\mathcal{L}$ be
a sheaf of differential graded Lie algebras on $X$, such that the DGLAs
$\mathcal{L}(U_{i_{0}\ldots i_{k}})$ has no negative cohomology. Then, every
refinement ${\mathcal{V}}\geq{\mathcal{U}}$ of open covers of $X$ induces a
natural morphism of deformation functors
$\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{U}}))}\to\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{V}}))}$.
In particular, the direct limit
$\operatorname{Def}_{[{\mathcal{L}}]}=\lim_{\stackrel{{\scriptstyle\longrightarrow}}{{\mathcal{U}}}}\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{U}}))}$
is well defined and there is natural isomorphism of functors of Artin rings
$H_{\rm
Ho}^{1}(X;\exp{\mathcal{L}})\cong\operatorname{Def}_{[{\mathcal{L}}]}.$
Moreover, if acyclic open covers for $\mathcal{L}$ are cofinal in the directed
family of all open covers of $X$, then
$H_{\rm Ho}^{1}(X;\exp\mathcal{L})\cong H_{\rm
sc}^{1}(\exp\mathcal{L}(\mathcal{U}))\qquad\text{and}\qquad\operatorname{Def}_{[{\mathcal{L}}]}\cong\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{U}}))},$
for every ${\mathcal{L}}$-acyclic open cover ${\mathcal{U}}$ of $X$.
###### Proof.
Let ${\mathcal{V}}\geq{\mathcal{U}}$ be a refinement of open covers of $X$,
and let $\tau$ be a refinement function, it induces a natural morphism of
semicosimplicial Lie algebras
${\mathcal{L}}({\mathcal{U}})\to{\mathcal{L}}({\mathcal{V}})$ and so a
commutative diagram of natural transformations
$\textstyle{\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{U}}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}_{\rm
sc}(\exp{\mathcal{L}}({\mathcal{U}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{V}}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}_{\rm
sc}(\exp{\mathcal{L}}({\mathcal{V}})).}$
Horizontal arrows are isomorphisms by Theorem 7.6, and the right vertical
arrow is independent of the refinement function $\tau$, as observed in Lemma
2.7. Hence, also the left morphism is independent of $\tau$, then the direct
limit
$\operatorname{Def}_{[{\mathcal{L}}]}=\lim_{\stackrel{{\scriptstyle\longrightarrow}}{{\mathcal{U}}}}\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{U}}))}$
is well defined and we have a natural isomorphism
$\operatorname{Def}_{[{\mathcal{L}}]}\cong H_{\rm
Ho}^{1}(X;\exp{\mathcal{L}})$. Assume now that acyclic open covers for
${\mathcal{L}}$ are cofinal in the family of all open covers of $X$. Then, for
any refinement ${\mathcal{V}}\geq{\mathcal{U}}$ of acyclic open covers, the
DGLAs-morphism
$\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{U}}))\to\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{V}}))$
is a quasi-isomorphism by Leray’s theorem. Therefore, we have a commutative
diagram of natural transformations
$\textstyle{\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{U}}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\scriptstyle{\sim}$$\textstyle{H^{1}_{\rm
sc}(\exp{\mathcal{L}}({\mathcal{U}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{V}}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{H^{1}_{\rm
sc}(\exp{\mathcal{L}}({\mathcal{V}})),}$
where also the right vertical arrow is forced to be an isomorphism. Taking the
direct limit over ${\mathcal{L}}$-acyclic covers, we obtain that, if
${\mathcal{U}}$ is an ${\mathcal{L}}$-acyclic open cover of $X$, then $H_{\rm
Ho}^{1}(X;\exp\mathcal{L})\cong H^{1}_{\rm sc}(\exp\mathcal{L}(\mathcal{U}))$
and
$\operatorname{Def}_{[{\mathcal{L}}]}\cong\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal{L}}({\mathcal{U}}))}$.
∎
## 8\. Conclusions and further developements
We can now sum up our results to obtain a DGLA description of infinitesimal
deformations of a coherent sheaf. In Section 1, we analised infinitesimal
deformations of a coherent sheaf $\mathcal{F}$ of $\mathcal{O}_{X}$-modules on
a ringed space $(X,\mathcal{O}_{X})$. If $\mathcal{E}^{\cdot}\to\mathcal{F}\to
0$ is a locally free resolution of $\mathcal{F}$ on $X$, we showed how
infinitesimal deformations of ${\mathcal{F}}$ can be expressed in terms of the
sheaf of DGLAs $\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})$. More precisely, in
Section 2, we showed that the functor of infinitesimal deformations of
${\mathcal{F}}$ is isomorphic to $H^{1}_{\rm
Ho}(X;\exp{\mathcal{E}nd}^{*}(\mathcal{E}^{\cdot}))$.
Since negative Ext-groups between coherent sheaves are always trivial, all
terms in the semicosimplicial DGLA
$\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})(\mathcal{U})$ have zero negative
cohomology. Therefore, Theorem 7.6 applies and we obtain that the functor of
infinitesimal deformations of ${\mathcal{F}}$ is isomorphic to
$\operatorname{Def}_{[\mathcal{E}nd^{*}(\mathcal{E}^{\cdot})]}$; in
particular, we recover the well known fact that the tangent space to
$\operatorname{Def}_{\mathcal{F}}$ is
$\operatorname{Ext}^{1}({\mathcal{F}},{\mathcal{F}})$ and that its
obstructions are contained in
$\operatorname{Ext}^{2}({\mathcal{F}},{\mathcal{F}})$.
Moreover, if $X$ is a smooth complex variety, then the DGLA controlling
infinitesimal deformations of $\mathcal{F}$ turns out to be not at all
mysterious. Indeed, let
${\mathcal{E}nd}^{*}(\mathcal{E}^{\cdot})\to\mathcal{A}^{0,*}_{X}(\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))$
be the Dolbeault resolution of ${\mathcal{E}nd}^{*}(\mathcal{E}^{\cdot})$.
Since this resolution is fine, by Theorem 3.2 the functor of infinitesimal
deformations of ${\mathcal{F}}$ is isomorphic to the deformation functor
associated with the DGLA $A^{0,*}_{X}(\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))$
of global sections of
$\mathcal{A}^{0,*}_{X}(\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))$. We can also
give an explicit description of this isomorphism of deformation functors.
Indeed, a natural isomorphism
$\operatorname{Def}_{A^{0,*}_{X}(\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))}(B)\to\operatorname{Def}_{\mathcal{F}}(B),\qquad\mbox{for}\
B\in\bf{Art}_{\mathbb{K}}$
is defined by associating with every Maurer-Cartan element $\xi$ of the DGLA
$A^{0,*}_{X}(\mathcal{E}nd^{*}(\mathcal{E}^{\cdot}))$ the cohomology sheaf of
$({\mathcal{A}}_{X}^{0,*}(\mathcal{E}^{\cdot})\otimes
B,\overline{\partial}+d_{\mathcal{E}^{\cdot}}+\xi)$. Note that, by
semicontinuity, this cohomology sheaf is concentrated in degree zero.
The techniques developed in this paper apply to a wide range of other
geometric examples. More explicitly, we can use them in all cases when local
deformations admit a simple DGLA description in terms of a resolution of the
object to be deformed, for instance, in the case of infinitesimal deformations
of a singular variety. Namely, let $X$ be a singular variety,
${\mathcal{O}}_{X}$ the sheaf of regular function of $X$ and
${\mathcal{R}}^{\cdot}\to{\mathcal{O}}_{X}$ its standard free resolution [12,
Section 1.5]. Then, the deformation functor of infinitesimal deformations of
$X$ is isomorphic to $H_{\rm
Ho}^{1}(X;\exp{{\mathcal{D}}er}^{*}({\mathcal{R}}^{\cdot}))$; see [5] for
details. From this, we also recover the classical result that the tangent
space to deformations of $X$ is
$\operatorname{Ext}^{1}({\mathbb{L}}_{X},{\mathcal{O}}_{X})$, and that
obstructions are contained in
$\operatorname{Ext}^{2}({\mathbb{L}}_{X},{\mathcal{O}}_{X})$, where
${\mathbb{L}}_{X}$ is the cotangent complex of $X$.
## References
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* [3] J. L. Dupont: _Simplicial de Rham cohomology and characteristic classes of flat bundles._ Topology, 15, (1976), 233-245.
* [4] J. L. Dupont: _Curvature and characteristic classes._ Lecture Notes in Mathematics, 640, Springer-Verlag, (1978).
* [5] D. Fiorenza, D. Iacono, E. Martinengo: _Infinitesimal deformations of singular varieties._ (in prepapartion).
* [6] D. Fiorenza, M. Manetti: _$L_{\infty}$ -structures on mapping cones._ Algebra & Number Theory, 1, (2007), 301-330.
* [7] D. Fiorenza, M. Manetti, E. Martinengo: _Semicosimplicial DGLAs in deformation theory_. arxiv:math.AG/08030399.
* [8] E. Getzler: _Lie theory for nilpotent $L_{\infty}$-algebras._ Ann. of Math., 170, (1), (2009), 271-301.
* [9] V. Hinich: _Descent of Deligne groupoids._ Int. Math. Res. Notices, (1997), 5, 223-239.
* [10] A. Hirschowitz, C. Simpson: _Descent pour les n-champs._ arXiv:9807049v3.
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* [12] L. Illusie: _Complexe cotangent et deformations I, II._ Lecture Notes in Mathematics, 239, 283, Springer-Verlag, New York/Berlin, (1971-1972).
* [13] K. Kodaira: _Complex Manifolds and Deformation of Complex Structures._ Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 283, Springer-Verlag, New York/Berlin, (1986).
* [14] K. Kodaira, D. C. Spencer: _On Deformations of Complex Analytic Structures, II._ Ann. of Math., 67 (2), (1958), 403-466.
* [15] M. Kuranishi: _Deformations of compact complex manifolds._ Séminaire de Mathematiques Supérieures, No. 39, (Été 1969), Les Presses de l’Université de Montreal, Montreal, (1971).
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* [17] M. Manetti: _Deformation theory via differential graded Lie algebras._ Seminari di Geometria Algebrica 1998-1999, Scuola Normale Superiore (1999).
* [18] M. Manetti: _Extended deformation functors._ Int. Math. Res. Not., 14, (2002), 719-756.
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* [20] J. P. Pridham: _Deformations via Simplicial Deformation Complexes._ arXiv:math/0311168v6.
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|
arxiv-papers
| 2009-04-08T10:11:51 |
2024-09-04T02:49:01.778954
|
{
"license": "Public Domain",
"authors": "Domenico Fiorenza, Donatella Iacono, Elena Martinengo",
"submitter": "Domenico Fiorenza",
"url": "https://arxiv.org/abs/0904.1301"
}
|
0904.1304
|
# Probing new physics in $B\to J/\Psi~{}\pi^{0}$ decay
Jing-Wu Li1111Email:lijw@xznu.edu.cn, Dong-Sheng
Du2222Email:duds@mail.ihep.ac.cn, Xiang-Yao Wu3333Email:wuxy2066@163.com
1Department of Physics, Xu Zhou Normal University, XuZhou 221116, China,
2Institute of High Energy Physics, P.O. Box 918(4), Beijing 100049,
3Institute of Physics, Jilin Normal University, Siping 136000, China
###### Abstract
We calculate the branching ratio of $B\to J/\Psi~{}\pi^{0}$ with a mixed
formalism that combines the QCD-improved factorization and the perturbative
QCD approaches. The result is consistent with experimental data. The quite
small penguin contribution in $B\to J/\Psi~{}\pi^{0}$ decay can be calculated
with this method. We suggest two methods to extract the weak phase $\beta$.
One is through the dependence of the mixing induced CP asymmetry
$S_{J/\Psi\pi^{0}}$ on the weak phase$\beta$ , the other is from the relation
of the total asymmetry $A_{CP}$ with the weak phase $\beta$. Our result shows
that the deviation $\bigtriangleup S_{J/\psi\pi^{0}}$ of the mixing induced CP
asymmetry from $Sin(-2\beta)$ is of $\mathcal{O}(10^{-3})$ and has much less
uncertainty. The above $\mathcal{O}(10^{-3})$ deviation can provide a good
reference for identifying new physics.
###### pacs:
13.25.Hw, 12.38.Bx
B physics is entering the era of precision measurement, It is not far from
revealing new physics beyond the Standard Model(SM). Many authors have studied
the topics and suggest some windows for looking for new physics(NP)npa1 -npa9
. Because falvour-changing neutral current (FCNC) processes only occur at the
loop-level in the SM , so they are particularly sensitive to NP interactions.
It was pointed out that $B^{0}_{q}-\bar{B}^{0}_{q}$ mixing and decays are good
places for new physics to enter through the exchange of new particles in the
box diagrams, or through new contributions at the tree level bmixing1
-bmixing3 , so $B^{0}_{q}-\bar{B}^{0}_{q}$ system has been studied in many
papers for probing new physicsbmixingatt1 ; bmixingatt5 . $B\to J/\Psi\pi^{0}$
decay is a good mode for looking for new physics and extracting the weak phase
$\beta$ . The direct CP asymmetry $C_{J/\psi\pi^{0}}$ and the deviation
$\bigtriangleup S_{J/\psi\pi^{0}}\equiv S_{J/\psi\pi^{0}}-\sin(-2\beta)$ of
the mixing -induced CP asymmetry from $\sin(-2\beta)$ in this decay arise from
quite small penguin contribution in the SM, so these quantities are sensitive
to new physics effect. Comparing the prediction of CP asymmetry in the SM with
the experimental data, one can find new physics signal. Thus it is essential
to calculate the $\bigtriangleup S_{J/\psi\pi^{0}}$ and $C_{J/\psi\pi^{0}}$ in
$B\to J/\Psi\pi^{0}$ in the SM accurately.
The deviation $\bigtriangleup
S_{J/\psi\pi^{0}}=S_{J/\psi\pi^{0}}-\sin(-2\beta)$ or direct CP asymmetry
$C_{J/\psi\pi^{0}}$ in $B\to J/\Psi~{}\pi^{0}$ decay have been studied in Ref.
deltasciu by fitting to the current experimental data, the result is
$C_{J/\psi\pi^{0}}=0.09\pm 0.19$ which has very large uncertainty. In that
case we can not say anything about new physics effects.
In order to reveal new physics effects, we need both better theoretical
prediction and experimental measurement with less uncertainties. That is the
aim of our present paper.
In what follows, we first evaluate the penguin pollution effect by a method
which have been used to explain many B decays into charmonia
successfullybtocharmn1 ; btocharmn2 . We find the penguin pollution in the
$B\to J/\Psi~{}\pi^{0}$ decay is quite small, the deviation $\bigtriangleup
S_{J/\psi\pi^{0}}=S_{J/\psi\pi^{0}}-\sin(-2\beta)$ in $B\to J/\Psi~{}\pi^{0}$
decay is $\mathcal{O}(10^{-3})$, which means that the measured deviation
$\bigtriangleup S_{J/\psi\pi^{0}}$ at $1\%$ will indicate the presence of new
physics.
The latest experimental data of $\bigtriangleup S_{J/\psi\pi^{0}}$ is
$S_{J/\psi\pi^{0}}=-0.4\pm 0.4$pdg2006 , which has large error, so we are
expecting to have more precise measurement in the near future.
The decay rate of of $B\to J/\Psi~{}\pi^{0}$ can be written as
$\Gamma=\frac{1}{32\pi
m_{B}}G_{F}^{2}(1-r_{2}^{2}+\frac{1}{2}r_{2}^{4}-r_{3}^{2})|{\cal A}|^{2}\;.$
(1)
with $r_{2}=m_{J/\psi}/m_{B}$, $r_{3}=m_{\pi}/m_{B}$.
The amplitude ${\cal A}$ consists of factorizable part and nonfactorizable
part. It can be written as
$\displaystyle{\cal A}$ $\displaystyle=$ $\displaystyle{\cal A}_{NF}+{\cal
A}_{VERT}+{\cal A}_{HS}\;,$ (2)
where ${\cal A}_{NF}$ denote the factorizable contribution in Naive
Factorization Assumption(NF), ${\cal A}_{VERT}$ is the vertex corrections from
Fig. 1.(a)-(d) , ${\cal A}_{HS}$ is the spectator correction from Fig.
1.(e)-(f).
Figure 1: Nonfactorizable contribution to the $B^{0}\to J/\psi~{}\pi^{0}$
decay
The factorizable part ${\cal A}_{NF}$ in Eq. (2) for $B\to J/\Psi\pi^{0}$
decay can not be calculated reliably in the pQCD approach, because its
characteristic scale is around 1 GeV. We parameterize the sum of the
factorizable part ${\cal A}_{NF}$ and the vertex corrections ${\cal A}_{VERT}$
as,
$\displaystyle{\cal A}_{NF}+{\cal
A}_{VERT}=a_{eff}m_{B}^{2}f_{J/{\psi}}F_{1}^{B\to\eta}(m_{J/\psi}^{2})(1-r_{2}^{2})\;,$
(3)
where $f_{J/{\psi}}$ is decay constant of $J/\psi$ meson,
For the $B\to\pi$ transition form factors, we employ the models derived from
the light-cone sum rules formbtopi , which have been parameterized as
$\displaystyle
F_{1}^{B\to\pi}(q^{2})=\frac{r_{1}}{1-q^{2}/m_{1}^{2}}+\frac{r_{2}}{1-q^{2}/m_{fit}^{2}}\;$
(4)
with $r_{1}=0.744$, $r_{2}=-0.486$, $m_{1}=5.32Gev$, $m_{fit}^{2}=40.73Gev$
for $B\to\pi$ transition.
The factorization and vertex correction from Fig. 1.(a)-(d) can be calculated
in the QCDFqcdf . Summing up the factorizable part and vertex correction , we
can get the Wilson coefficient $a_{eff}$,
$\displaystyle a_{eff}$ $\displaystyle=$ $\displaystyle
V_{c}^{\ast}\left[C_{1}+V_{c}^{\ast}\frac{C_{2}}{N_{c}}+\frac{\alpha_{s}}{4\pi}\frac{C_{F}}{N_{c}}C_{2}\left(-18+12\ln\frac{m_{b}}{\mu}+f_{I}\right)\right]$
(5) $\displaystyle-
V_{t}^{\ast}\Big{[}C_{3}+\frac{C_{4}}{N_{c}}+\frac{\alpha_{s}}{4\pi}\frac{C_{F}}{N_{c}}C_{4}\left(-18+12\ln\frac{m_{b}}{\mu}+f_{I}\right)$
$\displaystyle+C_{5}+\frac{C_{6}}{N_{c}}+\frac{\alpha_{s}}{4\pi}\frac{C_{F}}{N_{c}}C_{6}\left(6-12\ln\frac{m_{b}}{\mu}-f_{I}\right)+C_{7}+\frac{C_{8}}{N_{c}}+C_{9}+\frac{C_{1}0}{N_{c}}\Big{]}\;$
with the function,
$\displaystyle f_{I}=\frac{2\sqrt{2N_{c}}}{f_{J/\psi}}\int
dx_{3}\Psi^{L}(x_{2})\left[\frac{3(1-2x_{2})}{1-x_{2}}\ln x_{2}-3\pi
i+3\ln(1-r_{2}^{2})+\frac{2r_{2}^{2}(1-x_{2})}{1-r_{2}^{2}x_{2}}\right]\;,$
(6)
The spectator corrections ${\cal A}_{HS}$ from Fig. 1.(e)-(f), can be
calculated reliably in the pQCD as in Ref. btocharmn1 ; btocharmn2 ,
$\displaystyle{\cal A}_{HS}$ $\displaystyle=$ $\displaystyle V_{c}^{\ast}{\cal
M}_{1}^{(J/\psi\pi)}-V_{t}^{\ast}{\cal M}_{4}^{(J/\psi\pi)}-V_{t}^{\ast}{\cal
M}_{6}^{(J/\psi\pi)}\;,$ (7)
where the amplitudes ${\cal M}_{1,4}^{(J/\psi\pi)}$ and ${\cal
M}_{6}^{(J/\psi\pi)}$ result from the $(V-A)(V-A)$ and $(V-A)(V+A)$ operators
in the effective Hamiltonian, respectively. Their factorization formulas are
given by the pQCD approach. In the calculation of ${\cal
M}_{1,4}^{(J/\psi\eta)}$ and ${\cal M}_{6}^{(J/\psi\eta)}$, because $J/\psi$
is heavy, we reserve the power terms of $r_{2}$ up to
$\mathcal{O}(r^{4}_{2})$, the power terms of $r_{3}$ up to
$\mathcal{O}(r^{2}_{3})$ .
$\displaystyle{\cal M}_{1,4}^{(J/\psi\pi)}$ $\displaystyle=$ $\displaystyle
16\pi
m_{B}^{2}C_{F}\sqrt{2N_{c}}\int_{0}^{1}[dx]\int_{0}^{\infty}b_{1}db_{1}\Phi_{B}(x_{1},b_{1})$
(8)
$\displaystyle\times\Big{\\{}\Big{[}(1-2r^{2}_{2}+r^{4}_{2})(1-x_{2})\Phi_{\pi}(x_{3})\Psi^{L}(x_{2})+\frac{1}{2}(r^{2}_{2}-r^{4}_{2})\Phi_{\pi}(x_{3})\Psi^{t}(x_{2})$
$\displaystyle-
r_{\pi}(1-r^{2}_{2})x_{3}\Phi^{p}_{\pi}(x_{3})\Psi_{L}(x_{2})+r_{\pi}\left(2r^{2}_{2}(1-x_{2})+(1-r^{2}_{2})x_{3}\right)\Phi^{t}_{\pi}(x_{3})\Psi^{L}(x_{2})\Big{]}$
$\displaystyle\times E_{1,4}(t_{d}^{(1)})h_{d}^{(1)}(x_{1},x_{2},x_{3},b_{1})$
$\displaystyle-\Big{[}(x_{2}-x_{2}r^{4}_{2}+x_{3}-2r^{2}_{2}x_{3}+r^{4}_{2}x_{3})x_{3})\Phi_{\pi}(x_{3})\Psi^{L}(x_{2})$
$\displaystyle+r^{2}_{2}(2r_{\pi}\Phi^{t}_{\pi}(x_{3})-\frac{1}{2}(1-r^{2}_{2})\Phi_{\pi}(x_{3}))\Psi^{t}(x_{2})$
$\displaystyle-
r_{\pi}(1-r^{2}_{2})x_{3}\Phi^{p}_{\pi}(x_{3})\Psi_{L}(x_{2})-r_{\pi}\left(2r^{2}_{2}x_{2}+(1-r^{2}_{2})x_{3}\right)\Phi^{t}_{\pi}(x_{3})\Psi^{L}(x_{2})\Big{]}$
$\displaystyle\times
E_{1,4}(t^{(2)}_{d})h_{d}^{(2)}(x_{1},x_{2},x_{3},b_{1})\;,$
$\displaystyle{\cal M}_{6}^{(J/\psi\pi)}$ $\displaystyle=$ $\displaystyle
16\pi
m_{B}^{2}C_{F}\sqrt{2N_{c}}\int_{0}^{1}[dx]\int_{0}^{\infty}b_{1}db_{1}\Phi_{B}(x_{1},b_{1})$
(9)
$\displaystyle\times\Big{\\{}\Big{[}(1-x_{2}+r^{4}_{2}x_{2}+x_{3}-2r^{2}_{2}x_{3}+r^{4}_{2}x_{3}-r^{4}_{2})\Phi_{\pi}(x_{3})\Psi^{L}(x_{2})+$
$\displaystyle
r^{2}_{2}(2r_{\pi}\Phi^{t}_{\pi}(x_{3})-\frac{1}{2}(1-r^{2}_{2})\Phi_{\pi}(x_{3}))\Psi^{t}(x_{2})$
$\displaystyle-
r_{\pi}(1-r^{2}_{2})x_{3}\Phi^{p}_{\pi}(x_{3})\Psi^{L}(x_{2})-r_{\pi}\left(2r^{2}_{2}(1-x_{2})+(1-r^{2}_{2})x_{3}\right)\Phi^{t}_{\pi}(x_{3})\Psi^{L}(x_{2})\Big{]}$
$\displaystyle\times E_{6}(t_{d}^{(1)})h_{d}^{(1)}(x_{1},x_{2},x_{3},b_{1})$
$\displaystyle-\Big{[}(1-2r^{2}_{2}+r^{4}_{2})x_{2}\Phi_{\pi}(x_{3})\Psi^{L}(x_{2})+\frac{1}{2}(r^{2}_{2}-r^{4}_{2})r^{2}_{2}\Phi_{\pi}(x_{3})\Psi^{t}(x_{2})$
$\displaystyle-
r_{\pi}(1-r^{2}_{2})x_{3}\Phi^{p}_{\pi}(x_{3})\Psi^{L}(x_{2})+r_{\pi}\left(2r^{2}_{2}x_{2}+(1-r^{2}_{2})x_{3}\right)\Phi^{t}_{\pi}(x_{3})\Psi^{L}(x_{2})\Big{]}$
$\displaystyle\times
E_{6}(t^{(2)}_{d})h_{d}^{(2)}(x_{1},x_{2},x_{3},b_{1})\Big{\\}}\;,$
with the color factor $C_{F}=4/3$, the number of colors $N_{c}=3$, the symbol
$[dx]\equiv dx_{1}dx_{2}dx_{3}$ and the mass ratio
$r_{\pi}=m_{0}^{\pi}/m_{B}$, $m_{0}^{\pi}$ being the chiral scale associated
with the $\pi$ meson.
The evolution factor $E_{i}$ and hard function $h_{d}$ in Eq.(9) can be found
in Ref. btocharmn2 . In the derivation of spectator correction in the pQCD, we
need to take the wave function of relevant mesons, we list the wave functions
in appendix.
For the $B^{0}$ decay, the CP asymmetry is time dependent,
$\displaystyle A_{CP}(t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma({\bar{B}}^{0}(t)\to{J/\psi\pi^{0}})-\Gamma(B^{0}(t)\to{J/\psi\pi^{0}})}{\Gamma({\bar{B}}^{0}(t)\to{J/\psi\pi^{0}})+\Gamma(B^{0}(t)\to{J/\psi\pi^{0}})}\;,$
(10) $\displaystyle=$ $\displaystyle S_{J/\psi\pi^{0}}\sin(\Delta
Mt)-C_{J/\psi\pi^{0}}\cos(\Delta Mt)\;,$
Where the mixing-induced asymmetry $S_{J/\psi\pi^{0}}$ and direct CP asymmetry
is defined as
$\displaystyle S_{J/\psi\pi^{0}}=\frac{2\,{\rm
Im}\,\lambda_{J/\psi\pi^{0}}}{1+|\lambda_{J/\psi\pi^{0}}|^{2}}\;,$
$\displaystyle
C_{J/\psi\pi^{0}}=\frac{1-|\lambda_{J/\psi\pi^{0}}|^{2}}{1+|\lambda_{J/\psi\pi^{0}}|^{2}}\;,$
(11)
where
$\lambda_{CP}=\frac{V_{tb}^{*}V_{td}\langle
J/\psi\pi^{0}|H_{eff}|\overline{B}^{0}\rangle}{V_{tb}V_{td}^{*}\langle
J/\psi\pi^{0}|H_{eff}|B^{0}\rangle}.$ (12)
There are two ways to extract weak phase $\beta$ through $B^{0}\to
J/\Psi~{}\pi^{0}$ decay. The first way is through the dependence of the
mixing-induced CP asymmetry on weak phase $\beta$. The $S_{J/\psi\pi^{0}}$ is
not sensitive of input parameters, as shown in Fig. 4. That means that the
theoretical uncertainties of $S_{J/\psi\pi^{0}}$ is quite small. If we measure
the mixing-induced asymmetry $S_{J/\psi\pi^{0}}$, we can determine weak phase
$\beta$ through the dependence of $S_{J/\psi\pi^{0}}$ on $\beta$ as shown in
Fig. 3 and Table 1,
$\beta$(deg) | 18.0 | 18.3 | 18.6 | 18.9 | 19.2 | 19.5 | 19.8 | 20.1
---|---|---|---|---|---|---|---|---
$S_{J/\psi\pi^{0}}$ | -0.58515 | -0.59357 | -0.60192 | -0.61021 | -0.61843 | -0.62658 | -0.63467 | -0.64269
$\beta$ (deg) | 20.4 | 20.7 | 21 | 21.3 | 21.6 | 21.9 | 22.2 | 22.5
$S_{J/\psi\pi^{0}}$ | -0.65063 | -0.65851 | -0.66631 | -0.67404 | -0.68170 | -0.68929 | -0.69680 | -0.70424
$\beta$ (deg) | 22.8 | 23.1 | 23.4 | 23.7 | 24.0 | 24.3 | 24.6 | 24.9
$S_{J/\psi\pi^{0}}$ | -0.71160 | -0.71888 | -0.72608 | -0.73321 | -0.74025 | -0.74722 | -0.75410 | -0.76090
Table 1: Determination of weak phase $\beta$ through mixing-induced CP
asymmetry $S_{J/\psi\pi^{0}}$
Another way is to use the relation of the total asymmetry $A_{CP}$ with the
weak phase $\beta$. By integrating $A_{CP}(t)$with respect to the time
variable t, we can get the total asymmetry $A_{CP}$,
$A_{CP}=\frac{x}{1+x^{2}}S_{J/\psi\pi^{0}}-\frac{1}{1+x^{2}}C_{J/\psi\pi^{0}},$
(13)
with $x=\Delta m/\Gamma\simeq 0.723$ for the $B^{0}$-$\overline{B}^{0}$ mixing
in the SM pdg2006 .
Like the mixing-induced asymmetry, the total asymmetry is also not sensitive
to the input parameters, so we can determine the weak phase through the
relation of the total CP asymmetry with weak phase $\beta$ shown in Fig. 3.
The numerical calculation needs some parameters and meson distribution
amplitudes as input, we list them in the appendix.
With the parameters and meson distribution amplitude in the appendix, we get
the branching ratios of $B\to J/\Psi~{}\pi^{0}$ decays, $\Delta
S_{J/\psi\pi^{0}}$ and $C_{J/\psi\pi^{0}}$,
$\displaystyle Br(B^{0}\to J/\psi\pi^{0})$ $\displaystyle=$
$\displaystyle[1.89^{+0.182}_{-0.21}(\omega
b)^{+0.0496}_{-0.02}(\mu)^{+0.193}_{-0.171}(F_{1})^{+0.015}_{-0.014}(f_{J/\psi})^{+0.04}_{-0.059}(\lambda)^{+0.04}_{-0.068}(A)]\times
10^{-5}\,,$ $\displaystyle C_{J/\psi\pi^{0}}$ $\displaystyle=$
$\displaystyle[-9.936_{-3.093}^{+0.866}(\omega
b)_{-2.368}^{+1.173}(\gamma)_{-0.289}^{+6.914}(\mu)_{-1.18}^{+1.34}(F_{1})_{-0.56}^{+0.54}(\beta)]\times
10^{-3}\,,$ $\displaystyle\Delta S_{J/\psi\pi^{0}}$ $\displaystyle=$
$\displaystyle[2.84^{+4.07}_{-1.00}(\omega
b)^{+0.72}_{-0.35}(\gamma)^{+2.1}_{-0.17}(\mu)^{+0.29}_{-0.20}(F_{1})^{+0.03}_{-0.05}(\beta)]\times
10^{-3}\,.$ (14)
The main theoretical errors of the branching ratio are induced by the
uncertainties below. The first error is from $\omega b=0.4\pm 0.04GeV$, the
second one is due to renormalization scale $\mu$ taken from $mb/2$ to $mb$,
the third one is induced by $15\%$ uncertainty of $B\to\pi$ form factor
$F_{1}^{B\to\pi}$, the fourth one arise from decay constant
$f_{J/\psi}=0.405\pm 0.05GeV$, the fifth error is from CKM matrix parameter
$\lambda=0.2272\pm 0.001$, the sixth one is from CKM matrix parameter
$A=0.818^{+0.007}_{-0.017}$.
Compared with the experimental datapdg2006
$\displaystyle Br(B^{0}\to J/\psi\pi^{0})$ $\displaystyle=$
$\displaystyle(2.2\pm 0.4)\times 10^{-5}\,,$ (15)
our prediction of the branching ratio for $B\to J/\Psi~{}\pi^{0}$ is
consistent with it.
Unlike the branching ratio, $\Delta S_{J/\psi\pi^{0}}$ and $C_{J/\psi\pi^{0}}$
is not sensitive to CKM matrix parameter $\lambda$ or $A$, because these
parameter dependences cancel out. The independence of $\Delta
S_{J/\psi\pi^{0}}$ and $C_{J/\psi\pi^{0}}$ on some CKM parameters is shown in
Fig. 4(a),(b),and Fig. 5.(a),(b).
To find new physics and to extract the weak phase $\beta$, we need reliable
evaluation for the direct CP asymmetry $C_{J/\psi\pi^{0}}$ and $\Delta
S_{J/\psi\pi^{0}}$, so we now consider the dependence of the direct CP
asymmetry $C_{J/\psi\pi^{0}}$ and $\Delta S_{J/\psi\pi^{0}}$with all
parameters of input. The main uncertainties of $C_{J/\psi\pi^{0}}$ and $\Delta
S_{J/\psi\pi^{0}}$ are induced by uncertainties of shape parameter $\omega b$,
CKM matrix phase$\gamma$, renormalization scale $\mu$, $B\to\pi$ form factor
$F_{1}^{B\to\pi}$ and the weak phase $\beta$. The uncertainties of $\Delta
S_{J/\psi\pi^{0}}$ and $C_{J/\psi\pi^{0}}$ are shown in Fig. 4(c)-(f) and Fig.
5.(c)-(f).
Comparing with the result in Ref. deltasciu ,
$\displaystyle C_{J/\psi\pi^{0}}$ $\displaystyle=$ $\displaystyle 0.09\pm
0.19$ (16) $\displaystyle S_{J/\psi\pi^{0}}$ $\displaystyle=$
$\displaystyle-0.47\pm 0.30$ (17)
our results of $\Delta S_{J/\psi\pi^{0}}$and $C_{J/\psi\pi^{0}}$ has much less
theoretical uncertainties. So we conclude that if the measured deviation
$\Delta S_{J/\psi\pi^{0}}$ of the mixing-induced asymmetry is at $1\%$ or the
direct asymmetry $C_{J/\psi\pi^{0}}$ is at the level of percentage then we can
say that there should be new physics . We are expecting precise measurement to
the CP asymmetry of $B^{0}\to J/\psi\pi^{0}$ in the near future.
## Appendix A Input Parameters And Wave Functions
We use the following input parameters in the numerical calculations
$\displaystyle\Lambda_{\overline{\mathrm{MS}}}^{(f=4)}$ $\displaystyle=$
$\displaystyle 250{\rm MeV},\quad f_{\pi}=130{\rm MeV},\quad f_{B}=190{\rm
MeV},$ $\displaystyle m_{0}^{\pi}$ $\displaystyle=$ $\displaystyle 1.4{\rm
GeV},\quad M_{B}=5.2792{\rm GeV},\quad\tau_{B^{0}}=1.53\times 10^{-12}{\rm
s},$ (18)
For the CKM matrix elements, we adopt the wolfenstein parametrization for the
CKM matrix up to $\mathcal{O}$$(\lambda^{3})$pdg2006 ,
$V_{CKM}=\left(\begin{array}[]{ccc}1-\frac{\lambda^{2}}{2}&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda&1-\frac{\lambda^{2}}{2}&A\lambda^{2}\\\
A\lambda^{3}(1-\rho-i\eta)&-A\lambda^{2}&1\end{array}\right),$ (19)
with the parameters $\lambda=0.2272,A=0.818,\rho=0.221$ and $\eta=0.340$.
For the $B$ meson distribution amplitude, we adopt the modelkls01
$\displaystyle\phi_{B}(x,b)$ $\displaystyle=$ $\displaystyle
N_{B}x^{2}(1-x)^{2}\mathrm{exp}\left[-\frac{M_{B}^{2}\
x^{2}}{2\omega_{b}^{2}}-\frac{1}{2}(\omega_{b}b)^{2}\right],$ (20)
where $\omega_{b}$ is a free parameter and we take $\omega_{b}=0.4\pm 0.05$
GeV in numerical calculations, and $N_{B}=91.745$ is the normalization factor
for $\omega_{b}=0.4$.
The $J/\psi$ meson asymptotic distribution amplitudes are given by BC04
$\displaystyle\Psi^{L}(x)$ $\displaystyle=$
$\displaystyle\Psi^{T}(x)=9.58\frac{f_{J/\psi}}{2\sqrt{2N_{c}}}x(1-x)\left[\frac{x(1-x)}{1-2.8x(1-x)}\right]^{0.7}\;,$
$\displaystyle\Psi^{t}(x)$ $\displaystyle=$ $\displaystyle
10.94\frac{f_{J/\psi}}{2\sqrt{2N_{c}}}(1-2x)^{2}\left[\frac{x(1-x)}{1-2.8x(1-x)}\right]^{0.7}\;,$
$\displaystyle\Psi^{V}(x)$ $\displaystyle=$ $\displaystyle
1.67\frac{f_{J/\psi}}{2\sqrt{2N_{c}}}\left[1+(2x-1)^{2}\right]\left[\frac{x(1-x)}{1-2.8x(1-x)}\right]^{0.7}\;,$
(21)
For the light meson wave function, we neglect the $b$ dependant part, which is
not important in numerical analysis. We choose the wave function of $\pi$
meson ball3 :
$\displaystyle\Phi_{\pi}(x)$ $\displaystyle=$
$\displaystyle\frac{3}{\sqrt{6}}f_{\pi}x(1-x)\left[1+0.44C_{2}^{3/2}(2x-1)+0.25C_{4}^{3/2}(2x-1)\right],$
(22) $\displaystyle\Phi_{\pi}^{P}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{\pi}}{2\sqrt{6}}\left[1+0.43C_{2}^{1/2}(2x-1)+0.09C_{4}^{1/2}(2x-1)\right],$
(23) $\displaystyle\Phi_{\pi}^{t}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{\pi}}{2\sqrt{6}}(1-2x)\left[1+0.55(10x^{2}-10x+1)\right].$
(24)
The Gegenbauer polynomials are defined by
$\begin{array}[]{ll}C_{2}^{1/2}(t)=\frac{1}{2}(3t^{2}-1),&C_{4}^{1/2}(t)=\frac{1}{8}(35t^{4}-30t^{2}+3),\\\
C_{2}^{3/2}(t)=\frac{3}{2}(5t^{2}-1),&C_{4}^{3/2}(t)=\frac{15}{8}(21t^{4}-14t^{2}+1).\end{array}$
(25)
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Figure 2: The dependence of the mixing-induced asymmetry $S_{J/\psi\pi^{0}}$
for $B^{0}\to J/\Psi~{}\pi^{0}$ on the weak phase $\beta$ in diagram $(a)$.
The dependence of the deviation $\Delta S_{J/\psi\pi^{0}}$ of the mixing-
induced asymmetry from $\sin(-2\beta)$ on the weak phase $\beta$ in diagram
$(b)$
Figure 3: The dependence of the the mixing-induced asymmetry
$S_{J/\psi\pi^{0}}$ for $B^{0}\to J/\Psi~{}\pi^{0}$ on the weak phase $\beta$
in diagram $(a)$ can be used to extract the weak phase $\beta$ . The
dependence of total CP asymmetry $A_{CP}$ on the weak phase $\beta$ in
diagram$(b)$ can be used to extract the weak phase $\beta$ also.
Figure 4: The uncertainties of $\Delta S_{J/\psi\pi^{0}}$ of the mixing-
induced asymmetry from $\sin(-2\beta)$ are induced by that of renormalization
scale $\mu$ in $(c)$ , that of $B\to\pi$ form factor in $(d)$, that of the
weak phase $\gamma$ in $(e)$ and that of $\sin(2\beta)$ in $(f)$ . Figure 5:
The uncertainties of the direct CP asymmetry $C_{J/\psi\pi^{0}}$ are induced
by that of renormalization scale $\mu$ in $(c)$ , that of $B\to\pi$ form
factor in $(d)$, that of the weak phase $\gamma$ in $(e)$ and that of
$\sin(2\beta)$ in $(f)$ .
|
arxiv-papers
| 2009-04-08T10:55:59 |
2024-09-04T02:49:01.787665
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jing-Wu Li, Dong-Sheng Du, Xiang-Yao Wu",
"submitter": "JingWu Li",
"url": "https://arxiv.org/abs/0904.1304"
}
|
0904.1356
|
# Morphology and Interaction between Lipid Domains
Tristan S. Ursell1, William S. Klug2 and Rob Phillips1111Address
correspondence to: phillips@pboc.caltech.edu
1Department of Applied Physics, California Institute of Technology, Pasadena,
CA 91125
2Department of Mechanical and Aerospace Engineering, Program in Biomedical
Engineering,
and California NanoSystems Institute, University of California Los Angeles,
Los Angeles, CA 90095
Cellular membranes are a heterogeneous mix of lipids, proteins and small
molecules. Special groupings of saturated lipids and cholesterol form a
liquid-ordered phase, known as ‘lipid rafts,’ serving as platforms for
signaling, trafficking and material transport throughout the secretory
pathway. Questions remain as to how the cell maintains heterogeneity of a
fluid membrane with multiple phases, through time, on a length-scale
consistent with the fact that no large-scale phase separation is observed. We
have utilized a combination of mechanical modeling and in vitro experiments to
show that membrane morphology can be a key player in maintaining this
heterogeneity and organizing such domains in the membrane. We demonstrate that
lipid domains can adopt a flat or dimpled morphology, where the latter
facilitates a repulsive interaction that slows coalescence and tends to
organize domains. These forces, that depend on domain morphology, play an
important role in regulating lipid domain size and in the lateral organization
of lipids in the membrane.
The plasma and organelle membranes of cells are composed of a host of
different lipids, lipophilic molecules and membrane proteins [1]. Together,
they form a heterogeneous layer capable of regulating the flow of materials
and signals into and out of the cell. Lipid structure and sterol content play
a key role in membrane organization, where steric interactions and
energetically costly mismatch in the hydrophobic structure of lipid tails
result in lateral phase-separation. Saturated lipids and cholesterol are
sequestered into liquid-ordered ($L_{\mbox{\footnotesize o}}$) domains, often
known as ‘lipid rafts’, from an unsaturated liquid-disordered
($L_{\mbox{\footnotesize d}}$) phase [2, 3, 4]. Domains composed of saturated
sphingolipids and cholesterol, with sizes in the range of $\sim
50-500\,\mbox{nm}$, have been implicated in a range of biological processes
from lateral protein organization and virus uptake to signaling and plasma-
membrane tension regulation [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
How the cell maintains the lateral heterogeneity of lipids over time, and what
physical mechanism might be responsible for the spatial organization of these
domains, challenges classical theories of phase-separation and ‘domain
ripening’ (such as Cahn-Hilliard kinetics [18]). The maintenance of lateral
heterogeneity is thought to arise from a combination of lipid recycling and
energetic barriers to domain coalescence [19, 20, 21] (potentially provided by
transmembrane proteins [22]), resulting in a stable distribution of domain
sizes. The precise origin of this energy barrier and the nature of its
dependence on membrane elastic properties remains unclear.
The simplest physical model that describes the evolution of lipid domain size
and position predicts that domains diffuse and coalesce, such that the number
of domains constantly decreases, while the average domain size constantly
increases [18]. Indeed, models of two-dimensional phase separation have been
studied in detail for many physical systems [23, 24, 25, 26], and where the
phase boundary is unfavorable and characterized by an energy per unit length
[27], the domain size grows continuously ($\propto t^{1/3}$) [18, 28, 29].
However, membranes can adopt three-dimensional morphologies that affect the
kinetics of phase separation [30, 31, 32, 33, 34]. In those cases where
morphology is considered as part of the phase separation model, novel
coalescence kinetics emerge [31]. Experimentally, model membranes have shown
that nearly complete phase separation on the surface of a cell-sized vesicle
can be reached in as little as one minute [2]. This seems inconsistent with
the fact that on the cell surface, much smaller domains persist on that same
time-scale [21] and no large-scale phase separation is observed. With these
facts in mind, our central questions are: how can membranes that have phase-
separated maintain their lateral heterogeneity on long time scales and short
length scales? Are there membrane-mediated (i.e. elastic) forces that inhibit
coalescence and spatially organize domains?
We begin to answer these questions by examining the energetics of the membrane
using a linear elastic model. A phase-separated membrane is endowed with
bending stiffness, membrane tension, an energetic cost at the phase boundary,
and domains of a particular size. Membrane bending and tension establish a
natural length-scale over which a morphological instability develops that
switches domains from a flat to ‘dimpled’ shape, similar to classical Euler
buckling [35] (see Figure 1). The dimpling instability is size-selective and
‘turns on’ a membrane-mediated interaction that inhibits domain coalescence.
This transition is a precursor to budding, and is distinct from transitions
that require spontaneous curvature. While variations in membrane composition
may change specific parameter values, the mechanical effects we describe are
generic. Thus, these systems exhibit shape-dependent coarsening kinetics, that
are relevant for a broad class of two-dimensional binary fluid systems. The
interaction between domains is a mechanical effect, and we use a model
treating dimpled domains as curved rigid inclusions to distill the main
principles governing this interaction. The confluence of membrane properties
required for this morphological change and its attendant forces lies squarely
in the biological regime. Experimentally, we use a model mixture of lipids and
cholesterol to show that such an interaction exists between dimpled domains
and is well approximated by a simple model. Combined with lipid recycling
[19], we offer elastic interactions as a mechanism for the maintenance of
lipid lateral-heterogeneity and organization of domains in cellular membranes.
Figure 1: Three dimensional rendering of a dimpled lipid domain in
dimensionless coordinates. For a domain (shown in red), a competition between
bending, membrane tension, and phase boundary line tension results in a
morphological transition from a flat to a dimpled state as depicted above. The
dimple costs bending energy but relieves line tension by reducing the phase
boundary length (shown as a white line around the domain). This morphology
facilitates interactions between domains that significantly alter the kinetics
of coalescence and lateral lipid organization. The projected domain radius is
$\rho_{o}=r_{o}/\lambda_{2}$.
The first section of the paper outlines the energetic contributions to the
mechanical model, and predicts the conditions under which domain dimpling
occurs. The second section outlines how dimpled domains facilitate an elastic
interaction and compares the model interaction to our measurements made in
phase-separated giant unilamellar vesicles.
The Elastic Model and Morphological Transitions
The energetics of a lipid domain are dominated by a competition — on one hand
the applied membrane tension and bending stiffness both energetically favor a
flat domain; on the other hand the phase boundary line tension prefers any
domain morphology (in 3D) that reduces the boundary length. We use a continuum
mechanical model that couples these effects, relating the energetics of
membrane deformation to domain morphology. As we will show, this competition
results in a morphological transition from a flat to dimpled domain shape,
where two dimpled domains are then capable of interacting elastically.
Lipid domains in a liquid state naturally adopt a circular shape to minimize
the phase boundary length [2], allowing us to formulate our continuum
mechanical model in polar coordinates. We employ a Monge representation, where
the membrane mid-plane is described by a height function $h({\bf r})$ in the
limit of small membrane deformations (i.e. $|\nabla h|<1$). With this height
function, we characterize how membrane tension, bending, spontaneous curvature
and line tension all contribute to domain energetics.
Changes in membrane height alter the projected area of the membrane and hence
do work against the applied membrane tension, resulting in an increase in
energy written as
${G_{\mbox{\tiny tens}}=\pi\tau\left(\int_{0}^{r_{o}}(\nabla h_{1})^{2}r{\rm
d}r+\int_{r_{o}}^{\infty}(\nabla h_{2})^{2}r{\rm d}r\right),}$ (1)
where $\tau$ is the constant membrane tension, $r_{o}$ is the projected radius
of the domain, $h_{1}$ is the height function of the domain and $h_{2}$ is the
height function of the surrounding membrane [36, 37]. Membrane curvature is
penalized by the bending stiffness with a bending energy written as [39, 36]
${G_{\mbox{\tiny
bend}}=\pi\kappa_{b}^{\mbox{\tiny(2)}}\left(\sigma\int_{0}^{r_{o}}\left(\nabla^{2}h_{1}\right)^{2}r{\rm
d}r+\int_{r_{o}}^{\infty}\left(\nabla^{2}h_{2}\right)^{2}r{\rm d}r\right).}$
(2)
Our model allows the domain and surrounding membrane to have differing
stiffnesses, $\kappa_{b}^{\mbox{\tiny(1)}}$ and $\kappa_{b}^{\mbox{\tiny(2)}}$
respectively, characterized by the parameter
$\sigma=\kappa_{b}^{\mbox{\tiny(1)}}/\kappa_{b}^{\mbox{\tiny(2)}}$, and from
this point on we drop the superscript on $\kappa_{b}^{\mbox{\tiny(2)}}$.
Recent experiments suggest that the bending moduli of a cholesterol-rich
domain and the surrounding membrane are roughly equal [4, 38], and hence for
simplicity, we assume the bending moduli of the two regions are equal (i.e.
$\sigma=1$), unless otherwise noted. In addition to bending stiffness, the
domain may exhibit a preferred ‘spontaneous’ curvature due to lipid asymmetry
or protein binding [40, 34]. The contribution of domain spontaneous curvature
can be simplified to a boundary integral, which couples to the overall
curvature field by
${G_{\mbox{\tiny
spont}}=-2\pi\sigma\kappa_{b}c_{o}\int_{0}^{r_{o}}\left(\nabla^{2}h_{1}\right)r{\rm
d}r=-2\pi\sigma\kappa_{b}c_{o}r_{o}\epsilon,}$ (3)
where $c_{o}$ is the spontaneous curvature of the domain and $\epsilon$ is the
membrane slope at the phase boundary as shown in Figure 1. Further, we assume
the saddle-splay curvature moduli are equal in the two regions, yielding no
dependence on Gaussian curvature. In principle, this contribution could be
accounted for with a boundary term, explored in detail in the supplementary
information (SI). The phase boundary line tension is applied to the projected
circumference of the domain, as shown in Figure 1, by $G_{\mbox{\tiny
line}}=2\pi r_{o}\gamma$ where $\gamma$ is the energy per unit length at the
phase boundary.
Finally, a constraint must be imposed that relates the actual domain area,
$\mathcal{A}$, to the projected domain radius $r_{o}$. The energetic cost to
change the area per lipid molecule is high ($\sim
50-100\,k_{B}T/\mbox{nm}^{2}$ where $k_{B}=1.38\times 10^{-23}\,J/\mbox{K}$
and $T=300\,\mbox{K}$ [41]), hence we assume the domain area is conserved
during any morphological change (see SI for details). We impose this
constraint using a Lagrange multiplier, $\tau_{o}$, with units of tension by
${G_{\mbox{\tiny area}}=\tau_{o}\left(\pi\int_{0}^{r_{o}}(\nabla
h_{1})^{2}r{\rm d}r+\pi r_{o}^{2}-\mathcal{A}\right).}$ (4)
This results in an effective membrane tension within the domain
$\tau_{1}=\tau+\tau_{o}$, which must be negative to induce dimpling. Examining
the interplay between bending and membrane tension, we see that two natural
length scales emerge - within the domain we define
$\lambda_{1}=\sqrt{\sigma\kappa_{b}/\tau_{1}}$ and outside the domain we
define $\lambda_{2}=\sqrt{\kappa_{b}/\tau}$. These length scales allow us to
define the relevant dimensionless parameters in this system.
The total free energy of an elastic domain and its surrounding membrane is
then the sum of these five terms, $G=G_{\mbox{\tiny tens}}+G_{\mbox{\tiny
bend}}+G_{\mbox{\tiny spont}}+G_{\mbox{\tiny line}}+G_{\mbox{\tiny area}}$.
Details on all the terms in the free energy can be found in the SI. With this
free energy in hand, we examine how the morphology of a circular domain
evolves as we tune domain size and the elastic properties of the membrane.
The height field and radius can be rescaled by the elastic decay lengths such
that the Euler-Lagrange equation for the domain can be written in the
parameter-free form $\nabla^{2}(\nabla^{2}+\beta^{2})\eta_{1}=0$, while the
equation for the surrounding membrane is $\nabla^{2}(\nabla^{2}-1)\eta_{2}=0$,
where the dimensionless variables are defined by $\lambda_{2}\eta_{i}=h_{i}$,
$\lambda_{2}\rho=r$, $\lambda_{2}\rho_{o}=r_{o}$ and
$\beta=i\lambda_{2}/\lambda_{1}$. Using the same dimensionless notation, the
energy from line tension and spontaneous curvature can be written as
$G_{\mbox{\tiny line}}=2\pi\kappa_{b}\rho_{o}\chi$ and $G_{\mbox{\tiny
spont}}=-2\pi\sigma\kappa_{b}\epsilon\rho_{o}\upsilon_{o}$, with
$\upsilon_{o}=\lambda_{2}c_{o}$ and $\chi=\gamma\lambda_{2}/\kappa_{b}$. The
dimensionless line tension, $\chi$, is simply a rescaled version of the line
tension $\gamma$ and is one of two key parameters that characterize the
morphological transition; the dimensionless domain area,
$\alpha=\mathcal{A}/\lambda_{2}^{2}$, is the second key parameter.
The admissible solutions for $\eta_{1}(\rho)$ and $\eta_{2}(\rho)$ are zeroth
order Bessel functions $J_{0}(\beta\rho)$ and $K_{0}(\rho)$, respectively,
with the boundary conditions $|\nabla\eta_{1}(0)|=|\nabla\eta_{2}(\infty)|=0$
and $|\nabla\eta_{1}(\rho_{o})|=|\nabla\eta_{2}(\rho_{o})|=\epsilon$. The
boundary slope, $\epsilon$, is the parameter that indicates the morphology of
the domain; $\epsilon=0$ indicates a flat domain, while $0<|\epsilon|\lesssim
1$ indicates a dimpled domain. The five contributions to membrane deformation
energy yield a relatively simple expression for the total free energy, given
by
$\displaystyle G$ $\displaystyle=$
$\displaystyle\pi\kappa_{b}\rho_{o}\left[\epsilon^{2}\left(\sigma\beta\frac{J_{0}(\beta\rho_{o})}{J_{1}(\beta\rho_{o})}+\frac{K_{0}(\rho_{o})}{K_{1}(\rho_{o})}\right)+2(\chi-\epsilon\sigma\upsilon_{o})\right]$
$\displaystyle-\kappa_{b}(\sigma\beta^{2}+1)(\pi\rho_{o}^{2}-\alpha).$
Mechanical equilibrium is enforced by rendering the energy stationary with
respect to unknown parameters $\epsilon$, $\rho_{o}$, and $\beta$,
${\frac{\partial G}{\partial\epsilon}=0,\quad\frac{\partial
G}{\partial\rho_{o}}=0,\quad\frac{\partial G}{\partial\beta}=0.}$ (6)
These equilibrium equations physically correspond to torque balance at the
phase boundary, lateral force balance at the phase boundary and domain area
conservation, respectively.
Analysis of the equilibrium equations reveals a second-order transition at a
critical line-tension, $\chi_{c}$, as shown in Figure 2. For $\chi$ less than
this critical value, only the flat, trivial solution with $\epsilon=0$ exists.
At $\chi_{c}$ a non-trivial solution describing buckled or dimpled
morphologies emerges. For zero spontaneous curvature, the bifurcation is
defined by a transcendental characteristic equation
${\sigma\beta\frac{J_{0}(\beta\rho_{o})}{J_{1}(\beta\rho_{o})}+\frac{K_{0}(\rho_{o})}{K_{1}(\rho_{o})}=0,}$
(7)
with $\beta=\sqrt{(\chi_{c}/\rho_{o}-1)/\sigma}$ and
$\rho_{o}=\sqrt{\alpha/\pi}$. For a given dimensionless domain area, $\alpha$,
this defines the critical line tension required to dimple the domain. In
Figure 2a(inset), this relation is used to generate a morphological phase
diagram that shows where in the space of dimensionless domain area and line
tension we find the discontinuous transition (i.e. bifurcation) from a flat
domain, to a dimpled domain. Near the morphological transition the boundary
slope grows as $|\epsilon|\propto\sqrt{\chi/\chi_{c}-1}$, indicating that a
dimple rapidly deviates from the flat state. The transition is symmetric, in
that both possible dimple curvatures have the same energy, and hence the
domain is equally likely to dimple upwards or downwards. In the experimentally
relevant limit of small dimensionless domain area, the complexity of eqn. 7 is
reduced to
${\chi_{c}\sqrt{\alpha}=\frac{\gamma_{c}}{\kappa_{b}}\sqrt{\mathcal{A}}\simeq
8\sigma\sqrt{\pi}.}$ (8)
This leads to the conclusion that the dominant parameter governing domain
dimpling at zero spontaneous curvature is $\chi\sqrt{\alpha}$. For a small
domain, the dimpling transition is directly regulated by domain area, the
bending modulus, and line tension, but only weakly depends on applied membrane
tension. Intuitively, domains dimple when line tension or domain size increase
(subject to small $\alpha$), as shown in Figure 2a(inset). Likewise, a
decrease in bending stiffness, due, for instance, to changes in membrane
sterol content [42, 43], can also induce dimpling. The effects of applied
membrane tension are weak because the change in projected area upon dimpling
does not lead to a significant energy cost relative to the cost of bending and
line tension.
If membrane elastic properties are fixed (i.e. fixed $\kappa_{b}$, $\tau$ and
$\gamma$), the dimpling-induced interactions ‘turn on’ only after a critical
domain size is achieved. This scenario is encountered when two domains, too
small to dimple on their own, diffusively coalesce into a larger domain
capable of dimpling and hence interacting. Indeed, such a size-selective
coalescence mechanism was observed recently in model membrane vesicles [44].
This constitutes a distinct class of coarsening dynamics, where classical
diffusion-limited kinetics are obeyed until the domain size distribution has
matured past the critical size for dimpling - then domain coalescence is a
relatively slow, interaction-limited process.
Figure 2: Bifurcation diagram for dimpling transition at constant area
($\alpha=\pi/4$, $\kappa_{b}=25\,k_{B}T$, $\lambda_{2}=500\,\mbox{nm}$,
$\sigma=1$). Constant line tension and increasing area produces a
qualitatively similar graph. a) At zero spontaneous curvature
($\upsilon_{o}=0\rightarrow\mbox{black}$) the bifurcation is symmetric, the
upper and lower branches are at the same energy, and $\epsilon=0$ becomes
unstable above the critical point (horizontal black dashed line). With finite
spontaneous curvature ($\upsilon_{o}=2$,
$c_{o}=(250\,\mbox{nm})^{-1}\rightarrow\mbox{blue}$) the lower energy branch
(upper) has non-zero $\epsilon$ for all line tensions, asymptoting to the
$\upsilon_{o}=0$ branch. At a line tension slightly higher than $\chi_{c}$ for
the $\upsilon_{o}=0$ case, a bifurcation yields a higher energy dimple with
the opposite curvature as $\upsilon_{o}$ (indicated by the second vertical
dashed line). Inset: Equilibrium phase diagrams for $\sigma=0.5$(red),
$\sigma=1$(green), and $\sigma=2$(blue) (the dashed lines are the
approximation of eqn. 8) showing flat (F) and dimpled (D) domains. b) Energy
difference between the flat and dimpled state, normalized by $\kappa_{b}$, for
domains with and without spontaneous curvature
($\upsilon_{o}=0\rightarrow\mbox{black}$;
$\upsilon_{o}=2\rightarrow\mbox{blue}$).
For the model domain considered in Figure 2, with area $\alpha=\pi/4$
($r_{o}\simeq 250\,\mbox{nm}$), the critical dimensionless line tension is
$\chi_{c}\simeq 13$, corresponding to a critical line tension of
$\gamma_{c}\simeq 0.65\,k_{B}T/\mbox{nm}$. This value compares well with both
theoretical estimates of the line tension [27, 45], and the higher side of
experimentally measured values [4, 46, 47].
Spontaneous curvature does not affect the Euler-Lagrange equations, and hence
will not effect the class of equilibrium membrane shapes. However, domains
with zero and nonzero spontaneous curvature exhibit qualitatively different
behavior. Biological membranes can be asymmetric with respect to leaflet
composition [42, 5, 48], endowing a domain with potentially large spontaneous
curvature. The energetic contribution from spontaneous curvature takes the
form of an additional line tension depending linearly on the slope taken by
the domain boundary, $\epsilon$. This breaks the symmetry of the membrane,
giving an energetic preference to a dimple with the same curvature as the
spontaneous curvature, and eliminating the trivial $\epsilon=0$ solution even
at small line-tensions. As line tension increases, a bifurcation produces a
second, stable, higher-energy dimple of the opposite curvature as
$\upsilon_{o}$. The more energetically stable branch of this transition
corresponds to a dimpled state for all values of line tension and non-zero
values of domain area, as demonstrated in Figure 2a. This predicts that as
soon as a domain with finite spontaneous curvature forms, it dimples,
regardless of size, and begins to experience interactions with any nearby
dimpled domains. It is reasonable to expect that domains with similar
composition will have similar spontaneous curvature, and hence form dimples
whose curvature has the same sign. As we will show, dimples whose curvature
has the same sign tend to interact repulsively. Such a mechanism of
coalescence inhibition was observed recently in simulation [34].
This indicates that control of spontaneous curvature via domain composition or
protein binding can regulate dimpling and hence domain interaction [49, 48].
Indeed, recent experimental [50] and theoretical [51] work shows that protein
binding and lipid asymmetry, respectively, lead to precisely these kinds of
dimpled domains.
Figure 3: Theoretical and experimental dimpled domain shapes. Domains are
shown in red, surrounding membrane in blue. a) Minimum energy dimples with and
without spontaneous curvature (see legend, $\alpha=\pi/4$, $\sigma=1$). b)
Epi-fluorescence cross-section of a dimple on the surface of a GUV; the red
and blue lines are a guide to the eye. c) 1D model of interaction - dimples
maintain shape, but tilt ($\phi$) as a function of separation distance ($d$).
Dimples with the same sign of curvature repel, while dimples with opposite
sign attract. d) Epi-fluorescence cross-section of two dimpled domains
interacting on the surface of a GUV. Scale bars are $3\,\mu\mbox{m}$.
Calculated shapes of dimpled domains induced by line tension and spontaneous
curvature are shown in Figure 3a, alongside dimpled domains observed on giant
unilamellar vesicles, shown in Figure 3b and d.
Elastic Interactions of Dimpled Domains
Given two domains that have met the criteria for dimpling, the deformation in
the membrane surrounding the domains mediates an elastic interaction when they
are within a few elastic decay lengths ($\lambda_{2}$) of each other. This
equips us to begin addressing how short length-scale and long time-scale
membrane heterogeneity might be achieved. As previously stated, free diffusion
sets the maximum rate at which a quenched membrane can evolve into a fully
phase-separated membrane [18], where this evolution can happen in as little as
a minute on the surface of a cell-sized vesicle [2]. On the other hand,
recycling and hence homogenization of cellular membrane is a process that
takes place on the time-scale of an hour or more [52]. Our measurements of
domain interactions (detailed below and other data shown in SI) estimate the
coalescence barrier between dimpled domains at $\sim 5-10\,k_{B}T$. Hence,
given the diffusion-limited rate of phase separation, interactions slow this
process by approximately $e^{-5}\simeq 0.007$ to $e^{-10}\simeq 0.00005$. This
makes the time-scale of lipid heterogeneity comparable to the time-scale of
membrane recycling and even eukaryotic cell division.
The physical origin of domain interaction is explained by a simple model based
on the assumption that the dimpled domain shape is constant during
interaction, but the domains are free to tilt by an angle $\phi$, as shown in
Figure 3c. This assumption was, in part, inspired by experimental observations
of domain shapes on the surface of giant unilamellar vesicles, as shown, for
example, in Figure 3d. The interaction energy is roughly an order of magnitude
less than the free energy associated with the morphological transition itself
(see Figure 2b), thus interaction does not perturb the domain shape
significantly. Only allowing domains to rotate simplifies the interaction
between two domains to a change in the boundary conditions in the three
regions of interest, shown in blue in Figure 3c. Applying the small gradient
approximation, the boundary slope is given by $|\epsilon-\phi|$ in the outer
regions and by $|\epsilon+\phi|$ in the inner region. With the single domain
boundary slope, $\epsilon$, set by the energy minimization of the previous
section (i.e. eqn. 6), the pairwise energy is minimized at every domain
spacing, $d$, by $\partial G/\partial\phi=0$ to find the domain tilt angle
that minimizes the deformation energy (see SI for details). This results in
two qualitatively distinct scenarios: two domains whose curvatures have the
same sign repel each other, while two domains whose curvatures have the
opposite sign attract each other. Scaling arguments can be used to show that
the strength of interaction between two dimpled domains increases roughly
linearly with their area, so long as they are both larger than some critical
area (see SI for details). Mathematically, the assumption of rigidly rotating
dimpled domains on a membrane is identical to a previous 2D model of bending-
mediated interactions between intramembrane proteins represented by rigid
conical inclusions [53].
Independent of the effects of spontaneous curvature, slight osmolar imbalances
and constriction due to the lipid phase boundaries create small pressure
gradients across the membrane that tend to orient all dimples in a cell or
vesicle in the same direction, resulting in net repulsive interactions between
all domains. Transitions between ‘upward’ and ‘downward’ dimples are
infrequent, due to a large energy barrier. In the simplest case, where the
domains are the same size, the tilt angle $\phi$ monotonically increases as
two domains get closer, $\phi(d)\simeq-\epsilon e^{-d}$. Likewise, the
interaction energy, $V_{\mbox{\tiny int}}(d)\simeq
2\pi\kappa_{b}\epsilon^{2}\rho_{o}^{2}e^{-d}$, increases monotonically with
decreasing separation. For direct comparison, we fit both the 1D model
outlined here and the 2D inclusion model [53] to the data of Fig.4, showing
that they are experimentally indistinguishable, though with a slightly
different elastic decay length.
To quantitatively compare our interaction model with experiment, we examined
the thermal motion of small domains on the surface of giant unilamellar
vesicles, as described in ‘Materials and Methods.’ Membrane tension was
regulated by balancing the internal and external osmolarity, giving us coarse
control over the elastic decay length $\lambda_{2}$. Through time, the
distance between every domain pair was measured and the net results were used
to construct a histogram. The potential of mean force as a function of
distance between domains is shown in Figure 4b. We selected vesicles that had
a low density of approximately equal-sized domains, and thus generally the
interactions were described by a repulsive pairwise potential. Though areal
density of domains and generic data quality varied in our experiments (see
SI), all data sets exhibit the repulsive core of the elastic interaction.
Multi-body interactions occur, though infrequently; their effect can be seen
as a small variation in the baseline of Figure 4b, which is not captured by
the pairwise interaction model. At high membrane tension, when we would not
expect dimpled domains, we qualitatively verified that domains coalesce in a
rapid manner as compared to our low tension experiments (data not shown).
Other recent experiments have also observed repulsive interactions between
domains on low membrane tension vesicles and the lack thereof on taut vesicles
[44].
Figure 4: Measuring domain interactions on the surface of a vesicle. a) Three
images of dilute interacting domains on the surface of the same vesicle (scale
bar is $10\,\mu\mbox{m}$). b) The repulsive interaction potential of domains
on the surface of the same vesicle as (a). The energy is measured in $k_{B}T$
and distance is domain center-to-center. The blue dashed line is a fit to the
1D interaction model in this paper, $V_{\mbox{\tiny
int}}(r)=a_{1}e^{-r/\lambda_{2}^{\mbox{\tiny(1D)}}}+a_{2}$, with elastic decay
length $\lambda_{2}^{\mbox{\tiny(1D)}}=240\,\mbox{nm}$. The orange dashed line
is a fit to the model, $V_{\mbox{\tiny
int}}(r)=2\pi\kappa_{b}\left[(a_{1}a_{2})^{2}K_{0}(r/\lambda_{2}^{\mbox{\tiny(2D)}})+a_{2}^{2}a_{3}^{4}K_{2}^{2}(r/\lambda_{2}^{\mbox{\tiny(2D)}})\right]+a_{4}$,
with elastic decay length $\lambda_{2}^{\mbox{\tiny(2D)}}=270\,\mbox{nm}$,
based on the theory of Weikl et al [53]. Both elastic decay lengths indicate a
membrane tension of $\sim 4\times 10^{-4}\,k_{B}T/\mbox{nm}^{2}$. Errors bars
are shown in green on the $x$-axis.
Our measurement of the pairwise potential allows us to estimate elastic
properties of the membrane. The elastic decay length was fit to the 1D and 2D
interactions models described above, and found to be
$\lambda_{2}^{\mbox{\tiny(1D)}}\simeq 240\,\mbox{nm}$ and
$\lambda_{2}^{\mbox{\tiny(2D)}}\simeq 270\,\mbox{nm}$, respectively. Taken
with a nominal bending modulus of $25\,k_{B}T$, we estimate the membrane
tension to be $\sim 4\times 10^{-4}\,k_{B}T/\mbox{nm}^{2}$. From the images,
we measure the size of the domains at $r_{o}\simeq 350-400\,\mbox{nm}$, and
hence $\rho_{o}\simeq 1.5$. We estimate the line tension, $\gamma$, using eqn.
8, based on the fact that the domains are dimpled, and find a lower bound of
$\gamma\simeq 0.49\,k_{B}T/\mbox{nm}$. This is in good agreement with
theoretical estimates and values determined from AFM measurements [47], though
somewhat higher than the value of $\gamma\simeq 0.22\,k_{B}T/\mbox{nm}$
measured via shape analysis of fully phase separated vesicles [4] and
$\gamma\simeq 0.40\,k_{B}T/\mbox{nm}$ from micropipette aspiration experiments
[46]. Finally, viewing the repulsive core of the interaction as an effective
activation barrier to coalescence, a simple Arrhenius argument suggests a
decrease in coalescence kinetics by two to three orders of magnitude. Indeed,
such a slowing of coalescence was recently observed in a similar model
membrane system [44].
Discussion
Comparing biologically relevant domain sizes ($\sim 50-500\,\mbox{nm}$) with
the elastic decay length ($\lambda_{2}$), we expect physiologically relevant
domains to be small (i.e. small $\alpha$), as presumed in eqn. 8. Estimating
the elastic decay length requires knowledge of the membrane tension and
bending stiffness. We note that in vitro experiments of osmotically balanced
single and multicomponent vesicles, and measurements of the plasma membrane of
unstressed cells suggest membrane tensions of
$10^{-4}-10^{-2}\,k_{B}T/\mbox{nm}^{2}$ [4, 41, 54, 55]. The typical bending
modulus of a phosphocholine bilayer is $\sim 10-50\,k_{B}T$, depending on the
exact lipid and cholesterol content [41, 56, 57]. Choosing a nominal membrane
tension of $10^{-4}\,k_{B}T/\mbox{nm}^{2}$ and nominal bending modulus of
$25\,k_{B}T$ [41, 4] corresponds to an elastic decay length of
$\lambda_{2}\simeq 500\,\mbox{nm}$, suggesting that for lipid domains on the
order of $50-500\,\mbox{nm}$, small $\alpha$ is an appropriate approximation.
Our experiments on the surface of GUVs have three potentially confounding
effects, all due to the spherical curvature of the vesicle. First, the surface
metric is not entirely flat with respect to the image plane. Thus,
measurements of distance are underestimated the farther they are made from the
projected vesicle center. This problem is ameliorated by concentrating on
domains which are at the bottom (or top) of the vesicle where the surface is
nearly flat and demanding that our tracking software exclude domains that are
out of focus; see SI for a more detailed explanation. The second potential
complication is that we use a flat 2D coordinate system for our theoretical
analysis, however domains reside on a curved surface. Given that the domain
deformation, and hence energy density, decays exponentially with
$\lambda_{2}$, as long as $\lambda_{2}$ is small with respect to the vesicle
radius, the energetics that govern morphology converge on an essentially flat
surface metric. The final complication is that the circular area of focus
creates a fictitious confining potential for the domains, such that the
effective measured potential of mean force is the sum of the elastic pairwise
potential and a fictitious potential, $V_{\mbox{\tiny eff}}=V_{\mbox{\tiny
int}}+V_{\mbox{\tiny fict}}$. The fictitious potential is removed by
simulating non-interacting particles in a circle the same size as the radius
of focus (see SI for details).
The constant tension ensemble used in our theoretical analysis has a range of
validity, determined by the excess area available on a thermally fluctuating
membrane with conserved volume and total surface area $\mathcal{A}_{o}$ (i.e.
a vesicle). In the limit where the morphological transitions use only a small
portion ($\Delta\mathcal{A}$) of this excess area, defined by
$k_{B}T/8\pi\kappa_{b}\gg\Delta\mathcal{A}/\mathcal{A}_{o}$, the tension is
constant. Outside this regime the tension rises exponentially with reduction
in excess area, tending to stabilize dimples from fully budding (see SI for
details).
In addition to the elastic mechanism of interaction, described herein, there
may be other organizing forces at work in a phase-separated membrane: those of
elastic [27], entropic [58, 59] and electrostatic origin [60], however their
putative length-scale, on the order of ten nanometers or less, is not
accessible to the spatial and temporal resolution of our experiments, and not
consistent with our measurement of an interaction length-scale of hundreds of
nanometers.
Conclusion
We have shown that lipid domains are subject to a morphological dimpling
transition that depends on the bilayer elastic properties and domain size.
Dimpling allows two domains in proximity to repulsively interact due to the
deformation in the surrounding membrane. Our model makes some key predictions:
at zero spontaneous curvature the domain size distribution reaches a critical
point where coalescence is arrested by repulsive interactions; domains with
finite spontaneous curvature are always subject to interaction and hence
should always coalesce at a rate slower than the diffusion-limited rate.
Additionally, the strength of elastic interactions is augmented by increasing
line tension or domain area, with an approximately linear scaling. The domain
size and bilayer elastic parameters necessary to induce the dimpled morphology
are consistent with physiological conditions. Further, careful regulation of
membrane cholesterol in cells may be related to the membrane mechanical
properties necessary for morphological transitions. Combined with lipid
recycling, our work offers a mechanism working against diffusion-driven
coalescence, to maintain fine-scale lateral heterogeneity of lipids over time.
We proposed a simple 1D model of an elastic interaction that mediates dimpled-
domain repulsion, and then used a standard ternary membrane system to verify
the existence of dimpled domains and their subsequent repulsive interaction.
Finally, it follows that the morphologies and elastic forces which organize
lipid domains might play an important role in the binding and lateral
organization of proteins in the membrane.
Materials and Methods
Giant unilamellar vesicles (GUVs) were prepared from a mixture of DOPC
(1,2-Dioleoyl-sn-Glycero-3-Phosphocholine), DPPC (1,2-Dipalmitoyl-sn-
Glycero-3-Phosphocholine) and cholesterol (Avanti Polar Lipids)
(25:55:20/molar) that exhibits liquid-liquid phase coexistence [2].
Fluorescence contrast between the two lipid phases is provided by the
rhodamine head-group labeled lipids: DOPE (1,2-Dioleoyl-sn-
Glycero-3-Phosphoethanolamine-N- (Lissamine Rhodamine B Sulfonyl)) or DPPE
(1,2-Dipalmitoyl-sn-Glycero-3-Phosphoethanolamine-N- (Lissamine Rhodamine B
Sulfonyl)), at a molar fraction of $\sim 0.005$. The leaflet compositions are
presumed symmetric and hence $\upsilon_{o}=0$.
GUVs were formed via electroformation [2, 61]. Briefly, $3-4\,\mu\mbox{g}$ of
lipid in chloroform were deposited on an indium-tin oxide coated slide and
dessicated for $\sim 2\,\mbox{hrs}$ to remove excess solvent. The film was
then hydrated with a $100\,\mbox{mM}$ sucrose solution and heated to $\sim
50\,\mbox{C}$ to be above the miscibility transition temperature. An
alternating electric field was applied; $10\,\mbox{Hz}$ for 120 minutes,
$2\,\mbox{Hz}$ for 50 minutes, at $\sim 500\,\mbox{Volts/m}$ over $\sim
2\,\mbox{mm}$. Low membrane tensions were achieved by careful osmolar
balancing with sucrose ($\sim 100\,\mbox{mM}$) inside the vesicles, and
glucose ($\sim 100-108\,\mbox{mM}$) outside.
Domains were induced by a temperature quench (see SI) and imaged using
standard TRITC epi-fluorescence microscopy at 80x magnification with a cooled
(-30 C) CCD camera (Roper Scientific, $6.7\times 6.7\,\mu\mbox{m}^{2}$ per
pixel, 20 MHz digitization). Images were taken from the top or bottom of a GUV
where the surface metric is approximately flat (see SI). Data sets contained
$\sim 500-1500$ frames collected at 10-20 Hz with a varying number of domains
(usually $5-10$). The frame rate was chosen to minimize exposure-time blurring
of the domains, while allowing sufficiently large diffusive domain motion.
Software was written to track the position of each well-resolved domain and
calculate the radial distribution function. The raw radial distribution
function was corrected for the fictitious confining potential of the circular
geometry (see SI). In the dilute interaction limit, pairwise interactions
dominate, and the negative natural logarithm of the radial distribution
function is the interaction potential (potential of mean force) plus a
constant, as shown in Figure 4b.
We thank Patricia Bassereau, Ben Freund, Kerwyn Huang, Greg Huber, Sarah
Keller and Udo Seifert for stimulating discussion and comments on the
manuscript, and Jenny Hsaio for help with experiments. TU and RP acknowledge
the support of the National Science Foundation award No. CMS-0301657, NSF
CIMMS award No. ACI-0204932, NIRT award No. CMS-0404031 and the National
Institutes of Health Director’s Pioneer Award. WK acknowledges support from
NSF CAREER Award CMMI-0748034.
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|
arxiv-papers
| 2009-04-08T15:15:09 |
2024-09-04T02:49:01.793893
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tristan S. Ursell, William S. Klug, Rob Phillips",
"submitter": "Tristan Ursell",
"url": "https://arxiv.org/abs/0904.1356"
}
|
0904.1391
|
# Coronal Loop Models and Those Annoying Observations!
James A. Klimchuk
###### Abstract
It was once thought that all coronal loops are in static equilibrium, but
observational and modeling developments over the past decade have shown that
this is clearly not the case. It is now established that warm ($\sim 1$ MK)
loops observed in the EUV are explainable as bundles of unresolved strands
that are heated impulsively by storms of nanoflares. A raging debate
concerning the multi-thermal versus isothermal nature of the loops can be
reconciled in terms of the duration of the storm. We show that short and long
storms produce narrow and broad thermal distributions, respectively. We also
examine the possibility that warm loops can be explained with thermal
nonequilibrium, a process by which steady heating produces dynamic behavior
whenever the heating is highly concentrated near the loop footpoints. We
conclude that this is not a viable explanation for monolithic loops under the
conditions we have considered, but that it may have application to multi-
stranded loops. Serious questions remain, however.
NASA Goddard Space Flight Center, Code 671, Greenbelt, MD 20771, USA
## 1\. Introduction
The unusual title of this paper is meant to indicate the emotional aspects of
being a coronal loops modeler. Whenever we start to feel confident that the
problem is solved, new observations come along and force us to modify our
thinking. It can be frustrating, but it is also very rewarding when we gain
improved physical understanding of this fascinating phenomenon. The coronal
loops problem is an outstanding example of how the greatest progress is made
when observation and theory work together, one feeding off of the other.
The loops problem can be thought of as a puzzle, with the pieces of the puzzle
being observational constraints. The goal is to fit the pieces together into a
physically consistent picture (there may be more than one solution). Five key
pieces are: (1) density, (2) lifetime, (3) thermal distribution, (4) flows,
and (5) intensity profile. For the density, we are particularly interested in
how the observed density compares with the density that is expected for static
equilibrium. Thermal distribution refers to whether and how the temperature
varies over the loop cross section, i.e., across the loop axis, and intensity
profile refers to the variation of brightness along the loop axis.
For many years, our picture of coronal loops was relatively simple and the
puzzle seemed easy to solve. The observational constraints came primarily from
soft X-ray (SXR) observations of hot ($>2$ MK) loops. These loops were found
to be long-lived (e.g., Porter & Klimchuk 1995) and to satisfy static
equilibrium scaling laws (e.g., Rosner, Tucker, & Vaiana 1978; Kano & Tsuneta
1996). The most straightforward explanation was that these loops are heated in
a steady fashion.
The picture became much more confused with new observations of warm ($\sim 1$
MK) loops made in the EUV by SOHO/EIT and TRACE. These warm loops can appear
to occupy the same volume as hot loops—though not necessarily at the same
time—but their properties are fundamentally different. Besides the obvious
temperature difference, EUV loops are over dense relative to static
equilibrium, they have super-hydrostatic scale heights, and they have
exceptionally flat temperature profiles when measured with the filter ratio
technique (e.g., Aschwanden et al. 1999; Lenz et al. 1999; Aschwanden,
Schrijver, & Alexander 2001; Winebarger, Warren, & Mariska 2003). These loops
are clearly not in static equilibrium.
This paper describes the logical progression that has been followed by the
loops community in attempting to explain the observations, especially those of
the more challenging EUV loops. We represent this progression with the
flowchart in Figure 1, which is in many ways a recent history of how the
discipline has evolved.
Figure 1.: Flow chart showing the logical progression used to infer the
physical nature and heating of a coronal loop. Some boxes indicate
observational questions and others indicate conclusions that are drawn from
the answers.
## 2\. Density
Suppose we wish to investigate an observed loop. We can start by asking the
question “Is the loop over dense relative to static equilibrium?” Given the
observed temperature and length, static equilibrium theory predicts a unique
density. We want to know whether the observed density is larger than this
value? If it is not, and if the loop does not evolve rapidly, then steady
heating is a possible, though not unique, explanation. This was essentially
where things stood through the Yohkoh mission in the 1990’s.
As we have already indicated, however, most EUV loops are indeed over dense.
This is indicated in Figure 2 (reproduced from Klimchuk 2006), which reveals
the physics of what is going on. The figure shows the ratio of the radiative
to conductive cooling times plotted against temperature for a large sample of
loops. The warm loops were observed by TRACE, and the hot loops were observed
by Yohkoh/SXT. The ratio of the cooling times is determined from the measured
temperature, density, and length according to
$\tau_{rad}/\tau_{cond}=T^{4}/(n^{2}L^{2})$, although the power of temperature
depends weakly on the radiative loss function and is slightly different in
different temperature regimes.
Figure 2.: Ratio of radiative to conductive cooling times versus temperature
for many observed loops. Solid line is the cooling track of an impulsively
heated loop strand simulation. From Klimchuk (2006).
Coronal energy losses from radiation and thermal conduction are comparable for
loops that are in static equilibrium (Vesecky, Antiochos, & Underwood 1979),
and such loops would fall along a horizontal line near 0 in the plot. Loops
that lie above the line are under dense, and loops that lie below the line are
over dense. The observed loops follow a clear trend ranging from hot and under
dense in the upper-right to warm and over dense in the lower-left. Note,
however, that the densities used for the cooling time ratios were measured
using emission measures and loop diameters and assuming a filling factor of
unity, $n=[EM/(df)]^{1/2}$, so they are lower limits. Smaller filling factors
would shift the points downward in the plot. Thus, the hot loops could be in
static equilibrium, and the warm loops could be even more over dense than
indicated.
It is abundantly clear that static equilibrium cannot explain warm loops. An
explanation relying on steady end-to-end flows is also not viable
(Patsourakos, Klimchuk, & MacNeice 2004). Thermal nonequilibrium is a
possibility that we return to later. The most promising explanation for the
observed over densities of warm loops is implusive heating. This can also
explain the under densities of hot loops, if they are real. The solid curve
that fits the points so well in Figure 2 is the evolutionary track from a 1D
hydrodynamic simulation of a loop that has been heated impulsively by a
nanoflare. Cooling begins at the upper-right end of the track and progresses
downward and to the left. The early stages are dominated by thermal conduction
and are characterized by under densities, while the late stages are dominated
by radiation and are characterized by over densities. The ability of nanoflare
models to reproduce the observed densities of loops is well established
(Klimchuk 2002; Warren, Winebarger, & Hamilton 2002; Winebarger, Warren, &
Mariska 2003; Cargill & Klimchuk 2004; Klimchuk 2006).
## 3\. Lifetime
If a loop is heated impulsively, then we might expect it to exist for
approximately a cooling time (combining the effects of conduction and
radiation), as determined from the observed temperature, density, and length.
This is the next question in the flowchart. If the lifetime and cooling time
are similar, we can conclude that the loop is a monolithic structure that
heats and cools as a homogeneous unit, with uniform temperature over the cross
section. Observations show that this is not case, however. The vast majority
of loops live longer than a cooling time and sometimes much longer (e.g.,
Winebarger, Warren, & Seaton 2003; López Fuentes, Klimchuk, & Mandrini 2007).
If these loops contain cooling plasma, then they cannot be monolithic. Rather,
they must be bundles of thin, unresolved strands that are heated impulsively
at different times. Although each strand cools rapidly, the composite bundle
appears to evolve slowly (e.g., Winebarger, Warren, & Mariska 2003). Multi-
stranded bundles of this type can explain a number of observed properties of
warm loops: over density, long lifetime, super-hydrostatic scale height, and
flat temperature profile. They can also explain the observed under density of
hot loops. Realizing this was a time of rejoicing in the modeling community!
But….
## 4\. Thermal Distribution
An important prediction of the multi-strand model is that loops should have
multi-thermal cross-sections. Since the unresolved strands are heated at
different times, they will be in different stages of cooling and out of phase
with each other. A critical question became “Are loops multi-thermal?” An
intense debate ensued and continues to this day. Some have answered with a
resounding yes (the “Schmelz camp,” e.g., Schmelz & Martens 2006) and others
have answered with a resounding no (the “Aschwanden camp,” e.g., Aschwanden &
Nightingale 2005). As we now demonstrate, however, it is not especially useful
to phrase the multi-thermal question in a way that requires a binary response.
Imagine that a loop bundle is heated by a “storm” of nanoflares that occur
randomly over a finite window in time. It is easy to see that the range of
strand temperatures that are present at any given moment depends on the
duration of the storm. For a very short storm, all of strands will be heated
at about the same time and will cool together. The instantaneous thermal
distribution of the loop will be narrow. In contrast, a storm that lasts
longer than a cooling time will produce a much wider thermal distribution.
Some strands will have just been heated and will be very hot; others will have
cooled to intermediate temperatures; and still others will have had time to
cool to much lower temperatures. The flowchart in Figure 1 therefore asks the
more meaningful question “How multi-thermal is the loop?” A broad thermal
distribution implies a long-duration nanoflare storm, and a narrow
distribution implies a short-duration storm. It now appears that the multi-
thermal and isothermal camps may both be correct.
The duration of the nanoflare storm also determines the lifetime of the loop
bundle, so the thermal width and lifetime will be closely related. Figure 3
shows results for simulated nanoflare storms lasting 500, 2500, and 5000 s,
top to bottom. The left column has light curves (intensity versus time) as
would be observed in the 195 channel of TRACE, with sensitivity peaking near 1
MK. The right column has emission measure distributions, EM($T$) =
$T\times\\!$DEM($T$) cm-5, at the time of peak 195 intensity. Only the coronal
part of the loop is included; the transition region footpoints are neglected.
All three of the storms are comprised of identical nanoflares that have
triangular heating profiles lasting 500 s. They were simulated with our “0D”
hydro code EBTEL and are the same as example 4 in Klimchuk, Patsourakos, &
Cargill (2008). In actuality, Figure 3 was produced with only one simulation.
The light curves and EM distributions were constructed using sliding time
windows that correspond to the storm durations.
Figure 3.: Simulated 195 light curves (left) and emission measure
distributions (right) for nanoflare storms lasting 500, 2500, and 5000 s, top
to bottom. The instantaneous EM distributions are from the times of peak 195
intensity (t = 3958, 4705, 5445 s for the three storms).
As expected, both the lifetime and thermal width increase as the storms get
longer. The full widths at half maximum (FWHM) of the light curves are 1098,
2579, and 5008 sec for the 500, 2500, and 5000 s storms, respectively. The
FWHM of the EM distributions are 0.13, 0.23, and 0.36 in $\log T$. The full
widths at the 1% levels are 0.24, 0.62, and 1.14 in $\log T$.
It may seem surprising at first that the EM distributions do not all reach the
same maximum temperature, since the nanoflares are the same in all three
storms. This is not because individual strands are reheated multiple times in
the longer storms; all strands are heated only once. Rather, it is because the
distributions are from the time of peak 195 intensity. In the short duration
storm, all of the strands have cooled appreciably by the time the peak
intensity is reached. Had we chosen to plot the distribution at an earlier
time, it would still been narrow, but it would be shifted to higher
temperature.
Warren et al. (2008) have made Gaussian fits to EM distributions observed by
Hinode/EIS. They find a typical central temperature of 1.4 MK and a typical
Gaussian half width of 0.3 MK. This corresponds to a FWHM in $\log T$ of
roughly 0.24, which by Figure 3 implies a 195 lifetime of roughly 2500 s.
Although Warren et al. did not measure the lifetimes of their loops, this
value is consistent with the small number of 195 lifetimes that have been
reported for other cases (Winebarger & Warren 2005; Ugarte-Urra, Winebarger, &
Warren 2006). To our knowledge, there does not exist a single published
example where both the thermal width and lifetime have been measured for the
same loop. Making such measurements should be a high priority. It is a crucial
consistency check of the nanoflare concept. Density measurements should be
made at the same time.
## 5\. Very Hot and Very Faint Plasma
The nanoflare model makes two observational predictions in addition to the
ones we have already discussed. First, it predicts that small amounts of very
hot ($>5$ MK) plasma should be present. Figure 4 shows two examples of long
(infinite) duration storms, one comprised of relatively weak nanoflares and
the other comprised of nanoflares that are ten times stronger. The solid curve
in each case is the EM distribution for the whole loop, while the dashed and
dot-dashed curves are the contributions from the coronal section and
footpoints, respectively. We see that the EM of the hottest plasma is 1.5-2
orders of magnitude smaller than that of the most prevalent plasma. The reason
is two-fold. First, the initial cooling after the nanoflare has occurred is
very rapid, so the hottest plasma persists for a relatively brief period.
Second, the densities are low during this early phase, because chromospheric
evaporation has only just begun to fill the loop strand with plasma.
Figure 4.: Emission measure distributions for long (infinite) duration
nanoflare storms comprised of weak (left) and strong (right) nanoflares:
coronal section (dashed), transition region footpoints (dot-dashed), and whole
loop (solid).
As a consequence of the small emission measures, the intensities of hot
spectral lines and channels are predicted to be very faint. The intensities
may be reduced still further by ionization nonequilibrium effects (Bradshaw &
Cargill 2006; Reale & Orlando 2008). Low levels of super-hot emission have
nonetheless been detected recently by the CORONAS, RHESSI, and Hinode missions
(Zhitnik et al. 2006; McTiernan 2009; Patsourakos & Klimchuk 2009; Ko et al.
2009). In particular, EM distributions inferred from multi-filter XRT
observations of two active regions suggest that the distributions may have two
distinct components (Schmelz et al. 2009; Reale et al. 2009). The implications
are considerable, since this would rule out a simple power-law energy
distribution for the responsible nanoflares. Detailed modeling is now
underway.
## 6\. Flows
High-speed upflows that reach or exceed 100 km s-1 are predicted during the
early evaporation phase of a nanoflare event. Depending on the geometry of the
observations, these can produce highly blue-shifted emission. The emission
will be very faint, however, for the reasons given above. A composite spectral
line profile from a bundle of unresolved strands will be dominated by the
weakly red-shifted emission produced during the much longer radiative cooling
phase, when the plasma slowly drains and condenses back onto the chromosphere.
Signatures of evaporation take the form of blue wing enhancements on this main
component (Patsourakos & Klimchuk 2006). They can be very subtle, and they
only appear in lines that are well tuned to the temperature of the evaporating
plasma. Significantly hotter and cooler lines are not expected to show
evidence of evaporation.
We have performed sit-and-stare observations with Hinode/EIS and find blue
wing asymmetries in Fe XVII ($T\approx 5$ MK) similar to those predicted by
our nanoflare models (Patsourakos & Klimchuk 2006). The measurements are very
challenging, however, due to the faint nature of the line. Hara et al. (2008)
also report blue-wing asymmetries that are suggestive of nanoflares.
## 7\. Thermal Nonequilibrium
We have worked our way down the flowchart of Figure 1 and concluded that the
observed properties of many loops can be explained by storms of nanoflares
occurring within bundles of unresolved strands. There remains the possibility,
indicated in the upper right, that many loops can also be explained by thermal
nonequilibrium. We consider this possibility now.
Thermal nonequilibrium is a fascinating phenomenon in which dynamic behavior
is produced by perfectly steady heating (Antiochos & Klimchuk 1991; Karpen et
al. 2001; Mueller, Peter, & Hansteen 2004; Karpen, Antiochos, & Klimchuk
2006). No equilibrium exists if the steady heating is sufficiently highly
concentrated near the loop footpoints. Instead, the loop goes through periodic
convulsions as it searches for a nonexistent equilibrium. Cold, dense
condensations form, slide down the loop leg, and later reform in a cycle that
repeats with periods of several tens of minutes to several hours.
We have recently explored whether thermal nonequilibrium can explain the
observed properties of EUV loops (Klimchuk & Karpen 2009). We first considered
a monolithic loop, which we simulated with our 1D hydro code ARGOS (Antiochos
et al. 1999). The code uses adaptive mesh refinement, which is critical for
resolving the thin transition regions that exist on either side of the dynamic
condensations. We imposed a steady heating that decreases exponentially with
distance from both footpoints. The heating scale length of 5 Mm is one-
fifteenth of loop halflength. We introduced a small asymmetry by making the
amplitude of the heating on the right side only 75% that on the left.
Figure 5 shows the evolution of temperature, density, and intensity as would
be observed in the 171 channel of TRACE. These are averages over the upper 80%
of the loop. The behavior is typical of the several cycles that we simulated.
The loop is visible in 171 for only about 1000 s. This is a factor of 2-4
shorter than observed lifetimes (Winebarger & Warren 2005; Ugarte-Urra,
Winebarger, & Warren 2006). A more serious problem is the distribution of
emission along the loop (the intensity profile), which disagrees dramatically
with observations. Figure 6 shows 171 intensity and temperature as a function
of position along the loop at $t=5000$ s. The emission is strongly
concentrated in transition region layers at the loop footpoints ($s=45$ and
$203$ Mm) and to either side of a cold condensation at $s=163$ Mm. In stark
contrast, most observed 171 loops have a fairly uniform brightness along their
length.
Figure 5.: Evolution of temperature (dashed), density (dotted), and 171
intensity (solid) for a monolithic loop undergoing thermal nonequilibrium. All
quantities are normalized. The steady heating is 75% as strong in the right
leg as in the left. Figure 6.: Temperature (dashed, MK) and 171 intensity
(solid, arbitrary units) as a function of position along the loop at $t=5000$
s in the simulation of Figure 5.
The maximum temperature in the loop is 4.4 MK and occurs before the
condensation forms. We performed another simulation with a reduced heating
rate that has a maximum temperature of only 1.8 MK. Neither the light curve
nor the intensity profile are consistent with observations. We conclude that
EUV loops are not monolithic structures undergoing thermal nonequilibrium, at
least not under conditions that lead to cold condensations. We note, however,
that Mok et al. (2008) report a different type of nonequilibrium behavior. One
prominent loop in their 3D simulation of an active region exhibits a cooling
and heating cycle, but the temperature never drops to the point where a
condensation forms. The reasons for the differing behavior are yet to be
understood. Whether the loop has properties matching observed loops (density,
lifetime, thermal width) is unknown.
The simulation of Figures 5 and 6 may nonetheless have some relevance to the
Sun. The condensation falls onto the right footpoint at $t=5600$ s. Falling
condensations have been seen in the C IV channel of TRACE (Schrijver 2001).
They are relatively rare, however, and occur in only a small fraction of
loops.
We next considered the possibility of a multi-stranded loop bundle in which
the individual strands undergo thermal nonequilibrium in an out-of-phase
fashion. To approximate such a loop, we performed two additional simulations,
similar to the first but with heating imbalances of 50% and 90% instead of
75%. We then averaged all three simulations in time and added them together
along with their mirror images to form a composite loop. The resulting 171
intensity profile is shown in Figure 7. It is reasonably uniform except for
the very intense spikes at the footpoints (note the logarithmic scale). A more
realistic loop bundle with a wider variety of heating imbalances would be even
more uniform. We tentatively conclude that the intensity profile is consistent
with observations, although we are concerned because bright 171 moss emission
is generally observed at the footpoints of SXR loops rather than the
footpoints of EUV loops.
Figure 7.: Logarithm of 171 intensity as a function of position along a
composite loop bundle comprised of individual strands undergoing thermal
nonequilibrium. See text for details. Figure 8.: Temperature as a function of
position along the composite loop bundle of Figure 7. Solid is the actual mean
temperature, while dashed and dotted are the temperatures inferred from TRACE
and Yohkoh/SXT filter ratios, respectively.
Figure 8 shows three temperature profiles for the composite loop: the average
of the actual temperatures in the individual strands (solid), the temperature
that would be inferred from TRACE 171/195 intensity ratios (dashed), and the
temperature that would be inferred from Yohkoh/SXT Al12/AlMg intensity ratios
(dotted). They are different because Yohkoh/SXT is more sensitive to the
hotter plasma and TRACE is more sensitive to the warmer plasma. Notice that
the profiles is very flat. This is a well-know property of EUV loops. We have
also inferred densities from the simulated TRACE observations using exactly
the same procedure that was used for the real loops in Figure 2. The model
loop is over dense by a factor of 23, consistent with observed values. We have
repeated this excise using reduced heating in the strands and find that the
over density is a factor of 10 in this case.
Although there is some reason for encouragement, it is not obvious that
bundles of unresolved strands undergoing thermal nonequilibrium can explain
all the salient properties of observed EUV loops. Reproducing the lifetimes is
especially challenging. The condensations in the different strands must be
sufficiently out of phase to give a uniform intensity profile, but they cannot
be so out of phase as to produce a composite loop lifetime longer than 1 hour.
Even if the phasing is correct for one condensation cycle, it is likely to be
incorrect for subsequent cycles because the interval between condensations
depends on both the amplitude of the heating and its left-right imbalance. The
imbalance determines the location where the condensation forms, and it must be
appreciably different among the strands in order to get a uniform intensity
profile. Note that the results shown in Figures 7 and 8 make use of temporal
averages over complete cycles, and therefore the lifetime of the equivalent
loop bundle is effectively infinite.
## 8\. Conclusions
We have described how a combination of observational and modeling work has led
to the conclusion that warm ($\sim 1$ MK) EUV loops can be explained as
bundles of unresolved strands that are heated by storms of nanoflares. Static
equilibrium is out of the question. The observed lifetimes and thermal
distributions of the plasma indicate that the storms last for typically
2-4$\times 10^{3}$ s. Additional support for this picture is provided by the
shapes of hot spectral line profiles and by the observation that line
intensities peak at slightly later times for lines of progressively cooler
temperature (Ugarte-Urra, Warren, & Brooks 2009). Also, there is now good
evidence for very hot and very faint plasma, as predicted by the nanoflare
models.
It is not clear whether most hot ($>2$ MK) SXR loops are also heated by
nanoflares. If they are, the storms must be long duration in order to explain
the observed lifetimes. The loops would then be expected to have co-spatial
EUV counterparts, and it is not obvious that they do. One possibility is that
the frequency of nanoflares is much higher in long-lived SXR loops, so that
the plasma in a strand never cools to EUV temperatures before being reheated.
It is worth noting that virtually all of the proposed coronal heating
mechanisms predict impulsive energy release on individual magnetic field lines
(Klimchuk 2006).
We considered the possibility that EUV loops can be explained by thermal
nonequilibrium. We concluded that this is not a viable mechanism for
monolithic loops under the conditions we have considered—although the results
of Mok et al. (2008) are very intriguing—but that it may have application in
multi-stranded bundles. Serious questions remain that require further
investigation.
We close by pointing out that distinct loops are only one component of the
corona and that the diffuse component contributes at least as much emission.
It is not generally appreciated that the intensity of EUV and SXR loops is
typically much less than that of the background (of order 10-40%). The diffuse
component may also be made up of individual strands, but we must explain why
the strands have a higher concentration in loops.
### Acknowledgments.
I am very pleased to acknowledge useful discussions with many people, but I
especially wish to thank Spiros Patsourakos, Harry Warren, and Judy Karpen, my
collaborator on the thermal nonequilibrium study that is being published here
for the first time. I benefited greatly from participation in the Coronal
Loops Workshop Series and the International Space Science Institute team led
by Susanna Parenti. Financial support came primarily from the NASA Living With
a Star program.
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|
arxiv-papers
| 2009-04-08T17:47:08 |
2024-09-04T02:49:01.800728
|
{
"license": "Public Domain",
"authors": "James A. Klimchuk",
"submitter": "James Klimchuk",
"url": "https://arxiv.org/abs/0904.1391"
}
|
0904.1425
|
# N-body simulations for testing the stability of triaxial galaxies in MOND
Xufen Wu1,HongSheng Zhao1,2,Yougang Wang3,Claudio Llinares4,Alexander Knebe4,5
1SUPA, University of St Andrews, North Haugh, Fife, KY16 9SS, UK
2Sterrewacht Leiden, P.O. Box 9513, 2300 RA Leiden, Netherlands
3National Astronomical Observatories, Chinese Academy of Sciences, Beijing,
100012, P.R. China
4Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam
5Departamento de Física Teórica, Módulo C-XI, Univ. Autónoma de Madrid,
E-28049 Madrid, Spain
###### Abstract
We perform a stability test of triaxial models in MOdified Newtonian Dynamics
(MOND) using N-body simulations. The triaxial models considered here have
densities that vary with $r^{-1}$ in the center and $r^{-4}$ at large radii.
The total mass of the model varies from $10^{8}M_{\odot}$ to
$10^{10}M_{\odot}$, representing the mass scale of dwarfs to medium-mass
elliptical galaxies, respectively, from deep MOND to quasi-Newtonian gravity.
We build triaxial galaxy models using the Schwarzschild technique, and evolve
the systems for 200 Keplerian dynamical times (at the typical length scale of
1.0 kpc). We find that the systems are virial overheating, and in quasi-
equilibrium with the relaxation taking approximately 5 Keplerian dynamical
times (1.0 kpc). For all systems, the change of the inertial (kinetic) energy
is less than 10% (20%) after relaxation. However, the central profile of the
model is flattened during the relaxation and the (overall) axis ratios change
by roughly 10% within 200 Keplerian dynamical times (at 1.0kpc) in our
simulations. We further find that the systems are stable once they reach the
equilibrium state.
###### keywords:
galaxies: kinematics and dynamics- methods: N-body simulations
## 1 Introduction
Elliptical galaxies are often triaxial and appear stable. A triaxial
equilibrium is non-trivial to build dynamically especially for a system with a
cuspy profile of the light and/or the dark halo. The main objective of this
work is to test whether triaxial models of galaxies are stable in Modified
Newtonian dynamics (MOND, Milgrom 1983). Extensive studies about the stability
of triaxial models have been performed in standard Newtonian gravity (see
below), however, there is no literature on this topic in MOND.
For Newtonian physics, it has been three decades of studies on constructing a
self-consistent model for triaxial galaxies since Schwarzschild numerically
presented the triaxial Hubble profile in 1979 (Schwarzschild 1979; 1982).
Despite its original application to a modified Hubble profile, the method of
Schwarzschild (1979) is still widely used for testing the self-consistency of
various models for the density distribution in galaxies. For instance, Statler
(1987) showed that the perfect triaxial Kuzmin (1973) profile and the de Zeeuw
& Lynden-Bell (1985) profile are also self-consistent. Those models have
constant density cores, however, observations showed that elliptical galaxies
have non-constant cores (Moller, Stiavelli, & Zeilinger 1995; Crane et al.
1993; Jaffe et al. 1994; Ferrarese et al. 1994, Lauer et al. 1995), i.e. the
surface brightness increases quickly towards the central region of the
galaxies. Almost all elliptical galaxies have power-law cusps $\rho\sim
r^{-\gamma}$ with $\gamma$ ranging from 1 to 2 for High Surface Brightness to
Low Surface Brightness elliptical galaxies in the central region. Spherical
models with a fixed value of $\gamma$ have been proposed, e.g. a $\gamma=2$
model by Jaffe (1983) and a $\gamma=1$ model by Hernquist (1990). Today such
models are rather discussed within a family of density distributions with
$\gamma$ being a free parameter (Dehnen 1993, Carollo 1993 and Tremaine et al.
1994). In this regard, Merritt & Fridman (1996) tested the modified Dehnen
profile,
$\rho(r)={(3-\gamma)M\over 4\pi abc}{1\over
r^{\gamma}(1+r)^{4-\gamma}},0\leq\gamma<3,$ (1)
where $r=\sqrt{({x\over a})^{2}+({y\over b})^{2}+({z\over c})^{2}},(c\leq
b\leq a)$, $a$, $b$ and $c$ is the long, intermediate, and short axis of the
ellipsoids. They found that triaxial galaxies with central density cusps
$(\gamma=1)$ were in equilibrium and self-consistent in Newtonian dynamics.
The subsequent work by Capuzzo-Dolcetta et al. (2007) proved that a two-
component triaxial Hernquist system, including a baryonic component plus a
Cold Dark Matter (CDM) halo are also self-consistent.
Modified Newtonian Dynamics (MOND) – proposed by Milgrom (1983a,b) as an
alternative gravity theory – was initially designed to abandon the need for
that yet-to-be-discovered dark matter that (possibly) accounts for as much as
85% of all matter in the Universe. MOND, on the other hand, perfectly predicts
the rotation curves of galaxies as well as the Tully-Fisher relation in the
absence of CDM (McGaugh et al. 2000; McGaugh, 2005). Indeed, MOND successfully
matches the observations on a wide range of scales, from globular clusters
(Angus & McGaugh 2008, in preparation) to different types of galaxies
including dwarfs and giants, spirals and ellipticals (Milgrom 2007; Gentile et
al. 2007; Milgrom & Sanders 2007; Famaey & Binney 2005; Sanders & Noordermeer
2007; Angus 2008a). The development of several frameworks for a relativistic
formulation of MOND (Bekenstein 2004; Sanders 2005; Bruneton & Esposito-Farése
2007; Zhao 2007; Skordis 2008) enabled the study of the Cosmic Microwave
Background (CMB) (Skordis et al. 2006; Li et al. 2008), cosmological structure
formation (Halle & Zhao 2008; Skordis 2008), strong gravitational lensing of
galaxies (Zhao et al 2006; Chen & Zhao 2006, Shan et al. 2008) and weak
lensing of clusters of galaxies (Angus 2007; Famaey et al. 2007a). As a
dynamically selected reference frame, external fields break the Strong
Equivalence Principle (Bekenstein & Milgrom 1984; Zhao & Tian 2006; Famaey,
Bruneton & Zhao 2007b, Feix et al. 2008a,b). Consequently, the rotation curve,
escape speed and morphology of galaxies are determined by both the background
and the internal gravity (Famaey et al. 2007b; Wu et al. 2007, 2008). Despite
its great success we need to accept that even MOND cannot do well without dark
matter completely: a recent study utilizing a combination of strong and weak
lensing by galaxy clusters showed that MOND requires neutrinos of mass $5-7$eV
(Natarajan & Zhao 2008). And to be consistent with (dark) matter estimates of
galaxy clusters and observataions of the CMB anisotropic spectrum (as well as
the matter power spectrum), MOND requires neutrino masses of up to $11$eV
(Angus 2008b). One theory capable of accommodating both these requirements is
that of a mass-varying neutrino by Zhao (2008).
In this paper, we utilize a numerical solver for the MONDian analog to
Poisson’s equation to study the stability of triaxial galaxies in MOND. The
code named NMODY has been widely applied to different problems: it has been
applied to study dissipationless collapses, showing that the end-products are
consistent with several observations (Nipoti, Londrillo & Ciotti 2006;
Nipotti, Londrillo & Ciotti 2007a). The code has also been used to study
various important aspects of galaxy formation. Nipoti et al. (2007b) and
Ciotti et al. (2007a,b) found that phase mixing is less effective and the
timescale of galaxy mergers is longer for MOND than for CDM. Recently, Jordi
et al. (2009) and Haghi et al. (2009) applied the external fields into the
NMODY code and studied the internal dynamics of distant star clusters.
Further, MOND also produces stronger bars than CDM (Tiret & Combes 2007), and
hydrodynamical simulations of spherical bulges indicated that there are tight
correlations between bulge mass, central black hole and stellar velocity
dispersion in MOND (Zhao et al. 2008). These differences and similarities to
CDM simulations immediately lead to the question of the stability of triaxial
systems in MOND as realistic galaxies are not spherically symmetric objects.
Wang et al. (2008) recently found that the self-consistency of a triaxial
cuspy centre $\gamma=1$ also exists for MOND. By extending the original
Schwarzschild method and weighting the orbits during the generation of the
Initial Conditions (ICs) for N-body simulations it is possible to study the
stability and future evolution of these density models (Zhao 1996). This
method proved successful in, for instance, creating equilibrium ICs for a
fast-rotating, triaxial, double-exponential bar reminiscent of a steady-state
Galactic bar (Zhao 1996) when evolved forward in time using a Self-Consistent-
Field code (Hernquist & Ostriker 1992).
Whether there are stable galaxy models in MOND is a lacuna in the studies. It
is important to build stable galaxy models for dynamical studies. Here we will
expand upon previous work by studying the stability and evolution utilizing
direct N-body simulations. Our target of study will be an isolated triaxial
galaxy with a mild cusp of $\gamma=1$ in the centre within the Bekenstein-
Milgrom MOND theory (1984). We use the same density models applied in Wang et
al. (2008), with total mass ranging from $10^{10}M_{\odot}$ to
$10^{8}M_{\odot}$, respectively, representing medium-mass elliptical galaxies
down to dwarf ellipsoidals, which are in quasi-Newtonian to deep MONDian
gravity. We generate the ICs utilizing the method outlined in Zhao (1996) and
our N-body simulations confirm that these systems are (initially) in quasi-
equilibrium and relax on a rather short time scale of only a few Keplerian
dynamical times (1.0 kpc) (see below, sub-section 3.1). The systems quickly
reach a state of equilibrium, consistent with the results of Wang et al.
(2008). The inertial energy changes by less than 10% and the kinetic energy by
less than 20% during the relaxation process. At the same time, the initial
$\gamma=1$ cusps are flattened. After the relaxation, the systems remain
stable. We further note that the triaxialities of the systems do not change
significantly during 200 Keplerian dynamical times (1.0 kpc). Moreover, the
scalar Virial theorem is valid at any time.
## 2 Models, Schwarzschild technique, and ICs for N-body
### 2.1 Poisson’s equation in MONDian
The MONDian Poisson’s equation can be written as (Bekenstein-Milgrom 1984):
$\nabla\cdot\left[\mu\left({|\nabla\Phi|\over
a_{0}}\right)\nabla\Phi\right]=4\pi G\rho,$ (2)
where $\Phi$ is the MONDian potential generated by the matter density $\rho$.
For the gravity acceleration constant, we use $a_{0}=3600$ km2s-2kpc-1, which
is same as adopted by Milgrom (1983a,b), Sanders & McGaugh (2002) and
Bekenstein (2006). The so-called MONDian interpolation function $\mu$ is
approaching 1 for $|\nabla\Phi|>>a_{0}$ (Newtonian limit) and
$\mu\to{|\nabla\Phi|\over a_{0}}$ for $|\nabla\Phi|<<a_{0}$ (deep MOND
regime), and the gravity acceleration is then given by $\sqrt{a_{0}g_{N}}$,
taking the place of the Newtonian acceleration $g_{N}=\nabla\Phi_{N}$ at the
same limit. For our simulations we chose the ’simple’ $\mu$-function in the
form of (Famaey & Binney 2005; Zhao & Famaey 2006; Sanders & Noordermeer 2007)
$\mu(x)={x\over 1+x}.$ (3)
Furthermore, we use the density distribution given by Eq. 1, choosing
$\gamma=1$, and $M$ being the total mass of the system. For our simulations,
we choose the ratios $a:b:c=1:0.86:0.7$, with $a=1$kpc.
### 2.2 Initial Potential
A very important step in our calculation is to solve Poisson’s equation in
MOND (cf. Eq. 2). This is achieved via numerical integration utilizing the
N-body code NMODY (Ciotti et al. 2006; Nipoti et al. 2007a) on a spherical
grid of coordinates $(r,\theta,\psi)$. To this extent we applied a grid of
$256\times 64\times 128$ cells. Note that we yet do not evolve the system
forward in time; we simply extract the potential of our (static) density
distribution and use it for the Schwarzschild method detailed below.
### 2.3 Schwarzschild technique
Since Schwarzschild (1979, 1982) pioneered the orbit-superposition method to
construct self-consistent models of galaxies, this technique has been widely
applied in dynamical studies (e.g., Zhao 1996; Rix et al. 1997; van der Marel
et al. 1998; Kuijken 2004; Binney 2005; Capuzzo-Dolcetta et al. 2007). The
essence of this method is to sample phase-space with a large number of orbits.
Properly assigning weights to different orbits can then give rise to the mass
distribution we are interested in.
Specifically, let $N_{\rm orbits}$ be the total number of orbits and $N_{\rm
cells}$ be the number of spatial cells (both will be specified below). For
each orbit $j\in N_{\rm orbits}$, we count the fraction of time, denoted by
$O_{ij}$, that it spends in each of the cells $i\in N_{\rm cells}$. The
occupation number $W_{j}$ for each orbit $j$ is then determined by the
following set of linear equations
$\sum_{j=1}^{N_{\rm
orbits}}W_{j}O_{ij}=M_{i},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1,...,N_{\rm cells}\
,$ (4)
where $M_{i}$ is the mass in cell $i$ expected from the given mass
distribution. We have checked the sum $\sum_{i=1}^{N_{\rm cells}}O_{ij}$
equals unity for each orbit. There are now various choices of how to actually
solve Eq. 4: liner programming (Schwarzschild 1979; 1982; 1993), Lucy’s method
(Lucy 1974; Statler 1987), maximum entropy methods (Richstone & Tremaine 1988;
Statler 1991; Gebhardt et al. 2003), or least-square solvers (Lawson & Hanson
1974; Merritt & Fridman 1996; Capuzzo-Dolcetta et al. 2007). We chose the
least-square method (cf. Wang et al. 2008).
Because of the symmetry of the mass distribution specified by Eq. (1), it is
sufficient to only consider mass cells in the first octant in our analyses.
Following Merri & Fridmann (1996), we divide the first octant into cells of
equal masses, i.e. the first octant of each initial system is divided into 21
sectors by 20 shells, where every sector has the same amount of mass inside.
The first octant is further divided into three parts by the planes $z=cx/a$,
$y=bx/a$ and $z=cy/b$ (see Fig. 1, upper left panel). Each of these three
parts is sub-divided into 16 cells by the planes $ay/bx=1/5$, $2/5$, $2/3$ and
$az/cx=1/5$, $2/5$, $2/3$ (see Fig. 1, upper right panel). Therefore, the
total number of cells in the first octant is $16\times 3\times 21=1008$.
However, we only consider the innermost 912 cells (the inner 19 sectors) when
solving Eq. 4 since only a few orbits supply densities in the outermost sector
of grid cells; we simply discard the cells of the outermost sector.
Figure 1: The first octant is divided by planes $z=cx/a$, $y=bx/a$ and
$z=cy/b$ (upper left panel) into three parts. Each part is subdivided by
planes $ay/bx=1/5$, $2/5$, $2/3$ and $az/cx=1/5$, $2/5$, $2/3$ into 16 cells
(upper right panel). In the lower panel, curve A is the circle of the minimal
radius of 1:1 resonant orbit at the energy $E_{k}$, curve B is the zero-
velocity surface of the energy $E_{k}$. There are 10 dotted lines
$x=z\tan\theta$ divide the values of $\theta$ from $2.25^{\circ}$ to
$87.75^{\circ}$. The 15 diamonds equally divide the radius into 16 parts. The
diamonds are the initial positions from which the $x-z$ orbits are launched.
In a spherical system, the total energy and the three components of the
angular momentum are integrals-of-motion. However, in a triaxial system only
the energy remains constant (Merritt 1980; Valluri & Merritt 1998;
Papaphilippou & Laskar 1998). We therefore use the 7/8 order Runge-Kutta
algorithm (Fehlberg 1968) for orbital calculations, with 100 orbital times
(see below Section 3.1) for the full integration time of each orbit. We
followed Schwarzschild (1993) and Merritt & Fridman (1996) in assigning
initial conditions from one of two sets of starting points (cf. also Wang et
al. 2008): stationary orbits with zero initial velocity, and orbits in the
$x-z$ plane with $v_{x}=v_{z}=0$, and $v_{y}=\sqrt{2(E-\Phi)}\neq 0$ in the
first octant. Note that there is quite a large number of non-symmetric orbits
that will lead to artifacts in the procedure if only being considered in the
first octant. To circumvent this problem and to keep the symmetry of the
system, we reflect the orbits from the boundaries of each octant. Note that by
our method the computational workload is not increased as would be the case
when calculating eight octants.
As mentioned above, there are $16\times 3$ cells in each sector. To obtain
enough orbits for the library, we sub-divide the cells by the midplanes of the
cells once again, i.e., the midlines of each grid as seen in the upper right
panel of Fig. 1; these midlines equally divide the cell at $x=az/c$ and
$x=ay/b$. Thus we have $4\times 16\times 3=192$ sub-cells in each sector. The
central points on the outer shell surfaces of the sub-cells are the launching
points. Hence there are 192 stationary starting orbits in each sector. The
total energy on the $k$th sector is defined as the outer boundary shell
$E_{k}$, and for the stationary orbits inside the $k$th sector this amounts to
$E_{k}=\Phi(x,y,z)$. For the orbits launched from the $x-z$ plane the initial
energy is $E_{k}=\Phi(x,0,z)$, as shown in Fig. 1 (lower panel). This figure
further shows that the radius of the inner shell (marked as curve A) is the
minimal radius of 1:1 resonant orbits (x:y), and the outer shell (marked as
curve B) is the zero velocity surface. We define 10 lines satisfying
$x=z\tan\theta$ where $\theta$ lies within the range $2.25^{\circ}$ to
$87.75^{\circ}$. Along the radial direction, we equally divide the radius
between two boundaries into 16 parts with 15 points, where those 15 points are
the initial positions for the orbits launched from the $x-z$ plane. Hence
there are 150 $x-z$ plane starting orbits.
In summary, we have 192 stationary and 150 $x-z$ plane orbits in each sector
amounting to a total of $N_{\rm orbits}=6840$ and we use $N_{\rm
cells}=16\times 3\times 19=912$ cells for the generation of our orbit library.
The energy in each sector is a constant which equals the potential energy on
the outer shell surface. As a result, the energies for the systems can be
considered ’quantized’ with each system having 19 ’energy levels’. In Fig. 2
there are some examples of asymmetric orbits: the upper four panels are
stationary starting orbits, and the lower four panels are orbits launched from
the $x-z$ plane. In order to generate equilibrium Initial Conditions for the
$N$-body simulations to be presented in Section 3, we need to symmetrize the
orbits by using ’mirror particles’.
Figure 2: The asymmetric non-zero weights orbits in the orbit library. The
upper four panels: stationary starting orbits and the lower four panels: x-z
plane launched orbits. All the left panels are orbits projected on x-y plane
and right panels are projected on x-z plane.
Figure 3: The accumulated of energy distribution of different orbit families
in the intermediate model with a total mass of $M=10^{9}M_{\odot}$. The
horizontal axis is dimensionless energy of ${E\over GM/1.0kpc}$, and vertical
axis is the integration of mass as a function of energy. The dashed, dotted,
dot-dashed lines are for box, chaotic and loop orbits. The solid line is for
all of the orbits.
Finally, in Fig. 3 we show the integration of mass as a function of energy for
the model with a mass of $10^{9}M_{\odot}$. There we find that more than half
of the mass stems from loop orbits with chaotic orbits contributing more than
$1/3$ of the mass; box orbits therefore do not play an important role in the
model. We further like to note that the self-consistency of the model in MOND
has been examined in Wang et al. (2008), and Antonov’s third law was applied
to check the stability of the models initially. However, it is unknown whether
or not Antonov’s third law is also valid in MOND so far. The most direct way
to check for the stability and investigate the evolution N-body of the system
is by means of N-body simulation to be elaborated upon in the following sub-
section.
### 2.4 ICs for N-body systems
In order to study the stability and evolution of the systems, we need to
convert the orbits into an N-body model. According to Zhao (1996), the number
of particles $n_{j}$ on the $j$th orbit is proportional to the weight of the
orbit, i.e., for an $\mathit{N}$ particle system there are $W_{j}N$ particles
on the $j$th orbit. Here we sample the particles on the $j$th orbit
isochronously at $t_{j}={T_{j}\over n_{j}}\times(i+0.5),i=0,1,2,...,n_{j}-1$
where $T_{j}$ is the total integration time of the $j$th orbit. To this extend
we interpolate the positions and velocities from the 6-dimensional output data
of the Schwarzschild orbits. We generate
$n_{j}=W_{j}N$ (5)
particles on the j$\mathit{th}$ orbit and symmetrize the particles in phase-
space. The remaining particles are kept as our Initial Conditions of the
N-body system.
## 3 N-body simulations in MOND
All results presented in this section were obtained by evolving our systems
forward in time with using the N-body particle-mesh code NMODY (Ciotti et al.
2006; Nipoti et al. 2007a).
### 3.1 Technical Details
In our simulations, we have $\mathit{N}=8\times 10^{5}$ particles for each
model, and choose a grid for the numerical integration of Eq. 2 with $64\times
32\times 64$ cells in the spherical coordinates $(r,\theta,\psi)$, where the
radial grids are defined by $r_{i}=2.0\tan[(i+0.5)0.5\pi/(256+1)]$kpc. The
density is obtained via a quadratic particle-mesh interpolation and the time
integration is performed by the classical two order leap-frog scheme. As our
time unit for all subsequent plots we use the following definition (cf. Wang
et al. 2008)
$T_{\rm simu}=\left({GM\over a^{3}}\right)^{-1/2}=4.7\times
10^{6}yr\left({M\over 10^{10}M_{\odot}}\right)^{-1/2}\left({a\over
1kpc}\right)^{3/2}.$ (6)
which represents the Newtonian (or Keplerian) dynamical time at the radius of
$r=a$ without the factor of $2\pi$. We remind the reader that the parameter
$T_{\rm simu}$ is neither the dynamical time nor the orbital time in general
MOND simulations. The orbital time in our MONDian systems is defined as the
period of the 1:1 resonant orbit in the $x-y$ plane (Wang et al. 2008). Fig. 4
shows the periods of circular orbits at different radii for the three models
presented here. Fig. 4 as well as Eq. 6 imply that the MONDian dynamical time
at the radius of 1 kpc is about $7T_{\rm simu}$, $5T_{\rm simu}$ and
$3.5T_{\rm simu}$ for the three models whose masses are $10^{10}M_{\odot}$,
$10^{9}M_{\odot}$ and $10^{8}M_{\odot}$. We further like to note that the
internal time step used by the code NMODY to integrate the equations-of-motion
is ${0.3\over\sqrt{\max|\nabla\cdot\bf g|}}$, where the factor $0.3$ is a
typical number used in N-body simulations, and $\nabla\cdot\bf{g=4\pi
G\rho_{eff}}$, where $\rho_{eff}$ is the effective dynamical density of the
system, i.e. the sum of the baryon and (phantom) dark matter density in the
Newtonian force law to produce the gravity or potential of baryons in MOND
(see We et al. 2008). The time steps here are determined by the maximum values
of $\nabla\cdot\bf{g}$, which means the densest dynamical region of the
models, where gravity changes most sharply. Note that all particles share a
common time step that typically is $0.005\sim 0.03T_{simu}$.
A flowchart of the technical steps involved in the process prior to the
analysis stage can be viewed in Fig. 5. This figure summarizes the methodology
of how to generate and evolve the $N$-body systems. In Table 1 we present the
total times each systems has been evolved for.
Table 1: Total simulation times = $200\times T_{\rm simu}$. Model | N-body run duration $T$ | unit time $T_{simu}$
---|---|---
$10^{10}M_{\odot}$ | 0.94 Gyrs | 4.7 Myrs
$10^{9}M_{\odot}$ | 3.0 Gyrs | 14.9 Myrs
$10^{8}M_{\odot}$ | 9.4 Gyrs | 47.0 Myrs
Figure 4: The period of 1:1 resonant circular orbits on the $x-y$ plane as a
function of radius. The solid, dotted, and dashed lines are for models with
mass of $10^{10}M_{\odot}$, $10^{9}M_{\odot}$ and $10^{8}M_{\odot}$,
respectively.
Figure 5: Flowchart of the simulations.
### 3.2 Virial Theorem
The scalar Virial theorem, $W+2K=0$, is valid for systems in equilibrium,
where $W$ is the Clausius integral,
$W=\int\rho\vec{x}\cdot\nabla\Phi d^{3}x,$ (7)
and $K$ is the kinetic energy of the system (Binney & Tremaine 1987). In the
left panel of Fig. 6, we show that the evolution of $-2K/W$ for all models is
always about unity, as expected for an equilibrium system. We though note that
during the first circa five Keplerian dynamical times (1.0 kpc) all systems
are moving from a quasi-equilibrium state with $-2K/W\approx 1.1-1.2$ to
$-2K/W=1.0\pm 0.1$ afterwards (marginally oscillating about unity). This
figure demonstrates that our $N$-body ICs start off in quasi-equilibrium and
after approximately five Keplerian dynamical times (1.0 kpc) can be considered
fully relaxed. These ’hot’ $N$-body ICs could be due to a number of reasons
including the resolution of the simulation and chaotic orbits, respectively.
Regarding the latter, we need to mention that we compute the orbit library for
100 orbital times, and the time integration may not be long enough to ensure a
relaxation of those chaotic orbits; particles coming from the chaotic orbits
could lead to higher pressure ’overheating’ the system.
In Fig. 6, we plot the velocity dispersion $v_{\rm rms}$ for all systems as a
function of the simulation time unit. The plot indicates that each $v_{\rm
rms}$ decreases by about 10% during the relaxation process and stays constant
afterwards (with tiny variations though).111There are typos in Wang et al.
(2008) about the total mass of models and $v_{rms}$.
Figure 6: The evolution of $2K/|W|$ for all three systems. The solid, dotted,
and dashed lines are for models with mass of $10^{10}M_{\odot}$,
$10^{9}M_{\odot}$ and $10^{8}M_{\odot}$, respectively. The evolution is shown
for 200 Keplerian dynamical times (1.0 kpc).
Note that (as inferred from the right panel of Fig. 6) the kinetic energy of
the systems decrease nearly one quarter for the maximal evolved case after the
relaxation.222The kinetic energy $K$ is proportional to $v_{\rm rms}^{2}$.
However, the Virial theorem is still satisfied. That does not mean the energy
conservation law is broken: $W+K$ is not the total energy of a MONDian system,
and it isnot conserved either. The total energy is still the conserved
quantity but for a MONDian system it is given by (Bekenstein & Milgrom 1984):
$E=-L+K$ (8)
where $L$ is the Lagrangian of the MONDian system, defined by
$L=\int d^{3}r\left\\{\rho\Phi+{1\over 8\pi
G}a_{0}^{2}\mathcal{F}\left[{(\nabla\Phi)^{2}\over
a_{0}^{2}}\right]\right\\},$ (9)
and $\mathcal{F}(x^{2})$ is an arbitrary function with
$\mu(x)=\mathcal{F}^{\prime}(x^{2})$. For an isolated system in MOND, the
potential is logarithmic thus the potential energy is infinite. Therefore, the
only meaningful quantity is the difference in energies between different
systems (Bekenstein & Milgrom, 1984; Nipoti et al. 2007). However, the evident
evolution of $W+K$ at the very beginning (i.e. the first 5 Keplerian dynamical
times (1.0 kpc)) shows that the N-body ICs are not accurately in equilibrium,
and hence referred to as quasi-equilibrium.
### 3.3 Energy Distribution
One of the characteristic quantities to describe relaxation processes is the
so-called differential energy distribution, i.e. the quotient of mass $dM$
over the energy band interval $[E,E+dE]$ (Binney & Tremaine 1987). The energy
of a unit mass element is $E={1\over 2}v^{2}+\phi(\vec{x})$, where $\phi$ is
logarithmically infinite in MOND and hence all particles are bound. But since
the absolute value of potential energies is meaningless, we can define the
zero point as the last point of the radial grid. Hence, there are positive
relative energies for part of the particles though all of them are bound to
the system.
The left panels of Fig. 7 show the evolution of ${dM\over dE}$ over 200
Keplerian dynamical times (1.0 kpc) for all three models (upper to lower) and
we find that all distributions are rather similar. And the most pronounced
evolution of the energy distribution is at the low-$E$ end, where particles
are most strongly bound to the system. All of the differential energy
distributions have 19 peaks, as can be seen in left panels of Fig. 7 That is
due to the energy definition of the Schwarzschild technique outlined in §2.3:
Inside every sector, the energy (kinetic plus potential energy) is a constant,
while the outer shell is the zero-velocity surface of this sector. The
adjacent two sectors have energy jumps at the shell. Therefore there are 19
’quantized energy levels’ for our models, and for each model there are no mass
distributions outside these 19 constant ’energy levels’ and hence they appear
as ’valleys’ in the left panels of Fig. 7. That explains why the curves appear
noisy. We note that after the initial relaxation of about five Keplerian
dynamical times (1.0 kpc)333Even though we do not show the curves for 5
Keplerian dynamical times (1.0 kpc) we acknowledge that the drop happens
during that initial relaxation phase. Here we care about the long-term
evolution within 200 Keplerian dynamical times (1.0 kpc) and hence decided to
rather focus on the late evolution of the systems. the low-$E$ end of the
distribution becomes devoid of particles, i.e., particles are leaving the
central regions where the potential well is deepest. This actually hints at a
possible flattening of the initially present density cusp $\gamma=1$! We
return to this issue later in sub-section 3.5.
Comparing the three left panels in Fig. 7, we observe that the system in the
mild MOND regime (i.e. the model with a mass of $1\times 10^{10}M_{\odot}$:
upper-left panel) has the most significant evolution, whereas the model in
deep MOND evolves least (lower-left panel). We therefore conclude that our ICs
are most stable for the deep MOND regime.
Figure 7: Left panels: Evolution of the differential energy distribution
${dM\over dE}$. The panels (from upper to lower) correspond to our models with
total mass of $10^{10}M_{\odot}$, $10^{9}M_{\odot}$ and $10^{8}M_{\odot}$,
respectively. Right panels: The accumulation of energy distribution. The black
lines denote the ICs, and the violet, blue, yellow and green lines show the
differential energy after 50, 100, 150, 200 Keplerian dynamical times (1.0
kpc). Both ${dM\over dE}$ and the Energy are given in units where G=1 and M=1.
As seen in the right panels of Figure 7, the accumulation of energy
distribution clearly confirms the previous conclusion from the differential
distributions. After the relaxation, the mass in the inner region escapes to
the outer, while the outer part is nearly unchanged. The mass distribution
obviously does not evolve after relaxation.
### 3.4 Kinetics
To further check upon the stability of our systems, we calculate the radial
velocity dispersion profiles $\sigma_{r}(r)$ as well as the anisotropy
parameter
$\beta(r)\equiv 1-{\sigma_{\theta}^{2}+\sigma_{\psi}^{2}\over
2\sigma_{r}^{2}}.$ (10)
Here $r$ is the spheroidal radius, the same as in Eq. 1, and
$\sigma_{\theta}$, $\sigma_{\psi}$ are the tangential and azimuthal velocity
dispersions.
The results can be viewed in the left panels of Fig. 8. We find that, for all
three models, $\sigma_{r}$ drops within the central 2 kpc during the
relaxation at the beginning of the simulation, i.e. the first five Keplerian
dynamical times (1.0 kpc) (though not shown for clarity). Afterwards, there is
only very little evolution noticeable. The reduction of $\sigma_{r}$ in the
core means that the ICs are too hot in the radial direction to sustain
equilibrium. We also note that the drop is more pronounced the less MONDian
the ICs are. In the mild MONDian model with a mass of $M=10^{10}M_{\odot}$,
(upper-most panel) the model obviously appears to be hotter inside than
outside. This trend is weakened for the deep-MOND model with a total mass of
$M=10^{8}M_{\odot}$ where the self-gravity of the system is much weaker than
$a_{0}$. The slope of $\sigma_{r}(r)$ oscillates around a constant value of
approximately 20 km/s. In the intermediate model with $M=10^{9}M_{\odot}$ the
slope is between the most massive and least massive ones.
Figure 8: Left panels: Evolution of the radial velocity dispersion
$\sigma_{r}(r)$. Right panels: Evolution of the velocity dispersion anisotropy
$\beta(r)$. The upper, middle and lower panels are corresponding to models of
$M=1.0\times 10^{10}M_{\odot}$, $M=1.0\times 10^{9}M_{\odot}$ and $M=1.0\times
10^{8}M_{\odot}$. The ordering of the panels corresponds to Fig. 7 as does the
colouring of the lines.
However, after the relaxation, the slope of $\sigma_{r}(r)$ has the same
behavior in all three models, radially cooling down towards the cores. The
curves of velocity dispersion after relaxation look like the rotation curves
of disc galaxies at the similar mass range (Milgrom & Sanders 2007; Gentile
2008). In the centres, the tangential velocity dispersion
$\sigma_{\theta}^{2}+\sigma_{\psi}^{2}$ plays an important role for the Virial
Theorem and keeps the cores in equilibrium. Therefore, the systems prefer more
isotropic velocity dispersions in the cuspy centres. To confirm this, we also
present the anisotropy parameter $\beta(r)$ in the right panels of Fig. 8. We
do find the expected small values of $\beta$ in the centres as well as a
radial increase of $\beta(r)$. Inside 1 kpc the $\beta$-profiles oscillate
during the whole evolution while they remain stable in the outer parts.
Furthermore, the anisotropy increases with radius in all three models out to
about 25 kpc where it turns approximately constant, $\beta=0.6$, i.e., the
velocities are distributed hyper-radially.
Nevertheless, within 2 kpc, there should be a more substantial redistribution
of kinetic energies in the tangential direction after relaxation due to the
evolution of $\sigma_{r}$ seen in the left panels of Fig. 8; otherwise, the
systems lose quite a lot of kinetic energy in the core. Unfortunately, the
evolution of $\beta$ in the central region, seen in right panels of Fig. 8, is
not as large as expected. Thus, there are outflows of kinetic energy from the
centres of the systems. The values of velocity dispersion and anisotropic
parameters in the deep MOND regions of our systems fit pretty well with the
analytical predictions of isothermal spheres by Milgrom (1984; 1994).
### 3.5 Mass distributions
Due to parts of the kinetic energies spilling out of the cores (cf. sub-
section 3.3 and 3.4), the mass densities could redistribute at the same time.
Indeed, we find that there are outflows of mass: the cuspy centers with an
initial value of $\gamma=1$ are flattened. This can be viewed in Fig. 9 where
we show the densities along the major axis (left panels) and cumulative (right
panels) mass distributions for our MOND models. With regards to the density
panels, the three models show a similar behavior. It is clear that the mass is
redistributed during the relaxation with losses in the very central region of
$r<0.5$ kpc and gains outside. The density curves are ocsillating around the
initial analytical density as given by Eq. 1. Therefore, the system becomes
slightly less cuspy and keeps the triaxial density after reaching equilibrium.
Note that the density distribution still remains triaxial after the system is
in equilibrium, and there is no obvious evolution within 200 Keplerian
Dynamical times (at the typical scale a=1.0 kpc). The right panels of Figure 9
show the total mass inside the radial direction $r$. The black dashed straight
lines in right panels are defined by $M_{0}={a_{0}r^{2}\over G}$, the mass to
produce the gravity acceleration $a_{0}$ in a point mass approximation.
$M_{0}$ is the watershed of the enclosed mass producing MOND and Newton
dominating gravities. At a certain radius $r$, when the enclosed mass is
smaller than $M_{0}$, there occurs a transition to MONDian gravity. We find
that in all of the three models MONDian effects cannot be ignored. Even for
$1.0\times 10^{10}M_{\odot}$, the MONDian gravity dominates the regions of
$r>10^{0.3}\sim 2$ kpc. Obviously, the model $M=1.0\times 10^{8}M_{\odot}$ is
in deep MOND region. The colours show the evolution of the systems. Mass in
the inner part of the system is lost during the density re-distribution, while
in the outer part, beyond 4 kpc, the total mass is not affected. We further
note that after the redistribution (during the relaxation) the mass
distribution has stabilized.
Figure 9: The evolution of the mass distribution for the models with
$M=10^{10}M_{\odot}$ (upper), $M=10^{9}M_{\odot}$ (middle),
$M=10^{8}M_{\odot}$ (lower). The left panels show the density distributions on
the major axis the density information can be obtained from the axis ratios of
Figure 10. The right panels show the accumulated mass inside the radius $r$.
The dashed black lines in the right panels are defined as ${a_{0}r^{2}\over
G}$, which are the watersheds of enclosed mass producing MONDian dominating
gravity (below the lines) and Newtonian dominating gravity (upon the lines).
The colouring of the lines is representative of the evolutionary stage of the
model and corresponds to Fig. 7.
### 3.6 Shape
As confirmed in §3.5, the mass redistributes inside our systems during
relaxation. Hence, the question arises whether the shape (i.e., the initial
triaxiality) remains stable or undergoes changes. To address this, we show the
evolution of the axis ratios of the eigenvalues $\sqrt{I_{yy}/I_{xx}}$ and
$\sqrt{I_{zz}/I_{xx}}$ of the moment of inertia tensor $m_{ij}x_{i}x_{j}$,
($m_{ij}=M/N$) in Fig. 10. The three models give similar results, hence we
highlight the model with a total mass of $10^{9}M_{\odot}$ in the upper panel.
We find that the axial ratios (as a function of radius) merely evolve about
10% within 200 Keplerian dynamical times (1.0 kpc). The system is rounder in
the center, but the whole system keeps the triaxial shape during the long
stage of evolution. It is clear that the axis ratio between the minor and
major axes is more stable than that of the intermediate and major axes. Not
only the ratios are almost constant, but also the absolute values of $I_{xx}$,
$I_{yy}$ and $I_{zz}$ seem in dynamic equilibrium and stable, changing less
than $20\%$ during the oscillation (cf. Fig 11). However, we note that at the
time $t=0$ the ratios of $\sqrt{I_{yy}/I_{xx}}$ and $\sqrt{I_{zz}/I_{xx}}$ do
not accurately equal $b:a=0.86$ and $c:a=0.7$ in most of the inner regions,
which is caused by the numerical effects in generating the $N$-body ICs.
However, at the edge of the galaxy (i.e., including more than 80% of the total
mass), the axis ratios are close to the suggested ones. We conclude that the
ICs generated by our application of the Schwarzschild technique roughly lead
to a 5% error for the axis ratios.
A study of the tensor kinetic energies $K_{xx}$, $K_{yy}$ and $K_{zz}$,
defined as $K_{xx}=0.5<v_{x}\cdot v_{x}>$, shows a similar behaviour to the
moment of inertia tensor analysis presented above. The ratios remain constant
although the absolute values change by at most 20%. This can again be verified
in Figure 11 where we plot the inertial (left panel) and kinetic energy (right
panel) components for the three models.
Figure 10: Evolution of axis ratios with the median model of a total mass of
$1.0\times 10^{9}M_{\odot}$ (Upper panel), $1.0\times 10^{10}M_{\odot}$ (lower
left panel) and $1.0\times 10^{8}M_{\odot}$ (lower right panel). The lower and
upper series of lines are for the ratios of minor : major axis and
intermediate : major axis, i.e., $\sqrt{I_{zz}/I_{xx}}$ and
$\sqrt{I_{yy}/I_{xx}}$. The different line symbols are defined the same as in
the figure: solid, dotted, short dashed, dot-dashed and long dashed lines are
for system evolving 0, 50, 100, 150 and 200 $T_{simu}$, where $T_{simu}$ is
the Newtonian orbital time at 1.0 kpc .
Figure 11: Upper, middle and lower panels are different mass models of
$10^{10}M_{\odot}$, $10^{9}M_{\odot}$ and $10^{8}M_{\odot}$. The left panels
are the evolution of the inertial tensor $I_{xx}$ (solid line), $I_{yy}$
(dotted) and $I_{zz}$ (dashed). The right panels are the evolution of kinetics
energy $K_{xx}$ (solid), $K_{yy}$ (dotted) and $K_{zz}$ (dashed). The total
simulation time is 200 $T_{simu}$.
As a final note, considering existing Schwarzschild plus N-body simulations in
the literature, we find that the evolution seen in our MONDian cuspy
elliptical models is comparable to that seen in Fig.5 of Poon & Merritt (2004,
ApJ 606, 774) for triaxial ellipticals in Newtonian gravity. Our simulations
are much longer than 10 crossing times, which provides a typical scale for
checking stability in Newtonian Schwarzschild simulations in the literature
(Poon & Merritt 2004, Zhao 1996).
## 4 Conclusions and Discussion
We explored the stability and evolution of the triaxial Dehnen model (Dehnen
1993; Merritt & Fridman 1996; Capuzzo-Dolcetta et al. 2007) with a $\gamma=1$
central cusp using MOND. We utilized the Schwarzschild method (Schwarzschild
1979) to build orbit models which were in turn used to generate initial
conditions (ICs) for N-body simulations using the method outlined in Zhao
(1996). These ICs were evolved forward in time for 200 Keplerian dynamical
times (at the typical length scale of 1.0 kpc) by the numerical integrator
NMODY developed by the Bologna group (Ciotti et al. 2006; Nipoti et al. 2007)
and designed to include the effects of MOND. We additionally ran the same
simulations with a second MONDian gravity solver AMIGA (Llinares, Knebe & Zhao
2008, cf. Appendix B) based upon an entirely different grid-geometry to
confirm the credibility of our results.
In our simulations, the virial theorem was satisfied at all times. We showed
that the systems start in quasi-equilibrium with a short relaxation phase of
approximately less than five Keplerian dynamical times (1.0 kpc). We found
outflows of energy and mass from the centres of the systems under
investigation. Hence, during the relaxation stage, there is a flattening of
the initially present $\gamma=1$ cusp to a core. Despite the obvious mass
redistribution, we need to acknowledge that the shape of the systems remained
unchanged in the course of the simulations; the axis ratios of the eigenvalues
of the moment of inertia tensor (as well as the kinetic energy tensor) stayed
constant.
The effects of resolution of the simulations should not remain unmentioned. We
found that the potential calculated from the N-body ICs differs by 10%
compared to the analytical potential. Furthermore, the analytically predicted
velocity dispersions at the initial time are $107.3$ km/s, $54.2$ km/s and
$29.3$ km/s for the models $M=10^{10},10^{9},10^{8}M_{\odot}$, respectively.
However, they do not match the (numerical) values plotted in right panel of
Fig. 6. Hence we use the Clausius integral $|W|$ in Equation 7, calculate the
analytical densities for the systems at $t=0$, to minimize the errors.
Moreover, we have found that due to the resolution limitation of the NMODY
code, the errors accumulate during the simulations, which is insensitive to
more massive systems, but becomes an issue when the mass of the system
decreases. This causes small non-zero net velocities in the systems. The
simulation centres of the systems with $10^{9}M_{\odot}$ and $10^{8}M_{\odot}$
move significantly after 200 Keplerian dynamical times (1.0 kpc) and hence we
restrict our analysis to this time frame. To further check the credibility of
our results and the dependence on the code, we ran the simulations again with
a technically substantially different code (AMIGA), which is also capable of
integrating the analog to Poisson’s equation (cf. equation 2 in Appendix B).
The results are practically indistinguishable reassuring their tenability.
We like to close with the notation that our systems are isolated systems,
corresponding to the cases of field galaxies. The self-potentials of the
systems in MOND are logarithmic at large radii, therefore no stars can escape
from such systems. However, for any system embedded in external fields, the
potential is truncated when the strength of the external field becomes
comparable to the internal field (Milgrom 1984; Wu et al. 2007). Therefore,
Poisson’s equation should be modified to
$\nabla\cdot\left[\mu\left({|\nabla\Phi_{int}-\vec{g}_{ext}|\over
a_{0}}\right)(\nabla\Phi_{int}-\vec{g}_{ext})\right]=4\pi G\rho_{b},$ (11)
where the $\mu$-function is determined by both the internal and external
gravitational accelerations. Hence the strong equivalence principle is
violated, and the directions along and against the external field, the
$\mu$-function has different values even though the mass density distributions
are the same. A direct result is that the potentials become non-symmetric
along and against the directions of the external field, i.e., a symmetric
system is not in equilibrium due to the non-symmetry of self-potential.
Therefore, MOND predicts that there are no real symmetric systems within the
external gravity backgrounds. This will be explored in greater detail in a
future paper (Wang et al. in preparation).
## 5 acknowledgments
We thank the anonymous referee for helpful suggestions and comments to the
earlier version of the manuscript. We thank Luca Ciotti, Pasquale Londrillo,
Carlo Nipoti for generously sharing their code, Martin Feix for polishing the
English writing and Victor Debattista, Mordehai Milgrom and Christos Siopis
for nice comments in the earlier version of the paper. We thank the Mordehai
Milgrom and Francoise Combes for the comments to the paper. XW and HSZ
acknowledges the Dark Cosmology Center of Copenhagen University and
Sterrewacht of Leiden University. XW acknowledges the support of SUPA
studentship. HSZ acknowledges partial support from UK PPARC Advanced
Fellowship and National Natural Science Foundation of China (NSFC under grant
No. 10428308). YGW acknowledges the support of the 973 Program
(No.2007CB815402), the CAS Knowledge Innovation Program (Grant No.
KJCX3-SYW-N2), and the NSFC grant 10503010. CLL and AK acknowledge funding by
the DFG under grant KN 755/2. AK further acknowledges funding through the Emmy
Noether programme of the DFG (KN 755/1).
## Appendix A Symmetry and Numerical Challenges
Evolving the systems without filtering the high frequent components of mass
during the Legendre transformation in time for up to 200 Keplerian dynamical
times (1.0 kpc) we find that they appear ’unstable’. However, a detailed
investigation revealed that is a numerical effect rather than a physical
instability. There is a small, uneven force in the $z$ direction which comes
from the asymmetry of the density distribution of ICs for N-body and systems
during the simulations and the errors obviously accumulate when the numbers of
particles are not big enough. Further, the total momentum of the systems is
not conserved giving a non-zero net velocity along the minor axis. Hence the
code requires a large number of particles for smooth density distributions to
make sure the tiny asymmetry of some particles does not affect the whole
system.
Note that this effect is more serious when the systems are not symmetrized by
utilizing ’mirror particles’ inside the systems. It is known that particles
generated from asymmetric orbits (e.g. the ’banana orbits’) could break the
symmetry of the systems in phase space. Furthermore, there are a couple of
hundred of chaotic orbits with positive weights, and, during the Schwarzschild
process, the time integration of 100 orbital times may not be long enough to
obtain symmetry. We therefore show in Fig. 12 the average value of positions
and velocities along the $z$-axis for every orbit of the $N$-body ICs of the
model with mass $10^{9}M_{\odot}$. We plot $\bar{z}$ vs. $\bar{v}_{z}$ since
the $z$-direction displays the most serious shifting. For a perfectly
symmetric system the values of $\bar{z}$ and $\bar{v}_{z}$ should be close to
zero, while we find that they are not. Hence we need to symmetrize the systems
prior to the N-body procedure. The simplest way to achieve this is by placing
’mirror particles’ into the system, i.e. using a minus sign in front of the
6-dimensional components. Therefore, the total numbers of particles increases
to $N\times 2^{6}=64N$.
Figure 12: The average values of positions ($\bar{z}$) vs. velocities
($\bar{v}_{z}$) projected on the z axis for each non-zero weight orbit. The
model has the total mass of $10^{9}M_{\odot}$.
To illustrate (and quantify) this effect we plot in Fig. 13 the centre-shifts
along the three axes (left panel) as well as the evolution of the net
velocities along the same axes (right panel) during the first 40 Keplerian
dynamical times (1.0 kpc) for the models without the 64 mirrors. We find that
the most massive system (i.e. $10^{10}M_{\odot}$, which is in mild-MOND
gravity) is least affected by these numerical artifacts. As a matter of fact,
this particular system shows credible signs of stability even after 200
Keplerian dynamical times (1.0 kpc). We need to acknowledge that this is
partly due to our definition of the time unit (cf. Eq. 6): it is shorter for
more massive systems. However, all the analysis presented in this paper
indicates that the system is stable despite the apparent numerical artifacts
of the code. We though cannot evolve the system further in time as the
accumulation of errors would lead to substantial deviations from the system’s
equilibrium state; but this ’instability’ is caused by numerics rather than
physics! Given the technical particulars of the NMODY code such a centre-shift
will lead to a decrease in resolution as the code utilizes a spherical grid.
,
Figure 13: Left panel: The shifted positions of the centre of mass of the
three systems. The solid, dotted, dashed lines are for the systems with their
total mass of $10^{10}M_{\odot}$, $10^{9}M_{\odot}$ and $10^{8}M_{\odot}$. The
colours of black, yellow and blue correspond to the z, y and x axis. The right
panel: net velocities of the systems in the three axis directions.
## Appendix B Comparison with another MOND solver
As just highlighted in Section A, there are numerical challenges to evolving
our systems under MONDian gravity using the N-body code NMODY. In order to
confirm that the results are not unique to this one code we therefore decided
to also use another novel solver for the MONDian analog to Poisson’s equation,
namely the AMIGA code (Llinares, Knebe & Zhao 2008). AMIGA is the successor to
MLAPM (Knebe, Green & Binney 2001) that has recently been adapted to also
solve Eq. 2.444We like to note in passing that MLAPM has already been
successfully applied to study cosmological structure formation under MOND
(Knebe & Gibson 2004) under certain assumptions. The code utilizes adaptive
meshes in Cartesian coordinates in a cubical volume as opposed to the
spherical grid of NMODY. The solution is obtained via multi-grid relaxation
and we refer the interested reader to Llinares et al. (2008) for more details.
However, here we need to elaborate upon the boundary constraints as we cannot
assume that the potential on the boundary will be a constant: the box is a
cube and not a sphere. We decided to use the solution for a point mass in the
center of the box
$\Phi(r)=-\frac{GM}{2}\left(\frac{1}{r}-\frac{1}{r_{0}}\right)+\left(f(r)-f(r_{0})\right),$
(12)
with
$\begin{array}[]{rcl}f(r)&=&\displaystyle-\sqrt{GMa_{0}}\\\
&&\displaystyle\left[\frac{-1}{2r}\sqrt{q^{2}+4r^{2}}+\ln\left(2r+\sqrt{q^{2}+4r^{2}}\right)\right]\\\
\\\ q^{2}&=&\displaystyle\frac{GM}{a_{0}},\end{array}$ (13)
where, $M$ is the total mass in the box and $r_{0}$ is a length scale (a
constant of integration). For $a_{0}\rightarrow 0$ we recover the Newtonian
solution and for $a_{0}$ finite and $r\rightarrow\infty$ we have $\ln(r)$,
which is the typical behaviour for any MONDian solution. In the case that we
use $r_{0}=B$, with $B$ being the size of the cubical box, we end up with
$\Phi=0$ in a sphere of radius $B$, that is equivalent to the conditions used
in NMODY.
We now run simulations with the same Initial Conditions for N-body as used
with NMODY utilizing a domain grid with $128^{3}$ cells. Each of these domain
grid cells is refined and split into eight sub-cells once the number of
particles inside that cell is in excess of 6. The box size is $B=165.5152$kpc
and the scale for the boundary conditions is $r_{0}=82.7576$ Kpc (half of the
box).
The results obtained are similar to the NMODY simulations. The system is
stable with a normal secular evolution. We observe the same kind of evolution.
We confirm that all other quantities behave in a similar manner too, and hence
are confident that the results presented in the previous section 3 are not
dominated and/or contaminated by numerical artifacts.
## Appendix C Longer time evolution of the $10^{10}M_{\odot}$ model
We initially ran the models for 200 simulation time units (i.e. 200 circular
orbital times at the length of 1.0 kpc, see Table 1). Hence the most massive
model has been evolved for the least time, about 1 Gyr. For brighter galaxies
because they are also bigger galaxies, the dynamical time is longer. This is
the opposite to our trend in Table 1. This can be understood because our toy
galaxies do not sit on the fundamental plane. To see this, we estimate
$r_{h}$, the characteristic length of an elliptical galaxy on the fundamental
plane (e.g., Eq. 9 in Zhao, Xu, Dobbs 2008, which follows Faber et al. 1997 ).
$\log{GMr_{h}^{-2}\over 350\times 10^{-10}m/s^{2}}=-1.52\log{M\over 1.5\times
10^{11}M_{\odot}}\pm 0.5,$ (14)
we find $r_{h}=0.082^{+0.049}_{-0.032}$ kpc for the model with
$10^{10}M_{\odot}$. This is one order of magnitude smaller than our assumed
size $1$ kpc. The dynamical time for a $10^{10}M_{\odot}$ galaxy sitting on
the fundamental plane is about 0.1 Million years.
To be on the safe side, we re-ran the model for about 3 Gyrs ($650T_{simu}$),
and show its virial ratio (Fig. 14). And our conclusion in the §3.2 does not
change. The virial ratio oscillates around 1 within 10% at most.
Figure 14: The virial ratio of model with $10^{10}M_{\odot}$, simulating 3
Gyrs (650 circular orbital time at typical length of 1.0kpc).
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|
arxiv-papers
| 2009-04-08T20:39:25 |
2024-09-04T02:49:01.807701
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xufen Wu, HongSheng Zhao, Yougang Wang, Claudio Llinares, Alexander\n Knebe",
"submitter": "Xufen Wu",
"url": "https://arxiv.org/abs/0904.1425"
}
|
0904.1883
|
# On the subgroup structure of the full Brauer group of Sweedler Hopf algebra
Giovanna Carnovale Juan Cuadra Dipartimento di Matematica Pura Universidad de
Almería ed Applicata Dpto. Álgebra y Análisis Matemático via Trieste 63
E-04120 Almería, Spain I-35121 Padua, Italy jcdiaz@ual.es
carnoval@math.unipd.it
###### Abstract
We introduce a family of three parameters $2$-dimensional algebras
representing elements in the Brauer group $BQ(k,H_{4})$ of Sweedler Hopf
algebra $H_{4}$ over a field $k$. They allow us to describe the mutual
intersection of the subgroups arising from a quasitriangular or
coquasitriangular structure. We also define a new subgroup of $BQ(k,H_{4})$
and construct an exact sequence relating it to the Brauer group of Nichols
$8$-dimensional Hopf algebra with respect to the quasitriangular structure
attached to the $2\times 2$-matrix with $1$ in the $(1,2)$-entry and zero
elsewhere.
MSC:16W30, 16K50
## Introduction
The Brauer group of a Hopf algebra is an extremely complicated invariant that
reflects many aspects of the Hopf algebra: its automorphisms group, its Hopf-
Galois theory, its second lazy cohomology group, (co)quasitriangularity, etc.
It is very difficult to describe all its elements and to find their
multiplication rules. For the most studied case, that of a commutative and
cocommutative Hopf algebra, these are the results known so far: the first
explicit computation was done by Long in [14] for the group algebra
$k{\mathbb{Z}}_{n},$ where $n$ is square-free and $k$ algebraically closed
with $char(k)\nmid n$; DeMeyer and Ford [12] computed it for
$k{\mathbb{Z}}_{2}$ with $k$ a commutative ring containing $2^{-1}$. Their
result was extended by Beattie and Caenepeel in [2] for $k{\mathbb{Z}}_{n},$
where $n$ is a power of an odd prime number and some mild assumptions on $k$.
In [4] Caenepeel achieved to compute the multiplication rules for a subgroup,
the so-called split part, of the Brauer group for a faithfully projective
commutative and cocommutative Hopf algebra $H$ over any commutative ring $k$.
These results were improved in [6] and allowed him to compute the Brauer group
of Tate-Oort algebras of prime rank. For a unified exposition of these results
the profuse monograph [5] is recommended.
Since the Brauer group was defined for any Hopf algebra with bijective
antipode ([7], [8]), it was a main goal to compute it for the smallest
noncommutative noncocommutative Hopf algebra: Sweedler’s four dimensional Hopf
algebra $H_{4}$, which is generated over the field $k$ ($char(k)\neq 2$) by
the group-like $g$, the $(g,1)$-primitive element $h$ and relations
$g^{2}=1,h^{2}=0,gh=-hg$. A first step was the calculation in [20] of the
subgroup $BM(k,H_{4},R_{0})$ induced by the quasitriangular structure
$R_{0}=2^{-1}(1\otimes 1+g\otimes 1+1\otimes g-g\otimes g).$ It was shown to
be isomorphic to the direct product of $(k,+)$, the additive group of $k$, and
$BW(k)$, the Brauer-Wall group of $k$. It was later proved in [9] that the
subgroups $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ arising from all the
quasitriangular structures $R_{t}$ and the coquasitriangular structures
$r_{s}$ of $H_{4}$ respectively, with $s,t\in k$, are all isomorphic.
In this paper we introduce a family of three parameters $2$-dimensional
algebras $C(a;t,s)$, for $a,t,s\in k,$ that represent elements in
$BQ(k,H_{4})$. They will allow us to shed a ray of light on the subgroup
structure of $BQ(k,H_{4})$ and will provide some evidences about the
difficulty of the computation of this group. The algebra $C(a;t,s)$ is
generated by $x$ with relation $x^{2}=a$ and has a $H_{4}$-Yetter-Drinfeld
module algebra structure with action and coaction:
$g\cdot x=-x,\quad h\cdot x=t,\quad\rho(x)=x\otimes g+s\otimes h.$
We list the main properties of these algebras in Section 2 (Lemma 2.1) and we
show that $C(a;t,s)$ is $H_{4}$-Azumaya if and only if $2a\neq st$. When
$s=lt$ they represent elements in $BM(k,H_{4},R_{l})$ and this subgroup is
indeed generated by the classes of $C(a;1,t)$ with $2a\neq t$ together with
$BW(k)$, Proposition 2.6. The same statement holds true for
$BC(k,H_{4},r_{l})$ when $t=sl$ replacing $C(a;1,t)$ by $C(a;s,1)$,
Proposition 2.5.
Using the description of $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ in terms
of these algebras, Section 3 is devoted to analyze the intersection of these
subgroups inside $BQ(k,H_{4})$. Let $i_{t}$ and $\iota_{s}$ denote the
inclusion map of the former and the latter respectively. It is known that
$BW(k)$ is contained in any of the above subgroups. Theorem 3.5 states that:
1. (1)
$Im(i_{t})\cap Im(\iota_{s})\neq BW(k)$ iff $ts=1$. If this is the case,
$Im(i_{t})=Im(\iota_{s})$;
2. (2)
$Im(i_{t})\cap Im(i_{s})\neq BW(k)$ if and only if $t=s$;
3. (3)
$Im(\iota_{t})\cap Im(\iota_{s})\neq BW(k)$ if and only if $t=s$.
A remarkable property of our algebras is that they represent the same class in
$BQ(k,H_{4})$ if and only if they are isomorphic, Corollary 3.4.
A morphism from the automorphism group of $H_{4}$ to $BQ(k,H_{4})$ was
constructed in [19], allowing to consider $k^{\cdot 2}$ as a subgroup of
$BQ(k,H_{4})$. In Section 4 we show that the subgroup $BM(k,H_{4},R_{l})$ is
conjugated to $BM(k,H_{4},R_{l\alpha^{2}})$ inside $BQ(k,H_{4})$, for
$\alpha\in k^{\cdot}$, by a suitable representative of $k^{\cdot 2}$, Lemma
4.1.
Any $H_{4}$-Azumaya algebra possesses two natural ${\mathbb{Z}}_{2}$-gradings:
one stemming from the action of $g$ and one from the coaction (after
projection) of $g$. In Section 6 we introduce the subgroup
$BQ_{grad}(k,H_{4})$ consisting of those classes of $BQ(k,H_{4})$ that can be
represented by $H_{4}$-Azumaya algebras for which the two
${\mathbb{Z}}_{2}$-gradings coincide. On the other hand, the Drinfeld double
of $H_{4}$ admits a Hopf algebra map $T$ onto Nichols $8$-dimensional Hopf
algebra $E(2)$. This map is quasitriangular as $E(2)$ is equipped with the
quasitriangular structure $R_{N}$ corresponding to the $2\times 2$-matrix $N$
with $1$ in the $(1,2)$-entry and zero elsewhere, see (5.1). If we consider
the associated Brauer group $BM(k,E(2),R_{N})$, then Theorem 5.2 claims that
$T$ induces a group homomorphism $T^{*}$ fitting in the following exact
sequence
$\begin{array}[]{l}\begin{CD}1\longrightarrow{\mathbb{Z}}_{2}@>{}>{}>BM(k,E(2),R_{N})@>{T^{*}}>{}>BQ_{grad}(k,H_{4})\longrightarrow
1.\end{CD}\end{array}$
So in order to compute $BQ(k,H_{4})$ one should first understand
$BM(k,E(2),R_{N})$. This new problem cannot be attacked with the available
techniques for computations of groups of type BM, [20], [10], [11]. Those
computations were achieved by finding suitable invariants for a class by means
of a Skolem-Noether-like theory. In the Appendix we underline some obstacles
to the application of these techniques to the computation of
$BM(k,E(2),R_{N})$: the set of elements represented by algebras for which the
action of one of the standard nilpotent generators of $E(2)$ is inner
coincides with the set of classes represented by ${\mathbb{Z}}_{2}$-graded
central simple algebras and this is not a subgroup of $BM(k,E(2),R_{N}),$
Theorems 6.1, 6.3. Moreover, $BM(k,E(2),R_{N})$ seems to be much more complex
than the groups of type BM treated until now since, according to Proposition
5.3, each group $BM(k,H_{4},R_{t})$ may be viewed as a subgroup of it.
## 1 Preliminaries
In this paper $k$ is a field, $H$ will denote a Hopf algebra over $k$ with
bijective antipode $S$, coproduct $\Delta$ and counit $\varepsilon$. Tensor
products $\otimes$ will be over $k$ and, for vector spaces $V$ and $W$, the
usual flip map is denoted by $\tau:V\otimes W\to W\otimes V$. We shall adopt
the Sweedler-like notations $\Delta(h)=h_{(1)}\otimes h_{(2)}$ and
$\rho(m)=m_{(0)}\otimes m_{(1)}$ for coproducts and right comodule structures
respectively. For $H$ coquasitriangular (resp. quasitriangular), the set of
all coquasitriangular (resp. quasitriangular) structures will be denoted by
$\cal U$ (resp. $\cal T$).
Yetter-Drinfeld modules. Let us recall that if $A$ is a left $H$-module with
action $\cdot$ and a right $H$-comodule with coaction $\rho$ the two
structures combine to a left module structure for the Drinfeld double
$D(H)=H^{*,cop}\bowtie H$ of $H$ (cfr. [15]) if and only if they satisfy the
so-called Yetter-Drinfeld compatibility condition:
$\rho(l\cdot b)=l_{(2)}\cdot b_{(0)}\otimes
l_{(3)}b_{(1)}S^{-1}(l_{(1)}),\quad\forall l\in H,b\in A.$ (1.1)
Modules satisfying this condition are usually called Yetter-Drinfeld modules.
If $A$ is a left $H$-module algebra and a right $H^{op}$-comodule algebra
satisfying (1.1) we shall call it a Yetter-Drinfeld $H$-module algebra.
The Brauer group (see [7], [8]). Suppose that $A$ is a Yetter-Drinfeld
$H$-module algebra. The $H$-opposite algebra of $A$, denoted by
$\overline{A}$, is the underlying vector space of $A$ endowed with product
$a\circ c=c_{(0)}(c_{(1)}\cdot a)$ for every $a,c\in A$. The same action and
coaction of $H$ on $A$ turn $\overline{A}$ into a Yetter-Drinfeld $H$-module
algebra. Given two Yetter-Drinfeld $H$-module algebras $A$ and $B$ we can
construct a new Yetter-Drinfeld module $A\\#B$ whose underlying vector space
is $A\otimes B$, with action $h\cdot(a\otimes b)=h_{(1)}\cdot a\otimes
h_{(2)}\cdot b$ and with coaction $a\otimes b\mapsto a_{(0)}b_{(0)}\otimes
b_{(1)}a_{(1)}$. This object becomes a Yetter-Drinfeld module algebra if we
provide it with the multiplication
$(a\\#b)(c\\#d)=ac_{(0)}\\#(c_{(1)}\cdot b)d.$
For every finite dimensional Yetter-Drinfeld module $M$ the algebras ${\rm
End}(M)$ and ${\rm End}(M)^{op}$ can be naturally provided of a Yetter-
Drinfeld module algebra structure through (1.2) and (1.3) below respectively:
$\begin{array}[]{l}(h\cdot f)(m)=h_{(1)}\cdot f(S(h_{(2)})\cdot
m),\vspace{2pt}\\\ \rho(f)(m)=f(m_{(0)})_{(0)}\otimes
S^{-1}(m_{(1)})f(m_{(0)})_{(1)},\end{array}$ (1.2) $\begin{array}[]{l}(h\cdot
f)(m)=h_{(2)}\cdot f(S^{-1}(h_{(1)})\cdot m),\vspace{2pt}\\\
\rho(f)(m)=f(m_{(0)})_{(0)}\otimes f(m_{(0)})_{(1)}S(m_{(1)}),\end{array}$
(1.3)
where $h\in H,f\in End(M),m\in M.$ A finite dimensional Yetter-Drinfeld module
algebra $A$ is called $H$-Azumaya if the following module algebra maps are
isomorphisms:
$\begin{array}[]{ll}F\colon A\\#{\overline{A}}\rightarrow{\rm
End}(A),&F(a\\#b)(c)=ac_{(0)}(c_{(1)}\cdot b),\vspace{2pt}\\\
G\colon\overline{A}\\#{{A}}\rightarrow{\rm
End}(A)^{op},&G(a\\#b)(c)=a_{(0)}(a_{(1)}\cdot c)b.\end{array}$ (1.4)
The algebras ${\rm End}(M)$ and ${\rm End}(M)^{op}$, for a finite dimensional
Yetter-Drinfeld module $M$, provided with the preceding structures are
$H$-Azumaya.
The following relation $\sim$ established on the set of isomorphism classes of
$H$-Azumaya algebras is an equivalence relation: $A\sim B$ if there exist
finite dimensional Yetter-Drinfeld modules $M$ and $N$ such that $A\\#{\rm
End}(M)\cong B\\#{\rm End}(N)$ as Yetter-Drinfeld module algebras. The set of
equivalence classes of $H$-Azumaya algebras, denoted by $BQ(k,H)$, is a group
with product $[A][B]=[A\\#B]$, inverse element $[\overline{A}]$ and identity
element $[End(M)]$ for finite dimensional Yetter-Drinfeld modules $M$. This
group is called the full Brauer group of $H$. The adjective full is used to
distinguish it from the subgroups presented next, that receive the same name
in the literature.
Given a left $H$-module algebra $A$ with action $\cdot$ and a quasitriangular
structure $R=R^{(1)}\otimes R^{(2)}$ on $H$, a right $H^{op}$-comodule algebra
structure $\rho$ on $A$ is determined by
$\rho(a)=(R^{(2)}\cdot a)\otimes R^{(1)},\quad\forall a\in A.$
We will call this coaction the coaction induced by $\cdot$ and $R$. It is
well-known that $(A,\cdot,\rho)$ satisfies the Yetter-Drinfeld condition. This
allows the definition of the subgroup $BM(k,H,R)$ of $BQ(k,H)$ whose elements
are equivalence classes of $H$-Azumaya algebras with coaction induced by $R$
([8, §1.5]). To underline that a representative $A$ of a given class in
$BQ(k,H)$ represents a class in $BM(k,H,R)$ we shall say that $A$ is an
$(H,R)$-Azumaya algebra. The inclusion map will be denoted by $i\colon
BM(k,H,R)\to BQ(k,H)$. For $H$ finite dimensional $BQ(k,H)=BM(k,D(H),{\cal
R}),$ where ${\cal R}$ is the natural quasitriangular structure on the
Drinfeld double $D(H)$.
Dually, given a right $H^{op}$-comodule algebra $A$ with coaction $\varrho$
and a coquasitriangular structure $r$ on $H$, a $H$-module algebra structure
$\cdot$ on $A$ is determined by
$h\cdot a=a_{(0)}r(h\otimes a_{(1)}),\quad\forall a\in A,h\in H,$
and $(A,\cdot,\varrho)$ becomes a Yetter-Drinfeld module algebra. We will call
this action the action induced by $\chi$ and $r$. The subset $BC(k,H,r)$ of
$BQ(k,H)$ consisting of those classes admitting a representative whose action
is induced by $r$ is a subgroup ([8, §1.5]). To stress that a representative
$A$ of a class in $BQ(k,H)$ represents a class in $BC(k,H,r)$ we shall say
that $A$ is an $(H,r)$-Azumaya algebra. The inclusion of $BC(k,H,r)$ in
$BQ(k,H)$ will be denoted by $\iota\colon BC(k,H,r)\to BQ(k,H)$.
On Sweedler Hopf algebra. In the sequel we will assume that $char(k)\neq 2.$
Let $H_{4}$ be Sweedler Hopf algebra, that is, the Hopf algebra over $k$
generated by a grouplike element $g$ and an element $h$ with relations,
coproduct and antipode:
$g^{2}=1,\quad h^{2}=gh+hg=0,\quad\Delta(h)=1\otimes h+h\otimes g,\quad
S(g)=g,\quad S(h)=gh.$
The Hopf algebra $H_{4}$ has a family of quasitriangular (indeed triangular)
structures. They were classified in [18] and are given by:
$R_{t}=\frac{1}{2}(1\otimes 1+1\otimes g+g\otimes 1-g\otimes
g)+\frac{t}{2}(h\otimes h+h\otimes gh+gh\otimes gh-gh\otimes h),$
where $t\in k$. It is well-known that $H_{4}$ is self-dual so that $H_{4}$ is
also cotriangular. Let $\\{1^{*},g^{*},h^{*},(gh)^{*}\\}$ be the basis of
$H^{*}_{4}$ dual to $\\{1,g,h,gh\\}$. We will often make use of the Hopf
algebra isomorphism
$\begin{array}[]{rl}\phi\colon H_{4}&\to H_{4}^{*}\\\ 1&\mapsto
1^{*}+g^{*}=\varepsilon\\\ h&\mapsto h^{*}+(gh)^{*}\\\ g&\mapsto
1^{*}-g^{*}\\\ gh&\mapsto h^{*}-(gh)^{*}.\end{array}$
So, the cotriangular structures of $H_{4}$ can be obtained applying the
isomorphism $\phi\otimes\phi$ to the $R_{t}$’s. They are:
$\begin{array}[]{c|rrrr}r_{t}&1&g&h&gh\\\ \hline\cr 1&1&1&0&0\\\ g&1&-1&0&0\\\
h&0&0&t&-t\\\ gh&0&0&t&t\\\ \end{array}$
The Drinfeld double $D(H_{4})=H_{4}^{*,cop}\bowtie H_{4}$ of $H_{4}$ is
isomorphic to the Hopf algebra generated by $\phi(h)\bowtie 1$,
$\phi(g)\bowtie 1$, $\varepsilon\bowtie g$ and $\varepsilon\bowtie h$ with
relations:
$\begin{array}[]{l}(\phi(h)\bowtie 1)^{2}=0;\\\ (\phi(g)\bowtie
1)^{2}=\varepsilon\bowtie 1;\\\ (\phi(h)\bowtie 1)(\phi(g)\bowtie
1)+(\phi(g)\bowtie 1)(\phi(h)\bowtie 1)=0;\\\ (\varepsilon\bowtie h)^{2}=0;\\\
(\varepsilon\bowtie h)(\varepsilon\bowtie g)+(\varepsilon\bowtie
g)(\varepsilon\bowtie h)=0;\\\ (\varepsilon\bowtie g)^{2}=\varepsilon\bowtie
1;\\\ (\phi(h)\bowtie 1)(\varepsilon\bowtie g)+(\varepsilon\bowtie
g)(\phi(h)\bowtie 1)=0;\\\ (\phi(g)\bowtie 1)(\varepsilon\bowtie
h)+(\varepsilon\bowtie h)(\phi(g)\bowtie 1)=0;\\\ (\varepsilon\bowtie
g)(\phi(g)\bowtie 1)=(\phi(g)\bowtie 1)(\varepsilon\bowtie g);\\\
(\phi(h)\bowtie 1)(\varepsilon\bowtie h)-(\varepsilon\bowtie h)(\phi(h)\bowtie
1)=(\phi(g)\bowtie 1)-(\varepsilon\bowtie g)\end{array}$
and with coproduct induced by the coproducts in $H_{4}$ and $H_{4}^{*,cop}$.
For $l\in H_{4}$ we will sometimes write $\phi(l)$ instead of $\phi(l)\bowtie
1$ and $l$ instead of $1\bowtie l$ for simplicity.
Let us recall that a Yetter-Drinfeld $H_{4}$-module $M$ with action $\cdot$
and coaction $\rho$ becomes a $D(H_{4})$-module by letting $1\bowtie l$ act as
$l$ for every $l\in H_{4}$ and $(\phi(l)\bowtie
1).m=m_{(0)}(\phi(l)(m_{(1)}))$ for $m\in M$. Conversely, a $D(H_{4})$-module
$M$ becomes naturally a Yetter-Drinfeld module with $H_{4}$-action obtained by
restriction and $H_{4}$-coaction given by
$\rho(m)=\frac{1}{2}(\phi(1+g).m\otimes 1+\phi(1-g).m\otimes
g+\phi(h+gh).m\otimes h+\phi(h-gh)\otimes gh).$
We will often switch from one notation to the other according to convenience.
Centers and centralizers. If $A$ is a Yetter-Drinfeld $H$-module algebra, and
$B$ is a Yetter-Drinfeld submodule algebra of $A$, the left and the right
centralizer of $B$ in $A$ are defined to be:
$C^{l}_{A}(B):=\\{a\in A~{}|~{}ba=a_{(0)}(a_{(1)}\cdot b)\ \forall b\in B\\},$
$C^{r}_{A}(B):=\\{a\in A~{}|~{}ab=b_{(0)}(b_{(1)}\cdot a)\ \forall b\in B\\}.$
For the particular case $B=A$ we have the right center $Z^{r}(A)$ and the left
center $Z^{l}(A)$ of $A$. Both are trivial when $A$ is $H$-Azumaya, [8,
Proposition 2.12].
## 2 Some low dimensional representatives in $BQ(k,H_{4})$
In this section we shall introduce a family of 2-dimensional representatives
of classes in $BQ(k,H_{4})$ that will turn out to be easy to compute with.
They appeared for the first time in [16] and a particular case of them is
treated in [1, Section 1.5].
Let $a,\,t,\,s\in k$. The algebra $C(a)$ generated by $x$ with relation
$x^{2}=a$ is acted upon by $H_{4}$ by
$g\cdot 1=1,\quad g\cdot x=-x,\quad h\cdot 1=0,\qquad h\cdot x=t,$
and it is a right $H_{4}$-comodule via
$\rho_{s}(1)=1\otimes 1,\quad\quad\rho_{s}(x)=x\otimes g+s\otimes h.$
It is not hard to check that $C(a)$ with this action and coaction is a left
$H_{4}$-module algebra and a right $H^{op}$-comodule algebra. We shall denote
it by $C(a;t,s)$.
###### Lemma 2.1
Let notation be as above.
1. (1)
$C(a;t,s)$ is a Yetter-Drinfeld module algebra with the preceding structures.
2. (2)
As a module algebra $C(a;t,s)\cong C(a^{\prime};t^{\prime},s^{\prime})$ if and
only if there is $\alpha\in k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$ and
$t=\alpha t^{\prime}$.
3. (3)
As a comodule algebra $C(a;t,s)\cong C(a^{\prime};t^{\prime},s^{\prime})$ if
and only if there is $\alpha\in k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$
and $s=\alpha s^{\prime}$.
4. (4)
As a Yetter-Drinfeld module algebra $C(a;t,s)\cong
C(a^{\prime};t^{\prime},s^{\prime})$ if and only if there exists $\alpha\in
k^{\cdot}$ such that $a=\alpha^{2}a^{\prime}$, $t=\alpha t^{\prime}$ and
$s=\alpha s^{\prime}$.
5. (5)
The module structure on $C(a;t,s)$ is induced by its comodule structure and a
cotriangular structure $r_{l}$ if and only if $t=sl$.
6. (6)
The comodule structure on $C(a;t,s)$ is induced by its module structure and a
triangular structure $R_{l}$ if and only if $s=lt$.
7. (7)
The $H_{4}$-opposite algebra of $C(a;t,s)$ is $C(st-a;t,s)$.
8. (8)
$C(a;t,s)$ is an $H_{4}$-Azumaya algebra if and only if $2a\neq st$.
Proof: Let $x$ and $y$ be algebra generators in $C(a;t,s)$ and
$C(a^{\prime};t^{\prime},s^{\prime})$ respectively with $x^{2}=a$ and
$y^{2}=a^{\prime}$.
(1) We verify condition (1.1) for $b=x$ and $l=h$. The other cases are easier
to check.
$\begin{array}[]{l}h_{(2)}\cdot x_{(0)}\otimes
h_{(3)}x_{(1)}S^{-1}(h_{(1)})\\\ \hskip 48.36958pt=g\cdot x\otimes(-gh)+g\cdot
s\otimes(gh)(-gh)+h\cdot x\otimes g^{2}\\\ \hskip 56.9055pt+h\cdot s\otimes
gh+x\otimes hg+s\otimes h^{2}\\\ \hskip 48.36958pt=x\otimes gh+t\otimes
1-x\otimes gh\\\ \hskip 48.36958pt=\rho_{s}(h\cdot x).\end{array}$
(2) An algebra isomorphism $f\colon C(a;t,s)\to
C(a^{\prime};t^{\prime},s^{\prime})$ must map $x$ to $\alpha y$ for some
$\alpha\in k^{\cdot}$. Then $a=x^{2}=(\alpha y)^{2}=\alpha^{2}a^{\prime}$.
Besides, $h.f(x)=f(h.x)$ implies $t^{\prime}\alpha=t$. It is easy to verify
that the condition is also sufficient.
(3) In the above setup $\rho_{s^{\prime}}(f(x))=(f\otimes{\rm id})\rho_{s}(x)$
implies $s^{\prime}\alpha=s$. It is not hard to check that this condition is
also sufficient.
(4) It follows from the preceding statements.
(5) If the module structure on $C(a;t,s)$ is induced by its comodule structure
$\rho_{s}$ and some $r_{l}\in{\cal U},$ then $t=h\cdot x=xr_{l}(h\otimes
g)+sr_{l}(h\otimes h)=sl.$ Conversely, if $t=sl$, then
$\begin{array}[]{l}g\cdot 1=1=1r_{l}(g\otimes 1);\qquad h\cdot
1=0=1r_{l}(h\otimes 1);\vspace{2pt}\\\ g\cdot x=-x=xr_{l}(g\otimes
g)+sr_{l}(g\otimes h)=x_{(0)}r_{l}(g\otimes x_{(1)});\vspace{2pt}\\\ h\cdot
x=t=xr_{l}(h\otimes g)+sr_{l}(h\otimes h)=x_{(0)}r_{l}(h\otimes
x_{(1)}).\end{array}$
Therefore the action is induced by the coaction and $r_{l}$.
(6) If the comodule structure on $C(a;t,s)$ is induced by the action and some
$R_{l}\in{\cal T},$ then
$x\otimes g+s\otimes h=\rho_{s}(x)=(R_{l}^{(2)}\cdot x)\otimes
R_{l}^{(1)}=\frac{1}{2}(2x\otimes g)+\frac{l}{2}(2t\otimes h)=x\otimes
g+lt\otimes h$
hence $s=lt$. Conversely, if $s=lt$ then
$\begin{array}[]{l}\rho_{s}(1)=1\otimes 1=(R_{l}^{(2)}\cdot 1)\otimes
R_{l}^{(1)},\\\ \rho_{s}(x)=x\otimes g+s\otimes h=(R_{l}^{(2)}\cdot x)\otimes
R_{l}^{(1)},\end{array}$
so the comodule structure is induced by the action and $R_{l}$.
(7) $\overline{C(a;t,s)}$ has $1,\,x$ as a basis and $1$ is the unit. The
action and coaction on $1$ and $x$ are as for $C(a;t,s)$. By direct
computation, $x\circ x=x(g\cdot x)+s(h\cdot x)=-a+st,$ so
$\overline{C(a;t,s)}=C(st-a;t,s)$.
(8) The algebra $C(a;t,s)$ is $H_{4}$-Azumaya if and only if the maps $F$ and
$G$ defined in (1.4) are isomorphisms. The space $C(a;t,s)\\#C(a;t,s)$ has
ordered basis $1\\#1,\,1\\#x,\,x\\#1,\,x\\#x$ while ${\rm End}(C(a;t,s))$ has
basis $1^{*}\otimes 1,\,1^{*}\otimes x,\,x^{*}\otimes 1,\,x^{*}\otimes x$ with
the usual identification $C(a;t,s)^{*}\otimes C(a;t,s)\cong{\rm
End}(C(a;t,s))$. Then for every $b,c\in C(a;t,s)$ we have
$\begin{array}[]{l}F(b\\#c)(1)=bc,\quad F(b\\#c)(x)=bx(g\cdot c)+sb(h\cdot
c),\vspace{2pt}\\\ G(1\\#b)(c)=cb,\quad G(x\\#b)(c)=x(g\cdot c)b+s(h\cdot
c)b.\end{array}$
The matrices associated with $F$ and $G$ with respect to the given bases are
respectively
$\left(\begin{array}[]{cccc}1&0&0&a\\\ 0&1&1&0\\\ 0&st-a&a&0\\\
1&0&0&st-a\end{array}\right)\qquad\left(\begin{array}[]{cccc}1&0&0&a\\\
0&1&1&0\\\ 0&a&st-a&0\\\ 1&0&0&st-a\end{array}\right)$
whose determinants $-(st-2a)^{2}$ and $(st-2a)^{2}$ are nonzero if and only if
$2a\neq st$. $\Box$
We have seen so far that the algebras $C(a;s,t)$ can be viewed as
representatives of classes in $BM(k,H_{4},R_{l})$ or in $BC(k,H_{4},r_{l})$
for suitable $l\in k$. It is known that these groups are all isomorphic to
$(k,+)\times BW(k),$ where $BW(k)$ is the Brauer-Wall group of $k$. We aim to
find to which pair $(\beta,[A])\in(k,+)\times BW(k)$ do the class of
$C(a;t,s)$ correspond. The group $BM(k,H_{4},R_{0})$ was computed in [20]. The
computation of $BC(k,H_{4},r_{0})$ follows from self-duality of $H_{4}$. It
was shown in [9] that all groups $BC(k,H_{4},r_{t})$ (hence, dually, all
$BM(k,H_{4},R_{t})$) are isomorphic. We shall use the description of
$BM(k,E(1),R_{t})$ given in [11] beause this might allow generalizations. In
the mentioned paper the Brauer group $BM(k,E(n),R_{0})$ is computed for the
family of Hopf algebras $E(n)$, where $E(1)=H_{4}$. We shall recall first
where do the isomorphism of the different Brauer groups $BC$ and $BM$ stem
from. The notion of lazy cocycle plays a key role here.
We recall from [3] that a lazy cocycle on $H$ is a left 2-cocycle $\sigma$
such that twisting $H$ by $\sigma$ does not modify the product in $H$. In
other words: for every $h,l,m\in H,$
$\sigma(h_{(1)}\otimes l_{(1)})\sigma(h_{(2)}l_{(2)}\otimes
m)=\sigma(l_{(1)}\otimes m_{(1)})\sigma(h\otimes l_{(2)}m_{(2)})$ (2.1)
$\sigma(h_{(1)}\otimes
l_{(1)})h_{(2)}l_{(2)}=h_{(1)}l_{(1)}\sigma(h_{(2)}\otimes l_{(2)})$ (2.2)
It turns out that a lazy left cocycle is also a right cocycle. Given a lazy
cocycle $\sigma$ for $H$ and a $H^{op}$-comodule algebra $A$, we may construct
a new $H^{op}$-comodule algebra $A_{\sigma}$, which is equal to $A$ as a
$H^{op}$-comodule, but with product defined by:
$a\bullet b=a_{(0)}b_{(0)}\sigma(a_{(1)}\otimes b_{(1)}).$
The group of lazy cocycles for $H_{4}$ is computed in [3]. Lazy cocycles are
parametrized by elements $t\in k$ as follows:
$\begin{array}[]{c|rrrr}\sigma_{t}&1&g&h&gh\\\ \hline\cr 1&1&1&0&0\\\
g&1&1&0&0\\\ h&0&0&\frac{t}{2}&\frac{t}{2}\\\
gh&0&0&\frac{t}{2}&-\frac{t}{2}\\\ \end{array}$
We have the following group isomorphisms:
1. (2.3)
$\Psi_{t}:BC(k,H_{4},r_{0})\rightarrow
BC(k,H_{4},r_{t}),[A]\mapsto[A_{\sigma_{t}}],$ constructed in [9, Proposition
3.1].
2. (2.4)
$\Phi_{t}:BM(k,H_{4},R_{t})\rightarrow BC(k,H_{4},r_{t}),[A]\mapsto[A^{op}].$
We explain how $A^{op}$ is equipped with the corresponding structure. The left
$H_{4}$-module algebra $A$ becomes a right $H_{4}^{*}$-comodule algebra. Then
$A^{op}$ is a right $H_{4}^{*,op}$-comodule algebra. The quasitriangular
structure $R_{t}$ is a coquasitriangular structure in $H_{4}^{*}$. Then
$A^{op}$ may be endowed with the left $H_{4}^{*}$-action stemming from the
comodule structure and $R_{t}$. On the other hand, $A^{op}$ may be viewed as
an $H_{4}^{op}$-comodule algebra through the isomorphism
$\phi:H_{4}\rightarrow H_{4}^{*}$. The coquasitriangular structure $R_{t}$ on
$H_{4}^{*}$ corresponds to the coquasitriangular structure $r_{t}$ on $H_{4}$
via $\phi.$
An isomorphism between $BM(k,H_{4},R_{0})$ and $BM(k,H_{4},R_{t})$ can be
constructed combining the above ones. Thus, the crucial step is to analyze the
sought correspondence for $BM(k,H_{4},R_{0})$.
The Brauer group $BM(k,H_{4},R_{0})$ is computed in [20] through the split
exact sequence (see also [1, Theorem 3.8] for an alternative approach):
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k,+)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BM(k,H_{4},R_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{BW(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{1.}$
The map $j^{*}:BM(k,H_{4},R_{0})\to BW(k),[A]\mapsto[A]$ is obtained by
restricting the $H_{4}$-action of $A$ to a $k{\mathbb{Z}}_{2}$-action via the
inclusion map $j:k{\mathbb{Z}}_{2}\rightarrow H_{4}$. This map is split by
$\pi^{*}:BW(k)\rightarrow BM(k,H_{4},R_{0}),[B]\mapsto[B]$, where $B$ is
considered as an $H_{4}$-module by restriction of scalars via the algebra
projection $\pi:H_{4}\rightarrow k{\mathbb{Z}}_{2},g\mapsto g,h\mapsto 0$. A
class $[A]$ lying in the kernel of $j^{*}$ is a matrix algebra with an inner
action of $H_{4}$ such that the restriction to $k{\mathbb{Z}}_{2}$ is strongly
inner. Thus there exist uniquely determined $u,w\in A$ such that
$g\cdot a=uau^{-1},\quad h\cdot a=w(g\cdot a)-aw\quad\forall a\in A,$ (2.5)
$u^{2}=1,\quad wu+uw=0,\quad w^{2}=\beta,$ (2.6)
for certain $\beta\in k$. Mapping $[A]\mapsto\beta$ defines a group
isomorphism $\chi\colon Ker(j^{*})\\\ \cong(k,+)$. We will determine
$j^{*}([C(a;t,s)])$ and $\chi([C(a;t,s)]\pi^{*}j^{*}([C(a;t,s)]^{-1}))$
whenever this is well-defined. To this purpose, we will first describe all
products of two algebras of type $C(a;t,s)$.
###### Lemma 2.2
Let $x,y$ be generators for $C(a;t,s)$ and
$C(a^{\prime};t^{\prime},s^{\prime})$ respectively, with relations,
$H_{4}$-actions and coactions as above. The product
$C(a;t,s)\\#C(a^{\prime};t^{\prime},s^{\prime})$ is isomorphic to the
generalized quaternion algebra with generators $X=x\\#1$ and $Y=1\\#y$,
relations, $H_{4}$-action and and $H_{4}$-coaction:
$X^{2}=a,\quad Y^{2}=a^{\prime},\quad XY+YX=st^{\prime},$ $g\cdot X=-X,\quad
g\cdot Y=-Y,\quad h\cdot X=t,\quad h\cdot Y=t^{\prime},$ $\rho(X)=X\otimes
g+s\otimes h,\quad\rho(Y)=Y\otimes g+s^{\prime}\otimes h.$
Proof: By direct computation:
$X^{2}=(x\\#1)(x\\#1)=a\\#1,\quad Y^{2}=(1\\#y)(1\\#y)=a^{\prime}\\#1,\quad
XY=x\\#y,$ $YX=(1\\#y)(x\\#1)=x\\#(g\cdot y)+s\\#(h\cdot
y)=-XY+st^{\prime}\\#1.$
The formulas for the action and the coaction follow immediately from the
definition of action and coaction on a $\\#$-product. $\Box$
Elements in $BW(k)$ are represented by graded tensor products of the following
three type of algebras: $C(1)$ generated by the odd element $x$ with
$x^{2}=1$; classically Azumaya algebras having trivial
${\mathbb{Z}}_{2}$-action; and $C(a)\\#C(1),$ where $C(a)$ is generated by the
odd element $y$ with $y^{2}=a\in k^{\cdot}$ ([13, Theorem IV.4.4]).
###### Proposition 2.3
For $a\neq 0$ let $[C(a;t,0)]\in BM(k,H_{4},R_{0})$ denote the class of
$C(a;t,0).$ Then
$[C(a;t,0)]=(t^{2}(4a)^{-1},[C(a)])\in(k,+)\times BW(k),$
so the group $BM(k,H_{4},R_{0})$ is generated by $BW(k)$ and the classes
$[C(a;1,0)]$.
Proof: It is clear that if $a\neq 0$ then $j^{*}([C(a;t,0)])=[C(a)]$ and that
$\pi^{*}([C(a)])=[C(a;0,0)]$. Thus, $[C(a;t,0)\\#C(-a;0,0)]\in{\rm
Ker}(j^{*})$. We shall compute its image through $\chi$. By Lemma 2.2,
$C(a;t,0)\\#C(-a;0,0)$ is generated by $X$ and $Y$ with relations,
$H_{4}$-action and $H_{4}$-coaction:
$X^{2}=a,\quad Y^{2}=-a,\quad XY+YX=0,$ $g\cdot X=-X,\quad g\cdot Y=-Y,\quad
h\cdot X=t,\quad h\cdot Y=0,$ $\rho(X)=X\otimes g,\quad\rho(Y)=Y\otimes g.$
We look for the element $w$ satisfying (2.5) and (2.6). This element must be
odd with respect to the ${\mathbb{Z}}_{2}$-grading induced by the $g$-action,
hence $w=\lambda X+\mu Y$ for some $\lambda,\mu\in k$. Condition $h\cdot
X=-wX-Xw$ implies $t=-2\lambda a$ and condition $h\cdot Y=-wY-Yw$ implies
$0=-2\mu a$ so $w^{2}=a\lambda^{2}=t^{2}(4a)^{-1}$. Thus
$[C(a;t,0)]=(t^{2}(4a)^{-1},[C(a)])$ and we have the first statement. For the
second one, let $(\beta,[A])\in(k,+)\times BW(k)$. If $\beta=0$ there is
nothing to prove. If $\beta\neq 0$, the class
$[C((4\beta)^{-1}t^{2};t,0)]=[C((4\beta)^{-1};1,0)]=(\beta,[C((4\beta)^{-1})])$,
so $BM(k,H_{4},R_{0})\cong(k,+)\times BW(k)$ is generated by $BW(k)$ and the
$[C(a;1,0)]$ for $a\neq 0.$ $\Box$
###### Lemma 2.4
Let $A$ be a $D(H_{4})$-module algebra.
1. (1)
If the $h$-action on $A$ is trivial, then $A$ is $(H_{4},R_{0})$-Azumaya if
and only if it is $(H_{4},R_{t})$-Azumaya for every $t\in k$.
2. (2)
If the $\phi(h)$-action on $A$ is trivial, then $A$ is $(H_{4},r_{0})$-Azumaya
if and only if it is $(H_{4},r_{t})$-Azumaya for every $t\in k$.
3. (3)
The representatives of $BW(k)$ in $BC(k,H_{4},r_{t})$ and $BM(k,H_{4},R_{s})$
all coincide when viewed inside $BQ(k,H_{4})$.
Proof: (1) It follows from the form of the elements in ${\cal T}$ that if $A$
is $(H_{4},R_{0})$-Azumaya and the action of $h$ on $A$ is trivial (i.e., if
it lies in $BW(k)$), then its comodule structure $\rho_{t}$ induced by $R_{t}$
coincides with the comodule structure $\rho_{0}$ induced by $R_{0}$. Hence,
the maps $F$ and $G$ with respect to the action and $\rho_{t}$ are the same as
the maps $F$ and $G$ with respect to the action and $\rho_{0}$, so $A$ is
$(H_{4},R_{t})$-Azumaya for every $t\in k$.
(2) It is proved as (1).
(3) The first statement shows that the representatives of $BW(k)$ inside the
different $BM(k,H_{4},R_{t})$ coincide. The second statement shows the same
for $BC(k,H_{4},r_{t})$. Therefore we may assume $s=t=0$. The elements of this
copy of $BW(k)$ consist of ${\mathbb{Z}}_{2}$-graded Azumaya algebras $A$
where the grading is induced by the action of $g$. The $h$-action is trivial.
If the coaction $\rho$ is induced by $R_{0}$, then $a\in A$ is odd if and only
if $\rho(a)=a\otimes g$. The action $\rightharpoonup$ induced on $A$ by
$r_{0}$ and $\rho$ is as follows: $h\rightharpoonup a=0$ for every $a\in A$
and $g\rightharpoonup a=-a$ if and only if $\rho(a)=a\otimes g$, that is, the
original action on $A$ and $\rightharpoonup$ coincide. Thus, the maps $F$ and
$G$ coincide in all cases and $A$ represents an element in $BW(k)\subset
BM(k,H_{4},R_{0})$ if and only if it represents an element in $BW(k)\subset
BC(k,H_{4},r_{0})$. $\Box$
###### Proposition 2.5
The group $BC(k,H_{4},r_{s})$ is generated by the Brauer-Wall group and the
classes $[C(a;s,1)]$ for $2a\neq s$.
Proof: We will first deal with the case $s=0$. We will show that the
isomorphism $\Phi_{0}:BM(k,H_{4},R_{0})\rightarrow BC(k,H_{4},r_{0}),$
$[A]\mapsto[A^{op}]$ in (2.4) maps $[C(a;1,0)]$ to $[C(a;0,1)]$ and
$BW(k)\subset BM(k,H_{4},R_{0})$ to $BW(k)\subset BC(k,H_{4},r_{0})$. The
class $[C(a;1,0)]$ is mapped to the class of the algebra $C(a)^{op}$ with
comodule structure
$\rho(x)=x\otimes(1^{*}-g^{*})+1\otimes(h^{*}+(gh)^{*})=x\otimes\phi(g)+1\otimes\phi(h)$
and $H_{4}$-action induced by the cotriangular structure $r_{0}$, that is,
$g\cdot x=-x$ and $h\cdot x=0$. The algebra $C(a)^{op}$ with these structures
is just $C(a;0,1)$.
Let $A$ be a representative of a class in $BW(k)\subset BM(k,H_{4},R_{0})$
with action $\cdot$ for which $h\cdot a=0$ for all $a\in A$. The class $[A]$
is mapped by $\Phi_{0}$ to the class of $A^{op}$ with coaction
$\rho(a)=a\otimes 1^{*}+(g\cdot a)\otimes g^{*}+(h\cdot a)\otimes
h^{*}+(gh\cdot a)\otimes(gh)^{*}\in A\otimes\phi(k{\mathbb{Z}}_{2}).$
Therefore $[A^{op}]\in BW(k)\subset BC(k,H_{4},r_{0})$.
We now take $s\in k$ arbitrary and use the isomorphism
$\Psi_{s}:BC(k,H_{4},r_{0})\rightarrow BC(k,H_{4},r_{s})$ in (2.3) to prove
the statement. We will show that $[C(a;0,1)]$ is mapped to
$[C(a+2^{-1}s;s,1)]$ through $\Psi_{s}$. Recall that $\Psi_{s}$ maps the class
of $C(a;0,1)$ to the class of the algebra $C(a;0,1)_{\sigma_{s}}$. It is
generated by $x$ with relation
$x\bullet x=x^{2}\sigma_{s}(g\otimes g)+x\sigma_{s}(h\otimes
g)+x\sigma_{s}(g\otimes h)+\sigma_{s}(h\otimes h)=a+\frac{s}{2},$
with (same) coaction $\rho(x)=x\otimes g+1\otimes h$ and action induced by
$\rho$ and $r_{s}$, that is:
$g\cdot x=r_{s}(g\otimes g)x+r_{s}(g\otimes h)=-x,\quad h\cdot
x=r_{s}(h\otimes g)x+r_{s}(h\otimes h)=s.$
Then $\Psi_{s}([C(a;0,1)])=[C(a+\frac{s}{2};s,1)]$.
Since the coaction is not changed by $\Psi_{s}$ the class of an element $A$
for which the image of the coaction is in $A\otimes k{\mathbb{Z}}_{2}$ is
again of this form. Hence the classes in $BW(k)\subset BC(k,H_{4},r_{0})$
correspond to the classes in $BW(k)\subset BC(k,H_{4},r_{s})$. $\Box$
###### Proposition 2.6
The group $BM(k,H_{4},R_{t})$ is generated by the Brauer-Wall group and the
classes $[C(a;1,t)]$ for $2a\neq t$.
Proof: Through the isomorphism $\Phi_{t}:BM(k,H_{4},R_{t})\rightarrow
BC(k,H_{4},r_{t})$ in (2.4), the class $[C(a;1,t)]$ is mapped to $[C(a;t,1)]$
and the classes in $BW(k)\subset BM(k,H_{4},R_{t})$ correspond to the classes
in $BW(k)\subset BC(k,H_{4},r_{t})$. The $H_{4}$-comodule structure on the
algebra $C(a)^{op}$ is:
$\rho(x)=x\otimes(1^{*}-g^{*})+1\otimes(h^{*}+(gh)^{*})=x\otimes\phi(g)+1\otimes\phi(h)$
The $H_{4}$-action induced by the cotriangular structure $r_{t}$ on $H_{4}$
gives $h\cdot x=t$. Therefore this algebra is $C(a;t,1)$. Finally, the
statement concerning $BW(k)$ is proved as in the preceding theorem. $\Box$
###### Remark 2.7
That $BM(k,H_{4},R_{t})$ is generated by $BW(k)$ and the classes $[C(a;1,t)]$
for $2a\neq t$ was first discovered in [1, Theorem 3.8 and Page 392] as a
consequence of the Structure Theorems for $(H_{4},R_{t})$-Azumaya algebras.
Since we will strongly use Proposition 2.6 later, for the reader’s convenience
we offered this alternative and self-contained approach. Notice that it mainly
relies on Lemma 2.2 that will be another key result for us in the sequel.
## 3 Fitting $BM(k,H_{4},R_{t})$ and $BC(k,H_{4},r_{s})$ into $BQ(k,H_{4})$
As groups $BM(k,H_{4},R_{t})\cong BC(k,H_{4},r_{s})$ for every $s,t\in k$.
However, their images in $BQ(k,H_{4})$ through the natural embeddings
$i_{t}\colon BM(k,H_{4},R_{t})\to
BQ(k,H_{4})\quad\textrm{and}\quad\iota_{s}\colon BC(k,H_{4},r_{s})\to
BQ(k,H_{4})$
do not coincide in general. In this section we will describe the mutual
intersections of these images.
###### Proposition 3.1
Let $0\neq t\in k$ then $Im(i_{t})=Im(\iota_{t^{-1}})$
Proof: Given $t\neq 0$, by Lemma 2.1, $[C(a;1,t)]\in Im(i_{t})\cap
Im(\iota_{t^{-1}})$ for every $a\neq 2t.$ Besides, by Lemma 2.4,
$i_{t}(BW(k))=\iota_{s}(BW(k))$ for any $s\in k$. Since the elements of
$BW(k)$ and the $[C(a;1,t)]$’s generate $BM(k,H_{4},R_{t})$ and
$BC(k,H_{4},r_{t^{-1}})$ we are done. $\Box$
Given $[A]$ in $BQ(k,H_{4})$, there are two natural
${\mathbb{Z}}_{2}$-gradings on $A$, the one coming from the $g$-action, for
which $|a|=1$ iff $g\cdot a=-a$ for $0\neq a\in A$ and the one arising from
the coaction, for which $\deg(a)=1$ if and only if $({\rm
id}\otimes\pi)\rho(a)=a\otimes g$ where $\pi$ is the projection onto
$k{\mathbb{Z}}_{2}$. If we view $A$ as a $D(H_{4})$-module, the grading
$|\cdot|$ is associated with the $1\bowtie g$-action whereas the grading
$\deg$ is associated with the $\phi(g)\bowtie 1$-action. Let us observe that
for the classes $C(a;t,s)$ the two natural gradings coincide, for every
$a,\,t,\,s\in k$.
###### Lemma 3.2
Let $[A]\in BQ(k,H_{4})$ and $[B]$ in $i_{0}(BW(k))$. As a $H_{4}$-module
algebra,
* (1)
$A\\#B\cong A\widehat{\otimes}B$, the ${\mathbb{Z}}_{2}$-graded tensor product
with respect to the $\deg$-grading on $A$ and the natural $|\cdot|$-grading on
$B$.
* (2)
$B\\#A\cong B\widehat{\otimes}A$, the ${\mathbb{Z}}_{2}$-graded tensor product
with respect to the $|\cdot|$-grading on $A$ and the natural $|\cdot|$-grading
on $B$.
Proof: The two gradings on $B$ coincide and we have, for homogeneous $b\in B$
and $c\in A$ (for the $\deg$-grading):
$(a\\#b)(c\\#d)=ac_{(0)}\\#(c_{(1)}\cdot b)d=ac\\#(g^{\deg(c)}\cdot
b)d=(-1)^{\deg(c)|b|}ac\\#bd.$
For homogeneous $b\in B$ and $c\in A$ (for the $|\cdot|$-grading):
$(d\\#c)(b\\#a)=db_{(0)}\\#(b_{(1)}\cdot c)a=db\\#(g^{|b|}\cdot
c)a=(-1)^{|c||b|}db\\#ca.$
$\Box$
It follows from Propositions 2.5, 2.6 and Lemma 3.2 that all elements in
$Im(i_{t})$ and $Im(\iota_{t})$ can be represented by algebras for which the
two ${\mathbb{Z}}_{2}$-gradings coincide, since this property is respected by
the $\\#$-product. Indeed, this kind of representatives give rise to a
subgroup that we will study in Section 5.
We will show now that groups of type $BC$ or $BM$ either intersect only in
$BW(k)$ or coincide and that the latter happens only in the situation of
Proposition 3.1.
###### Theorem 3.3
Consider the class of $C(a;t,s)$ in $BQ(k,H_{4})$. Then:
1. (1)
$[C(a;t,s)]\in Im(i_{l})$ if and only if $s=lt$;
2. (2)
$[C(a;t,s)]\in Im(\iota_{l})$ if and only if $sl=t$.
Proof: (1) We know from Lemma 2.1 that if the action (resp. coaction) of
$C(a;t,s)$ comes from the cotriangular (resp. triangular) structure, then the
indicated relations among the parameters hold. We only need to show that the
condition is still necessary if we change representative in the class.
Let us assume that $[C(a;t,s)]\in Im(i_{l})$ for some $l\in k$. Then
$[C(a;t,s)]=[C(b;1,l)][A]=[A][C(b;1,l)]$ for some $[A]\in i_{l}(BW(k))$ and
$b\in k$ with $2b\neq l$. Hence $[C(a;t,s)\\#C(l-b;1,l)]=[A]\in i_{l}(BW(k))$.
We may choose $A$ so that the $h$-action and the $\phi(h)$-action on $A$ are
trivial.
Since $[C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}]$ is trivial in $BQ(k,H_{4})$,
there is a $D(H_{4})$-module $P$ such that
$C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}\cong{\rm End(P)}$ as $D(H_{4})$-module
algebras. Then ${\rm End(P)}$ has a strongly inner $D(H_{4})$-action. In other
words, there is a convolution invertible algebra map $\nu\colon
D(H_{4})\to{\rm End}(P)$ such that
$(m\bowtie n)\cdot f=\nu(m_{(2)}\bowtie n_{(1)})f\nu^{-1}(m_{(1)}\bowtie
n_{(2)})$
for every $m\bowtie n\in D(H_{4}),f\in{\rm End(P)}$, where $\nu^{-1}$ denotes
the convolution inverse of $\nu$. In particular, for $u=\nu(\varepsilon\bowtie
g)$ and $w=\nu(\varepsilon\bowtie h)u$ we have
$\begin{array}[]{l}g\cdot f=ufu^{-1},\quad h\cdot f=w(g\cdot
f)-fw,\vspace{5pt}\\\ u^{2}=1,\quad w^{2}=0,\quad uw+wu=0.\end{array}$
We should be able to find $U,\,W\in C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ such
that
$\begin{array}[]{l}U^{2}=1,\quad g\cdot Z=UZU^{-1},\vspace{5pt}\\\ g\cdot
W=-W,\quad W^{2}=0,\quad h\cdot Z=W(g\cdot Z)-ZW\end{array}$
for all $Z$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}.$
Using the presentation of $C(a;t,s)\\#C(l-b;1,l)$ in Lemma 2.2 we may write
$W=\sum_{0\leq i,j\leq 1}X^{i}Y^{j}\\#\alpha_{ij}$ with
$\alpha_{ij}\in\overline{A}$ homogeneous of degree $i+j+1\;{\rm mod}\ 2$ with
respect to the $g$-grading. Since the action of $h$ on $1\\#\overline{A}$ is
trivial we have, for homogeneous $\gamma\in\overline{A}$:
$\begin{array}[]{ll}0&=h\cdot(1\\#\gamma)\vspace{2pt}\\\
&=W(g\cdot(1\\#\gamma))-(1\\#\gamma)W\vspace{2pt}\\\
&=(-1)^{|\gamma|}\sum_{0\leq i,j\leq
1}X^{i}Y^{j}\\#\alpha_{ij}\gamma-\sum_{0\leq i,j\leq
1}(X^{i}Y^{j})_{(0)}\\#((X^{i}Y^{j})_{(1)}\cdot\gamma)\alpha_{ij}\vspace{2pt}\\\
&=(-1)^{|\gamma|}[1\\#\alpha_{00}\gamma+Y\\#\alpha_{01}\gamma+X\\#\alpha_{10}\gamma+XY\\#\alpha_{11}\gamma]\vspace{2pt}\\\
&\phantom{=}-1\\#\gamma\alpha_{00}-Y\\#(-1)^{|\gamma|}\gamma\alpha_{01}-X\\#(-1)^{|\gamma|}\gamma\alpha_{10}-XY\\#\gamma\alpha_{11}.\end{array}$
From here we deduce that the odd elements $\alpha_{00},\alpha_{11}$ and the
even elements $\alpha_{10},\alpha_{01}$ belong to the
${\mathbb{Z}}_{2}$-center of $\overline{A}.$ Hence $\alpha_{00},\alpha_{11}$
are zero and $\alpha_{10},\alpha_{01}$ are scalars. So, we can write $W=\alpha
X\\#1+\beta Y\\#1$ for some $\alpha,\beta\in k$ and we will get:
$\begin{array}[]{rl}\alpha t+\beta&=h\cdot W=-2W^{2}=0,\\\
t&=h\cdot(X\\#1)=\alpha(-2a+ts),\\\ 1&=h\cdot(Y\\#1)=-\alpha
s-2\beta(l-b)=\alpha(-s+2t(l-b)).\end{array}$
Combining the second equation with the third one multiplied by $t$ and using
$\alpha\neq 0$ we obtain
$a=ts-t^{2}(l-b).$ (3.1)
The $|\cdot|$-grading and the $\deg$-grading on
$C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ coincide. Therefore:
$\nu(\phi(g)\bowtie 1)f\nu(\phi(g)\bowtie 1)^{-1}=\phi(g)\cdot f=g\cdot
f=ufu^{-1}\qquad\forall f\in{\rm End}(P).$
Since ${\rm End}(P)$ is central and $\nu$ is an algebra morphism,
$u^{\prime}:=\nu(\phi(g)\bowtie 1)=\lambda u$ with $\lambda=\pm 1$ (both
possibilities will be analyzed later). The element
$w^{\prime}:=\nu(\phi(h)\bowtie 1)$ satisfies
$\phi(h)\cdot f=w^{\prime}f-(\phi(g)\cdot f)w^{\prime}\qquad\forall f\in{\rm
End}(P).$
Thus, we can take $W^{\prime}$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}$ such
that
$W^{\prime}U+UW^{\prime}=0,\quad(W^{\prime})^{2}=0\quad\phi(h)\cdot
Z=W^{\prime}Z-(g\cdot Z)W^{\prime}$
for all $Z$ in $C(a;t,s)\\#C(l-b;1,l)\\#\overline{A}.$ Arguing as for $W$
before, we see that $W^{\prime}=\gamma X\\#1+\delta Y\\#1$ for some
$\gamma,\delta\in k$. It follows from the last relation of $D(H_{4})$ in §1
that
$\nu(\varepsilon\bowtie hg)\nu(\phi(h)\bowtie 1)+\nu(\phi(h)\bowtie
1)\nu(\varepsilon\bowtie hg)=\nu(\phi(g)\bowtie 1)\nu(\varepsilon\bowtie
g)-\nu(\varepsilon\bowtie g)^{2}.$
This implies $WW^{\prime}+W^{\prime}W=\lambda-1.$ Besides,
$0=\phi(h)\cdot W^{\prime}=2(W^{\prime})^{2}=s\gamma+\delta l.$
Now, by direct computation:
$\begin{array}[]{rl}\lambda-1&=WW^{\prime}+W^{\prime}W\vspace{2pt}\\\
&=\alpha((X-tY)(\gamma X+\delta Y)+(\gamma X+\delta Y)(X-tY))\vspace{2pt}\\\
&=\alpha\gamma(2a-ts)+\alpha\delta(s-2t(l-b))\\\
&=-t\gamma-\delta.\end{array}$
Let us first assume $\lambda=1$. Then, $\gamma(s-tl)=0$. If $\gamma=0,$ then
$\delta=0$ and so $W^{\prime}=0$. This means that the $\phi(h)$-action is
identically zero, yielding $s=l=0$. Otherwise, $s=tl$ and we are done.
We finally show that the possibility $\lambda=-1$ can not occur. If
$\lambda=-1$, then $\delta=2-t\gamma$ and $s\gamma=-(2-t\gamma)l$. On the
other hand,
$l=\phi(h)\cdot(Y\\#1)=W^{\prime}(Y\\#1)+(Y\\#1)W^{\prime}=s\gamma+2(2-t\gamma)(l-b)$
(3.2)
Moreover,
$\begin{array}[]{rl}0&=(W^{\prime})^{2}\vspace{2pt}\\\
&=\gamma^{2}a+\delta^{2}(l-b)+\gamma\delta s\vspace{2pt}\\\
&\stackrel{{\scriptstyle(\ref{a=})}}{{=}}\gamma^{2}(ts-t^{2}(l-b))+(2-t\gamma)^{2}(l-b)+\gamma(2-t\gamma)s\vspace{2pt}\\\
&=2(l-b)(2-2t\gamma)+2\gamma s\end{array}$
From here, $s\gamma=(2t\gamma-2)(l-b).$ Substituting this in (3.2) we get
$l=2b$, contradicting the fact that $C(b;1,l)$ is $(H_{4},R_{l})$-Azumaya.
(2) If $l\not=0$, then $Im(\iota_{l})=Im(i_{l^{-1}})$ by Proposition 3.1 and
the statement follows from (1). It remains to show that $[C(a;t,s)]\in
Im(\iota_{0})$ implies $t=0$. If $[C(a;t,s)]\in Im(\iota_{0}),$ there exists
$b\in k^{\cdot}$ and an $H_{4}$-Azumaya algebra $A$ with trivial $h$-action
and trivial $\phi(h)$-action such that $[C(a;t,s)]=[A\\#C(b;0,1)].$ Then
$C(a;t,s)\\#C(-b;0,1)\\#\overline{A}\cong{\rm End}(P)$ for some
$D(H_{4})$-module $P$. Arguing as in (1) we see that there is $W=\alpha
X\\#1+\beta Y\\#1\in(C(a;t,s)\\#C(-b;0,1))\\#\overline{A}$ for some
$\alpha,\beta\in k$ such that
$\begin{array}[]{ll}&\hskip 11.0pth\cdot Z=W(g\cdot Z)-ZW,\\\ 0&=h\cdot
W=-2W^{2}=\alpha t+\beta,\\\ t&=h\cdot(X\\#1)=-2a\alpha,\\\
0&=h\cdot(Y\\#1)=2b\beta.\end{array}$
From here if follows that $t=0$. $\Box$
###### Corollary 3.4
Let $[C(a;t,s)]$, $[C(b;p,q)]$ be in $BQ(k,H_{4})$. Then
$[C(a;t,s)]=[C(b;p,q)]$ if and only if $C(a;t,s)\cong C(b;p,q)$.
Proof: We analyze the case $t\neq 0$, the other cases are treated similarly.
If $[C(a;t,s)]=[C(b;p,q)]$ and $p=0$ then $[C(a;t,s)]\in Im(\iota_{0}),$
contradicting Theorem 3.3. Then $tp\neq 0$ and we may reduce to the case
$[C(a;1,s)]=[C(b;1,q)]\in Im(i_{q})$. Applying again Theorem 3.3 we see that
$s=q$ and the equality of classes is an equality in $BM(k,H_{4},R_{q})$.
Applying $\Phi_{0}^{-1}\Psi_{q}^{-1}\Phi_{q}$ we obtain the equality
$[C(a-2^{-1}q;1,0)]=[C(b-2^{-1}q;1,0)]$ in $BM(k,H_{4},R_{0})$. From
Proposition 2.3, we obtain $(4a-2q)^{-1}=(4b-2q)^{-1}$ and we have the
statement. $\Box$
###### Theorem 3.5
Let $i_{t}:BM(k,H_{4},R_{t})\to BQ(k,H_{4})$ and
$\iota_{s}:BC(k,H_{4},r_{s})\to BQ(k,H_{4})$ be the natural embeddings in
$BQ(k,H_{4})$. Then:
1. (1)
$Im(i_{t})\cap Im(\iota_{s})\neq i_{0}(BW(k))$ if and only if $ts=1$. If this
is the case, then $Im(i_{t})=Im(\iota_{s})$;
2. (2)
$Im(i_{t})\cap Im(i_{s})\neq i_{0}(BW(k))$ if and only if $t=s$;
3. (3)
$Im(\iota_{t})\cap Im(\iota_{s})\neq i_{0}(BW(k))$ if and only if $t=s$.
Proof: This is a consequence of Propositions 2.3, 2.5, 2.6, 3.1 and Theorem
3.3. $\Box$
## 4 The action of $Aut(H_{4})$ on $Im(i_{t})$ and $Im(\iota_{s})$
For a Hopf algebra $H$, a group morphism from ${\rm Aut}_{\rm Hopf}(H)$ to
$BQ(k,H_{4})$ has been constructed in [8], where the case of $H_{4}$ was also
analized. The image of an automorphism $\alpha$ can be represented as follows.
Let us denote by $H_{\alpha}$ the right $H$-comodule $H$ with left $H$-action
$l\cdot m=\alpha(l_{(2)})mS^{-1}(l_{(1)})$. Then $A_{\alpha}={\rm
End}(H_{\alpha})$ can be endowed of the $H$-Azumaya algebra structure:
$\begin{array}[]{l}(l\cdot f)(m)=l_{(1)}\cdot f(S(l_{(2)})\cdot
m),\vspace{2pt}\\\ \rho(f)(m)=\sum f(m_{(0)})_{(0)}\otimes
S^{-1}(m_{(1)})f(m_{(0)})_{(1)}.\end{array}$
The assignment $\alpha\mapsto[A_{\alpha^{-1}}]$ defines a group morphism ${\rm
Aut}_{\rm Hopf}(H)\to BQ(k,H)$. The image of ${\rm Aut}_{\rm Hopf}(H)$ acts on
$BQ(k,H)$ by conjugation. An easy description of
$[B(\alpha)]:=[A_{\alpha}][B][A_{\alpha}]^{-1}$ for any representative $B$ has
been given in [8, Theorem 4.11]. As an algebra $B(\alpha)$ coincides with $B$,
while the $H$-action and $H$-coaction are:
$h\cdot_{\alpha}b=\alpha(h)\cdot
b,\quad\rho_{\alpha}(b)=b_{(0)}\otimes\alpha^{-1}(b_{(1)}).$ (4.1)
When $H=H_{4}$ the Hopf automorphism group is ${\rm Aut}_{\rm
Hopf}(H_{4})\cong k^{\cdot}$ and consists of the morphisms that are the
identity on $g$ and multiply $h$ by a nonzero scalar $\alpha$. The module
$H_{\alpha}$ has action
$\begin{array}[]{l}g\cdot g=g,\quad g\cdot h=-h,\vspace{2pt}\\\ h\cdot
g=\alpha hg+g^{2}S^{-1}(h)=-(1+\alpha)gh,\quad h\cdot h=0,\end{array}$
and the kernel of the group morphism consists of $\\{\pm 1\\}$. We may thus
embed $(k^{\cdot})^{2}\cong k^{\cdot}/\\{\pm 1\\}$ into $BQ(k,H_{4})$ (cf.
[19]). We shall denote by $K$ the image of this group morphism.
We analyze this action on the classes and subgroups described in the previous
sections.
###### Lemma 4.1
Let $\alpha\in k^{\cdot}$. Then:
1. (1)
$[A_{\alpha}][C(a;t,s)][A_{\alpha}]^{-1}=[C(a;\alpha t,s\alpha^{-1})]$.
2. (2)
$K$ acts trivially on $i_{0}(BW(k))$.
In particular, $BM(k,H_{4},R_{l\alpha^{2}})$ is conjugate to
$BM(k,H_{4},R_{l})$ in $BQ(k,H_{4})$ while $BM(k,H_{4},R_{0})$ and
$BC(k,H_{4},r_{0})$ are normalized by $K$.
Proof: (1) It follows from direct computation that
$h\cdot_{\alpha}x=\alpha t,\quad g\cdot_{\alpha}x=-x,\quad\rho(x)=x\otimes
g+s\alpha^{-1}\otimes h.$
(2) Since: the action of an automorphism of $H_{4}$ is trivial on $g$; the
action of $h$ is trivial on a representative of a class in $BW(k)$; and the
comodule map on a representative $A$ of a class in $BW(k)$ has image in
$A\otimes k{\mathbb{Z}}_{2}$, the formulas in (4.1) do not modify the action
and coaction on $A$ therefore $[A]=[A_{\alpha}][A][A_{\alpha}]^{-1}$ for every
$[A]\in i_{0}(BW(k))$.
Since $Im(i_{l})$ is generated by $i_{0}(BW(k))$ and the classes $[C(a;1,l)]$,
we see that $Im(i_{l})$ is conjugate to $Im(i_{\alpha^{2}l})$ in
$BQ(k,H_{4})$. If $l=0$ we get the statement concerning $Im(i_{0})$. The
statement concerning $BC(k,H_{4},r_{0})$ follows because this group is
generated by $i_{0}(BW(k))$ and the classes $[C(a;0,1)]$. $\Box$
###### Remark 4.2
The observation that $Im(i_{0})$ is normalized by $K$ has already been proved
in [21, §4]. Lemma 4.1 should be seen as a generalization of that result.
It is shown in [18] that $(H_{4},R_{t})$ is equivalent to $(H_{4},R_{s})$ if
and only if $t=\alpha^{2}s$ for some $\alpha\in k^{\cdot}$. The above lemma
shows that the Brauer groups of type $BM$ are conjugate in $BQ(k,H_{4})$ if
the corresponding triangular structures are equivalent. This is a general
fact:
###### Proposition 4.3
Let $R$ and $R^{\prime}$ be two equivalent quasitriangular structures on $H$
and let $\alpha\in{\rm Aut}_{\rm Hopf}(H)$ be such that
$(\alpha\otimes\alpha)(R^{\prime})=R$. Then the images of $BM(k,H,R)$ and
$BM(k,H,R^{\prime})$ are conjugate by the image of $\alpha$ in $BQ(k,H)$.
Proof: If $B$ represents an element in $BM(k,H,R)$ then there will be an
action $\cdot$ on $B$ such that the coaction $\rho$ is given by
$\rho(b)=(R^{(2)}\cdot b)\otimes R^{(1)}$ for all $b\in B$. The image of
$\alpha$ in $BQ(k,H)$ is represented by $A_{\alpha^{-1}}$. A representative of
$[A_{\alpha}]^{-1}[B][A_{\alpha}]$ is given by the algebra $B$ with action
$h\cdot_{\alpha^{-1}}b=\alpha^{-1}(h)\cdot b$. The coaction is given by
$\rho_{\alpha}(b)=(R^{(2)}\cdot
b)\otimes\alpha(R^{(1)})=(\alpha(R^{(2)})\cdot_{\alpha}b)\otimes\alpha(R^{(1)})=R^{\prime(2)}\cdot_{\alpha}b\otimes
R^{\prime(1)},$
so the coaction on $[A_{\alpha}]^{-1}[B][A_{\alpha}]$ is induced by
$R^{\prime}$ and $\cdot_{\alpha}$. $\Box$
For the dual statement, the proof is left to the reader.
###### Proposition 4.4
Let $r$ and $r^{\prime}$ be two equivalent coquasitriangular structures on $H$
and let $\alpha\in{\rm Aut}_{\rm Hopf}(H)$ be such that
$r^{\prime}(\alpha\otimes\alpha)=r$. Then the images of $BC(k,H,r)$ and
$BM(k,H,r^{\prime})$ are conjugate by the image of $\alpha$ in $BQ(k,H)$.
## 5 The subgroup $BQ_{grad}(k,H_{4})$
In this section we shall analyze the classes that can be represented by
$H_{4}$-Azumaya algebras for which the gradings coming from the $g$-action and
the comodule structure coincide. They form a subgroup that will be related to
the Brauer group $BM(k,E(2),R_{N})$ of Nichols $8$-dimensional Hopf algebra
$E(2)$ with respect to the quasitriangular structure $R_{N}$ attached to the
$2\times 2$-matrix $N$ with $1$ in the $(1,2)$-entry and zero elsewhere.
Let $BQ_{grad}(k,H_{4})$ be the set of classes that can be represented by a
$H_{4}$-Azumaya algebra $A$ for which the $|\cdot|$-grading and the
$\deg$-grading coincide. In other words, the classes in $BQ_{grad}(k,H_{4})$
can be represented by $D(H_{4})$-module algebras on which the actions of $g$
and $\phi(g)$ coincide. The last defining relation of $D(H_{4})$ in Section 1
implies that the action of $h$ and $\phi(h)$ on such representatives commute.
Clearly, $BQ_{grad}(k,H_{4})$ is a subgroup of $BQ(k,H_{4})$.
###### Proposition 5.1
$BQ_{grad}(k,H_{4})$ is normalized by $K$.
Proof: Let $[A]\in BQ_{grad}(k,H_{4})$ with $|a|=\deg(a)$ for every $a\in A$
and let $[A_{\alpha}]\in K$. Then $[A_{\alpha}\\#A\\#\overline{A_{\alpha}}]$
is represented by $A$ with action and coaction determined by (4.1). Since $g$
is fixed by all Hopf automorphisms of $H_{4}$ we have
$g\cdot_{\alpha}a=g\cdot a,\quad({\rm id}\otimes\pi)\rho_{\alpha}(a)=({\rm
id}\otimes\pi)\rho(a),$
so the two gradings are not modified by conjugation by $[A_{\alpha}]$. $\Box$
The subgroup $BQ_{grad}(k,H_{4})$ consists of those classes that can be
represented by module algebras for the quotient of $D(H_{4})$ by the Hopf
ideal $I$ generated by $\phi(g)\bowtie 1-\varepsilon\bowtie g$. Let us denote
by $\pi_{I}$ the canonical projection onto $D(H_{4})/I$.
Let $E(2)$ be the Hopf algebra with generators $c,\,x_{1},\,x_{2},$ with
relations
$c^{2}=1,\quad x_{i}^{2}=0,\quad cx_{i}+x_{i}c=0,\ i=1,2,\quad
x_{1}x_{2}+x_{2}x_{1}=0,$
coproduct
$\Delta(c)=c\otimes c,\quad\Delta(x_{i})=1\otimes x_{i}+x_{i}\otimes c,$
and antipode
$S(c)=c,\quad S(x_{i})=cx_{i}.$
The Hopf algebra morphism
$\begin{array}[]{rl}T\colon D(H_{4})&\longrightarrow E(2)\\\ \phi(g)\bowtie
1&\mapsto c\\\ \varepsilon\bowtie g&\mapsto c\\\ \varepsilon\bowtie h&\mapsto
x_{1}\\\ \phi(h)\bowtie 1&\mapsto cx_{2}\end{array}$
determines a Hopf algebra isomorphism $D(H_{4})/I\cong E(2)$. The canonical
quasitriangular structure ${\cal R}$ on $D(H_{4})$ is
$\begin{array}[]{rl}{\cal R}&=\frac{1}{2}[\varepsilon\bowtie(1\otimes
1^{*}+g\otimes g^{*}+h\otimes h^{*}+gh\otimes(gh)^{*})\bowtie
1]\vspace{3pt}\\\ &\hskip
3.0pt+\frac{1}{2}[\varepsilon\bowtie(1\otimes\varepsilon+g\otimes\varepsilon+1\otimes\phi(g)-g\otimes\phi(g)\vspace{3pt}\\\
&\hskip
20.0pt+h\otimes\phi(h)+h\otimes\phi(gh)+gh\otimes\phi(h)-gh\otimes\phi(gh))\bowtie
1]\end{array}$
so $(\pi_{I}\otimes\pi_{I})({\cal R})$ is a quasitriangular structure for
$D(H_{4})/I\cong E(2)$. Applying $T\otimes T$ to ${\cal R}$ we have:
$\begin{array}[]{rl}(T\otimes T)({\cal R})&=\frac{1}{2}(1\otimes 1+1\otimes
c+c\otimes 1-c\otimes c\\\ &\hskip 20.0pt+x_{1}\otimes cx_{2}+x_{1}\otimes
x_{2}+cx_{1}\otimes cx_{2}-cx_{1}\otimes x_{2})\end{array}$ (5.1)
The quasitriangular structures on $E(n)$ were computed in [17]. They are in
bijection with $n\times n$-matrices with entries in $k$. For a given matrix
$M$ the corresponding quasitriangular structure is denoted by $R_{M}$. The map
$T$ induces a quasitriangular morphism from $(D(H_{4}),{\cal R})$ onto
$(E(2),R_{N}),$ where $N$ is the $2\times 2$-matrix with $1$ in the
$(1,2)$-entry and zero elsewhere. If $A$ is a representative of a class in
$BQ_{grad}(k,H_{4})$ on which the ideal $I$ acts trivially, then $A$ is an
$E(2)$-module algebra and the maps $F$ and $G$ on $A\otimes A$ are the same as
those induced by $R_{N}$, so $A$ is $(E(2),R_{N})$-Azumaya.
###### Theorem 5.2
The group $BM(k,E(2),R_{N})$ fits into the following exact sequence
$\begin{array}[]{l}\begin{CD}1\longrightarrow{\mathbb{Z}}_{2}@>{}>{}>BM(k,E(2),R_{N})@>{T^{*}}>{}>BQ_{grad}(k,H_{4})\longrightarrow
1.\end{CD}\end{array}$
Proof: Restriction of scalars through $T$ provides a group morphism $T^{*}$
from $BM(k,E(2),R_{N})$ to $BQ(k,H)$ whose image is $BQ_{grad}(k,H_{4})$. The
kernel of $T^{*}$ consists of those classes $[A]$ such that $A\cong{\rm
End}(P)$ as $D(H_{4})$-module algebras, for some $D(H_{4})$-module $P$. The
class $[A]$ may be non-trivial only if $g$ and $\phi(g)$ act differently on
$P$ even though they act equally on ${\rm End}(P)$. The $\phi(g)$\- and
$g$-action on ${\rm End}(P)$ are strongly inner, hence there are elements $U$
and $u$ in ${\rm End}(P)$ such that $\phi(g)\cdot f=UfU^{-1}=ufu^{-1}=g\cdot
f$ for every $f\in{\rm End}(P).$ Since ${\rm End}(P)$ is a central algebra,
$U^{2}=u^{2}=1$, $uU=Uu$. From here, $U=\pm u,$ and if $[{\rm End}(P)]\neq 1$
in $BM(k,E(2),R_{N})$ we necessarily have $U=-u$. The actions of $g$ and
$\phi(g)$ on $P$ are given by the element $u$ and $U$ respectively, so for
every non-trivial $[A]$ in ${\rm Ker}(T^{*})$ we have $A\cong{\rm End}(P)$ for
some $D(H_{4})$-module $P$ for which $g$ acts as $-\phi(g)$. We claim that
there is at most one non-trivial element in ${\rm Ker}(T^{*})$.
Given any pair of such elements ${\rm End}(P)$ and ${\rm End}(Q)$ representing
classes in ${\rm Ker}(T^{*})$ we have ${\rm End}(P)\\#{\rm End}(Q)\cong{\rm
End}(P\otimes Q)$ as $D(H_{4})$-module algebras by [7, Proposition 4.3], where
$P\otimes Q$ is a $D(H_{4})$-module. Then, the actions of $g$ and $\phi(g)$ on
$P\otimes Q$ coincide, so $P\otimes Q$ is an $E(2)$-module. Thus, $[{\rm
End}(P)][{\rm End}(Q)]$ is trivial in $BM(k,E(2),R_{N})$ for every choice of
$P$ and $Q$. Therefore, ${\rm Ker}(T^{*})$ is either trivial or isomorphic to
${\mathbb{Z}}_{2}$. The proof is completed once we provide a non-trivial
element. Let us consider $P=k^{2}$ on which $g,\,h,\,\phi(g)$ and $\phi(h)$
act via the following matrices $u,\,w,\,U,\,W$, respectively:
$u=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\quad
w=\left(\begin{array}[]{cc}0&0\\\ -2&0\end{array}\right),\quad U=-u,\,\quad
W=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right).$
Then $P$ is a $D(H_{4})$-module but not an $E(2)$-module. On the other hand,
the $D(H_{4})$-module algebra structure on ${\rm End}(P)$ is in fact an
$E(2)$-module algebra structure:
$g\cdot f=ufu^{-1}=UfU^{-1}=\phi(g)\cdot f;$ (5.2) $h\cdot
f=wfu^{-1}+fuw,\quad\phi(h)\cdot f=Wf-UfU^{-1}W.$ (5.3)
Moreover, ${\rm End}(P)$ is $(E(2),R_{N})$-Azumaya because it is
$H_{4}$-Azumaya. We claim that the class of ${\rm End}(P)$ is not trivial in
$BM(k,E(2),R_{N})$. Indeed, if it were trivial, then the $E(2)$-action on
${\rm End}(P)$ given by $c.f=g.f$, $x_{1}.f=h.f$ and $(cx_{2}).f=\phi(h).f$
would be strongly inner. In other words, there would exist a convolution
invertible algebra morphism $p\colon E(2)\to{\rm End}(P)$ for which $l\cdot
f=\sum p(l_{(1)})fp^{-1}(l_{(2)})$ for every $l\in E(2)$. Putting
$u^{\prime}=p(c)$ we have $c.f=u^{\prime}f(u^{\prime})^{-1}=ufu^{-1}$. Since
${\rm End}(P)$ is a central simple algebra, we necessarily have
$u^{\prime}=\lambda u$ and since $(u^{\prime})^{2}=1$ we have $\lambda=\pm 1$.
Putting $w^{\prime}=p(x_{1})$ we have
$x_{1}.f=w^{\prime}fu^{\prime}-fw^{\prime}u^{\prime}$ and since
$u^{\prime}w^{\prime}=-w^{\prime}u^{\prime}$, we have $\lambda
w^{\prime}fu+\lambda fuw^{\prime}=x_{1}.f=h.f=wfu+fuw$ for every $f\in{\rm
End}(P)$. Using $uw=-wu$ we see that $(\lambda w^{\prime}-w)f=f(\lambda
w^{\prime}-w)$ so $w=\lambda w^{\prime}+\mu$ for some $\mu\in k$. Using once
more skew-commutativity of $u$ with $w$ and $w^{\prime}$ we see that $\mu=0$.
Putting $W^{\prime}=p(cx_{2})$ and using that
$u^{\prime}W^{\prime}=-W^{\prime}u^{\prime}$ we see that
$W^{\prime}f-ufuW^{\prime}=(cx_{2}).f=\phi(h).f=Wf-ufuW$ for every $f\in{\rm
End}(P)$. From here, we deduce that $u(W^{\prime}-W)=\nu\in k$. Using skew-
commutativity of $u$ with $W$ and $W^{\prime}$ we conclude that $\nu=0$ so
$W^{\prime}=W$. Then $W^{\prime}w^{\prime}-w^{\prime}W^{\prime}=\lambda(Ww-
wW)\neq 0$ so that relation $(cx_{2})x_{1}-x_{1}(cx_{2})=0$ in $E(2)$ cannot
be respected. Hence, $[{\rm End}(P)]\neq 1$ in $BM(k,E(2),R_{N})$ and ${\rm
Ker}(T^{*})\cong{\mathbb{Z}}_{2}$. $\Box$
The following proposition shows that the groups $BM(k,H_{4},R_{l})$ may be
viewed inside $BM(k,E(2),R_{N})$ and it also describes the image through
$T^{*}$ of them.
###### Proposition 5.3
For every $(\lambda,\mu)\in k\times k$ there is a group homomorphism
$\Theta_{\lambda,\mu}\colon BM(k,H_{4},R_{\lambda\mu})\to BM(k,E(2),R_{N})$
satisfying:
1. (1)
The image of $\Theta_{0,0}$ is the subgroup isomorphic to $BW(k)$ represented
by elements with trivial $x_{1}$\- and $x_{2}$-action and
$Ker(\Theta_{0,0})\cong(k,+)$.
2. (2)
$\Theta_{\lambda,\mu}$ is injective if and only if $(\lambda,\mu)\neq(0,0)$.
3. (3)
For $(\lambda,\mu)\neq(0,0),$ the image of $T^{*}\Theta_{\lambda,\mu}$ is
$Im(i_{\mu\lambda^{-1}})$ if $\lambda\neq 0$ and $Im(\iota_{\mu^{-1}\lambda})$
if $\mu\neq 0$.
Proof: For every $(\lambda,\mu)\in k\times k$ the map
$\theta_{\lambda,\mu}\colon E(2)\to H_{4}$ mapping $c\to g$, $x_{1}\to\lambda
h$ and $x_{2}\to\mu h$ is a Hopf algebra projection. A direct computation
shows that
$(\theta_{\lambda,\mu}\otimes\theta_{\lambda,\mu})(R_{N})=R_{\lambda\mu}$ so
the pull-back of $\theta_{\lambda,\mu}$ induces the desired homomorphism
$\Theta_{\lambda,\mu}$.
(1) Let $(\lambda,\mu)=(0,0)$. Then any element in $BM(k,H_{4},R_{0})$ can be
written as a pair of the form $([C(a;t,0)],[B])$ for $[B]\in BW(k)$. The image
through $\Theta_{0,0}$ of such an element is $[C(a)][B]\in BW(k)$ with trivial
$x_{i}$-action on $C(a)$. Clearly, $BW(k)=Im(\Theta_{0,0})$. That
$Ker(\Theta_{0,0})$ is isomorphic to $(k,+)$ follows from the isomorphism
$BM(k,H_{4},R_{0})\cong(k,+)\times BW(k)$ and the fact that $(k,+)$ is
realized as classes admitting a representative that is trivial when viewed as
a $k{\mathbb{Z}}_{2}$-module algebra.
(2) Let $(\lambda,\mu)\neq(0,0)$. If $\Theta_{\lambda,\mu}([A])=1$ then $A$ is
isomorphic to an endomorphism algebra with strongly inner $E(2)$-action. In
other words, $A\cong{\rm End}(P)$ and there is a convolution invertible
algebra map $p\colon E(2)\to A$ such that $l\cdot a=\sum
p(l_{(1)})ap^{-1}(l_{(2)})$ for every $l\in E(2),a\in A$. There are elements
$u,v,w\in A$ with $u$ invertible such that $c\cdot a=g\cdot a=uau^{-1}$,
$x_{1}\cdot a=(va-av)u=\lambda h\cdot a$ and $x_{2}\cdot a=(wa-aw)u=\mu h\cdot
a$. Then
$0=\mu x_{1}\cdot a-\lambda x_{2}\cdot a=((\mu v-\lambda w)a-a(\mu v-\lambda
w))u\quad\forall a\in A,$
and since $u$ is invertible and $A$ is central we have $\mu v-\lambda w=\eta$
for some $\eta\in k$. The relation between $v$ and $w$ gives $\eta=0$ and so
$\mu v=\lambda w.$ Thus, the same elements $u,v$ and $w$ ensure that the
$H_{4}$-action on $A$ is strongly inner. Therefore $[A]=1$ in
$BM(k,H_{4},R_{\lambda\mu})$. The converse follows from (1).
(3) Let us now assume that $(\lambda,\mu)\neq(0,0).$ It is immediate to see
that if $[A]\in BW(k)\subset BM(k,H_{4},R_{\lambda\mu})$ is represented by an
algebra with trivial $h$-action, then $\Theta_{\lambda,\mu}([A])$ is
represented by an algebra with trivial $x_{1}$\- and $x_{2}$-action. Hence
$T^{*}\Theta_{\lambda,\mu}(BM(k,H_{4},R_{\lambda\mu}))\subset i_{0}(BW(k))$
and the restriction of $T^{*}\Theta_{\lambda,\mu}$ to $BW(k)$ is an
isomorphism onto $i_{0}(BW(k))$. Let us now consider the class
$[C(a;1,\lambda\mu)]\in BM(k,H_{4},R_{\lambda\mu})$. Its image through
$\Theta_{\lambda,\mu}$ is the algebra generated by $x$ with $x^{2}=a$, with
$c\cdot x=-x$, $x_{1}\cdot x=\lambda$ and $x_{2}\cdot x=\mu$. A direct
verification shows that
$T^{*}\Theta_{\lambda,\mu}([C(a;1,\lambda\mu)])=[C(a;\lambda,\mu)]$. Then the
image of $T^{*}\Theta_{\lambda,\mu}$ is $Im(i_{\mu\lambda^{-1}})$ if
$\lambda\neq 0$ and $Im(\iota_{\mu^{-1}\lambda})$ if $\mu\neq 0$. $\Box$
Theorem 5.2 shows that one should understand $BM(k,E(2),R_{N})$ in order to
compute $BQ(k,H_{4})$. In view of Proposition 5.3, $BM(k,E(2),R_{N})$ seems to
be much more complex that the groups of type BM treated in [10, 11, 20].
## 6 Appendix
This last section is devoted to the analysis of some difficulties occurring in
the study of the structure of $(E(2),R_{N})$-Azumaya algebras. We show that
the set of classes represented by ${\mathbb{Z}}_{2}$-graded central simple
algebras (with respect to the grading induced by the $c$-action) is not a
subgroup of $BM(k,E(2),R_{N})$.
Let us consider the braiding $\psi_{VW}$ determined by $R_{N}$ between two
left $E(2)$-modules $V$ and $W$. Let $v\in V$ and $w\in W$ be homogeneous
elements with respect to the ${\mathbb{Z}}_{2}$-grading induced by the
$c$-action. By direct computation it is:
$\begin{array}[]{l}\psi_{VW}(v\otimes w)=\sum R_{N}^{(2)}\cdot w\otimes
R_{N}^{(1)}\cdot v\vspace{2pt}\\\ \hskip 48.36958pt=(-1)^{|v||w|}w\otimes
v+(-1)^{|w|+1}(-1)^{(|v|+1)(|w|+1)}(x_{2}\cdot w)\otimes(x_{1}\cdot
v).\end{array}$
If we denote by $\psi_{0}$ the braiding associated with the
${\mathbb{Z}}_{2}$-grading we have
$\psi_{VW}(v\otimes w)=\psi_{0}(v\otimes w)+(-1)^{|w|+1}\psi_{0}(x_{1}\cdot
v\otimes x_{2}\cdot w).$ (6.1)
Let $F$ and $G$ be the maps in (1.4) defining an $(E(2),R_{N})$-Azumaya
algebra $A$ and let $F_{0}$ and $G_{0}$ be the maps defining an
$(E(2),R_{0})$-Azumaya algebra, that is, the maps determining when an
$E(2)$-module algebra is ${\mathbb{Z}}_{2}$-graded central simple. It is not
hard to verify by direct computation that, for homogeneous $a,\,b,\,d\in A$
with respect to the $c$-action we have:
$F(a\\#b)(d)=F_{0}(a\\#b)(d)+(-1)^{|d|+1}F_{0}(a\\#x_{1}\cdot b)(x_{2}\cdot
d)$ (6.2) $G(a\\#b)(d)=G_{0}(a\\#b)(d)+(-1)^{|a|+1}F_{0}(x_{2}\cdot
a\\#b)(x_{1}\cdot d)$ (6.3)
Notice that if either $x_{1}$ or $x_{2}$ acts trivially, then $F=F_{0}$ and
$G=G_{0}$. So in this case, $A$ is $(E(2),R_{N})$-Azumaya if and only if it is
${\mathbb{Z}}_{2}$-graded central simple (i.e. $A$ is $(E(2),R_{0})$-Azumaya).
We will say that the $x_{i}$-action on an $E(2)$-module algebra $A$ is inner
if there exists an odd element $v\in A$ such that $x_{i}\cdot a=v(c\cdot
a)-av$ for every $a\in A$.
###### Theorem 6.1
Let $A$ be an $(E(2),R_{N})$-Azumaya algebra. The following assertions are
equivalent:
1. (1)
The $x_{1}$-action on $A$ is inner;
2. (2)
The $x_{2}$-action on $A$ is inner;
3. (3)
$A$ is a ${\mathbb{Z}}_{2}$-graded central simple algebra.
In addition, the $E(2)$-action on $A$ is inner if and only if $A$ is a central
simple algebra.
Proof: (1) $\Rightarrow$ (3) Let $v_{1}\in A$ be an odd element such that
$x_{1}\cdot a=v_{1}(c\cdot a)-av_{1}$ for all $a\in A$. Applying equality
(6.2) to any homogeneous $b$ and $d$ in $A$ gives:
$\begin{array}[]{ll}F(a\\#b)(d)&=F_{0}(a\\#b)(d)+F_{0}(a\\#b)((x_{2}\cdot
d)v_{1})\vspace{2pt}\\\ &\hskip 10.0pt+(-1)^{|d|}F_{0}(a\\#bv_{1})(x_{2}\cdot
d)\end{array}$ (6.4)
This equality extends to all elements $a$ and $b$ in $A$. If $A$ were not
${\mathbb{Z}}_{2}$-graded central simple, there would exist an element
$0\neq\sum_{i}a_{i}\\#b_{i}$ in $Ker(F_{0})$. Then
$(\sum_{i}a_{i}\\#b_{i})(1\\#v_{1})=\sum_{i}a_{i}\\#b_{i}v_{1}\in Ker(F_{0})$
and for every $f$ in $A$ we would have
$F_{0}(\sum_{i}a_{i}\\#b_{i})(f)=F_{0}(\sum_{i}a_{i}\\#b_{i}v_{1})(f)=0$. It
follows from (6.4) that $\sum_{i}a_{i}\\#b_{i}\in Ker(F),$ contradicting the
injectivity of $F$.
(2) $\Rightarrow$ (3) Similarly to (1) $\Rightarrow$ (3) replacing $F$ by $G$.
(3) $\Rightarrow$ (1), (2) Suppose that $A$ is a ${\mathbb{Z}}_{2}$-graded
central simple algebra. If $A$ is a central simple algebra then the
$E(2)$-action on $A$ is inner by the Skolem-Noether theorem. If $A$ is not
central simple then it is of odd type ([13, Theorem 3.4, Definition 3.5]) and
it is $(H_{4},R_{0})$-Azumaya for the subalgebra of $E(2)$, isomorphic to
$H_{4}$ generated by $c$ and $x_{i}$. By [1, Theorem 3.4] the $x_{i}$-action
is inner.
Let us finally assume that the $E(2)$-action on $A$ is inner. Then $A$ is a
${\mathbb{Z}}_{2}$-graded central simple algebra. Since $E(2)$ acts innerly on
$A$ then it acts trivially on its center $Z(A)$. Besides it is immediately
seen that $Z(A)$ is contained in the right and left $E(2)$-center, that are
trivial because $A$ is assumed to be $E(2)$-Azumaya. Hence $Z(A)$ must be
trivial and so $A$ is also a central algebra. By the structure theorems of
${\mathbb{Z}}_{2}$-graded central simple algebras ([13, Theorem IV.3.4]), $A$
is central simple. $\Box$
###### Proposition 6.2
Let $A$ and $B$ be two equivalent $(E(2),R_{N})$-Azumaya algebras. Then the
$x_{i}$-action on $A$ is inner if and only if it is so on $B$.
Proof: Let $P$ and $Q$ be finite dimensional $E(2)$-modules for which
$A\\#{\rm End}(P)\cong B\\#{\rm End}(Q)$. If the $x_{i}$-action on $A$ is
inner then it is so on $A\\#{\rm End}(P)$ by [11, Proposition 4.6], hence it
is so on $B\\#{\rm End}(Q)$, which is a ${\mathbb{Z}}_{2}$-graded central
simple algebra by Theorem 6.1. For $i=1,2$, let $W_{i},v_{i}$ be odd elements
in $B\\#{\rm End}(Q)$ and ${\rm End}(Q)$ respectively inducing the
$x_{i}$-action. We recall that $x_{j}\cdot v_{i}=0$ because the action on
${\rm End}(Q)$ is strongly inner, while $x_{j}\cdot W_{i}$ is a scalar for
every pair $i,j$ because $x_{j}\cdot W_{i}$ belongs to the graded center of
$B\\#{\rm End}(Q)$. The odd elements $T_{i}=W_{i}-1\\#v_{i}-(x_{2}\cdot
W_{i})(1\\#v_{1})\in B\\#{\rm End}(Q)$ for $i=1,2$ are such that $x_{j}\cdot
T_{i}=x_{j}\cdot W_{i}$ for every $i$ and $j$. Moreover, for every homogeneous
$f\in{\rm End}(Q)$ with respect to the $c$-action we have:
$\begin{array}[]{l}(-1)^{|f|}T_{i}(1\\#f)=W_{i}(c\cdot 1\\#c\cdot
f)-1\\#v_{i}(c\cdot f)-(x_{2}\cdot W_{i})(1\\#v_{1}(c\cdot f))\vspace{2pt}\\\
\hskip 36.98866pt=(1\\#f)W_{i}-(1\\#fv_{i})-(x_{2}\cdot
W_{i})(1\\#fv_{1})-(x_{2}\cdot W_{i})(x_{1}\cdot(1\\#f))\vspace{2pt}\\\ \hskip
36.98866pt=(1\\#f)[W_{i}-1\\#v_{i}-(x_{2}\cdot W_{i})(1\\#v_{1})]-(x_{2}\cdot
W_{i})(x_{1}\cdot(1\\#f))\vspace{2pt}\\\ \hskip
36.98866pt=(1\\#f)T_{i}-(x_{2}\cdot W_{i})(x_{1}\cdot(1\\#f)).\end{array}$
In other words,
$(1\\#f)T_{i}=(-1)^{|f||T_{i}|}T_{i}(1\\#f)+(x_{2}\cdot
T_{i})(x_{1}\cdot(1\\#f)),$
so by (6.1) the element $T_{i}\in C^{l}_{B\\#{\rm End}(Q)}({\rm End}(Q)),$ the
left centralizer of ${\rm End}(Q)$ in $B\\#{\rm End}(Q)$, that is, $T_{i}\in
B\\#1$ by the double centralizer theorem [1, Theorem 2.3]. Besides, for every
homogeneous $b\in B$ we have:
$\begin{array}[]{rl}T_{i}(c\cdot
b\\#1)-(b\\#1)T_{i}&=(-1)^{|b|}W_{i}(b\\#1)-(b\\#v_{i})-(x_{2}\cdot
W_{i})(b\\#v_{1})\vspace{2pt}\\\ &\hskip
10.0pt-(b\\#1)W_{i}+(b\\#v_{i})+(x_{2}\cdot W_{i})(b\\#v_{1})\vspace{2pt}\\\
&=x_{i}\cdot(b\\#1).\end{array}$
Hence the $x_{i}$-action on $B$ is inner. $\Box$
We conclude by showing that, contrarily to the cases treated in the literature
([10, 11, 20]), a Skolem-Noether-like approach is probably not appropriate for
the computation of $BM(k,E(2),R_{N})$ because the set of classes admitting a
representative with inner action is not a subgroup.
###### Theorem 6.3
The classes in $BM(k,E(2),R_{N})$ that are represented by
${\mathbb{Z}}_{2}$-graded central simple algebras do not form a subgroup.
Proof: Let $t\neq 0,1$ and $q\neq 2$ be in $k$. We consider the representative
$C(1;t,2)$ generated by $x$ with $x^{2}=1$, $c\cdot x=-x$, $x_{1}\cdot x=t$
and $x_{2}\cdot x=2$ and the representative $C(1;1,q)$ generated by $y$ with
$y^{2}=1$, $c\cdot y=-y$, $x_{1}\cdot y=1$ and $x_{2}\cdot y=q$. Both are
$(E(2),R_{N})$-Azumaya because $C(1;1,2t)$ is $(H_{4},R_{2t})$-Azumaya,
$C(1;1,q)$ is $(H_{4},R_{q})$-Azumaya and $C(1;t,2),C(1;1,q)$ are obtained
from these ones respectively by pulling back through $\theta_{\lambda,\mu}$.
They are also ${\mathbb{Z}}_{2}$-graded central simple algebras. Their product
$C(1;t,2)\\#C(1;1,q)$ is generated by the odd elements $X$ and $Y$ with
$X^{2}=1$, $Y^{2}=1$ and $XY+YX=2$. The element $X-Y$ is easily seen to lie in
the ${\mathbb{Z}}_{2}$-graded center, so $C(1;t,2)\\#C(1;1,q)$ is not a
${\mathbb{Z}}_{2}$-graded central simple algebra. If $B$ were another
representative of $[C(1;t,2)\\#C(1;1,q)]$ that is a ${\mathbb{Z}}_{2}$-graded
central simple algebra, then by Theorem 6.1, the $x_{1}$-action on it would be
inner. By Proposition 6.2, $x_{1}$ would act innerly on $C(1;t,2)\\#C(1;1,q)$.
Applying again Theorem 6.1, $C(1;t,2)\\#C(1;1,q)$ would be
${\mathbb{Z}}_{2}$-graded central simple. $\Box$
Acknowledgements
This research was partially supported by the Azioni Integrate Italia-España
AIIS05E34A Algebre, coalgebre, algebre di Hopf e loro rappresentazioni. The
second named author is also supported by projects MTM2008-03339 from MCI and
FEDER and P07-FQM-03128 from Junta de Andalucía.
## References
* [1] Armour A.; Chen H.-X.; Zhang Y. Structure theorems of $H_{4}$-Azumaya algebras. J. Algebra 305 (2006), 360-393.
* [2] Beattie, M.; Caenepeel, S. The Brauer-Long group of ${\mathbb{Z}}/p^{t}{\mathbb{Z}}$-dimodule algebras. J. Pure Appl. Algebra 60 (1989), 219-236.
* [3] Bichon, J.; Carnovale G. Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras. J. Pure Appl. Algebra 204 no. 3 (2006), 627-665.
* [4] Caenepeel, S. Computing the Brauer-Long group of a Hopf algebra I: the cohomological theory. Israel J. Math. 72 Nos. 1-2 (1990), 38-83.
* [5] Caenepeel, S. Brauer groups, Hopf algebras and Galois Theory. K-Monographs in Mathematics 4. Kluwer Academic Publishers, Dordrecht, 1998.
* [6] Caenepeel, S. The Brauer-Long group revisited: the multiplication rules. Algebra and Number Theory (Fez), 61-86. Lecture Notes in Pure Appl. Math 208. Marcel-Dekker, New York, 2000.
* [7] Caenepeel, S.; Van Oystaeyen, F.; Zhang, Y. Quantum Yang-Baxter Module Algebras. K-theory 8 no. 3 (1994), 231-255.
* [8] Caenepeel, S.; Van Oystaeyen, F.; Zhang, Y. The Brauer group of Yetter-Drinfeld module algebras. Trans. Amer. Math. Soc. 349 no. 9 (1997), 3737-3771.
* [9] Carnovale, G. Some isomorphisms for the Brauer groups of a Hopf algebra. Comm. Algebra 29 no. 11 (2001), 5291-5305.
* [10] Carnovale, G.; Cuadra, J. The Brauer group of some quasitriangular Hopf algebras. J. Algebra 259 no. 2 (2003), 512-532.
* [11] Carnovale, G.; Cuadra, J. Cocycle twisting of $E(n)$-module algebras and applications to the Brauer group. K-Theory 33 (2004), 251–276.
* [12] DeMeyer, F.; Ford, T. Computing the Brauer-Long group of ${\mathbb{Z}}_{2}$-dimodule algebras. J. Pure Appl. Algebra 54 (1988), 197-208.
* [13] Lam, T. Y. Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67. American Mathematical Society, Providence, RI, 2005.
* [14] Long, F.W. A generalization of the Brauer group of graded algebras. Proc. London Math. Soc. 29 no. 3 (1974), 237-256.
* [15] Majid, S. Doubles of quasitriangular Hopf algebras. Comm. Algebra 19 (1991), 3061-3073.
* [16] Montgomery, S.; Schneider, H.-J. Skew derivations of finite-dimensional algebras and actions of the Taft Hopf algebra. Tsukuba J. Math. 25 no. 2 (2001), 337-358.
* [17] Panaite, F; Van Oystaeyen, F. Quasitriangular structures for some pointed Hopf algebras of dimension $2^{n}$. Comm. Algebra 27 no. 10 (1999), 4929-4942.
* [18] Radford, D.E. Minimal quasitriangular Hopf algebras. J. Algebra 157 no. 2 (1993), 285-315.
* [19] Van Oystaeyen, F.; Zhang, Y. Embedding the Hopf automorphism group into the Brauer group. Can. Math. Bull. 41 (1998), 359-367.
* [20] Van Oystaeyen, F.; Zhang, Y. The Brauer group of Sweedler’s Hopf algebra $H_{4}$. Proc. Amer. Math. Soc. 129 no. 2 (2001), 371-380.
* [21] Van Oystaeyen, F.; Zhang, Y. Computing subgroups of the Brauer group of $H_{4}$. Comm. Algebra 30 no. 10 (2002), 4699-4709.
|
arxiv-papers
| 2009-04-12T21:22:37 |
2024-09-04T02:49:01.823117
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Giovanna Carnovale, Juan Cuadra",
"submitter": "Carnovale Giovanna",
"url": "https://arxiv.org/abs/0904.1883"
}
|
0904.2012
|
# Simplicial Databases
David I. Spivak
###### Abstract.
In this paper, we define a category ${\bf DB}$, called the category of
simplicial databases, whose objects are databases and whose morphisms are
data-preserving maps. Along the way we give a precise formulation of the
category of relational databases, and prove that it is a full subcategory of
${\bf DB}$. We also prove that limits and colimits always exist in ${\bf DB}$
and that they correspond to queries such as select, join, union, etc.
One feature of our construction is that the schema of a simplicial database
has a natural geometric structure: an underlying simplicial set. The geometry
of a schema is a way of keeping track of relationships between distinct
tables, and can be thought of as a system of foreign keys. The shape of a
schema is generally intuitive (e.g. the schema for round-trip flights is a
circle consisting of an edge from $A$ to $B$ and an edge from $B$ to $A$), and
as such, may be useful for analyzing data.
We give several applications of our approach, as well as possible advantages
it has over the relational model. We also indicate some directions for further
research.
This project was supported in part by the Office of Naval Research.
###### Contents
1. 1 Introduction
2. 2 The category of Tables
3. 3 Constructions and formal properties of Tables
4. 4 Schemas and databases
5. 5 Constructions and formal properties of Simplicial Databases
6. 6 Applications, advantages, and further research
## 1\. Introduction
The theory of relational databases is generally formulated within mathematical
logic. We provide a more modern and more flexible approach using methods from
category theory and algebraic topology. Category theory is useful both as a
language and as a tool, and has been successfully applied to many areas of
computer science. Using an inefficient language can hamper ones ability to
implement, work with, and reason about a subject. This can be seen as one
reason that SQL implements tables, rather than relational databases in their
pure form: perhaps mathematical logic is not a sufficiently flexible language
for discussing databases as they are used in practice.
One reason that relational databases have been so successful is that their
definition can be phrased within a precise mathematical language. The
definition we provide in this paper is just as precise, if not more so (see
the discussion at the beginning of Section 4). However, we go beyond simply
defining the objects of study (databases), but instead continue on to define
morphisms between databases. With these definitions, we have a category of
databases.
There are many categories whose objects are databases (the difference being in
their morphisms); what makes one definition better than another? First, a good
definition should make sense – the morphisms should somehow preserve the
structure of the databases. Second, applying common categorical constructions
(colimits, limits, etc.) to the category of databases should result in common
database constructions, such as unions, joins, etc. Third, the categorical
approach should make reasoning about databases, such as that needed for
maintaining and restructuring databases, easier.
Our formulation accomplishes these three goals (see Remark 4.3.8, and Sections
5 and 6, respectively). As an added bonus, the schemas for our databases have
geometric structure (more precisely, the structure of a simplicial set). In
other words, the schema is given as a geometric object which one should think
of as a kind of Entity-Relationship diagram for the schema. This approach may
lead to improvements in query optimization because one can adjust the “shape”
of the schema to fit with the purposes of the queries to be taken. The ability
to visualize data should also prove useful, because these visualizations seem
to “make sense” in practice. Examples of this phenomenon are given in 6.1.1
and 6.1.2, where we respectively discuss round trip flights and a sociological
experiment involving 4-cycles in high school partnerships.
The data on a given schema is given by a sheaf of sets on that schema. Sheaves
are ubiquitous in modern mathematics because they generalize sets and
functions and because they have good formal properties. Classical operations
on sheaves (such as direct images) allow one to transport data from one schema
to another in a functorial way. One of the main purposes of this paper is to
provide a good language for discussing databases mathematically, and the
consideration of data as a sheaf on a given schema helps to accomplish that
goal.
Other researchers have formulated databases in terms of category theory (for
example, see [RW92],[JRW02],[PS95],[Ber01],[DK94],[Dis96],[GB92]). Of note is
work by Cadish and Diskin, and work by Rosebrugh and Wood. There are many
differences between previous viewpoints and our own. Most notably, our work
uses simplicial methods to give a geometric structure to the schemas of
databases and uses sheaves over these spaces to model the data itself. Both of
these approaches appear to be new.
We assume throughout this paper that the reader has a basic knowledge of
category theory which includes knowing the definition of category, functor,
limit, and colimit, as well as basic facts such as Yoneda’s lemma. Good
references for this material include [ML98],[BW90], and [Bor94a]. We do not
assume that the reader has a prior knowledge of sheaves or of simplicial sets.
We begin by defining the category of tables, in Section 2. In Section 3, we
prove that the category of tables is closed under limits and certain colimits,
and that these constructions correspond to joins and unions. We also prove
that projections and deletions are easily defined under our formulation. In
Section 4, we first give a brief description of simplicial sets. We then
proceed to define the category of simplicial databases. In Section 5, we prove
that the category of simplicial databases is closed under all limits and
colimits and prove that they again correspond to joins and unions. Finally in
Section 6, we discuss some applications of our model and directions for future
research.
### 1.1. Acknowledgments
I would like to thank Paea LePendu for explaining relational databases to me,
for suggesting that databases should be categorified, and for his advice and
encouragement throughout the process. I would also like to thank Chris Wilson
for several useful conversations.
## 2\. The category of Tables
It is no accident that SQL uses tables instead of relations: Tables are
inherently more useful, yet just as easy to implement. They are disliked by
the purists of relational database theory not because they are bad, but
because they do not fit in with that theory. In this section we provide a
categorical structure to the set of tables, thus firmly grounding it in
rigorous mathematics.
### 2.1. Data types
In order to define schemas, records, and tables of a given type, we need to
define what we mean by “type.”
###### Definition 2.1.1.
A type specification is simply a function between sets $\pi\colon
U\rightarrow{\bf DT}$. The set ${\bf DT}$ is called the set of data types for
$\pi$, and the set $U$ is called the domain bundle for $\pi$. Given any
element $T\in{\bf DT}$, the preimage $\pi^{-1}(T)\subset U$ is called the
domain of $T$, and an element $x\in\pi^{-1}(T)$ is called an object of type
$T$.
###### Example 2.1.2.
Let $U$ denote the disjoint union
$U\colon=({\mathbb{Z}}\amalg{\mathbb{R}}\amalg{\bf Strings})$ and let ${\bf
DT}$ denote the three element set
$\\{`{\mathbb{Z}}\textnormal{'},`{\mathbb{R}}\textnormal{'},`{\bf
Strings}\textnormal{'}\\}$. Let $\pi\colon U\rightarrow{\bf DT}$ denote the
obvious function, which send all of ${\mathbb{Z}}$ to the element
$`{\mathbb{Z}}\textnormal{'}$, all of ${\mathbb{R}}$ to
$`{\mathbb{R}}\textnormal{'}$, and all of ${\bf Strings}$ to $`{\bf
Strings}\textnormal{'}$. The preimage $\pi^{-1}(`{\bf
Strings}\textnormal{'})\subset U$, which we have called the domain of the type
$`{\bf Strings}\textnormal{'}$, is indeed the set of strings.
As another example, the mod 2 function
$\pi\colon{\mathbb{Z}}\rightarrow\\{\textnormal{`even'},\textnormal{`odd'}\\}$
is a type specification in which the objects of type ‘even’ are the even
integers.
### 2.2. Schemas
We quickly recall the definition of fiber product (for sets).
###### Definition 2.2.1.
Let $A,B,$ and $C$ be sets, and suppose $f\colon A\rightarrow B$ and $g\colon
C\rightarrow B$ are functions with the same codomain. The fiber product of $A$
and $C$ over $B$, denoted $A\times_{B}C$, is the set
$A\times_{B}C\colon=\\{(a,c)\in A\times C|f(a)=g(c)\in B\\}.$
The fiber product moreover comes equipped with obvious projection maps making
the diagram
$\textstyle{A\times_{B}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$$\textstyle{\lrcorner}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B}$
commute. The corner symbol $\lrcorner$ serves to remind the reader that the
object in the upper left is a fiber product. We sometimes call
$g^{\prime}\colon A\times_{B}C\rightarrow A$ the pullback of $g$ along $f$;
similarly $f^{\prime}$ is the pullback of $f$ along $g$.
###### Remark 2.2.2.
The fiber product of the diagram $A\xrightarrow{f}B\xleftarrow{g}C$ above
should probably be denoted $f\times_{B}g$ instead of $A\times_{B}C$, since it
depends on the maps $f$ and $g$, not just their domains. However, this is not
often done, and in this paper the maps will be clear from context.
###### Definition 2.2.3.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification. A simple
schema of type $\pi$ consists of a pair $(C,\sigma)$, where $C$ is a finite
(totally) ordered set and $\sigma\colon C\rightarrow{\bf DT}$ is a function.
We sometimes denote the simple schema $(C,\sigma)$ by $\sigma$. We refer to
$C$ as the column set or set of attributes for $\sigma$ and $\pi$ as the type
specification for $\sigma$.
Let $U_{\sigma}\colon=\sigma^{-1}(U)$ denote the fiber product $U\times_{\bf
DT}C$. We call the pullback $\pi_{\sigma}\colon U_{\sigma}\rightarrow C$, i.e.
the left hand map in the diagram
$\textstyle{U_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\sigma}}$$\textstyle{\lrcorner}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{{\bf
DT},}$
the domain bundle on $C$ induced by $\sigma$.
###### Remark 2.2.4.
We do not worry much about the ordering on $C$, as evidenced by the fact that
we do not record it in the notation $(C,\sigma)$ for the simple schema. In
fact the ordering requirement can be dropped from the definition if one so
chooses.
The reason we include it is first because the columns of a displayed table
naturally come with an order (left to right), and second because it results in
a more commonly used mathematical object down the road in Section 4. See
Remark 4.1.1.
###### Example 2.2.5.
Let $\pi\colon U\rightarrow{\bf DT}$ denote the type specification of Example
2.1.2. Let $C=(\textnormal{`First Name', `Last Name',`Age'})$, and define
$\sigma\colon C\rightarrow{\bf DT}$ by
$\displaystyle\sigma(\textnormal{`First Name'})$ $\displaystyle=`{\bf
Strings}\textnormal{'}$ $\displaystyle\sigma(\textnormal{`Last Name'})$
$\displaystyle=`{\bf Strings}\textnormal{'}$
$\displaystyle\sigma(\textnormal{`Age'})$
$\displaystyle=`{\mathbb{Z}}^{\prime}$
We see that $C$ is a set of attributes for the simple schema $\sigma$. We call
$C$ the column set because, once we arrange data in terms of tables, the
columns of these tables will each be headed by an element of $C$.
One can check that the domain bundle $U_{\sigma}\rightarrow C$ induced by
$\sigma$ is the obvious function
$({\bf Strings}\amalg{\bf Strings}\amalg{\mathbb{Z}})\longrightarrow C.$
Thus an object of type ‘First Name’ is a string in this example.
###### Definition 2.2.6.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification. A morphism
of simple schemas (of type $\pi$), written
$f\colon(C,\sigma)\rightarrow(C^{\prime},\sigma^{\prime})$, is an order-
preserving function $f\colon C\rightarrow C^{\prime}$ such that the triangle
---
$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\sigma}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{\prime}}$$\textstyle{\bf
DT}$
commutes.
The category of simple schemas on $\pi$, denoted $\mathcal{S}^{\pi}$ is the
category whose objects are simple schemas and whose morphisms are morphisms
thereof.
###### Remark 2.2.7.
Let ${\bf\Delta}$ denote the category of finite ordered sets. Let
$({\bf\Delta}\downarrow{\bf DT})$ denote the category for which an object is a
finite ordered set with a map to ${\bf DT}$ and for which a morphism is an
order-preserving function, over ${\bf DT}$. One can easily see that the
category $\mathcal{S}^{\pi}$ is isomorphic to $({\bf\Delta}\downarrow{\bf
DT})$, regardless of $\pi$. However, we should think of $\pi$ as part of the
data for a simple schema.
Note that the symbol ${\bf\Delta}$ typically refers to the category of non-
empty finite ordered sets; one typically denotes the category of all finite
ordered sets as ${\bf\Delta}_{+}$. For typographical reasons, we do not follow
the standard convention in this paper.
### 2.3. Records and Tables
###### Definition 2.3.1.
Let $(C,\sigma)$ be a simple schema. A record on $(C,\sigma)$ is a function
$r\colon C\rightarrow U_{\sigma}$ such that $\pi_{\sigma}\circ
r=\textnormal{id}_{C}$, i.e. a section of the domain bundle for $\sigma$. We
denote the set of records on $\sigma$ by $\Gamma^{\pi}(\sigma)$, or simply by
$\Gamma(\sigma)$ if $\pi$ is understood.
In other words, a record must produce, for each attribute $c\in C$, an object
of type $\sigma(c)\in{\bf DT}$.
###### Example 2.3.2.
Let $\pi$ and $(C,\sigma)$ be as in Example 2.2.5. A record on that simple
schema is a section $r$ as depicted in the diagram
$\textstyle{{\bf Strings}\amalg{\bf
Strings}\amalg{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\sigma}}$$\textstyle{\\{\textnormal{`First
Name', `Last
Name',`BYear'}\\}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$
That is, a record is a way to designate a first name and a last name (in ${\bf
Strings}$) and an age (in ${\mathbb{Z}}$). For example (Barack; Obama; 1961)
denotes a record on this simple schema; that is, it defines a section of
$\pi_{\sigma}$.
The set $\Gamma(\sigma)$ of records on $(C,\sigma)$ is simply the set of all
possible such sections. In this example $\Gamma(\sigma)={\bf
Strings}\times{\bf Strings}\times{\mathbb{Z}}$.
###### Definition 2.3.3.
Let $\pi\colon U\rightarrow{\bf DT}$ be a type specification. A table of type
$\pi$ consists of a sequence $(K,C,\sigma,\tau)$, where $K$ is a set,
$(C,\sigma)$ is a simple schema of type $\pi$, and $\tau\colon
K\rightarrow\Gamma(\sigma)$ is a function. We sometimes denote the table
$(K,C,\sigma,\tau)$ simply by $\tau$. The set $K$ is called the set of keys of
$\tau$, and $(C,\sigma)$ is called the simple schema of $\tau$.
###### Remark 2.3.4.
Given a table $(K,C,\sigma,\tau)$, those familiar with SQL should think of the
set $K$ of keys as the set of row identifiers for a table. These row ids are
always unique identifiers and serve as an internal key system for the table;
they are generally not considered as part of the data.
###### Remark 2.3.5.
We do not require our tables to have finitely many rows. One could easily
enforce such a restriction if desired, and follow the rest of the paper with
that restriction in mind. The resulting category would be a full subcategory
of the one we present in Definition 2.4.1, it would still be closed under
finite limits (etc.), and queries would be taken in precisely the same way as
they are here.
###### Example 2.3.6.
Given a simple schema $(C,\sigma)$, a table on it is simply a collection of
records indexed by a set $K$. The records need not be distinct because the set
$K$ keeps track of the distinctions. Continuing with $\pi$ and $(C,\sigma)$ as
in Example 2.3.2, we could have $K=\\{1,2,`foo^{\prime}\\}$ and let
$\tau\colon K\rightarrow\Gamma(\sigma)$ be the assignment
$\displaystyle 1$ $\displaystyle\mapsto\textnormal{(Barack; Obama; 1961)}$
$\displaystyle 2$ $\displaystyle\mapsto\textnormal{(Michelle; Obama; 1964)}$
$\displaystyle`foo^{\prime}$ $\displaystyle\mapsto\textnormal{(Barack; Obama;
1961)}$
This table can be written in more standard form as:
K | ‘First Name’ | ‘Last Name’ | ‘BYear’
---|---|---|---
1 | Barack | Obama | 1961
2 | Michelle | Obama | 1964
‘foo’ | Barack | Obama | 1961
We indicate with the double vertical line the fact that this table corresponds
to a function whose domain is $K$.
###### Lemma 2.3.7.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification, let
$(C_{1},\sigma_{1})$ and $(C_{2},\sigma_{2})$ denote simple schemas on $\pi$,
and let $f\colon(C_{2},\sigma_{2})\rightarrow(C_{1},\sigma_{1})$ denote a
morphism of simple schemas. There is an induced map on record sets
$f^{*}\colon\Gamma(\sigma_{1})\rightarrow\Gamma(\sigma_{2})$.
###### Proof.
Consider the diagram
$\textstyle{U_{\sigma_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\textstyle{\lrcorner}$$\textstyle{U_{\sigma_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{\lrcorner}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\sigma_{2}}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{1}}$$\textstyle{{\bf
DT}.}$
Note that the left hand square is a fiber product square. This follows by
applying basic category theory (specifically the “pasting lemma” for fiber
products; see [ML98]) to the fact that the right hand square and the big
rectangle are fiber product squares. We must show that a section $r_{1}\colon
C_{1}\rightarrow U_{\sigma_{1}}$ of $\pi_{1}$ induces a section $r_{2}\colon
C_{2}\rightarrow U_{\sigma_{2}}$ of $\pi_{2}$, because this assignment will
constitute $f^{*}\colon\Gamma(\sigma_{1})\rightarrow\Gamma(\sigma_{2})$.
Suppose given $r_{1}$ with $\pi_{1}\circ r_{1}=\textnormal{id}_{C_{1}}$. We
have a map $r_{1}\circ f\colon C_{2}\rightarrow U_{\sigma_{1}}$ and a map
$\textnormal{id}_{C_{2}}\colon C_{2}\rightarrow C_{2}$ such that
$f\circ\textnormal{id}_{C_{2}}=f=\pi_{1}\circ(r_{1}\circ f)$. By the universal
property, these two maps define a map $r_{2}\colon C_{2}\rightarrow
U_{\sigma_{2}}$ such that, in particular $\pi_{2}\circ
r_{2}=\textnormal{id}_{C_{2}}$. This is the desired section of $\pi_{2}$.
∎
Given a morphism $f\colon\sigma_{2}\rightarrow\sigma_{1}$ of simple schemas,
the function $f^{*}\colon\Gamma(\sigma_{1})\rightarrow\Gamma(\sigma_{2})$
defined in the above lemma is said to be induced by $f$.
###### Definition 2.3.8.
Let $\pi\colon U\rightarrow{\bf DT}$ be a type specification, and let
$(K_{1},C_{1},\sigma_{1},\tau_{1})$ and $(K_{2},C_{2},\sigma_{2},\tau_{2})$
denote tables. A morphism of tables $\varphi\colon\tau_{1}\rightarrow\tau_{2}$
consists of a pair $(g,f)$, where $g\colon K_{1}\rightarrow K_{2}$ is a
function and $f\colon(C_{2},\sigma_{2})\rightarrow(C_{1},\sigma_{1})$ is a
morphism of simple schema such that the diagram of sets
(5)
commutes, where $f^{*}\colon\Gamma(\sigma_{1})\rightarrow\Gamma(\sigma_{2})$
is the function induced by $f$.
###### Example 2.3.9.
Let us continue with Example 2.3.6, except for a slight renaming of objects:
$C_{1}\colon=C,\sigma_{1}\colon=\sigma,K_{1}\colon=K,$ and
$\tau_{1}\colon=\tau$. Let $C_{2}=\\{\textnormal{`First', `Last'}\\}$ and let
$\sigma_{2}$ send both elements to the data type ${\bf Strings}\in{\bf DT}$;
thus $\Gamma(\sigma_{2})={\bf Strings}\times{\bf Strings}$.
Let $K_{2}=\\{5,6,`bar^{\prime}\\}$ and $\tau_{2}$ be the assignment
$\displaystyle 5$ $\displaystyle\mapsto\textnormal{(Barack; Obama)}$
$\displaystyle 6$ $\displaystyle\mapsto\textnormal{(Michelle; Obama)}$
$\displaystyle`bar^{\prime}$ $\displaystyle\mapsto\textnormal{(George;
Bush)}.$
A morphism of tables $\varphi\colon\tau_{1}\rightarrow\tau_{2}$ should consist
of a map $g\colon K_{1}\rightarrow K_{2}$ and a map
$f^{*}\colon\Gamma(C_{1})\rightarrow\Gamma(C_{2})$. We have an obvious map of
simple schema $f\colon C_{2}\rightarrow C_{1}$, namely
$\textnormal{`First'}\mapsto\textnormal{`First name'}$ and
$\textnormal{`Last'}\mapsto\textnormal{`Last name'}$. Then
$f^{*}\colon\Gamma(\sigma_{1})\rightarrow\Gamma(\sigma_{2})$ is just the
projection ${\bf Strings}\times{\bf Strings}\times{\mathbb{Z}}\rightarrow{\bf
Strings}\times{\bf Strings}$.
Now, to define a morphism of tables
$\varphi\colon\tau_{1}\rightarrow\tau_{2}$, our choice of $g$ must send both
of the records $(\textnormal{Barack; Obama; 1961})$ in $\tau_{1}$ to the
record $(\textnormal{Barack; Obama})$ and send the record
$(\textnormal{Michelle; Obama; 1964})$ to the record $(\textnormal{Michelle;
Obama})$. There is a unique such morphism $\phi$ in this case.
For a variety of reasons, there does not exist a morphism of tables
$\tau_{2}\rightarrow\tau_{1}$.
###### Remark 2.3.10.
The morphism of tables in Example 2.3.9 has a common form. As in the example,
a morphism of tables often is composed of a projection (in the columns)
together with an inclusion (in the rows). The requirement that the square (5)
in Definition 2.3.8 commutes is simply the requirement that morphisms preserve
the integrity of the data.
### 2.4. The category of tables
We have now defined tables and morphisms between tables. Given morphisms
depicted
$\textstyle{K_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{1}}$$\textstyle{\Gamma(\sigma_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{2}}$$\textstyle{\Gamma(\sigma_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{3}}$$\textstyle{\Gamma(\sigma_{3})}$
it is easy to see how composition is defined. It is also easy to understand
the identity morphism on a table $\tau\colon K\rightarrow\Gamma(C)$. Thus we
have a category.
###### Definition 2.4.1.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification. The category
whose objects are tables $K\rightarrow\Gamma(\sigma)$ and whose morphisms are
commutative squares as in Definition 2.3.8 is called the category of tables on
$\pi$ and is denoted ${\bf Tables}^{\pi}$, or simply ${\bf Tables}$, if $\pi$
is understood.
###### Example 2.4.2.
Suppose $\pi\colon U\rightarrow{\bf DT}$ is as in Example 2.2.5. Suppose that
$C=\\{c_{1},c_{2}\\}$ and $C^{\prime}=\\{c_{1}^{\prime}\\}$, and that
$\sigma\colon C\rightarrow{\bf DT}$ and $\sigma^{\prime}\colon
C^{\prime}\rightarrow{\bf DT}$ are the unique maps such that
$\Gamma(\sigma)={\mathbb{Z}}\times{\mathbb{Z}}$ and
$\Gamma(\sigma^{\prime})={\mathbb{Z}}$. Let $K$ and $K^{\prime}$ be any two
sets and $\tau\colon K\rightarrow\Gamma(\sigma)$ and $\tau^{\prime}\colon
K^{\prime}\rightarrow\Gamma(\sigma^{\prime})$ be any two tables.
For a morphism $\tau_{1}\rightarrow\tau_{2}$ in the category of tables, we are
allowed any kind of function between key sets $K\rightarrow K^{\prime}$, but
the only permitted maps
${\mathbb{Z}}\times{\mathbb{Z}}\longrightarrow{\mathbb{Z}}$ are the two
projections, because they are the only maps which are induced by morphisms of
simple schema.
###### Definition 2.4.3.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification and let
$\sigma\colon C\rightarrow{\bf DT}$ denote a simple schema. The category of
tables on $\sigma$ of type $\pi$, denoted ${\bf Tables}^{\pi}_{\sigma}$ is the
category whose objects are tables $\tau\colon K\rightarrow\Gamma(\sigma)$ and
whose morphisms are triangles
| |
---|---|---
$\textstyle{K_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{1}}$$\scriptstyle{g}$$\textstyle{\Gamma(\sigma)}$$\textstyle{K_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{2}}$
denoted by $g\colon\tau_{1}\rightarrow\tau_{2}$.
### 2.5. Relational tables
The most common formulation of databases used today is the relational model,
invented by E.F. Codd (see [Cod70]). It is based on the theory of mathematical
logic, and more specifically on relations. One can find a modern treatment of
the subject in [Dat05]. We define a relation in Definition 2.5.1 as a type of
table, where the map $\tau\colon K\rightarrow\Gamma(\sigma)$ is required to be
an injection.
###### Definition 2.5.1.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification, and let
$\sigma\colon C\rightarrow{\bf DT}$ denote a simple schema on $\pi$. A
relation on $\sigma$ is a table $\tau\colon K\rightarrow\Gamma(\sigma)$ for
which $\tau$ is an injective function.
A morphism of relations is a morphism of tables, for which the source and
target tables are relations. That is, the category of relations, denoted ${\bf
Rel}^{\pi}$ is the full subcategory of ${\bf Tables}^{\pi}$ spanned by the
relations. Similarly, given a simple schema $\sigma$, the category of
relations on $\sigma$ is the full subcategory of ${\bf Tables}^{\pi}_{\sigma}$
spanned by the relations. As usual the superscript $\pi$ can be dropped if it
is understood.
There is a functor ${\bf Rel}\rightarrow{\bf Tables}$ and a functor ${\bf
Rel}_{\sigma}\rightarrow{\bf Tables}_{\sigma}$, both of which are simply
inclusions of full subcategories.
## 3\. Constructions and formal properties of Tables
Our definition for the category of tables (Definition 2.4.1) is sensible
because objects are tables and morphisms are data-preserving maps. In this
section we show that category-theoretic operations on tables correspond to
operations on databases, such as joins and other queries. Fix a type
specification $\pi\colon U\rightarrow{\bf DT}$ for the remainder of the
section. We will drop $\pi$ as a superscript in this section; for example the
category $\mathcal{S}^{\pi}$ of simple schema on $\pi$ will be denoted simply
by $\mathcal{S}$.
We sometimes refer to the underlying keys or underlying simple schema of a
table, so we record these trivial constructions in a remark.
###### Remark 3.1.1.
There is a forgetful functor ${\bf Tables}\rightarrow{\bf Sets}$ given by
sending a table $\tau\colon K\rightarrow\Gamma(\sigma)$ to the key set $K$ and
a morphism of tables to the underlying map of keys. There is another forgetful
functor ${\bf Tables}\rightarrow\mathcal{S}^{\textnormal{op}}$ which sends the
table $\tau$ to its simple schema $\sigma$ and a morphism $\varphi=(g,f)$ of
tables to the underlying morphism of simple schema $f$.
###### Lemma 3.1.2.
There exists a final object and an initial object in ${\bf Tables}$.
###### Proof.
One checks immediately that if we take $K$ to be a terminal object in ${\bf
Sets}$ (i.e. any set $K$ with cardinality 1) and $\sigma$ to be the inital
object $\emptyset\rightarrow{\bf DT}$ in $\mathcal{S}$, then there is exactly
one table with these as its underlying keys and simple schema, and this table
is the terminal object in ${\bf Tables}$.
One also checks immediately that if we take $K=\emptyset$ to be the initial
object in ${\bf Sets}$ and $\sigma=\textnormal{id}_{{\bf DT}}\colon{\bf
DT}\rightarrow{\bf DT}$ to be the final object in $\mathcal{S}$, then there is
exactly one table with these as its underlying keys and simple schema, and
this table is the initial object in ${\bf Tables}$.
∎
Certain colimits exist in ${\bf Tables}$; namely colimits of diagrams that are
constant in the underlying simple schema.
###### Construction 3.1.3.
Let $\tau_{1}\colon K_{1}\rightarrow\Gamma(\sigma)$ and $\tau_{2}\colon
K_{2}\rightarrow\Gamma(\sigma)$ be two tables with the same simple schema. By
taking the disjoint union of $K_{1}$ and $K_{2}$ we get a new table
$\tau\colon K_{1}\amalg K_{2}\rightarrow\Gamma(\sigma)$. This query is called
UNION ALL in SQL.
We can also take the (non-disjoint) union of these two tables, if we know how
they overlap. That is, if there is some set $K$ with maps $g_{1}\colon
K\rightarrow K_{1}$ and $g_{2}\colon K\rightarrow K_{2}$ in such a way that
$\tau_{1}\circ g_{1}=\tau_{2}\circ g_{2}$, then we can obtain a new table
$\tau\colon K_{1}\amalg_{K}K_{2}\rightarrow\Gamma(\sigma)$. This query is
called UNION in SQL.
We will see that limits in the category of tables correspond to generalized
joins.
###### Proposition 3.1.4.
All finite limits exist in ${\bf Tables}$.
###### Proof.
It suffices (see, for example, [MLM94, p. 30]) to show that ${\bf Tables}$ has
a terminal object and is closed under taking fiber products; the first of
these facts was shown in Lemma 3.1.2. For the second, suppose we have a
diagram
(12)
in ${\bf Tables}$, where $\sigma\colon C\rightarrow{\bf DT}$ and
$\sigma_{i}\colon C_{i}\rightarrow{\bf DT}$ for $i=1,2$ are simple schemas. As
indicated, the maps $\Gamma(\sigma_{i})\rightarrow\Gamma(\sigma)$ are induced
by morphisms of simple schema $f_{i}\colon\sigma\rightarrow\sigma_{i}$, for
$i=1,2$.
Consider the simple schema
$(\sigma_{1}\amalg_{\sigma}\sigma_{2})\colon
C_{1}\amalg_{C}C_{2}\longrightarrow{\bf DT}$
induced by taking the colimit of the column sets. We would like to show that
the natural function
(13)
$\displaystyle\Gamma(\sigma_{1}\amalg_{\sigma}\sigma_{2})\longrightarrow\Gamma(\sigma_{1})\times_{\Gamma(\sigma)}\Gamma(\sigma_{2})$
is a bijection.
Let us first calculate the set $\Gamma(\sigma_{1}\amalg_{\sigma}\sigma_{2})$.
It is the set of all sections $r$ of the map $\pi^{\prime}$ in the diagram
$\textstyle{(\sigma_{1}\amalg_{\sigma}\sigma_{2})^{-1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lrcorner}$$\scriptstyle{\pi^{\prime}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C_{1}\amalg_{C}C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{\sigma_{1}\amalg_{\sigma}\sigma_{2}}$$\textstyle{{\bf
DT}.}$
To give such a section is to give, for each $c_{1}\in C_{1}$ an element of
$\pi^{-1}(\sigma_{1}(c_{1}))$, and for each $c_{2}\in C_{2}$ an element of
$\pi^{-1}(\sigma_{2}(c_{2}))$, in such a way that for all $c\in C$, the
induced elements in $\pi^{-1}(\sigma_{i}(f_{i}(c)))$ are the same for $i=1,2$.
This is precisely the data needed for a unique element of the set
$\Gamma(\sigma_{1})\times_{\Gamma(\sigma)}\Gamma(\sigma_{2})$; this proves the
claim that the map in (13) is a bijection.
It now follows that the fiber product of Diagram (12) is the table
$\tau_{1}\times_{\tau}\tau_{2}\colon
K_{1}\times_{K}K_{2}\longrightarrow\Gamma(\sigma_{1}\amalg_{\sigma}\sigma_{2})$
obtained by taking the fiber product of sources and targets in (12), and the
induced map between them.
∎
Proposition 3.1.4 gives the formula for the join of two tables over a third.
As one sees from the construction, the columns of the join are the union of
the columns of the given tables, and the key set is the fiber product of the
key sets of the given tables.
###### Lemma 3.1.5.
Let $\sigma\colon C\rightarrow{\bf DT}$ denote a simple schema. The category
${\bf Tables}_{\sigma}$ of tables on $\sigma$ is closed under small limits and
colimits.
###### Proof.
The category of sets is closed under small limits and colimits. To take the
limit or colimit of a diagram $X\colon I\rightarrow{\bf Tables}_{\sigma}$,
simply take the limit or colimit (respectively) of the underlying diagram of
key sets – see Definition 3.1.1. This set comes with a natural map to
$\Gamma(\sigma)$, and one shows easily that it is the limit or colimit
(respectively) of $X$.
∎
###### Example 3.1.6.
Let $\sigma\colon C\rightarrow{\bf DT}$ denote a simple schema. The initial
and final objects in ${\bf Tables}_{\sigma}$ are
$\emptyset\rightarrow\Gamma(\sigma)$ and
$\textnormal{id}_{\Gamma(\sigma)}\colon\Gamma(\sigma)\rightarrow\Gamma(\sigma)$,
respectively.
###### Construction 3.1.7.
Let $\tau\colon K\rightarrow\Gamma(\sigma)$ be a table with simple schema
$\sigma\colon C\rightarrow{\bf DT}$, and let $C^{\prime}\subset C$ be a subset
of its column set. There is an induced table
$\tau|_{C^{\prime}}\colon K\rightarrow\Gamma(\sigma|_{C^{\prime}}).$
In SQL this construction is called the projection of $\tau$ onto the subset
$C^{\prime}\subset C$ of columns.
Using the projection query, one can realize a SELECT query as a limit of
databases.
###### Construction 3.1.8.
Let us construct the SELECT query. One begins with a table $\tau\colon
K\rightarrow\Gamma(\sigma)$ with simple schema $\sigma\colon C\rightarrow{\bf
DT}$, from which to select. Let $f\colon C^{\prime}\subset C$ be a subset of
its columns, and let $\sigma^{\prime}=\sigma|_{C^{\prime}}\colon
C^{\prime}\rightarrow{\bf DT}$ be the restricted simple schema. One may select
from $\tau$ all records whose restriction to $C^{\prime}$ is a member of some
list. We encode this list as a table $\tau^{\prime}\colon
K^{\prime}\rightarrow\Gamma(\sigma^{\prime})$ on $\sigma^{\prime}$.
In order to select from $\tau$ all records whose restriction to $C^{\prime}$
is in the table $\tau^{\prime}$, take the limit of the diagram
$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{f^{*}\circ\tau}$$\textstyle{\Gamma(\sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{\Gamma(\sigma^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id$\textstyle{\Gamma(\sigma^{\prime})}$$\textstyle{K^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{\prime}}$$\scriptstyle{\tau^{\prime}}$$\textstyle{\Gamma(\sigma^{\prime}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id
This limit is the desired SELECT query.
###### Example 3.1.9.
Let $\tau\colon K\rightarrow\Gamma(\sigma)$ be the table from Example 2.3.6.
To select all instances for which the first name is Barack, let
$C^{\prime}=\\{\textnormal{`First Name'}\\}$. Let $\tau^{\prime}$ denote the
one-row table
K’ | ‘First Name’
---|---
k’ | Barack
Both $\tau$ and $\tau^{\prime}$ have a canonical map to the terminal table on
$C^{\prime}$, the table with one column (‘First Name’) and with a row for each
element of ${\bf Strings}$. Of course, this terminal table is too big to write
down, but we do not need it. The fiber product is easily computed to be the
table
K | ‘First Name’ | ‘Last Name’ | ‘BYear’
---|---|---|---
1 | Barack | Obama | 1961
‘foo’ | Barack | Obama | 1961
We conclude this section by a quick remark on the category-theoretic
properties of the relational tables.
###### Remark 3.1.10.
Relations behave much like ordinary tables. Limits exist in ${\bf Rel}$ and
${\bf Rel}_{\sigma}$. The functor ${\bf Rel}\rightarrow{\bf Tables}$ preserves
limits, and the functor ${\bf Rel}_{\sigma}\rightarrow{\bf Tables}_{\sigma}$
preserves limits but does not preserve colimits.
We take the viewpoint that the “correct” way to take a colimit of a diagram
$X\colon I\rightarrow{\bf Rel}_{\sigma}$ is to pass to the diagram
$I\rightarrow{\bf Tables}_{\sigma}$ and take its colimit instead. This claim,
in particular, says that sometimes UNION ALL is more appropriate than UNION
is. Since UNION ALL is not legal in the strict relational database theory (or
it would be the same as UNION), our viewpoint could be seen as controversial
to purists of the relational model.
## 4\. Schemas and databases
A relational database is a set of relations, together with a system of keys
and foreign keys which link the relations together. The definition of
relations themselves is, of course, quite mathematically precise. However, the
precise way in which these relations are allowed to be linked together is
rarely written down as a mathematical structure in its own right, either in
research papers or textbooks (we could not find it in [Dat05] or [EN07], for
example). For example, ER diagrams are exemplified or even defined, but not as
a mathematical object (like relations are). There are exceptions, such as
[RW92, 2.1], but as far as we know, these definitions are not actually the
ones used, either by practitioners or by theorists.
In this section we will define simplicial databases in a rigorous way (see
Definition 4.3.3). Although examples will be plentiful, they will never stand
in for precise definitions. We will also define morphisms of databases, thus
making explicit the idea of “data-preserving maps.” Providing a precise
definition of the category of databases may be useful to database theorists,
as well as to people interested in studying mathematical informatics.
### 4.1. Schemas
Roughly, a simplicial set is a picture that can be drawn with vertices, edges,
solid triangles, solid tetrahedra, and solid “higher-dimensional tetrahedra.”
For any integer $n\geq 0$, an $n$-dimensional solid tetrahedron, or
$n$-simplex, is the “diagonal triangle” shape in ${\mathbb{R}}^{n+1}$ given by
the algebraic equation $x_{1}+x_{2}+\cdots+x_{n+1}=1$ and the inequalities
$x_{i}\geq 0$ for $1\leq i\leq n+1$. To draw with these shapes is to connect
various tetrahedra together along their faces (or subfaces). For example, one
could connect four triangles together along various faces to obtain an empty
tetrahedron, the boundary of the 3-simplex.
Simplicial sets are a fundamental tool in algebraic topology, and are
important in many other fields within mathematics, such as combinatorial
commutative algebra. See [Fri08] or [GJ99] for details.
A database is a system of tables which are connected together via foreign
keys. This information is part of the schema for the database. In our
formulation, we keep track of this information using (something akin to)
simplicial sets as our schema. Tables are connected together when the
corresponding simplices are connected.
We use a slight variant of simplicial sets, which we will define in Definition
4.1.2. Namely, since columns can only take entries in a given data type, we
must keep track of this information. For this reason, the simplicial sets we
use as schema have labeled vertices, where each label is an element of ${\bf
DT}$. We do not define schemas exactly this way, however, because a more
generalizable way to phrase it may be useful for future generalizations.
###### Remark 4.1.1.
As mentioned in Remark 2.2.4, some prefer the columns of each table in a
database to be unordered, whereas we have chosen to consider them as an
ordered set. Simply using symmetric simplicial sets, a variant of simplicial
sets in which vertices are unordered, will solve any such issue. See [Gra01]
for details on symmetric simplicial sets.
###### Definition 4.1.2.
Let ${\bf\Delta}$ denote the category of finite ordered sets, let $\pi\colon
U\rightarrow{\bf DT}$ be a type specification, and let
$\mathcal{S}\cong({\bf\Delta}\downarrow{\bf DT})$
denote the category of simple schema on $\pi$ (see Definition 2.2.3 and Remark
2.2.7). We define the category of schema on $\pi$, denoted ${\bf Sch}^{\pi}$
to be the category whose objects are functors
$X\colon\mathcal{S}^{\textnormal{op}}\rightarrow{\bf Sets}$ and whose
morphisms are natural transformations of functors.
Let $X\in{\bf Sch}^{\pi}$ denote a schema. Given a simple schema $\sigma\colon
C\rightarrow{\bf DT}$, the $\sigma$-simplices of $X$ are the elements of the
set $X(\sigma)$, and we write $X_{\sigma}$ to denote $X(\sigma)$.
###### Remark 4.1.3.
Given a category $\mathcal{C}$, the category whose objects are functors
$\mathcal{C}^{\textnormal{op}}\rightarrow{\bf Sets}$ and whose morphisms are
natural transformations of functors is called the category of presheaves on
$\mathcal{C}$ and denoted ${\bf Pre}(\mathcal{C})$. It is a common
mathematical construction which “formally adds all colimits to $\mathcal{C}$.”
That is, ${\bf Pre}(\mathcal{C})$ is closed under taking colimits, and for any
functor $\mathcal{C}\rightarrow\mathcal{D}$ to a category $\mathcal{D}$ which
is closed under taking colimits, there is a unique colimit-preserving functor
${\bf Pre}(\mathcal{C})\rightarrow\mathcal{D}$ over $\mathcal{C}$. See, for
example, [MLM94, I.5.4].
Thus, we have ${\bf Sch}^{\pi}={\bf Pre}(\mathcal{S}^{\pi})$. Since
$\mathcal{S}^{\pi}$ signifies the category of ways to set up columns of a
tables, ${\bf Pre}(\mathcal{S}^{\pi})$ is the category of ways to glue such
things together.
###### Remark 4.1.4.
The category of (augmented) simplicial sets is the category ${\bf
Pre}({\bf\Delta})$. The only difference between it and ${\bf
Pre}(\mathcal{S}^{\pi})\cong{\bf Pre}({\bf\Delta}\downarrow{\bf DT})$ is that
each simplex in ${\bf Sch}^{\pi}$ has labeled vertices, whereas simplices in
${\bf Pre}({\bf\Delta})$ do not. In the introduction to this section we
described simplicial sets in terms of tetrahedra. After making the necessary
modifications, we see that a schema is constructed by gluing together labeled
tetrahedra along their faces, where we only allow these tetrahedra to be glued
if their labels match.
If $X$ is a schema, we sometimes refer to the simplices of its underlying
simplicial set as simplices of $X$. Thus, the $n$-simplices of $X$ is the
union of all $\sigma$-simplices of $X$, where $\sigma\colon C\rightarrow{\bf
DT}$ is a simple schema with cardinality $\textnormal{card}(C)=n+1$. That is,
we write
$X_{n}=\coprod_{\\{\sigma\colon C\rightarrow{\bf
DT}|\textnormal{card}(C)=n+1\\}}X_{\sigma}.$
There is a classifying map $s\colon X_{0}=\amalg_{a\in{\bf
DT}}(X_{a})\rightarrow{\bf DT}$ which sends all of $X_{a}$ to $a$, for each
$a\in{\bf DT}$.
One of the best features of the schema we are presenting here is their
geometric nature, as described in the first paragraph of this section.
Unfortunately, Definition 4.1.2 does not make the geometry explicit at all.
Hopefully the next few examples will help make it more clear.
###### Example 4.1.5.
Let $\sigma\colon C\rightarrow{\bf DT}$ denote a simple schema. It naturally
defines a schema $X=\Delta^{\sigma}$ as the functor which sends a simple
schema $\sigma^{\prime}\colon C^{\prime}\rightarrow{\bf DT}$ to the set
$X_{\sigma^{\prime}}=\textnormal{Hom}_{\mathcal{S}}(\sigma^{\prime},\sigma)$.
If $C$ has $n+1$ elements, one visualizes $\Delta^{\sigma}$ as an
$n$-dimensional tetrahedron whose vertices are labeled by elements in the
image of $\sigma$.
This is not just a heuristic: there is a geometric realization functor
$Re:{\bf Sch}\rightarrow{\bf Top}$ which realizes every schema as a
topological space in a natural way, and behaves as we have described for
simplices $\Delta^{\sigma}$.
As an example, suppose $C$ has two elements and their images under $\sigma$
are $a,b\in{\bf DT}$. We imagine $\Delta^{\sigma}$ as a line segment, whose
vertices are labeled $a$ and $b$. If $C^{\prime}$ has three elements and
$\sigma^{\prime}$ sends two of them to $a$ and one of them to $b$, we imagine
$\Delta^{\sigma^{\prime}}$ as a filled-in triangle, whose vertices are labeled
$a,a,$ and $b$. The figures we have imagined are the images of $\sigma$ and
$\sigma^{\prime}$ under $Re$.
###### Definition 4.1.6.
Let $\sigma\in\mathcal{S}$ denote a simple schema. The schema
$\Delta^{\sigma}\in{\bf Sch}$ defined in Example 4.1.5 is called the
$\sigma$-simplex and, as a functor
$\mathcal{S}^{\textnormal{op}}\rightarrow{\bf Sets}$, is said to be
represented by $\sigma$.
###### Example 4.1.7.
We have mentioned that every object in ${\bf Sch}^{\pi}$ can be obtained by
gluing together simplices. This is proven in [Bor94a, 2.15.6]. Let us explain
how we would construct the union $X$ of two edges along a common vertex.
Suppose that the common vertex is labeled $b$ and the other vertices are
labeled $a$ and $c$. The schema $X$ is obtained as the colimit of the diagram
$\Delta^{(a,b)}\leftarrow\Delta^{(b)}\rightarrow\Delta^{(b,c)}$
taken in ${\bf Sch}^{\pi}$.
We will now write down this schema explicitly as a presheaf on
$\mathcal{S}^{\pi}$, i.e. as a functor $X\colon({\bf\Delta}\downarrow{\bf
DT})^{\textnormal{op}}\rightarrow{\bf Sets}$. Given $\sigma\colon
C\rightarrow{\bf DT}$, we let $X_{\sigma}$ be a single element if the image of
$\sigma$ is contained in $\\{a,b\\}$ or contained in $\\{b,c\\}$. Otherwise we
take $X_{\sigma}$ to be the empty set.
###### Example 4.1.8.
A basic example of a schema is that of a set of labeled vertices with no edges
or higher simplices connecting them. This is obtained as a coproduct of
$0$-simplices (see Remark 4.1.3), and it is called a discrete schema.
### 4.2. Sheaves on a schema
###### Definition 4.2.1.
Let $X\in{\bf Sch}^{\pi}$ denote a schema. A subschema of $X$ consists of a
schema $X^{\prime}\in{\bf Sch}^{\pi}$ such that for every
$\sigma\in\mathcal{S}^{\pi}$ we have $X^{\prime}_{\sigma}\subset X_{\sigma}$.
The subschemas of $X$ form a category ${\bf Sub}(X)$, in which there is a
morphism $X^{\prime\prime}\rightarrow X^{\prime}$ in ${\bf Sub}(X)$ if and
only if $X^{\prime\prime}$ is a subschema of $X^{\prime}$.
We will soon be discussing colimits in the category ${\bf Sub}(X)$. One should
note that ${\bf Sub}(X)$ is particularly nice, in that the colimit of any
diagram $D\colon I\rightarrow{\bf Sub}(X)$ is the smallest subschema
$X^{\prime}\subset X$ which contains $D(i)$ for all $i\in I$. In the language
of lattices or locales, one writes
$\mathop{\textnormal{colim}}(D)=\bigvee_{i\in I}D(i)$. See [Bor94b, 1.3].
###### Definition 4.2.2.
We define a sheaf on $X$ to be a functor $\mathcal{K}\colon{\bf
Sub}(X)^{\textnormal{op}}\rightarrow{\bf Sets}$ such that, for every diagram
$D\colon I\rightarrow{\bf Sub}(X)$, the natural map
$\mathcal{K}(\mathop{\textnormal{colim}}(D))\longrightarrow\lim(\mathcal{K}(D))$
is an isomorphism. That is, $\mathcal{K}$ must send colimits of subschema to
corresponding limits of sets.
A morphism of sheaves on $X$ is a natural transformation of functors ${\bf
Sub}(X)^{\textnormal{op}}\rightarrow{\bf Sets}$. Let ${\bf Shv}(X)$ denote the
category of sheaves on $X$.
###### Remark 4.2.3.
Category theory experts will recognize ${\bf Shv}(X)$ as the category of
sheaves on a certain Grothendieck site (the locale of subobjects of $X$). It
is well known that ${\bf Shv}(X)$ is therefore closed under small limits and
colimits. Moreover, there is an adjunction
$\textstyle{{\bf
Pre}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Sh}$$\textstyle{{\bf
Shv}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
for which the right adjoint is the forgetful functor and the left adjoint is
called sheafification. Roughly, one sheafifies a presheaf on a schema by
replacing its value on each union of simplices by the fiber product of its
values on those simplices. See [MLM94] for details.
###### Example 4.2.4.
For any schema $X$, there is an object $\emptyset\in{\bf Sub}(X)$, which is
the colimit of the empty diagram on ${\bf Sub}(X)$. Hence if $\mathcal{K}$ is
to be a sheaf on $X$, one must have $\mathcal{K}(\emptyset)\cong\\{*\\}$.
If $X$ is a discrete schema (see Example 4.1.8), then $X$ is the coproduct its
$0$-simplices. Thus, if $\mathcal{K}$ is to be a sheaf on $X$, we must have
$\mathcal{K}(X)=\prod_{x\in X_{0}}\mathcal{K}(x).$
###### Example 4.2.5.
Suppose that $X\in{\bf Sch}^{\pi}$ is the schema $\Delta^{(`{\bf
Str}\textnormal{'},`{\mathbb{Z}}\textnormal{'})}$, which looks like this:
$\textstyle{~{}^{`{\bf
Str}\textnormal{'}}\\!\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet^{`{\mathbb{Z}}\textnormal{'}}.}$
The category ${\bf Sub}(X)$ is a partially ordered set with five objects:
$\emptyset$, $\bullet^{`{\bf
Str}\textnormal{'}}$,$\bullet^{`{\mathbb{Z}}\textnormal{'}}$, $(\bullet^{`{\bf
Str}\textnormal{'}},\bullet^{`{\mathbb{Z}}\textnormal{'}})$, and $X$ itself;
${\bf Sub}(X)$ has inclusions as morphisms.
A sheaf $\mathcal{K}\in{\bf Shv}(X)$ assigns a set to each of these five
objects, and functions to each inclusion. However, by Example 4.2.4, it must
assign to $\emptyset$ the terminal set, $\mathcal{K}(\emptyset)=\\{*\\}$, and
it must assign to $(\bullet^{`{\bf
Str}\textnormal{'}},\bullet^{`{\mathbb{Z}}\textnormal{'}})$ the product
$\mathcal{K}(\bullet^{`{\bf
Str}\textnormal{'}})\times\mathcal{K}(\bullet^{`{\mathbb{Z}}\textnormal{'}})$.
Thus, to specify a sheaf, we need only specify two values, and one morphism,
namely $\mathcal{K}(X)\rightarrow\mathcal{K}(\bullet^{`{\bf
Str}\textnormal{'}})\times\mathcal{K}(\bullet^{`{\mathbb{Z}}\textnormal{'}})$.
For example we may choose on objects the assignments
$\mathcal{K}(X)=\\{4,cc,10\\}$, $\mathcal{K}(\bullet^{`{\bf
Str}\textnormal{'}})=\\{1,2\\}$, and
$\mathcal{K}(\bullet^{`{\mathbb{Z}}\textnormal{'}})=\\{x,y,z\\}$; this implies
$\mathcal{K}((\bullet^{`{\bf
Str}\textnormal{'}},\bullet^{`{\mathbb{Z}}\textnormal{'}}))$ is isomorphic to
$\\{1x,1y,1z,2x,2y,2z\\}$. Any function from $\\{4,cc,10\\}$ to this six
element set, say $4\mapsto 1x,cc\mapsto 2z,10\mapsto 2z$, defines the
restriction maps in our sheaf $\mathcal{K}$. These restriction maps can be
thought of as “foreign keys.”
###### Definition 4.2.6.
Given a schema $X\in{\bf Sch}^{\pi}$, we have been working with the category
${\bf Sub}(X)$ of subschemas of $X$. There is a related category, called the
category of nonempty non-degenerate simple schemas over $X$ and denoted ${\bf
ND}(X)$, whose objects are monomorphisms $\Delta^{\sigma}\hookrightarrow X$ in
${\bf Sch}^{\pi}$, where $\sigma\colon C\rightarrow{\bf DT}$ is a schema with
$C\neq\emptyset$ (see Example 4.1.5), and whose morphisms are commutative
triangles.
Every simplex in a schema has a unique underlying non-degenerate simplex (of
which it is the degeneracy), so one can define a functor ${\bf ND}\colon{\bf
Sch}^{\pi}\rightarrow{\bf Cat}$.
Since every injection $\Delta^{\sigma}\hookrightarrow X$ is in particular a
subschema, there is an obvious functor
${\bf ND}(X)\rightarrow{\bf Sub}(X).$
This induces an adjunction $\textstyle{{\bf Pre}({\bf
ND}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
Pre}({\bf Sub}(X)).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ No
nontrivial unions exist in ${\bf ND}(X)$, so this adjunction becomes
$\textstyle{{\bf Pre}({\bf
ND}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L}$$\textstyle{{\bf
Shv}({\bf
Sub}(X)),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$
where ${\bf Pre}({\bf ND}(X))$ is the category of presheaves ${\bf
ND}(X)^{\textnormal{op}}\rightarrow{\bf Sets}$. See [Joh02, C.1.4.3] for more
details on this type of construction.
###### Proposition 4.2.7.
Let $X\in{\bf Sch}^{\pi}$ be a schema, and let ${\bf ND}(X)$ denote the
category of non-degenerate nonempty simple schema over $X$. The adjunction
$\textstyle{{\bf Pre}({\bf
ND}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L}$$\textstyle{{\bf
Shv}({\bf
Sub}(X)),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$
is an equivalence of categories.
###### Proof.
It is an easy exercise to show that the composition $L\circ R$ is equal to the
identity on ${\bf Pre}({\bf ND}(X))$ and that $K\circ L$ is canonically
isomorphic to the identity on ${\bf Shv}({\bf Sub}(X))$.
∎
Proposition 4.2.7 says that one does not have to worry about sheaves: the
category ${\bf Shv}(X)$ is equivalent to a category of functors (without
“sheaf” requirements).
###### Lemma 4.2.8.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification and let
$f\colon X\rightarrow Y$ denote a morphism of schema on $\pi$. There is an
adjunction
$\textstyle{{\bf Shv}({\bf
Sub}(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{{\bf
Shv}({\bf
Sub}(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$
defined as follows for sheaves $\mathcal{K}_{X}\in{\bf Shv}({\bf Sub}(X))$ and
$\mathcal{K}_{Y}\in{\bf Shv}({\bf Sub}(Y))$. For any $U\in{\bf Sub}(X)$ we
take
$f^{*}\mathcal{K}_{Y}(U)\colon=\mathcal{K}_{Y}(f(U)),$
where $f(U)\in{\bf Sub}(Y)$ is the image of $U$ in $Y$. For any $V\in{\bf
Sub}(Y)$ we take
$f_{*}\mathcal{K}_{X}(V)\colon=\mathcal{K}_{X}(f^{-1}(V)),$
where $f^{-1}(V)$ is the preimage of $V$ in $X$.
###### Proof.
Colimits of presheaves are computed objectwise, and it follows from
Proposition 4.2.7 that the functor $f^{*}$, defined above, preserves colimits.
Hence, it suffices to show that for any representable sheaf
$rY^{\prime}=\textnormal{Hom}_{{\bf Sub}(Y)}(-,Y^{\prime})\in{\bf Shv}({\bf
Sub}(Y))$ and sheaf $T\in{\bf Shv}({\bf Sub}(X))$, one has an isomorphism
$\textnormal{Hom}(f^{*}(rY^{\prime}),T)\cong^{?}\textnormal{Hom}(rY^{\prime},f_{*}T).$
To begin, note that for any $U\in{\bf Sub}(X)$ one has a chain of natural
isomorphisms
$\displaystyle f^{*}(rY^{\prime})(U)\colon=(rY^{\prime})(f(U))$
$\displaystyle\cong\textnormal{Hom}_{{\bf Sub}(Y)}(f(U),Y^{\prime})$
$\displaystyle\cong\textnormal{Hom}_{{\bf Sub}(X)}(U,f^{-1}(Y^{\prime}))\cong
r(f^{-1}(Y^{\prime}))(U).$
That is, $f^{*}(rY^{\prime})\cong r(f^{-1}(Y^{\prime})).$ By another chain of
natural isomorphisms, we have
$\displaystyle\textnormal{Hom}(f^{*}(rY^{\prime}),T)$
$\displaystyle\cong\textnormal{Hom}(r(f^{-1}(Y^{\prime})),T)$
$\displaystyle\cong T(f^{-1}(Y^{\prime}))$
$\displaystyle=:f_{*}T(Y^{\prime})=\textnormal{Hom}(rY^{\prime},f_{*}T).$
This proves the lemma.
∎
### 4.3. Simplicial databases
We think of a schema as a way of organizing the data in a database. Before we
say what a database is, let us give one more example of a schema. In some
sense it will be the fundamental example of a schema; however, it should not
really be thought of as a way to organize the data, but as the meaning of the
data itself.
###### Example 4.3.1.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification, and let
$\mathcal{S}=\mathcal{S}^{\pi}$ denote the category of simple schema on $\pi$.
Let $\Gamma^{\pi}\colon\mathcal{S}^{\textnormal{op}}\rightarrow{\bf Sets}$
denote the functor which assigns to a schema $\sigma\colon C\rightarrow{\bf
DT}$ the set $\Gamma^{\pi}(\sigma)$ of records on $\sigma$ (see Definition
2.3.1).
By Lemma 2.3.7, a map $\sigma\rightarrow\sigma^{\prime}$ induces a function
$\Gamma^{\pi}(\sigma^{\prime})\rightarrow\Gamma^{\pi}(\sigma)$, so
$\Gamma^{\pi}$ is indeed a contravariant functor. By definition we can
consider $\Gamma^{\pi}$ as a schema on $\pi$ and write $\Gamma^{\pi}\in{\bf
Sch}^{\pi}$.
We call $\Gamma^{\pi}$ the universal record on $\pi$, for reasons which will
be clear soon. If the type specification $\pi\colon U\rightarrow{\bf DT}$ is
obvious from context, we may denote $\Gamma^{\pi}$ simply by $\Gamma$.
###### Definition 4.3.2.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification, let
$\Gamma^{\pi}$ denote the universal record on $\pi$, and let $X\in{\bf
Sch}^{\pi}$ denote a schema on $\pi$. The universal sheaf on $X$ of type $\pi$
is the sheaf $\mathcal{U}^{\pi}$ whose value on a subschema $X^{\prime}\subset
X$ is the set
$\mathcal{U}^{\pi}(X^{\prime})=\textnormal{Hom}_{{\bf
Sch}^{\pi}}(X^{\prime},\Gamma^{\pi}).$
Each element of $\mathcal{U}^{\pi}(X^{\prime})$ is called a record on
$X^{\prime}$ of type $\pi$. If $\pi$ is clear from context, we may write
$\mathcal{U}$ to denote $\mathcal{U}^{\pi}$.
Now let $X,Y\in{\bf Sch}^{\pi}$ be schema and let $\mathcal{U}_{X}$ and
$\mathcal{U}_{Y}$ denote the universal sheaf of type $\pi$ on $X$ and $Y$,
respectively. A map of schema $f\colon Y\rightarrow X$ induces a morphism
$\mathcal{U}_{f}\colon f^{*}\mathcal{U}_{X}\rightarrow\mathcal{U}_{Y}$ as
follows. Let $Y^{\prime}\subset Y$ denote an object in ${\bf Sub}(Y)$; then
composing with $f$ induces a natural map
$f^{*}\mathcal{U}_{X}(Y^{\prime})=\textnormal{Hom}_{{\bf
Sch}^{\pi}}(f(Y^{\prime}),\Gamma^{\pi})\longrightarrow\textnormal{Hom}_{{\bf
Sch}^{\pi}}(Y^{\prime},\Gamma^{\pi})=\mathcal{U}_{Y}(Y^{\prime}),$
which we denote $\mathcal{U}_{f}$; it is similarly defined on morphisms.
###### Definition 4.3.3.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification. A simplicial
database (or simply database) of type $\pi$ is a triple $(X,\mathcal{K},\tau)$
where $X\in{\bf Sch}^{\pi}$ is a schema of type $\pi$, $\mathcal{K}\in{\bf
Shv}(X)$ is a sheaf of sets on ${\bf Sub}(X)$, and
$\tau\colon\mathcal{K}\rightarrow\mathcal{U}_{X}$ is a morphism of sheaves on
$X$ (see Definition 4.3.2). We refer to $X$ as the schema, $\mathcal{K}$ as
the sheaf of keys, and $\tau$ as the data of the database
$(X,\mathcal{K},\tau)$.
###### Remark 4.3.4.
Given a set of ways to measure objects, it often happens that we have several
objects with the same measurements. For example, we may have three green
apples, or two 1999 Toyota Corollas. In relational databases, if two objects
have the same attributes, then they are taken to be the same instance. To keep
them distinct, one introduces a unique identifier, an artificial key, which
becomes part of the data. This causes problems with database integration,
because the arbitrarily-chosen artificial keys in one database will generally
not match with those in another.
In our definition, the keys for the data are kept separate, as the sheaf of
sets $\mathcal{K}$. Different names for the keys in no way affect the data
itself and therefore do not interfere with database integration. We say more
about this in Section 5.3.
###### Example 4.3.5.
In Example 4.2.5, we wrote down a sheaf $\mathcal{K}\in{\bf Shv}(X)$ on the
schema
$X=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
11.36163pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-11.36163pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{~{}^{`{\bf
Str}\textnormal{'}}\\!\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
35.36163pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{\bullet^{`{\mathbb{Z}}\textnormal{'}},}$}}}}}}}\ignorespaces}}}}\ignorespaces$
and we will continue to use it in this example. To specify a database on $X$
of type $\pi$, we must give a morphism
$\tau\colon\mathcal{K}\rightarrow\mathcal{U}^{\pi}$ of sheaves on $X$.
We defined the universal sheaf $\mathcal{U}_{X}$ of type $\pi$ on $X$ in
Definition 4.3.2. We have
$\displaystyle\mathcal{U}_{X}(X)=\mathcal{U}_{X}((\bullet^{`{\bf
Str}\textnormal{'}},\bullet^{`{\mathbb{Z}}\textnormal{'}}))$
$\displaystyle={\bf Str}\times{\mathbb{Z}}$
$\displaystyle\mathcal{U}_{X}(\bullet^{`{\bf Str}\textnormal{'}})$
$\displaystyle={\bf Str}$
$\displaystyle\mathcal{U}_{X}(\bullet^{`{\mathbb{Z}}\textnormal{'}})$
$\displaystyle={\mathbb{Z}}$ $\displaystyle\mathcal{U}_{X}(\emptyset)$
$\displaystyle=\\{*\\}.$
To define a map $\tau\colon\mathcal{K}\rightarrow\mathcal{U}_{X}$, we must
give maps
$\begin{array}[]{lll}\tau(\bullet^{`{\bf
Str}\textnormal{'}})\colon\mathcal{K}(\bullet^{`{\bf
Str}\textnormal{'}})\rightarrow\mathcal{U}_{X}(\bullet^{`{\bf
Str}\textnormal{'}}),&&\tau(\bullet^{`{\mathbb{Z}}\textnormal{'}})\colon\mathcal{K}(\bullet^{`{\mathbb{Z}}\textnormal{'}})\rightarrow\mathcal{U}_{X}(\bullet^{`{\mathbb{Z}}\textnormal{'}})\end{array}$
and
$\tau(X)\colon\mathcal{K}(X)\rightarrow\mathcal{U}_{X}(X)$
that compose correctly with the restriction maps. We arbitrarily assign
$\begin{array}[]{lllllll}\tau(1)&=&\textnormal{Barack}&&\tau(x)&=&1961\\\
\tau(2)&=&\textnormal{Michelle}&&\tau(y)&=&1946\\\
&&&&\tau(z)&=&1964.\end{array}$
Now $\mathcal{K}(X)=\\{4,cc,10\\}$, and the restriction map sends $4\mapsto
1x$, $cc\mapsto 2z$, and $10\mapsto 2z$. This forces
$\tau(4)=\textnormal{(Barack; 1961)}$ and
$\tau(cc)=\tau(10)=\textnormal{(Michelle; 1964)}$. The other values and
restriction maps for $\mathcal{K}$ are now also forced.
###### Example 4.3.6.
In Example 4.3.5, we followed the definitions very closely, perhaps to the
detriment of the big ideas. In this example, we write down how the sheaf
“looks” as a collection of tables.
Let us first change the schema $X$ very slightly, by instead using the schema
$\sigma\colon\\{\textnormal{First, BYear}\\}\rightarrow{\bf DT}$, where
$\sigma(\textnormal{First})=\textnormal{`Str'}$ and
$\sigma(\textnormal{BYear})=`{\mathbb{Z}}$’, and now taking
$X=\Delta^{\sigma}$. The only difference is that we have labeled our columns
by more specific attribute names. We write
$\tau(X)\colon\mathcal{K}(X)\rightarrow\mathcal{U}_{X}(X)$ as the table
$\tau(X)=\begin{tabular}[]{|l||l|l|}\hline\cr$\mathcal{K}(X)$&First&BYear\\\
\hline\cr\hline\cr 4&Barack&1961\\\ \hline\cr cc&Michelle&1964\\\ \hline\cr
10&Michelle&1964\\\ \hline\cr\end{tabular}$
We write $\tau(\bullet^{\textnormal{First}})$ and
$\tau(\bullet^{\textnormal{BYear}})$ as the tables
$\tau(\bullet^{\textnormal{First}})=\begin{tabular}[]{|l||l|}\hline\cr$\mathcal{K}(\bullet^{\textnormal{First}})$&First\\\
\hline\cr\hline\cr 1&Barack\\\ \hline\cr 2&Michelle\\\
\hline\cr\end{tabular}\hskip
36.135pt\tau(\bullet^{\textnormal{BYear}})=\begin{tabular}[]{|l||l|}\hline\cr$\mathcal{K}(\bullet^{\textnormal{BYear}})$&BYear\\\
\hline\cr\hline\cr x&1961\\\ \hline\cr y&1946\\\ \hline\cr z&1964\\\
\hline\cr\end{tabular}$
We can consider the restriction maps
$\mathcal{K}(X)\rightarrow\mathcal{K}(\bullet^{\textnormal{First}})$ and
$\mathcal{K}(X)\rightarrow\mathcal{K}(\bullet^{\textnormal{BYear}})$ as
foreign keys attached to the $\tau(X)$ table. The way things are set up, this
foreign key information is kept in the restriction maps of the sheaf
$\mathcal{K}$. See Example 4.2.5.
###### Definition 4.3.7.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification, let
$\mathcal{X}=(X,\mathcal{K}_{X},\tau_{X})$ and
$\mathcal{Y}=(Y,\mathcal{K}_{Y},\tau_{Y})$ denote databases of type $\pi$, and
let $\mathcal{U}_{X}$ and $\mathcal{U}_{Y}$ denote the universal sheaf on $X$
and $Y$ (see Definition 4.3.2). A morphism of databases, denoted
$(f,f^{\sharp})\colon\mathcal{X}\rightarrow\mathcal{Y},$
consists of a map $f\colon Y\rightarrow X$ of schema (see Definition 4.1.2)
and a morphism of sheaves $f^{\sharp}\colon
f^{*}\mathcal{K}_{X}\rightarrow\mathcal{K}_{Y}$ on $Y$ such that the diagram
of sheaves
(18)
commutes.
The category of simplicial databases on $\pi$, whose objects are simplicial
databases as defined in Definition 4.3.3 and whose morphisms have just been
defined, is denoted ${\bf DB}^{\pi}$, or simply ${\bf DB}$ if $\pi$ is
understood. Fixing a schema $X$, the category of databases on $X$, denoted
${\bf DB}_{X}$, is the category whose objects are databases with schema $X$
and whose morphisms are identity on $X$.
###### Remark 4.3.8.
A database is roughly a bunch of tables glued together by foreign key
mappings. A morphism of databases is a way to coherently assign to each table
in one database, a table in another database, and a morphism between the two
tables. Recall that a morphism of tables is a “data-preserving map” (see
Definition 2.3.8, Example 2.3.9, and Remark 2.3.10). Thus, a morphism of
databases should be thought of as a coherent system of data-preserving maps.
We might make the following definition. A morphism without integrity is a pair
$(f,f^{\sharp})\colon\mathcal{X}\rightarrow\mathcal{Y}$ as above, but without
the requirement that diagram (18) commute.
###### Remark 4.3.9.
Let $Y$ be a schema and let $\mathcal{U}_{Y}$ denote the universal database on
$Y$. One can identify ${\bf DB}_{Y}$ with the category ${\bf
Shv}(Y)_{/\mathcal{U}_{Y}}$ of sheaves over $\mathcal{U}_{Y}$. Explicitly,
this is the category whose objects are arrows
$\mathcal{K}\rightarrow\mathcal{U}_{Y}$ and whose morphisms are commutative
triangles.
### 4.4. Relational simplicial databases
In this subsection, we present a category of relational databases as a full
subcategory of the category ${\bf DB}$ of simplicial databases. We also give
an adjunction which allows one to convert a database in our sense to a
relational database in a functorial way.
###### Definition 4.4.1.
Let $\pi$ denote a type specification. A simplicial database
$\mathcal{X}=(X,\mathcal{K},\tau)$ on $\pi$ is called relational if
$\tau\colon\mathcal{K}\rightarrow\mathcal{U}_{X}$ is a monomorphism of
sheaves. The category of relational simplicial databases, denoted
${\bf\mathcal{R}el}^{\pi}$ is the full subcategory of ${\bf DB}^{\pi}$ spanned
by the relational simplicial databases.
Note the precise similarity of this definition with Definition 2.5.1: the
schema $X$ is a gluing together of simple schema $\sigma$, the sheaf
$\mathcal{U}_{X}$ evaluated on a simplex $\Delta^{\sigma}\subset X$ is
$\Gamma(\sigma)$, and a monomorphism of sheaves is a morphism which restricts
to an injective function on each simplex.
Every function $f\colon A\rightarrow B$ between sets has an image
$\textnormal{im}(f)\subset B$ and an injection
$f^{m}\colon\textnormal{im}(f)\rightarrow B$; similarly, given a schema $X$,
every morphism $f\colon\mathcal{A}\rightarrow\mathcal{B}$ of sheaves of sets
on $X$ has an image sheaf denoted $\textnormal{im}(f)\subset\mathcal{B}$ and a
monomorphism of sheaves $f^{m}\colon\textnormal{im}(f)\rightarrow\mathcal{B}$.
If $\mathcal{X}=(X,\mathcal{K},\tau)$ is a database, we can take the image
sheaf $\textnormal{im}(\tau)$ of
$\tau\colon\mathcal{K}\rightarrow\mathcal{U}_{X}$, and the database
$(X,\textnormal{im}(\tau),\tau^{m})$ will be a relational simplicial database.
###### Lemma 4.4.2.
Let $\pi$ denote a type specification. There is an adjunction
$\textstyle{{\bf
DB}^{\pi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf\mathcal{R}el}^{\pi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
in which the left adjoint is given by
$(X,\mathcal{K},\tau)\mapsto(X,\textnormal{im}(\tau),\tau^{m})$ and the right
adjoint is the forgetful functor which realizes a relational simplicial
database as a simplicial database.
###### Proof.
This is a simple exercise that reduces to the fact that the image functor,
which sends the category of sets and functions to the category of sets and
injections, is a left adjoint to the forgetful functor.
∎
Since the forgetful functor ${\bf\mathcal{R}el}^{\pi}\rightarrow{\bf
DB}^{\pi}$ is fully faithful, the counit of the adjunction in Lemma 4.4.2 is
the identity functor on ${\bf\mathcal{R}el}^{\pi}$. Another way to say this is
that one does not lose information when considering a relational database as a
simplicial database, but one often does lose information when converting a
simplicial database to a relational database. Strictly “more information” can
be contained in a simplicial database than in a relational database.
### 4.5. Tables vs. simplicial databases
In this last subsection we present the functor $F\colon{\bf
Tables}\rightarrow{\bf DB}$ which realizes a table as a simplicial database.
We will also present the “global table” construction, which roughly takes a
database and joins everything together to make one big (unnormalized!) table.
###### Construction 4.5.1.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification and
$(K,C,\sigma,\tau)$ a table on $\pi$ (see Definition 2.3.3). Let
$X=\Delta^{\sigma}\in{\bf Sch}^{\pi}$ be the associated schema, let
$\mathcal{U}_{X}$ denote the universal database on $X$, and let
$\mathcal{K}_{X}$ denote the constant sheaf on ${\bf Sub}(X)$ which takes each
subschema to the set $K$. Define
$\tau_{X}\colon\mathcal{K}_{X}\rightarrow\mathcal{U}_{X}$ in the unique way
such that $\tau_{X}(X)\colon\mathcal{K}_{X}(X)\rightarrow\mathcal{U}_{X}(X)$
is the function $\tau\colon K\rightarrow\Gamma(\sigma)$. We are ready to
assign
$F((K,C,\sigma,\tau))\colon=(X,\mathcal{K}_{X},\tau_{X}).$
Given a map of tables
$\varphi\colon(K_{1},C_{1},\sigma_{1},\tau_{1})\rightarrow(K_{2},C_{2},\sigma_{2},\tau_{2})$,
we will now show that there is a canonical map of simplicial databases
$(X_{1},\mathcal{K}_{1},\tau_{1})\rightarrow(X_{2},\mathcal{K}_{2},\tau_{2})$.
Recall from Definition 2.3.8 that $\varphi=(g,f)$ where $g\colon
K_{1}\rightarrow K_{2}$ is a function and
$f\colon\sigma_{2}\rightarrow\sigma_{1}$ is a morphism of simple schema such
that Diagram (5), rewritten for the readers convenience here:
$\textstyle{K_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{1}}$$\scriptstyle{g}$$\textstyle{\Gamma(\sigma_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{K_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{2}}$$\textstyle{\Gamma(\sigma_{2}),}$
commutes.
The morphism $f\colon\sigma_{2}\rightarrow\sigma_{1}$ of simple schema induces
a morphism $\Delta^{\sigma_{2}}\rightarrow\Delta^{\sigma_{1}}$ of schema, i.e.
a map $f\colon X_{2}\rightarrow X_{1}$. The sheaf $f^{*}\mathcal{K}_{1}$ on
$X_{2}$ is the constant sheaf with value $K_{1}$, so $g$ gives a map
$f^{\sharp}\colon f^{*}\mathcal{K}_{1}\rightarrow\mathcal{K}_{2}$. We will
skip some details, but one can easily show that the commutativity of the
Diagram (18) is equivalent to the commutativity of Diagram (5), completing the
construction.
We can also extract a single table from a simplicial database, by looking at
its global sections. This requires a functor called $f_{+}$ defined in Section
5.1. We include the construction here, rather than later, in order to keep
like topics together, and conclude nicely with Remark 4.5.3.
###### Construction 4.5.2.
Let $\mathcal{X}=(X,\mathcal{K},\tau)$ denote a simplicial database. Recall
from Remark 4.1.4 that there is an induced classification map $s\colon
X_{0}\rightarrow{\bf DT}$. Assuming that $X$ has finitely many vertices, we
can construct a table whose simple schema is $s$.
To do so, we need only note that there is a unique map of schema $f\colon
X\rightarrow\Delta^{s}$. Indeed, given any simplex in $X$, its set of vertices
classifies a unique simplex in $\Delta^{s}$; this defines $f$. If we write
$K=\mathcal{K}(X)=f_{+}\mathcal{K}(\Delta^{s})$ and
$t=f_{+}\tau_{X}(\Delta^{s})\colon K\rightarrow\Gamma(s)$, then we are ready
to construct the table
$(K,X_{0},s,t)\in{\bf Tables}.$
Its columns are given by the vertices $X_{0}$ of $X$; its rows are difficult
to describe in general, but in specific cases are quite sensible.
###### Remark 4.5.3.
It is not hard to show that the two above constructions establish an
adjunction
$\textstyle{{\bf
Tables}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf
DB}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Given a database $\mathcal{X}$, the table obtained by the right adjoint will
be called the global table on $\mathcal{X}$.
## 5\. Constructions and formal properties of Simplicial Databases
The point of the formalism in Section 4 is to find a language in which to
describe databases such that the typical operations performed when working
with databases are sensible in the language. In other words, queries of
databases should make sense as categorical constructions, as they did in
Section 3 for tables.
### 5.1. Changing the schema
Let us begin with some ways that one can import data from one schema into
another. In Lemma 4.2.8 we discussed the adjunction
(21)
induced by a map of schema $f\colon Y\rightarrow X$. Given a database
$\mathcal{X}=(X,\mathcal{K}_{X},\tau_{X})$ on $X$ there is an induced database
$(Y,f^{*}\mathcal{K}_{X},\mathcal{U}_{f}\circ(f^{*}\tau_{X}))$, denoted
$f^{*}\mathcal{X}$; see Definition 4.3.2 and refer to the diagram
$\textstyle{f^{*}\mathcal{K}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\tau_{X}}$$\textstyle{f^{*}\mathcal{U}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{U}_{f}}$$\textstyle{\mathcal{U}_{Y}.}$
A slightly more complicated construction creates a database on $X$ from a
database $\mathcal{Y}=(Y,\mathcal{K}_{Y},\tau_{Y})$ on $Y$ and a map of schema
$f\colon Y\rightarrow X$. By the adjunction (21), we have the diagram
(26)
but since there is no canonical map
$f_{*}\mathcal{K}_{Y}\rightarrow\mathcal{U}_{X}$, we have not yet constructed
a database on $X$.
To do so, let $f_{+}(\mathcal{K}_{Y})$ denote the limit of Diagram (26). This
sheaf comes with a canonical map to $\mathcal{U}_{X}$, which we denote
$f_{+}\tau_{Y}\colon f_{+}\mathcal{K}_{Y}\rightarrow\mathcal{U}_{X}$. The
triple
$(X,f_{+}\mathcal{K}_{Y},f_{+}\tau_{Y})$
is a database on $X$, which we denote $f_{+}\mathcal{Y}$.
###### Proposition 5.1.1.
Let $\pi$ denote a type specification, and let $f\colon Y\rightarrow X$ be a
morphism of schema of type $\pi$. The functors $f^{*}$ and $f_{+}$ define an
adjunction
$\textstyle{{\bf
DB}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{{\bf
DB}_{Y}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{+}}$
###### Proof.
Let $\mathcal{X}=(X,\mathcal{K}_{X},\tau_{X})$ and
$\mathcal{Y}=(Y,\mathcal{K}_{Y},\tau_{Y})$ be databases. Giving a morphism
$f^{*}\mathcal{X}\rightarrow\mathcal{Y}$ of databases over $Y$ amounts to a
giving a map $\alpha$ of sheaves making the diagram
$\textstyle{f^{*}\mathcal{K}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}\tau_{X}}$$\scriptstyle{\alpha}$$\textstyle{f^{*}\mathcal{U}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{U}_{f}}$$\textstyle{\mathcal{K}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{Y}}$$\textstyle{\mathcal{U}_{Y}}$
commute. By the adjunction (21) this diagram is equivalent to the diagram
$\textstyle{\mathcal{K}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{X}}$$\scriptstyle{\alpha}$$\textstyle{\mathcal{U}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{U}_{f}}$$\textstyle{f_{*}\mathcal{K}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}\tau_{Y}}$$\textstyle{f_{*}\mathcal{U}_{Y},}$
by Lemma 4.2.8. Supplying a morphism $\alpha$ making this diagram commute is
equivalent to supplying a morphism $\mathcal{K}_{X}\rightarrow
f_{+}\mathcal{K}_{Y}$ over $\mathcal{U}_{X}$, because $f_{+}\mathcal{K}_{Y}$
is the limit of Diagram 26. The proof now follows from Remark 4.3.9.
∎
###### Definition 5.1.2.
Let $\pi$ denote a type specification, and let $f\colon Y\rightarrow X$ be a
morphism of schema of type $\pi$. The functor $f^{*}\colon{\bf
DB}_{X}\rightarrow{\bf DB}_{Y}$, defined above, is called the pullback
functor, and the functor $f_{+}\colon{\bf DB}_{Y}\rightarrow{\bf DB}_{X}$,
defined above, is called the push-forward functor.
Given a sheaf of sets $\mathcal{K}_{X}$ on $X$, we also refer to
$f^{*}\mathcal{K}_{X}\in{\bf Shv}(Y)$ as the pullback of $\mathcal{K}_{X}$,
and given a sheaf of sets $\mathcal{K}_{Y}$ on $Y$, we also refer to
$f_{+}\mathcal{K}_{Y}\in{\bf Shv}(X)$ as the push-forward of
$\mathcal{K}_{Y}$.
###### Example 5.1.3.
Let $X$ and $Y$ be the schema
$X\colon=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
11.36163pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-11.36163pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{~{}^{`{\bf
Str}\textnormal{'}}\\!\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
35.36163pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{\bullet^{`{\mathbb{Z}}\textnormal{'}},}$}}}}}}}\ignorespaces}}}}\ignorespaces\hskip
36.135ptY\colon=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
11.36163pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-11.36163pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{~{}^{`{\bf
Str}\textnormal{'}}\\!\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
35.36163pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{`{\bf
Str}\textnormal{'}},}$}}}}}}}\ignorespaces}}}}\ignorespaces$
and let $f\colon Y\rightarrow X$ be the unique morphism of schema between
them.
By Remark 4.3.9, a database on $X$ is given by a morphism of sheaves
$\tau_{X}\colon\mathcal{K}_{X}\rightarrow\mathcal{U}_{X}$, for some sheaf of
sets $\mathcal{K}_{X}$. We roughly think of it as a table of strings and
integers, with some values not filled in. (In fact, $\tau_{X}$ has more
information because, for example, two keys in $\mathcal{K}(X)$ might be sent
to the same key in $\mathcal{K}(\bullet^{`{\bf Str}\textnormal{'}})$).
The pullback database $f^{*}\tau_{X}\colon
f^{*}\mathcal{K}_{X}\rightarrow\mathcal{U}_{Y}$ is degenerate in the sense
that every row has the same string repeated in two columns. In some sense,
this is to be expected.
Now suppose that $\tau_{Y}\colon\mathcal{K}_{Y}\rightarrow\mathcal{U}_{Y}$ is
a database on $Y$. We roughly think of it as a table whose rows are pairs of
strings. The push-forward $f_{+}\tau_{Y}$ consists of three tables: one has
two columns (strings and integers) and the other two just have one column. The
one column table of integers
$f_{+}\tau_{Y}(\bullet^{`{\mathbb{Z}}\textnormal{'}})$ is empty. The one
column table of strings $f_{+}\tau_{Y}(\bullet^{`{\bf Str}\textnormal{'}})$
consists of those strings $S$ for which there is a row in $\tau_{Y}(Y)$
consisting of a repeated string $(S,S)$. Finally, the two column table
$f_{+}\tau_{Y}(X)$ consists of an element $(S,n)$ for every row $S$ in the
one-column table of strings and every integer $n\in{\mathbb{Z}}$.
One sees that by this example that if $f\colon Y\rightarrow X$ is not
surjective, then the pushforward functor $f_{+}$ results in huge tables. It is
not meant to be implemented as a hash table but as a theoretical construct.
Given a map of schemas $f\colon Y\rightarrow X$, there is one more important
way to send a database on $X$ to a database on $Y$, but only if $f$ is a
monomorphism of schema. A monomorphism of schema corresponds to the
relationship often known as “is a”, in which every object of type $x$ “is an”
object of type $y$. In this situation, there is a functor which takes as input
a database of $y$’s, and produces as output a database of $x$’s with all of
the $y$-information filled in, but nothing else. The functor that accomplishes
this task is denoted $f_{!}\colon{\bf DB}_{Y}\rightarrow{\bf DB}_{X}$ and is
called “extension by $\emptyset$,” meaning that on every simplex in $X$ that
is not in $f(Y)$, the value of the sheaf there is an empty table.
To define $f_{!}$ rigorously, we first notice that $f^{*}\colon{\bf
Shv}(X)\rightarrow{\bf Shv}(Y)$ not only has a right adjoint ($f_{*}$), but a
left adjoint as well, which we also denote $f_{!}\colon{\bf
Shv}(Y)\rightarrow{\bf Shv}(X)$. If $f$ is a monomorphism, then every
subschema $Y^{\prime}\subset Y$ is sent to a subschema $f(Y^{\prime})\subset
X$.
Let us define $f_{!}\mathcal{U}_{Y}$ and its canonical map to
$\mathcal{U}_{X}$. Every subschema $X^{\prime}\subset X$ is either of the form
$X^{\prime}=f(Y^{\prime})$ or not. If so, we set
$f_{!}\mathcal{U}_{Y}(X^{\prime})=\mathcal{U}_{Y}(Y^{\prime})=\mathcal{U}_{X}(X^{\prime})$.
If not, we set $f_{!}\mathcal{U}_{Y}(X^{\prime})=\emptyset$. There is a
canonical map $a_{f}\colon f_{!}\mathcal{U}_{Y}\rightarrow\mathcal{U}_{X}$
which is the identity map on $X^{\prime}=f(Y^{\prime})$ and which is
$\emptyset\rightarrow\mathcal{U}_{X}(X^{\prime})$ when
$X^{\prime}\not\in\textnormal{im}(f)$.
Now that we have a canonical map $a_{f}\colon
f_{!}\mathcal{U}_{Y}\rightarrow\mathcal{U}_{X}$ in the case that $f\colon
Y\rightarrow X$ is an inclusion, we can define $f_{!}\colon{\bf
DB}_{Y}\rightarrow{\bf DB}_{X}$ to be given by
$f_{!}(Y,\mathcal{K}_{Y},\tau_{Y})\colon=(X,f_{!}\mathcal{K}_{Y},a_{f}\circ\tau_{Y}).$
The functor $f_{!}$ is left adjoint to the functor $f^{*}\colon{\bf
DB}_{X}\rightarrow{\bf DB}_{Y}$ (but $f_{!}$ is defined only when $f\colon
Y\rightarrow X$ is an injection.)
### 5.2. Nulls
Nulls do not conform with the mathematical logic that underlies the strict
theoretical foundation of relational databases. They are easy enough to deal
with, however, by use of foreign keys. That is, for each column $c\in C$ of a
schema $\sigma\colon C\rightarrow{\bf DT}$ for which a table may contain a
null, one creates a new schema $\sigma^{\prime}$ on columns
$C^{\prime}=C-\\{c\\}$. By an easy use of foreign keys, one considers objects
classified by $\sigma$ to be also classified by $\sigma^{\prime}$. This is a
way to get around the problem of nulls. Other approaches can be found in
[JR03].
The same technique is done (automatically) in simplicial databases. Over a
simplex $\Delta^{\sigma}$, one puts objects for which the value on each column
is known. If the value on some set of columns is unknown for a certain object,
it is represented as a record on the subsimplex for which it is total.
If one so desired, he or she could implement simplicial databases so that
local sections of the database (records over subschema) appeared as global
sections of the database (records over the whole schema) by putting the value
“Null” in appropriate places. From our perspective it is preferable just to
leave local data as local data and not try to promote it to global data, at
least for theoretical purposes.
### 5.3. Duplicate records
SQL allows for a table to have the same record in two different rows.
Therefore, tables are not relations and SQL does not strictly implement
relational databases. One could argue that SQL is “wrong” in not conforming to
the theory (see [Dat05, p. 14]), but perhaps the pure relational theory is
overly strict; this is the position we take.
Simplicial database allow for duplicate entries. This should not be
threatening because internal keys ensure the integrity of the data. If
$\Gamma=A\times B\times C$, then relations on this simple schema are subsets
$K\subset\Gamma$. In the theory of simplicial databases, we allow non-
injective functions $\tau\colon K\rightarrow\Gamma$, called tables.
Philosophically, we see the relational model as “confusing the object with its
attributes.” A schema, or set of attributes, gives a set of ways to measure a
collection of objects. It is entirely possible that two objects in that
collection could have the same measurements according to the schema. In the
relational model, these two objects would be identified in the sense that only
one row of the table would be representing both. From now on, the database and
its users will have no choice but to consider these objects to be the same.
The only alternative to this is to introduce arbitrary identifiers. These
artificial keys are not part of the data being measured about the objects. In
our view, it is best to keep these arbitrary identifiers “internal” to the
database management system. Among several advantages, the most obvious is
database integration, in which it is important to know what aspects of the
data are “measured” and invariant, and what aspects are contrived. We will say
more about this in Section 6.5.3.
### 5.4. Limits and colimits of databases
We will see shortly that limits and colimits taken in the category of
simplicial databases have meaning in terms of the general theory of databases,
such as joins and unions.
###### Theorem 5.4.1.
Let $\pi\colon U\rightarrow{\bf DT}$ denote a type specification. The category
${\bf DB}^{\pi}$ of databases of type $\pi$ is closed under taking small
colimits and small limits.
###### Proof.
Let $I$ denote a small category and let $\mathcal{X}\colon I\rightarrow{\bf
DB}$ denote an $I$-shaped diagram in ${\bf DB}={\bf DB}^{\pi}$. There is a
functor ${\bf DB}\rightarrow{\bf Sch}^{\textnormal{op}}$ taking a database
$(A,\mathcal{K}_{A},\tau_{A})$ to its underlying schema $A$, and composing
this functor with $\mathcal{X}$ gives a functor which we denote $X\colon
I\rightarrow{\bf Sch}^{\textnormal{op}}$. For an object $i\in I$, we denote
the database $\mathcal{X}(i)$ by $\mathcal{X}_{i}$ and write
$\mathcal{X}_{i}=(X_{i},\mathcal{K}_{i},\tau_{i}).$
To define the colimit (respectively limit) of the diagram $\mathcal{X}$, we
must first specify its schema. Since ${\bf Sch}={\bf Pre}(\mathcal{S})$, where
$\mathcal{S}$ is the category of simple schema (see Definition 2.2.6), it is
closed under colimits and limits ([MLM94, p. 22]); hence so is ${\bf
Sch}^{\textnormal{op}}$. Let $C=\mathop{\textnormal{colim}}(X)$ (resp.
$L=\lim(X)$) denote the colimit (resp. limit) of the diagram $X\colon
I\rightarrow{\bf Sch}^{\textnormal{op}}$. Let $\mathcal{U}_{C}$ and
$\mathcal{U}_{L}$ denote the universal databases on $C$ and $L$, respectively.
As a colimit in ${\bf Sch}^{\textnormal{op}}$, the schema $C$ comes equipped
with morphisms in $c_{i}\colon C\rightarrow X_{i}$ in ${\bf Sch}$, for each
$i\in I$, making the appropriate diagrams commute. There is a pullback sheaf
$c_{i}^{*}\tau\colon c_{i}^{*}\mathcal{K}_{i}\rightarrow\mathcal{U}_{C}$. If
$f\colon i\rightarrow j$ is a morphism in $I$, then the map $X_{j}\rightarrow
X_{i}$ in ${\bf Sch}$ induces a morphism
$c_{i}^{*}\mathcal{K}_{i}\rightarrow c_{j}^{*}\mathcal{K}_{j}$
of pullback sheaves over $\mathcal{U}_{C}$ on $C$. Let $c^{*}\colon
I\rightarrow{\bf Shv}(C)_{/\mathcal{U}_{C}}$ denote the $I$-shaped diagram of
these pullback sheaves over $\mathcal{U}_{C}$. Define
$\tau_{C}\colon\mathcal{K}_{C}\rightarrow\mathcal{U}_{C}$ to be the colimit of
this diagram. Then the database
$\mathcal{C}=(C,\mathcal{K}_{C},\tau_{C})$
is our candidate for the colimit of the diagram $\mathcal{X}$. It is a matter
of tracing through the construction to show that $\mathcal{C}$ has the
necessary universal property.
Defining the limit of $\mathcal{X}$ is similar. As a limit in ${\bf
Sch}^{\textnormal{op}}$, the schema $L$ comes equipped with morphisms
$\ell_{i}\colon X_{i}\rightarrow L$ in ${\bf Sch}$, for each $i\in I$, making
the appropriate diagrams commute. There is a push-forward sheaf
$(\ell_{i})_{+}\mathcal{K}_{i}$ on $L$, which comes equipped with a map
$(\ell_{i})_{+}\tau\colon(\ell_{i})_{+}\mathcal{K}_{i}\rightarrow\mathcal{U}_{L}$.
If $f\colon i\rightarrow j$ is a morphism in $I$, then the map
$X_{j}\rightarrow X_{i}$ in ${\bf Sch}$ induces a morphism
$(\ell_{i})_{+}\mathcal{K}_{i}\rightarrow(\ell_{j})_{+}\mathcal{K}_{j}$
of push-forward sheaves over $\mathcal{U}_{L}$ on $L$. Let $(\ell_{+})\colon
I\rightarrow{\bf Shv}(L)_{/\mathcal{U}_{L}}$ denote the $I$-shaped diagram of
these push-forward sheaves over $\mathcal{U}_{L}$. Define
$\tau_{L}\colon\mathcal{K}_{L}\rightarrow\mathcal{U}_{L}$ to be the limit of
this diagram. Then the database
$\mathcal{L}=(L,\mathcal{K}_{L},\tau_{L})$
is our candidate for the limit of the diagram $\mathcal{X}$. Again, it is a
matter of tracing through the construction to show that $\mathcal{L}$ has the
necessary universal property.
This completes the proof.
∎
###### Remark 5.4.2.
The final object in the category ${\bf DB}^{\pi}$ of databases on $\pi\colon
U\rightarrow{\bf DT}$ is the empty database (with empty schema and trivial
sheaf). The initial object $(X,\mathcal{K},\tau)$ in ${\bf DB}^{\pi}$ has, as
its schema $X$, a single $n$-simplex for every map
$\sigma\colon\\{0,1,\ldots,n\\}\rightarrow{\bf DT}$; the sheaf is
$\mathcal{K}=\mathcal{U}_{X}$, and the map
$\tau\colon\mathcal{U}_{X}\rightarrow\mathcal{U}_{X}$ is the identity.
If one knows the $\check{\textnormal{C}}$ech nerve construction, one can
realize the initial object in those terms, by applying the
$\check{\textnormal{C}}$ech nerve functor to $\pi\colon U\rightarrow{\bf DT}$.
See [Spi08, 3.1] for details.
###### Corollary 5.4.3.
Let $X\in{\bf Sch}$ be a schema and let ${\bf DB}_{X}$ denote the category of
databases with schema $X$ and with morphisms which restrict to the identity on
$X$. Colimits and limits exist in ${\bf DB}_{X}$; in particular ${\bf DB}_{X}$
has an initial object and a final object.
###### Proof.
Given a non-empty diagram which restricts to the identity on a certain schema
$X$, one sees by the construction of limits and colimits in the proof of
Theorem 5.4.1 that the limit and the colimit of that diagram will also have
schema $X$.
The limit (respectively the colimit) of the empty diagram in ${\bf DB}_{X}$,
if it exists, is the final (resp. initial) object in ${\bf DB}_{X}$; we must
show it does exist. One immediately sees that the final object is
$(X,\mathcal{U}_{X},\textnormal{id}_{\mathcal{U}_{X}})$, and the initial
object is $(X,\emptyset,\emptyset\rightarrow\mathcal{U}_{X})$, where
$\emptyset$ here denotes the sheaf on $X$ whose value is constantly the empty
set, and where $\emptyset\rightarrow\mathcal{U}_{X}$ is the unique morphism of
sheaves.
∎
### 5.5. Projections
This query is built into the theory of simplicial databases. Given a database
$(X,\mathcal{K},\tau)$ and a subschema $X^{\prime}\subset X$, we have the
database $(X^{\prime},\mathcal{K}|_{X^{\prime}}\tau|_{X^{\prime}})$ given by
restricting the sheaf $\mathcal{K}$ and the map of sheaves
$\tau\colon\mathcal{K}\rightarrow\mathcal{U}$ to the subschema $X^{\prime}$.
One can view it as a table using Construction 4.5.2.
### 5.6. Unions and insertions
Given two databases with the same schema, one can apply the UNION query. To do
so, one keeps the same columns but takes the union of the rows. An insertion
is a special kind of union; namely it is a union of two databases on the same
schema, where one of the databases consists only of a single row.
We have a few more options in simplicial databases than one does in relational
databases; these differences are analogous to the difference between the UNION
query and the UNION ALL query in SQL. That is, since we allow duplicate
entries (see Section 5.3), the user can decide when an object in one database
is the same as an object with the same attributes stored in another database
and when it is different. Let us make all of this precise.
We can represent unions, insertions, and more by taking colimits of various
diagrams of databases. Let $\mathcal{X}=(X,\mathcal{K},\tau)$ denote a
simplicial database, and let
$\mathcal{X}^{\prime}=(X,\mathcal{K}^{\prime},\tau^{\prime})$ be a database
with the same schema, $X$. Both receive a map from the initial database on
$X$, and the coproduct will be
$(X,\mathcal{K}\amalg\mathcal{K}^{\prime},\tau\amalg\tau^{\prime})$ as
desired. (See the proof of Theorem 5.4.1 for details on the colimit
construction.)
The above construction gives a UNION ALL query: duplicated tuples will remain
distinct. There are two ways of having that not be the case. The first is to
simply eliminate the duplicates by converting the database to a relational
database; see Lemma 4.4.2. However, this may result in information loss if
there really were two entities with the same attributes, because these
duplicates will be eliminated.
The other way can occur if the user has more information about which instances
in the first database correspond to instances in the second database. This can
be accomplished by having a third database
$\mathcal{X}^{\prime\prime}=(X,\mathcal{K}^{\prime\prime},\tau^{\prime\prime})$
and maps from it to $\mathcal{X}$ and $\mathcal{X}^{\prime}$. The colimit of
this diagram,
$(X,\mathcal{K}\amalg_{\mathcal{K}^{\prime\prime}}\mathcal{K}^{\prime},\tau\amalg_{\tau^{\prime\prime}}\tau^{\prime})$,
will be the union of the records in $\mathcal{X}$ with those in
$\mathcal{X}^{\prime}$, and will identify two records if they agree in
$\mathcal{X}^{\prime\prime}$.
As mentioned above, inserting a row is a special case of taking the union of
databases.
We can take much more general colimits than those mentioned above, all of
which were constant in the schema. These constructions appear to be new;
perhaps they can provide useful ways to analyze and assemble data.
### 5.7. Join
Two databases can be joined together by specifying a common sub-schema of each
and “gluing together” along that sub-schema. If no common sub-schema is
mentioned we take the initial schema, which is empty, and join along that; the
result is called the natural join. The concept of gluing is rigorously
formulated as taking limits of certain diagrams in ${\bf
Sch}^{\textnormal{op}}$; thus the point we are making is that joining
databases in the usual sense can be accomplished by taking limits in the
category of simplicial databases. Let us make all of this precise.
Recall from Theorem 5.4.1 that the limit of the diagram of databases
$(X_{1},\mathcal{K}_{1},\tau_{1})\longrightarrow(X,\mathcal{K},\tau)\longleftarrow(X_{2},\mathcal{K}_{2},\tau_{2})$
has schema $X^{\prime}=X_{1}\amalg_{X}X_{2}$. This induces a diagram
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ulcorner}$
in ${\bf Sch}$. We can thus push-forward $\mathcal{K}_{1}$, $\mathcal{K}$, and
$\mathcal{K}_{2}$ to $X^{\prime}$ and get a diagram of push-forward sheaves
there (see Definition 5.1.2), all naturally mapping to
$\mathcal{U}_{X^{\prime}}$. For typographical reasons, we leave out the fact
that these are push-forwards and write the diagram
$\mathcal{K}_{1}\rightarrow\mathcal{K}\leftarrow\mathcal{K}_{2}$ over
$\mathcal{U}_{X^{\prime}}$. We are ready to write the limit database as
$(X_{1}\amalg_{X}X_{2},\mathcal{K}_{1}\times_{\mathcal{K}}\mathcal{K}_{2},\tau^{\prime}),$
where
$\tau^{\prime}\colon\mathcal{K}_{1}\times_{\mathcal{K}}\mathcal{K}_{2}\rightarrow\mathcal{U}_{X^{\prime}}$
is the structure map.
###### Example 5.7.1.
Suppose we have the two schemas pictured here:
$X_{1}\colon=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
12.89386pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-12.89386pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{~{}^{\textnormal{`First'}}\\!\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
36.89386pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{\bullet^{\textnormal{`Last'}}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\hskip
36.135ptX_{2}\colon=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.60385pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-16.60385pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{$\textstyle{~{}^{\textnormal{`L.Name'}}\\!\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.60385pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.0pt\raise
0.0pt\hbox{$\textstyle{\bullet^{\textnormal{`BYear'}},}$}}}}}}}\ignorespaces}}}}\ignorespaces$
and wish to join them together by equating ‘Last’ with ‘L.Name’ (both of which
have the same data type, namely ${\bf Str}$). To do so, we use the schema
$X=\bullet^{`{\bf Str}^{\prime}}$, which maps to each of $X_{1}$ and $X_{2}$
in an obvious way.
Now given any databases $\mathcal{X}_{1}=(X_{1},\mathcal{K}_{1},\tau_{1})$ and
$\mathcal{X}_{2}=(X_{2},\mathcal{K}_{2},\tau_{2})$ on $X_{1}$ and $X_{2}$, we
can join them by taking the limit of the solid arrow diagram
$\textstyle{\mathcal{X}_{1}\times_{\mathcal{X}}\mathcal{X}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}}$
where $\mathcal{X}=(X,\mathcal{U}_{X},\textnormal{id}_{\mathcal{U}_{X}})$ is
the final database on $X$. The schema of the resulting database is
‘First’‘Last’=‘LName’‘BYear’
This does not represent a table with three columns, but two tables, each with
two columns, and each projecting to a common 1-column table. However, its
global table does have three columns (see Remark 4.5.3). Its records are those
triples of the form (First,Last,BYear) for which there is a (First,Last) pair
in $\mathcal{X}_{1}$ and a (Last,BYear) pair in $\mathcal{X}_{2}$ with
matching values of Last. This is indeed their join.
###### Remark 5.7.2.
The “join” we are working with here could be thought of as a combination of
equi-join and outer join. Because databases are sheaves on a schema, they do
not have just one table but a system of tables, and the idea of nulls is built
into the theory (see Section 5.2).
More precisely, if
$\mathcal{X}_{1}\rightarrow\mathcal{X}\leftarrow\mathcal{X}_{2}$ is a diagram
of databases, the limit $\mathcal{X}^{\prime}$ represents the join of
$\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ along a shared set of columns (those
of $\mathcal{X}$). Its schema is roughly the union of the schemas of
$\mathcal{X}_{1}$ and $\mathcal{X}_{2}$. Its global table will be the equi-
join of the global tables for $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$.
The point of this remark, however, is that the new table
$\mathcal{X}^{\prime}$ does not only contain global information, but local
information as well. Much of the data of $\mathcal{X}_{1}$ (respectively
$\mathcal{X}_{2}$) is preserved upon passage to $\mathcal{X}^{\prime}$, and
that which cannot be extended to global data could still be viewed globally if
one uses Null values. It is in this sense that colimits in ${\bf DB}$ are
related to outer joins.
When joining databases together, one first chooses a set $C$ of columns to
equate. When two distinct objects have the same $C$-attributes, then the join
is “lossy” in the sense that there will be false information in the join. To
remedy this, one must be careful to distinguish between objects, even when
considered only in terms of $C$. The following example will hopefully make
this more clear.
###### Example 5.7.3.
Suppose one wants to join the following two tables:
$\tau_{1}$ Title LastName 1 Dr. Marx 2 Mr. Marx $\tau_{2}$ FirstName LastName
A Karl Marx B Groucho Marx
The outcome will be the following table:
Title | FirstName | LastName
---|---|---
Dr. | Karl | Marx
Dr. | Groucho | Marx
Mr. | Karl | Marx
Mr. | Groucho | Marx
This table has four entries, two of which are “accurate,” in that they
describe real instances, and two of which are not. This occurs because the
relational database cannot distinguish between the two instances of the last
name Marx.
Achieving a lossless join is easy, when databases are allowed to have
duplicate entries with the same attributes. Consider the table
$\tau$ | LastName
---|---
x | Marx
y | Marx
which accepts maps from both $\tau_{1}$ and $\tau_{2}$ by sending both $1$ and
$A$ to $x$, and sending both $2$ and $B$ to $y$ (see Definition 2.3.8). The
limit of this diagram is the table
Title | FirstName | LastName
---|---|---
Dr. | Karl | Marx
Mr. | Groucho | Marx
as desired.
In the example above, the table $\tau$ has two instances of the same string.
This is not superfluous because there are two people named Marx. They are
differentiated by their internal keys, but not by their attributes. Keeping
distinct objects distinct, even if they have the same attributes is very
useful in practice. It not only allows for lossless joins, but it is well-
suited for database integration as well.
### 5.8. Select
In Example 3.1.9, we selected from a table $\tau$ with columns
$C=\\{\textnormal{`First Name', `Last Name', `BYear'}\\}$ all instances for
which the value of ‘First Name’ was “Barack.” This was computed as follows.
First, we made a table $\tau^{\prime}$ whose column set $C^{\prime}$ consisted
of a single element, labeled ‘First Name’, and filled in $\tau^{\prime}$ with
a single entry, ‘Barack’. We might call this table the selection table. The
SELECT operation was performed by taking the fiber product
$\tau\rightarrow\textnormal{id}_{C^{\prime}}\leftarrow\tau^{\prime}$, where
$\textnormal{id}_{C^{\prime}}$ denotes the table of all possible values of
‘First Name’.
Performing SELECT operations in a general simplicial database has the same
flavor, in that it is always computed as a certain kind of fiber product.
Denote the database from which we are selecting as
$\mathcal{X}=(X,\mathcal{K}_{X},\tau_{X})$, let $S\subset X$ denote a
subschema and $\mathcal{S}=(S,\mathcal{K}_{S},\tau_{S})$ a relational table on
$S$, to serve as the selection table. That is, we will be selecting from $X$
all instances that have the designated $S$-attributes. Finally, we let
$1_{\mathcal{S}}=(S,\mathcal{U}_{S},\textnormal{id}_{\mathcal{U}_{S}})$ denote
the final database on the schema $S$. The fiber product
$\mathcal{X}_{\mathcal{S}}$ in the diagram
$\textstyle{\mathcal{X}_{\mathcal{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lrcorner}$$\textstyle{\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1_{\mathcal{S}}}$
is the desired result.
### 5.9. Deletions
Deletion can be subtle. If one deletes entries over a subschema, the action
must “cascade” up the hierarchy, deleting entries in larger schemas when they
refer or point to the deleted entries. To that end, we define the following
construction.
###### Definition 5.9.1.
Suppose given a schema $X$ and a subsheaf $\mathcal{K}_{1}\subset\mathcal{K}$
on $X$. Let $\overline{\mathcal{K}_{1}}\subset\mathcal{K}$ denote the presheaf
on $X$ with
$\overline{\mathcal{K}_{1}}(X^{\prime})\colon=\\{r\in\mathcal{K}(X^{\prime})|\exists
X^{\prime\prime}\subset
X^{\prime},X^{\prime\prime}\neq\emptyset,r_{X^{\prime\prime}}\in\mathcal{K}_{1}(X^{\prime\prime})\\}$
for subschema $X^{\prime}\in{\bf Sub}(X)$. Here $r_{X^{\prime\prime}}$ denotes
the image of $r$ under the restriction map
$\mathcal{K}(X^{\prime})\rightarrow\mathcal{K}(X^{\prime\prime})$. We call
$\overline{\mathcal{K}_{1}}$ the closure of $\mathcal{K}_{1}$ in
$\mathcal{K}$.
Suppose now we want to delete all entries of a given type from a database.
More concretely, suppose $\mathcal{X}=(X,\mathcal{K}_{X},\tau_{X})$ is a
database with schema $X$, that $i\colon S\subset X$ is a subschema, and that
$\mathcal{S}=(S,\mathcal{K}_{S},\tau_{S})$ is a relational database of objects
of this subtype, all of which we would like to delete from $X$. As explained
in Section 5.8, we can select the rows of $\mathcal{X}$ of the type specified
by $\mathcal{S}$ by defining $\mathcal{X}_{S}$ to be the limit as in the
diagram
$\textstyle{\mathcal{X}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lrcorner}$$\textstyle{\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(S,\mathcal{U}_{S},\textnormal{id}_{\mathcal{U}_{S}}).}$
We know that $\mathcal{X}_{\mathcal{S}}$ has schema $X=X\amalg_{S}S$ and we
momentarily invent notation and write
$\mathcal{X}_{\mathcal{S}}=(X,\mathcal{K}_{\mathcal{S}\subset\mathcal{X}},\tau_{\mathcal{S}\subset\mathcal{X}})$.
The map $\mathcal{X}_{\mathcal{S}}\rightarrow\mathcal{X}$ defines an inclusion
of sheaves $\mathcal{K}_{\mathcal{S}\subset\mathcal{X}}\subset\mathcal{K}_{X}$
on $X$, and we take its closure
$\overline{\mathcal{K}_{\mathcal{S}\subset\mathcal{X}}}\subset\mathcal{K}_{X}$.
By construction we can now delete this subsheaf objectwise on ${\bf Sub}(X)$.
That is, we define for $X^{\prime}\subset X$
$\mathcal{K}_{\mathcal{X}\backslash\mathcal{S}}(X^{\prime})=\mathcal{K}_{X}(X^{\prime})\backslash\mathcal{K}_{\mathcal{S}\subset\mathcal{X}}(X^{\prime}),$
where $A\backslash B$ denotes the maximal subset of $A$ which contains no
elements in $B$.
The database
$\mathcal{X}^{\prime}\colon=(X,\mathcal{K}_{\mathcal{X}\backslash\mathcal{S}},\tau),$
where $\tau$ is shorthand for
$\tau|_{\mathcal{K}_{\mathcal{X}\backslash\mathcal{S}}}\colon\mathcal{K}_{\mathcal{X}\backslash\mathcal{S}}\rightarrow\mathcal{U}_{X}$,
is the deletion of $\mathcal{S}$ from $\mathcal{X}$. There is a canonical map
$\mathcal{X}^{\prime}\rightarrow\mathcal{X}$ in ${\bf DB}$, and one can show
that $\mathcal{X}^{\prime}$ is the final object under $\mathcal{X}$ whose join
with $\mathcal{S}$ is empty.
## 6\. Applications, advantages, and further research
In this section, we discuss the applications of the category of simplicial
databases. First, simplicial databases can be used wherever relational
databases are used; though simplicial databases are more general, they are
still closed under applying the usual queries. On the other hand, there are
many advantages to using simplicial databases as opposed to relational ones.
In Section 6.1, we discuss how the geometry of a schema can provide an
intuitive picture for the content and layout of a database. As an example of
using category theory to reason about databases, we show in Section 6.2 that
query equivalences are trivially verified when one phrases them in categorical
language. In Section 6.3 we discuss how diagrams of databases can give various
users different privileges in terms of accessing and modifying data. In
Section 6.4 we address the issue of comparing our categorification of
databases to others versions. Finally, in Section 6.5, we discuss further
research on the subject and open questions.
### 6.1. Geometric intuition
In Section 4.1, we defined the category ${\bf Sch}^{\pi}$ of schemas for a
given type specification $\pi$. They are based on geometric objects called
simplicial sets. In this section, we show that the geometry of these objects
is intuitive and therefore useful in practice.
###### Example 6.1.1.
In this example, we consider a simplified situation in which one keeps track
of the cities from which airplane flights take off and those at which they
land. So suppose we have only one type, ${\bf DT}=\\{\textnormal{`City'}\\}$
and $U$ is the set of cities in the world that have airports. Let $X$ be the
schema
$\textstyle{~{}^{\textnormal{`City'}}\\!\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet^{\textnormal{`City'}}}$
For our sheaf of keys $\mathcal{K}$, we take
$\mathcal{K}(\textnormal{`City'})=U$. Over the 1-simplex $X$ take
$\mathcal{K}(X)$ to be the set of pairs $(c_{1},c_{2})$ for which $c_{1}$ is
the city of departure and $c_{2}$ is the city of arrival for some flight. Let
$\mathcal{X}$ denote this database of flights.
Now, joining this database with itself yields a database with schema
‘City’‘City’‘City’
whose global sections are “flights with layover,” i.e. pairs of flights with
the destination city of the first flight equal to the departing city of the
second flight. Similarly, the database of multi-city trips of a given length
$n$ is simply the union (colimit) of $n$ copies of the database of flights
$\mathcal{X}$ in this way.
Moreover, if we want to use $\mathcal{X}$ to find the set of available round-
trips, we simply join the ends of the schema in Diagram 6.1.1 to make a circle
‘City’‘City’
This is not just heuristic; we have literally taken the indicated limit of
databases. The result is a new database whose global sections are precisely
the pairs of flights which constitute a round-trip.
The point is that one can intuit this result by visualizing round-trips as
circles, and then applying that vision to the schemas themselves.
###### Example 6.1.2.
In 2004, Bearman et al. [BMS04] present data which shows that at a certain
high school called “Jefferson High,” there is a statistically small number of
sexual couples that later switch partners. That is, if $B_{1}$ and $G_{1}$ are
sexual partners and $B_{2}$ and $G_{2}$ are sexual partners, then it rarely
happens that later $B_{1}$ mates with $G_{2}$ and $B_{2}$ mates with $G_{1}$.
As they say “…we find many cycles of length 4 in the simulated networks, but
few in Jefferson…”
Suppose then that we take their raw data and put it on the schema
$\textstyle{~{}^{\textnormal{`Boyfriend'}}\\!\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet^{\textnormal{`Girlfriend'}}}$
which we denote $X$. Visually, we represent two boys and two girls who switch
partners as follows:
(33)
(where, say, horizontal lines represent the original partnerships and diagonal
lines represent the new partnerships). And indeed, we can take the union of
four copies of $X$ along various vertices to obtain a database with the above
4-cycle schema.
In other words, there is a way to take raw data over a line segment,
representing partnerships, and automatically generate data over the “switch
schema,” Diagram (33), just by taking the indicated limit of databases. The
global sections of this new “switched partners” database are precisely what is
being studied in Bearman’s paper.
As in Example 6.1.1, the point is that the shape of the schema is intuitive.
Using schemas that are geometrically intuitive may enhance the ability of
users to manipulate and make sense out of the raw data.
### 6.2. Query equivalences
It is well known that joining tables together is very costly. If one only
wishes to consider certain rows or columns of a join, he or she should isolate
those rows or columns before performing the join, not after. For that reason,
one is taught to “push selects and projects,” i.e. to do these operations
first.
How does one prove that projecting first and then joining will result in the
same database as will joining first and then projecting? The proofs of results
like these are generally tedious. In this section, we do not claim any new
results. We merely show that these simple query equivalences are obvious when
one uses the language of simplicial databases and knows basic category theory.
For example, it is a standard category-theoretic fact that, in any category
$\mathcal{C}$ with limits, there is a natural isomorphism
(34) $\displaystyle(A\times_{B}C)\times_{D}E\cong(C\times_{D}E)\times_{B}A.$
Note that both joins and selects are examples of such limits (see Sections 5.7
and 5.8). The formula (34) in particular applies to the category ${\bf DB}$ of
databases and proves that “selecting $E$ from a join of $A$ and $C$ gives the
same result as first selecting $E$ from $C$ and then joining the result with
$A$.
Projecting a database to a subschema is easy to describe in the theory of
simplicial databases: one simply restricts the sheaf $\mathcal{K}$ and the map
$\tau$ to that subschema (see Section 5.5). The fact that projects commute
with joins follows from basic sheaf theory, e.g. that the limit of a diagram
of sheaves is the same as the limit of the underlying diagram of presheaves.
### 6.3. Privileges
The sheaf-theoretic nature of our conception of databases lends itself nicely
to the idea of privileges. It often happens that one wishes to give a
particular user the ability to modify certain sections of the database but not
others. If $X$ is the schema for a database $\mathcal{X}$, perhaps we wish to
give a particular user the ability to modify data on the subschema $i\colon
X^{\prime}\subset X$.
To accomplish this, note that there is a map of databases
$\mathcal{X}=(X,\mathcal{K}_{X},\tau_{X})\longrightarrow(X^{\prime},i^{*}\mathcal{K}_{X},i^{*}\tau_{X})=\mathcal{X}^{\prime}$
We allow the user to see $\mathcal{X}^{\prime}$ as a database and make changes
to it (we could also limit the ways in which this user can modify
$\mathcal{X}^{\prime}$ – only allow insertions, for example). At any given
time, the user only sees the sub-database $\mathcal{X}^{\prime}$.
Suppose he or she adds a few lines to the sheaf $i^{*}\mathcal{K}_{X}$ to make
it $i^{*}\mathcal{K}_{X}\cup\mathcal{L}$. To update the main database, we take
the colimit of the diagram of sheaves
---
$\textstyle{i_{!}i^{*}\mathcal{K}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{!}(i^{*}\mathcal{K}_{X}\cup\mathcal{L})}$$\textstyle{\mathcal{K}_{X}}$
and the result will be a new sheaf on $X$ with the appropriate insertions.
Deletions are handled in a somewhat different way, but the idea is the same.
If the user deletes data from the sheaf $i^{*}\mathcal{K}_{X}$ to obtain the
sheaf $i^{*}\mathcal{K}_{X}\backslash\overline{\mathcal{D}}$, then to update
the main database may require us to delete entries from larger schemas (see
Section 5.9). The updated sheaf on $X$ will be the limit of the diagram
$\textstyle{\mathcal{K}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{+}(i^{*}\mathcal{K}_{X}\backslash\overline{\mathcal{D}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{i_{+}i^{*}\mathcal{K}_{X}.}$
Again, we are not claiming that privileges of this type are anything new. We
are claiming that they are naturally phrased in this categorical language,
thus bringing a new and powerful mathematical tool to bear on the problems of
the subject.
### 6.4. Comparison to other categories of databases
As mentioned in the introduction, many other categorifications of databases
have been presented over the years. One of the nice features of category
theory is that one can compare various categories using functors. Given
another categorical formulation of databases, we could try to produce a
functor from it to ${\bf DB}$ and from ${\bf DB}$ back to it. The way that
these functors behave (e.g. if they are adjoint, or if one or the other is
fully faithful) will tell us about the relative expressive power of the
models, as well as understand how to translate between them. We hope to work
on such a comparison in the future.
### 6.5. Further research
The category-theoretic and also geometric nature of simplicial databases opens
up many directions for future research. We present a few in this subsection
that we intend to pursue. Many of these ideas were suggested to us by Paea
LePendu.
#### 6.5.1. Topological methods
First, we would like to consider how we might use methods from algebraic
topology to study databases. Recall from Example 4.1.5 that there is a functor
${\bf Sch}\rightarrow{\bf Top}$ called topological realization that allows one
to naturally view any schema as a topological space. Furthermore, we already
saw in Example 6.1.2 that importing topological ideas can have real world
meaning: topological 4-cycles represented pairs of mating couples that
switched partners.
Another example of the usefulness of topological methods is given by “lifting
problems.” Problems of this sort include the famous question “are there three
foods, each pair of which taste good when eaten together, but the threesome of
which tastes bad when eaten together?”
To phrase this in terms of social networks, suppose that for any $n$ people,
either this group is said to be a friendship group or it is not. The above
lifting problem becomes: “are there three people, each pair of which is a
friendship group, but the triple is not?” These types of phenomena can be
represented geometrically, so having simplicial sets as schema may be useful
for their study.
Homotopical methods from algebraic topology may also be useful. When one
object “morphs” into another over the course of time (such as a child becoming
an adult), it is difficult to know how to treat that object in a database.
Homotopy theory is the study of gradual transformation through time, and the
author sees some potential for using it to study real-world phenomena.
Finally, the geometric nature of our schema may be useful for query
optimization. Schemas can be classified according to their geometric
structure. It may be that in performing many queries, a database management
system learns that some geometric structures are being used more often than
others. The patterns which emerge may be only visible when one uses schemas
that have this higher dimensional geometric nature.
#### 6.5.2. Functional dependencies and normal forms
In this paper we have not discussed functional dependencies or normal forms.
It is appealing to ask the following question:
###### Question 6.5.1.
Let $X\in{\bf Sch}$ denote a schema; it should be thought of as having a shape
(again, via the topological realization functor ${\bf Sch}\rightarrow{\bf
Top}$), namely a union of tetrahedra. We wonder:
1. (1)
Given a set of functional dependencies, is there a natural way to annotate the
shape $X$ so that these dependencies are made visual?
2. (2)
Given a schema $X$ that has been annotated in this way, can one easily
determine whether it is in a certain normal form?
3. (3)
If an annotated schema is not in normal form, do the annotations help in
finding the normalization?
If the answer to these questions is affirmative, we will have more evidence
that the geometric nature of our schema is useful for database design and
management.
We hope to address these questions in the near future.
#### 6.5.3. Database integration
We believe that having a rigorous definition for morphisms of databases (see
Definition 4.3.7) will be of use in the problem of database integration. The
morphisms of databases can account for simultaneous changes in schema and in
data. It is also easy to allow changes in data types as well, a topic we will
address in later work.
Also, as mentioned in Remark 4.3.4 and Section 5.3, the use of internal keys
should prove immensely valuable. Instead of including an arbitrarily chosen
identifier for an object as part of the data for that object, as required in
the theory of relational databases, our theory keeps these arbitrary
identifiers separate. When attempting to integrate databases, it is imperative
that one know which sections of the data are observed and invariant properties
of the objects being classified, and which sections of the data are
arbitrarily assigned for management reasons. Our theory keeps these sections
of the data distinct, by use of a sheaf of keys $\mathcal{K}$ that is not
considered part of the data.
In future research, we hope to show that database integration is made
substantially easier when one works with a rigorous and geometric model like
the one we present here. Before we do so, we need to explain how to work with
a change in type specifications, which is not hard, and how to deal with
constraints in the data. See Section 6.5.5 for our plans in this direction.
#### 6.5.4. Ontologies and networks
One intuitively knows that there is a connection between databases and
ontologies. An ontology is meant for organizing knowledge, a database is meant
for organizing information, and there is a strong correlation between the two.
In order to make this correlation precise, one must first find precise
definitions of ontologies and databases. Further, these definitions should be
phrased in the same language so that they can be compared. Category theory was
invented for the purposes of comparing different mathematical structures, and
as such provides a good setting for this project.
Our plan (see [Spi09])) for a categorical definition of communication networks
involves annotating the simplices of a simplicial set with databases. That is,
each node in a network has access to a database of “what it knows,” and
connections between nodes allows communication via a common language and set
of shared knowledge. In order to make this precise, we need a precise
definition for a category of databases, for which Definition 4.3.7 suffices.
#### 6.5.5. More exotic types
Throughout this paper, we have fixed a type specification $\pi\colon
U\rightarrow{\bf DT}$, where ${\bf DT}$ is a set of data types, and $U$ is the
disjoint union of the corresponding domains. This allows for types like
strings, characters, dates, integers, etc. It also allows for more general
types like “functions from $A$ to $B$” or “probability distributions on a
space.”
However, as flexible as our type specifications may be, the situation can be
generalized considerably by allowing $\pi$ to be a functor between categories,
rather than a function between sets. The simplest application is one that is
already implicitly used, namely sorting data. The set of strings is in fact an
ordered set, and so can be represented as a category (with a morphism from $A$
to $B$ if $B$ is lexicographically larger than $A$). Another application comes
from putting constraints in the data, like if we only allow (city, state)
pairs for which the city is within the state.
By generalizing type specifications to include categories rather than sets, we
open up many new possibilities for making sense of data. Causal relationships
can be represented, as can processes. In short, morphisms make the theory more
dynamic.
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* [PS95] Frank Piessens and Eric Steegmans, _Categorical data-specifications_ , Theory Appl. Categ. 1 (1995), No. 8, 156–173 (electronic). MR MR1356700 (97b:18001)
* [RW92] Robert Rosebrugh and R. J. Wood, _Relational databases and indexed categories_ , Category theory 1991 (Montreal, PQ, 1991), CMS Conf. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 1992, pp. 391–407. MR MR1192160 (93i:68054)
* [Spi08] David Spivak, _Geometric databases_ , Algebraic Topological Methods in Computer Science, application pending, 2008.
* [Spi09] by same author, _Geometric networks: A higher-dimensional approach to networks and databases._ , Technical Proposal for ONR grant, available at
http://www.uoregon.edu/$\sim$dspivak/technical.pdf, 2009.
|
arxiv-papers
| 2009-04-13T21:20:57 |
2024-09-04T02:49:01.836186
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David I. Spivak",
"submitter": "David Spivak",
"url": "https://arxiv.org/abs/0904.2012"
}
|
0904.2047
|
# $Z^{\prime}$ Boson Mixings with $Z\\!-\\!\gamma$ and Charge Assignments
Ying Zhang1, Qing Wang2,3111Corresponding author at: Department of Physics,
Tsinghua University, Beijing 100084, P.R.China
Email address: wangq@mail.tsinghua.edu.cn (Q.Wang). 1School of Science, Xi’an
Jiaotong University, Xi’an, 710049, P.R.China
2Center for High Energy Physics, Tsinghua University, Beijing 100084,
P.R.china
3Department of Physics,Tsinghua University,Beijing 100084,P.R.China
(May 17, 2009)
###### Abstract
Based on the general description for $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixing
as derived from the electroweak chiral Lagrangian, we characterize and
classify the various new physics models involving the $Z^{\prime}$ boson that
have appeared in the literature into five classes: 1. Models with minimal
$Z^{\prime}\\!-\\!Z$ mass mixing; 2. Models with minimal $Z^{\prime}\\!-\\!Z$
kinetic mixing; 3\. Models with general $Z^{\prime}\\!-\\!Z$ mixing; 4. Models
with $Z^{\prime}\\!-\\!\gamma$ kinetic and $Z^{\prime}\\!-\\!Z$ mixing; and 5.
Models with Stueckelberg-type mixing. The corresponding mixing matrices are
explicitly evaluated for each of these classes. We constrain and classify the
$Z^{\prime}$ boson charges with respect to quark-leptons by anomaly
cancellation conditions.
PACS numbers: 12.60.Cn; 14.70.Pw; 11.30.Ly; 12.39.Fe
## I Introduction
With the running of the LHC at CERN Geneva, a TeV energy era begins and
researchers are anxiously expecting a possible new revolution in particle
physics. There are various predictions from both the Standard Model (SM) and
new physics models. Among these the appearance of possible new underlying
interactions beyond conventional strong/weak/electromagnetic gauge
interactions is of special interest. From knowledge accumulated in resent
years in particle physics, we know that the expected new interactions at least
must govern the electroweak symmetry breaking that result in the massive
$W^{\pm}$ and $Z^{0}$ bosons and may further be responsible for the origin of
masses for ordinary quarks and leptons. Theoreticians have also touted various
ambitious alternative sources of these new interactions, such as unifications,
supersymmetries, and extra dimensions. With the exception of the well-known
scalar-type interactions which suffer unnaturalness, triviality and hierarchy
problems, the typical new interaction that avoids the shortcomings of
elementary scalar fields is a gauge interaction and minimal such kind of
interaction involves an additional so-called $U(1)^{\prime}$ gauge
interaction. In most instances this extra $U(1)^{\prime}$ gauge force is a
”relic” of some larger underlying new physics gauge interactions such as those
occurring in GUT models, string theories, left-right symmetric models and
models deconstructed from extra space dimensions. Alternatively, in some
special models, the $U(1)^{\prime}$ gauge force takes on a special role: for
example 1) in little Higgs type models, it can partially remove the quadratic
divergence from the SM Higgs mass at the one loop levelRizzoARXIV2006 ; 2) in
topcolor-assisted technicolor (TC2) models, it ensures top quark condensation
while not for the bottom quark Hill ; Lane ; Chiv ; 3) in SUSY models, it can
mediate SUSY breakingZ'SUSY ; and 4) in models based on string theory, it
mediates particles communicating between the hidden and visible sectors
CasselARXIV2009 . This represents but a sampling of new physics models
involving additional $U(1)^{\prime}$ factors: a recent review of others can be
found in Ref.Langacker .
Phenomenologically, we are interested in the possibility of experimentally
finding the carrier, an electrically-neutral color singlet spin-one boson
$Z^{\prime}$, of this additional gauge force especially at the LHC. As a
detection has not been made so far, this boson has to be massive and the
corresponding $U(1)^{\prime}$ gauge symmetry must be violated. The more
preferred and exciting experimental finding would be that the $Z^{\prime}$
mass is relatively light compared with the other new physics particles, for
then it might arise as a first signature of the new physics beyond SM at the
LHC. This prospect heightens the need for theoretical studies of such a light
$Z^{\prime}$ boson and its interactions with known particles would also be of
the special importance in new physics research.
Physically, one main effect of the $Z^{\prime}$ boson derives from its mixings
with conventional $Z$ boson and $\gamma$ photon; another stems from its gauge
couplings to ordinary quarks and leptons, which leads to various charge
assignments. There exist a diversity of new physics models involving the
$Z^{\prime}$ boson, each model has its own arrangement of
$Z^{\prime}-Z-\gamma$ mixings and $Z^{\prime}$ coupling to ordinary quarks and
leptons. To compare models, a model independent investigation is needed of
these Z’ boson interactions with known particles, particularly in classifying
and comparing the role of the Z’ boson within each model. The electroweak
chiral Lagrangian (EWCL) method provides such a platform to perform model
independent research. In our previous paper Z'our , we have written down the
bosonic part up to order $p^{4}$ of the most genral EWCL involving the
$Z^{\prime}$ boson222In the Lagrangian, terms involving a neutral Higgs boson
that only plays a passive role are also included to help in matching unitarity
requirements within the theory. and known particles. This EWCL alos describes
the most general $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings, and with it we can
further classify the various $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings that
appear in each model enabling us to compare and discriminate between the
different new physics models333It should be emphasized that a $p^{4}$ order
EWCL provides some special degrees of freedom for the
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings. For example, all kinetic mixings
are from $p^{4}$ order terms in EWCL (see Eq.(II)), as a $p^{2}$ order EWCL
only cannot offer the most general $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings.
. Here the classification categorizes the general
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings into several simplifying cases that
appear in the new physics models in the literature. The reason in doing this
is because the general $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings is too
complex to be discussed analytically, while we will show that for all
simplifying cases presented in this paper, mixings can be diagonalized
exactly. This improves on the approximate diagonalization result usually used
in the literature and we can exhibit explicitly the relationship between the
various simplifying cases. The main purpose of this paper is to present these
finding s and moreover to generalize the EWCL given in Ref.Z'our to include
the $Z^{\prime}$ boson couplings to ordinary quarks and leptons for the most
general charge assignments. In terms of these charges, new physics models
involving the $Z^{\prime}$ boson can also be classified. Because most of the
experimental searches for the $Z^{\prime}$ boson depend heavily on these
charge assignments and on how $Z^{\prime}$ mixes with $Z$ and $\gamma$, we
combine a discussions on these two issues in present paper.
This paper is organized as follows. In Sec.II, we first give a short review of
the bosonic part of the EWCL involving the $Z^{\prime}$ boson and general
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings. In Sec.III, we classify the various
models involving the $Z^{\prime}$ boson that have appear in the literatures
according to their arrangements of the $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$
mixings. In Sec.IV, we set up the general $Z^{\prime}$ boson charge
assignments to the ordinary quarks and leptons in terms of the anomaly
cancellation conditions. Sec.V provides a summary of the paper.
## II The Bosonic part of the EWCL involving the $Z^{\prime}$ boson and
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings
As given in Ref.Z'our , the covariant derivative in the EWCL including the
$Z^{\prime}$ boson is
$\displaystyle
D_{\mu}\hat{U}=\partial_{\mu}\hat{U}+igW_{\mu}\hat{U}-i\hat{U}\frac{\tau_{3}}{2}g^{\prime}B_{\mu}-i\hat{U}(\tilde{g}^{\prime}B_{\mu}+g^{\prime\prime}X_{\mu})I\;,$
(1)
where the two by two unitary field $\hat{U}$ represents four Goldstone boson
degrees of freedom resulting from spontaneous symmetry breaking of
$SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)^{\prime}\rightarrow U(1)_{em}$, and
$\tilde{g}^{\prime}$ is a Stueckelberg-type coupling constant associated with
which is a special kind of $U(1)$. To help in understanding this choice of
covariant derivative, we denote $SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)^{\prime}$ group elements as
$(e^{i\theta^{a}t^{a}_{L}+i\theta^{\prime}t^{\prime}},e^{i\theta t})$ for
which the Hermitian matrices $t^{a}_{L}$ ($\theta^{a}$) with $a=1,2,3$, $t$
($\theta$) and $t^{\prime}$ ($\theta^{\prime}$) are generators (group
parameters) of $SU(2)_{L}$, $U(1)_{Y}$ and an extra $U(1)^{\prime}$
respectively. The electromagnetic $U(1)_{\mathrm{em}}$ group generator has now
been generalized from its traditional expression to $t_{\mathrm{em}}\equiv
t_{L}^{3}+t+ct^{\prime}$ depending on an additional arbitrary parameter $c$.
This generator results in the $U(1)_{\mathrm{em}}$ group element
$(e^{i\theta_{\mathrm{em}}(t^{3}_{L}+ct^{\prime})},e^{i\theta_{\mathrm{em}}t})$
and we can label the representative element for the corresponding coset by
$(\hat{U},1)$. Group theory tells us that each symmetry breaking generator
corresponds to a coset which can be represented by introducing a
representative element for each coset. Denoting the representative element by
$n$, its transformation rule to $n^{\prime}$ under the action of an arbitrary
group element $g$ is then $gn=n^{\prime}h$ where $h$ is an element belonging
to the un-broken subgroup. Specifically for the above gauge group, this
transformation rule then stipulates that
$\displaystyle(e^{i\theta^{a}t_{L}^{a}+i\theta^{\prime}t^{\prime}},~{}e^{i\theta
t})(\hat{U},1)\stackrel{{\scriptstyle
gn=n^{\prime}h}}{{=====}}(\underbrace{e^{i\theta^{a}t_{L}^{a}+i\theta^{\prime}t^{\prime}}\hat{U}e^{-i\theta(t_{L}^{3}+ct^{\prime})}}_{\hat{U}^{\prime}},~{}1)\underbrace{(e^{i\theta(t_{L}^{3}+ct^{\prime})},~{}e^{i\theta
t})}_{U(1)_{\mathrm{em}}}$ (2)
which yields the following transformation rule for the Goldstone field
$\hat{U}$ under $SU(2)_{L}\otimes U(1)\otimes U(1)^{\prime}$
$\displaystyle\hat{U}^{\prime}=e^{i\theta^{a}t_{L}^{a}+i\theta^{\prime}t^{\prime}}~{}\hat{U}~{}e^{-i\theta(t_{L}^{3}+ct^{\prime})}\;.$
(3)
The choice of the Goldstone field in the two dimensional internal space
corresponds in taking the generator $t_{L}^{a}=\tau^{a}/2$, $t=t^{\prime}=1$
(Note, according to our arrangement of group elements, $t$ and $t^{\prime}$
act on different spaces, so $t=t^{\prime}=1$ will not cause confusion). With
(3) and the standard $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)^{\prime}$
transformation rule for electroweak gauge fields $W_{\mu},B_{\mu}$ and the
extra $U(1)^{\prime}$ gauge field $X_{\mu}$, we derive the action of the
covariant derivative on the Goldstone field $\hat{U}$ as:
$D_{\mu}\hat{U}=\partial_{\mu}\hat{U}+i(gW_{\mu}+g_{X}X_{\mu})\hat{U}-i\hat{U}(\frac{\tau^{3}}{2}g^{\prime}+cg^{\prime})B_{\mu}$.
Further identifying $g_{X}\equiv-g"$ and
$cg^{\prime}\equiv\tilde{g}^{\prime}$, we obtain the result given in Eq.(1).
With symmetry breaking pattern $SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)^{\prime}\rightarrow U(1)_{em}$, the Higgs mechanism ensures that the
Goldstone bosons represented by the $\hat{U}$ field will be eaten out by the
electroweak gauge bosons $W^{\pm},Z^{0}$ and $Z^{\prime}$ which then acquire
mass. Here $W_{\mu}$, $B_{\mu}$ and $X_{\mu}$ are respectively the gauge
fields of $SU(2)_{L}$, $U(1)_{Y}$ and $U(1)^{\prime}$ before mixing.
The full bosonic part of the Lagrangian up to order $p^{4}$ is
$\displaystyle\mathcal{L}_{Stueck-SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)^{\prime}\rightarrow
U(1)_{em}}=\mathcal{L}_{0}+\mathcal{L}_{2}+\mathcal{L}_{4}\;,$ (4)
with each term in the Lagrangian defined as
$\displaystyle\mathcal{L}_{0}$ $\displaystyle=$ $\displaystyle-V(h)\;,$ (5)
$\displaystyle\mathcal{L}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}h)^{2}-\frac{1}{4}f^{2}\mathrm{tr}[\hat{V}_{\mu}\hat{V}^{\mu}]+\frac{1}{4}\beta_{1}f^{2}\mathrm{tr}[T\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]+\frac{1}{4}\beta_{2}f^{2}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]$
(6)
$\displaystyle+\frac{1}{4}\beta_{3}f^{2}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}^{\mu}]+\beta_{4}f(\partial^{\mu}h)\mathrm{tr}[\hat{V}_{\mu}]\;,$
$\displaystyle\mathcal{L}_{4}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{K}+\mathcal{L}_{B}+\mathcal{L}_{H}+\mathcal{L}_{A}\;,$
(7)
where $T\equiv\hat{U}\tau_{3}\hat{U}^{\dagger}$ and
$\hat{V}_{\mu}\equiv(\hat{D}_{\mu}\hat{U})\hat{U}^{\dagger}$. Here the Higgs
field $h$ is treated as $p^{0}$ order and
$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{2}\mathrm{tr}[W_{\mu\nu}W^{\mu\nu}]-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}\;$
$\displaystyle\mathcal{L}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha_{1}gg^{\prime}B_{\mu\nu}\mathrm{tr}[TW^{\mu\nu}]+\frac{i}{2}\alpha_{2}g^{\prime}B_{\mu\nu}\mathrm{tr}[T[\hat{V}^{\mu},\hat{V}^{\nu}]]+i\alpha_{3}g\mathrm{tr}[W^{\mu\nu}[\hat{V}^{\mu},\hat{V}^{\nu}]]+\ldots$
$\displaystyle\mathcal{L}_{H}$ $\displaystyle=$
$\displaystyle(\partial_{\mu}h)\Big{\\{}\alpha_{H,1}\mathrm{tr}[T\hat{V}^{\mu}]\mathrm{tr}[\hat{V}_{\nu}\hat{V}^{\nu}]+\alpha_{H,2}\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[\hat{V}^{\mu}\hat{V}^{\nu}]+\alpha_{H,3}\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[T[\hat{V}^{\mu},\hat{V}^{\nu}]]+\ldots\Big{\\}}\;.$
All coefficients in above Lagrangian are functions of Higgs field $h$.
Detailed expressions can be found in Ref.Z'our .
Mixings among $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ come from the gauge boson mass
term $\mathcal{L}_{M}$ and kinetic term $\mathcal{L}_{K}$. In the unitary
gauge $\hat{U}=1$, they become
$\displaystyle\mathcal{L}_{M,Z^{\prime}-Z-\gamma}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{8}(1\\!-\\!2\beta_{1})(gW^{3}_{\mu}\\!-g^{\prime}B_{\mu})^{2}+\frac{f^{2}}{2}(1\\!-\\!2\beta_{3})(g^{\prime\prime}X_{\mu}\\!+\tilde{g}^{\prime}B_{\mu})^{2}$
(8)
$\displaystyle+\frac{f^{2}}{2}\beta_{2}(g^{\prime\prime}X_{\mu}+\tilde{g}^{\prime}B_{\mu})(gW^{3,\mu}-g^{\prime}B^{\mu})\;,$
$\displaystyle\mathcal{L}_{K,Z^{\prime}-Z-\gamma}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}B_{\mu\nu}B_{\mu\nu}-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}-\frac{1}{4}(1\\!-\\!\alpha_{8}g^{2})(\partial_{\mu}W^{3}_{\nu}\\!-\partial_{\nu}W^{3}_{\mu})^{2}$
$\displaystyle+\frac{1}{2}\alpha_{1}gg^{\prime}B_{\mu\nu}(\partial_{\mu}W^{3}_{\nu}\\!-\partial_{\nu}W^{3}_{\mu})+gg^{\prime\prime}\alpha_{24}X^{\mu\nu}(\partial_{\mu}W^{3}_{\nu}-\partial_{\nu}W^{3}_{\mu})+g^{\prime}g^{\prime\prime}\alpha_{25}B_{\mu\nu}X^{\mu\nu}\;.$
Apart from the four gauge couplings
$g,g^{\prime},g^{\prime\prime},\tilde{g}^{\prime}$, seven extra dimensionless
parameters $\beta_{1},\beta_{2},\beta_{3}$ and
$\alpha_{1},\alpha_{8},\alpha_{24},\alpha_{25}$ determine the mixing terms. Of
these eleven, $\alpha_{8}$ can be absorbed into the redefinition of field
$W^{3}_{\mu}$ and coupling constant $g$ by
$\displaystyle
W^{3}_{\mu}\rightarrow\frac{W^{3}_{\mu}}{\sqrt{1-\alpha_{8}g^{2}}}\hskip
56.9055ptg\rightarrow g\sqrt{1-\alpha_{8}g^{2}}\;.$ (10)
Hence we are left with ten parameters, and on eliminating the three gauge
couplings $g,g^{\prime},g^{\prime\prime}$, leaves us seven independent
parameters
$\tilde{g}^{\prime},\beta_{1},\beta_{2},\beta_{3},\alpha_{1},\alpha_{24},\alpha_{25}$
that are related to mixings. However, the mixing masses and kinetic terms
given by (8) and (II) are so complex that to diagonalize them we must exploit
the general $3\times 3$ rotation matrix $U_{ij}$
$\displaystyle(W^{3}_{\mu},~{}B_{\mu},~{}X_{\mu})^{T}=U(Z_{\mu},~{}A_{\mu},~{}Z^{\prime}_{\mu})^{T}\;,$
(11)
which has nine matrix elements. The fact that no correction terms arise for
the kinetic terms $-\frac{1}{4}B_{\mu\nu}B_{\mu\nu}$ and
$-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}$ leads to two constraints on the matrix
elements of $U$,
$\displaystyle(U^{-1,T}U^{-1})_{22}=(U^{-1,T}U^{-1})_{33}=1\;,$ (12)
which imply that there are only seven independent matrix elements. This is
consistent with the earlier result that there are at most seven parameters
$\tilde{g}^{\prime},\beta_{1},\beta_{2},\beta_{3},\alpha_{1},\alpha_{24},\alpha_{25}$
related to mixings. In Ref.Z'our , we had obtained a set of relations between
matrix elements $U_{ij}$ and parameters
$g,g^{\prime},g^{\prime\prime},\tilde{g}^{\prime}$,
$\beta_{1},\beta_{2},\beta_{3}$,
$\alpha_{1},\alpha_{8},\alpha_{24},\alpha_{25}$ as follows
$\displaystyle
U\equiv\left(\begin{array}[]{ccc}\frac{1}{2g}c_{\alpha}&\frac{1}{2g}&-\frac{1}{2g}s_{\alpha}\\\
-\frac{1}{2g^{\prime}}c_{\alpha}&\frac{1}{2g^{\prime}}&\frac{1}{2g^{\prime}}s_{\alpha}\\\
\frac{1}{g^{\prime\prime}}(s_{\alpha}+\frac{\tilde{g}^{\prime}}{2g^{\prime}}c_{\alpha})&-\frac{\tilde{g}^{\prime}}{2g^{\prime\prime}g^{\prime}}&\frac{1}{g^{\prime\prime}}(c_{\alpha}-\frac{\tilde{g}^{\prime}}{2g^{\prime}}s_{\alpha})\end{array}\right)\left(\begin{array}[]{ccc}\frac{c_{\beta}}{A_{1}}&0&\frac{s_{\beta}}{A_{1}}\\\
ga&gb&gc\\\
-\frac{s_{\beta}}{A_{2}}&0&\frac{c_{\beta}}{A_{2}}\end{array}\right)\left(\begin{array}[]{ccc}\frac{M_{Z}}{f}&0&0\\\
0&1&0\\\ 0&0&\frac{M_{Z^{\prime}}}{f}\end{array}\right)\;,~{}~{}~{}~{}~{}$
(22)
where $c_{\alpha}\equiv\cos\alpha_{Z^{\prime}}$,
$s_{\alpha}\equiv\sin\alpha_{Z^{\prime}}$, $s_{\beta}=\sin\beta_{Z^{\prime}}$,
$c_{\beta}=\cos\beta_{Z^{\prime}}$ as well as the following definitions
$\displaystyle
A_{1}^{2}=\frac{1}{4}(1\\!-\\!2\beta_{1})c_{\alpha}^{2}+\beta_{2}s_{\alpha}c_{\alpha}+(1\\!-\\!2\beta_{3})s_{\alpha}^{2}\hskip
28.45274ptA_{2}^{2}=\frac{1}{4}(1\\!-\\!2\beta_{1})s_{\alpha}^{2}-\beta_{2}s_{\alpha}c_{\alpha}+(1-2\beta_{3})c_{\alpha}^{2}\;,~{}~{}~{}$
(23)
$\displaystyle\tan\alpha_{Z^{\prime}}=\frac{3+2\beta_{1}-8\beta_{3}-\sqrt{(3+2\beta_{1}-8\beta_{3})^{2}+16\beta_{2}^{2}}}{4\beta_{2}}\hskip
28.45274pt\tan\beta_{Z^{\prime}}=\frac{-G_{2}+\sqrt{G_{2}^{2}+4G_{0}^{2}}}{2G_{0}}~{}~{}~{}~{}~{}~{}$
(24) $\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{gA_{1}A_{2}[{g^{\prime}}^{2}{g^{\prime\prime}}^{2}-{g}^{2}{g^{\prime}}^{2}{g^{\prime\prime}}^{2}(2\alpha_{1}+\alpha_{8})+g^{2}{g^{\prime\prime}}^{2}-4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}(\alpha_{24}+\alpha_{25})+g^{2}\tilde{g}^{\prime
2}]}$
$\displaystyle\times\Big{\\{}[g^{2}{g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}-g^{\prime 2}{g^{\prime\prime}}^{2}+g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}\alpha_{8}+4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}\alpha_{25}](s_{\alpha}s_{\beta}A_{1}+c_{\alpha}c_{\beta}A_{2})$
$\displaystyle+[2g^{2}g^{\prime}\tilde{g}^{\prime}+4g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}(\alpha_{24}+\alpha_{25})](-c_{\alpha}s_{\beta}A_{1}+s_{\alpha}c_{\beta}A_{2})\Big{\\}}\;.$
$\displaystyle b^{2}$ $\displaystyle=$
$\displaystyle\frac{4{g^{\prime}}^{2}{g^{\prime\prime}}^{2}}{(g^{2}+{g^{\prime}}^{2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}-{g}^{2}{g^{\prime}}^{2}{g^{\prime\prime}}^{2}(2\alpha_{1}+\alpha_{8})+4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}(\alpha_{24}+\alpha_{25})}\;.$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{1}{gA_{1}A_{2}[{g^{\prime}}^{2}{g^{\prime\prime}}^{2}-{g}^{2}{g^{\prime}}^{2}{g^{\prime\prime}}^{2}(2\alpha_{1}+\alpha_{8})+g^{2}{g^{\prime\prime}}^{2}-4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}(\alpha_{24}+\alpha_{25})+g^{2}\tilde{g}^{\prime
2}]}$
$\displaystyle\times\Big{\\{}[g^{2}{g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}-g^{\prime 2}{g^{\prime\prime}}^{2}+g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}\alpha_{8}+4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}\alpha_{25}](-s_{\alpha}c_{\beta}A_{1}+c_{\alpha}s_{\beta}A_{2})$
$\displaystyle+[2g^{2}g^{\prime}\tilde{g}^{\prime}+4g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}(\alpha_{24}+\alpha_{25})](c_{\alpha}c_{\beta}A_{1}+s_{\alpha}s_{\beta}A_{2})\Big{\\}}\;.~{}~{}~{}~{}$
$\displaystyle G_{0}$ $\displaystyle=$ $\displaystyle-
A_{1}A_{2}\Big{\\{}(-g^{2}-g^{\prime
2}+{g^{\prime\prime}}^{2}+(\tilde{g}^{\prime})^{2})c_{\alpha}s_{\alpha}+g^{\prime}\tilde{g}^{\prime}(s_{\alpha}^{2}-c_{\alpha}^{2})+g^{2}[2g^{\prime
2}c_{\alpha}s_{\alpha}+g^{\prime}\tilde{g}^{\prime}(c_{\alpha}^{2}-s_{\alpha}^{2})]\alpha_{1}$
$\displaystyle+g^{2}[(g^{\prime
2}-{g^{\prime\prime}}^{2}-(\tilde{g}^{\prime})^{2})c_{\alpha}s_{\alpha}-g^{\prime}\tilde{g}^{\prime}(s_{\alpha}^{2}-c_{\alpha}^{2})]\alpha_{8}+2g^{2}{g^{\prime\prime}}^{2}(c_{\alpha}^{2}-s_{\alpha}^{2})(\alpha_{24}+g^{\prime
2}\alpha_{1}\alpha_{25})$
$\displaystyle+{g^{\prime\prime}}^{2}[-4g^{\prime}\tilde{g}^{\prime}c_{\alpha}s_{\alpha}+2g^{\prime
2}(c_{\alpha}^{2}-s_{\alpha}^{2})][g^{2}(\alpha_{8}\alpha_{25}-\alpha_{1}\alpha_{24})-\alpha_{25}]+g^{2}{g^{\prime\prime}}^{2}[8g^{\prime
2}s_{\alpha}c_{\alpha}$
$\displaystyle+4g^{\prime}\tilde{g}^{\prime}(c_{\alpha}^{2}-s_{\alpha}^{2})]\alpha_{24}\alpha_{25}+g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}s_{\alpha}c_{\alpha}(4\alpha_{25}^{2}-\alpha_{1}^{2})+4g^{2}{g^{\prime\prime}}^{2}(g^{\prime}s_{\alpha}+\tilde{g}^{\prime}c_{\alpha})(g^{\prime}c_{\alpha}-\tilde{g}^{\prime}s_{\alpha})\alpha_{24}^{2}\Big{\\}}$
$\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle
A_{1}^{2}\Big{\\{}(g^{2}+g^{\prime
2})c_{\alpha}^{2}+({g^{\prime\prime}}^{2}+(\tilde{g}^{\prime})^{2})s_{\alpha}^{2}(1-g^{2}\alpha_{8})-g^{2}g^{\prime
2}c_{\alpha}^{2}(2\alpha_{1}+\alpha_{8})+4g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}s_{\alpha}^{2}\alpha_{25}$
(25) $\displaystyle-4g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}c_{\alpha}^{2}(\alpha_{24}^{2}+\alpha_{25}^{2}+2\alpha_{24}\alpha_{25})-g^{2}{g^{\prime\prime}}^{2}s_{\alpha}^{2}[g^{\prime
2}\alpha_{1}^{2}+4(\tilde{g}^{\prime})^{2}\alpha_{24}^{2}+4g^{\prime}\tilde{g}^{\prime}(\alpha_{8}\alpha_{25}-\alpha_{1}\alpha_{24})]\Big{\\}}$
$\displaystyle-[A_{1}\rightarrow A_{2},c_{\alpha}\leftrightarrow
s_{\alpha}]+s_{\alpha}c_{\alpha}(A_{1}^{2}+A_{2}^{2})\Big{\\{}-2g^{\prime}\tilde{g}^{\prime}[1-g^{2}(\alpha_{1}+\alpha_{8})]$
$\displaystyle+4g^{2}{g^{\prime\prime}}^{2}[(\alpha_{24}-\alpha_{25})(1-{g^{\prime\prime}}^{2}\alpha_{1})+2g^{\prime}\tilde{g}^{\prime}\alpha_{24}^{2}+{g^{\prime\prime}}^{2}\alpha_{8}\alpha_{25}]\Big{\\}}\;.$
Finally the masses of $Z$ and $Z^{\prime}$ bosons are determined from
$\displaystyle\mathbf{K}_{11}=-\frac{1}{4}\hskip
28.45274pt\mathbf{K}_{33}=-\frac{1}{4}\;,$ (26)
with
$\displaystyle\mathbf{K}\equiv
U^{T}\left(\begin{array}[]{ccc}-\frac{1}{4}(1-\alpha_{8}g^{2})&\frac{1}{4}\alpha_{1}gg^{\prime}&\frac{1}{2}gg^{\prime\prime}\alpha_{24}\\\
\frac{1}{4}\alpha_{1}gg^{\prime}&-\frac{1}{4}&\frac{1}{2}g^{\prime}g^{\prime\prime}\alpha_{25}\\\
\frac{1}{2}gg^{\prime\prime}\alpha_{24}&\frac{1}{2}g^{\prime}g^{\prime\prime}\alpha_{25}&-\frac{1}{4}\\\
\end{array}\right)U\;.~{}~{}~{}~{}~{}~{}~{}~{}$ (30)
General expressions for the mixing matrix elements $U_{ij}$ are too
complicated to be written analytically. In Ref.Z'our , we listed results for
$U_{ij}$, $M_{Z}$ and $M_{Z^{\prime}}$ expanded up to order $p^{4}$ and linear
in $\tilde{g}^{\prime}$. In real new physics models appearing in the
literature, the $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings are often not so
complex. In the next section, we identify and discuss typical
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings appearing in various new physics
models.
## III Classification of models in terms of their
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings
In this section, we organize the various new physics models that can be found
in the literature involving the $Z^{\prime}$ boson according to their
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings. Unlike the most general case
reviewed in the last section, these mixings are special
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings for which the mixing matrix elements
$U_{ij}$ and $M_{Z}$, $M_{Z^{\prime}}$ can all be work out exactly. Below we
consider five situations.
1. 1.
Minimal $Z^{\prime}\\!-\\!Z$ mass mixing RizzoARXIV2006 ; FranziniPRD1987 ;
RizzoPRD1991 ; LangackerPRD1992 ; ChiappettaPRD1996 ; FramptonPRD1996 ;
ErlerPRL2000 ; AnokaNPB2004 ; KozlovPRD2005 ; BassoARXIV2008 ;
ChanowitzARXIV2008 ; AppelquistPRD2003 ; FerrogliaAP2007 ; CarenaPRD2004 :
This kind of model provides minimal mixing by ignoring all mixings in the
kinetic terms and $Z\\!-\\!\gamma$, $Z^{\prime}\\!-\\!\gamma$ mixings in the
mass terms. They correspond to the vanishing five parameters
$\displaystyle\tilde{g}^{\prime}=\alpha_{1}=\alpha_{8}=\alpha_{24}=\alpha_{25}=0\;.$
(31)
With the exception of gauge couplings $g,g^{\prime},g^{\prime\prime}$, the
remaining three nontrivial parameters are denoted by the $Z^{\prime}\\!-\\!Z$
mass matrix
$\displaystyle\mathcal{M}^{2}=\left(\begin{array}[]{cc}M_{Z_{0}}^{2}&M_{ZZ^{\prime}}^{2}\\\
M_{ZZ^{\prime}}^{2}&M_{Z^{\prime}_{0}}^{2}\end{array}\right)\hskip
28.45274ptZ_{0}^{\mu}\equiv\frac{gW^{3}_{\mu}\\!-g^{\prime}B_{\mu}}{\sqrt{g^{2}\\!+g^{\prime
2}}}\hskip
14.22636ptA_{0}^{\mu}\equiv\frac{g^{\prime}W^{3}_{\mu}\\!+gB_{\mu}}{\sqrt{g^{2}\\!+g^{\prime
2}}}\hskip 14.22636ptZ_{0}^{\prime\mu}\equiv X^{\mu},~{}~{}~{}~{}$ (34)
where mass parameters $M_{Z_{0}}^{2}$, $M_{Z^{\prime}_{0}}^{2}$ and
$M_{ZZ^{\prime}}^{2}$ are related to $\beta_{1},\beta_{2},\beta_{3}$ as
$\displaystyle\frac{f^{2}}{4}(1\\!-\\!2\beta_{1})(g^{2}\\!+\\!g^{\prime
2})\equiv M_{Z_{0}}^{2}\hskip
17.07182ptf^{2}(1\\!-\\!2\beta_{3})g^{\prime\prime 2}\equiv
M_{Z_{0}^{\prime}}^{2}\hskip
17.07182pt\frac{f^{2}}{2}\beta_{2}g^{\prime\prime}\sqrt{g^{2}\\!+\\!g^{\prime
2}}\equiv M_{ZZ^{\prime}}^{2}\;.~{}~{}~{}~{}~{}$ (35)
Refs.AppelquistPRD2003 ; FerrogliaAP2007 use an alternative expression which
corresponds to setting
$\displaystyle f=v_{H}\hskip 14.22636ptg^{\prime}=g_{Y}\hskip
14.22636ptg^{\prime\prime}=g_{z}\hskip 14.22636pt\beta_{1}=0\hskip
14.22636pt\beta_{2}=-\frac{1}{2}z_{H}\hskip
14.22636pt1-2\beta_{3}=\frac{1}{4}(z_{H}^{2}+\frac{v_{\phi}^{2}}{f^{2}})\;.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
Ref.CarenaPRD2004 further generalizes this which leads then to
$\displaystyle g^{\prime}\\!=g_{Y}\hskip
11.38092ptg^{\prime\prime}\\!\\!=g_{z}\hskip
11.38092pt1-2\beta_{1}=\frac{v_{H_{1}}^{2}+v_{H_{2}}^{2}}{f^{2}}~{}~{}~{}\beta_{2}\\!=-\frac{z_{H_{1}}v_{H_{1}}^{2}\\!\\!+\\!z_{H_{2}}v_{H_{2}}^{2}}{2f^{2}}\hskip
11.38092pt$ $\displaystyle
1\\!-\\!2\beta_{3}\\!=\frac{1}{4f^{2}}(z_{H_{1}}^{2}v_{H_{1}}^{2}\\!\\!+\\!z_{H_{2}}^{2}v_{H_{2}}^{2}\\!\\!+\\!v_{\phi}^{2})\;.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
In this kind of model, the key $Z^{\prime}\\!-\\!Z$ mixing parameter is
$\beta_{2}$ which yields a non-vanishing off-diagonal element
$M_{ZZ^{\prime}}^{2}$ in the $Z^{\prime}\\!-\\!Z$ mass matrix. This element
further generates the seesaw splitting between the original $Z$ and
$Z^{\prime}$ masses,
$\displaystyle M_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[M_{Z_{0}}^{2}\\!+M_{Z_{0}^{\prime}}^{2}\\!-\sqrt{(M_{Z_{0}}^{2}\\!-M_{Z_{0}^{\prime}}^{2})^{2}\\!+4M_{ZZ^{\prime}}^{4}}]\approx
M_{Z_{0}}^{2}-\frac{M_{ZZ^{\prime}}^{4}}{M_{Z_{0}^{\prime}}^{2}\\!-M_{Z_{0}}^{2}}$
(36) $\displaystyle M_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[M_{Z_{0}}^{2}\\!+M_{Z_{0}^{\prime}}^{2}\\!+\sqrt{(M_{Z_{0}}^{2}\\!-M_{Z_{0}^{\prime}}^{2})^{2}\\!+4M_{ZZ^{\prime}}^{4}}]\approx
M_{Z_{0}^{\prime}}^{2}+\frac{M_{ZZ^{\prime}}^{4}}{M_{Z_{0}^{\prime}}^{2}\\!-M_{Z_{0}}^{2}}\;.~{}~{}~{}~{}$
(37)
Meanwhile the $Z^{\prime}\\!-\\!Z$ mixing can be parameterized by mixing angle
$\theta^{\prime}$
$\displaystyle\left(\begin{array}[]{c}Z_{0}^{\mu}\\\
Z_{0}^{\prime\mu}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta^{\prime}&\sin\theta^{\prime}\\\
-\sin\theta^{\prime}&\cos\theta^{\prime}\end{array}\right)\left(\begin{array}[]{c}Z^{\mu}\\\
Z^{\prime\mu}\end{array}\right)\hskip 28.45274pt\tan
2\theta^{\prime}=\frac{2M_{ZZ^{\prime}}^{2}}{M_{Z_{0}^{\prime}}^{2}\\!-M_{Z_{0}}^{2}}\;.~{}~{}~{}~{}$
(44)
leading to a rotation matrix introduced in (11) of the form
$\displaystyle U_{\mbox{\tiny Minimal $Z^{\prime}\\!\\!\\!-\\!\\!Z$ mass
mixing}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cos\theta_{W}&\sin\theta_{W}&0\\\
-\sin\theta_{W}&\cos\theta_{W}&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{ccc}\cos\theta^{\prime}&0&\sin\theta^{\prime}\\\
0&1&0\\\ -\sin\theta^{\prime}&0&\cos\theta^{\prime}\end{array}\right)$ (51)
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cos\theta_{W}\cos\theta^{\prime}&\sin\theta_{W}&\cos\theta_{W}\sin\theta^{\prime}\\\
-\sin\theta_{W}\cos\theta&\cos\theta_{W}&-\sin\theta_{W}\sin\theta^{\prime}\\\
-\sin\theta^{\prime}&0&\cos\theta^{\prime}\end{array}\right)\;,$ (55)
with an electroweak mixing angle $\tan\theta_{W}=g^{\prime}/g$.
2. 2.
Minimal $Z^{\prime}\\!-\\!Z$ kinetic mixing RizzoARXIV1998 ; LangackerPRD2008
; LangackerPRL2008 :
This kind of model provides minimal mixing by ignoring all mixings in the mass
terms and $Z\\!-\\!\gamma$, $Z^{\prime}\\!-\\!\gamma$ mixings in the kinetic
terms leading to the vanishing of seven parameters
$\displaystyle\tilde{g}^{\prime}=\beta_{1}=\beta_{2}=\beta_{3}=\alpha_{1}=\alpha_{8}=\alpha_{24}=0\;.$
(56)
Again with the exception of gauge couplings $g,g^{\prime},g^{\prime\prime}$,
the one remaining nontrivial parameter is denoted by
$\displaystyle
g^{\prime}g^{\prime\prime}\alpha_{25}\equiv-\frac{\sin\chi}{2}\;.$ (57)
following Ref.RizzoARXIV1998 , we redefine the gauge fields as
$\displaystyle B^{\mu}=B_{0}^{\mu}-\tan\chi Z^{\prime\mu}_{0}\hskip
56.9055ptX^{\mu}=\frac{Z^{\prime\mu}_{0}}{\cos\chi}\;,$ (58)
in terms of the fields $B_{0}^{\mu},Z_{0}^{\prime\mu},W^{3\mu}$, the kinetic
term appears diagonalized and the model reduces to a minimal $Z^{\prime}-Z$
mass mixing model discussed above444This detail was not pointed out in
Ref.RizzoARXIV1998 . with
$\displaystyle M_{Z_{0}}^{2}=\frac{f^{2}}{4}(g^{2}\\!+\\!g^{\prime 2})\hskip
17.07182ptM_{Z_{0}^{\prime}}^{2}=\frac{f^{2}[g^{\prime
2}\sin^{2}\chi\\!+\\!4g^{\prime\prime 2}]}{4\cos^{2}\chi}\hskip
17.07182ptM_{ZZ^{\prime}}^{2}=\frac{f^{2}}{4}g^{\prime}\sqrt{g^{2}\\!+\\!g^{\prime
2}}\tan\chi\;.~{}~{}~{}~{}~{}$ (59)
The rotation matrix introduced in (11) takes the form
$\displaystyle U_{\mbox{\tiny Minimal $Z^{\prime}\\!\\!\\!-\\!\\!Z$ kinetic
mixing}}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&-\tan\chi\\\
0&0&\frac{1}{\cos\chi}\end{array}\right)\times U_{\mbox{\tiny Minimal
$Z^{\prime}\\!\\!\\!-\\!\\!Z$ mass mixing}}$ (63)
$\displaystyle=\left(\begin{array}[]{ccc}\cos\theta^{\prime}\cos\theta_{W}&\sin\theta_{W}&\cos\theta_{W}\sin\theta^{\prime}\\\
-\sin\theta_{W}\cos\theta^{\prime}+\tan\chi\sin\theta^{\prime}&\cos\theta_{W}&-\sin\theta_{W}\sin\theta^{\prime}-\tan\chi\cos\theta^{\prime}\\\
-\sin\theta^{\prime}/\cos\chi&0&\cos\theta^{\prime}/\cos\chi\end{array}\right)\;.~{}~{}~{}~{}~{}$
(67)
3. 3.
General $Z^{\prime}\\!-\\!Z$ mixing RizzoARXIV2006 ; Hill ; Lane ; Chiv ;
CasselARXIV2009 ; Holdom1986 ; PDG2006 ; BabuPRD1996 :
This kind of model is combinations of minimal $Z^{\prime}\\!-\\!Z$ mass mixing
model and minimal $Z^{\prime}\\!-\\!Z$ kinetic mixing model discussed above
which correspond to
$\displaystyle\tilde{g}^{\prime}=\alpha_{1}=\alpha_{8}=\alpha_{24}=0\hskip
28.45274ptg^{\prime}g^{\prime\prime}\alpha_{25}\equiv-\frac{\sin\chi}{2}\;.$
(68)
In a similar manner as for minimal $Z^{\prime}\\!-\\!Z$ kinetic mixing model,
we can use (58) to remove the mixing in the kinetic term and then, in terms of
the fields $B_{0}^{\mu},Z_{0}^{\prime\mu},W^{3\mu}$, the model can be changed
into a minimal $Z^{\prime}\\!-\\!Z$ mass mixing model with identifications
$\displaystyle M_{Z_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}(1-2\beta_{1})(g^{2}\\!+g^{\prime 2})$
$\displaystyle M_{Z_{0}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}[g^{\prime
2}(1-2\beta_{1})\sin^{2}\chi+4g^{\prime\prime
2}(1-2\beta_{3})+4\beta_{2}g^{\prime}g^{\prime\prime}\sin\chi]}{4\cos^{2}\chi}$
$\displaystyle M_{ZZ^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}\frac{(1-2\beta_{1})g^{\prime}\sin\chi+2\beta_{2}g^{\prime\prime}}{\cos\chi}\sqrt{g^{2}\\!+g^{\prime
2}}\;.~{}~{}~{}~{}~{}$ (69)
The resulting rotation matrix has the same form as in (67), the only change is
that now the $\theta^{\prime}$ as determined through (44) is different due to
the new expressions for
$M_{Z_{0}}^{2},M_{Z_{0}^{\prime}}^{2},M_{ZZ^{\prime}}^{2}$ given by (69). In
some dynamical models such as TC2 models, the general $Z^{\prime}\\!-\\!Z$
mixings are generated by technicolor and topcolor dynamics, as in Refs.Hill1 ;
Lane1 ; Chiv1 , while mixing parameters are given through dynamical
computations depending on the nature of the TC2 models and results in the
following expressions
$\displaystyle g^{\prime}g^{\prime\prime}\alpha_{25}=\frac{g^{\prime
2}\gamma}{2c_{Z^{\prime}}}\hskip
56.9055pt\frac{f^{2}}{2}\beta_{2}g^{\prime\prime}=\frac{g^{\prime}}{4c_{Z^{\prime}}}\times\left\\{\begin{array}[]{lll}(F_{0}^{\mathrm{TC2}})^{2}\tan\theta^{\prime}&{}{}{}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Hill,Hill1}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
3(F_{0}^{\mathrm{1D}})^{2}\tan\theta^{\prime}&{}{}{}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Lane,Lane1}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
-3(F_{0}^{\mathrm{1D}})^{2}\cot\theta^{\prime}&{}{}{}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Chiv,Chiv1}{\@@citephrase{(}}{\@@citephrase{)}}}}\end{array}\right.\;,$
(73)
where all symbols appearing on the right-hand side of these results are
parameters pertaining to the TC2 models.
4. 4.
$Z^{\prime}\\!-\\!\gamma$ kinetic and $Z^{\prime}\\!-\\!Z$ mixing
HoldomPLB1991 :
B. Holdom extends the conventional $Z^{\prime}\\!-\\!Z$ mixing by further
adding in model a $Z^{\prime}\\!-\\!\gamma$ kinetic mixing term. His model
corresponds to having
$\displaystyle\tilde{g}^{\prime}=\alpha_{1}=\alpha_{8}=0\hskip
14.22636pt\frac{f^{2}}{4}(1\\!-\\!2\beta_{1})=m_{Z}^{2}\hskip
14.22636pt\frac{f^{2}}{2}\beta_{2}g^{\prime\prime}\sqrt{g^{2}\\!+g^{\prime
2}}=xm_{Z}^{2}\hskip 14.22636ptf^{2}(1\\!-\\!2\beta_{3})g^{\prime\prime
2}=m_{X}^{2}$ $\displaystyle gg^{\prime\prime}\sqrt{g^{2}\\!+g^{\prime
2}}\alpha_{24}=-\frac{1}{2}(gy+g^{\prime}w)\hskip
28.45274ptgg^{\prime\prime}\sqrt{g^{2}\\!+g^{\prime
2}}\alpha_{25}=\frac{1}{2}(g^{\prime}y-gw)\;.$ (74)
We can diagonalize the kinetic terms by redefining the $B^{\mu}$ and
$W^{3\mu}$ fields as
$\displaystyle
B^{\mu}=B_{0}^{\mu}-\frac{\sin\chi}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}Z_{0}^{\prime\mu}\hskip
28.45274ptW^{3\mu}=W^{3\mu}_{0}-\frac{\sin\overline{\chi}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}Z_{0}^{\prime\mu}~{}~{}~{}~{}~{}~{}~{}$
(75) $\displaystyle
X^{\mu}=\frac{Z_{0}^{\prime\mu}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\hskip
28.45274pt-\frac{\sin\overline{\chi}}{2}\equiv
g^{\prime}g^{\prime\prime}\alpha_{24}\hskip
28.45274pt-\frac{\sin\chi}{2}\equiv g^{\prime}g^{\prime\prime}\alpha_{25}$
and then in terms of fields $B_{0}^{\mu},Z_{0}^{\prime\mu},W^{3\mu}_{0}$, the
model becomes the minimal $Z^{\prime}\\!-\\!Z$ mass mixing model with
$\displaystyle M_{Z_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}(1-2\beta_{1})(g^{2}\\!+g^{\prime 2})$
$\displaystyle M_{Z_{0}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}[\frac{1}{4}(1-2\beta_{1})(g^{\prime}\sin\chi-g\sin\overline{\chi})^{2}+(1-2\beta_{3})g^{\prime\prime
2}+\beta_{2}g^{\prime\prime}(g^{\prime}\sin\chi-g\sin\overline{\chi})]}{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}$
$\displaystyle M_{ZZ^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}\frac{[(1-2\beta_{1})(g^{\prime}\sin\chi-g\sin\overline{\chi})+2\beta_{2}g^{\prime\prime}]}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\sqrt{g^{2}\\!+g^{\prime
2}}\;.~{}~{}~{}~{}~{}$ (76)
for which the rotation matrix introduced in (11) takes the form
$\displaystyle U_{\mbox{\tiny$Z^{\prime}\\!\\!\\!-\\!\\!\gamma$ kinetic and
$Z^{\prime}\\!\\!\\!-\\!\\!Z$
mixing}}=\left(\begin{array}[]{ccc}1&0&-\frac{\sin\overline{\chi}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\\\
0&1&-\frac{\sin\chi}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\\\
0&0&\frac{1}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\end{array}\right)\times
U_{\mbox{\tiny Minimal $Z^{\prime}\\!\\!\\!-\\!\\!Z$ mass mixing}}$ (80)
$\displaystyle=\left(\begin{array}[]{ccc}\frac{g\cos\theta^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}+\frac{\sin\theta^{\prime}\sin\overline{\chi}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}&~{}~{}~{}\frac{g^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}{}{}{}&\frac{g\sin\theta^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}-\frac{\cos\theta^{\prime}\sin\overline{\chi}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\\\
-\frac{g^{\prime}\cos\theta^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}+\frac{\sin\theta^{\prime}\sin\chi}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}&\frac{g}{\sqrt{g^{2}+g^{\prime
2}}}&-\frac{g^{\prime}\sin\theta^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}-\frac{\cos\theta^{\prime}\sin\chi}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\\\
-\frac{\sin\theta^{\prime}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}&0&\frac{\cos\theta^{\prime}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\end{array}\right)\;.~{}~{}~{}~{}~{}$
(84)
5. 5.
Stueckelberg-type mixing KorsJHEP2005 ; FeldmanPRL2006 ; FeldmanPRD2007 :
This kind of model provides mixing through the nonzero coupling constant
$\tilde{g}^{\prime}$ and except for gauge coupling
$g,g^{\prime},g^{\prime\prime}$, a typical choice as given in
Refs.KorsJHEP2005 ; FeldmanPRL2006 is the vanishing of all other parameters
$\displaystyle\beta_{1}=\beta_{2}=\beta_{3}=\alpha_{1}=\alpha_{8}=\alpha_{24}=\alpha_{25}=0\;,$
(85)
leading to diagonal kinetic terms and mixing occurring only in the mass terms.
After rotating the standard electroweak mixing angle $\theta_{W}$, we can
redefine the gauge fields
$\displaystyle\bar{B}^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{{g^{\prime\prime}}\sqrt{g^{2}+g^{\prime
2}}}{(g^{2}+g^{\prime 2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}B_{0}^{\mu}+\frac{g\tilde{g}^{\prime}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime 2}}{Z^{\prime}}_{0}^{\mu}$
$\displaystyle\bar{Z}^{\prime\mu}$ $\displaystyle=$
$\displaystyle\frac{g\tilde{g}^{\prime}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}B_{0}^{\mu}+\frac{{g^{\prime\prime}}\sqrt{g^{2}+g^{\prime
2}}}{(g^{2}+g^{\prime 2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}{Z^{\prime}}_{0}^{\mu}$ (86)
thereby changing the present model to a minimal $Z^{\prime}-Z$ mass mixing
model with
$\displaystyle M_{Z_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}(g^{2}+g^{\prime 2}+\frac{4g^{\prime
2}\tilde{g}^{\prime 2}}{g^{2}+g^{\prime 2}})$ $\displaystyle
M_{Z^{\prime}_{0}}^{2}$ $\displaystyle=$ $\displaystyle
f^{2}({g^{\prime\prime}}^{2}+\frac{g^{2}\tilde{g}^{\prime 2}}{g^{2}+g^{\prime
2}})$ $\displaystyle M_{ZZ^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{f^{2}g^{\prime}\tilde{g}^{\prime}\sqrt{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime 2}}}{g^{2}+g^{\prime 2}}\;.$
(87)
The overall rotation matrix then becomes
$\displaystyle U_{\mbox{\tiny Stuekckelberg type mixing}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cos\theta_{W}&\sin\theta_{W}&0\\\
-\sin\theta_{W}&\cos\theta_{W}&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&-\frac{{g^{\prime\prime}}\sqrt{g^{2}+g^{\prime 2}}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}&\frac{g\tilde{g}^{\prime}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime 2}}\\\
0&\frac{g\tilde{g}^{\prime}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}&\frac{{g^{\prime\prime}}\sqrt{g^{2}+g^{\prime 2}}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime 2}}\end{array}\right)$ (98)
$\displaystyle\times\left(\begin{array}[]{ccc}\cos\theta^{\prime}&0&\sin\theta^{\prime}\\\
0&1&0\\\ -\sin\theta^{\prime}&0&\cos\theta^{\prime}\end{array}\right)$
with $\theta^{\prime}$ evaluated from the second equation of (44) and those of
(87). In Ref.FeldmanPRD2007 , the Stueckelberg-type mixing is further
generalized to include kinetic mixing by relaxing the original condition
$\alpha_{25}=0$. This kinetic mixing can be diagonalized by applying (58) and
following a similar procedure to that leading to (86) in diagonalizing the
mass terms.
## IV the $Z^{\prime}$ boson charges to quark and leptons
The charges for the $Z^{\prime}$ boson with respect to ordinary quarks and
leptons can be expressed in terms of the gauge interaction as
$\displaystyle\mathcal{L}_{\mathrm{gauge~{}coupling}}={g^{\prime\prime}}X_{\mu}J^{\mu}_{X}\hskip
56.9055ptJ^{\mu}_{X}=\sum_{i}\bar{f}_{i}\gamma^{\mu}[y^{\prime}_{iL}P_{L}+y^{\prime}_{iR}P_{R}]f_{i}\;,$
(99)
where index $i$ distinguishes the three generations associated with the six
quarks $u,c,t,d,s,b$ and six leptons
$e,\mu,\tau,\nu_{e},\nu_{\mu},\nu_{\tau}$, and
$y_{i,L}^{\prime},y_{i,R}^{\prime}$ are the corresponding left- and right-hand
charges555Phenomenologically, we need to further express the gauge interaction
given in Eq.(99) in terms of mass eigenstate of $Z^{\prime}$, for then the
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings discussed in the last section set
in.. The $SU(2)_{L}$ symmetry requires equating $U(1)$ charges of the two
components of the left-hand fermion doublet, i.e.
$y^{\prime}_{u,L}=y^{\prime}_{d,L}\equiv y^{\prime}_{q}$ for quark and
$y^{\prime}_{\nu,L}=y^{\prime}_{e,L}\equiv y^{\prime}_{l}$ for lepton. Thus,
we can parameterize the fermionic $U(1)^{\prime}$ charges by $y^{\prime}_{q}$,
$y^{\prime}_{u}$, $y^{\prime}_{d}$, $y^{\prime}_{l}$, $y^{\prime}_{e}$ and
$y^{\prime}_{\nu}$. In general, the assignments of $U(1)^{\prime}$ charges are
generation-dependent, but in its simplest form $U(1)^{\prime}$ charges can be
generation-independent, much like hypercharge assignments in SM. TABLE.1 lists
four sets of more common assignments for the generation-independent
$U(1)^{\prime}$ charges of fermions in new physics models involving
$Z^{\prime}$ boson CarenaPRD2004 ; PDG2008 . In the $U(1)_{B-xL}$ model (see
column 3 of TABLE.1), $Z^{\prime}$ charges are determined by the baryon number
and lepton number from $y^{\prime}_{i}=B_{i}-xL_{i}$ with a free rational
parameter $x$. Leptophobic and hadrophobic $Z^{\prime}$ models correspond to
$x=\infty$ and $x=0$, respectively. The second set of charges comes from grand
unified theories. Parameter $x$ establishes the mixing of the two extra $U(1)$
groups in the $E_{6}$ symmetry breaking patterns $E_{6}\rightarrow SU(5)\times
U(1)\times U(1)$. $Z_{\chi}$, $Z_{\psi}$ and $Z_{\eta}$ of Ref.GUTs
correspond to the special case with $x=-3$, $x=1$ and $x=-1/2$, respectively.
The third set, $U(1)_{d-xu}$ results in the vanishing of the left-hand quark
doublet charge and the ratio of right-hand up quark charges to down quark
charges is controlled by $-x$. In the last set, the free parameter $x$ is the
ratio of the charges of the left-hand quark doublet and right-hand up quark
singlet and reduces to the $U(1)_{B-L}$ model for $x=1$.
Table 1: generation-independent $U(1)^{\prime}$ charges for quarks and leptons models | $Z^{\prime}$ EWCL | $U(1)_{B-xL}$ | $U(1)_{10+x\bar{5}}$ | $U(1)_{d-xu}$ | $U(1)_{q+xu}$
---|---|---|---|---|---
$(u_{L},d_{L})$ | $y^{\prime}_{q}$ | $1/3$ | $1/3$ | $0$ | $1/3$
$u_{R}$ | $y^{\prime}_{u}$ | $1/3$ | $-1/3$ | $-x/3$ | $x/3$
$d_{R}$ | $y^{\prime}_{d}$ | $1/3$ | $-x/3$ | $1/3$ | $(2-x)/3$
$(\nu_{L},e_{L})$ | $y^{\prime}_{l}$ | $-x$ | $x/3$ | $(x-1)/3$ | $-1$
$e_{R}$ | $y^{\prime}_{e}$ | $-x$ | $-1/3$ | $x/3$ | $-(2+x)/3$
$\nu_{R}$ | $y^{\prime}_{\nu_{R}}$ | $-1$ | $(x-2)/3$ | $-x/3$ | $(x-4)/3$
Theoretically, the charges of quarks and leptons must satisfy the anomaly
cancellation conditions to preserve the gauge symmetry. We now examine the
constraints on generation-independent $U(1)^{\prime}$ charges arising as a
consequence of these anomaly cancellation conditions. Davidson et.al.
DavidsonPRD1979 have studied anomaly cancellation for additional
$U(1)^{\prime}$ gauge group and derived the following anomaly cancellation
conditions for $U(1)_{Y}\otimes U(1)^{\prime}$ gauge groups
$\displaystyle\sum y^{\alpha}_{L}=\sum
Q^{2}(y^{\alpha}_{L}\\!-\\!y^{\alpha}_{R})=0\hskip 17.07182pt\sum
Q(y^{\alpha}_{L}y^{\beta}_{L}\\!-\\!y^{\alpha}_{R}y^{\beta}_{R})=0\hskip
17.07182pt\sum(y^{\alpha}_{L}y^{\beta}_{L}y^{\gamma}_{L}\\!-\\!y^{\alpha}_{R}y^{\beta}_{R}y^{\gamma}_{R})=0\;,~{}~{}~{}~{}$
(100)
where $\alpha,\beta,\gamma$ indexes $U(1)_{Y}$ and $U(1)^{\prime}$ charges.
Substituting the $U(1)_{Y}$ charges for ordinary quarks and leptons and
assuming the generation-independence of $U(1)^{\prime}$ charges, we find that
above equations imply
$\displaystyle\left\\{\begin{array}[]{l}y^{\prime}_{l}+3y^{\prime}_{q}=0\\\
3y^{\prime}_{l}+5y^{\prime}_{q}-3y^{\prime}_{e}-4y^{\prime}_{u}-y^{\prime}_{d}=0\\\
-{y^{\prime}_{l}}^{2}+{y^{\prime}_{q}}^{2}+{y^{\prime}_{e}}^{2}-2{y^{\prime}_{u}}^{2}+{y^{\prime}_{d}}^{2}=0\\\
3y^{\prime}_{l}+y^{\prime}_{q}-6y^{\prime}_{e}-8y^{\prime}_{u}-2y^{\prime}_{d}=0\\\
2{y^{\prime}_{l}}^{3}+6{y^{\prime}_{q}}^{3}-{y^{\prime}_{e}}^{3}-3{y^{\prime}_{u}}^{3}-3{y^{\prime}_{d}}^{3}-{y^{\prime}_{\nu_{R}}}^{3}=0\end{array}\right.\;.$
(106)
The last equation in (106) can be satisfied by assigning
$y^{\prime}_{\nu_{R}}$ a proper value or adding in our theory some other new
fermions. Solving the above equations, we obtain two sets of solutions which
satisfy the anomaly cancellation conditions
$\displaystyle\left\\{\begin{array}[]{l}y^{\prime}_{l}=-3y^{\prime}_{q}\\\
y^{\prime}_{d}=2y^{\prime}_{q}-y^{\prime}_{u}\\\
y^{\prime}_{e}=-2y^{\prime}_{q}-y^{\prime}_{u}\\\
y^{\prime}_{\nu_{R}}=-4y^{\prime}_{q}+y^{\prime}_{u}\end{array}\right.\hskip
56.9055pt{\rm or}\hskip
56.9055pt\left\\{\begin{array}[]{l}y^{\prime}_{l}=-3y^{\prime}_{q}\\\
y^{\prime}_{d}=-\frac{14}{5}y^{\prime}_{q}+\frac{1}{5}y^{\prime}_{u}\\\
y^{\prime}_{e}=-\frac{2}{5}y^{\prime}_{q}-\frac{7}{5}y^{\prime}_{u}\\\
y^{\prime}_{\nu_{R}}=\frac{\sqrt[3]{35}}{5}(4y^{\prime}_{q}-y^{\prime}_{u})\end{array}\right.\;.$
(115)
Of the six of $U(1)^{\prime}$ charges, only two of them $y^{\prime}_{q}$ and
$y^{\prime}_{u}$ are independent; the other four being linear combinations of
these two. In addition, there are two kinds of linear combinations: the first
of Eq.(115) which was given and discussed in detail in Ref.AppelquistPRD2003 ,
while the second is a new solution having not yet appeared in the literature.
We can utilize the values of $y^{\prime}_{q}$ and $y^{\prime}_{u}$ to classify
the new physics models and in the following we list some typical cases:
1. 1.
Left Handed:
$y^{\prime}_{u}=y^{\prime}_{d}=y^{\prime}_{e}=y^{\prime}_{\nu_{R}}=0~{}\Rightarrow~{}y^{\prime}_{q}=y^{\prime}_{l}=0$
2. 2.
Right Handed:
$y^{\prime}_{q}=y^{\prime}_{l}=0~{}\Rightarrow~{}y^{\prime}_{d}\\!=-y^{\prime}_{u}\\!=y^{\prime}_{e}\\!=-y^{\prime}_{\nu_{R}}$
or
$y^{\prime}_{d}\\!=\frac{1}{5}y^{\prime}_{u}\\!=-\frac{1}{7}y^{\prime}_{e}\\!=-\frac{1}{\sqrt[3]{35}}y^{\prime}_{\nu_{R}}$
3. 3.
Left-Right symmetric:
$y^{\prime}_{q}=y^{\prime}_{u}=y^{\prime}_{d}~{}\Rightarrow~{}y^{\prime}_{l}=y^{\prime}_{e}=y^{\prime}_{\nu_{R}}=-3y^{\prime}_{q}$
4. 4.
$\nu_{R}$ decouple:
$y^{\prime}_{\nu_{R}}=0~{}\Rightarrow~{}y^{\prime}_{u}=4y^{\prime}_{q},~{}y^{\prime}_{e}=2y^{\prime}_{l}=3y^{\prime}_{d}=-6y^{\prime}_{q}$
Checking the assignments given in TABLE.1 against the two solutions in (115),
we find that the $U(1)_{B-xL}$, $U(1)_{d-xu}$ and $U(1)_{q+xu}$ models are
anomaly-free when parameter $x=1$ with the right-hand neutrino charge
$y^{\prime}_{\nu_{R}}=-1$, $y^{\prime}_{\nu_{R}}=-\frac{1}{3}$ and
$y^{\prime}_{\nu_{R}}=-1$, respectively. Furthermore, the $U(1)_{10+x\bar{5}}$
model is anomaly-free when $x=-3$ with $y^{\prime}_{\nu_{R}}=-5/3$. Even
though the anomaly cancellation condition can not be satisfied with the
present quarks and leptons, we still have the possibility of canceling the
anomalies by adding some extra fermions into theory.
If we relax the generation-independence criterion on the $U(1)^{\prime}$
charges, we need to add generation indices to each of the charges in Eq.(106)
and sum over the generations on the left-hand side of Eq.(106). In this case,
there are too many free parameters and solutions. We list several possible
solutions in TABLE.2, in which the first and last columns are the two
solutions given in Ref.PDG2008 , and the remaining solutions can be seen to be
some kind of generation-dependent generalization of charge assignments given
in the third, fourth and fifth columns in TABLE.1. The typical feature of
these solutions is that for the solutions given in the first four columns of
TABLE.2, the charges for the first two generations are parameterized in a like
manner as those in the generation-independent situation by $x$ or $y$
separately, and differences appear only in the third generation of quarks and
leptons. Of special note is that for the solution to $U(1)_{q+xu+yc+zt}$, the
anomaly cancellation condition is satisfied for each generation independently.
Table 2: generation-dependent charge models | $U(1)_{B-xL_{e}\\!-yL_{\mu}\\!}$ | $U(1)_{10+x\bar{5}}~{}\mathrm{\tiny gen\\!\\!-\\!dep}$ | $U(1)_{d-xu}~{}\mathrm{\tiny gen\\!\\!-\\!dep}$ | $U(1)_{q+xu+yc+zt}$ | $2\\!+\\!1~{}\mathrm{\tiny leptocratic}$
---|---|---|---|---|---
$q_{1,L}$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $1/3$
$u_{R}$ | $1/3$ | $-1/3$ | $-x/3$ | $x/3$ | $x/3$
$d_{R}$ | $1/3$ | $-x/3$ | $1/3$ | $(2-x)/3$ | $(2-x)/3$
$q_{2,L}$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $1/3$
$c_{R}$ | $1/3$ | $-1/3$ | $-y/3$ | $y/3$ | $x/3$
$s_{R}$ | $1/3$ | $-y/3$ | $1/3$ | $(2-y)/3$ | $(2-x)/3$
$q_{3,L}$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $1/3$
$t_{R}$ | $1/3$ | $-1/3$ | $2\\!-\\!\frac{2}{3}(x\\!\\!+\\!y)\\!\pm\\!\\!\sqrt{3\\!-\\!x^{2}\\!\\!-\\!y^{2}}$ | $z/3$ | $x/3$
$b_{R}$ | $1/3$ | $3+\frac{x+y}{3}$ | $1/3$ | $(2-z)/3$ | $(2-x)/3$
$(\nu^{e}_{L},e_{L})$ | $-x$ | $x/3$ | $(x-1)/3$ | $-1$ | $-1-2y$
$e_{R}$ | $-x$ | $-1/3$ | $x/3$ | $-(2+x)/3$ | $-(2\\!+\\!x)/3-2y$
$(\nu^{\mu}_{L},\mu_{L})$ | $-y$ | $y/3$ | $(y-1)/3$ | $-1$ | $y-1$
$\mu_{R}$ | $-y$ | $-1/3$ | $y/3$ | $-(2+y)/3$ | $-(2\\!+\\!x)/3+y$
$(\nu^{\tau}_{L},\tau_{L})$ | $x+y-3$ | $3+\frac{x+y}{3}$ | $\frac{2}{3}-\frac{1}{3}(x+y)$ | $-1$ | $y-1$
$\tau_{R}$ | $x+y-3$ | $-1/3$ | $x\\!+\\!y\\!-\\!3\\!\mp\\!\frac{4}{3}\sqrt{3\\!-\\!x^{2}\\!-\\!y^{2}}$ | $-(2+z)/3$ | $-(2+x)/3+y$
## V Summary
In this paper, we have classified various new physics models involving the
$Z^{\prime}$ boson in two different ways: one according to $Z^{\prime}$ boson
mixings with $Z$ and $\gamma$, and the other according to $Z^{\prime}$ boson
charges with respect to quarks and leptons. In regard to the former, we based
the general description for the $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixing
derived from the EWCL on our previous workZ'our , characterizing these new
physics models into five classes: 1. Models with minimal $Z^{\prime}\\!-\\!Z$
mass mixing; 2.Models with minimal $Z^{\prime}\\!-\\!Z$ kinetic mixing;
3.Models with general $Z^{\prime}\\!-\\!Z$ mixing; 4.Models with
$Z^{\prime}\\!-\\!\gamma$ kinetic and $Z^{\prime}\\!-\\!Z$ mixing; and
5.Models with Stueckelberg-type mixing. Although the general
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixing is complicated and there is no exact
analytical expression for the mixing matrix $U$ and masses
$M_{Z},M_{Z^{\prime}}$, we obtain explicit analytical expressions for each of
our five simplifying classes. We find that the most elementary mixing is the
minimal $Z^{\prime}\\!-\\!Z$ mass mixing, the other four classes of mixings
can be transformed into the minimal $Z^{\prime}\\!-\\!Z$ mass mixing through
field transformations. In regard to the latter classification, we exploit the
anomaly cancellation conditions to constrain the $U(1)^{\prime}$ charges. For
generation-independent $U(1)^{\prime}$ charges, there are six charges
$y^{\prime}_{q}$,$y^{\prime}_{u}$,$y^{\prime}_{d}$,$y^{\prime}_{l}$,$y^{\prime}_{e}$,$y^{\prime}_{\nu}$
for which anomaly cancellation requires that only two are independent
parameters while the other four can depend on these two parameters in two
different ways. While one appears already in the literature, the other is new.
For generation-dependent $U(1)^{\prime}$ charges, we have listed some possible
special solutions.
## Acknowledgments
This work was supported by National Science Foundation of China (NSFC) under
Grant No. 10875065.
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|
arxiv-papers
| 2009-04-14T05:00:57 |
2024-09-04T02:49:01.848150
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ying Zhang, Qing Wang",
"submitter": "Wang Qing",
"url": "https://arxiv.org/abs/0904.2047"
}
|
0904.2051
|
# Joint-sparse recovery from multiple measurements††thanks: Department of
Computer Science, University of British Columbia, Vancouver V6T 1Z4, BC,
Canada ({ewout78,mpf}@cs.ubc.ca). Research partially supported by the Natural
Sciences and Engineering Research Council of Canada.
Ewout van den Berg Michael P. Friedlander
###### Abstract
The joint-sparse recovery problem aims to recover, from sets of compressed
measurements, unknown sparse matrices with nonzero entries restricted to a
subset of rows. This is an extension of the single-measurement-vector (SMV)
problem widely studied in compressed sensing. We analyze the recovery
properties for two types of recovery algorithms. First, we show that recovery
using sum-of-norm minimization cannot exceed the uniform recovery rate of
sequential SMV using $\ell_{1}$ minimization, and that there are problems that
can be solved with one approach but not with the other. Second, we analyze the
performance of the ReMBo algorithm [M. Mishali and Y. Eldar, IEEE Trans. Sig.
Proc., 56 (2008)] in combination with $\ell_{1}$ minimization, and show how
recovery improves as more measurements are taken. From this analysis it
follows that having more measurements than number of nonzero rows does not
improve the potential theoretical recovery rate.
## 1 Introduction
A problem of central importance in compressed sensing [1, 10] is the
following: given an $m\times n$ matrix $A$, and a measurement vector
$b=Ax_{0}$, recover $x_{0}$. When $m<n$, this problem is ill-posed, and it is
not generally possible to uniquely recover $x_{0}$ without some prior
information. In many important cases, $x_{0}$ is known to be sparse, and it
may be appropriate to solve
$\displaystyle\mathop{\hbox{minimize}}_{x\in\mathbb{R}^{n}}\quad\|x\|_{0}\quad\mathop{\hbox{subject
to}}\quad Ax=b,$ (1.1)
to find the sparsest possible solution. (The $\ell_{0}$-norm $\|\cdot\|_{0}$
of a vector counts the number of nonzero entries.) If $x_{0}$ has fewer than
$s/2$ nonzero entries, where $s$ is the number of nonzeros in the sparsest
null-vector of $A$, then $x_{0}$ is the unique solution of this optimization
problem [12, 19]. The main obstacle of this approach is that it is
combinatorial [24], and therefore impractical for all but the smallest
problems. To overcome this, Chen et al. [6] introduced basis pursuit:
$\displaystyle\mathop{\hbox{minimize}}_{x\in\mathbb{R}^{n}}\quad\|x\|_{1}\quad\mathop{\hbox{subject
to}}\quad Ax=b.$ (1.2)
This convex relaxation, based on the $\ell_{1}$-norm $\|x\|_{1}$, can be
solved much more efficiently; moreover, under certain conditions [2, 11], it
yields the same solution as the $\ell_{0}$ problem (1.1).
A natural extension of the single-measurement-vector (SMV) problem just
described is the multiple-measurement-vector (MMV) problem. Instead of a
single measurement $b$, we are given a set of $r$ measurements
$b^{(k)}=Ax_{0}^{(k)},\quad k=1,\ldots,r,$
in which the vectors $x_{0}^{(k)}$ are jointly sparse—i.e., have nonzero
entries at the same locations. Such problems arise in source localization
[22], neuromagnetic imaging [8], and equalization of sparse-communication
channels [7, 15]. Succinctly, the aim of the MMV problem is to recover $X_{0}$
from observations $B=AX_{0}$, where $B=[b^{(1)},\ b^{(2)},\ldots,\ b^{(r)}]$
is an $m\times r$ matrix, and the $n\times r$ matrix $X_{0}$ is row
sparse—i.e., it has nonzero entries in only a small number of rows. The most
widely studied approach to the MMV problem is based on solving the convex
optimization problem
$\displaystyle\mathop{\hbox{minimize}}_{X\in\mathbb{R}^{n\times
r}}\quad\|X\|_{p,q}\quad\mathop{\hbox{subject to}}\quad AX=B,$
where the mixed $\ell_{p,q}$ norm of $X$ is defined as
$\|X\|_{p,q}=\Big{(}\sum_{j=1}^{n}\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{q}^{p}\Big{)}^{1/p},$
and $X^{{j}{\scalebox{0.6}{$\rightarrow$}}}$ is the (column) vector whose
entries form the $j$th row of $X$. In particular, Cotter et al. [8] consider
$p=2$, $q\leq 1$; Tropp [28, 29] analyzes $p=1$, $q=\infty$; Malioutov et al.
[22] and Eldar and Mishali [14] use $p=1$, $q=2$; and Chen and Huo [5] study
$p=1$, $q\geq 1$. A different approach is given by Mishali and Eldar [23], who
propose the ReMBo algorithm, which reduces MMV to a series of SMV problems.
In this paper we study the sum-of-norms problem and the conditions for uniform
recovery of all $X_{0}$ with a fixed row support, and compare this against
recovery using $\ell_{1,1}$. We then construct matrices $X_{0}$ that cannot be
recovered using $\ell_{1,1}$ but for which $\ell_{1,2}$ does succeed, and vice
versa. We then illustrate the individual recovery properties of $\ell_{1,1}$
and $\ell_{1,2}$ with empirical results. We further show how recovery via
$\ell_{1,1}$ changes as the number of measurements increases, and propose a
boosted-$\ell_{1}$ approach to improve on the $\ell_{1,1}$ approach. This
analysis provides the starting point for our study of the recovery properties
of ReMBo, based on a geometrical interpretation of this algorithm.
We begin in Section 2 by summarizing existing $\ell_{0}$-$\ell_{1}$
equivalence results, which give conditions under which the solution of the
$\ell_{1}$ relaxation (1.2) coincides with the solution of the $\ell_{0}$
problem (1.1). In Section 3 we consider the $\ell_{1,2}$ mixed-norm and sum-
of-norms formulations and compare their performance against $\ell_{1,1}$. In
Sections 4 and 5 we examine two approaches that are based on sequential
application of (1.2).
##### Notation.
We assume throughout that $A$ is a full-rank matrix in $\mathbb{R}^{m\times
n}$, and that $X_{0}$ is an $s$ row-sparse matrix in $\mathbb{R}^{n\times r}$.
We follow the convention that all vectors are column vectors. For an arbitrary
matrix $M$, its $j$th column is denoted by the column vector
$M^{\scalebox{0.6}{$\downarrow$}{j}}$; its $i$th row is the transpose of the
column vector $M^{{i}{\scalebox{0.6}{$\rightarrow$}}}$. The $i$th entry of a
vector $v$ is denoted by $v_{i}$. We make exceptions for
$e_{i}=I^{\scalebox{0.6}{$\downarrow$}{i}}$ and for $x_{0}$ (resp., $X_{0}$),
which represents the sparse vector (resp., matrix) we want to recover. When
there is no ambiguity we sometimes write $m_{i}$ to denote
$M^{\scalebox{0.6}{$\downarrow$}{i}}$. When concatenating vectors into
matrices, $[a,b,c]$ denotes horizontal concatenation and $[a;b;c]$ denotes
vertical concatenation. When indexing with $\mathcal{I}$, we define the vector
$v_{\mathcal{I}}:=[v_{i}]_{i\in\mathcal{I}}$, and the $m\times|\mathcal{I}|$
matrix
$A_{\mathcal{I}}:=[A^{\scalebox{0.6}{$\downarrow$}{j}}]_{j\in\mathcal{I}}$.
Row or column selection takes precedence over all other operators.
## 2 Existing results for $\ell_{1}$ recovery
The conditions under which (1.2) gives the sparsest possible solution have
been studied by applying a number of different techniques. By far the most
popular analytical approach is based on the restricted isometry property,
introduced by Candès and Tao [3], which gives sufficient conditions for
equivalence. Donoho [9] obtains necessary and sufficient (NS) conditions by
analyzing the underlying geometry of (1.2). Several authors [13, 19, 12]
characterize the NS-conditions in terms of properties of the kernel of $A$:
$\textrm{Ker}(A)=\\{x\mid Ax=0\\}.$
Fuchs [16] and Tropp [27] express sufficient conditions in terms of the
solution of the dual of (1.2):
$\displaystyle\mathop{\hbox{maximize}}_{y}\quad
b^{T}\\!y\quad\mathop{\hbox{subject to}}\quad\|A^{T}\\!y\|_{\infty}\leq 1.$
(2.1)
In this paper we are mainly concerned with the geometric and kernel
conditions. We use the geometrical interpretation of the problems to get a
better understanding, and resort to the null-space properties of $A$ to
analyze recovery. To make the discussion more self-contained, we briefly
recall some of the relevant results in the next three sections.
### 2.1 The geometry of $\ell_{1}$ recovery
The set of all points of the unit $\ell_{1}$-ball,
$\\{x\in\mathbb{R}^{n}\mid\|x\|_{1}\leq 1\\}$, can be formed by taking convex
combinations of $\pm e_{j}$, the signed columns of the identity matrix.
Geometrically this is equivalent to taking the convex hull of these vectors,
giving the cross-polytope $\mathcal{C}=\mathrm{conv}\\{\pm e_{1},\pm
e_{2},\ldots,\pm e_{n}\\}$. Likewise, we can look at the linear mapping
$x\mapsto Ax$ for all points $x\in\mathcal{C}$, giving the polytope
$\mathcal{P}=\\{Ax\mid x\in\mathcal{C}\\}=A\mathcal{C}$. The faces of
$\mathcal{C}$ can be expressed as the convex hull of subsets of vertices, not
including pairs that are reflections with respect to the origin (such pairs
are sometimes erroneously referred to as antipodal, which is a slightly more
general concept [21]). Under linear transformations, each face from the cross-
polytope $\mathcal{C}$ either maps to a face on $\mathcal{P}$ or vanishes into
the interior of $\mathcal{P}$.
The solution found by (1.2) can be interpreted as follows. Starting with a
radius of zero, we slowly “inflate” $\mathcal{P}$ until it first touches $b$.
The radius at which this happens corresponds to the $\ell_{1}$-norm of the
solution $x^{*}$. The vertices whose convex hull is the face touching $b$
determine the location and sign of the non-zero entries of $x^{*}$, while the
position where $b$ touches the face determines their relative weights. Donoho
[9] shows that $x_{0}$ can be recovered from $b=Ax_{0}$ using (1.2) if and
only if the face of the (scaled) cross-polytope containing $x_{0}$ maps to a
face on $\mathcal{P}$. Two direct consequences are that recovery depends only
on the sign pattern of $x_{0}$, and that the probability of recovering a
random $s$-sparse vector is equal to the ratio of the number of $(s-1)$-faces
in $\mathcal{P}$ to the number of $(s-1)$-faces in $\mathcal{C}$. That is,
letting $\mathcal{F}_{d}(\mathcal{P})$ denote the collection of all $d$-faces
[21] in $\mathcal{P}$, the probability of recovering $x_{0}$ using $\ell_{1}$
is given by
$P_{\ell_{1}}(A,s)=\frac{|\mathcal{F}_{s-1}(A\mathcal{C})|}{|\mathcal{F}_{s-1}(\mathcal{C})|}.$
When we need to find the recoverability of vectors restricted to a support
$\mathcal{I}$, this probability becomes
$P_{\ell_{1}}(A,\mathcal{I})=\frac{|\mathcal{F}_{\mathcal{I}}(A\mathcal{C})|}{|\mathcal{F}_{\mathcal{I}}(\mathcal{C})|},$
(2.2)
where $\mathcal{F}_{\mathcal{I}}(\mathcal{C})=2^{|\mathcal{I}|}$ denotes the
number of faces in $\mathcal{C}$ formed by the convex hull of $\\{\pm
e_{j}\\}_{i\in\mathcal{I}}$, and $\mathcal{F}_{\mathcal{I}}(A\mathcal{C})$ is
the number of faces on $A\mathcal{C}$ generated by $\\{\pm
A^{\scalebox{0.6}{$\downarrow$}{j}}\\}_{j\in\mathcal{I}}$.
### 2.2 Null-space properties and $\ell_{1}$ recovery
Equivalence results in terms of null-space properties generally characterize
equivalence for the set of all vectors $x$ with a fixed support, which is
defined as
$\textrm{Supp}(x)=\\{j\mid x_{j}\neq 0\\}.$
We say that $x$ can be uniformly recovered on
$\mathcal{I}\subseteq\\{1,\ldots,n\\}$ if all $x$ with
$\textrm{Supp}(x)\subseteq\mathcal{I}$ can be recovered. The following theorem
illustrates conditions for uniform recovery via $\ell_{1}$ on an index set;
more general results are given by Gribonval and Nielsen [20].
###### Theorem 2.1 (Donoho and Elad [12], Gribonval and Nielsen [19]).
Let $A$ be an $m\times n$ matrix and $\mathcal{I}\subseteq\\{1,\ldots,n\\}$ be
a fixed index set. Then all $x_{0}\in\mathbb{R}^{n}$ with
$\textrm{Supp}(x_{0})\subseteq\mathcal{I}$ can be uniquely recovered from
$b=Ax_{0}$ using basis pursuit (1.2) if and only if for all
$z\in\textrm{Ker}(A)\setminus\\{0\\}$,
$\sum_{j\in\mathcal{I}}|z_{j}|<\sum_{j\not\in\mathcal{I}}|z_{j}|.$ (2.3)
That is, the $\ell_{1}$-norm of $z$ on $\mathcal{I}$ is strictly less than the
$\ell_{1}$-norm of $z$ on the complement $\mathcal{I}^{c}$.
### 2.3 Optimality conditions for $\ell_{1}$ recovery
Sufficient conditions for recovery can be derived from the first-order
optimality conditions necessary for $x^{*}$ and $y^{*}$ to be solutions of
(1.2) and (2.1) respectively. The Karush-Kuhn-Tucker (KKT) conditions are also
sufficient in this case because the problems are convex. The Lagrangian
function for (1.2) is given by
$\mathcal{L}(x,y)=\|x\|_{1}-y^{T}\\!(Ax-b);$
the KKT conditions require that
$Ax=b\text{and}0\in\partial_{x}\mathcal{L}(x,y),$ (2.4)
where $\partial_{x}\mathcal{L}$ denotes the subdifferential of $\mathcal{L}$
with respect to $x$. The second condition reduces to
$0\in\mathop{\hbox{\rm sgn}}(x)-A^{T}\\!y,$
where the signum function
$\mathop{\hbox{\rm sgn}}(\gamma)\in\begin{cases}\mathop{\hbox{\rm
sign}}(\gamma)&\hbox{if $\gamma\neq 0$,}\\\
[-1,1]&\hbox{otherwise},\end{cases}$
is applied to each individual component of $x$. It follows that $x^{*}$ is a
solution of (1.2) if and only if $Ax^{*}=b$ and there exists an $m$-vector $y$
such that $|a_{j}^{T}\\!y|\leq 1$ for $j\not\in\textrm{Supp}(x)$, and
$a_{j}^{T}\\!y=\mathop{\hbox{\rm sign}}(x_{j}^{*})$ for all
$j\in\textrm{Supp}(x)$. Fuchs [16] shows that $x^{*}$ is the unique solution
of (1.2) when $[a_{j}]_{j\in\textrm{Supp}(x)}$ is full rank and, in addition,
$|a_{j}^{T}\\!y|<1$ for all $j\not\in\textrm{Supp}(x)$. When the columns of
$A$ are in general position (i.e., no $k+1$ columns of $A$ span the same $k-1$
dimensional hyperplane for $k\leq n$) we can weaken this condition by noting
that for such $A$, the solution of (1.2) is always unique, thus making the
existence of a $y$ that satisfies (2.4) for $x_{0}$ a necessary and sufficient
condition for $\ell_{1}$ to recover $x_{0}$.
## 3 Recovery using sums-of-row norms
Our analysis of sparse recovery for the MMV problem of recovering $X_{0}$ from
$B=AX_{0}$ begins with an extension of Theorem 2.1 to recovery using the
convex relaxation
$\displaystyle\mathop{\hbox{minimize}}_{X}\quad\sum_{j=1}^{n}\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|\quad\mathop{\hbox{subject
to}}\quad AX=B;$ (3.1)
note that the norm within the summation is arbitrary. Define the row support
of a matrix as
$\textrm{Supp}_{\mathrm{row}}(X)=\\{j\mid\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|\neq
0\\}.$
With these definitions we have the following result. (A related result is
given by Stojnic et al. [26].)
###### Theorem 3.1.
Let $A$ be an $m\times n$ matrix, $k$ be a positive integer,
$\mathcal{I}\subseteq\\{1,\ldots,n\\}$ be a fixed index set, and let
$\|\cdot\|$ denote any vector norm. Then all $X_{0}\in\mathbb{R}^{n\times r}$
with $\textrm{Supp}_{\mathrm{row}}(X_{0})\subseteq\mathcal{I}$ can be uniquely
recovered from $B=AX_{0}$ using (3.1) if and only if for all $Z$ with columns
$Z^{\scalebox{0.6}{$\downarrow$}{k}}\in\textrm{Ker}(A)\setminus\\{0\\}$,
$\sum_{j\in\mathcal{I}}\|Z^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|<\sum_{j\not\in\mathcal{I}}\|Z^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|.$
(3.2)
###### Proof.
For the “only if” part, suppose that there is a $Z$ with columns
$Z^{\scalebox{0.6}{$\downarrow$}{k}}\in\textrm{Ker}(A)\setminus\\{0\\}$ such
that (3.2) does not hold. Now, choose
$X^{{j}{\scalebox{0.6}{$\rightarrow$}}}=Z^{{j}{\scalebox{0.6}{$\rightarrow$}}}$
for all $j\in\mathcal{I}$ and with all remaining rows zero. Set $B=AX$. Next,
define $V=X-Z$, and note that $AV=AX-AZ=AX=B$. The construction of $V$ implies
that
$\sum_{j}\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|\geq\sum_{j}\|V^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|$,
and consequently $X$ cannot be the unique solution of (3.1).
Conversely, let $X$ be an arbitrary matrix with
$\textrm{Supp}_{\mathrm{row}}(X)\subseteq\mathcal{I}$, and let $B=AX$. To show
that $X$ is the unique solution of (3.1) it suffices to show that for any $Z$
with columns
$Z^{\scalebox{0.6}{$\downarrow$}{k}}\in\textrm{Ker}(A)\setminus\\{0\\}$,
$\sum_{j}\|(X+Z)^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|>\sum_{j}\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|.$
This is equivalent to
$\sum_{j\not\in\mathcal{I}}\|Z^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|+\sum_{j\in\mathcal{I}}\|(X+Z)^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|-\sum_{j\in\mathcal{I}}\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|>0.$
Applying the reverse triangle inequality, $\|a+b\|-\|b\|\geq-\|a\|$, to the
summation over $j\in\mathcal{I}$ and reordering exactly gives condition (3.2).
∎
In the special case of the sum of $\ell_{1}$-norms, i.e., $\ell_{1,1}$,
summing the norms of the columns is equivalent to summing the norms of the
rows. As a result, (3.1) can be written as
$\displaystyle\mathop{\hbox{minimize}}_{X}\quad\sum_{k=1}^{r}\|X^{\scalebox{0.6}{$\downarrow$}{k}}\|_{1}\quad\mathop{\hbox{subject
to}}\quad
AX^{\scalebox{0.6}{$\downarrow$}{k}}=B^{\scalebox{0.6}{$\downarrow$}{k}},\quad
k=1,\ldots,r.$
Because this objective is separable, the problem can be decoupled and solved
as a series of independent basis pursuit problems, giving one
$X^{\scalebox{0.6}{$\downarrow$}{k}}$ for each column
$B^{\scalebox{0.6}{$\downarrow$}{k}}$ of $B$. The following result relates
recovery using the sum-of-norms formulation (3.1) to $\ell_{1,1}$ recovery.
###### Theorem 3.2.
Let $A$ be an $m\times n$ matrix, $r$ be a positive integer,
$\mathcal{I}\subseteq\\{1,\ldots,n\\}$ be a fixed index set, and $\|\cdot\|$
denote any vector norm. Then uniform recovery of all $X\in\mathbb{R}^{n\times
r}$ with $\textrm{Supp}_{\mathrm{row}}(X)\subseteq\mathcal{I}$ using sums of
norms (3.1) implies uniform recovery on $\mathcal{I}$ using $\ell_{1,1}$.
###### Proof.
For uniform recovery on support $\mathcal{I}$ to hold it follows from Theorem
3.1 that for any matrix $Z$ with columns
$Z^{\scalebox{0.6}{$\downarrow$}{k}}\in\textrm{Ker}(A)\setminus\\{0\\}$,
property (3.2) holds. In particular it holds for $Z$ with
$Z^{\scalebox{0.6}{$\downarrow$}{k}}={\bar{z\mkern 2.8mu}\mkern-2.8mu}{}$ for
all $k$, with ${\bar{z\mkern
2.8mu}\mkern-2.8mu}{}\in\textrm{Ker}(A)\setminus\\{0\\}$. Note that for these
matrices there exist a norm-dependent constant $\gamma$ such that
$|{\bar{z\mkern
2.8mu}\mkern-2.8mu}{}_{j}|=\gamma\|Z^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|.$
Since the choice of ${\bar{z\mkern 2.8mu}\mkern-2.8mu}{}$ was arbitrary, it
follows from (3.2) that the NS-condition (2.3) for independent recovery of
vectors $B^{\scalebox{0.6}{$\downarrow$}{k}}$ using $\ell_{1}$ in Theorem 2.1
is satisfied. Moreover, because $\ell_{1,1}$ is equivalent to independent
recovery, we also have uniform recovery on $\mathcal{I}$ using $\ell_{1,1}$. ∎
An implication of Theorem 3.2 is that the use of restricted isometry
conditions—or any technique, for that matter—to analyze uniform recovery
conditions for the sum-of-norms approach necessarily lead to results that are
no stronger than uniform $\ell_{1}$ recovery. (Recall that the $\ell_{1,1}$
and $\ell_{1}$ norms are equivalent).
### 3.1 Recovery using $\ell_{1,2}$
Figure 1: Recovery rates for fixed, randomly drawn $20\times 60$ matrices $A$,
averaged over 1,000 trials at each row-sparsity level $s$. The nonzero entries
in the $60\times r$ matrix $X_{0}$ are sampled i.i.d. from the normal
distribution. The solid and dashed lines represent $\ell_{1,2}$ and
$\ell_{1,1}$ recovery, respectively.
In this section we take a closer look at the $\ell_{1,2}$ problem
$\displaystyle\mathop{\hbox{minimize}}_{X}\quad\|X\|_{1,2}\quad\mathop{\hbox{subject
to}}\quad AX=B,$ (3.3)
which is a special case of the sum-of-norms problem. Although Theorem 3.2
establishes that uniform recovery via $\ell_{1,2}$ is no better than uniform
recovery via $\ell_{1,1}$, there are many situations in which it recovers
signals that $\ell_{1,1}$ cannot. Indeed, it is evident from Figure 1 that the
probability of recovering individual signals with random signs and support is
much higher for $\ell_{1,2}$. The reason for the degrading performance or
$\ell_{1,1}$ with increasing $k$ is explained in Section 4.
In this section we construct examples for which $\ell_{1,2}$ works and
$\ell_{1,1}$ fails, and vice versa. This helps uncover some of the structure
of $\ell_{1,2}$, but at the same time implies that certain techniques used to
study $\ell_{1}$ can no longer be used directly. Because the examples are
based on extensions of the results from Section 2.3, we first develop
equivalent conditions here.
#### 3.1.1 Sufficient conditions for recovery via $\ell_{1,2}$
The optimality conditions of the $\ell_{1,2}$ problem (3.3) play a vital role
in deriving a set of sufficient conditions for joint-sparse recovery. In this
section we derive the dual of (3.3) and the corresponding necessary and
sufficient optimality conditions. These allow us to derive sufficient
conditions for recovery via $\ell_{1,2}$.
The Lagrangian for (3.3) is defined as
$\mathcal{L}(X,Y)=\|X\|_{1,2}-\Braket{Y,AX-B},$ (3.4)
where $\Braket{V,W}\mathrel{\mathop{:}}=\mathop{\hbox{\rm trace}}(V^{T}\\!W)$
is an inner-product defined over real matrices. The dual is then given by
maximizing
$\inf_{X}\mathcal{L}(X,Y)=\inf_{X}\left\\{\|X\|_{1,2}-\Braket{Y,AX-B}\right\\}=\Braket{B,Y}-\sup_{X}\left\\{\Braket{A^{T}\\!Y,X}-\|X\|_{1,2}\right\\}$
(3.5)
over $Y$. (Because the primal problem has only linear constraints, there
necessarily exists a dual solution $Y^{*}$ that maximizes this expression [25,
Theorem 28.2].) To simplify the supremum term, we note that for any convex,
positively homogeneous function $f$ defined over an inner-product space,
$\sup_{v}\ \\{\Braket{w,v}-f(v)\\}=\begin{cases}0&\hbox{if $w\in\partial
f(0)$,}\\\ \infty&\hbox{otherwise.}\end{cases}$
To derive these conditions, note that positive homogeneity of $f$ implies that
$f(0)=0$, and thus $w\in\partial f(0)$ implies that $\Braket{w,v}\leq f(v)$
for all $v$. Hence, the supremum is achieved with $v=0$. If on the other hand
$w\not\in\partial f(0)$, then there exists some $v$ such that
$\Braket{w,v}>f(v)$, and by the positive homogeneity of $f$, $\Braket{w,\alpha
v}-f(\alpha v)\to\infty$ as $\alpha\to\infty$. Applying this expression for
the supremum to (3.5), we arrive at the necessary condition
$A^{T}\\!Y\in\partial\|0\|_{1,2},$ (3.6)
which is required for dual feasibility.
We now derive an expression for the subdifferential $\partial\|X\|_{1,2}$. For
rows $j$ where $\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{2}>0$, the
gradient is given by
$\nabla\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{2}=X^{{j}{\scalebox{0.6}{$\rightarrow$}}}/\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{2}$.
For the remaining rows, the gradient is not defined, but
$\partial\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{2}$ coincides with the
set of unit $\ell_{2}$-norm vectors
$\mathcal{B}_{\ell_{2}}^{r}=\\{v\in\mathbb{R}^{r}\ \mid\|v\|_{2}\leq 1\\}$.
Thus, for each $j=1,\ldots,n$,
$\partial_{X^{{j}{\scalebox{0.6}{$\rightarrow$}}}}\|X\|_{1,2}\in\begin{cases}X^{{j}{\scalebox{0.6}{$\rightarrow$}}}/\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{2}&\hbox{if
$\|X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{2}>0$,}\\\\[4.0pt]
\mathcal{B}_{\ell_{2}}^{r}&\hbox{otherwise.}\end{cases}$ (3.7)
Combining this expression with (3.6), we arrive at the dual of (3.3):
$\displaystyle\mathop{\hbox{maximize}}_{Y}\quad\mathop{\hbox{\rm
trace}}(B^{T}\\!Y)\quad\mathop{\hbox{subject
to}}\quad\|A^{T}\\!Y\|_{\infty,2}\leq 1.$ (3.8)
The following conditions are therefore necessary and sufficient for a primal-
dual pair $(X^{*},Y^{*})$ to be optimal for (3.3) and its dual (3.8):
$\displaystyle AX^{*}$ $\displaystyle=B$ (primal feasibility); (3.9a)
$\displaystyle\|A^{T}\\!Y^{*}\|_{\infty,2}$ $\displaystyle\leq 1$ (dual
feasibility); (3.9b) $\displaystyle\|X^{*}\|_{1,2}$
$\displaystyle=\mathop{\hbox{\rm trace}}(B^{T}\\!Y^{*})$ (zero duality gap).
(3.9c)
The existence of a matrix $Y^{*}$ that satisfies (3.9) provides a certificate
that the feasible matrix $X^{*}$ is an optimal solution of (3.3). However, it
does not guarantee that $X^{*}$ is also the unique solution. The following
theorem gives sufficient conditions, similar to those in Section 2.3, that
also guarantee uniqueness of the solution.
###### Theorem 3.3.
Let $A$ be an $m\times n$ matrix, and $B$ be an $m\times r$ matrix. Then a set
of sufficient conditions for $X$ to be the unique minimizer of (3.3) with
Lagrange multiplier $Y\in\mathbb{R}^{m\times r}$ and row support
$\mathcal{I}=\textrm{Supp}_{\mathrm{row}}(X)$, is that
$\displaystyle AX=B,$ (3.10a)
$\displaystyle(A^{T}\\!Y)^{\scalebox{0.6}{$\downarrow$}{j}}=(X^{*})^{{j}{\scalebox{0.6}{$\rightarrow$}}}/\|(X^{*})^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{2},$
$\displaystyle\qquad j$ $\displaystyle\in\mathcal{I}$ (3.10b)
$\displaystyle\|(A^{T}\\!Y)^{\scalebox{0.6}{$\downarrow$}{j}}\|_{2}<1,$
$\displaystyle\qquad j$ $\displaystyle\not\in\mathcal{I}$ (3.10c)
$\displaystyle\mathop{\hbox{\rm rank}}(A_{\mathcal{I}})=|\mathcal{I}|.$
(3.10d)
###### Proof.
The first three conditions clearly imply that $(X,Y)$ primal and dual
feasible, and thus satisfy (3.9a) and (3.9b). Conditions (3.10b) and (3.10c)
together imply that
$\mathop{\hbox{\rm
trace}}(B^{T}\\!Y)\equiv\sum_{j=1}^{n}[(A^{T}\\!Y)^{\scalebox{0.6}{$\downarrow$}{j}}]^{T}X^{{j}{\scalebox{0.6}{$\rightarrow$}}}=\sum_{j=1}^{n}X^{{j}{\scalebox{0.6}{$\rightarrow$}}}\equiv\|X\|_{1,2}.$
The first and last identities above follow directly from the definitions of
the matrix trace and of the norm $\|\cdot\|_{1,2}$, respectively; the middle
equality follows from the standard Cauchy inequality. Thus, the zero-gap
requirement (3.9c) is satisfied. The conditions (3.10a)–(3.10c) are therefore
sufficient for $(X,Y)$ to be an optimal primal-dual solution of (3.3). Because
$Y$ determines the support and is a Lagrange multiplier for every solution
$X$, this support must be unique. It then follows from condition (3.10d) that
$X$ must be unique. ∎
### 3.2 Counter examples
Using the sufficient and necessary conditions developed in the previous
section we now construct examples of problems for which $\ell_{1,2}$ succeeds
while $\ell_{1,1}$ fails, and vice versa. Because of its simplicity, we begin
with the latter.
##### Recovery using $\ell_{1,1}$ where $\ell_{1,2}$ fails.
Let $A$ be an $m\times n$ matrix with $m<n$ and unit-norm columns that are not
scalar multiples of each other. Take any vector $x\in\mathbb{R}^{n}$ with at
least $m+1$ nonzero entries. Then $X_{0}=\mathop{\hbox{\rm diag}}(x)$,
possibly with all identically zero columns removed, can be recovered from
$B=AX_{0}$ using $\ell_{1,1}$, but not with $\ell_{1,2}$. To see why, note
that each column in $X_{0}$ has only a single nonzero entry, and that, under
the assumptions on $A$, each one-sparse vector can be recovered individually
using $\ell_{1}$ (the points $\pm
A^{\scalebox{0.6}{$\downarrow$}{j}}\in\mathbb{R}^{m}$ are all $0$-faces of
$\mathcal{P}$) and therefore that $X_{0}$ can be recovered using $\ell_{1,1}$.
On the other hand, for recovery using $\ell_{1,2}$ there would need to exist a
matrix $Y$ satisfying the first condition of (3.9) for all
$j\in\mathcal{I}=\\{1,\ldots,n\\}$. For this given $X_{0}$ this reduces to
$A^{T}Y=M$, where $M$ is the identity matrix, with the same columns removed as
$X$. But this equality is impossible to satisfy because $\mathop{\hbox{\rm
rank}}(A)\leq m<m+1\leq\mathop{\hbox{\rm rank}}(M)$. Thus, $X_{0}$ cannot be
the solution of the $\ell_{1,2}$ problem (3.3).
##### Recovery using $\ell_{1,2}$ where $\ell_{1,1}$ fails.
For the construction of a problem where $\ell_{1,2}$ succeeds and $\ell_{1,1}$
fails, we consider two vectors, $f$ and $s$, with the same support
$\mathcal{I}$, in such a way that individual $\ell_{1}$ recovery fails for
$f$, while it succeeds for $s$. In addition we assume that there exists a
vector $y$ that satisfies
$y^{T}\\!A^{\scalebox{0.6}{$\downarrow$}{j}}=\mathop{\hbox{\rm
sign}}(s_{j})\quad\hbox{for all $j\in\mathcal{I}$,}\hbox{\qquad
and\qquad}|y^{T}\\!A^{\scalebox{0.6}{$\downarrow$}{j}}|<1\quad\hbox{for all
$j\not\in\mathcal{I}$;}$
i.e., $y$ satisfies conditions (3.10b) and (3.10c). Using the vectors $f$ and
$s$, we construct the 2-column matrix $X_{0}=[(1-\gamma)s,\ \gamma f]$, and
claim that for sufficiently small $\gamma>0$, this gives the desired
reconstruction problem. Clearly, for any $\gamma\neq 0$, $\ell_{1,1}$ recovery
fails because the second column can never be recovered, and we only need to
show that $\ell_{1,2}$ does succeed.
For $\gamma=0$, the matrix $Y=[y,0]$ satisfies conditions (3.10b) and (3.10c)
and, assuming (3.10d) is also satisfied, $X_{0}$ is the unique solution of
$\ell_{1,2}$ with $B=AX_{0}$. For sufficiently small $\gamma>0$, the
conditions that $Y$ need to satisfy change slightly due to the division by
$\|X_{0}^{{j}{\scalebox{0.6}{$\rightarrow$}}}\|_{2}$ for those rows in
$\textrm{Supp}_{\mathrm{row}}(X)$. By adding corrections to the columns of $Y$
those new conditions can be satisfied. In particular, these corrections can be
done by adding weighted combinations of the columns in $\bar{Y}$, which are
constructed in such a way that it satisfies $A_{\mathcal{I}}^{T}{\bar{Y\mkern
2.0mu}\mkern-2.0mu}{}=I$, and minimizes
$\|A_{\mathcal{I}^{c}}^{T}\bar{Y}\|_{\infty,\infty}$ on the complement
$\mathcal{I}^{c}$ of $\mathcal{I}$.
Note that on the above argument can also be used to show that $\ell_{1,2}$
fails for $\gamma$ sufficiently close to one. Because the support and signs of
$X$ remain the same for all $0<\gamma<1$, we can conclude the following:
###### Corollary 3.4.
Recovery using $\ell_{1,2}$ is generally not only characterized by the row-
support and the sign pattern of the nonzero entries in $X_{0}$, but also by
the magnitude of the nonzero entries.
A consequence of this conclusion is that the notion of faces used in the
geometrical interpretation of $\ell_{1}$ is not applicable to the $\ell_{1,2}$
problem.
### 3.3 Experiments
To get an idea of just how much more $\ell_{1,2}$ can recover in the above
case where $\ell_{1,1}$ fails, we generated a $20\times 60$ matrix $A$ with
entries i.i.d. normally distributed, and determined a set of vectors $s_{i}$
and $f_{i}$ with identical support for which $\ell_{1}$ recovery succeeds and
fails, respectively. Using triples of vectors $s_{i}$ and $f_{j}$ we
constructed row-sparse matrices such as $X_{0}=[s_{1},f_{1},f_{2}]$ or
$X_{0}=[s_{1},s_{2},f_{2}]$, and attempted to recover from $B=AX_{0}W$, where
$W=\mathop{\hbox{\rm diag}}(\omega_{1},\omega_{2},\omega_{3})$ is a diagonal
weighting matrix with nonnegative entries and unit trace, by solving (3.3).
For problems of this size, interior-point methods are very efficient and we
use SDPT3 [30] through the CVX interface [18, 17]. We consider $X_{0}$ to be
recovered when the maximum absolute difference between $X_{0}$ and the
$\ell_{1,2}$ solution $X^{*}$ is less than $10^{-5}$. The results of the
experiment are shown in Figure 2. In addition to the expected regions of
recovery around individual columns $s_{i}$ and failure around $f_{i}$, we see
that certain combinations of vectors $s_{i}$ still fail, while other
combinations of vectors $f_{i}$ may be recoverable. By contrast, when using
$\ell_{1,1}$ to solve the problem, any combination of $s_{i}$ vectors can be
recovered while no combination including an $f_{i}$ can be recovered.
| | |
---|---|---|---
$|\mathcal{I}|=5$ | $|\mathcal{I}|=5$ | $|\mathcal{I}|=5$ | $|\mathcal{I}|=7$
| | |
$|\mathcal{I}|=10$ | $|\mathcal{I}|=10$ | $|\mathcal{I}|=10$ | $|\mathcal{I}|=10$
Figure 2: Generation of problems where $\ell_{1,2}$ succeeds, while
$\ell_{1,1}$ fails. For a $20\times 60$ matrix $A$ and fixed support of size
$|\mathcal{I}|=5,7,10$, we create vectors $f_{i}$ that cannot be recovered
using $\ell_{1}$, and vectors $s_{i}$ than can be recovered. Each triangle
represents an $X_{0}$ constructed from the vectors denoted in the corners. The
location in the triangle determines the weight on each vector, ranging from
zero to one, and summing up to one. The dark areas indicates the weights for
which $\ell_{1,2}$ successfully recovered $X_{0}$.
## 4 Boosted $\ell_{1}$
As described in Section 3, recovery using $\ell_{1,1}$ is equivalent to
individual $\ell_{1}$ recovery of each column
$x_{k}:=X_{0}^{\scalebox{0.6}{$\downarrow$}{k}}$ based on
$b_{k}\mathrel{\mathop{:}}=B^{\scalebox{0.6}{$\downarrow$}{k}}$, for
$k=1,\ldots,r$:
$\displaystyle\mathop{\hbox{minimize}}_{x}\quad\|x\|_{1}\quad\mathop{\hbox{subject
to}}\quad Ax=b_{k}.$ (4.1)
Assuming that the signs of nonzero entries in the support of each $x_{k}$ are
drawn i.i.d. from $\\{1,-1\\}$, we can express the probability of recovering a
matrix $X_{0}$ with row support $\mathcal{I}$ using $\ell_{1,1}$ in terms of
the probability of recovering vectors on that support using $\ell_{1}$. To see
how, note that $\ell_{1,1}$ recovers the original $X_{0}$ if and only if each
individual problem in (4.1) successfully recovers each $x_{k}$. For the above
class of matrices $X_{0}$ this therefore gives a recovery rate of
$P_{\ell_{1,1}}(A,\mathcal{I},k)=\left[P_{\ell_{1}}(A,\mathcal{I})\right]^{r}.$
Using $\ell_{1,1}$ to recover $X_{0}$ is clearly not a good idea. Note also
that uniform recovery of $X_{0}$ on a support $\mathcal{I}$ remains unchanged,
regardless of the number of observations, $r$, that are given. As a
consequence of Theorem 3.2, this also means that the uniform-recovery
properties for any sum-of-norms approach cannot increase with $r$. This
clearly defeats the purpose of gathering multiple observations.
In many instances where $\ell_{1,1}$ fails, it may still recover a subset of
columns $x_{k}$ from the corresponding observations $b_{k}$. It seems wasteful
to discard this information because if we could recognize a single correctly
recovered $x_{k}$, we would immediately know the row support
$\mathcal{I}=\textrm{Supp}_{\mathrm{row}}(X_{0})=\textrm{Supp}(x_{k})$ of
$X_{0}$. Given the correct support we can recover the nonzero part $\bar{X}$
of $X_{0}$ by solving
$\displaystyle\mathop{\hbox{minimize}}_{\bar{X}}\quad\|A_{\mathcal{I}}\bar{X}-B\|_{F}.$
(4.2)
In practice we obviously do not know the correct support, but when a given
solution $x_{k}^{*}$ of (4.1) that is sufficiently sparse, we can try to solve
(4.2) for that support and verify if the residual at the solution is zero. If
so, we construct the final $X^{*}$ using the non-zero part and declare
success. Otherwise we simply increment $k$ and repeat this process until there
are no more observations and recovery was unsuccessful. We refer to this
algorithm, which is reminiscent of the ReMBo approach [23], as boosted
$\ell_{1}$; its sole aim is to provide a bridge to the analysis of ReMBo. The
complete boosted $\ell_{1}$ algorithm is outlined in Figure 4.
The recovery properties of the boosted $\ell_{1}$ approach are opposite from
those of $\ell_{1,1}$: it fails only if all individual columns fail to be
recovered using $\ell_{1}$. Hence, given an unknown $n\times r$ matrix $X$
supported on $\mathcal{I}$ with its sign pattern uniformly random, the boosted
$\ell_{1}$ algorithm gives an expected recovery rate of
$P_{\ell_{1}^{B}}(A,\mathcal{I},r)=1-\left[1-P_{\ell_{1}}(A,\mathcal{I})\right]^{r}.$
(4.3)
To experimentally verify this recovery rate, we generated a $20\times 80$
matrix $A$ with entries independently sampled from the normal distribution and
fixed a randomly chosen support set $\mathcal{I}_{s}$ for three levels of
sparsity, $s=8,9,10$. On each of these three supports we generated vectors
with all possible sign patterns and solved (1.2) to see if they could be
recovered or not (see Section 3.3). This gives exactly the face counts
required to compute the $\ell_{1}$ recovery probability in (2.2), and the
expected boosted $\ell_{1}$ recovery rate in (4.3)
For the empirical success rate we take the average over 1,000 trials with
random coefficient matrices $X$ supported on $\mathcal{I}_{s}$, and its
nonzero entries independently drawn from the normal distribution. To reduce
the computational time we avoid solving $\ell_{1}$ and instead compare the
sign pattern of the current solution $x_{k}$ against the information computed
to determine the face counts (both $A$ and $\mathcal{I}_{s}$ remain fixed).
The theoretical and empirical recovery rates using boosted $\ell_{1}$ are
plotted in Figure 4.
given $A$, $B$ for _$k=1,\ldots,r$_ do solve (1.2) with $b_{k}=B^{\scalebox{0.6}{$\downarrow$}{k}}$ to get $x$ $\mathcal{I}\leftarrow\textrm{Supp}(x)$ if _$|\mathcal{I}| <m/2$_ then solve (4.2) to get $X$ if _$A_{\mathcal{I}}X=B$_ then $X^{*}=0$ $(X^{*})^{{j}{\scalebox{0.6}{$\rightarrow$}}}\leftarrow X^{{j}{\scalebox{0.6}{$\rightarrow$}}}$ for $j\in\mathcal{I}$ return solution $X^{*}$ return failure |
---|---
Figure 3: The boosted $\ell_{1}$ algorithm | Figure 4: Theoretical (dashed) and experimental (solid) performance of boosted $\ell_{1}$ for three problem instances with different row support $s$.
## 5 Recovery using ReMBo
The boosted $\ell_{1}$ approach can be seen as a special case of the ReMBo
[23] algorithm. ReMBo proceeds by taking a random vector $w\in\mathbb{R}^{r}$
and combining the individual observations in $B$ into a single weighted
observation $b\mathrel{\mathop{:}}=Bw$. It then solves a single measurement
vector problem $Ax=b$ for this $b$ (we shall use $\ell_{1}$ throughout) and
checks if the computed solution $x^{*}$ is sufficiently sparse. If not, the
above steps are repeated with a different weight vector $w$; the algorithm
stops when a maximum number of trials is reached. If the support $\mathcal{I}$
of $x^{*}$ is small, we form
$A_{\mathcal{I}}=[A^{\scalebox{0.6}{$\downarrow$}{j}}]_{j\in\mathcal{I}}$, and
check if (4.2) has a solution $\bar{X}$ with zero residual. If this is the
case we have the nonzero rows of the solution $X^{*}$ in $\bar{X}$ and are
done. Otherwise, we simply proceed with the next $w$. The ReMBo algorithm
reduces to boosted $\ell_{1}$ by limiting the number of iterations to $r$ and
choosing $w=e_{i}$ in the $i$th iteration. We summarize the ReMBo-$\ell_{1}$
algorithm in Figure 6. The formulation given in [23] requires a user-defined
threshold on the cardinality of the support $\mathcal{I}$ instead of the fixed
threshold $m/2$. Ideally this threshold should be half of the spark [12] of A,
where
$\textrm{Spark}(A)\mathrel{\mathop{:}}=\min_{z\in\textrm{Ker}(A)\setminus\\{0\\}}\
\|z\|_{0}$
which is the number of nonzeros of the sparsest vector in the kernel of $A$;
any vector $x_{0}$ with fewer than $\textrm{Spark}(A)/2$ nonzeros is the
unique sparsest solution of $Ax=Ax_{0}=b$ [12]. Unfortunately, the spark is
prohibitively expensive to compute, but under the assumption that $A$ is in
general position, $\textrm{Spark}(A)=m+1$. Note that choosing a higher value
can help to recover signals with row sparsity exceeding $m/2$. However, in
this case it can no longer be guaranteed to be the sparsest solution.
given $A$, $B$. Set $\mathrm{Iteration}\leftarrow 0$ while _$\mathrm{Iteration} <\mathrm{MaxIteration}$_ do $w\leftarrow\mathrm{Random}(n,1)$ solve (1.2) with $b=Bw$ to get $x$ $\mathcal{I}\leftarrow\textrm{Supp}(x)$ if _$|\mathcal{I}| <m/2$_ then solve (4.2) to get $X$ if _$A_{\mathcal{I}}X=B$_ then $X^{*}=0$ $(X^{*})^{{j}{\scalebox{0.6}{$\rightarrow$}}}\leftarrow X^{{j}{\scalebox{0.6}{$\rightarrow$}}}$ for $j\in\mathcal{I}$ return solution $X^{*}$ $\mathrm{Iteration}\leftarrow\mathrm{Iteration}+1$ return failure |
---|---
Figure 5: The ReMBo-$\ell_{1}$ algorithm | Figure 6: Theoretical performance model for ReMBo on three problem instances with different sparsity levels $s$.
To derive the performance analysis of ReMBo, we fix a support $\mathcal{I}$ of
cardinality $s$, and consider only signals with nonzero entries on this
support. Each time we multiply $B$ by a weight vector $w$, we in fact create a
new problem with an $s$-sparse solution $x_{0}=X_{0}w$ corresponding with a
right-hand side $b=Bw=AX_{0}w=Ax_{0}$. As reflected in (2.2), recovery of
$x_{0}$ using $\ell_{1}$ depends only on its support and sign pattern.
Clearly, the more sign patterns in $x_{0}$ that we can generate, the higher
the probability of recovery. Moreover, due to the elimination of previously
tried sign patterns, the probability of recovery goes up with each new sign
pattern (excluding negation of previous sign patterns). The maximum number of
sign patterns we can check with boosted $\ell_{1}$ is the number of
observations $r$. The question thus becomes, how many different sign patterns
we can generate by taking linear combinations of the columns in $X_{0}$? (We
disregard the situation where elimination occurs and
$|\textrm{Supp}(X_{0}w)|<s$.) Equivalently, we can ask how many orthants in
$\mathbb{R}^{s}$ (each one corresponding to a different sign pattern) can be
properly intersected by the hyperplane given by the range of the $s\times r$
matrix $\bar{X}$ consisting of the nonzero rows of $X_{0}$ (with proper we
mean intersection of the interior). In Section 5.1 we derive an exact
expression for the maximum number of proper orthant intersections in
$\mathbb{R}^{n}$ by a hyperplane generated by $d$ vectors, denoted by
$C(n,d)$.
Based on the above reasoning, a good model for the recovery rate of $n\times
r$ matrices $X_{0}$ with $\textrm{Supp}_{\mathrm{row}}(X_{0})=\mathcal{I}<m/2$
using ReMBo is given by
$P_{\scriptscriptstyle
R}(A,\mathcal{I},r)=1-\prod_{i=1}^{C(|\mathcal{I}|,r)/2}\left[1-\frac{\mathcal{F}_{\mathcal{I}}(A\mathcal{C})}{\mathcal{F}_{\mathcal{I}}(\mathcal{C})-2(i-1)}\right].$
(5.1)
The term within brackets denotes the probability of failure and the fraction
represents the success rate, which is given by the ratio of the number of
faces $\mathcal{F}_{\mathcal{I}}(A\mathcal{C})$ that survived the mapping to
the total number of faces to consider. The total number reduces by two at each
trial because we can exclude the face $f$ we just tried, as well as $-f$. The
factor of two in $C(|\mathcal{I}|,r)/2$ is also due to this
symmetry111Henceforth we use the convention that the uniqueness of a sign
pattern is invariant under negation..
This model would be a bound for the average performance of ReMBo if the sign
patterns generated would be randomly sampled from the space of all sign
patterns on the given support. However, because it is generated from the
orthant intersections with a hyperplane, the actual pattern is highly
structured. Indeed, it is possible to imagine a situation where the
$(s-1)$-faces in $\mathcal{C}$ that perish in the mapping to $A\mathcal{C}$
have sign patterns that are all contained in the set generated by a single
hyperplane. Any other set of sign patterns would then necessarily include some
faces that survive the mapping and by trying all patterns in that set we would
recover $X_{0}$. In this case, the average recovery over all $X_{0}$ on that
support could be much higher than that given by (5.1). We do not yet fully
understand how the surviving faces of $\mathcal{C}$ are distributed. Due to
the simplicial structure of the facets of $\mathcal{C}$, we can expect the
faces that perish to be partially clustered (if a $(d-2)$-face perishes, then
so will the two $(d-1)$-faces whose intersection gives this face), and
partially unclustered (the faces that perish while all their sub-faces
survive). Note that, regardless of these patterns, recovery is guaranteed in
the limit whenever the number of unique sign patterns tried exceeds half the
number of faces lost,
$(|\mathcal{F}_{\mathcal{I}}(\mathcal{C})|-|\mathcal{F}_{\mathcal{I}}(\mathcal{AC})|)/2$.
Figure 6 illustrates the theoretical performance model based on $C(n,d)$, for
which we derive the exact expression in Section 5.1. In Section 5.2 we discuss
practical limitations, and in Section 5.3 we empirically look at how the
number of sign patterns generated grows with the number of normally
distributed vectors $w$, and how this affects the recovery rates. To allow
comparison between ReMBo and boosted $\ell_{1}$, we used the same matrix $A$
and support $\mathcal{I}_{s}$ used to generate Figure 4.
### 5.1 Maximum number of orthant intersections with subspace
###### Theorem 5.1.
Let $C(n,d)$ denote the maximum attainable number of orthant interiors
intersected by a hyperplane in $\mathbb{R}^{n}$ generated by $d$ vectors. Then
$C(n,1)=2$, $C(n,d)=2^{n}$ for $d\geq n$. In general, $C(n,d)$ is given by
$C(n,d)=C(n-1,d-1)+C(n-1,d)=2\sum_{i=0}^{d-1}{n-1\choose i}.$ (5.2)
###### Proof.
The number of intersected orthants is exactly equal to the number of proper
sign patterns (excluding zero values) that can be generated by linear
combinations of those $d$ vectors. When $d=1$, there can only be two such sign
patterns corresponding to positive and negative multiples of that vector, thus
giving $C(n,1)=2$. Whenever $d\geq n$, we can choose a basis for
$\mathbb{R}^{n}$ and add additional vectors as needed, and we can reach all
points, and therefore all $2^{n}=C(n,d)$ sign patterns.
For the general case (5.2), let $v_{1},\ldots,v_{d}$ be vectors in
$\mathbb{R}^{n}$ such that the affine hull with the origin,
$S=\mathrm{aff}\\{0,v_{1},\ldots,v_{d}\\}$, gives a hyperplane in
$\mathbb{R}^{n}$ that properly intersects the maximum number of orthants,
$C(n,d)$. Without loss of generality assume that vectors $v_{i}$,
$i=1,\ldots,d-1$ all have their $n$th component equal to zero. Now, let
$T=\mathrm{aff}\\{0,v_{1},\ldots,v_{d-1}\\}\subseteq\mathbb{R}^{n-1}$ be the
intersection of $S$ with the $(n-1)$-dimensional subspace of all points
$\mathcal{X}=\\{x\in\mathbb{R}^{n}\mid x_{n}=0\\}$, and let $C_{T}$ denote the
number of $(n-1)$-orthants intersected by $T$. Note that $T$ itself, as
embedded in $\mathbb{R}^{n}$, does not properly intersect any orthant.
However, by adding or subtracting an arbitrarily small amount of $v_{d}$, we
intersect $2C_{T}$ orthants; taking $v_{d}$ to be the $n$th column of the
identity matrix would suffice for that matter. Any other orthants that are
added have either $x_{n}>0$ or $x_{n}<0$, and their number does not depend on
the magnitude of the $n$th entry of $v_{d}$, provided it remains nonzero.
Because only the first $n-1$ entries of $v_{d}$ determine the maximum number
of additional orthants, the problem reduces to $\mathbb{R}^{n-1}$. In fact, we
ask how many new orthants can be added to $C_{T}$ taking the affine hull of
$T$ with $v$, the orthogonal projection $v_{d}$ onto $\mathcal{X}$. Since the
maximum orthants for this $d$-dimensional subspace in $\mathbb{R}^{n-1}$ is
given by $C(n-1,d)$, this number is clearly bounded by $C(n-1,d)-C_{T}$.
Adding this to $2C_{T}$, we have
$\displaystyle C(n,d)$ $\displaystyle\leq
2C_{T}+[C(n-1,d)-C_{T}]=C_{T}+C(n-1,d)$ (5.3) $\displaystyle\leq
C(n-1,d-1)+C(n-1,d)$ $\displaystyle\leq 2\sum_{i=0}^{d-1}{n-1\choose i}.$
The final expression follows by expanding the recurrence relations, which
generates (a part of) Pascal’s triangle, and combining this with $C(1,j)=2$
for $j\geq 1$. In the above, whenever there are free orthants in
$\mathbb{R}^{n-1}$, that is, when $d<n$, we can always choose the
corresponding part of $v_{d}$ in that orthant. As a consequence we have that
no hyperplane supported by a set of vectors can intersect the maximum number
of orthants when the range of those vectors includes some $e_{i}$.
We now show that this expression holds with equality. Let $U$ denote an
$(n-d)$-hyperplane in $\mathbb{R}^{n}$ that intersects the maximum $C(n,n-d)$
orthants. We now claim that in the interior of each orthant not intersected by
$U$ there exists a vector that is orthogonal to $U$. If this were not the case
then $T$ must be aligned with some $e_{i}$ and can therefore not be optimal.
The span of these orthogonal vectors generates a $d$-hyperplane $V$ that
intersects $C_{V}=2^{n}-C(n,n-d)$ orthants, and it follows that
$\displaystyle C(n,d)$ $\displaystyle\geq C_{V}=2^{n}-C(n,n-d)$
$\displaystyle\geq 2^{n}-2\sum_{i=0}^{n-d-1}{n-1\choose
i}=2\sum_{i=0}^{n-1}{n-1\choose i}-2\sum_{i=0}^{n-d-1}{n-1\choose i}$
$\displaystyle=2\sum_{n-d}^{n-1}{n-1\choose i}=2\sum_{i=0}^{d-1}{n-1\choose
i}\geq C(n,d),$
where the last inequality follows from (5.3). Consequently, all inequalities
hold with equality. ∎
###### Corollary 5.2.
Given $d\leq n$, then $C(n,d)=2^{n}-C(n,n-d)$, and $C(2d,d)=2^{2d-1}$.
###### Corollary 5.3.
A hyperplane $\mathcal{H}$ in $\mathbb{R}^{n}$, defined as the range of
$V=[v_{1},\ v_{2},\ldots,\ v_{d}]$, intersects the maximum number of orthants
$C(n,d)$ whenever $\mathop{\hbox{\rm rank}}(V)=n$, or when
$e_{i}\not\in\mathop{\hbox{\rm range}}(V)$ for $i=1,\ldots,n$.
### 5.2 Practical considerations
In practice it is generally not feasible to generate all of the
$C(|\mathcal{I}|,r)/2$ unique sign patterns. This means that we would have to
replace this term in (5.1) by the number of unique patterns actually tried.
For a given $X_{0}$ the actual probability of recovery is determined by a
number of factors. First of all, the linear combinations of the columns of the
nonzero part of $\bar{X}$ prescribe a hyperplane and therefore a set of
possible sign patterns. With each sign pattern is associated a face in
$\mathcal{C}$ that may or may not map to a face in $A\mathcal{C}$. In
addition, depending on the probability distribution from which the weight
vectors $w$ are drawn, there is a certain probability for reaching each sign
pattern. Summing the probability of reaching those patterns that can be
recovered gives the probability $P(A,\mathcal{I},X_{0})$ of recovering with an
individual random sample $w$. The probability of recovery after $t$ trials is
then of the form
$1-[1-P(A,\mathcal{I},X_{0})]^{t}.$
To attain a certain sign pattern $\bar{e}$, we need to find an $r$-vector $w$
such that $\mathop{\hbox{\rm sign}}(\bar{X}w)=\bar{e}$. For a positive sign on
the $j$th position of the support we can take any vector $w$ in the open
halfspace $\\{w\mid\bar{X}^{{j}{\scalebox{0.6}{$\rightarrow$}}}w>0\\}$, and
likewise for negative signs. The region of vectors $w$ in $\mathbb{R}^{r}$
that generates a desired sign pattern thus corresponds to the intersection of
$|\mathcal{I}|$ open halfspaces. The measure of this intersection as a
fraction of $\mathbb{R}^{r}$ determines the probability of sampling such a
$w$. To formalize, define $\mathcal{K}$ as the cone generated by the rows of
$-\mathop{\hbox{\rm diag}}(\bar{e})\bar{X}$, and the unit Euclidean
$(k-1)$-sphere $\mathcal{S}^{k-1}=\\{x\in\mathbb{R}^{r}\mid\|x\|_{2}=1\\}$.
The intersection of halfspaces then corresponds to the interior of the polar
cone of $\mathcal{K}$: $\mathcal{K}^{\circ}=\\{x\in\mathbb{R}^{r}\mid
x^{T}\\!y\leq 0,\ \forall y\in\mathcal{K}\\}$. The fraction of
$\mathbb{R}^{r}$ taken up by $\mathcal{K}^{\circ}$ is given by the
$(k-1)$-content of $\mathcal{S}^{k-1}\cap\mathcal{K}^{\circ}$ to the
$(k-1)$-content of $\mathcal{S}^{k-1}$ [21]. This quantity coincides precisely
with the definition of the external angle of $\mathcal{K}$ at the origin.
### 5.3 Experiments
In this section we illustrate the theoretical results from Section 5 and
examine some practical considerations that affect the performance of ReMBo.
For all experiments that require the matrix $A$, we use the same $20\times 80$
matrix that was used in Section 4, and likewise for the supports
$\mathcal{I}_{s}$. To solve (1.2), we again use CVX in conjunction with SDPT3.
We consider $x_{0}$ to be recovered from $b=Ax_{0}=AX_{0}w$ if
$\|x^{*}-x_{0}\|_{\infty}\leq 10^{-5}$, where $x^{*}$ is the computed
solution.
The experiments that are concerned with the number of unique sign patterns
generated depend only on the $s\times r$ matrix $\bar{X}$ representing the
nonzero entries of $X_{0}$. Because an initial reordering of the rows does not
affect the number of patterns, those experiments depend only on $\bar{X}$,
$s=|\mathcal{I}|$, and the number of observations $r$; the exact indices in
the support set $\mathcal{I}$ are irrelevant for those tests.
#### 5.3.1 Generation of unique sign patterns
The practical performance of ReMBo depends on its ability to generate as many
different sign patterns using the columns in $X_{0}$ as possible. A natural
question to ask then is how the number of such patterns grows with the number
of randomly drawn samples $w$. Although this ultimately depends on the
distribution used for generating the entries in $w$, we shall, for sake of
simplicity, consider only samples drawn from the normal distribution. As an
experiment we take a $10\times 5$ matrix $\bar{X}$ with normally-distributed
entries, and over $10^{8}$ trials record how often each sign-pattern (or
negation) was reached, and in which trial they were first encountered. The
results of this experiment are summarized in Figure 7. From the distribution
in Figure 7(b) it is clear that the occurrence levels of different orthants
exhibits a strong bias. The most frequently visited orthant pairs were reached
up to $7.3\times 10^{6}$ times, while others, those hard to reach using
weights from the normal distribution, were observed only four times over all
trials. The efficiency of ReMBo depends on the rate of encountering new sign
patterns. Figure 7(c) shows how the average rate changes over the number of
trials. The curves in Figure 7(d) illustrate the theoretical probability of
recovery in (5.1), with $C(n,d)/2$ replaced by the number of orthant pairs at
a given iteration, and with face counts determined as in Section 4, for three
instances with support cardinality $s=10$, and observations $r=5$.
|
---|---
(a) | (b)
|
(c) | (d)
Figure 7: Sampling the sign patterns for a $10\times 5$ matrix $\bar{X}$, with
(a) number of unique sign patterns versus number of trials, (b) relative
frequency with which each orthant is sampled, (c) average number of new sign
patterns per iteration as a function of iterations, and (d) theoretical
probability of recovery using ReMBo for three instances of $X_{0}$ with row
sparsity $s=10$, and $r=5$ observations.
#### 5.3.2 Role of $\bar{X}$.
Although the number of orthants that a hyperplane can intersect does not
depend on the basis with which it was generated, this choice does greatly
influence the ability to sample those orthants. Figure 8 shows two ways in
which this can happen. In part (a) we sampled the number of unique sign
patterns for two different $9\times 5$ matrices $\bar{X}$, each with columns
scaled to unit $\ell_{2}$-norm. The entries of the first matrix were
independently drawn from the normal distribution, while those in the second
were generated by repeating a single column drawn likewise and adding small
random perturbations to each entry. This caused the average angle between any
pair of columns to decrease from $65$ degrees in the random matrix to a mere
$8$ in the perturbed matrix, and greatly reduces the probability of reaching
certain orthants. The same idea applies to the case where $d\geq n$, as shown
in part (b) of the same figure. Although choosing $d$ greater than $n$ does
not increase the number of orthants that can be reached, it does make reaching
them easier, thus allowing ReMBo to work more efficiently. Hence, we can
expect ReMBo to have higher recovery on average when the number of columns in
$X_{0}$ increases and when they have a lower mutual coherence
$\mu(X)=\min_{i\neq j}|x_{i}^{T}x_{j}|/(\|x_{i}\|_{2}\cdot\|x_{j}\|_{2})$.
|
---|---
(a) | (b)
Figure 8: Number of unique sign patterns for (a) two $9\times 5$ matrices
$\bar{X}$ with columns scaled to unit $\ell_{2}$-norm; one with entries drawn
independently from the normal distribution, and one with a single random
column repeated and random perturbations added, and (b) $10\times r$ matrices
with $r=10,12,15$.
#### 5.3.3 Limiting the number of iterations
The number of iterations used in the previous experiments greatly exceeds that
what is practically feasible: we cannot afford to run ReMBo until all possible
sign patterns have been tried, even if there was a way detect that the limit
had been reached. Realistically, we should set the number of iterations to a
fixed maximum that depends on the computational resources available, and the
problem setting.
In Figure 7 we show the unique orthant count as a function of iterations and
the predicted recovery rate. When using only a limited number of iterations it
is interesting to know what the distribution of unique orthant counts looks
like. To find out, we drew 1,000 random $\bar{X}$ matrices for each size
$s\times r$, with $s=10$ nonzero rows fixed, and the number of columns ranging
from $r=1,\ldots,20$. For each $\bar{X}$ we counted the number of unique sign
patterns attained after respectively 1,000 and 10,000 iterations. The
resulting minimum, maximum, and median values are plotted in Figure 9(a) along
with the theoretical maximum. More interestingly of course is the average
recovery rate of ReMBo with those number of iterations. For this test we again
used the $20\times 80$ matrix $A$ with predetermined support $\mathcal{I}$,
and with success or failure of each sign pattern on that support precomputed.
For each value of $r=1,\ldots,20$ we generated random matrices $X$ on
$\mathcal{I}$ and ran ReMBo with the maximum number of iterations set to 1,000
and 10,000. To save on computing time, we compared the on-support sign pattern
of each combined coefficient vector $Xw$ to the known results instead of
solving $\ell_{1}$. The average recovery rate thus obtained is plotted in
Figures 9(b)–(c), along with the average of the predicted performance using
(5.1) with $C(n,d)/2$ replaced by orthant counts found in the previous
experiment.
| |
---|---|---
(a) | (b) | (c)
Figure 9: Effect of limiting the number of weight vectors $w$ on (a) the
distribution of unique orthant counts for $10\times k$ random matrices
$\bar{X}$, solid lines give the median number and the dashed lines indicate
the minimum and maximum values, the top solid line is the theoretical maximum;
(b–c) the average performance of the ReMBo-$\ell_{1}$ algorithm (solid) for
fixed $20\times 80$ matrix $A$ and three different support sizes $r=8,9,10$,
along with the average predicted performance (dashed). The support patterns
used are the same as those used for Figure 4.
## 6 Conclusions
The MMV problem is often solved by minimizing the sum-of-row norms of the
unknown coefficients $X$. We show that the (local) uniform recovery
properties, i.e., recovery of all $X_{0}$ with a fixed row support
$\mathcal{I}=\textrm{Supp}_{\mathrm{row}}(X_{0})$, cannot exceed that of
$\ell_{1,1}$, the sum of $\ell_{1}$ norms. This is despite the fact that
$\ell_{1,1}$ reduces to solving the basis pursuit problem (1.2) for each
column separately, which does not take advantage of the fact that all vectors
in $X_{0}$ are assumed to have the same support. A consequence of this
observation is that the use of restricted isometry techniques to analyze
(local) uniform recovery using sum-of-norm minimization can at best give
improved bounds on $\ell_{1}$ recovery.
Empirically, minimization with $\ell_{1,2}$, the sum of $\ell_{2}$ norms,
clearly outperforms $\ell_{1,1}$ on individual problem instances: for supports
where uniform recovery fails, $\ell_{1,2}$ recovers more cases than
$\ell_{1,1}$. We construct cases where $\ell_{1,2}$ succeeds while
$\ell_{1,1}$ fails, and vice versa. From the construction where only
$\ell_{1,2}$ succeeds it also follows that the relative magnitudes of the
coefficients in $X_{0}$ matter for recovery. This is unlike $\ell_{1,1}$
recovery, where only the support and the sign patterns matter. This implies
that the notion of faces, so useful in the analysis of $\ell_{1}$, disappears.
We show that the performance of $\ell_{1,1}$ outside the uniform-recovery
regime degrades rapidly as the number of observations increases. We can turn
this situation around, and increase the performance with the number of
observations by using a boosted-$\ell_{1}$ approach. This technique aims to
uncover the correct support based on basis pursuit solutions for individual
observations. Boosted-$\ell_{1}$ is a special case of the ReMBo algorithm
which repeatedly takes random combinations of the observations, allowing it to
sample many more sign patterns in the coefficient space. As a result, the
potential recovery rates of ReMBo (at least in combination with an $\ell_{1}$
solver) are a much higher than boosted-$\ell_{1}$. ReMBo can be used in
combination with any solver for the single measurement problem $Ax=b$,
including greedy approaches and reweighted $\ell_{1}$ [4]. The recovery rate
of greedy approaches may be lower than $\ell_{1}$ but the algorithms are
generally much faster, thus giving ReMBo the chance to sample more random
combinations. Another advantage of ReMBo, even more so than
boosted-$\ell_{1}$, is that it can be easily parallelized.
Based on the geometrical interpretation of ReMBo-$\ell_{1}$ (cf. Figure 6), we
conclude that, theoretically, its performance does not increase with the
number of observations after this number reaches the number of nonzero rows.
In addition we develop a simplified model for the performance of
ReMBo-$\ell_{1}$. To improve the model we would need to know the distribution
of faces in the cross-polytope $\mathcal{C}$ that map to faces on
$A\mathcal{C}$, and the distribution of external angles for the cones
generated by the signed rows of the nonzero part of $X_{0}$.
It would be very interesting to compare the recovery performance between
$\ell_{1,2}$ and ReMBo-$\ell_{1}$. However, we consider this beyond the scope
of this paper.
All of the numerical experiments in this paper are reproducible. The scripts
used to run the experiments and generate the figures can be downloaded from
http://www.cs.ubc.ca/~mpf/jointsparse.
## Acknowledgments
The authors would like to give their sincere thanks to Özgür Yılmaz and Rayan
Saab for their thoughtful comments and suggestions during numerous
discussions.
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|
arxiv-papers
| 2009-04-14T05:54:33 |
2024-09-04T02:49:01.855644
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ewout van den Berg and Michael P. Friedlander",
"submitter": "Michael Friedlander",
"url": "https://arxiv.org/abs/0904.2051"
}
|
0904.2088
|
# Dynamics of Particles in Non Scaling FFAG Accelerators
James K. Jones james.jones@stfc.ac.uk Bruno D. Muratori
bruno.muratori@stfc.ac.uk Susan L. Smith susan.smith@stfc.ac.uk Stephan I.
Tzenov stephan.tzenov@stfc.ac.uk STFC Daresbury Laboratory, Daresbury,
Warrington, Cheshire, WA4 4AD, United Kingdom
###### Abstract
Non scaling Fixed-Field Alternating Gradient (FFAG) accelerators have an
unprecedented potential for muon acceleration, as well as for medical purposes
based on carbon and proton hadron therapy. They also represent a possible
active element for an Accelerator Driven Subcritical Reactor (ADSR). Starting
from first principle the Hamiltonian formalism for the description of the
dynamics of particles in non scaling FFAG machines has been developed. The
stationary reference (closed) orbit has been found within the Hamiltonian
framework. The dependence of the path length on the energy deviation has been
described in terms of higher order dispersion functions. The latter have been
used subsequently to specify the longitudinal part of the Hamiltonian. It has
been shown that higher order phase slip coefficients should be taken into
account to adequately describe the acceleration in non scaling FFAG
accelerators. A complete theory of the fast (serpentine) acceleration in non
scaling FFAGs has been developed. An example of the theory is presented for
the parameters of the Electron Machine with Many Applications (EMMA), a
prototype electron non scaling FFAG to be hosted at Daresbury Laboratory.
###### pacs:
29.20.-c, 29.20.D-, 41.85.-p
## I Introduction
Fixed-Field Alternating Gradient (FFAG) accelerators were proposed half
century ago KL ; Kol ; Sy ; Ke , when acceleration of electrons was first
demonstrated. These machines, which were intensively studied in the 1950s and
1960s but never progressed beyond the model stage, have in recent years become
the focus of renewed attention. Acceleration of protons has been recently
achieved at the KEK Proof-of-Principle (PoP) proton FFAG Ai .
To avoid the slow crossing of betatron resonances associated with a typical
low energy-gain per turn, the first FFAGs designed and constructed so far have
been based on the ”scaling” principle. The latter implies that the orbit shape
and betatron tunes must be kept fixed during the acceleration process. Thus,
magnets must be built with constant field index, while in the case of spiral-
sector designs the spiral angle must be constant as well. Machines of this
type use conventional magnets with the bending and focusing field being kept
constant during acceleration. The latter alternate in sign, providing a more
compact radial extension and consequently smaller aperture as compared to the
AVF cyclotrons. The ring essentially consists of a sequence of short cells
with very large periodicity.
Non scaling FFAG machines have until recently been considered as an
alternative. The bending and the focusing is provided simultaneously by
focusing and defocusing quadrupole magnets repeating in an alternating
sequence. There is a number of advantages of the non scaling FFAG lattice as
compared to the scaling one, among which are the relatively small transverse
magnet aperture (tending to be much smaller than the one for scaling machines)
and the lower field strength. Unfortunately this lattice leads to a large
betatron tune variation across the required energy range for acceleration as
opposed to the scaling lattice. As a consequence several resonances are
crossed during the acceleration cycle, some of them nonlinear created by the
magnetic field imperfections, as well as half-integer and integer ones. A
possible bypass to this problem is the rapid acceleration (of utmost
importance for muons), which allows betatron resonances no time to essentially
damage beam quality.
Because non scaling FFAG accelerators have otherwise very desirable features,
it is important to investigate analytically and numerically some of the
peculiarities of the beam dynamics, the new type of fast acceleration regime
(so-called serpentine acceleration) and the effects of crossing of linear as
well as nonlinear resonances. Moreover, it is important to examine the most
favorable phase at which the cavities need to be set for the optimal
acceleration. Some of these problems will be discussed in the present paper.
An example of the theory developed here is presented for the parameters of the
Electron Machine with Many Applications (EMMA) emma , a prototype electron non
scaling FFAG to be hosted at Daresbury Laboratory. The Accelerators and Lasers
In Combined Experiments (ALICE) accelerator alice is used as an injector to
the EMMA ring. The energy delivered by this injector can vary from a $10$ to
$20$ MeV single bunch train with a bunch charge of $16$ to $32$ pC at a rate
of $1$ to $20$ Hz. ALICE is presently designed to deliver bunches which are
around $4$ ps and $8.35$ MeV from the exit of the booster of its injector
line. These are then accelerated to $10$ or $20$ MeV in the main ALICE linac
after which they are sent to the EMMA injection line. The EMMA injection line
ends with a septum for injection into the EMMA ring itself followed by two
kickers so as to direct the beam onto the correct, energy dependent,
trajectory. After circulation in the EMMA ring, the electron bunches are
extracted using what is almost a mirror image of the injection setup with two
kickers followed by an extraction septum. The beam is then transported to a
diagnostic line whose purpose it is to analyze in as much detail as possible
the effect the non scaling FFAG has had on the bunch.
The paper is organized as follows. Firstly, we review some generalities and
first principles of the Hamiltonian formalism Tzenov suitably modified to
cover the case of a non scaling FFAG lattice. Subsequently the synchrobetatron
framework is applied to determine the energy dependent reference orbit.
Stability of motion about the stationary reference orbit is described in terms
of betatron oscillations with energy dependent Twiss parameters and betatron
tunes. Dispersion, measuring the effect of energy variation on the path length
along the reference orbit is an essential feature of non scaling FFAGs. Within
the developed synchrobetatron formalism higher order dispersion functions have
been introduced and their contribution to the longitudinal dynamics has been
further analyzed. Finally, a complete description of the so-called serpentine
acceleration in non scaling lepton FFAGs is given together with conclusions.
The calculations of the reference orbit and phase stability are detailed in
the appendices.
## II Generalities and First Principles
Let the ideal (design) trajectory of a particle in an accelerator be a planar
curve with curvature $K$. The Hamiltonian describing the motion of a particle
in a natural coordinate system attached to the orbit thus defined is Tzenov :
$H=-{\left(1+Kx\right)}{\sqrt{{\frac{{\left({\mathcal{H}}-q\varphi\right)}^{2}}{c^{2}}}-m_{p0}^{2}c^{2}-{\left(P_{x}-qA_{x}\right)}^{2}-{\left(P_{z}-qA_{z}\right)}^{2}}}-q{\left(1+Kx\right)}A_{s},$
(1)
where $m_{p0}$ is the rest mass of the particle. The guiding magnetic field
can be represented as a gradient of a function $\psi{\left(x,z;s\right)}$
$\mathbf{B}=\nabla\psi,$ (2)
where the latter satisfies the Laplace equation
$\nabla^{2}\psi=0.$ (3)
Using the median symmetry of the machine, it is straightforward to show that
$\psi$ can be written in the form
$\psi={\left(a_{0}+a_{1}x+{\frac{a_{2}x^{2}}{2!}}+\dots\right)}z$
$-{\left(b_{0}+b_{1}x+{\frac{b_{2}x^{2}}{2!}}+\dots\right)}{\frac{z^{3}}{3!}}+{\left(c_{0}+c_{1}x+\dots\right)}{\frac{z^{5}}{5!}}+\dots.$
(4)
Inserting the above expression into the Laplace equation (3), one readily
finds relations between the coefficients $b_{k}$ and $c_{k}$ on one hand and
$a_{k}$ on the other:
$b_{0}=a_{0}^{\prime\prime}+Ka_{1}+a_{2},$ (5)
$b_{1}=-2Ka_{0}^{\prime\prime}-K^{\prime}a_{0}^{\prime}+a_{1}^{\prime\prime}-K^{2}a_{1}+Ka_{2}+a_{3},$
(6)
$b_{2}=6K^{2}a_{0}^{\prime\prime}+6KK^{\prime}a_{0}^{\prime}-4Ka_{1}^{\prime\prime}-2K^{\prime}a_{1}^{\prime}$
$+a_{2}^{\prime\prime}+2K^{3}a_{1}-2K^{2}a_{2}+Ka_{3}+a_{4},$ (7)
$c_{0}=b_{0}^{\prime\prime}+Kb_{1}+b_{2}.$ (8)
Prime in the above expressions implies differentiation with respect to the
longitudinal coordinate $s$. The coefficients $a_{k}$ have a very simple
meaning:
$a_{0}={\left(B_{z}\right)}_{x,z=0},\qquad a_{1}={\left({\frac{\partial
B_{z}}{\partial x}}\right)}_{x,z=0},$
$a_{2}={\left({\frac{\partial^{2}B_{z}}{\partial x^{2}}}\right)}_{x,z=0}.$ (9)
In other words, this implies that, provided the vertical component $B_{z}$ of
the magnetic field and its derivatives with respect to the horizontal
coordinate $x$ are known in the median plane, one can in principle reconstruct
the entire field chart.
The vector potential $\mathbf{A}$ can be represented as
$A_{x}=-z\overline{F}{\left(x,z;s\right)},\quad
A_{z}=x\overline{F}{\left(x,z;s\right)},\quad
A_{s}=\overline{G}{\left(x,z;s\right)},$ (10)
where the Poincar${\grave{\rm e}}$ gauge condition
$xA_{x}+zA_{z}=0,$ (11)
written in the natural coordinate system has been used. From Maxwell’s
equation
$\mathbf{B}=\nabla\times\mathbf{A},$ (12)
we obtain
$2\overline{F}+{\left(x\partial_{x}+z\partial_{z}\right)}\overline{F}=B_{s},$
(13)
${\frac{Kx}{1+Kx}}\overline{G}+{\left(x\partial_{x}+z\partial_{z}\right)}\overline{G}=zB_{x}-xB_{z}.$
(14)
Applying Euler’s theorem for homogeneous functions, we can write
$\overline{F}={\frac{1}{2}}B_{s}^{(0)}+{\frac{1}{3}}B_{s}^{(1)}+{\frac{1}{4}}B_{s}^{(2)}+\dots,$
(15)
${\overline{G}}_{u}={\left(1+{\frac{Kx}{2}}\right)}B_{u}^{(0)}+{\left({\frac{1}{2}}+{\frac{Kx}{3}}\right)}B_{u}^{(1)}$
$+{\left({\frac{1}{3}}+{\frac{Kx}{4}}\right)}B_{u}^{(2)}+\dots,$ (16)
${\overline{G}}={\frac{z{\overline{G}}_{x}-x{\overline{G}}_{z}}{1+Kx}}.$ (17)
Here $u=(x,z)$ and $B_{\alpha}^{(k)}$ denotes homogeneous polynomials in $x$
and $z$ of order $k$, representing the corresponding parts of the components
of the magnetic field $\mathbf{B}={\left(B_{x},B_{z},B_{s}\right)}$. Thus,
having found the magnetic field represented by equation (4), it is
straightforward to calculate the vector potential $\mathbf{A}$.
The accelerating field in AVF cyclotrons and FFAG machines can be represented
by a scalar potential $\varphi$ (the corresponding vector potential
$\mathbf{A}=0$). Due to the median symmetry, we have
$\varphi=A_{0}+A_{1}x+{\frac{A_{2}x^{2}}{2!}}+\dots$
$-{\left(B_{0}+B_{1}x+{\frac{B_{2}x^{2}}{2!}}+\dots\right)}{\frac{z^{2}}{2!}}$
$+{\left(C_{0}+C_{1}x+\dots\right)}{\frac{z^{4}}{4!}}+\dots.$ (18)
Inserting the above expansion into the Laplace equation for $\varphi$, we
obtain similar relations between $B_{k}$ and $C_{k}$ on one hand and $A_{k}$
on the other, which are analogous to those relating $b_{k}$, $c_{k}$ and
$a_{k}$.
We consider the canonical transformation, specified by the generating function
$S_{2}{\left(x,z,{\cal
T},{\widehat{P}}_{x},{\widehat{P}}_{z},E;s\right)}=x{\widehat{P}}_{x}+z{\widehat{P}}_{z}+{\cal
T}E$ $+q\int{\rm d}{\cal T}\varphi{\left(x,z,{\cal T};s\right)},$ (19)
where
${\cal T}=-t,$ (20)
is a canonical variable canonically conjugate to $\mathcal{H}$. The relations
between the new and the old variables are
$\widehat{u}={\frac{\partial S_{2}}{\partial{\widehat{P}}_{u}}}=u,\qquad
u={\left(x,z\right)},\qquad\widehat{\cal T}={\frac{\partial S_{2}}{\partial
E}}={\cal T},$ (21) $P_{u}={\frac{\partial S_{2}}{\partial
u}}={\widehat{P}}_{u}-q\int{\rm d}{\cal T}E_{u}{\left(x,z,{\cal T};s\right)}$
$={\widehat{P}}_{u}-q{\widetilde{E}}_{u}{\left(x,z,{\cal T};s\right)},\qquad
E_{u}=-{\frac{\partial\varphi}{\partial u}},$ (22)
$\mathcal{H}={\frac{\partial S_{2}}{\partial{\cal
T}}}=E+q\varphi{\left(x,z,{\cal T};s\right)}=m_{p0}\gamma
c^{2}+q\varphi{\left(x,z,{\cal T};s\right)}.$ (23)
The new Hamiltonian acquires now the form
$\widehat{H}=-{\left(1+Kx\right)}{\sqrt{{\frac{E^{2}}{c^{2}}}-m_{p0}^{2}c^{2}-{\left(\widehat{P}_{x}-q\widetilde{E}_{x}-qA_{x}\right)}^{2}-{\left(\widehat{P}_{z}-q\widetilde{E}_{z}-qA_{z}\right)}^{2}}}-q{\left(1+Kx\right)}{\left(A_{s}+\widetilde{E}_{s}\right)},$
(24)
where
$\widetilde{E}_{s}=\int{\rm d}{\cal T}E_{s}{\left(x,z,{\cal T};s\right)}$
$=-{\frac{1}{1+Kx}}\int{\rm d}{\cal T}{\frac{\partial\varphi{\left(x,z,{\cal
T};s\right)}}{\partial s}}.$ (25)
We introduce the new scaled variables
$\widetilde{P}_{u}={\frac{\widehat{P}_{u}}{p_{0}}}={\frac{\widehat{P}_{u}}{m_{p0}c}},\quad\Theta=c{\cal
T},\quad\gamma={\frac{E}{E_{p}}}={\frac{E}{m_{p0}c^{2}}}.$ (26)
The new scaled Hamiltonian can be expressed as
$\widetilde{H}={\frac{\widehat{H}}{p_{0}}}=-{\left(1+Kx\right)}{\sqrt{\gamma^{2}-1-{\left(\widetilde{P}_{x}-\widetilde{q}\widetilde{E}_{x}-\widetilde{q}A_{x}\right)}^{2}-{\left(\widetilde{P}_{z}-\widetilde{q}\widetilde{E}_{z}-\widetilde{q}A_{z}\right)}^{2}}}-\widetilde{q}{\left(1+Kx\right)}{\left(A_{s}+\widetilde{E}_{s}\right)},$
(27)
where
$\widetilde{q}={\frac{q}{p_{0}}}.$ (28)
The quantities $\widetilde{E}_{x}$ and $\widetilde{E}_{z}$ can be neglected as
compared to the components of the vector potential $\mathbf{A}$, so that
$\widetilde{H}=\beta\gamma{\left(1+Kx\right)}{\left[-{\sqrt{1-{\left(\overline{P}_{x}-\overline{q}A_{x}\right)}^{2}-{\left(\overline{P}_{z}-\overline{q}A_{z}\right)}^{2}}}-\overline{q}A_{s}\right]}-\widetilde{q}{\left(1+Kx\right)}\widetilde{E}_{s},$
(29)
where now
$\overline{q}={\frac{q}{p}}={\frac{q}{\beta\gamma
p_{0}}},\qquad\overline{P}_{u}={\frac{\widehat{P}_{u}}{p}}={\frac{\widehat{P}_{u}}{\beta\gamma
p_{0}}},\qquad u=(x,z).$ (30)
Since $\overline{P}_{u}$ and $u$ are small deviations, we can expand the
square root in power series in the canonical variables $x$, $\overline{P}_{x}$
and $z$, $\overline{P}_{z}$. Tedious algebra yields
$\widetilde{H}=\widetilde{H}_{0}+\widetilde{H}_{1}+\widetilde{H}_{2}+\widetilde{H}_{3}+\widetilde{H}_{4}+\dots,$
(31)
$\widetilde{H}_{0}=-\beta\gamma-\widetilde{q}{\left(1+Kx\right)}\widetilde{E}_{s},$
(32) $\widetilde{H}_{1}=\beta\gamma{\left(\overline{q}a_{0}-K\right)}x,$ (33)
$\widetilde{H}_{2}={\frac{\beta\gamma}{2}}{\left({\overline{P}}_{x}^{2}+{\overline{P}}_{z}^{2}\right)}+{\frac{\widetilde{q}}{2}}{\left[{\left(Ka_{0}+a_{1}\right)}x^{2}-a_{1}z^{2}\right]},$
(34)
$\widetilde{H}_{3}={\frac{\beta\gamma}{2}}Kx{\left({\overline{P}}_{x}^{2}+{\overline{P}}_{z}^{2}\right)}+{\frac{\widetilde{q}a_{0}^{\prime}z}{3}}{\left(z\overline{P}_{x}-x\overline{P}_{z}\right)}+{\frac{\widetilde{q}}{3}}{\left[{\left(Ka_{1}+{\frac{a_{2}}{2}}\right)}x^{3}-{\left(Ka_{1}+a_{2}+{\frac{b_{0}}{2}}\right)}xz^{2}\right]},$
(35)
$\widetilde{H}_{4}={\frac{\beta\gamma}{8}}{\left({\overline{P}}_{x}^{2}+{\overline{P}}_{z}^{2}\right)}^{2}+{\frac{\widetilde{q}xz}{12}}{\left(Ka_{0}^{\prime}+3a_{1}^{\prime}\right)}{\left(z\overline{P}_{x}-x\overline{P}_{z}\right)}+{\frac{\overline{q}^{2}\beta\gamma
a_{0}^{\prime 2}z^{2}}{18}}{\left(x^{2}+z^{2}\right)}$
$+{\frac{\widetilde{q}}{4}}{\left[{\left({\frac{Ka_{2}}{2}}+{\frac{a_{3}}{6}}\right)}x^{4}-{\left(Ka_{2}+{\frac{a_{3}}{3}}+{\frac{Kb_{0}}{2}}+{\frac{b_{1}}{2}}\right)}x^{2}z^{2}+{\frac{b_{1}}{6}}z^{4}\right]}.$
(36)
The Hamiltonian decomposition (31) represents the milestone of the
synchrobetatron formalism. For instance, ${\widetilde{H}_{0}}$ governs the
longitudinal motion, ${\widetilde{H}_{1}}$ describes linear coupling between
longitudinal and transverse degrees of freedom and is the basic source of
dispersion. The part ${\widetilde{H}_{2}}$ is responsible for linear betatron
motion and chromaticity, while the remainder describes higher order
contributions.
## III The Synchro-Betatron Formalism and the Reference Orbit
In the present paper we consider a FFAG lattice with polygonal structure. To
define and subsequently calculate the stationary reference orbit, it is
convenient to use a global Cartesian coordinate system whose origin is located
in the center of the polygon. To describe step by step the fraction of the
reference orbit related to a particular side of the polygon, we rotate each
time the axes of the coordinate system by the polygon angle
$\Theta_{p}=2\pi/N_{L}$, where $N_{L}$ is the number of sides of the polygon.
Let $X_{e}$ and $P_{e}$ denote the reference orbit and the reference momentum,
respectively. The vertical component of the magnetic field in the median plane
of a perfectly linear machine can be written as
$B_{z}{\left(X_{e};s\right)}=a_{1}(s){\left[X_{e}-X_{c}-d(s)\right]},$
$a_{0}{\left(X_{e};s\right)}=B_{z}{\left(X_{e};s\right)},$ (37)
where $s$ is the distance along the polygon side, and $X_{c}$ is the distance
of the side of the polygon from the center of the machine
$X_{c}={\frac{L_{p}}{2\tan(\Theta_{p}/2)}}.$ (38)
Here $L_{p}$ is the length of the polygon side which actually represents the
periodicity parameter of the lattice. Usually $X_{c}$ is related to an
arbitrary energy in the range from injection to extraction energy. In the case
of EMMA it is related to the 15 MeV orbit. The quantity $d(s)$ in equation
(37) is the relative offset of the magnetic center in the quadrupoles with
respect to the corresponding side of the polygon. In what follows [see
equations (47) and (50)] $d_{F}$ corresponds to the offset in the focusing
quadrupoles and $d_{D}$ corresponds to the one in the defocusing quadrupoles.
Similarly, $a_{F}$ and $a_{D}$ stand for the particular value of $a_{1}$ in
the focusing and the defocusing quadrupoles, respectively.
A design (reference) orbit corresponding to a local curvature
$K{\left(X_{e};s\right)}$ can be defined according to the relation
$K{\left(X_{e};s\right)}={\frac{q}{p_{0}\beta_{e}\gamma_{e}}}B_{z}{\left(X_{e};s\right)},$
(39)
where $\gamma_{e}$ is the energy of the reference particle. In terms of the
reference orbit position $X_{e}(s)$ the equation for the curvature can be
written as
$X_{e}^{\prime\prime}={\frac{q}{p_{0}\beta_{e}\gamma_{e}}}{\left(1+X_{e}^{\prime
2}\right)}^{3/2}B_{z}{\left(X_{e};s\right)},$ (40)
where the prime implies differentiation with respect to $s$.
To proceed further, we notice that equation (40) parameterizing the local
curvature can be derived from an equivalent Hamiltonian
$H_{e}{\left(X_{e},P_{e};s\right)}=-{\sqrt{\beta_{e}^{2}\gamma_{e}^{2}-P_{e}^{2}}}-{\widetilde{q}}\int{\rm
d}X_{e}B_{z}{\left(X_{e};s\right)}.$ (41)
Taking into account Hamilton’s equations of motion
$X_{e}^{\prime}={\frac{P_{e}}{\sqrt{\beta_{e}^{2}\gamma_{e}^{2}-P_{e}^{2}}}},\qquad\qquad
P_{e}^{\prime}={\widetilde{q}}B_{z}{\left(X_{e};s\right)},$ (42)
and using the relation
$P_{e}={\frac{\beta_{e}\gamma_{e}X_{e}^{\prime}}{\sqrt{1+X_{e}^{\prime 2}}}},$
(43)
we readily obtain equation (40). Note also that the Hamiltonian (41) follows
directly from the scaled Hamiltonian (27) with $x=0$,
${\widetilde{P}}_{x}=P_{e}$, ${\widetilde{P}}_{z}=0$, $A_{x}=A_{z}=0$ and the
accelerating cavities being switched off respectively.
Hamilton’s equations of motion (42) can be linearized and subsequently solved
approximately by assuming that
$P_{e}\ll\beta_{e}\gamma_{e}.$ (44)
Thus, assuming electrons ($q=-e$), we have
$P_{e}=\beta_{e}\gamma_{e}X_{e}^{\prime},\qquad
X_{e}^{\prime\prime}=-{\frac{ea_{1}(s)}{p_{0}\beta_{e}\gamma_{e}}}{\left(X_{e}-X_{c}-d(s)\right)}.$
(45)
The three types of solutions to equations (45) are as follows:
Drift Space
$X_{e}=X_{0}+{\frac{P_{0}}{\beta_{e}\gamma_{e}}}{\left(s-s_{0}\right)},\qquad\qquad
P_{e}=P_{0},$ (46)
where $X_{0}$ and $P_{0}$ are the initial position and reference momentum and
$s$ is the distance in longitudinal direction.
Focusing Quadrupole
$X_{e}=X_{c}+d_{F}+{\left(X_{0}-X_{c}-d_{F}\right)}\cos\omega_{F}{\left(s-s_{0}\right)}$
$+{\frac{P_{0}}{\beta_{e}\gamma_{e}\omega_{F}}}\sin\omega_{F}{\left(s-s_{0}\right)},$
(47)
$P_{e}=-\beta_{e}\gamma_{e}\omega_{F}{\left(X_{0}-X_{c}-d_{F}\right)}\sin\omega_{F}{\left(s-s_{0}\right)}$
$+P_{0}\cos\omega_{F}{\left(s-s_{0}\right)},$ (48)
where
$\omega_{F}^{2}={\frac{ea_{F}}{p_{0}\beta_{e}\gamma_{e}}}.$ (49)
Defocusing Quadrupole
$X_{e}=X_{c}+d_{D}+{\left(X_{0}-X_{c}-d_{D}\right)}\cosh\omega_{D}{\left(s-s_{0}\right)}$
$+{\frac{P_{0}}{\beta_{e}\gamma_{e}\omega_{D}}}\sinh\omega_{D}{\left(s-s_{0}\right)},$
(50)
$P_{e}=\beta_{e}\gamma_{e}\omega_{D}{\left(X_{0}-X_{c}-d_{D}\right)}\sinh\omega_{D}{\left(s-s_{0}\right)}$
$+P_{0}\cosh\omega_{D}{\left(s-s_{0}\right)},$ (51)
where
$\omega_{D}^{2}={\frac{ea_{D}}{p_{0}\beta_{e}\gamma_{e}}}.$ (52)
In addition to the above, the coordinate transformation at the polygon bend
when passing to the new rotated coordinate system needs to be specified. The
latter can be written as
$X_{e}=X_{c}+{\frac{X_{0}-X_{c}}{\cos\Theta_{p}-P_{0}\sin\Theta_{p}/\beta_{e}\gamma_{e}}},$
$P_{e}=\beta_{e}\gamma_{e}\tan{\left[\Theta_{p}+\arctan{\left({\frac{P_{0}}{\beta_{e}\gamma_{e}}}\right)}\right]}.$
(53)
Once the reference trajectory has been found the corresponding contributions
to the total Hamiltonian (31) can be written as follows
$\widetilde{H}_{0}=-\beta\gamma+{\frac{Z}{AE_{p}}}{\left({\frac{{\rm d}\Delta
E}{{\rm d}s}}\right)}\int{\rm d}\Theta\sin\phi(\Theta),$ (54)
$\widetilde{H}_{1}=-{\left(\beta\gamma-\beta_{e}\gamma_{e}\right)}K\widetilde{x},$
(55)
$\widetilde{H}_{2}={\frac{1}{2\beta\gamma}}{\left({\widetilde{P}}_{x}^{2}+{\widetilde{P}}_{z}^{2}\right)}+{\frac{1}{2}}{\left[{\left(g+\beta_{e}\gamma_{e}K^{2}\right)}{\widetilde{x}}^{2}-g{\widetilde{z}}^{2}\right]},$
(56)
$\widetilde{H}_{3}={\frac{K{\widetilde{x}}}{2\beta\gamma}}{\left({\widetilde{P}}_{x}^{2}+{\widetilde{P}}_{z}^{2}\right)}+{\frac{Kg}{6}}{\left(2{\widetilde{x}}^{3}-3\widetilde{x}{\widetilde{z}}^{2}\right)},$
(57)
$\widetilde{H}_{4}={\frac{{\left({\widetilde{P}}_{x}^{2}+{\widetilde{P}}_{z}^{2}\right)}^{2}}{8\beta^{3}\gamma^{3}}}-{\frac{K^{2}g}{24}}{\widetilde{z}}^{4}.$
(58)
Here, we have introduced the following notation
$g={\frac{qa_{1}}{p_{0}}}.$ (59)
Moreover, $Z$ is the charge state of the accelerated particle, $A$ is the mass
ratio with respect to the proton mass in the case of ions, and $\phi(\Theta)$
is the phase of the RF. For a lepton accelerator like EMMA, $A=Z=1$. In
addition, $({\rm d}\Delta E/{\rm d}s)$ is the energy gain per unit
longitudinal distance $s$, which in thin lens approximation scales as $\Delta
E/\Delta s$, where $\Delta s$ is the length of the cavity. It is convenient to
pass to new scaled variables as follows
${\widetilde{p}}_{u}={\frac{{\widetilde{P}}_{u}}{\beta_{e}\gamma_{e}}},\qquad
h={\frac{\gamma}{\beta_{e}^{2}\gamma_{e}}},$ (60)
$\tau=\beta_{e}\Theta,\qquad\Gamma_{e}={\frac{\beta\gamma}{\beta_{e}\gamma_{e}}}={\sqrt{\beta_{e}^{2}h^{2}-{\frac{1}{\beta_{e}^{2}\gamma_{e}^{2}}}}}.$
(61)
Thus, expressions (54) – (58) become
$\widetilde{H}_{0}=-\Gamma_{e}+{\frac{Z}{A\beta_{e}^{2}E_{e}}}{\left({\frac{{\rm
d}\Delta E}{{\rm d}s}}\right)}\int{\rm d}\tau\sin\phi(\tau),$ (62)
$\widetilde{H}_{1}=-{\left(\Gamma_{e}-1\right)}K\widetilde{x},$ (63)
$\widetilde{H}_{2}={\frac{1}{2\Gamma_{e}}}{\left({\widetilde{p}}_{x}^{2}+{\widetilde{p}}_{z}^{2}\right)}+{\frac{1}{2}}{\left[{\left(g_{e}+K^{2}\right)}{\widetilde{x}}^{2}-g_{e}{\widetilde{z}}^{2}\right]},$
(64)
$\widetilde{H}_{3}={\frac{K{\widetilde{x}}}{2\Gamma_{e}}}{\left({\widetilde{p}}_{x}^{2}+{\widetilde{p}}_{z}^{2}\right)}+{\frac{Kg_{e}}{6}}{\left(2{\widetilde{x}}^{3}-3\widetilde{x}{\widetilde{z}}^{2}\right)},$
(65)
$\widetilde{H}_{4}={\frac{{\left({\widetilde{p}}_{x}^{2}+{\widetilde{p}}_{z}^{2}\right)}^{2}}{8\Gamma_{e}^{3}}}-{\frac{K^{2}g_{e}}{24}}{\widetilde{z}}^{4},$
(66) $E_{p}=m_{p0}c^{2},\qquad\qquad g_{e}={\frac{g}{\beta_{e}\gamma_{e}}}.$
(67)
The longitudinal part of the reference orbit can be isolated via a canonical
transformation
$F_{2}{\left(\widetilde{x},\widetilde{\widetilde{p}}_{x},\widetilde{z},\widetilde{\widetilde{p}}_{z},\tau,\eta;s\right)}=\widetilde{x}\widetilde{\widetilde{p}}_{x}+\widetilde{z}\widetilde{\widetilde{p}}_{z}+{\left(\tau+s\right)}{\left(\eta+{\frac{1}{\beta_{e}^{2}}}\right)},$
(68) $\sigma=\tau+s,\qquad\qquad\eta=h-{\frac{1}{\beta_{e}^{2}}},$ (69)
where $\sigma$ is the new longitudinal variable and $\eta$ is the energy
deviation with respect to the energy $\gamma_{e}$ of the reference particle.
## IV Dispersion and Betatron Motion
The (linear and higher order) dispersion can be introduced via a canonical
transformation aimed at canceling the first order Hamiltonian
${\widetilde{H}}_{1}$ in all orders of $\eta$. The explicit form of the
generating function is
$G_{2}{\left({\widetilde{x}},{\widehat{p}}_{x},{\widetilde{z}},{\widehat{p}}_{z},\sigma,\widehat{\eta};s\right)}=\sigma{\widehat{\eta}}+{\widetilde{z}}{\widehat{p}}_{z}+{\widetilde{x}}{\widehat{p}}_{x}$
$+\sum\limits_{k=1}^{\infty}{\widehat{\eta}}^{k}{\left[{\widetilde{x}}{\cal
X}_{k}(s)-{\widehat{p}}_{x}{\cal P}_{k}(s)+{\cal S}_{k}(s)\right]},$ (70)
${\widetilde{x}}=\widehat{x}+\sum\limits_{k=1}^{\infty}{\widehat{\eta}}^{k}{\cal
P}_{k},\qquad\qquad{\widetilde{p}}_{x}={\widehat{p}}_{x}+\sum\limits_{k=1}^{\infty}{\widehat{\eta}}^{k}{\cal
X}_{k},$ (71)
$\sigma=\widehat{\sigma}+\sum\limits_{k=1}^{\infty}k{\widehat{\eta}}^{k-1}{\left({\cal
P}_{k}{\widehat{p}}_{x}-{\cal X}_{k}{\widehat{x}}\right)}$
$-\sum\limits_{k=1}^{\infty}k{\widehat{\eta}}^{k-1}{\left({\cal S}_{k}+{\cal
X}_{k}\sum\limits_{m=1}^{\infty}{\widehat{\eta}}^{m}{\cal P}_{m}\right)}.$
(72)
Equating terms of the form ${\widehat{x}}{\widehat{\eta}}^{n}$ and
${\widehat{p}}_{x}{\widehat{\eta}}^{n}$ in the new transformed Hamiltonian, we
determine order by order the conventional (first order) and higher order
dispersions. The first order in ${\widehat{\eta}}$ (terms proportional to
${\widehat{x}}{\widehat{\eta}}$ and ${\widehat{p}}_{x}{\widehat{\eta}}$)
yields the well-known result
${\cal P}_{1}^{\prime}={\cal X}_{1},\qquad\qquad{\cal
X}_{1}^{\prime}+{\left(g_{e}+K^{2}\right)}{\cal P}_{1}=K.$ (73)
Since in the case of vanishing betatron motion
${\left({\widehat{x}}=0,\quad{\widehat{p}}_{x}=0\right)}$ the new longitudinal
coordinate $\widehat{\sigma}$ should not depend on the new longitudinal
canonical conjugate variable $\widehat{\eta}$, the second sum in equation (72)
must be identically zero. We readily obtain ${\cal S}_{1}=0$, and
${\cal S}_{2}=-{\frac{{\cal X}_{1}{\cal P}_{1}}{2}}.$ (74)
In second order we have
${\cal P}_{2}^{\prime}={\cal X}_{2}-{\cal X}_{1}+K{\cal X}_{1}{\cal P}_{1},$
(75) ${\cal X}_{2}^{\prime}+{\left(g_{e}+K^{2}\right)}{\cal
P}_{2}=-Kg_{e}{\cal P}_{1}^{2}-{\frac{K{\cal
X}_{1}^{2}}{2}}-{\frac{K}{2\gamma_{e}^{2}}},$ (76)
and in addition the function ${\cal S}_{3}(s)$ is expressed as
${\cal S}_{3}=-{\frac{1}{3}}{\left({\cal X}_{1}{\cal P}_{2}+2{\cal X}_{2}{\cal
P}_{1}\right)}.$ (77)
Close inspection of equations (73), (75) and (76) shows that ${\cal P}_{1}$ is
the well-known linear dispersion function, ${\cal P}_{2}$ stands for a second
order dispersion and so on. Up to third order in ${\widehat{\eta}}$ the new
Hamiltonian describing the longitudinal motion and the linear transverse
motion acquires the form
$\widehat{H}_{0}=-{\frac{{\widetilde{\cal
K}}_{1}{\widehat{\eta}}^{2}}{2}}+{\frac{{\widetilde{\cal
K}}_{2}{\widehat{\eta}}^{3}}{3}}+{\frac{Z}{A\beta_{e}^{2}E_{e}}}{\left({\frac{{\rm
d}\Delta E}{{\rm d}s}}\right)}\int{\rm d}\tau\sin\phi(\tau),$ (78)
$\widehat{H}_{2}={\frac{1}{2}}{\left({\widehat{p}}_{x}^{2}+{\widehat{p}}_{z}^{2}\right)}+{\frac{1}{2}}{\left[{\left(g_{e}+K^{2}\right)}{\widehat{x}}^{2}-g_{e}{\widehat{z}}^{2}\right]},$
(79)
where
${\widetilde{\cal K}}_{1}=K{\cal
P}_{1}-{\frac{1}{\gamma_{e}^{2}}}\qquad{\widetilde{\cal K}}_{2}={\frac{K{\cal
P}_{1}}{\gamma_{e}^{2}}}-K{\cal P}_{2}-{\frac{{\cal
X}_{1}^{2}}{2}}-{\frac{3}{2\gamma_{e}^{2}}}.$ (80)
For the sake of generality, let us consider a Hamiltonian of the type
$\widehat{H}_{b}=\sum\limits_{u=(x,z)}{\left[{\frac{{\cal
F}_{u}}{2}}{\widehat{p}}_{u}^{2}+{\cal
R}_{u}{\widehat{u}}{\widehat{p}}_{u}+{\frac{{\cal
G}_{u}}{2}}{\widehat{u}}^{2}\right]}.$ (81)
A generic Hamiltonian of the type (81) can be transformed to the normal form
${\mathcal{H}}_{b}=\sum\limits_{u=(x,z)}{\frac{\chi_{u}^{\prime}}{2}}{\left({\overline{P}}_{u}^{2}+{\overline{U}}^{2}\right)},$
(82)
by means of a canonical transformation specified by the generating function
${\mathcal{F}}_{2}{\left(\widehat{x},{\overline{P}}_{x},\widehat{z},{\overline{P}}_{z};s\right)}=\sum\limits_{u=(x,z)}{\left({\frac{\widehat{u}{\overline{P}}_{u}}{\sqrt{\beta_{u}}}}-{\frac{\alpha_{u}{\widehat{u}}^{2}}{2\beta_{u}}}\right)}.$
(83)
Here the prime implies differentiation with respect to the longitudinal
variable $s$. The old and the new canonical variables are related through the
expressions
$\widehat{u}={\overline{U}}\sqrt{\beta_{u}},\qquad\qquad{\widehat{p}}_{u}={\frac{1}{\sqrt{\beta_{u}}}}{\left({\overline{P}}_{u}-\alpha_{u}{\overline{U}}\right)}.$
(84)
The phase advance $\chi_{u}(s)$ and the generalized Twiss parameters
$\alpha_{u}(s)$, $\beta_{u}(s)$ and $\gamma_{u}(s)$ are defined as
$\chi_{u}^{\prime}={\frac{{\rm d}\chi_{u}}{{\rm d}s}}={\frac{{\cal
F}_{u}}{\beta_{u}}},$ (85) $\alpha_{u}^{\prime}={\frac{{\rm d}\alpha_{u}}{{\rm
d}s}}={\cal G}_{u}\beta_{u}-{\cal F}_{u}\gamma_{u},$ (86)
$\beta_{u}^{\prime}={\frac{{\rm d}\beta_{u}}{{\rm d}s}}=-2{\cal
F}_{u}\alpha_{u}+2{\cal R}_{u}\beta_{u}.$ (87)
The third Twiss parameter $\gamma_{u}(s)$ is introduced via the well-known
expression
$\beta_{u}\gamma_{u}-\alpha_{u}^{2}=1.$ (88)
The corresponding betatron tunes are determined according to the expression
$\nu_{u}={\frac{N_{p}}{2\pi}}\int\limits_{s}^{s+L_{p}}{\frac{{\rm
d}\theta{\cal F}_{u}(\theta)}{\beta_{u}(\theta)}}.$ (89)
Typical dependence of the horizontal and vertical betatron tunes on energy in
the EMMA non scaling FFAG is shown in Figures 1 and 2.
Figure 1: Horizontal betatron tune for the EMMA ring as a function of energy.
Figure 2: Vertical betatron tune for the EMMA ring as a function of energy.
## V Acceleration in a Non Scaling FFAG Accelerator
The process of acceleration in a non scaling FFAG accelerator can be studied
by solving Hamilton’s equations of motion for the longitudinal degree of
freedom. The latter are obtained from the Hamiltonian (41) supplemented by an
additional term [similar to that in equation (54)], which takes into account
the electric field of the RF cavities. They read as
${\frac{{\rm d}\Theta}{{\rm
d}s}}=-{\frac{\gamma}{\sqrt{\beta^{2}\gamma^{2}-P^{2}}}},$ (90) ${\frac{{\rm
d}\gamma}{{\rm
d}s}}=-{\frac{ZeU_{c}}{2AE_{p}}}\sum\limits_{k=1}^{N_{c}}\delta_{p}{\left(s-s_{k}\right)}\sin{\left({\frac{\omega_{c}\Theta}{c}}-\varphi_{k}\right)}.$
(91)
Here $U_{c}$ is the cavity voltage, $\omega_{c}$ is the RF frequency, $N_{c}$
is the number of cavities and $\varphi_{k}$ is the corresponding cavity phase.
One could use the results obtained in the previous section with the additional
requirement that the phase slip coefficient ${\widetilde{\cal K}}_{1}$
averaged over one period vanishes. Instead, we shall use an equivalent but
more illustrative approach. The path length in a FFAG arc and therefore the
time of flight $\Theta$ is often well approximated as a quadratic function of
energy. The acceleration process is then described by a longitudinal
Hamiltonian, which contains terms proportional to the zero-order (conventional
phase slip) factor and first-order phase slip factor. It usually suffices to
take into account only terms to second order in the energy deviation
$\Theta=\Theta_{0}+2{\cal A}\gamma_{m}\gamma-{\cal A}\gamma^{2},$ (92)
as suggested by Figure 3.
Figure 3: Time of flight as a function of energy for a single 0.394481 meter
EMMA cell.
Here $\gamma_{m}$ corresponds to the reference energy with a minimum time of
flight. Provided the time of flight $\Theta_{i}$ at injection energy
$\gamma_{i}$ and the time of flight $\Theta_{m}$ at reference energy
$\gamma_{m}$ are known, the constants entering equation (92) can be expressed
as
${\cal
A}={\frac{\Theta_{m}-\Theta_{i}}{{\left(\gamma_{m}-\gamma_{i}\right)}^{2}}},\qquad\qquad\Theta_{0}=\Theta_{m}-{\cal
A}\gamma_{m}^{2}.$ (93)
Next, we pass to a new variable
${\widehat{\gamma}}=\gamma-\gamma_{m},\qquad\qquad\Theta=\Theta_{m}-{\cal
A}{\widehat{\gamma}}^{2},$ (94)
similar to the variable ${\widehat{\eta}}$ introduced in the previous section.
Then, Hamilton’s equation of motion (90) can be rewritten in an equivalent
form
${\frac{{\rm d}\Theta}{{\rm d}s}}={\frac{\Theta_{m}}{L_{p}}}-{\frac{{\cal
A}{\widehat{\gamma}}^{2}}{L_{p}}},$ (95)
In what follows, it is convenient to introduce a new phase
${\widetilde{\varphi}}$ and the azimuthal angle $\theta$ along the machine
circumference as an independent variable according to the relations
${\rm d}s=R{\rm
d}\theta,\qquad{\widetilde{\varphi}}={\frac{\omega_{c}\Theta}{c}},\qquad
R={\frac{N_{L}L_{p}}{2\pi}}.$ (96)
It is straightforward to verify (see the averaging procedure below) that the
necessary condition to have acceleration is
${\frac{\omega_{c}N_{L}{\left|\Theta_{m}\right|}}{2\pi c}}=h,$ (97)
where $h$ is an integer (a harmonic number). Averaging Hamilton’s equations of
motion
${\frac{{\rm d}{\widetilde{\varphi}}}{{\rm
d}\theta}}=-h-ha{\widehat{\gamma}}^{2},\qquad\qquad a={\frac{\cal
A}{\left|\Theta_{m}\right|}},$ (98) ${\frac{{\rm d}{\widehat{\gamma}}}{{\rm
d}\theta}}=-{\frac{ZeU_{c}}{2AE_{p}}}\sum\limits_{k=1}^{N_{c}}\delta_{p}{\left(\theta-\theta_{k}\right)}\sin{\left({\widetilde{\varphi}}-\varphi_{k}\right)},$
(99)
we rewrite them in a simpler form as
${\frac{{\rm d}\varphi}{{\rm
d}\theta}}=ha{\widehat{\gamma}}^{2},\qquad\qquad{\frac{{\rm
d}{\widehat{\gamma}}}{{\rm d}\theta}}=\lambda\sin\varphi,$ (100)
where
$\varphi=-{\widetilde{\varphi}}-h\theta+\psi_{0},\qquad\qquad\lambda={\frac{ZeU_{c}{\cal
D}}{4\pi AE_{p}}},$ (101) ${\cal D}={\sqrt{{\cal A}_{c}^{2}+{\cal
A}_{s}^{2}}},\qquad\qquad\psi_{0}=\arctan{\left({\frac{{\cal A}_{s}}{{\cal
A}_{c}}}\right)},$ (102) ${\cal
A}_{c}=\sum\limits_{k=1}^{N_{c}}\cos{\left(h\theta_{k}+\varphi_{k}\right)},\qquad{\cal
A}_{s}=\sum\limits_{k=1}^{N_{c}}\sin{\left(h\theta_{k}+\varphi_{k}\right)}.$
(103)
The effective longitudinal Hamiltonian, which governs the equations of motion
(100) can be written as
$H_{0}={\frac{ha}{3}}{\widehat{\gamma}}^{3}+\lambda\cos\varphi.$ (104)
Since the Hamiltonian (104) is a constant of motion, the second Hamilton
equation (100) can be written as
${\frac{{\rm d}{\widehat{\gamma}}}{{\rm
d}\theta}}=\pm\lambda{\sqrt{1-{\frac{1}{\lambda^{2}}}{\left(H_{0}-{\frac{ha}{3}}{\widehat{\gamma}}^{3}\right)}^{2}}}.$
(105)
Figure 4: An example of the so-called serpentine acceleration for the EMMA
ring for the central trajectory, where the longitudinal $H_{0}=0$. The
harmonic number is assumed to be 11, with the RF wavelength 0.405m. The
parameter $a$ from Eq. (98) is taken to be $2.686310^{-5}$.
Let us first consider the case of the central trajectory, where $H_{0}=0$. It
is of utmost importance for the so called gutter acceleration. Equation (105)
can be solved in a straightforward manner to give
$\theta={\frac{J}{b}}\,{}_{2}F_{1}{\left({\frac{1}{6}},{\frac{1}{2}};{\frac{7}{6}};J^{6}\right)}-{\frac{\cal
C}{b}},$ (106)
where
$J={\widehat{\gamma}}\,{\sqrt[3]{\frac{ha}{3\lambda}}},\qquad\qquad
b=\lambda\,{\sqrt[3]{\frac{ha}{3\lambda}}},$ (107) ${\cal
C}={}_{2}F_{1}{\left({\frac{1}{6}},{\frac{1}{2}};{\frac{7}{6}};J_{i}^{6}\right)}J_{i}.$
(108)
In the above expressions ${}_{2}F_{1}{\left(\alpha,\beta;\gamma;x\right)}$
denotes the Gauss hypergeometric function of the argument $x$. This case is
illustrated in Figure 4.
In the general case where $H_{0}\neq 0$, we have
$\theta={\frac{J}{b{\sqrt{a_{1}c}}}}\,F_{1}{\left({\frac{1}{3}};{\frac{1}{2}},{\frac{1}{2}};{\frac{4}{3}};{\frac{J^{3}}{a_{1}}},-{\frac{J^{3}}{c}}\right)}-{\frac{{\cal
C}_{1}}{b}},$ (109)
where
$a_{1}=1+{\frac{H_{0}}{\lambda}},\qquad\qquad c=1-{\frac{H_{0}}{\lambda}},$
(110) ${\cal
C}_{1}={\frac{J_{i}}{\sqrt{a_{1}c}}}\,F_{1}{\left({\frac{1}{3}};{\frac{1}{2}},{\frac{1}{2}};{\frac{4}{3}};{\frac{J_{i}^{3}}{a_{1}}},-{\frac{J_{i}^{3}}{c}}\right)}.$
(111)
Here now, $F_{1}{\left(\alpha;\beta,\gamma;\delta;x,y\right)}$ denotes the
Appell hypergeometric function of the arguments $x$ and $y$. The phase
portrait corresponding to the general case for a variety of values of the
longitudinal Hamiltonian $H_{0}$ is illustrated in Figure 5.
## VI Concluding Remarks
Based on the Hamiltonian formalism, the synchro-betatron approach for the
description of the dynamics of particles in non scaling FFAG machines has been
developed. Its starting point is the specification of the static reference
(closed) orbit for a fixed energy as a solution of the equations of motion in
the machine reference frame. The problem of dynamical stability and
acceleration is sequentially studied in the natural coordinate system
associated with the reference orbit thus determined.
It has been further shown that the dependence of the path length on the energy
deviation can be described in terms of higher order (nonlinear) dispersion
functions. The method provides a systematic tool to determine the dispersion
functions to every desired order, and represents a natural definition through
constitutive equations for the resulting Twiss parameters.
The formulation thus developed has been applied to the electron FFAG machine
EMMA. The transverse and longitudinal dynamics are explored and an initial
attempt is made at understanding the limits of longitudinal stability of such
a machine.
Unlike the conventional synchronous acceleration, the acceleration process in
FFAG accelerators is an asynchronous one in which the reference particle
performs nonlinear oscillations around the crest of the RF waveform. To the
best of our knowledge, it is the first time that such a fully analytic theory
describing the acceleration in non scaling FFAGs has been developed.
Figure 5: Examples of serpentine acceleration for the EMMA ring, with varying
value of the longitudinal Hamiltonian. The limits of stability are given at
values of the longitudinal Hamiltonian of $\pm 0.31272$, corresponding to
either a 0 phase at 10MeV, or a $\pi$ phase at 20MeV.
## Appendix A Calculation of the Reference Orbit
The explicit solutions of the linearized equations of motion (45) can be used
to calculate approximately the reference orbit. To do so, we introduce a state
vector
${\bf Z}_{e}={\left(\begin{array}[]{cc}X_{e}\\\ \\\ P_{e}\end{array}\right)}.$
(112)
The effect of each lattice element can be represented in a simple form as
${\bf Z}_{out}={\widehat{\cal M}}_{el}{\bf Z}_{in}+{\bf A}_{el}.$ (113)
Here ${\bf Z}_{in}$ is the initial value of the state vector, while ${\bf
Z}_{out}$ is its final value at the exit of the corresponding element. The
transfer matrix ${\widehat{\cal M}}_{el}$ and the shift vector ${\bf A}_{el}$
for various lattice elements are given as follows:
1\. Polygon Bend.
Within the approximation (44) considered here we can linearize the second of
equations (53) and write
${\widehat{\cal M}}_{p}={\left(\begin{array}[]{cc}1/\cos\Theta_{p}\ \
-X_{c}\tan\Theta_{p}/{\left(\beta_{e}\gamma_{e}\cos\Theta_{p}\right)}\\\ \\\
0\ \ \ \ \ \ \ \ \ \ 1/\cos^{2}\Theta_{p}\end{array}\right)},$ ${\bf
A}_{p}={\left(\begin{array}[]{cc}X_{c}{\left(1-1/\cos\Theta_{p}\right)}\\\ \\\
\beta_{e}\gamma_{e}\tan\Theta_{p}\end{array}\right)}.$ (114)
2\. Drift Space.
${\widehat{\cal M}}_{O}={\left(\begin{array}[]{cc}1\
L_{O}/\beta_{e}\gamma_{e}\\\ \\\ 0\ \ \ \ \ \ \ \ \
1\end{array}\right)},\qquad\qquad{\bf A}_{O}=0,$ (115)
where $L_{O}$ is the length of the drift. Every cell of the EMMA lattice
includes a short drift of length $L_{0}$ and a long one of length $L_{1}$.
3\. Focusing Quadrupole.
The transfer matrix can be written in a straightforward manner as
${\widehat{\cal
M}}_{F}={\left(\begin{array}[]{cc}\cos{\left(\omega_{F}L_{F}\right)}\ \ \
\sin{\left(\omega_{F}L_{F}\right)}/{\left(\beta_{e}\gamma_{e}\omega_{F}\right)}\\\
\\\ -\beta_{e}\gamma_{e}\omega_{F}\sin{\left(\omega_{F}L_{F}\right)}\ \ \
\cos{\left(\omega_{F}L_{F}\right)}\end{array}\right)},$ (116) ${\bf
A}_{F}={\left(\begin{array}[]{cc}{\left(X_{c}+d_{F}\right)}{\left[1-\cos{\left(\omega_{F}L_{F}\right)}\right]}\\\
\\\
\beta_{e}\gamma_{e}\omega_{F}{\left(X_{c}+d_{F}\right)}\sin{\left(\omega_{F}L_{F}\right)}\end{array}\right)},$
(117)
where $L_{F}$ is the length of the focusing quadrupole.
4\. Defocusing Quadrupole.
The transfer matrix in this case can be written in analogy to the above one as
${\widehat{\cal
M}}_{D}={\left(\begin{array}[]{cc}\cosh{\left(\omega_{D}L_{D}\right)}\ \ \
\sinh{\left(\omega_{D}L_{D}\right)}/{\left(\beta_{e}\gamma_{e}\omega_{D}\right)}\\\
\\\ \beta_{e}\gamma_{e}\omega_{D}\sinh{\left(\omega_{D}L_{D}\right)}\ \ \
\cosh{\left(\omega_{D}L_{D}\right)}\end{array}\right)},$ (118) ${\bf
A}_{D}={\left(\begin{array}[]{cc}{\left(X_{c}+d_{D}\right)}{\left[1-\cosh{\left(\omega_{D}L_{D}\right)}\right]}\\\
\\\
-\beta_{e}\gamma_{e}\omega_{D}{\left(X_{c}+d_{D}\right)}\sinh{\left(\omega_{D}L_{D}\right)}\end{array}\right)},$
(119)
where $L_{D}$ is the length of the defocusing quadrupole.
Since the reference orbit must be a periodic function of $s$ with period
$L_{p}$, it clearly satisfies the condition
${\bf Z}_{out}={\bf Z}_{in}={\bf Z}_{e}.$ (120)
Thus, the equation for determining the reference orbit becomes
${\bf Z}_{e}={\widehat{\cal M}}{\bf Z}_{e}+{\bf A},\qquad{\rm or}\qquad{\bf
Z}_{e}={\left(1-{\widehat{\cal M}}\right)}^{-1}{\bf A}.$ (121)
Here ${\widehat{\cal M}}$ and ${\bf A}$ are the transfer matrix and the shift
vector for one period, respectively. The inverse of the matrix
$1-{\widehat{\cal M}}$ can be expressed as
${\left(1-{\widehat{\cal
M}}\right)}^{-1}={\frac{\cos^{3}\Theta_{p}}{1+{\left(1-{\rm Sp}{\widehat{\cal
M}}\right)}\cos^{3}\Theta_{p}}}$ $\times{\left(\begin{array}[]{cc}1-{\cal
M}_{22}\ \ \ {\cal M}_{12}\\\ \\\ {\cal M}_{21}\ \ \ 1-{\cal
M}_{11}\end{array}\right)}.$ (122)
For the EMMA lattice in particular, the components of the one period transfer
matrix and shift vector can be written explicitly as
${\cal
M}_{11}={\frac{1}{c_{p}}}{\left[c_{F}c_{D}+{\left({\frac{\omega_{D}}{\omega_{F}}}-L_{0}L_{1}\omega_{F}\omega_{D}\right)}s_{F}s_{D}+{\left(L_{0}+L_{1}\right)}\omega_{D}c_{F}s_{D}-L_{1}\omega_{F}s_{F}c_{D}\right]},$
(123) $\displaystyle{\cal
M}_{12}={\frac{1}{\beta_{e}\gamma_{e}c_{p}}}{\left\\{{\left({\frac{L_{0}+L_{1}}{c_{p}}}-X_{c}t_{p}\right)}c_{F}c_{D}+{\left[{\left(L_{0}L_{1}\omega_{F}\omega_{D}-{\frac{\omega_{D}}{\omega_{F}}}\right)}X_{c}t_{p}-{\frac{\omega_{F}L_{1}}{\omega_{D}c_{p}}}\right]}s_{F}s_{D}\right.}$
${\left.+{\left[{\frac{1}{\omega_{D}c_{p}}}-{\left(L_{0}+L_{1}\right)}\omega_{D}X_{c}t_{p}\right]}c_{F}s_{D}+{\left({\frac{1}{\omega_{F}c_{p}}}+L_{1}\omega_{F}X_{c}t_{p}-{\frac{L_{0}L_{1}\omega_{F}}{c_{p}}}\right)}s_{F}c_{D}\right\\}},$
(124) ${\cal
M}_{21}=-{\frac{\beta_{e}\gamma_{e}}{c_{p}}}{\left(\omega_{F}s_{F}c_{D}+L_{0}\omega_{F}\omega_{D}s_{F}s_{D}-\omega_{D}c_{F}s_{D}\right)},$
(125) ${\cal
M}_{22}={\frac{1}{c_{p}}}{\left[{\frac{c_{F}c_{D}}{c_{p}}}+{\left(L_{0}\omega_{F}\omega_{D}X_{c}t_{p}-{\frac{\omega_{F}}{\omega_{D}c_{p}}}\right)}s_{F}s_{D}+\omega_{F}{\left(X_{c}t_{p}-{\frac{L_{0}}{c_{p}}}\right)}s_{F}c_{D}-\omega_{D}X_{c}t_{p}c_{F}s_{D}\right]},$
(126) $\displaystyle
A_{1}=X_{c}+d_{F}+{\left(d_{D}-d_{F}\right)}{\left(c_{F}-L_{1}\omega_{F}s_{F}\right)}+{\left({\frac{X_{c}}{c_{p}}}+d_{D}\right)}$
$\displaystyle\times{\left[L_{1}\omega_{F}s_{F}c_{D}-c_{F}c_{D}-{\left(L_{0}+L_{1}\right)}\omega_{D}c_{F}s_{D}-{\frac{\omega_{D}s_{F}s_{D}}{\omega_{F}}}+L_{0}L_{1}\omega_{F}\omega_{D}s_{F}s_{D}\right]}$
$+t_{p}{\left[{\left(L_{0}+L_{1}\right)}c_{F}c_{D}+{\frac{c_{F}s_{D}}{\omega_{D}}}+{\frac{s_{F}c_{D}}{\omega_{F}}}-{\frac{L_{1}\omega_{F}s_{F}s_{D}}{\omega_{D}}}-L_{0}L_{1}\omega_{F}s_{F}c_{D}\right]},$
(127) $\displaystyle
A_{2}=-\beta_{e}\gamma_{e}\omega_{F}{\left(d_{D}-d_{F}\right)}s_{F}+\beta_{e}\gamma_{e}{\left({\frac{X_{c}}{c_{p}}}+d_{D}\right)}{\left(\omega_{F}s_{F}c_{D}+\omega_{F}\omega_{D}L_{0}s_{F}s_{D}-\omega_{D}c_{F}s_{D}\right)}$
$+\beta_{e}\gamma_{e}t_{p}{\left(c_{F}c_{D}-{\frac{\omega_{F}s_{F}s_{D}}{\omega_{D}}}-\omega_{F}L_{0}s_{F}c_{D}\right)}.$
(128)
For the sake of brevity, the following notations
$c_{p}=\cos\Theta_{p},\quad c_{F}=\cos{\left(\omega_{F}L_{F}\right)},\quad
c_{D}=\cosh{\left(\omega_{D}L_{D}\right)},$ (129) $t_{p}=\tan\Theta_{p},\quad
s_{F}=\sin{\left(\omega_{F}L_{F}\right)},\quad
s_{D}=\sinh{\left(\omega_{D}L_{D}\right)},$ (130)
have been introduced in the final expressions for the components of the one
period transfer matrix and shift vector.
## Appendix B Phase Stability in FFAGs
To study the stability of the serpentine acceleration in FFAG accelerators, we
write the longitudinal Hamiltonian (104) in an equivalent form
$H_{0}=\lambda{\left(J^{3}+\cos\varphi\right)}.$ (131)
Hamilton’s equations of motion can be written as
${\frac{{\rm d}\varphi}{{\rm d}\theta}}=3bJ^{2},\qquad\qquad{\frac{{\rm
d}J}{{\rm d}\theta}}=b\sin\varphi.$ (132)
Let $\varphi_{a}(\theta)$ and $J_{a}(\theta)$ be the exact solution of
equations (132) described already in Section V. Let us further denote by
$\varphi_{1}$ and $J_{1}$ a small deviation about this solution such that
$\varphi=\varphi_{a}+\varphi_{1}$ and $J=J_{a}+J_{1}$. Then, the linearized
equations of motion governing the evolution of $\varphi_{1}$ and $J_{1}$ are
${\frac{{\rm d}\varphi_{1}}{{\rm
d}\theta}}=6bJ_{a}J_{1},\qquad\qquad{\frac{{\rm d}J_{1}}{{\rm
d}\theta}}=b\varphi_{1}\cos\varphi_{a}.$ (133)
The latter should be solved provided the constraint
$3J_{a}^{2}J_{1}-\varphi_{1}\sin\varphi_{a}=0,$ (134)
following from the Hamiltonian (131) holds. Differentiating the second of
equations (133) with respect to $\theta$ and eliminating $\varphi_{1}$, we
obtain
${\frac{{\rm d}^{2}J_{1}}{{\rm
d}\theta^{2}}}-{\frac{6b^{2}H_{0}}{\lambda}}J_{a}J_{1}+15b^{2}J_{a}^{4}J_{1}=0.$
(135)
Next, we examine the case of separatrix acceleration with $H_{0}=0$. In
Section V we showed that to a good accuracy the energy gain
${\left[J_{a}(\theta)=b\theta+J_{i}\right]}$ is linear in the azimuthal
variable $\theta$. Therefore, equation (135) can be written as
${\frac{{\rm d}^{2}J_{1}}{{\rm d}J_{a}^{2}}}+15J_{a}^{4}J_{1}=0.$ (136)
Figure 6: Phase stability of the standard EMMA ring, for the central
trajectory at $H_{0}=0$. The errors are given as 0.1MeV in energy and
$1.3^{\rm o}$ in phase.
The latter possesses a simple solution of the form
$J_{1}={\sqrt{\left|J_{a}\right|}}{\left[C_{1}{\cal
J}_{1/6}{\left({\sqrt{\frac{5}{3}}}{\left|J_{a}\right|}^{3}\right)}+C_{2}{\cal
Y}_{1/6}{\left({\sqrt{\frac{5}{3}}}{\left|J_{a}\right|}^{3}\right)}\right]},$
(137)
where ${\cal J}_{\alpha}(z)$ and ${\cal Y}_{\alpha}(z)$ stand for the Bessel
functions of the first and second kind, respectively. In addition the
constants $C_{1}$ and $C_{2}$ should be determined taking into account the
initial conditions
${\frac{{\rm d}J_{1}{\left(J_{i}\right)}}{{\rm
d}J_{a}}}=\varphi_{1}{\left(J_{i}\right)}\cos\varphi_{i},\qquad
J_{1}{\left(J_{i}\right)}=J_{1i}.$ (138)
## References
* (1) A. A. Kolomensky and A. N. Lebedev 1966, “Theory of Cyclic Accelerators”, North-Holland Publishing Company.
* (2) A. A. Kolomensky et al. 1955, “Some questions of the theory of cyclic accelerators”, Edition AN SSSR, page 7, PTE, N0. 2, 26(1956).
* (3) K. R. Symon et al. 1956, Phys. Rev. 103 (1956) 1837.
* (4) D. W. Kerst et al. 1960 , Review of Science Instruments 31 1076\.
* (5) M. Aiba et al. 2000, “Development of a FFAG proton synchrotron”, Proceedings of EPAC 2000, p. 581.
* (6) R. Edgecock et al., ”EMMA - the World’s First Non-scaling FFAG”, Proceedings of EPAC 2008, p. 3380.
* (7) S. L. Smith, ”The Status of the Daresbury Energy Recovery Linac Prototype (ERLP)”, Proceedings of ERL 2007, p. 6.
* (8) S. I. Tzenov 2004, “Contemporary Accelerator Physics”, World Scientific.
|
arxiv-papers
| 2009-04-14T13:03:33 |
2024-09-04T02:49:01.863957
|
{
"license": "Public Domain",
"authors": "James K. Jones, Bruno D. Muratori, Susan L. Smith, Stephan I. Tzenov",
"submitter": "Stephan Tzenov",
"url": "https://arxiv.org/abs/0904.2088"
}
|
0904.2231
|
# On quantum optical properties of single-walled carbon nanotube
Z. L. Guo School of Physics, Peking University, Beijing, 100871, China Z. R.
Gong Institute of Theoretical Physics, The Chinese Academy of Sciences,
Beijing, 100080, China C. P. Sun Institute of Theoretical Physics, The
Chinese Academy of Sciences, Beijing, 100080, China
###### Abstract
We study quantum optical properties of the single-walled carbon nanotube
(SWCNT) by introducing the effective interaction between the quantized
electromagnetic field and the confined electrons in the SWCNT. Our purpose is
to explore the quantum natures of electron transport in the SWCNT by probing
its various quantum optical properties relevant to quantum coherence, such as
the interference of the scattered and emitted photons, and the bunching and
anti-bunching of photons which are characterized by the higher order coherence
functions. In the strong field limit, we study the interband Rabi oscillation
of electrons driven by a classical light. We also investigate the possible
lasing mechanism in superradiation of coherent electrons in a SWCNT driven by
a light pump or electron injection, which generate electron population
inversion in the higher energy-band of SWCNT.
###### pacs:
78.67.Ch, 78.55.-m, 81.07.-b
## I INTRODUCTION
Carbon nanotubes (CNTs) have been under great focus these years because of
their promising thermal and electrical conductivities, and other unusual
features that may lead to new applications Carbon1 ; Carbon2 ; Carbon3 . In
recent years, individual single-walled carbon nanotubes (SWCNTs) have
experimentally become available for the design of future quantum devices
SWCNT1 ; SWCNT2 ; SWCNT3 ; SWCNT4 . Through putting such a SWCNT between
electrodes while maintaining a low contact resistance, novel CMOS devices can
be made from this novel material field1 ; field2 ; field3 . Surpassing the
current silicon-based CMOS devices, CNT-based CMOS devices appear to have the
potential for wide applications. To this end, a broad research is required on
various aspects of its characteristics beforehand.
The conventional investigation for a new material is to explore its
photoluminescence optical1 ; optical2 ; optical3 ; optical4 ; optical5 ;
optical6 ; optical7 ; optical8 ; optical9 . We usually study the
characteristic spectroscopy of the light scattered by or emitted from this
material. Meanwhile, since ballistic transport–a motion of electrons with
negligible electrical resistivity due to scattering in the process of
transportation–happens in a SWCNT at low temperature field1 ; ballistic ,
SWCNT should be treated beyond the classical scenario, and pure quantum
effects should be taken into account. As a result, not only should the
classical optical properties (e.g., the intensity, the spectrum, etc,) of the
SWCNT be considered, but also the quantum optical properties (e.g., the
bunching and antibunching phenomena, etc,) need to be studied in details. In
this paper we develop a fully quantum approach for the SWCNT-light interaction
to address the quantum effects relevant to the higher order quantum coherence.
Our investigation is oriented by the great potential to implement the quantum
optical devices based on current carbon nanotube technology, which works in
the quantum regime, or at a level of single quantum state.
Starting from the minimal coupling theorem, we derive the effective
Hamiltonian of the SWCNT interacting with a fully quantized light field. The
interband Rabi oscillation is first studied for the light field whose
intensity is sufficiently strong to be treated classically. We explore the
full quantum features of the transporting electron in the SWCNT which is
displayed by its quantum optical properties. To this end, we quantize the
light field interacting with the confined electrons in SWCNT, and calculate
and analyze the higher order coherence functions of the photons scattered or
emitted from the SWCNT. It is shown that the total population inversion of
electrons, the first order and the second order coherence functions strongly
depends on the chiral vector of the SWCNT, while this dependence does not
exist in the generic graphene. Additionally, the anti-bunching feature of the
light field is predicted with detailed calculations based on the long time
approximation. A similar discovery has been made in an experiment ballistic ,
but to our best knowledge no microscopic theoretical explanation has been
given.
This paper is organized as follows. In Sec. II, the interaction between the
quantized light field and the SWCNT based on the tight binding approach is
derived from the the minimal coupling theorem. In Sec. III, we study the
interband Rabi oscillation of the electrons in the SWCNT induced by strong
light when the driving light can be treated classically, the reason of which
is generally proved in App. A. The interference of the scattered light from
the SWCNT and the second order correlation of the emitted photons are
investigated in Sec. IV and Sec. V, respectively. Additionally, the possible
lasing mechanism of the SWCNT through a light pump or electron injection is
discussed in Sec. VI. The conclusions are presented in Sec. VII.
## II MODEL SETUP
Figure 1: Schematic illustration of the $2$-D hexagonal lattice of the SWCNT,
which contain two sets of sublattices A and B. The pair numbers $(n,m)$
denotes the chiral vector.
The difference between carbon nanotubes and graphene is that carbon nanotubes
allow merely discrete wave vectors along their specific chiral vector while
graphene allows continuous ones, as long as we neglect such effects as
distortion of the lattice in carbon nanotubes. Thus, to simplify the modeling
of the system in consideration, we can take the tight banding model of
graphene into account, and then apply discrete wave vector restriction to
demonstrate the properties of the nanotube. The honeycomb lattice of graphene
is divided into two triangular sublattices $A$ and $B$ (see Fig. 1). Here, the
chiral vector of the SWCNT is denoted as a pair of numbers $(n,m).$ The
discrete wave vectors for carbon nanotubes will be directly introduced by
boundary conditions later.
Since electrons in graphene approximately hop from one site to the nearest
neighbor one, a tight binding model
$H_{e}=-J\sum\limits_{\mathbf{r}\in
A}\sum_{\alpha=1}^{3}[a^{\dagger}(\mathbf{r})b(\mathbf{r}+\mathbf{r}_{\alpha})+h.c.]$
(1)
is applied to describe the motion of the electrons in the graphene. Here, $J$
is the hopping constant; $a\ (a^{\dagger})$ and $b\ (b^{\dagger})$ annihilates
(creates) an electron at sublattice $A$ and $B$, respectively. And
$\mathbf{r}_{\alpha}$ $(\alpha=1,2,3)$ are the real space vectors pointing
from one site to its nearest neighbors. Usually they are chosen as
$\displaystyle\mathbf{r}_{1}$ $\displaystyle=$
$\displaystyle\frac{l}{\sqrt{3}}(0,-1),$ (2a) $\displaystyle\mathbf{r}_{2}$
$\displaystyle=$
$\displaystyle\frac{l}{\sqrt{3}}(\frac{\sqrt{3}}{2},\frac{1}{2}),$ (2b)
$\displaystyle\mathbf{r}_{3}$ $\displaystyle=$
$\displaystyle\frac{l}{\sqrt{3}}(-\frac{\sqrt{3}}{2},\frac{1}{2})$ (2c) , and
are schematically plotted in Fig. 1. Here, $l$ is the lattice constant of both
the triangular sublattice $A$ and $B.$
To diagonalize the above tight banding Hamiltonian, a 2D Fourier
transformation
$c_{\mathbf{k}}=\sum\limits_{\mathbf{r}\in
C}c(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{r}},(c=a\text{ or }b,C=A\text{ or
}B).$ (3)
is used to give the momentum space- representation of the Hamiltonian (1)
$H_{e}=\sum\limits_{\mathbf{k}}\left(\Phi_{\mathbf{k}}a_{\mathbf{k}}^{\dagger}b_{\mathbf{k}}+h.c.\right).$
(4)
Here the transition energy
$\Phi_{\mathbf{k}}\equiv-J\sum\limits_{\mathbf{\delta}\in\\{\mathbf{r}_{\alpha}\\}}e^{i\mathbf{k}\cdot\mathbf{\delta}}$
is a summation over all the directions of nearest neighbors. It is explicitly
written as
$\Phi_{\mathbf{k}}=-Je^{i\frac{k_{x}l}{\sqrt{3}}}\left(1+2\cos\frac{k_{y}l}{2}e^{-i\frac{\sqrt{3}k_{x}l}{2}}\right).$
(5)
and corresponds to the transition of electrons between two sublattices $A$ and
$B$. Further, this Hamiltonian (4) is diagonalized as
$H_{e}=\sum\limits_{\mathbf{k}}E_{\mathbf{k}}\left(\alpha_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k}}-\beta_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k}}\right)$
(6)
through a unitary transformation
$\displaystyle\alpha_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(e^{-i\varphi_{\mathbf{k}}}a_{\mathbf{k}}+e^{i\varphi_{\mathbf{k}}}b_{\mathbf{k}}\right),$
(7a) $\displaystyle\beta_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(e^{-i\varphi_{\mathbf{k}}}a_{\mathbf{k}}-e^{i\varphi_{\mathbf{k}}}b_{\mathbf{k}}\right).$
(7b) Here, the single particle spectrum is
$E_{\mathbf{k}}=J\sqrt{1+4\cos(\frac{k_{x}l}{2})\left[\cos(\frac{\sqrt{3}}{2}k_{y}l)+\cos(\frac{k_{x}l}{2})\right]}$
(8)
with the phase $\varphi_{\mathbf{k}}$ determined by
$\tan
2\varphi_{\mathbf{k}}=-\frac{2\cos\left(k_{x}l/2\right)\sin\left(\sqrt{3}k_{y}l/2\right)}{1+2\cos\left(k_{x}l/2\right)\cos\left(\sqrt{3}k_{y}l/2\right)}.$
(9)
We have to point out that the energy $2E_{\mathbf{k}}$ of single electron
excitation actually has six Dirac points on the six vertices of the first
Brillouin Zone in the momentum space. It has been discovered that in the
vicinity of Dirac points, the effective motion of the electrons accords with
the relativistic theory, which is described by the massless or massive Dirac
equation with an effective light velocity .
In order to study the quantum optical properties of the nanotubes, it is
necessary to introduce a quantized light field
$H_{p}=\sum\limits_{\mathbf{q}}\hbar\Omega_{\mathbf{q}}d_{\mathbf{q}}^{\dagger}d_{\mathbf{q}},$
(10)
where $\Omega_{\mathbf{q}}$ is the frequency of photons with momentum
$\mathbf{q}$. $d_{\mathbf{q}}^{\dagger}$ and $d_{\mathbf{q}}$ creates and
annihilates a photon with momentum $\mathbf{q},$ respectively. We choose
$\hbar=1$ and only one polarization direction for each mode of light denoted
by $\mathbf{q}$ in the following discussions.
The interaction between the carbon nanotube and the light field is obtained
according to the minimal coupling principle of electromagnetic field. By
replacing the mechanical momentum of the electrons with canonical ones and
neglecting the multi-photon interactions, the interaction Hamiltonian is
obtained as
$H_{I}=-\frac{e}{mc}\sum\limits_{\mathbf{k},\mathbf{q}}\mathbf{k}\cdot\mathbf{A}_{\mathbf{q}}\left(a_{\mathbf{k}}^{\dagger}+b_{\mathbf{k}}^{\dagger}\right)\left(a_{\mathbf{k-q}}+b_{\mathbf{k-q}}\right).$
(11)
Here, the vector potential of the quantized light field is
$\mathbf{A}_{\mathbf{q}}=-i\sqrt{\frac{1}{2\epsilon_{0}V\Omega_{\mathbf{q}}}}\mathbf{e}_{\mathbf{q}}\left(d_{\mathbf{q}}-d_{-\mathbf{q}}^{\dagger}\right),$
(12)
where $\mathbf{e}_{\mathbf{q}}$ is the unit polarization vector of mode
$\mathbf{q}.$ $\epsilon_{0}$ is the vacuum electric permittivity and $V$ is
the volume effectively occupied by the light field.
So far, we have obtained the quantized mode of the SWCNT interacting with a
light field, whose Hamiltonian is $H=H_{e}+H_{p}+H_{I}$, with
$\displaystyle H_{e}$ $\displaystyle=$
$\displaystyle\sum\limits_{\mathbf{k}}E_{\mathbf{k}}\left(\alpha_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k}}-\beta_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k}}\right),$
(13a) $\displaystyle H_{p}$ $\displaystyle=$
$\displaystyle\sum\limits_{\mathbf{q}}\hbar\Omega_{\mathbf{q}}d_{\mathbf{q}}^{\dagger}d_{\mathbf{q}},$
(13b) $\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k},\mathbf{q}}D_{\mathbf{k,q}}\left(d_{\mathbf{q}}\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k}-\mathbf{q}}+h.c.\right),$
(13c) where we have made the rotating wave approximation to eliminate the fast
varying terms, such as
$d_{-\mathbf{q}}^{\dagger}\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}$,
$d_{\mathbf{q}}\beta_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k-q}}$,
$d_{-\mathbf{q}}^{\dagger}\alpha_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k-q}}$,
$d_{\mathbf{q}}\alpha_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k-q}}$,
$d_{-\mathbf{q}}^{\dagger}\beta_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}$,
and $d_{\mathbf{q}}\beta_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}$, and the
coefficient $D_{\mathbf{k,q}}$ for electron-photon interaction is
$\displaystyle D_{\mathbf{k,q}}$ $\displaystyle=$
$\displaystyle-\frac{e}{\sqrt{2}mc}\mathbf{k}\cdot\mathbf{e}_{\mathbf{q}}\sqrt{\frac{\hbar}{2\epsilon_{0}V\Omega_{\mathbf{q}}}}(\cos\varphi_{\mathbf{k}}\sin\varphi_{\mathbf{k-q}}$
(14) $\displaystyle+\cos\varphi_{\mathbf{k-q}}\sin\varphi_{\mathbf{k}}).$
We note that when interaction between the light field and the SWCNT is
significant, the momentum of photons in the light field is approximately
$\left|\mathbf{q}\right|\sim 10^{7}m^{-1}$, which is much smaller than the
momentum of the electron near the boundary of the first Brillouin Zone of
graphene $\left|\mathbf{k}\right|\sim 10^{10}m^{-1}.$ Thus we neglect the
momentum $\mathbf{q}$ of photons so that
$\cos\varphi_{\mathbf{k}}\sin\varphi_{\mathbf{k-q}}\approx\cos\varphi_{\mathbf{k-q}}\sin\varphi_{\mathbf{k}}\approx\sin[2\varphi_{\mathbf{k}}]/2$,
and the coefficient $D_{\mathbf{k,q}}$ is approximately
$D_{\mathbf{k,q}}=-\frac{e}{\sqrt{2}mc}\mathbf{k}\cdot\mathbf{e}_{\mathbf{q}}\sqrt{\frac{\hbar}{2\epsilon_{0}V\Omega_{\mathbf{q}}}}\sin
2\varphi_{\mathbf{k}}.$ (15)
Specially, $D_{\mathbf{k,q}}$ is taken average over all polarization
directions of the light field to obtain the final $D_{\mathbf{k,q}}$ we use in
calculations.
Figure 2: (a)The energy spectrum $E(k)$ of graphene versus $k$. (b)The
interaction intensity $D(k)$ between electrons in graphene and single-mode
light, in which we take the average over all the possible directions for e(q).
The single quasi-particle energy $E_{\mathbf{k}}$ and the interaction
coefficient $D_{\mathbf{k,q}}$ are plotted versus the momentum $\mathbf{k}$ of
the electrons in Fig. 2. The six Dirac points are clear to be found at the
degeneracy points of upper and lower bands in Fig. 2(a). For the photon
momentum $\left|\mathbf{q}\right|\mathbf{\ll\left|\mathbf{k}\right|}$ chosen
in Fig. 2(b), the absolute value of the interaction coefficient
$D_{\mathbf{k,q}}$ becomes large when $\mathbf{k}$ is near the boundary of the
first Brillouin Zone and decreases rapidly as $\mathbf{k}$ deviate from that
boundary.
## III Interband Rabi Oscillation Induced by Strong Light Field
The general photon-electron interaction contains multi-mode light field, which
case is too complex to be analytically treated in revealing the essential
properties. Thus, we simplify the Hamiltonian by making the reasonable
assumption that only one particular quantum mode of the light field would
dominate the dynamics. This could be experimentally realized by adding a high-
finesse microcavity to the system to pick out a single mode of quantized light
under consideration. In this sense, the model Hamiltonian is reduced to
$H=H_{0}+H_{1},$ where
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum\limits_{\mathbf{k}}E_{\mathbf{k}}\left(\alpha_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k}}-\beta_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k}}\right)+\Omega
d^{\dagger}d,$ (16a) $\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}D_{\mathbf{k}}\left(d\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}+h.c.\right),$
(16b) indicates that a single-mode light field would induce the coherent
transitions of electrons between the upper band and the lower band. The output
of the electronic flow would display an obvious resonance, namely, Rabi
oscillation, which is experimentally observable.
In the strong light limit, the light field can be treated as a classical one,
where the creation and annihilation operators $d^{\dagger}$ and $d$ are
replaced by C-numbers, namely
$d\rightarrow\sqrt{N}e^{-i\Omega
t},d^{\dagger}\rightarrow\left(d\right)^{\ast}.$ (17)
with $N$ the total number of photons. This approximation is valid since in a
strong light field only the intensity of the light plays an important role. We
can generally prove this classical approximation in App. A.
Then we obtain the semi-classical Hamiltonian
$H(t)=\sum\limits_{\mathbf{k}}h_{\mathbf{k}}(t),$ in which the single momentum
Hamiltonian is
$\displaystyle h_{\mathbf{k}}(t)$ $\displaystyle=$ $\displaystyle
E_{\mathbf{k}}\left(\alpha_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k}}-\beta_{\mathbf{k}-\mathbf{q}}^{\dagger}\beta_{\mathbf{k}-\mathbf{q}}\right)+$
(18)
$\displaystyle\sqrt{N}D_{\mathbf{k}}\left(\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k}-\mathbf{q}}e^{-i\Omega
t}+h.c.\right).$
for electrons with momentum $\mathbf{k}$. Here, we have neglected the constant
$N\Omega$ in the total energy of the light field and the difference between
$E_{\mathbf{k}}$ and $E_{\mathbf{k-q}}$ for the reason mentioned at the end of
Sec. II. In terms of the quasi-spin operators
$\displaystyle S_{\mathbf{k}}^{z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\alpha_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k}}-\beta_{\mathbf{k}-\mathbf{q}}^{\dagger}\beta_{\mathbf{k}-\mathbf{q}}\right),$
(19a) $\displaystyle S_{\mathbf{k}}^{+}$ $\displaystyle=$
$\displaystyle\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k}-\mathbf{q}},S_{\mathbf{k}}^{-}=\left(S_{\mathbf{k}}^{+}\right)^{\dagger},$
(19b) which obviously satisfy the commutation relations of the regular
spin-$1/2$ operators, the above single momentum Hamiltonian is rewritten as
$h_{\mathbf{k}}(t)=E_{\mathbf{k}}S_{\mathbf{k}}^{z}+\sqrt{N}D_{\mathbf{k}}\left(S_{\mathbf{k}}^{+}e^{-i\Omega
t}+h.c.\right).$ (20)
It describes a quasi-spin precession in a time-dependent effective magnetic
field
$\mathbf{B=(}\sqrt{N}D_{\mathbf{k}}\cos\Omega
t,\sqrt{N}D_{\mathbf{k}}\sin\Omega t,E_{\mathbf{k}}\mathbf{).}$ (21)
Such spin precession is just the Rabi oscillation between bands.
To solve the dynamic equation governed by $h_{\mathbf{k}}(t),$ a time-
dependent unitary transformation
$U(t)=\exp(i\Omega S_{\mathbf{k}}^{z}t),$ (22)
is used to transform the Hamiltonian above into a time-independent one
$h_{\mathbf{k}}^{\prime}=U^{\dagger}h_{\mathbf{k}}(t)U-i\partial_{t}U^{\dagger}U$
or
$h_{\mathbf{k}}^{\prime}=-\Delta_{\mathbf{k}}S_{\mathbf{k}}^{z}+\sqrt{N}D_{\mathbf{k}}S_{\mathbf{k}}^{+}+h.c$
(23)
Here,
$\Delta_{\mathbf{k}}=\Omega-2E_{\mathbf{k}}.$ (24)
is the detuning between the energy of the light field and that of the quasi-
spin.
The Heisenberg equations of the system
$\displaystyle i\frac{\partial}{\partial t}S_{\mathbf{k}}^{z}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{\mathbf{k}}\sin\theta_{\mathbf{k}}[S_{\mathbf{k}}^{+}-S_{\mathbf{k}}^{-}],$
(25a) $\displaystyle i\frac{\partial}{\partial t}S_{\mathbf{k}}^{\pm}$
$\displaystyle=$
$\displaystyle\pm\varepsilon_{\mathbf{k}}[\cos\theta_{\mathbf{k}}S_{\mathbf{k}}^{\pm}+\sin\theta_{\mathbf{k}}S_{\mathbf{k}}^{z}],$
(25b) determine the Rabi oscillation of the electrons with momentum
$\mathbf{k}$ between the upper and lower bands. Here the mixing angle
$\theta_{\mathbf{k}}$ is defined by $\displaystyle\cos\theta_{\mathbf{k}}$
$\displaystyle=$
$\displaystyle\frac{\Delta_{\mathbf{k}}}{\varepsilon_{\mathbf{k}}},$ (26a)
$\displaystyle\sin\theta_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{N}D_{\mathbf{k}}}{\varepsilon_{\mathbf{k}}},$ (26b)
$\displaystyle\varepsilon_{\mathbf{k}}^{2}$ $\displaystyle=$
$\displaystyle\Delta_{\mathbf{k}}^{2}+4ND_{\mathbf{k}}^{2}.$ (26c) The above
first order partial differential equations (25a)-(25b) with initial operators
$S_{\mathbf{k}}^{z}(0)$ and $S_{\mathbf{k}}^{\pm}(0)$ is solved through the
Laplace transformation
$\lambda(p)=\int\limits_{0}^{+\infty}\lambda(t)e^{-pt}dt,$ (27)
which gives
$\displaystyle pS_{\mathbf{k}}^{z}-S_{\mathbf{k}}^{z}(0)$ $\displaystyle=$
$\displaystyle-i\frac{1}{2}\varepsilon_{\mathbf{k}}\sin\theta_{\mathbf{k}}[S_{\mathbf{k}}^{+}-S_{\mathbf{k}}^{-}],$
(28a) $\displaystyle pS_{\mathbf{k}}^{\pm}-S_{\mathbf{k}}^{\pm}(0)$
$\displaystyle=$ $\displaystyle\mp
i\varepsilon_{\mathbf{k}}[\cos\theta_{\mathbf{k}}S_{\mathbf{k}}^{\pm}+\sin\theta_{\mathbf{k}}S_{\mathbf{k}}^{z}].$
(28b) In terms of the normalized Laplacian parameter
$p^{\prime}=p/\varepsilon_{\mathbf{k}},$ the above equation is solved as
$\displaystyle S_{\mathbf{k}}^{z}(p^{\prime})$ $\displaystyle=$
$\displaystyle\frac{S_{\mathbf{k}}^{z}(0)}{\varepsilon_{\mathbf{k}}}\frac{[p^{\prime
2}+\cos^{2}\theta_{\mathbf{k}}]}{p^{\prime}[p^{\prime
2}+1]}+\frac{S_{\mathbf{k}}^{y}(0)}{\varepsilon_{\mathbf{k}}}\frac{\sin\theta_{\mathbf{k}}}{p^{\prime
2}+1}$ (29a)
$\displaystyle-\frac{S_{\mathbf{k}}^{x}(0)}{\varepsilon_{\mathbf{k}}}\frac{\sin\theta_{\mathbf{k}}\cos\theta_{\mathbf{k}}}{p^{\prime}[p^{\prime
2}+1]},$ $\displaystyle S_{\mathbf{k}}^{\pm}(p^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon_{\mathbf{k}}}\frac{S_{\mathbf{k}}^{\pm}(0)\mp
i\sin\theta_{\mathbf{k}}S_{\mathbf{k}}^{z}(p^{\prime})}{p^{\prime}\pm
i\cos\theta_{\mathbf{k}}},$ (29b) where $\displaystyle S_{\mathbf{k}}^{x}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(S_{\mathbf{k}}^{+}+S_{\mathbf{k}}^{-}),$ (30a)
$\displaystyle S_{\mathbf{k}}^{y}$ $\displaystyle=$
$\displaystyle\frac{1}{2i}(S_{\mathbf{k}}^{+}-S_{\mathbf{k}}^{-}).$ (30b) In
the SWCNT, the electrons fill up the lower band when the system stays at its
ground state at zero temperature. As a consequence, we may simply set
$S_{\mathbf{k}}^{x}(0)$ and $S_{\mathbf{k}}^{y}(0)$ as zero for convenience in
the following discussions. The inverse Laplace transformation gives the time
evolution of $S_{\mathbf{k}}^{z}(t)$ and $S_{\mathbf{k}}^{\pm}(t)$
respectively
$S_{\mathbf{k}}^{z}(t)=S_{\mathbf{k}}^{z}(0)\left[\cos^{2}\theta_{\mathbf{k}}+\sin^{2}\theta_{\mathbf{k}}\cos\left(\varepsilon_{\mathbf{k}}t\right)\right].$
(31)
and
$\displaystyle S_{\mathbf{k}}^{\pm}(t)$ $\displaystyle=$ $\displaystyle-
S_{\mathbf{k}}^{z}(0)\sin\theta_{\mathbf{k}}\cos\theta_{\mathbf{k}}[1-\cos\left(\varepsilon_{\mathbf{k}}t\right)]$
(32) $\displaystyle\mp
iS_{\mathbf{k}}^{z}(0)\sin\theta_{\mathbf{k}}\sin\left(\varepsilon_{\mathbf{k}}t\right).$
Finally, the total population inversion
$W(t)=\sum\limits_{\mathbf{k}}\left\langle
S_{\mathbf{k}}^{z}(t)+\frac{1}{2}\right\rangle$ (33)
is calculated as the summation over those of single momentum, which reads
$W(t)=\sum\limits_{\mathbf{k}}\left\\{\left\langle
S_{\mathbf{k}}^{z}(0)\right\rangle\left[1-2\sin^{2}\theta_{\mathbf{k}}\sin^{2}\left(\frac{\varepsilon_{\mathbf{k}}t}{2}\right)\right]+\frac{1}{2}\right\\}.$
(34)
When the temperature is zero, the system stays at its ground state and thus
$\left\langle S_{\mathbf{k}}^{z}(t=0)\right\rangle=-1/2$ is valid for all
$\mathbf{k}$. Then the total population inversion is obtained
$W(t)=\sum\limits_{\mathbf{k}}\frac{1}{2}\sin^{2}\theta_{\mathbf{k}}\\{1-\cos\left(\varepsilon_{\mathbf{k}}t\right)\\}$
(35)
If we consider the continuous momentum in a 2-D graphene and the
inhomogeneously-broadened system in which different quasi-spins have different
momentums by introducing the distribution $g(\varepsilon_{\mathbf{k}})$
centered on $\varepsilon_{\mathbf{0}}$ as
$g(\varepsilon_{\mathbf{k}})=2\sqrt{\pi}T\exp\left[-T^{2}\left(\varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{0}}\right)^{2}\right],$
(36)
which satisfies
$\frac{1}{2\pi}\int_{\infty}^{-\infty}g(\varepsilon_{\mathbf{k}})d\varepsilon_{\mathbf{k}}=1$.
When $2\sqrt{N}D_{\mathbf{k}}\gg\Delta_{\mathbf{k}}$ results in
$\sin\theta_{\mathbf{k}}\simeq 1$, the total population inversion can be
calculated as
$\displaystyle W(t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int_{\infty}^{-\infty}g(\varepsilon_{\mathbf{k}})\\{1-\cos\left(\varepsilon_{\mathbf{k}}t\right)\\}d\varepsilon_{\mathbf{k}}\mbox{}$
(37) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[1-\cos\left(\varepsilon_{\mathbf{0}}t\right)\right]\exp\left(-\frac{t^{2}}{4T^{2}}\right).$
It must be pointed out that the time dependence of the total population
inversion includes two aspects when the energy distribution is Gaussian type.
One is the periodic factor as $(1-\cos\left(\varepsilon_{\mathbf{0}}t\right))$
resulting from the central frequency of the Gaussian distribution. The other
is the exponential decay $\exp\left(-\frac{t^{2}}{4T^{2}}\right)$ resulted
from the broadening of the Gaussian distribution. The randomness of the energy
spectrum of the quasi-spins actually induces these effects, which can be
considered as a kind of spin echo.
Figure 3: Population inversion of electrons in the SWCNT is plotted versus
time. The parameters for the SWCNT are respectively: (1)chiral vector is
$(6,4)$, $\Omega=0.4$, $\tau=5$ for the black short dotted line; (2)chiral
vector is $(8,0)$, $\Omega=2$, $\tau=5$ for the red solid line; (3)chiral
vector is $(8,0)$, $\Omega=2$, $\tau=70$ for the blue short dashed line. Here,
$\tau$ is the time scale.
From Fig. 3 for the population inversion of the $(2n,0)$ SWCNTs with the
incurring light frequency of $\Omega=2J$, we may see that a considerable
proportion of the electrons are excited to the upper band (more than $1/15$
for $(8,0)$ SWCNT), and exhibits collapse and revival in a long period of
time. The explanation for it is straightforward: in the $(2n,0)$ SWCNTs, there
are large degeneracies of possible states onto the equi-energy lines $E=J$ of
the $2$-D graphene energy bands. Thus, the $(2n,0)$ SWCNTs are potential
experimental candidates for the demonstration of Rabi oscillation in solids.
## IV First Order Coherence of Scattered and Emitted Photons
The strong light field only couples the upper and lower bands of electrons
through its intensity, which essentially cancels the quantum optical features
of the SWCNT characterized by the higher order quantum coherence. To save
curiosity of quantized light field interacting with SWCNT, we return to the
Hamiltonian Eq.(16a)-(16b)
The first order correlation function of the light field
$G^{(1)}(\tau)=\left\langle d^{+}(t)d(t+\tau)\right\rangle$ (38)
to characterize the interference of the electrons in SWCNT is independent of
$t$ after long time evolution $t\rightarrow+\infty$, which corresponds to the
steady solution for the light-SWCNT coupling system. In the interaction
picture with respect to
$H_{0}=\Omega d^{\dagger}d+\sum\limits_{\mathbf{k}}\Omega S_{\mathbf{k}}^{z},$
(39)
the Langevin equations read as
$\displaystyle\frac{\partial}{\partial t}d$ $\displaystyle=$
$\displaystyle-i\sum\limits_{\mathbf{k}}D_{\mathbf{k}}S_{\mathbf{k}}^{-},$
(40a) $\displaystyle\frac{\partial}{\partial t}S_{\mathbf{k}}^{-}$
$\displaystyle=$
$\displaystyle\left(i\Delta_{\mathbf{k}}-\gamma_{\mathbf{k}}\right)S_{\mathbf{k}}^{-}+2iD_{\mathbf{k}}S_{\mathbf{k}}^{z}d,$
(40b) $\displaystyle\frac{\partial}{\partial t}S_{\mathbf{k}}^{z}$
$\displaystyle=$
$\displaystyle-2\gamma_{\mathbf{k}}(S_{\mathbf{k}}^{z}+\frac{1}{2})+iD_{\mathbf{k}}(d^{\dagger}S_{\mathbf{k}}^{-}-S_{\mathbf{k}}^{+}d).$
(40c) Here, we phenomenologically add decay terms of the SWCNT part to the
Langevin equations, while neglect the decay of light field since an ideal
probe is considered. We also assume that the SWCNT system reaches its
equilibrium state with the light field before there is considerable change in
the light field. Actually, this assumption is very crucially used in Haken’s
theory of laser Laser . By setting the time derivatives of the $S$ operators
as zero, the steady solution of the total system can be obtained with steady
quasi-spin operators $\displaystyle S_{\mathbf{k}}^{z}$ $\displaystyle=$
$\displaystyle-\frac{\gamma_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}}{2(\Delta_{\mathbf{k}}^{2}+2d^{\dagger}dD_{\mathbf{k}}^{2}+\gamma_{\mathbf{k}}^{2})},$
(41a) $\displaystyle S_{\mathbf{k}}^{-}$ $\displaystyle=$
$\displaystyle-\frac{iD_{\mathbf{k}}(\gamma_{\mathbf{k}}-i\Delta_{\mathbf{k}})d}{\Delta_{\mathbf{k}}^{2}+2d^{\dagger}dD_{\mathbf{k}}^{2}+\gamma_{\mathbf{k}}^{2}}.$
(41b) Therefore, if the number of photons does not fluctuate intensively long
time after the light is turned on, we could simply set the particle number
operator $d^{\dagger}d=N$ as a constant.
In order to study the first order coherence of the light field, we use the
mean field approach for the Langevin equations of the above system by setting
$S_{\mathbf{k}}^{z}d\approx\left.\left\langle
S_{\mathbf{k}}^{z}(t)\right\rangle\right|_{t\rightarrow\infty}d(\tau)\equiv
S_{\mathbf{k}}^{z}(\infty)d(\tau)$ (42)
for long time evolution. Here we can analytically calculate the first order
correlation function through the partial differential equations (40a-40c).
After applying Laplace transformation to Eq. (40a-40c), we have
$\displaystyle pd-d(0)$ $\displaystyle=$
$\displaystyle-i\sum\limits_{\mathbf{k}}D_{\mathbf{k}}S_{\mathbf{k}}^{-},$
(43a)
$\displaystyle\left(p-i\Delta_{\mathbf{k}}^{\prime}\right)S_{\mathbf{k}}^{-}$
$\displaystyle=$ $\displaystyle 2iD_{\mathbf{k}}\left.\left\langle
S_{\mathbf{k}}^{z}(t)\right\rangle\right|_{t\rightarrow\infty}d+S_{\mathbf{k}}^{-}(0),$
(43b) for the effective detuning
$\Delta_{\mathbf{k}}^{\prime}=\Delta_{\mathbf{k}}+i\gamma_{\mathbf{k}}.$ This
gives the solution of $d(p)$ as
$d(p)=\frac{d(0)+\Lambda^{-}(p)}{p-\Lambda^{z}(p)}.$ (44)
Here,
$\Lambda^{-}(p)=-i\sum\limits_{\mathbf{k}}\frac{D_{\mathbf{k}}S_{\mathbf{k}}^{-}(0)}{p-i\Delta_{\mathbf{k}}^{\prime}}$
(45)
represents the contribution from $S_{\mathbf{k}}^{-}(0)$, while contribution
from the long time evolution of $\left\langle S_{\mathbf{k}}^{z}\right\rangle$
is given by
$\Lambda^{z}(p)=\sum\limits_{\mathbf{k}}\frac{2D_{\mathbf{k}}^{2}\left.\left\langle
S_{\mathbf{k}}^{z}(t)\right\rangle\right|_{t\rightarrow\infty}}{p-i\Delta_{\mathbf{k}}^{\prime}}.$
(46)
Since the electron-photon interaction serves as a perturbation term in the
Hamiltonian, the singularities of the $d(p)$ is mainly determined by the
denominator $p-\Lambda^{z}(p)$. Under the Wigner-Weisskopf approximation, the
$0$-th order zero point of the denominator is $p=0,$ and to the $1$-st order
it is
$p=-i\Omega^{\prime}-\Gamma^{\prime},$ (47)
where the renormalized frequency and the effective decay are
$\displaystyle\Omega^{\prime}$ $\displaystyle=$
$\displaystyle-\mathrm{Im}\Lambda^{z}(0),$ (48a)
$\displaystyle\Gamma^{\prime}$ $\displaystyle=$
$\displaystyle-\mathrm{Re}\Lambda^{z}(0).$ (48b) Applying the inverse Laplace
transformation, we obtain an expression for $d(\tau)$ as
$d(\tau)=\exp\left(-i\Omega^{\prime}\tau-\Gamma^{\prime}\tau\right)\left[d(0)+F(\tau)\right],$
(49)
where the contribution from $S_{\mathbf{k}}^{-}(0)$ is
$F(\tau)=-i\sum\limits_{\mathbf{k}}\frac{D_{\mathbf{k}}S_{\mathbf{k}}^{-}(0)}{\mu_{\mathbf{k}}+i\nu_{\mathbf{k}}}\left[1-e^{-i\mu_{\mathbf{k}}\tau-\nu_{\mathbf{k}}\tau}\right],$
(50)
with $\mu_{\mathbf{k}}=\left(-\Delta_{\mathbf{k}}-\Omega^{\prime}\right)$ and
$\nu_{\mathbf{k}}=\left(\gamma_{\mathbf{k}}-\Gamma^{\prime}\right).$ If we
compare a quasi-spin system to a heat bath, the term $F(\tau)$ represents its
induced quantum fluctuation. The couplings of the light field to SWCNT is
characterized by
$\displaystyle\Omega^{\prime}$ $\displaystyle=$
$\displaystyle\sum\limits_{\mathbf{k}}\frac{D_{\mathbf{k}}^{2}}{\Delta_{\mathbf{k}}^{2}+2ND_{\mathbf{k}}^{2}+\gamma_{\mathbf{k}}^{2}}\Delta_{\mathbf{k}},$
(51a) $\displaystyle\Gamma^{\prime}$ $\displaystyle=$
$\displaystyle\sum\limits_{\mathbf{k}}\frac{D_{\mathbf{k}}^{2}}{\Delta_{\mathbf{k}}^{2}+2ND_{\mathbf{k}}^{2}+\gamma_{\mathbf{k}}^{2}}\gamma_{\mathbf{k}},$
(51b) where the explicit steady solution for $S_{\mathbf{k}}^{z}$ that has
been used.
The contribution $\left\langle d^{+}(0)F(\tau)\right\rangle$ from
$S_{\mathbf{k}}^{-}(0)$ in the first order correlation
$G^{(1)}(\tau)=\exp\left(-i\Omega^{\prime}\tau-\Gamma^{\prime}\tau\right)\left(G^{(1)}(0)+\left\langle
d^{+}(0)F(\tau)\right\rangle\right)$ (52)
vanishes since the average on the photon number states reduces to zero due to
the photon number conservation. Then the normalized first order coherence
function $g^{(1)}(\tau)\equiv G^{(1)}(\tau)/G^{(1)}(0)$ is explicitly written
as
$g^{(1)}(\tau)=\exp\left(-i\Omega^{\prime}\tau-\Gamma^{\prime}\tau\right),$
(53)
which is used to measure the interference of the scattered and emitted
photons. It is clear that long time first order correlation of the light field
vanishes exponentially with $\tau$. In most cases, considering that the decay
rates $\gamma_{\mathbf{k}}$ are much smaller than the detuning
$\Delta_{\mathbf{k}}$, $\Gamma^{\prime}\ll\Omega^{\prime}$ is obviously
satisfied. Thus, neglecting the decay effect in the first order coherence, the
existence of SWCNT contributes to the first order coherence a shift
$\Omega^{\prime}$ in the frequency of light.
## V Second Order Correlation of The Scattered and Emitted Light
The first order coherence function only demonstrates the interference of the
scattered and emitted photon. To distinguish the fully quantum optical
properties of the SWCNT, e.g. , the bunching and anti-bunching of the photons,
the second order coherence function
$G^{(2)}(\tau)=\left\langle
d^{\dagger}(t)d^{\dagger}(t+\tau)d(t+\tau)d(t)\right\rangle$ (54)
is needed. According to Eq. (47), we calculate
$\displaystyle G^{(2)}(\tau)$ $\displaystyle=$
$\displaystyle\exp\left(-2\Gamma^{\prime}\tau\right)G^{(2)}(0)$ (55)
$\displaystyle+\exp\left(-2\Gamma^{\prime}\tau\right)\left\langle
d^{\dagger}(0)F^{\dagger}(\tau)F(\tau)d(0)\right\rangle,$
where we have neglected terms $\left\langle
d^{\dagger}(0)d^{\dagger}(0)F(\tau)d(0)\right\rangle$ and $\left\langle
d^{\dagger}(0)F^{\dagger}(\tau)d(0)d(0)\right\rangle$ because of the photon
number conservation for the light field. Neglecting correlation between
different quasi-spins, only terms with the same momentum can survive in
$F^{\dagger}(\tau)$ and $F(\tau).$ Therefore, the non-vanishing second term is
calculated as
$\displaystyle\left\langle
d^{\dagger}(0)F^{\dagger}(\tau)F(\tau)d(0)\right\rangle$ (56)
$\displaystyle\approx$
$\displaystyle\sum\limits_{\mathbf{k}}f_{\mathbf{k}}(\tau)\left\langle
d^{\dagger}(0)S_{\mathbf{k}}^{+}(0)S_{\mathbf{k}}^{-}(0)d(0)\right\rangle$
$\displaystyle\approx$
$\displaystyle\sum\limits_{\mathbf{k}}f_{\mathbf{k}}(\tau)\left\langle
d^{\dagger}(0)d(0)\right\rangle\left(\left.\left\langle
S_{\mathbf{k}}^{z}(t)\right\rangle\right|_{t\rightarrow\infty}+\frac{1}{2}\right),$
where the time dependent coefficients are
$f_{\mathbf{k}}(\tau)=\frac{2D_{\mathbf{k}}^{2}}{\mu_{\mathbf{k}}^{2}+\nu_{\mathbf{k}}^{2}}\left[\cosh\left(\nu_{\mathbf{k}}\tau\right)-\cos\left(\mu_{\mathbf{k}}\tau\right)\right]e^{-\nu_{\mathbf{k}}\tau}.$
(57)
Accordingly, the normalized second order coherence function
$g^{(2)}(\tau)\equiv G^{(2)}(\tau)/\left|G^{(1)}(0)\right|^{2}$ is written as
$\displaystyle g^{(2)}(\tau)$ $\displaystyle=$
$\displaystyle\exp\left(-2\Gamma^{\prime}\tau\right)\left[\frac{G^{(2)}(0)}{G^{(1)}(0)^{2}}\right.+$
(58)
$\displaystyle\left.\sum\limits_{\mathbf{k}}\frac{f_{\mathbf{k}}(\tau)}{G^{(1)}(0)}\left(\left.\left\langle
S_{\mathbf{k}}^{z}(t)\right\rangle\right|_{t\rightarrow\infty}+\frac{1}{2}\right)\right]$
Here, the second item in $g^{(2)}(\tau)$ is non-negative for any $\tau$, and
returns to zero when $\tau\rightarrow 0$, thus the explicit effect of the
anti-bunching of the light coupled with the SWCNT is illustrated in Fig. 4.
Figure 4: The second order correlation of the light $g^{(2)}(t)-g^{(2)}(0)$ is
plotted, in which the antibunching feature is obviously displayed. The two
chiral vectors $(6,4)$ and $(8,0)$ are chosen to represent significant anti-
bunching effect due to different reasons. Here, we have chosen the frequency
of the light field $\omega=2J$.
Due to the divergence near the resonance area for $g^{(2)}(t)$, the anti-
bunching feature is significant where the light and the energy gap between
upper and lower bands reach resonance while the interaction intensity
$D_{\mathbf{k}}$ is comparatively high. In this case, the SWCNT is equivalent
to one or several 2-level atoms that interact strongly with the incurring
light, just as the case in the $(6,4)$ SWCNT when the incurring light
frequency is $2J$. Similar to Sec. III concerning Rabi oscillation, here we
still have a distinct effect for the $(2n,0)$ SWCNTs, when the incurring light
frequency is really close to $2J$. In the case for $(8,0)$ SWCNT, the strong
anti-bunching feature is instead caused by the large degeneracy on the $E=J$
line in the first Brillouin zone. Unlike the case for the $(6,4)$ SWCNT, in
which merely several electron states are involved, the significant anti-
bunching here is caused by the excitation in the $E=J$ band of the SWCNT,
where thousands of possible states participate in at the same time.
## VI Possible Lasing Mechanism of Carbon Nanotube
The above investigations imply that the light emitted from or scattered by the
SWCNT is strongly correlated in time domain, thus explicitly displays quantum
effects. It is straight forward to imagine that if electrons in the SWCNT
experience a population inversion, the emitted light would be amplified. This
observation may enable a possible lasing mechanism. In this section, we will
explore this mechanism for the SWCNT by using Haken’s laser theory Laser .
The Heisenberg equations(25a,25b) without dissipation usually have no steady
solution. Thus we phenomenologically introduce decays on both the light field
and the quasi-spin operators to make the physical observables reach the stable
results. In order to obtain the steady solution, we neglect the fluctuations
because the time average of them vanishes. This simplification results in the
laser-like equations
$\displaystyle\frac{\partial}{\partial t}\widetilde{d}^{\dagger}$
$\displaystyle=$
$\displaystyle-\kappa\widetilde{d}^{\dagger}+i\sum\limits_{\mathbf{k}}D_{\mathbf{k}}\widetilde{S}_{\mathbf{k}}^{+}e^{-i\Delta_{\mathbf{k}}t},$
(59a) $\displaystyle\frac{\partial}{\partial t}\widetilde{S}_{\mathbf{k}}^{+}$
$\displaystyle=$
$\displaystyle-\gamma_{\mathbf{k}}\widetilde{S}_{\mathbf{k}}^{+}-2iD_{\mathbf{k}}\widetilde{d}^{\dagger}S_{\mathbf{k}}^{z}e^{i\Delta_{\mathbf{k}}t},$
(59b) $\displaystyle\frac{\partial}{\partial t}S_{\mathbf{k}}^{z}$
$\displaystyle=$
$\displaystyle-2\gamma_{\mathbf{k}}(S_{\mathbf{k}}^{z}+\frac{1}{2})-iD_{\mathbf{k}}(\widetilde{S}_{\mathbf{k}}^{+}\widetilde{d}e^{-i\Delta_{\mathbf{k}}t}-h.c.),$
(59c)
where we have removed the higher frequency factors by defining
$\widetilde{d}^{\dagger}=d^{\dagger}\exp(-i\Omega t)$ and
$\widetilde{S}_{\mathbf{k}}^{+}=S_{\mathbf{k}}^{+}\exp[-i2E(\overrightarrow{k})t].$
This approach changes the observation from a laboratory frame of reference
into some rotating one. Equation (59b) can be formally integrated as
$\widetilde{S}_{\mathbf{k}}^{+}(t)=\widetilde{S}_{\mathbf{k}}^{+}(0)e^{-\gamma_{\mathbf{k}}t}-2iD_{\mathbf{k}}\int\limits_{0}^{t}e^{-\gamma_{\mathbf{k}}(t-\tau)}\widetilde{d}^{\dagger}S_{\mathbf{k}}^{z}e^{i\Delta_{\mathbf{k}}\tau}d\tau.$
(60)
According to Haken’s laser theory, if
$\widetilde{d}^{\dagger}S_{\mathbf{k}}^{z}$ varies with time much slower than
$\widetilde{S}_{\mathbf{k}}^{+}(t)$, it could be regarded as a time-
independent one and then the above integral becomes
$\widetilde{S}_{\mathbf{k}}^{+}(t)=\widetilde{S}_{\mathbf{k}}^{+}(0)e^{-\gamma_{\mathbf{k}}t}-2iD_{\mathbf{k}}\widetilde{d}^{\dagger}S_{\mathbf{k}}^{z}\frac{\left(e^{i\Delta_{\mathbf{k}}t}-e^{-\gamma_{\mathbf{k}}t}\right)}{\gamma_{\mathbf{k}}+i\Delta_{\mathbf{k}}},$
(61)
After a long time, the first term in the above solution Eq.(60), which is
totally determined by the initial polarization
$\widetilde{S}_{\mathbf{k}}^{+}(0)$, will vanish. Thus, when $\gamma_{k}t\gg
1$, only the initial state-independent part
$\widetilde{S}_{\mathbf{k}}^{+}(t)\approx-2iD_{\mathbf{k}}\widetilde{d}^{\dagger}S_{\mathbf{k}}^{z}\frac{e^{i\Delta_{\mathbf{k}}t}}{\gamma_{\mathbf{k}}+i\Delta_{\mathbf{k}}},$
(62)
remains. In this case the motion equation of the $z-$direction spin operators
becomes
$\frac{\partial}{\partial
t}S_{\mathbf{k}}^{z}\approx-2\gamma_{\mathbf{k}}(S_{\mathbf{k}}^{z}+\frac{1}{2})-\theta_{\mathbf{k}}\widetilde{d}^{\dagger}\widetilde{d}S_{\mathbf{k}}^{z},$
(63)
where
$\theta_{\mathbf{k}}=4\gamma_{\mathbf{k}}D_{\mathbf{k}}^{2}/(\gamma_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2})$.
Then we obtain the effective motion equation of the light field
$\frac{\partial}{\partial
t}\widetilde{d}^{\dagger}=-\widetilde{d}^{\dagger}\left(\kappa-\sum\limits_{\mathbf{k}}\frac{2D_{\mathbf{k}}^{2}e^{i\Omega
t}}{\gamma_{\mathbf{k}}+i\Delta_{\mathbf{k}}}S_{\mathbf{k}}^{z}\right).$ (64)
In the following discussions we will demonstrate a lasing-like phenomenon by
considering the solution of Eq.(64)
Usually, a lasing process requires population inversion. To realize such
population inversion in our setup, a pump of electrons is needed to inject
electrons with specific state into the carbon nanotube. Phenomenologically, we
add a pump term $c_{\mathbf{k}}>0$ to each term $S_{\mathbf{k}}^{z}$, then
$\frac{\partial}{\partial
t}S_{\mathbf{k}}^{z}=c_{\mathbf{k}}-2\gamma_{\mathbf{k}}(S_{\mathbf{k}}^{z}+\frac{1}{2})-\theta(\mathbf{k})\widetilde{d}^{\dagger}\widetilde{d}S_{\mathbf{k}}^{z},$
(65)
The population inversion is obtained from Eq. (65) as
$\displaystyle S_{\mathbf{k}}^{z}$ $\displaystyle=$ $\displaystyle
S_{\mathbf{k}}^{z}(0)\exp\left(-\int\limits_{0}^{t}\left[\theta_{\mathbf{k}}\widetilde{d}^{\dagger}\widetilde{d}+2\gamma_{\mathbf{k}}\right]d\tau^{\prime}\right)+$
(66)
$\displaystyle(c_{\mathbf{k}}-\gamma_{\mathbf{k}})\int\limits_{0}^{t}\exp\left(\int\limits_{0}^{\tau}\left[\theta_{\mathbf{k}}\widetilde{d}^{\dagger}\widetilde{d}+2\gamma_{\mathbf{k}}\right]d\tau^{\prime}\right)d\tau\times$
$\displaystyle\exp\left(-\int\limits_{0}^{t}\left[\theta_{\mathbf{k}}\widetilde{d}^{\dagger}\widetilde{d}+2\gamma_{\mathbf{k}}\right]d\tau^{\prime}\right).$
After a long time evolution $\left(\gamma_{\mathbf{k}}t\gg 1\right)$, this
solution becomes
$S_{\mathbf{k}}^{z}=(c_{\mathbf{k}}-\gamma_{\mathbf{k}})\int\limits_{0}^{t}\exp\left(-\int\limits_{\tau}^{t}\theta_{\mathbf{k}}\widetilde{d}^{\dagger}\widetilde{d}d\tau^{\prime}\right)e^{-2\gamma_{\mathbf{k}}(t-\tau)}d\tau.$
(67)
It follows from Eq.(67) that the main contribution of the integral comes from
the accumulation of the weighted photon numbers in the time $\tau\sim t$. In
this sense we can assume that
$\int\limits_{\tau}^{t}\theta_{\mathbf{k}}\widetilde{d^{\dagger}}\widetilde{d}d\tau^{\prime}=\theta_{\mathbf{k}}\widetilde{d^{\dagger}}\widetilde{d}(t-\tau)$
Then the population inversion is integrated as
$S_{\mathbf{k}}^{z}\approx\frac{(c_{\mathbf{k}}-\gamma_{\mathbf{k}})}{\theta_{\mathbf{k}}\widetilde{d}^{\dagger}\widetilde{d}+2\gamma_{\mathbf{k}}}.$
(68)
Eventually, the motion equation of the light field is obtained as
$\frac{\partial}{\partial
t}\widetilde{d}^{\dagger}\approx(\kappa^{\prime}-i\delta\omega)\widetilde{d}^{\dagger}-\eta\widetilde{d}^{\dagger}\widetilde{d}^{\dagger}\widetilde{d},$
(69)
where
$\delta\omega=\sum\limits_{\mathbf{k}}D_{\mathbf{k}}^{2}\frac{(c_{\mathbf{k}}-\gamma_{\mathbf{k}})}{\gamma_{\mathbf{k}}}\frac{\Delta_{\mathbf{k}}}{\gamma_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}},$
(70a) appears as the Lamb shift of photons, and
$\kappa^{\prime}=-\kappa+\sum\limits_{\mathbf{k}}D_{\mathbf{k}}^{2}\frac{(c_{\mathbf{k}}-\gamma_{\mathbf{k}})}{\gamma_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}},$
(70b) represents a dissipation or amplification of the optical mode together
with
$\eta=\sum\limits_{\mathbf{k}}2D_{\mathbf{k}}^{4}\frac{c_{\mathbf{k}}-\gamma_{\mathbf{k}}}{(\gamma_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2})^{2}}.$
(70c) describing the extent of nonlinearity of the light field induced by the
SWCNT. Here, we have expanded the second item on the right hand side of
Eq.(69) up to the first order of
$2D_{\mathbf{k}}^{2}\widetilde{d}^{\dagger}\widetilde{d}$.
Obviously, Eq.(69) is typical to describe the lasing process in an
amplification medium. When electrons are injected into the SWCNT to realize a
population inversion,
$\kappa^{\prime}=-\kappa+\sum\limits_{\mathbf{k}}D_{\mathbf{k}}^{2}\frac{(c_{\mathbf{k}}-\gamma_{\mathbf{k}})}{\gamma_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}}>0$
(71)
with $\eta>0$, we obtain a lasing equation
$\frac{\partial}{\partial
t}\widetilde{d^{\dagger}}=\kappa^{\prime}\widetilde{d^{\dagger}}-\eta\widetilde{d^{\dagger}}\widetilde{d^{\dagger}}\widetilde{d}.$
(72)
Then the effect of the coherently injected electrons the SWCNT on the light
field is equivalent to that of a double-well potential formed as
$V(\left|d\right|)=-\kappa^{\prime}\left|d\right|^{2}+\frac{\eta}{2}\left|d\right|^{4},$
(73)
Thus there exists a symmetry breaking based instability for laser
amplification. When $\kappa^{\prime}<0$, $d=0$ is the unique stable point for
the effective potential $V(\left|d\right|)$. In this case we may safely
neglect the nonlinearity, and the system is only affected by stochastic
processes. However, when $\kappa^{\prime}$ passes through zero, the point
$d=0$ is no longer the stable point. Instead, the photon amplitude $d$
acquires its new stable points with nonzero amplitude
$\left|d\right|=\sqrt{\frac{\kappa^{\prime}}{\eta}}$ (74)
indicating a phase transition in the system. The above phenomenon that nonzero
stable points of $V(\left|d\right|)$appear means that a coherent light field
with non-vanishing amplitude is produced by the radiation of electrons
confined in the SWCNT.
## VII Conclusion
In summary, our investigation in this paper is oriented by the needs of
designing the quantum devices in future. We theoretically studied a solid
state based quantum optical system, namely, the SWCNT interacting with
quantized light field. The ballistic transport of electrons in SWCNT means
quantum coherence of electrons in terminology of quantum optics. Thus, the
emitted and scattered light from such coherent electrons could be quantum
coherent as well, and then we use the higher order coherence function to
describe it. On the other hand, SWCNT with different chirality $(n,m)$ have
different properties in their Rabi oscillations of the electrons when driven
by a strong single-mode light field. The anti-bunching features of the light
scattered by or emitted from them is also studied in details here. The reason
for such distinction of chirality is that different sets of wave vectors
$\mathbf{k}$ are allowed in SWCNT with different chiral vectors, which may
lead to different energy structures in the SWCNT. Such effect is especially
significant on the $(2n,0)$ type SWCNT, where large degeneracy of possible
electron states onto $E=J$ occurs. This is a characteristic property absent in
2D graphene. The possible lasing mechanism in the SWCNT is also investigated
theoretically, which may promise the realization of nanoscale laser devices.
## Appendix A Semi-classical
It is noticed that the semi-classical approximation applied in Sec. III is
valid only for the quasi-classical case in which the initial state possesses a
very large number of single frequency photons. We will justify this
approximation with necessary details in this appendix.
The complete dynamics of the SWCNT interacting with a strong light field is
displayed through the Schrodinger equations governed by the Hamiltonian
$H=H_{0}+H_{1},$ in the interaction picture, where
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum\limits_{\mathbf{k}}E_{\mathbf{k}}\left(\alpha_{\mathbf{k}}^{\dagger}\alpha_{\mathbf{k}}-\beta_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k}}\right),$
(75a) $\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}D_{\mathbf{k}}\left(de^{-i\Omega
t}\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}+h.c.\right)$ (75b) And the
initial condition of the system
$\left|\Psi\left(0\right)\right\rangle=\left|\xi=\sqrt{N}e^{i\theta}\right\rangle\otimes\left|\phi\left(0\right)\right\rangle,$
(76)
where the coherent state
$\left|\xi\right\rangle=\exp\left(\xi
d^{\dagger}-\xi^{\ast}d\right)\left|0\right\rangle\equiv
D\left(\xi\right)\left|0\right\rangle$ (77)
represents the state of the light field while
$\left|\phi\left(0\right)\right\rangle$ stands for the initial state of the
electrons in the SWCNT. We note that $\left|\alpha\right|\simeq\sqrt{N}.$Since
there is no broken global phase symmetry, the arbitrary $\theta$ is chosen as
$0.$ The main reason for choosing the initial photon state as a coherent one
is that the average number
$\left\langle\sqrt{N}\right|d^{\dagger}d\left|\sqrt{N}\right\rangle=N$ should
be satisfied.
We introduce the photon vacuum picture, similar to the approach for the semi-
classical approximation of photon-atom system [cite P.L.Kingt Concept of
Quantum Optics], defined by
$\left|\Phi\left(t\right)\right\rangle=D\left(\xi\right)^{-1}\left|\Psi\left(t\right)\right\rangle,\left|\Phi\left(0\right)\right\rangle=\left|0\right\rangle\otimes\left|\phi\left(0\right)\right\rangle$
(78)
which satisfies the Schrodinger equation (in the interaction picture) with the
effective Hamiltonian
$H_{e}=D\left(\xi\right)^{-1}HD\left(\xi\right)=H_{0}+V_{q}+H_{q}$
where
$\displaystyle V_{q}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}D_{\mathbf{k}}\left(\sqrt{N}e^{-i\Omega
t}\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}+h.c.\right)$ (79a)
$\displaystyle H_{q}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}D_{\mathbf{k}}\left(de^{-i\Omega
t}\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}+h.c.\right),$ (79b)
Here $|0\rangle$ can be understood as a displaced vacuum. It should be noticed
that the above derivation is exact for the initial condition (78).
For a very large $N$, $H_{q}$ in the above Hamiltonian is very small with
respect to the $V_{q}$, and it can be neglected in the first order
approximation. Under this approximation, the state of photons is subjected to
a collective evolution governed by the effective Hamiltonian
$\displaystyle H_{e}$ $\displaystyle=$ $\displaystyle H_{0}+V_{q},$ (80a)
$\displaystyle V_{q}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}D_{\mathbf{k}}\left(\sqrt{N}e^{-i\Omega
t}\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}+h.c.\right).$ (80b)
Transforming back to the original picture, one proves the conclusion: If $N$
is a macroscopic number, namely, it is large enough, the total system will
evolve with a factorizable wave function
$|\Psi(t)\rangle=|\sqrt{N}e^{i\theta}\rangle\otimes|\phi(t)\rangle,$ where
$|\phi(t)\rangle$ obeys the Schrödinger equation governed by the effective
Hamiltonian $H_{e}$.
The next question is the effects of the neglected term, $H_{q}$, on the
dynamics in the photons vacuum picture. In the framework of the perturbation
theory, the role of $H_{q}$ relies on the coupling to the vacuum, that is
$\displaystyle H_{q}|\Phi(0)\rangle$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}D_{\mathbf{k}}\left(de^{-i\Omega
t}\alpha_{\mathbf{k}}^{\dagger}\beta_{\mathbf{k-q}}+h.c.\right)|0\rangle\otimes|\phi(0)\rangle$
(81) $\displaystyle=$ $\displaystyle e^{i\Omega
t}|1\rangle\otimes\sum_{\mathbf{k}}D_{\mathbf{k}}\beta_{\mathbf{k-q}}^{\dagger}\alpha_{\mathbf{k}}|\phi(0)\rangle$
which leads to a single-particle excitation of the vacuum. Finally we reach
the following conclusions: (1) In the large $N$ limit, this excitation is weak
compared with the collective motion; (2) If there is initially no collective
excitation or single excited electrons in the SWCNT, the system will be stable
and remain in the displaced vacuum state even when $H_{q}$ is taken into
account.
###### Acknowledgements.
The authors thank H. Dong for his schematic diagram of the graphene. This work
is supported by NSFC No.10474104, No.60433050, and No.10704023, NFRPC
No.2006CB921205 and 2005CB724508.
## References
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* (3) M. S. Dresselhaus, G. Dresselhaus, Ph. Avouris, Carbon Nanotubes (Springer-Verlag, Berlin, 2110).
* (4) M. J. O’Connell, S. M. Bachilo, C. B. Huffman, V. C. Moore, M. S. Strano, E. H. Haroz, K. L. Rialon, P. J. Boul, W. H. Noon, C. Kittrell, Jianpeng Ma, R. H. Hauge, R. Bruce Weisman, and R. E. Smalley, Science 297, 593 (2002).
* (5) V. C. Moore, M. S. Strano, E. H. Haroz, R. H. Hauge, and R. E. Smalley, Nano Lett. 3, 1379 (2003).
* (6) Chongwu Zhou, Jing Kong, and Hongjie Dai, Appl. Phys. Lett. 76, 1597 (2000).
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* (9) Xiaolei Liu, Chenglung Lee, and Chongwu Zhou, Appl. Phys. Lett. 79, 3329 (2001).
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* (11) S. M. Bachilo, M. S. Strano, C. Kittrell, R. H. Hauge, R. E. Smalley, R. B. Weisman, Science 298, 2361 (2002).
* (12) H. Katauraa, Y. Kumazawaa, Y. Maniwaa, I. Umezub, S. Suzukic, Y. Ohtsukac and Y. Achiba, Synthetic Metals 103, 2555 (1999).
* (13) E. Chang, G. Bussi, A. Ruini, and E. Molinari, Phys. Rev. Lett. 92, 196401 (2004).
* (14) C. D. Spataru, S. Ismail-Beigi, L. X. Benedict, and S. G. Louie, Phys. Rev. Lett. 92, 077402 (2004).
* (15) V. Perebeinos, J. Tersoff, and Ph. Avouris, Phys. Rev. Lett. 92, 257402 (2004).
* (16) H. Zhao and S. Mazumdar, Phys. Rev. Lett. 93, 157402 (2004).
* (17) F. Wang, G. Dukovic, L. E. Brus, T. F. Heinz, Science 308, 838 (2005).
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|
arxiv-papers
| 2009-04-15T02:34:08 |
2024-09-04T02:49:01.874275
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Z. L. Guo, Z. R. Gong, and C. P. Sun",
"submitter": "Zelei Guo",
"url": "https://arxiv.org/abs/0904.2231"
}
|
0904.2240
|
###### Abstract
A theorem related to the Newman-Penrose constants is proven. The theorem
states that all the Newman-Penrose constants of asymptotically flat,
stationary, asymptotically algebraically special electrovacuum spacetimes are
zero. Straightforward application of this theorem shows that all the Newman-
Penrose constants of the Kerr-Newman spacetime must vanish.
On Newman-Penrose constants of stationary electrovacuum spacetimes
Xiangdong Zhanga,111e-mail : zhangxiangdong@mail.bnu.edu.cn Xiaoning
Wub,222e-mail : wuxn@amss.ac.cn and Sijie Gao a,333e-mail : sijie@bnu.edu.cn
a. Department of Physics,
Beijing Normal University,
Beijing, China, 100080.
b. Institute of Applied Mathematics.
Academy of Mathematics and System Science.
Chinese Academy of Sciences,
P.O.Box 2734, Beijing, China, 100080.
PACS number : 04.20.-q, 04.20.Ha
Keywords : Newman-Penrose constants, stationary electrovacuum condition, Kerr-
Newman solution.
## 1 Introduction
Newmen-Penrose(N-P) constants are very interesting and useful quantities in
the study of asymptotic flat space-times. They were first found by E.T.Newman
and R.Penrose in 1968[1] and then discussed by many other authors[2, 3, 4, 5,
6, 7]. Although the N-P constants have been found for forty years, their
physical interpretation remains an open question. One reason is that the
computation of these constants for a general asymptotically flat spacetime is
not easy. In stationary vacuum cases, these constants can be viewed as
combination of multi-pole moments of space-times[8]-[11]. Calculations of the
NP constants for vacuum solutions have been made by many authors [12, 13, 14,
15, 16, 17]. People used to conjecture that the algebraically special
condition (ASC) leads to the vanishing of NP constants. However, Kinnersley
and Walker[12] provided a counterexample. Recently, some authors[15] proposed
the asymptotically algebraically special condition (AASC), and proved that the
N-P constants vanish for vacuum, stationary, asymptotic algebraically special
space-times. In fact, the two conditions are closely related. It is well known
that the ASC implies that the Weyl curvature possesses a multiple principle
null direction. This condition can be expressed in terms of two geometric
invariants $I$ and $J$, defined by
$I=\Psi_{0}\Psi_{4}-4\Psi_{1}\Psi_{3}+3(\Psi_{2})^{2}$ and
$J=\Psi_{4}\Psi_{2}\Psi_{0}+2\Psi_{3}\Psi_{2}\Psi_{1}-(\Psi_{2})^{3}-(\Psi_{3})^{2}\Psi_{0}-(\Psi_{1})^{2}\Psi_{4}$
[18, 21]. A spacetime is said to be algebraically special if
$I^{3}-27J^{2}=0$. It has been shown that a general asymptotically flat,
stationary spacetime satisfies $I^{3}-27J^{2}\sim O(r^{-21})$ near future null
infinity[15]. Thus, $I^{3}-27J^{2}$ will peel off very quickly for a general
stationary vacuum asymptotically flat spacetime although the spacetime may not
be algebraically special. A spacetime is said to be “asymptotically
algebraically special” if $I^{3}-27J^{2}\sim O(r^{-22})$ [15], i.e. one order
faster than general cases. From the geometric point of view, this indicates
that one pair of principle null directions coincides near null infinity. By
imposing this condition, authors of [15] showed that the NP constants vanish
for vacuum, stationary spacetimes. Based on the result of [15], N-P constants
can bee seen as combination of Janis-Newman multi-poles of gravitational
field[24]. An intriguing question is whether the Janis-Newman multi-poles of
matter field will contribute to the N-P constants. In this paper, we extend
the discussion to the electrovacuum case. By imposing the AACS, we show that
the NP constants still vanish in the presence of a stationary Maxwell field.
If the Maxwell field is not stationary, the multi-pole moments of the Maxwell
field will contribute to the N-P constants.
This paper is organized as follows : In section II, we apply the method of
Taylor expansion to a stationary electrovacuum space-time. With the help of
the Killing equation, we reduce the dynamical freedom of gravitational field
into a set of arbitrary constants. Detailed expressions are given up to order
$O(r^{-6})$. We then prove that all the N-P constants of a stationary
asymptotically algebraically special electrovacuum space-time are zero.
Finally, we make some concluding remarks in section III.
## 2 The Newman-Penrose constants of stationary asymptotically algebraically
special electrovacuum space-times
In an asymptotically flat spacetime, the Newman-Penrose constants are defined
by [18]
$\displaystyle G_{m}=\int_{S_{\infty}}{}_{2}Y_{2,m}\Psi^{1}_{0}dS,$
where ${}_{2}Y_{2,m}$ is a spin-weight harmonic function and $\Psi^{1}_{0}$ is
a component of the Weyl tensor. Since the integral is performed on a two-
sphere at infinity, we only need the asymptotic form of the Weyl tensor in the
calculation. According to the peeling off theorem given by Sachs [18], we may
express the Weyl tensor and Maxwell field as :
$\displaystyle\Psi_{n}\sim O(r^{n-5})\,\quad n=0,1,2,3,4,$
$\displaystyle\phi_{m}\sim O(r^{m-3})\quad m=0,1,2.$ (1)
The vacuum case has been studied previously[7, 13, 14, 15]. An interesting
issue is to consider the effect of matter fields on N-P constants. In this
paper, we shall concentrate on the electromagnetic field. Like in the vacuum
case, we require the space-time to be stationary. Obviously, there is no Bondi
energy flux in such a space-time, i.e. ${\dot{\sigma}}^{0}=0$. In this case we
can choose some suitable coordinates, such that the asymptotic shear
$\sigma^{0}$ is zero. Similarly, the stationary condition has eliminated the
freedom of the news function. We also demand the Weyl tensor satisfy the
asymptotically algebraically special condition, which has been discussed
above. The main purpose of this paper is to prove the following theorem:
###### Theorem 1
All the N-P constants of an asymptotically flat, stationary, asymptotically
algebraically special electrovacuum space-time are zero.
Note that Kerr-Newman solution satisfies all the conditions in the theorem. It
follows immediately that the all N-P constants in a Kerr-Newman spacetime must
vanish.
Proof of the theorem. We choose the standard Bondi-Sachs’ coordinates and
construct the standard Bondi null tetrad [15, 22]. With the gauge choice in
[18, 19], we can write down the N-P coefficients and null tetrad of the
stationary electrovacuum spacetime. Some low order terms have been calculated
and can be found in [18]. Calculation of the N-P constants requires higher
order terms in the expansions. Consider the following N-P equations
$\displaystyle\delta\lambda-{\bar{\delta}}\mu=\bar{\tau}\mu+(\bar{\alpha}-3\beta)\lambda-\Psi_{3}+\Phi_{21},$
(2)
$\displaystyle\Delta\lambda-{\bar{\delta}}\nu=2\alpha\nu+(\bar{\gamma}-3\gamma-\mu-\bar{\mu})\lambda-\Psi_{4},$
(3)
where $\Phi_{ij}=8\pi\phi_{i}{\bar{\phi}}_{j}$ is the Maxwell stress tensor.
The coefficient of $r^{-2}$ in equation (2) yields $\Psi_{3}^{0}=0$. Expanding
equation (3) up to $O(r^{-3})$, we obtain
$\Psi_{4}^{0}=\Psi_{4}^{1}=\Psi_{4}^{2}=0$.
Now we shall use the Killing equation to reduce other dynamical freedoms and
get a general asymptotic expansion of the stationary electrovacuum space-time.
Write down the time-like Killing vector as
$\displaystyle t^{a}=Tl^{a}+n^{a}+{\bar{A}}m^{a}+A{\bar{m}}^{a}.$
The Killing equations are given by
$\displaystyle-DT+(\gamma+{\bar{\gamma}})+{\bar{\tau}}A+\tau{\bar{A}}=0,$ (4)
$\displaystyle DA+\tau+{\bar{\rho}}A+{\sigma}{\bar{A}}=0,$ (5)
$\displaystyle-{D^{\prime}}T-(\gamma+{\bar{\gamma}})T-\nu
A-{\bar{\nu}}{\bar{A}}=0,$ (6) $\displaystyle-\tau
T+{\bar{\nu}}+{D^{\prime}}A+({\bar{\gamma}}-\gamma)A-\delta T-\tau T-\mu
A-{\bar{\lambda}}{\bar{A}}=0,$ (7)
$\displaystyle-{\sigma}T+{\bar{\lambda}}+\delta A+({\bar{\alpha}}-\beta)A=0,$
(8) $\displaystyle-\rho
T+\mu+\delta{\bar{A}}-({\bar{\alpha}}-\beta){\bar{A}}-{\bar{\rho}}T+{\bar{\mu}}+{\bar{\delta}}A-(\alpha-{\bar{\beta}})A=0.$
(9)
Similarly to the analysis in [15], assuming the asymptotic behaviors of $T$
and $A$ as
$\displaystyle T$ $\displaystyle=$ $\displaystyle
T^{0}+\frac{T^{1}}{r}+\cdots,$ $\displaystyle A$ $\displaystyle=$
$\displaystyle A^{0}+\frac{A^{1}}{r}+\cdots,$ (10)
we can solve the Killing equations order by order. The stationary condition
implies ${\dot{\sigma}}^{0}=0$. It has been found that the Maxwell field does
not change the lowest two powers of $1/r$ in the Killing equations. So the
constant terms in the Killing equations yield the same result as in the vacuum
case, i.e., $T^{0}=\frac{1}{2}$, ${\dot{T}}^{1}=0$, ${\dot{A}}^{1}=0$. The
coefficients of $r^{-1}$ in the Killing equations give rise to
${\sigma}^{0}=0$, $A^{1}=0$, ${\dot{T}}^{2}=0$,
$\Psi^{0}_{2}={\bar{\Psi}}^{0}_{2}$,
$T^{1}=\frac{1}{2}(\Psi^{0}_{2}+{\bar{\Psi}}^{0}_{2})$ ,
${\dot{A}}^{2}=-\frac{1}{2}\eth\Psi^{0}_{2}+\frac{1}{2}\delta_{0}(\Psi^{0}_{2}+{\bar{\Psi}}^{0}_{2})$
and
${\dot{\Psi}}^{0}_{2}=0.$ (11)
From the $r^{-2}$ terms in the N-P equation
$\displaystyle\delta\nu-\Delta\mu=\gamma\mu-2\nu\beta+\bar{\gamma}\mu+\mu^{2}+|\lambda|^{2}+\Phi_{22}\,,$
(12)
we find $8\pi|\phi_{2}^{0}|^{2}={\dot{\Psi}}_{2}^{0}=0$. Hence
$\phi_{2}^{0}=0$. From the $r^{-5}$ terms in the N-P equation
$\displaystyle\delta\rho-\bar{\delta}\sigma=\tau\rho+(\bar{\beta}-3\alpha)\sigma+(\rho-\bar{\rho})\tau-\Psi_{1}+\Phi_{01}\,,$
(13)
we have
$\displaystyle\frac{1}{6}({\bar{\eth}}\Psi_{0}^{0}-40\pi\phi_{0}^{0}{\bar{\phi}}_{1}^{0})+\frac{1}{2}{\bar{\eth}}\Psi_{0}^{0}=-\frac{1}{3}({\bar{\eth}}\Psi_{0}^{0}-40\pi\phi_{0}^{0}{\bar{\phi}}_{1}^{0})+{\bar{\eth}}\Psi_{0}^{0}-16\pi\phi_{0}^{0}{\bar{\phi}}_{1}^{0}$
(14)
which implies
$\displaystyle\phi_{0}^{0}{\bar{\phi}}_{1}^{0}=0$ (15)
This equation will play an important role in our proof, which gives
$\phi_{0}^{0}=0$ or $\phi_{1}^{0}=0$. Now we discuss the two cases
respectively.
1) $\phi_{0}^{0}=0$. Consider the Maxwell equations
$\displaystyle D\phi_{1}-\bar{\delta}\phi_{0}$ $\displaystyle=$
$\displaystyle-2\alpha\phi_{0}+2\rho\phi_{1},$ (16) $\displaystyle
D\phi_{2}-\bar{\delta}\phi_{1}$ $\displaystyle=$
$\displaystyle-\lambda\phi_{0}+\rho\phi_{2}.$ (17)
The coefficients of $r^{-4}$ in these equations yield
$\phi_{1}^{1}=\phi_{2}^{2}=0$.
Consider the other two Maxwell equations
$\displaystyle\delta\phi_{1}-\Delta\phi_{0}$ $\displaystyle=$
$\displaystyle(\mu-2\lambda)\phi_{0}+2\tau\phi_{1}-\sigma\phi_{2},$ (18)
$\displaystyle\delta\phi_{2}-\Delta\phi_{1}$ $\displaystyle=$
$\displaystyle-\nu\phi_{0}+2\mu\phi_{1}+(\tau-2\beta)\phi_{2}.$ (19)
The $r^{-2}$ terms in equation (18) and the $r^{-3}$ terms in equation (19)
yield
$\displaystyle\dot{\phi_{1}^{0}}$ $\displaystyle=$ $\displaystyle 0$ (20)
$\displaystyle\dot{\phi_{0}^{0}}$ $\displaystyle=$
$\displaystyle\eth\phi_{1}^{0}=0$ (21)
where “$\cdot$” denotes ${\frac{\partial}{\partial u}}$. Combining these two
equations, we have $\phi_{1}^{0}=constant$. So from the $r^{-3}$ terms of Eq.
(17), we obtain $\phi_{2}^{1}=-{\bar{\eth}}\phi_{1}^{0}=0$.
2) $\phi_{1}^{0}=0$. Again, from the $r^{-3}$ terms of Eq.(17), we have
$\phi_{2}^{1}=-{\bar{\eth}}\phi_{1}^{0}=0$.
Thus in both cases we have $\phi_{2}^{1}=0$. Note that it is $\Phi_{ij}$,
instead of $\phi_{i}$, that appear in the N-P equations. The fact that
$\phi_{i}=O(r^{-3})$ (except $\phi_{1}\sim O(r^{-2})$ in case 1)) shows that
the presence of the electromagnetic field does not contribute to $r^{-1}$ and
$r^{-2}$ terms. The electromagnetic field makes contribution only to order
$r^{-3}$ and higher orders in the expansions. Combining these results, we
obtain the reduced N-P coefficients
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{1}{r}+\frac{8\pi\phi^{0}_{0}\bar{\phi}^{0}_{0}}{3r^{5}}+O(r^{-6}),$
$\displaystyle{\sigma}$ $\displaystyle=$
$\displaystyle-\frac{\Psi^{0}_{0}}{2r^{4}}-\frac{\Psi^{1}_{0}}{3r^{5}}+O(r^{-6}),$
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{\alpha^{0}}{r}-\frac{{\bar{\alpha}}^{0}{\bar{\Psi}}^{0}_{0}}{6r^{4}}+\frac{\alpha^{0}8\pi\phi^{0}_{0}\bar{\phi}^{0}_{0}-\bar{\alpha}^{0}\bar{\Psi}^{1}_{0}-24\pi(\phi_{1}^{0}{\bar{\phi}}_{0}^{1}+\phi^{1}_{1}\bar{\phi}^{0}_{0})}{12r^{5}}+O(r^{-6}),$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle-\frac{{\bar{\alpha}}^{0}}{r}-\frac{\Psi^{0}_{1}}{2r^{3}}+\frac{\alpha^{0}\Psi^{0}_{0}+2{\bar{\eth}}\Psi^{0}_{0}}{6r^{4}}-\frac{3\Psi^{2}_{1}+8\pi\bar{\alpha}^{0}\phi^{0}_{0}\bar{\phi}^{0}_{0}-\alpha^{0}\Psi^{1}_{0}}{12r^{5}}+O(r^{-6})\,,$
$\displaystyle\tau$ $\displaystyle=$
$\displaystyle-\frac{\Psi^{0}_{1}}{2r^{3}}+\frac{{\bar{\eth}}\Psi^{0}_{0}}{3r^{4}}+\frac{{\bar{\eth}}\Psi^{1}_{0}-8\pi\eth(\phi^{0}_{0}\bar{\phi}^{0}_{0})-48\pi(\phi_{0}^{1}{\bar{\phi}}_{1}^{0}+\phi^{0}_{0}\bar{\phi}^{1}_{1})}{8r^{5}}+O(r^{-6}),$
$\displaystyle{\lambda}$ $\displaystyle=$
$\displaystyle-\frac{{\bar{\Psi}}^{0}_{0}}{12r^{4}}-\frac{3{\bar{\Psi}}^{0}_{0}\Psi^{0}_{2}+{\bar{\Psi}}^{1}_{0}+48\pi\phi^{2}_{2}\bar{\phi}^{0}_{0}}{24r^{5}}+O(r^{-6}),$
$\displaystyle\mu$ $\displaystyle=$
$\displaystyle-\frac{1}{2r}-\frac{\Psi^{0}_{2}}{r^{2}}+\frac{{\bar{\eth}}\Psi^{0}_{1}-16\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0}}{2r^{3}}-\frac{{\bar{\eth}}^{2}\Psi^{0}_{0}}{6r^{4}}-\frac{6\Psi^{3}_{2}+8\pi\phi^{0}_{0}\bar{\phi}^{0}_{0}}{24r^{5}}+O(r^{-6}),$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle-\frac{\Psi^{0}_{2}}{2r^{2}}+\frac{2{\bar{\eth}}\Psi^{0}_{1}-48\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0}+\alpha^{0}\Psi^{0}_{1}-{\bar{\alpha}}^{0}{\bar{\Psi}}^{0}_{1}}{6r^{3}}$
$\displaystyle-\frac{1}{24}\left[2\left(\alpha^{0}{\bar{\eth}}\Psi^{0}_{0}-{\bar{\alpha}}^{0}\eth{\bar{\Psi}}^{0}_{0}\right)+3{\bar{\eth}}^{2}\Psi^{0}_{0}\right]r^{-4}$
$\displaystyle+\frac{1}{20}[\alpha^{0}8\pi(\phi^{0}_{0}\bar{\phi}^{1}_{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})+\alpha^{0}\Psi^{2}_{1}-{\bar{\alpha}}^{0}8\pi(\phi_{1}^{0}{\bar{\phi}}_{0}^{1}+\phi_{1}^{1}\bar{\phi}^{0}_{0})-{\bar{\alpha}}^{0}\bar{\Psi}^{2}_{1}$
$\displaystyle-|\Psi^{0}_{1}|^{2}-4\Psi^{3}_{2}-32\pi(\phi_{1}^{0}{\bar{\phi}}_{1}^{2}+\phi_{1}^{1}\bar{\phi}_{1}^{1}+\phi_{1}^{2}{\bar{\phi}}_{1}^{0})]r^{-5}+O(r^{-6}),$
$\displaystyle\nu$ $\displaystyle=$
$\displaystyle-\frac{1}{12}\left[{\bar{\Psi}}^{0}_{1}+2{\bar{\eth}}^{2}\Psi^{0}_{1}\right]r^{-3}+\frac{1}{24}\left[\eth\bar{\Psi}_{0}^{0}+{\bar{\eth}}^{3}\Psi_{0}^{0}\right]r^{-4}$
(22)
$\displaystyle-\frac{1}{120}[6\Psi_{2}^{1}\bar{\Psi}_{1}^{0}-8\Psi_{2}^{0}\eth\bar{\Psi}_{0}^{0}+24\pi(\phi_{1}^{0}\bar{\phi}_{0}^{1}+\phi_{1}^{1}\bar{\phi}_{0}^{0})+3\bar{\Psi}_{1}^{2}+24\Psi_{3}^{4}$
$\displaystyle+192\pi\phi_{2}^{2}\bar{\phi}_{1}^{1}]r^{-5}+O(r^{-6}).$
and the null tetrad
$\displaystyle l^{a}$ $\displaystyle=$ $\displaystyle{\frac{\partial}{\partial
r}}\,,$ $\displaystyle n^{a}$ $\displaystyle=$
$\displaystyle{\frac{\partial}{\partial
u}}+\left[-\frac{1}{2}-\frac{\Psi^{0}_{2}}{r}+\frac{{\bar{\eth}}\Psi^{0}_{1}+\eth{\bar{\Psi}}^{0}_{1}+64\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0}}{6r^{2}}\right.-\frac{{\bar{\eth}}^{2}\Psi^{0}_{0}+\eth^{2}{\bar{\Psi}}^{0}_{0}}{24r^{3}}$
$\displaystyle-\frac{1}{20}\left(3|\Psi_{1}^{0}|^{2}+\Psi_{2}^{3}+\bar{\Psi}_{2}^{3}+16\pi(\phi^{0}_{1}\bar{\phi}^{2}_{1}+\phi^{1}_{1}{\bar{\phi}}^{1}_{1}+\phi^{2}_{1}{\bar{\phi}}^{0}_{1})r^{-4}+O(r^{-5})\right]{\frac{\partial}{\partial
r}}$
$\displaystyle+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{3}}\Psi^{0}_{1}-\frac{1+\zeta{\bar{\zeta}}}{12\sqrt{2}r^{4}}{\bar{\eth}}\Psi^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial\zeta}}$
$\displaystyle+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{3}}{\bar{\Psi}}^{0}_{1}-\frac{1+\zeta{\bar{\zeta}}}{12\sqrt{2}r^{4}}\eth{\bar{\Psi}}^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\,,$
$\displaystyle m^{a}$ $\displaystyle=$
$\displaystyle\left[-\frac{\Psi^{0}_{1}}{2r^{2}}+\frac{{\bar{\eth}}\Psi^{0}_{0}}{6r^{3}}-\frac{\Psi_{1}^{2}+8\pi(\phi_{0}^{1}{\bar{\phi}}_{1}^{0}+\phi^{0}_{0}\bar{\phi}^{1}_{1})}{12r^{4}}+O(r^{-5})\right]{\frac{\partial}{\partial
r}}$ (23)
$\displaystyle+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{4}}\Psi^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial\zeta}}+\left[\frac{1+\zeta{\bar{\zeta}}}{\sqrt{2}r}+O(r^{-5})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\,,$
where
$\delta_{0}=\frac{(1+\zeta{\bar{\zeta}})}{\sqrt{2}}\frac{\partial}{\partial{\bar{\zeta}}}$,
$\zeta=e^{i\phi}\cot\frac{\theta}{2}$, $\eth
f=(\delta_{0}+2s{\bar{\alpha}}^{0})f$ ( $s$ is the spin-weight of $f$). The
differential operators $\eth$ and ${\bar{\eth}}$ are defined in [18, 20].
Then the components of the Weyl curvature and the electromagnetic tensor
reduce to
$\displaystyle\Psi_{0}=\frac{\Psi^{0}_{0}}{r^{5}}+\frac{\Psi^{1}_{0}}{r^{6}}+O(r^{-7}),$
$\displaystyle\Psi_{1}=\frac{\Psi^{0}_{1}}{r^{4}}+\frac{\Psi^{1}_{1}}{r^{5}}+\frac{\Psi^{2}_{1}}{r^{6}}+O(r^{-7}),$
$\displaystyle\Psi_{2}=\frac{\Psi^{0}_{2}}{r^{3}}+\frac{\Psi^{1}_{2}}{r^{4}}+\frac{\Psi^{2}_{2}}{r^{5}}+\frac{\Psi^{3}_{2}}{r^{6}}+O(r^{-7}),$
$\displaystyle\Psi_{3}=\frac{\Psi^{2}_{3}}{r^{4}}+\frac{\Psi^{3}_{3}}{r^{5}}+\frac{\Psi^{4}_{3}}{r^{6}}+O(r^{-7}),$
$\displaystyle\Psi_{4}=\frac{\Psi_{4}^{3}}{r^{4}}+\frac{\Psi^{4}_{4}}{r^{5}}+\frac{\Psi^{5}_{4}}{r^{6}}+O(r^{-7}).$
$\displaystyle\phi_{0}=\frac{\phi_{0}^{0}}{r^{3}}+\frac{\phi_{0}^{1}}{r^{4}}+\frac{\phi_{0}^{2}}{r^{5}}+O(r^{-6}),$
$\displaystyle\phi_{1}=\frac{\phi_{1}^{0}}{r^{2}}+\frac{\phi_{1}^{1}}{r^{3}}+\frac{\phi_{1}^{2}}{r^{4}}+\frac{\phi_{1}^{3}}{r^{5}}+O(r^{-6}),$
$\displaystyle\phi_{2}=\frac{\phi_{2}^{2}}{r^{3}}+\frac{\phi_{2}^{3}}{r^{4}}+\frac{\phi_{2}^{4}}{r^{5}}+O(r^{-6}).$
(24)
The Bianchi identity takes the form
$\displaystyle\bar{\delta}\Psi_{0}-D\Psi_{1}+D\Phi_{01}-\delta\Phi_{00}=4\alpha\Psi_{0}-4\rho\Psi_{1}-2\tau\Phi_{00}+2\rho\Phi_{01}+2\sigma\Phi_{10}\,.$
(25)
The coefficient of $r^{-6}$ in equation (25) yields
$\Psi^{1}_{1}=-{\bar{\eth}}\Psi^{0}_{0}$.
Similarly, the other components of the Bianchi identity and the Maxwell
equations lead to
$\displaystyle\phi_{1}^{1}=-{\bar{\eth}}\phi_{0}^{0},\quad\phi_{1}^{2}=-\frac{1}{2}{\bar{\eth}}\phi_{0}^{1},\quad\phi_{1}^{3}=-\frac{1}{3}{\bar{\eth}}\phi_{0}^{2}-\frac{1}{2}\bar{\Psi}_{1}^{0}\phi_{0}^{0}.$
$\displaystyle\phi_{2}^{2}=\frac{1}{2}{\bar{\eth}}^{2}\phi_{0}^{0},\quad\phi_{2}^{3}=\frac{1}{6}{\bar{\eth}}^{2}\phi_{0}^{1},$
$\displaystyle\phi_{2}^{4}=\frac{1}{12}{\bar{\eth}}^{2}\phi_{0}^{2}+\frac{1}{12}\eth\bar{\Psi}_{0}^{0}+\frac{1}{2}\bar{\Psi}_{1}^{0}{\bar{\eth}}\phi_{0}^{0}$
$\displaystyle\Psi^{1}_{1}=-{\bar{\eth}}\Psi^{0}_{0},\quad\Psi^{2}_{1}=-\frac{1}{2}{\bar{\eth}}\Psi^{1}_{0}+16\pi(\phi_{0}^{0}\bar{\phi}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})+4\pi\eth(\phi_{0}^{0}\bar{\phi}_{0}^{0}),$
$\displaystyle\Psi^{1}_{2}=-{\bar{\eth}}\Psi^{0}_{1}+16\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0},\quad\Psi^{2}_{2}=\frac{1}{2}{\bar{\eth}}^{2}\Psi^{0}_{0},$
$\displaystyle\Psi^{3}_{2}=-\frac{2}{3}|\Psi^{0}_{1}|^{2}-\frac{1}{3}{\bar{\eth}}\Psi^{2}_{1}+\frac{16}{9}\pi\eth(\phi_{1}^{0}{\bar{\phi}}_{0}^{1}+\phi_{1}^{1}\bar{\phi}_{0}^{0})-\frac{8}{9}\pi{\bar{\eth}}(\phi_{0}^{0}\bar{\phi}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})-\frac{20}{9}\pi\phi_{0}^{0}\bar{\phi}_{0}^{0}$
$\displaystyle\quad\quad\quad+\frac{80}{9}\pi(\phi_{1}^{0}{\bar{\phi}}_{1}^{2}+\phi_{1}^{1}{\bar{\phi}}_{1}^{1}+\phi_{1}^{2}{\bar{\phi}}_{1}^{0})+\frac{8}{9}\pi{\frac{\partial}{\partial
u}}(\phi_{0}^{0}{\bar{\phi}}_{0}^{1}+\phi_{0}^{1}{\bar{\phi}}_{0}^{0}),$
$\displaystyle\Psi^{2}_{3}=\frac{1}{2}{\bar{\eth}}^{2}\Psi^{0}_{1},\quad\Psi^{3}_{3}=-\frac{1}{2}{\bar{\Psi}}^{0}_{1}\Psi^{0}_{2}-\frac{1}{6}{\bar{\eth}}^{3}\Psi^{0}_{0},$
$\displaystyle\Psi^{4}_{3}=-\frac{1}{4}{\bar{\eth}}\Psi_{2}^{3}+\frac{1}{8}\Psi_{2}^{0}\eth\Psi_{0}^{0}+\frac{1}{2}{\bar{\Psi}}_{1}^{0}{\bar{\eth}}\Psi_{1}^{0}+\frac{1}{12}k\eth(\phi_{2}^{2}{\bar{\phi}}_{0}^{0})$
$\displaystyle\quad\quad\quad-\frac{4}{3}\pi{\bar{\eth}}(\phi_{1}^{0}{\bar{\phi}}_{1}^{2}+\phi_{1}^{1}{\bar{\phi}}_{1}^{1}\phi_{1}^{2}{\bar{\phi}}_{1}^{0})+4\pi(\phi_{2}^{2}{\bar{\phi}}_{1}^{1}+\phi_{2}^{3}{\bar{\phi}}_{1}^{0})+4\pi(\phi_{1}^{0}{\bar{\phi}}_{0}^{1}+\phi_{1}^{1}{\bar{\phi}}_{0}^{0})$
$\displaystyle\quad\quad\quad+4\pi\bar{\Psi}_{1}^{0}\phi_{1}^{0}{\bar{\phi}}_{1}^{0}+\frac{4}{3}\pi{\frac{\partial}{\partial
u}}(\phi_{1}^{1}{\bar{\phi}}_{0}^{1}+\phi_{1}^{2}{\bar{\phi}}_{0}^{0}),$
$\displaystyle\Psi^{3}_{4}=-\frac{1}{6}{\bar{\eth}}^{3}\Psi_{1}^{0},\quad\Psi^{4}_{4}=-\frac{1}{24}{\bar{\eth}}^{4}\Psi_{0}^{0},$
$\displaystyle\Psi^{5}_{4}=-\frac{1}{5}{\bar{\eth}}\Psi_{3}^{4}-\frac{8}{5}\pi{\bar{\eth}}(\phi_{2}^{2}{\bar{\phi}}_{1}^{1}+\phi_{2}^{3}{\bar{\phi}}_{1}^{0})-\frac{1}{5}{\bar{\Psi}}_{1}^{0}{\bar{\eth}}^{2}\Psi_{1}^{0}-\frac{1}{20}\Psi_{2}^{0}\bar{\Psi}_{0}^{0}$
$\displaystyle\quad\quad\quad+4\pi(\phi_{2}^{2}{\bar{\phi}}_{0}^{0})+\frac{8}{5}\pi{\frac{\partial}{\partial
u}}(\phi_{2}^{2}{\bar{\phi}}_{0}^{1}+\phi_{2}^{3}{\bar{\phi}}_{0}^{0}).$ (26)
Similarly to the treatment in [15], the $r^{-3}$ terms in the Killing
equations lead to $\eth\Psi_{1}^{0}=0$. Thus we have
$\displaystyle\Psi^{0}_{1}$ $\displaystyle=$
$\displaystyle\sum^{1}_{m=-1}B_{m}{\ }_{1}Y_{1,m},$
$\displaystyle\Psi^{0}_{2}$ $\displaystyle=$ $\displaystyle C.$ (27)
The coefficient of $r^{-3}$ in Eq. (2) gives
$\Psi_{3}^{1}={\bar{\delta}}\Psi_{2}^{0}=0$.
In order to find more restrictions on $\Psi_{0}$, we need to compute higher
order terms of the Killing equations. The terms of order $r^{-4}$ of the
Killing equations yield
$\displaystyle 3T^{3}+(\gamma^{4}+{\bar{\gamma}}^{4})=0,$ (28) $\displaystyle
4A^{3}=\frac{1}{3}{\bar{\eth}}\Psi^{0}_{0},$ (29)
$\displaystyle{\dot{T}}^{4}+\frac{8}{3}\pi\phi_{1}^{0}{\bar{\phi}}_{1}^{0}=0,$
(30)
$\displaystyle\frac{1}{2}\Psi^{0}_{1}T^{1}-\tau^{4}+{\bar{\nu}}^{4}+{\dot{A}}^{4}+(\Psi^{0}_{2}+{\bar{\Psi}}^{0}_{2})A^{2}+2A^{3}-\delta_{0}T^{3}+\Psi^{0}_{2}A^{2}=0,$
(31) $\displaystyle\frac{1}{6}\Psi^{0}_{0}+\eth A^{3}=0,$ (32) $\displaystyle
2T^{3}+\mu^{4}+{\bar{\mu}}^{4}+\eth{\bar{A}}^{3}+{\bar{\eth}}A^{3}=0\,.$ (33)
Eq.(29) and (32) imply
$\displaystyle\Psi^{0}_{0}=\sum^{2}_{m=-2}A_{m}(u){\ }_{2}Y_{2,m},$ (34)
Eq.(27) and ${\dot{T}}^{3}=0$ ( which comes from the $r^{-3}$ terms in the
Killing equations) imply that $\Psi^{0}_{0}$ is independent of $u$.
Combining Eqs.(24),(26),(27) and (34), one finds
$\displaystyle I^{3}-27J^{2}\sim O(r^{-21}).$ (35)
This result holds for a general asymptotically flat stationary spacetime. As
mentioned in the introduction, the AASC requires
$\displaystyle I^{3}-27J^{2}\sim O(r^{-22}),$ (36)
which is just one order faster than the falloff rate of a general
asymptotically flat spacetime. This means that the AASC is a weak requirement
and as demonstrated at the end of this section, there exist many asymptotic
flat space-times which satisfy this condition.
Our purpose is to calculate the Newman-Penrose constants, which are contained
in the coefficients of $\Psi^{1}_{0}$. From the $r^{-5}$ terms in the Killing
equations, we have
$\displaystyle
4T^{4}+(\gamma^{5}+{\bar{\gamma}}^{5})-\frac{1}{2}{\bar{\Psi}}^{0}_{1}A^{2}-\frac{1}{2}\Psi^{0}_{1}{\bar{A}}^{2}=0,$
(37) $\displaystyle
A^{4}=\frac{1}{5}\tau^{5}=\frac{1}{40}\left[{\bar{\eth}}\Psi_{0}^{1}-48\pi(\phi_{0}^{0}{\bar{\phi}}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})-8\pi\eth(\phi_{0}^{0}{\bar{\phi}}_{0}^{0})\right],$
(38)
$\displaystyle\frac{1}{8}\Psi_{0}^{1}+\frac{3}{8}\Psi_{0}^{0}\Psi_{2}^{0}-2\pi\phi_{0}^{0}{\bar{\phi}}_{2}^{2}-\frac{1}{4}(\Psi_{1}^{0})^{2}+\eth
A^{4}=0,$ (39)
$\displaystyle-\rho^{5}+2T^{4}+(\mu^{5}+{\bar{\mu}}^{5})+\frac{3}{2}\Psi^{0}_{1}{\bar{A}}^{2}+\frac{3}{2}{\bar{\Psi}}^{0}_{1}A^{2}+\eth{\bar{A}}^{4}+{\bar{\eth}}A^{4}=0\,.$
(40)
Eqs. (38) and (39) yield:
$\displaystyle\eth{\bar{\eth}}\Psi^{1}_{0}+5\Psi^{1}_{0}=10(\Psi^{0}_{1})^{2}-15\Psi^{0}_{0}\Psi^{0}_{2}+80\pi\phi_{0}^{0}{\bar{\phi}}_{2}^{2}+48\pi\eth(\phi_{0}^{0}{\bar{\phi}}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})+8\pi\eth^{2}(\phi_{0}^{0}{\bar{\phi}}_{0}^{0}).$
(41)
The terms of $\phi^{i}_{j}$ on the right-hand side of Eq. (41) are the
contribution from the Maxwell field [15]. To simplify this equation, we need
to investigate the electromagnetic field in more detail.
Since the electromagnetic field is stationary, we have $\pounds_{t}F_{ab}=0$,
where $t^{c}$ is the Killing vector. Noting that $\phi_{0}=F_{lm}$ and using
the expansion of $t^{c}$, we have
$\displaystyle\pounds_{t}\phi_{0}$ $\displaystyle=$
$\displaystyle\pounds_{t}F_{ab}l^{a}m^{b}$ (42) $\displaystyle=$
$\displaystyle(Tl^{c}+n^{c}+{\bar{A}}m^{c}+A{\bar{m}}^{c})\phi_{0}$
$\displaystyle=$ $\displaystyle F_{ab}l^{a}[t,\quad m]^{b}+F_{ab}m^{b}[t,\quad
l]^{a}$ $\displaystyle=$
$\displaystyle(\gamma+\bar{\gamma}+\bar{A}\bar{\tau}+A\tau)\phi_{0}-(\tau+\bar{A}\sigma+A\rho)(\phi_{1}-{\bar{\phi}}_{1})$
$\displaystyle+\left[T\bar{\varrho}-\mu+\gamma+\bar{\gamma}-A(\bar{\beta}-\alpha)\right]\phi_{0}+\left[T\sigma-\bar{\lambda}-A(\bar{\alpha}-\beta)\right]{\bar{\phi}}_{0}$
where $[t,\ m]^{b}$ denotes the commutator of $t^{c}$ and $m^{b}$. So we
obtain
$\displaystyle(Tl^{c}+n^{c}+{\bar{A}}m^{c}+A{\bar{m}}^{c})\phi_{0}$ (43)
$\displaystyle=$
$\displaystyle(\gamma+\bar{\gamma}+\bar{A}\bar{\tau}+A\tau)\phi_{0}-(\tau+\bar{A}\sigma+A\rho)(\phi_{1}-{\bar{\phi}}_{1})+\left[T\bar{\varrho}-\mu+\gamma+\bar{\gamma}-A(\bar{\beta}-\alpha)\right]\phi_{0}$
$\displaystyle+\left[T\sigma-\bar{\lambda}-A(\bar{\alpha}-\beta)\right]{\bar{\phi}}_{0}$
Substituting (23) into (43) yields:
$\displaystyle{\frac{\partial}{\partial
u}}\phi_{0}+\left[-\frac{1}{2}-\frac{\Psi^{0}_{2}}{r}+O(r^{-2})\right]{\frac{\partial}{\partial
r}}\phi_{0}+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{3}}\Psi^{0}_{1}+O(r^{-4})\right]{\frac{\partial}{\partial\zeta}}\phi_{0}$
(44)
$\displaystyle+\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{3}}{\bar{\Psi}}^{0}_{1}+O(r^{-4})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\phi_{0}+T{\frac{\partial}{\partial
r}}\phi_{0}-\bar{A}\left[\frac{\Psi^{0}_{1}}{2r^{2}}+O(r^{-3})\right]{\frac{\partial}{\partial
r}}\phi_{0}$
$\displaystyle+\bar{A}\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{4}}\Psi^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial\zeta}}\phi_{0}+\bar{A}\left[\frac{1+\zeta{\bar{\zeta}}}{\sqrt{2}r}+O(r^{-5})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\phi_{0}$
$\displaystyle-A\left[\frac{{\bar{\Psi}}^{0}_{1}}{2r^{2}}+O(r^{-3})\right]{\frac{\partial}{\partial
r}}\phi_{0}$
$\displaystyle+A\left[\frac{1+\zeta{\bar{\zeta}}}{6\sqrt{2}r^{4}}{\bar{\Psi}}^{0}_{0}+O(r^{-5})\right]{\frac{\partial}{\partial{\bar{\zeta}}}}\phi_{0}+A\left[\frac{1+\zeta{\bar{\zeta}}}{\sqrt{2}r}+O(r^{-5})\right]{\frac{\partial}{\partial\zeta}}\phi_{0}$
$\displaystyle=$
$\displaystyle(\gamma+\bar{\gamma}+\bar{A}\bar{\tau}+A\tau)\phi_{0}-(\tau+\bar{A}\sigma+A\rho)(\phi_{1}-{\bar{\phi}}_{1})$
$\displaystyle+\left[T\bar{\varrho}-\mu+\gamma+\bar{\gamma}-A(\bar{\beta}-\alpha)\right]\phi_{0}+\left[T\sigma-\bar{\lambda}-A(\bar{\alpha}-\beta)\right]{\bar{\phi}}_{0}$
Again, we compute the $\phi^{i}_{j}$ terms in Eq.(41) in the two cases.
For case 1) $\phi_{0}^{0}=0$, computing the coefficient of $r^{-5}$ in Eq.
(44), we obtain
$\displaystyle\dot{\phi_{0}^{2}}=-3\Psi_{2}^{0}\phi_{0}^{0}=0$ (45)
The coefficient of $r^{-5}$ of equation (18) gives
$\displaystyle\eth\phi_{1}^{1}-\dot{\phi_{0}^{2}}-2\phi_{0}^{1}-3\Psi_{2}^{0}\phi_{0}^{0}=-\frac{1}{2}\phi_{0}^{1}-\Psi_{1}^{0}\phi_{1}^{0}\,.$
(46)
Using $\phi_{0}^{0}=0$ and $\dot{\phi_{0}^{2}}=0$, we get
$\displaystyle\phi_{0}^{1}=\frac{2}{3}\phi_{1}^{0}\Psi_{1}^{0}\,.$ (47)
By taking $\eth$ on both sides and using $\eth\Psi_{1}^{0}=0$, we have
immediately
$\displaystyle\eth\phi_{0}^{1}=\frac{2}{3}\phi_{1}^{0}\eth\Psi_{1}^{0}=0.$
(48)
Then the $\phi^{i}_{j}$ terms in Eq.(41) become
$\displaystyle
80\pi\phi_{0}^{0}{\bar{\phi}}_{2}^{2}+48\pi\eth(\phi_{0}^{0}{\bar{\phi}}_{1}^{1}+\phi_{0}^{1}{\bar{\phi}}_{1}^{0})+8\pi\eth^{2}(\phi_{0}^{0}{\bar{\phi}}_{0}^{0})$
(49) $\displaystyle=$ $\displaystyle
48\pi\eth(\phi_{0}^{1}{\bar{\phi}}_{1}^{0})$ $\displaystyle=$
$\displaystyle(48\pi\eth\phi_{0}^{1}){\bar{\phi}}_{1}^{0}+\phi_{0}^{1}(48\pi\eth{\bar{\phi}}_{1}^{0})$
$\displaystyle=$ $\displaystyle 0\,,$
where Eqs. (21) and (48) have been used in the last step.
For case 2) $\phi_{1}^{0}=0$, the coefficient of $r^{-4}$ in Eq. (44) leads to
$\displaystyle
0=\dot{\phi_{0}^{1}}-3T^{0}\phi_{0}^{0}+\frac{3}{2}\phi_{0}^{0}=\dot{\phi_{0}^{1}}\,.$
(50)
Because the spinweight of $\phi_{0}$ is 1, we can expand $\phi_{0}^{0}$ as
$\phi_{0}^{0}=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}d_{l,m}{\ }_{1}Y_{l,m}$, where
$d_{l,m}$ are some constants. The $r^{-4}$ terms in (18) yield
$\displaystyle\dot{\phi_{0}^{1}}=-{\bar{\eth}}\eth\phi_{0}^{0}=\frac{1}{2}\sum_{l=1}^{\infty}(l+2)(l-1)\sum_{m=-l}^{l}d_{l,m}\
{}_{1}Y_{l,m}\,.$ (51)
Combining (50) and (51) and using the fact that spin-weight harmonic function
components are linearly independent, we obtain $l=1$. Consequently,
$\displaystyle\phi_{0}^{0}=\sum_{m=-1}^{1}d_{m}{\ }_{1}Y_{1,m}\,,$ (52)
where $d_{m}$ are constants. By expanding $\phi_{0}^{0}$, we find
$\eth\phi_{0}^{0}=0$. The contribution from the Maxwell field in Eq. (41) then
leads to:
$\displaystyle
80\pi\phi_{0}^{0}{\bar{\phi}}_{2}^{2}+48\pi\eth(\phi_{0}^{0}{\bar{\phi}}_{1}^{1})+8\pi\eth^{2}(\phi_{0}^{0}{\bar{\phi}}_{0}^{0})$
(53) $\displaystyle=$ $\displaystyle
40\pi\phi_{0}^{0}\eth^{2}{\bar{\phi}}_{0}^{0}-48\pi\eth(\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0})+8\pi\eth(\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0}+{\bar{\phi}}_{0}^{0}\eth\phi_{0}^{0})$
$\displaystyle=$ $\displaystyle
40\pi\phi_{0}^{0}\eth^{2}{\bar{\phi}}_{0}^{0}-48\pi\eth\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0}-48\pi\phi_{0}^{0}\eth^{2}{\bar{\phi}}_{0}^{0}+8\pi\eth\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0}$
$\displaystyle+8\pi\eth{\bar{\phi}}_{0}^{0}\eth\phi_{0}^{0}+8\pi\phi_{0}^{0}\eth^{2}{\bar{\phi}}_{0}^{0}+8\pi{\bar{\phi}}_{0}^{0}\eth^{2}\phi_{0}^{0}$
$\displaystyle=$
$\displaystyle-32\pi\eth\phi_{0}^{0}\eth{\bar{\phi}}_{0}^{0}+8\pi{\bar{\phi}}_{0}^{0}\eth^{2}\phi_{0}^{0}$
$\displaystyle=$ $\displaystyle 0$
where we have used $\phi_{1}^{1}=-{\bar{\eth}}\phi_{0}^{0}$ and
$\phi_{2}^{2}=\frac{1}{2}{\bar{\eth}}^{2}\phi_{0}^{0}$. Therefore, the
electromagnetic field makes no contribution to the equation of $\Psi_{0}^{1}$.
So in either case, the equation of $\Psi_{0}^{1}$ reduces to
$\displaystyle\eth{\bar{\eth}}\Psi^{1}_{0}+5\Psi^{1}_{0}=10(\Psi^{0}_{1})^{2}-15\Psi^{0}_{0}\Psi^{0}_{2}\,,$
(54)
which is exactly the same equation as that in the vacuum case. Then by
imposing the AASC, it is shown in [15] that Eq. (54) implies that all the
Newman-Penrose constants must be zero. This completes the proof of our
theorem.
Remark : The asymptotically algebraically special condition has played an
important role in the proof of this paper and in [15]. Obviously, this
condition is satisfied by the Kerr-Newman solution. The following arguments
show that the AASC is a rather weak condition imposed on a general
asymptotically flat spacetime. Note that the Kerr-Newman spacetime is
axisymmetric. Such symmetry is not required in our theorem. From Eq.(27), we
can see that $\Psi^{0}_{1}$ contains ${}_{1}Y_{1,1}$ and ${}_{1}Y_{1,-1}$
components that do not appear in the Kerr-Newman solution. Simple calculation
shows that $span\\{{}_{1}Y_{1,1},{}_{1}Y_{1,0},{}_{1}Y_{1,-1}\\}$ is not a
representative space of $SO(3)$. Thus we cannot cancel such components by a
rotation. Based on the characteristic initial value method[23], it is not
difficult to construct exact solutions with non-zero $B_{1}$ and $B_{-1}$.
Furthermore, the spin-weight components of $\Psi^{k}_{0}$ are just the Janis-
Newman multi-poles of gravitational field[24]. The AASC only gives a
restriction between Janis-Newman’s dipoles and quadrupoles[15]. Since there is
no restriction on higher order multi-poles, it is easy to see that there are
many solutions which satisfy the conditions of our theorem and are not
equivalent to the Kerr-Newman solution.
## 3 Concluding remarks
We have proven that all the N-P constants of an asymptotic flat, stationary,
asymptotically algebraically special electrovacuum space-time are zero. The
Kerr-Newman solution manifestly satisfies all the conditions. So our theorem
implies that all the N-P constants of the Kerr-Newman solution are zero. This
result has been obtained resently[25] by other authors. In the proof of the
theorem, we have assumed that the Maxwell field is stationary. If this
condition is not imposed, ${\dot{\phi}}^{1}_{0}$ will not be zero. Then
Eq.(51) tells us $\phi^{0}_{0}$ will contain other components of the spin-
weight spherical functions. These terms correspond to the Janis-Newman multi-
pole of Maxwell field[24]. In the presence of these terms, the N-P constants
may not vanish. Last but not least, an interesting issue is to single out the
Kerr-Newman solution from solutions which satisfy the conditions of our
theorem. From the discussion of the last section, we find that the AASC is not
enough to uniquely determine the Kerr-Newman solution. It seems that more
restrictions on the Maxwell field are needed. This will be discussed in our
future work.
## Acknowledgement
This work is supported by the Natural Science Foundation of China (NSFC) under
Grant Nos. 10705048, 10605006, 10731080. Authors would like to thank the
referees for helpful comments on the asymptotically algebraically special
condition.
## References
* [1] E. T. Newman and R. Penrose, Proc. Roy. Soc. Lond. A 305 (1968) 175.
* [2] R. H. Price, Phys. Rev. D 5 (1972) 2419\.
* [3] J.A. Valiente Kroon, Class.Quant.Grav. 16 (1999) 1653.
* [4] J.A. Valiente Kroon, J.Math.Phys. 41 (2000) 898.
* [5] H. Friedrich and J. Kánnár, J. Math. Phys. 41 (2000) 2195\.
* [6] W. B. Bonnor, Classical and Quantum Gravity, 18 (2001) 233.
* [7] J.A. Valiente Kroon, Class. Quant. Grav. 20 (2003) L53.
* [8] R. Geroch, J. Math. Phys. 11 (1970) 1955, 2580\.
* [9] R. Hansen, J. Math. Phys. 15 (1974) 46.
* [10] P. K. Kundu, J. Math. Phys. 29 (1988) 1866.
* [11] H. Friedrich, “Static vacuum solutions from convergent null dadta expensions at space-like infinity”, gr-qc/0606133.
* [12] W. Kinnersely and M. Walker, Phys. Rev. D 2 (1970) 1359.
* [13] R. Lazkoz and J. A. Valiente Kroon, Phys. Rev. D62 (2000) 084033.
* [14] S. Dain and J. A. Valiente Kroon, Class. Quant. Grav., 19(2002)811.
* [15] X. Wu and Y. Shang, Class. Quant. Grav. 24 (2007) 679.
* [16] S. Bai et.al., Phys. Rev. D 75 (2007) 044003\.
* [17] J. A. Valiente Kroon, Class. Quant. Grav. 24 (2007) 3037\.
* [18] R. Penrose and R. Rindler, Spinors and Space-Time Vol.I and II, Cambridge University Press, 1986.
* [19] E. T. Newman and T. W. J. Unti, J. Math. Phys. 3 (1962 ) 891.
* [20] J. Stewart, Advanced General Relativity, Cambridge University Press, 1990.
* [21] D. Kramer, H. Stephani, E. Herlt and M. MacCallum, Exact Solutions of Einstein’s Field Equations, Cambridge University Press, 1980.
* [22] X. Wu and S. Bai, “On local uniqueness of Kerr space-time”, Phys.Rev.D 78 (2008) 124009.
* [23] H. Friedrich, Proc. R. Soc. Lond. A 378 (1981) 169-184, 401-421.
* [24] A. I. Janis and E. T. Newman, J. Math. Phys. 6 (1965) 902.
* [25] X. Gong et.al., Phys.Rev.D 76 (2007) 107501.
|
arxiv-papers
| 2009-04-15T05:52:34 |
2024-09-04T02:49:01.881249
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiangdong Zhang, Xiaoning Wu and Sijie Gao",
"submitter": "Xiangdong Zhang",
"url": "https://arxiv.org/abs/0904.2240"
}
|
0904.2266
|
# NONLINEAR ITERATION SOLUTION FOR THE FULL GLUON PROPAGATOR AS A FUNCTION OF
THE MASS GAP
V. Gogokhia gogohia@rmki.kfki.hu HAS, CRIP, RMKI, Depart. Theor. Phys.,
Budapest 114, P.O.B. 49, H-1525, Hungary
###### Abstract
We have explicitly shown that QCD is the color gauge invariant theory at non-
zero mass gap as well. It has been defined as the value of the regularized
full gluon self-energy at some finite point. The mass gap is mainly generated
by the nonlinear interaction of massless gluon modes. All this allows one to
establish the structure of the full gluon propagator in the explicit presence
of the mass gap. In this case, the two independent general types of formal
solutions for the full gluon propagator as a function of the regularized mass
gap have been found. The nonlinear iteration solution at which the gluons
remain massless is explicitly present. The existence of the solution with an
effective gluon mass is also demonstrated.
###### pacs:
11.15.Tk, 12.38.Lg
## I Introduction
Quantum Chromodynamics (QCD) 1 ; 2 is widely accepted as a realistic quantum
field gauge theory of the strong interactions not only at the fundamental
(microscopic) quark-gluon level but at the hadronic (macroscopic) level as
well. This means that in principle it should describe the properties of
experimentally observed hadrons in terms of experimentally never seen colored
quarks and gluons (the color confinement phenomenon), i.e., to describe the
hadronic world from first principles – an ultimate goal of any fundamental
theory.
The Lagrangian of QCD, however, does not contain explicitly any of the mass
scale parameters which could have a physical meaning even after the
corresponding renormalization program is performed. This clearly shows that it
is not enough to know it in order to calculate the physical observables in
low-energy QCD from first principles. It is also important to know the true
dynamical structure of the QCD ground state especially at large distances,
which may be source of the above-mentioned mass scale parameter. If it will
survive the renormalization program, then QCD is a complete and self-
consistent theory without the need to introduce some extra degrees of freedom
in order to generate it. In this way it may become a mass gap so needed in
non-perturbative (NP) QCD in order to explain the above-mentioned color
confinement and other NP effects 3 . It will be responsible for the NP QCD
dynamics as $\Lambda_{QCD}$ is responsible for the nontrivial perturbative QCD
dynamics (scale violation, asymptotic freedom (AF) 1 ; 2 ).
The propagation of gluons is one of the main dynamical effects in the QCD
vacuum. In our previous work 4 it has been shown that the only place when the
mass gap may appear is the corresponding Schwinger-Dyson (SD) equation of
motion for the full gluon propagator. It should be complemented by the
corresponding Slavnov-Taylor (ST) identity (see next section). The importance
of this equation is due to the fact that its solutions reflect the quantum-
dynamical structure of the QCD ground state. It is highly nonlinear (NL)
equation, and therefore the number of independent solutions, which should be
considered on equal footing, is not fixed $a\ priori$. The color gauge
structure of this equation has been investigated in detail in the above-
mentioned paper 4 . We have explicitly shown that the color gauge invariance
of QCD is consistent with the mass gap, generated in the gluon sector of QCD.
Our primary goal in this investigation is to find formal solutions for the
full gluon propagator as a function of the regularized mass gap. However, for
the sake of completeness and further clarity, it is instructive to describe
briefly the main results of Ref. 4 in the subsequent section.
## II The color gauge invariance of QCD at non-zero mass gap
QCD is a $SU(3)$ color gauge invariant theory. As underlined above, its
dynamical context is determined by the corresponding equations of motion,
among which the SD equation for the full gluon propagator plays an important
role. It can be written as follows:
$D_{\mu\nu}(q)=D^{0}_{\mu\nu}(q)+D^{0}_{\mu\rho}(q)i\Pi_{\rho\sigma}(q;D)D_{\sigma\nu}(q),$
(1)
where
$D^{0}_{\mu\nu}(q)=i\left\\{T_{\mu\nu}(q)+\xi L_{\mu\nu}(q)\right\\}{1\over
q^{2}}$ (2)
is the free gluon propagator, and $\xi$ is the gauge-fixing parameter. Also,
here and everywhere below
$T_{\mu\nu}(q)=\delta_{\mu\nu}-(q_{\mu}q_{\nu}/q^{2})=\delta_{\mu\nu}-L_{\mu\nu}(q)$,
as usual. $\Pi_{\rho\sigma}(q;D)$ is the full gluon self-energy which depends
on the full gluon propagator due to the non-abelian character of QCD. Thus the
gluon SD equation is highly NL one. Evidently, we omit the color group
indices, since for the gluon propagator (and hence for its self-energy) they
factorize, for example $D^{ab}_{\mu\nu}(q)=D_{\mu\nu}(q)\delta^{ab}$.
Diagrammatic representation of the gluon SD equation (2.1) is shown in our
previous work 4 , as well as the detail description of the full gluon self-
energy $\Pi_{\rho\sigma}(q;D)$. It is the sum of a few terms which are
tensors, having the dimensions of mass squared. All these skeleton loop
integrals are therefore quadratically divergent in perturbation theory (PT),
and so they are assumed to be regularized, as discussed below. Let us note in
advance that here and below the signature is Euclidean, since it implies
$q_{i}\rightarrow 0$ when $q^{2}\rightarrow 0$ and vice-versa.
### II.1 The mass gap
Let us introduce the general mass scale parameter $\Delta^{2}(D)$, having the
dimensions of mass squared, by the subtraction from the full gluon self-energy
its value at $q=0$. Thus, one obtains
$\Pi^{s}_{\rho\sigma}(q;D)=\Pi_{\rho\sigma}(q;D)-\Pi_{\rho\sigma}(0;D)=\Pi_{\rho\sigma}(q;D)-\delta_{\rho\sigma}\Delta^{2}(D),$
(3)
which is nothing but the definition of the subtracted full gluon self-energy
$\Pi^{s}_{\rho\sigma}(q;D)$. Contrary to QED, QCD being a non-abelian gauge
theory can suffer from infrared (IR) singularities in the $q^{2}\rightarrow 0$
limit due to the self-interaction of massless gluon modes. Thus the initial
subtraction at zero in the definition (2.3) may be dangerous 1 . That is why
in all the quantities below the dependence on the finite (slightly different
from zero) dimensionless subtraction point $\alpha$ is to be understood. From
a technical point of view, however, it is convenient to put formally
$\alpha=0$ in all the derivations below, and to restore the explicit
dependence on non-zero $\alpha$ in all the quantities only at the final stage.
At the same time, in all the quantities where the dependence on $\lambda$
(which is the dimensionless ultraviolet (UV) regulating parameter) and
$\alpha$ is not shown explicitly, nevertheless, it should be assumed. For
example, $\Delta^{2}(D)\equiv\Delta^{2}(\lambda,\alpha;D)$ and similarly for
all other quantities. So all the expressions are regularized. For our purpose,
in principle, it is not important how $\lambda$ and $\alpha$ have been
introduced. They should be removed at the final stage only as a result of the
renormalization program.
By the mass gap we understand some fixed mass squared which is related to
$\Delta^{2}(D)$ as follows:
$\Delta^{2}(D)=\Delta^{2}\times c(D),$ (4)
where the dimensionless constant $c(D)$ depends on $D$, while the fixed mass
squared $\Delta^{2}$ does not depend on $D$. It will be called the mass gap.
As the general mass scale parameter itself and constant $c(D)$, it may depend
on all other dimensionless parameters of the theory, namely
$\Delta^{2}\equiv\Delta^{2}(\lambda,\alpha,\xi,g^{2})$, where $g^{2}$ is the
coupling constant squared and so on. In this section we will not distinguish
between $\Delta^{2}(D)$ and $\Delta^{2}$, calling both the mass gap, for
simplicity. From the subtraction (2.3) it follows that the mass gap
$\Delta^{2}$, having the dimensions of mass squared, is dynamically generated
in the QCD gluon sector. It is defined as the value of the full gluon self-
energy at some finite point (see discussion above). It is mainly due to the
nonlinear interaction of massless gluon modes. Let us remind that no
truncations/approximations/assumptions/, as well as no special gauge choice
are made for the regularized skeleton loop integrals, contributing to the full
gluon self-energy 4 .
### II.2 The transversality of the full gluon self-energy
Let us continue with the general decompositions of the full gluon self-energy
and its subtracted counterpart, which enter the subtraction (2.3), as follows:
$\displaystyle\Pi_{\rho\sigma}(q;D)$ $\displaystyle=$ $\displaystyle
T_{\rho\sigma}(q)q^{2}\Pi(q^{2};D)+q_{\rho}q_{\sigma}\tilde{\Pi}(q^{2};D),$
$\displaystyle\Pi^{s}_{\rho\sigma}(q;D)$ $\displaystyle=$ $\displaystyle
T_{\rho\sigma}(q)q^{2}\Pi^{s}(q^{2};D)+q_{\rho}q_{\sigma}\tilde{\Pi}^{s}(q^{2};D),$
(5)
where all the invariant functions of $q^{2}$ are dimensionless ones, while in
addition the invariant functions $\Pi^{s}(q^{2};D)$ and
$\tilde{\Pi}^{s}(q^{2};D)$ cannot have the pole-type singularities in the
$q^{2}\rightarrow 0$ limit, since $\Pi^{s}_{\rho\sigma}(0;D)=0$, by
definition; otherwise they remain arbitrary.
Contracting them with $q_{\rho}$ along with the subtraction (2.3), one obtains
$\tilde{\Pi}(q^{2};D)=\tilde{\Pi}^{s}(q^{2};D)+{\Delta^{2}(D)\over q^{2}},$
(6)
and
$\Pi(q^{2};D)=\Pi^{s}(q^{2};D)+{\Delta^{2}(D)\over q^{2}}.$ (7)
It is worth emphasizing that the full gluon self-energy has a massless single
particle singularity due to non-zero mass gap $\Delta^{2}(D)$, which is of the
non-perturbative (NP) origin. At the same time, its subtracted counterpart
cannot have such a singularity, as mentioned above. In other words, this means
that in the explicit presence of the mass gap both invariant functions of the
full gluon self-energy gain additional contributions due to it (of course, not
only at some finite subtraction point $q^{2}=\mu^{2}\neq 0$). If the mass gap
is welcome in the transversal invariant function $\Pi(q^{2};D)$, it is not
welcome in its longitudinal counterpart $\tilde{\Pi}(q^{2};D)$, since just it
violates the ST identity. Let us also note in advance that transversality of
the full gluon self-energy and its subtracted counterpart can be achieved only
in the formal $\Delta^{2}(D)=0$ limit (for a brief discussion of all these
preliminary remarks see subsections below). So in the general case of non-zero
$\Delta^{2}(D)$ only two possibilities remain.
(i). Both are not transversal and then
$\displaystyle q_{\rho}\Pi_{\rho\sigma}(q;D)$ $\displaystyle=$ $\displaystyle
q_{\sigma}q^{2}\tilde{\Pi}(q^{2};D)=q_{\sigma}[q^{2}\tilde{\Pi}^{s}(q^{2};D)+\Delta^{2}(D)]\neq
0,$ $\displaystyle q_{\rho}\Pi^{s}_{\rho\sigma}(q;D)$ $\displaystyle=$
$\displaystyle
q_{\sigma}q^{2}\tilde{\Pi}^{s}(q^{2};D)=q_{\sigma}[q^{2}\tilde{\Pi}(q^{2};D)-\Delta^{2}(D)].$
(8)
The last inequality in the first of the relations (2.8) follows from the fact
that $\tilde{\Pi}^{s}(q^{2};D)$ cannot have a single particle singularity
$-\Delta^{2}(D)/q^{2}$ in order to cancel $\Delta^{2}(D)$.
(ii). Transversality of the subtracted gluon self-energy is maintained, i.e.,
$\tilde{\Pi}^{s}(q^{2};D)=0$ and then
$q_{\rho}\Pi_{\rho\sigma}(q;D)=q_{\sigma}q^{2}\tilde{\Pi}(q^{2};D)=q_{\sigma}\Delta^{2}(D)\neq
0,\quad q_{\rho}\Pi^{s}_{\rho\sigma}(q;D)=0,$ (9)
Contrary to the first case, now we know how precisely the transversality of
the full gluon self-energy is violated. So it is always violated at non-zero
mass scale parameter $\Delta^{2}(D)$. In this connection one thing should be
made perfectly clear. It is the initial subtraction (2.3) which leaves the
subtracted gluon-self energy logarithmical divergent only, and hence the
invariant function $\Pi^{s}(q^{2};D)$ is free of the quadratic divergences,
but a logarithmic ones can be still present in it, at any $D$. Since the
transversality condition for the full gluon self-energy is violated in these
relations, that is why we cannot disregard $\Delta^{2}(D)$ from the very
beginning (compare with the pure quark case considered in our initial work 4
).
### II.3 The ST identity for the full gluon propagator
In order to calculate the physical observables in QCD from first principles,
we need the full gluon propagator rather than the full gluon self-energy. The
basic relation to which the full gluon propagator should satisfy is the
corresponding ST identity
$q_{\mu}q_{\nu}D_{\mu\nu}(q)=i\xi.$ (10)
It is a consequence of the color gauge invarince/symmetry of QCD, and
therefore ”is an exact constraint on any solution to QCD” 1 . This is true for
any other ST identities. Being a result of this exact symmetry, it is the
general one, and it is important for the renormalization of QCD. If some
equation, relation or the regularization scheme, etc. do not satisfy it
automatically, i.e., without any additional conditions, then they should be
modified and not this identity (identity is an equality, where both sides are
the same, i.e., there is no room for additional conditions). In other words,
all the relations, equations, regularization schemes, etc. should be adjusted
to it and not vice versa. It implies that the general tensor decomposition of
the full gluon propagator is
$D_{\mu\nu}(q)=i\left\\{T_{\mu\nu}(q)d(q^{2})+\xi
L_{\mu\nu}(q)\right\\}{1\over q^{2}},$ (11)
where the invariant function $d(q^{2})$ is the corresponding Lorentz structure
of the full gluon propagator (sometimes we will call it as the full effective
charge (”running”), for simplicity). Let us emphasize once more that these
basic relations are to be satisfied in any case, for example, whether the mass
gap or any other mass scale parameter is put formally zero or not.
On account of the exact relations (2.5), (2.6) and (2.7), the initial gluon SD
equation (2.1) can be equivalently re-written down as follows:
$D_{\mu\nu}(q)=D^{0}_{\mu\nu}(q)+D^{0}_{\mu\rho}(q)iT_{\rho\sigma}(q)[q^{2}\Pi^{s}(q^{2};D)+\Delta^{2}(D)]D_{\sigma\nu}(q)+D^{0}_{\mu\rho}(q)iL_{\rho\sigma}(q)q^{2}\tilde{\Pi}(q^{2};D)D_{\sigma\nu}(q).$
(12)
Contracting this equation with $q_{\mu}$ and $q_{\nu}$, one arrives at
$q_{\mu}q_{\nu}D_{\mu\nu}(q)=i\xi-i\xi^{2}\tilde{\Pi}(q^{2};D)$, so the ST
identity (2.10) is not automatically satisfied. In order to get from this
relation the ST identity, one needs to put $\tilde{\Pi}(q^{2};D)=0$, which is
equivalent to $\tilde{\Pi}^{s}(q^{2};D)=-(\Delta^{2}(D)/q^{2})$, as it follows
from the relation (2.6). This, however, is impossible since
$\tilde{\Pi}^{s}(q^{2};D)$ cannot have the power-type singularities at small
$q^{2}$, as underlined above. The only solution to the previous relation is to
disregard $\Delta^{2}(D)$ from the very beginning, i.e., put formally zero
$\Delta^{2}(D)=0$ everywhere. In this case from all the relations it follows
that the gluon full self-energy coincides with its subtracted counterpart, and
both quantities become purely transversal, i.e.,
$\Pi(q^{2};D)=\Pi^{s}(q^{2};D)$ and
$\tilde{\Pi}(q^{2};D)=\tilde{\Pi}^{s}(q^{2};D)=0$ (see relations (2.5)-(2.7)).
The one way to satisfy the ST identity and thus to maintain the color gauge
structure of QCD is to discard the mass gap $\Delta^{2}(D)$ from the very
beginning, i.e., put it formally zero $\Delta^{2}(D)=0$ in all the equations,
relations, etc. In this limit the initial gluon SD equation (2.12) is modified
to
$D^{PT}_{\mu\nu}(q)=D^{0}_{\mu\nu}(q)+D^{0}_{\mu\rho}(q)iT_{\rho\sigma}(q)q^{2}\Pi^{s}(q^{2};D^{PT})D^{PT}_{\sigma\nu}(q),$
(13)
and the corresponding Lorentz structure which appears in Eq. (2.11) becomes
$d^{PT}(q^{2})={1\over 1+\Pi^{s}(q^{2};D^{PT})}.$ (14)
It is easy to see that the gluon SD equation (2.13) automatically satisfies
the ST identity (2.10) now. Evidently, in the formal $\Delta^{2}(D)=0$ limit
we denote $D_{\mu\nu}(q)$ and $d(q^{2})$ as $D^{PT}_{\mu\nu}(q)$ and
$d^{PT}(q^{2})$, respectively (for reason see below). As it has been pointed
out in Ref. 4 , in this case there will be no problems for ghosts to
accomplish their role, namely to cancel the longitudinal component in the full
gluon propagator (2.13).
### II.4 The general structure of the full gluon propagator
The formal $\Delta^{2}(D)=0$ limit is a real way how to preserve the color
gauge invariance in QCD. Then a natural question arises why does the mass gap
$\Delta^{2}(D)$ exist in this theory at all? There is no doubt that the color
gauge invariance of QCD should be maintained at non-zero mass gap as well,
since it is explicitly present in the full gluon self-energy, and hence in the
full gluon propagator. However, by keeping it ”alive”, the two important
problems arise. The first problem is how to replace the original gluon SD
equation (2.12), since it is not consistent with the ST identity unless the
mass gap is discarded from the very beginning (see above). The second problem
is how to make the full gluon propagator purely transversal when the mass gap
is explicitly present.
By introducing the spurious technics we were able to show that the ST identity
(2.10) can be automatically satisfied at non-zero mass gap $\Delta^{2}(D)$ as
well. In other words, our aim is to save the mass gap in the transversal
invariant function (2.7), while removing it from the longitudinal invariant
function (2.6), but without going formally to the PT $\Delta^{2}=0$ limit. In
order to keep the mass gap ”alive”, and, at the same time, to satisfy the ST
identity (2.10), we introduced a temporary dependence on $\Delta^{2}(D)$ in
the free gluon propagator, thus making it an auxiliary (spurious) free gluon
propagator. Substituting it into the initial gluon SD equation (2.12) and
restoring again the dependence on the free gluon propagator, such obtained
gluon SD equation should satisfy the ST identity (2.10). After doing some
tedious algebra, one finally obtains 4
$D_{\mu\nu}(q)=D^{0}_{\mu\nu}(q)+D^{0}_{\mu\rho}(q)iT_{\rho\sigma}(q)[q^{2}\Pi^{s}(q^{2};D)+\Delta^{2}(D)]D_{\sigma\nu}(q).$
(15)
Such modified gluon SD equation (2.15) is satisfied by the same expression for
the Lorentz structure $d(q^{2})$ in Eq. (2.11) as the original gluon SD
equation (2.12), namely
$d(q^{2})={1\over 1+\Pi^{s}(q^{2};D)+(\Delta^{2}(D)/q^{2})},$ (16)
which is not surprising, since the original gluon SD equation (2.12) and its
modified version (2.15) differ from each other only by the longitudinal
(unphysical) part.
However, the important observation is that now it is not required to put the
mass gap $\Delta^{2}(D)$ formally zero everywhere. The spurious mechanism does
not affect the dynamical context of the original gluon SD equation. In other
words, it makes it possible to retain the mass gap in the transversal part of
the gluon SD equation, and, at the same time, to cancel the term in its
longitudinal part, which violates the ST identity. In this way, the modified
gluon SD equation (2.15) satisfies automatically the ST identity (2.10).
Due to AF in QCD the PT regime is realized at $q^{2}\rightarrow\infty$. In
this limit all the Green’s functions are possible to approximate by their free
PT counterparts (up to the corresponding PT logarithms). However, from the
relation (2.16) it follows that in this limit the mass gap term contribution
$\Delta^{2}(D)/q^{2}$ is only next-to-next-to-leading order one. The leading
order contribution is the subtracted gluon self-energy $\Pi^{s}(q^{2};D)$,
which behaves like $\ln q^{2}$ in this limit, as mentioned above. The constant
$1$ is the next-to-leading order term in the $q^{2}\rightarrow\infty$ limit.
Such a special structure of the relation (2.16), namely the mass gap enters it
through the combination $\Delta^{2}(D)/q^{2}$ in its denominator only,
explains immediately why the mass gap $\Delta^{2}(D)$ is not important in PT.
From this structure it follows that the PT regime at $q^{2}\rightarrow\infty$
is effectively equivalent to the formal $\Delta^{2}(D)=0$ limit and vice
versa. That is the reason why this limit can be called the PT limit. And that
is why we denote $D_{\mu\nu}(q;\Delta^{2}=0)=D_{\mu\nu}(q;0)\equiv
D^{PT}_{\mu\nu}(q)$, and hence $d(q^{2};\Delta^{2}=0)=d(q^{2};0)\equiv
d^{PT}(q^{2})$, etc., in accordance with the previous notations. Let us note,
however, that sometimes it is useful to distinguish between the asymptotic
suppression of the mass gap contribution $\Delta^{2}/q^{2}$ in the
$q^{2}\rightarrow\infty$ limit and the formal PT $\Delta^{2}=0$ limit (see our
subsequent paper).
Thus the formal PT $\Delta^{2}(D)=0$ limit exists, and it is a regular one. As
it follows from above, in this limit one recovers the PT QCD system of
equations (2.13)-(2.14) from the NP QCD one (2.15)-(2.16). So, we distinguish
between the PT and NP phases in QCD by the explicit presence of the mass gap.
Its aim is to be responsible for the NP QCD dynamics, since it dominates at
$q^{2}\rightarrow 0$ in the ”solution” (2.16). When it is put formally zero,
then the PT phase survives only. Evidently, when such a scale is explicitly
present then the QCD coupling constant plays no role in the NP QCD dynamics.
### II.5 Transversality of the relevant full gluon propagator
The NP QCD system of equations(2.15)-(2.16) depends explicitly on the mass gap
$\Delta^{2}$. As it has been discussed in detail in our previous work 4 , then
the ghosts are not able to cancel the longitudinal component in the full gluon
propagator, i.e., they are of no use in this case (the transversality
condition for the full gluon self-energy is always violated, see relations
(2.8) and (2.9)). This is the price we have paid to keep the mass gap ”alive”
in the full gluon propagator. Our aim here is to formulate a method which
allows one to make the gluon propagator, relevant for NP QCD, purely
transversal in a gauge invariant way, even if the mass gap is explicitly
present.
For this purpose let us define the truly NP (TNP) part of the full gluon
propagator as follows:
$D^{TNP}_{\mu\nu}(q;\Delta^{2})=D_{\mu\nu}(q;\Delta^{2})-D_{\mu\nu}(q;\Delta^{2}=0)=D_{\mu\nu}(q;\Delta^{2})-D^{PT}_{\mu\nu}(q),$
(17)
i.e., the subtraction is made with respect to the mass gap $\Delta^{2}$, and
therefore the separation between these two terms is exact. So it becomes
$D^{TNP}_{\mu\nu}(q;\Delta^{2})=iT_{\mu\nu}(q)\Bigr{[}d(q^{2};\Delta^{2})-d^{PT}(q^{2})\Bigl{]}{1\over
q^{2}}=iT_{\mu\nu}(q)d^{TNP}(q^{2};\Delta^{2}){1\over q^{2}},$ (18)
where the explicit expression for the TNP Lorentz structure
$d^{TNP}(q^{2};\Delta^{2})=d(q^{2};\Delta^{2})-d^{PT}(q^{2})$ can be obtained
from the relations (2.16) and (2.14) for $d(q^{2};\Delta^{2})$ and
$d^{PT}(q^{2})$, respectively.
The subtraction (2.17) is equivalent to
$D_{\mu\nu}(q;\Delta^{2})=D^{TNP}_{\mu\nu}(q;\Delta^{2})+D^{PT}_{\mu\nu}(q).$
(19)
The TNP gluon propagator (2.18) does not survive in the formal PT
$\Delta^{2}=0$ limit. This means that it is free of the PT contributions, by
construction. The full gluon propagator in this limit is reduced to its PT
counterpart. This means that the full gluon propagator, being also NP,
nevertheless, is ”contaminated” by them. The TNP gluon propagator is purely
transversal in a gauge invariant way (no special (Landau) gauge choice by
hand), while its full counterpart has a longitudinal component as well. There
is no doubt that the true NP dynamics of the full gluon propagator is
completely contained in its TNP part, since the subtraction (2.19) is nothing
but adding zero to the full gluon propagator. We can write
$D_{\mu\nu}(q;\Delta^{2})=i\left\\{T_{\mu\nu}(q)d(q^{2};\Delta^{2})+\xi
L_{\mu\nu}(q)\right\\}(1/q^{2})-iT_{\mu\nu}(q)d^{PT}(q^{2})(1/q^{2})+iT_{\mu\nu}(q)d^{PT}(q^{2})(1/q^{2})=D^{TNP}_{\mu\nu}(q;\Delta^{2})+D^{PT}_{\mu\nu}(q)$,
and so the true NP dynamics in the full gluon propagator is not affected, but
contrary exactly separated from its PT dynamics, indeed. In other words, the
TNP gluon propagator is the full gluon propagator but free of its PT ”tail”.
Taking this important observation into account, we propose instead of the full
gluon propagator to use its TNP counterpart (2.18) as the relevant gluon
propagator for NP QCD, i.e., to replace
$D_{\mu\nu}(q;\Delta^{2})\rightarrow
D^{TNP}_{\mu\nu}(q;\Delta^{2})=D_{\mu\nu}(q;\Delta^{2})-D^{PT}_{\mu\nu}(q),$
(20)
and hence $d(q^{2};\Delta^{2})\rightarrow
d^{TNP}(q^{2};\Delta^{2})=d(q^{2};\Delta^{2})-d^{PT}(q^{2})$.
The subtraction (2.20) plays effectively the role of ghosts in our proposal.
However, the ghosts cancel only the longitudinal component in the PT gluon
propagator, while our proposal leads to the cancellation of the PT
contribution in the full gluon propagator as well (and thus to an automatical
cancellation of its longitudinal component). Nevertheless, this is not a
problem, since the mass gap is not survived in the formal PT limit, anyway.
In fact, our proposal is reduced to a rather simple prescription. If one knows
a full gluon propagator, and is able to identify the mass scale parameter
responsible for the NP dynamics in it, then the full gluon propagator should
be replaced in accordance with the subtraction (2.20). The only problem with
it is that, being exact, it may not be unique. However, the uniqueness of such
kind of separation can be achieved only in the explicit solution for the full
gluon propagator as a function of the mass gap (see below). Anyway, this
subtraction is a first necessary step, which guarantees transversality of the
TNP gluon propagator $D^{TNP}_{\mu\nu}(q;\Delta^{2})$ without losing even one
bit of information on the true NP dynamics in the full gluon propagator
$D_{\mu\nu}(q;\Delta^{2})$. At the same time, its non-trivial PT dynamics is
completely saved in its PT part $D^{PT}_{\mu\nu}(q)$. So it is worth
emphasizing that the both terms in the subtraction (2.19) are valid in the
whole momentum range, i.e., they are not asymptotics.
The full gluon propagator (2.19), keeping the mass gap ”alive”, is not
”physical” in the sense that it cannot be made transversal by ghosts.
Therefore it cannot be used for numerical calculations of the physical
observables from first principles. However, our proposal makes it possible to
present it as the exact sum of the two ”physical” propagators. The TNP gluon
propagator is automatically transversal, by construction. It fully contains
all the information of the full gluon propagator on its NP context. Just it
should be used in accordance with the prescription (2.20) in order to
calculate the physical observables in low-energy QCD. In high-energy QCD the
PT gluon propagator (2.13) is to be used. It is free of the mass gap and the
ghosts can cancel its longitudinal component, making it thus transversal
(”physical”).
Concluding, in this section we have briefly remind how to preserve the color
gauge invariance/symmetry in QCD at non-zero mass gap. This means that from
now on we can forget the relations (2.8) and (2.9) at all, since there are no
any more their negative consequences for the truly NP QCD. In this connection
let us remind the initial subtraction (2.3) has been done in a gauge invariant
way (i.e., not in a separate propagators, which enter the skeleton loop
integrals, contributing to the full gluon self-energy).
## III Massive solution
One of the direct consequences of the explicit presence of the mass gap in the
full gluon propagator is that the gluon may acquire an effective mass, indeed
5 . From Eq. (2.16) it follows that
${1\over q^{2}}d(q^{2})={1\over
q^{2}+q^{2}\Pi^{s}(q^{2};\xi)+\Delta^{2}c(\xi)},$ (21)
where instead of the dependence on $D$ the dependence on $\xi$ is explicitly
shown, while here and below the dependence on all other parameters is not
shown, for simplicity. The full gluon propagator (2.11) may have a pole-type
solution at the finite point if and only if the denominator in Eq. (3.1) has a
zero at this point $q^{2}=-m^{2}_{g}$ (Euclidean signature), i.e.,
$-m^{2}_{g}-m^{2}_{g}\Pi^{s}(-m^{2}_{g};\xi)+\Delta^{2}c(\xi)=0,$ (22)
where $m^{2}_{g}\equiv m^{2}_{g}(\xi)$ is an effective gluon mass. The
previous equation is a transcendental equation for its determination.
Evidently, the number of its solutions is not fixed, $a\ priori$. Excluding
the mass gap, one obtains that the denominator in the full gluon propagator
becomes
$q^{2}+q^{2}\Pi^{s}(q^{2};\xi)+\Delta^{2}c(\xi)=q^{2}+m^{2}_{g}+q^{2}\Pi^{s}(q^{2};\xi)+m^{2}_{g}\Pi^{s}(-m^{2}_{g};\xi).$
(23)
Let us now expand $\Pi^{s}(q^{2};\xi)$ in a Taylor series near $m^{2}_{g}$:
$\Pi^{s}(q^{2};\xi)=\Pi^{s}(-m^{2}_{g};\xi)+(q^{2}+m^{2}_{g})\Pi^{\prime
s}(-m^{2}_{g};\xi)+O\Bigl{(}(q^{2}+m^{2}_{g})^{2}\Bigr{)}.$ (24)
Substituting this expansion into the previous relation and after doing some
tedious algebra, one obtains
$q^{2}+m^{2}_{g}+q^{2}\Pi^{s}(q^{2};\xi)+m^{2}_{g}\Pi^{s}(-m^{2}_{g};\xi)=(q^{2}+m^{2}_{g})[1+\Pi^{s}(-m^{2}_{g};\xi)-m^{2}_{g}\Pi^{\prime
s}(-m^{2}_{g};\xi)][1+\Pi^{s,R}(q^{2};\xi)],$ (25)
where $\Pi^{s,R}(q^{2};\xi)=0$ at $q^{2}=-m^{2}_{g}$ and it is regular at
small $q^{2}$; otherwise it remains arbitrary.
The full gluon propagator (2.11) thus now looks
$D_{\mu\nu}(q;m^{2}_{g})=iT_{\mu\nu}(q){Z_{3}(m_{g}^{2})\over(q^{2}+m^{2}_{g})[1+\Pi^{s,R}(q^{2};m^{2}_{g})]}+i\xi
L_{\mu\nu}(q){1\over q^{2}},$ (26)
where, for future purpose, in the invariant function
$\Pi^{s,R}(q^{2};m^{2}_{g})$ instead of the gauge-fixing parameter $\xi$ we
introduced the dependence on the gluon effective mass squared $m_{g}^{2}$,
which depends on $\xi$ itself. The gluon propagator’s renormalization constant
is
$Z_{3}(m_{g}^{2})={1\over 1+\Pi^{s}(-m^{2}_{g};\xi)-m^{2}_{g}\Pi^{\prime
s}(-m^{2}_{g};\xi)}.$ (27)
In the formal PT limit $\Delta^{2}=0$, an effective gluon mass is also zero,
$m_{g}^{2}(\xi)=0$, as it follows from Eq. (3.2). So an effective gluon mass
is the NP effect. At the same time, it cannot be interpreted as the ”physical”
gluon mass, since it remains explicitly gauge-dependent quantity (at least at
this stage). In other words, we were unable to renormalize it along with the
gluon propagator (3.6). In the formal PT $\Delta^{2}=m_{g}^{2}(\xi)=0$ limit
the gluon propagator’s renormalization constant (3.7) becomes the standard one
1 ; 2 , namely
$Z_{3}(0)={1\over 1+\Pi^{s}(0;\xi)}.$ (28)
It is interesting to note that Eq. (3.2) has a second solution in the formal
PT $\Delta^{2}=0$ limit. In this case an effective gluon mass remains finite,
but $1+\Pi^{s}(-m^{2}_{g};\xi)=0$. So a scale responsible for the NP dynamics
is not determined by an effective gluon mass itself, but by this condition.
Its interpretation from the physical point of view is not clear. The massive
solution (3.6) is difficult to use for the solution of the color confinement
problem, since it is smooth in the $q^{2}\rightarrow 0$ limit. However, its
existence shows the general possibility for a vector particles to acquire
masses dynamically, i.e., without so-called Higgs mechanism 6 , which requires
the existence of not yet discovered Higgs particle. Apparently, it can be also
useful in the generalization of QCD to non-zero temperature and density 7 ; 8
(and references therein), when the gluons may indeed acquire effective masses.
The above-mentioned possibility is due only to the internal dynamics and
symmetries of the corresponding gauge theory.
The general procedure described above in subsection E of section II can be
directly applied to the massive solution (3.6). So it becomes
$D_{\mu\nu}(q;m^{2}_{g})=D^{TNP}_{\mu\nu}(q;m^{2}_{g})+D^{PT}_{\mu\nu}(q),$
(29)
where
$D^{TNP}_{\mu\nu}(q;m^{2}_{g})=iT_{\mu\nu}(q)\left[{Z_{3}(m^{2}_{g})\over(q^{2}+m^{2}_{g})[1+\Pi^{s,R}(q^{2};m^{2}_{g})]}-{Z_{3}(0)\over
q^{2}[1+\Pi^{s,R}(q^{2};0)]}\right]$ (30)
and
$D^{PT}_{\mu\nu}(q)=i\left[T_{\mu\nu}(q){Z_{3}(0)\over[1+\Pi^{s,R}(q^{2};0)]}+\xi
L_{\mu\nu}(q)\right]{1\over q^{2}}.$ (31)
Let us remind that in the massive solution the role of the mass gap is played
by an effective gluon mass, so the formal PT limit is $m^{2}_{g}=0$. In
accordance with our prescription (2.20), we should finally replace the full
gluon propagator (3.6) as follows: $D_{\mu\nu}(q;m_{g}^{2})\rightarrow
D^{TNP}_{\mu\nu}(q;m_{g}^{2})$, where the latter is explicitly given in Eq.
(3.10).
## IV General NL iteration solution
In order to find another type of the general formal solution for the full
gluon propagator (2.15), let us begin again with its ”solution” (2.16) which
is
$d(q^{2})\equiv d(q^{2};\Delta^{2})={1\over
1+\Pi^{s}(q^{2};d)+c(d)(\Delta^{2}/q^{2})},$ (32)
where the dependence on $D$ is replaced by the equivalent dependence on $d$
and the relation (2.4) is already used. It is worth reminding that the
invariant function $\Pi^{s}(q^{2};d)$ and $c(d)$ are, in fact, the sum of the
corresponding skeleton loop integrals (see section II and our initial paper 4
). Let us introduce further the dimensionless variable $z=\Delta^{2}/q^{2}$.
The full Lorentz structure (4.1) regularly depends on the mass gap, and hence
on $z$. Thus it can be expand in a Taylor series in powers of $z$ around zero
$z$ as follows:
$d(q^{2};\Delta^{2})=d(q^{2};z)=\sum_{k=0}^{\infty}z^{k}f_{k}(q^{2}),$ (33)
where the functions $f_{k}(q^{2})$ are the corresponding derivatives of
$d(q^{2};z)$ with respect to $z$ at $z=0$, which is equivalent to the PT
$\Delta^{2}=0$ limit. For example,
$f_{0}(q^{2})=d(q^{2};z=0)=d^{PT}(q^{2})=[1+\Pi^{s}(q^{2};d^{PT}]^{-1}$,
$f_{1}(q^{2})=(\partial d(q^{2};z)/\partial
z)_{z=0}=\left[\partial[1+\Pi^{s}(q^{2};d)+c(d)z]^{-1}/\partial
z\right]_{z=0}=-\left[1+\Pi^{s}(q^{2};d^{PT})\right]^{-2}c(d^{PT})=-[d^{PT}(q^{2})]^{2}c(d^{PT})$,
and so on, i.e.,
$f_{k}(q^{2})=(-1)^{k}d^{PT}(q^{2})[d^{PT}(q^{2})c(d^{PT})]^{k}$. Fortunately,
these explicit expressions play no any role in what follows. In any case, they
depend on the unknown, in general, quantities $\Pi^{s}(q^{2};d)$ and $c(d)$,
which by themselves NL depend on $d$ and finally on $d^{PT}$ and $c(d^{PT})$.
So our expansion (4.2) is nothing but the NL iteration series in powers of the
mass gap (for the direct NL iteration procedure with $d^{(0)}=1$ as input
information see appendix A). To use also unknown functions $f_{k}(q^{2})$ much
more convenient from the technical point of view. However, it is worth
emphasizing that, contrary to the relation (4.1), the expansion (4.2) can be
considered now as a formal solution for $d(q^{2})$, since $f_{k}(q^{2})$
depend on $d^{PT}(q^{2})$, which is assumed to be ”known”.
The functions $f_{k}(q^{2})$ are regular functions of the variable $q^{2}$,
since they finally depend on $d^{PT}(q^{2})$ which is a regular function of
$q^{2}$. Therefore they can be expand in a Taylor series near $q^{2}=0$ (here
we can put the subtraction point $\alpha=0$, for simplicity, since all the
quantities are already regularized, i.e., they depend on $\alpha$ and so on,
see appendix A). Introducing the dimensionless variable $x=q^{2}/M^{2}$, where
$M^{2}$ is some fixed auxiliary mass squared, it is convenient to present this
expansion as a sum of the two terms, namely
$f_{k}(q^{2})=\sum_{n=0}^{k}x^{n}f_{kn}(0)+x^{k+1}B_{k}(x),$ (34)
where the coefficient $f_{kn}(0)$ are the corresponding derivatives of the
functions $f_{k}(q^{2})\equiv f_{k}(x)$ with respect to $x$ at $x=0$. Of
course, these coefficients depend on the parameters of the theory such as
$\lambda,\alpha,\xi,g^{2}$, and so on, which are not shown explicitly. The
dependence on these parameters will be restored at the final stage of our
derivations. The dimensionless functions $B_{k}(x)$ are regular functions of
$x$; otherwise they remain arbitrary.
So the general Lorentz structure (4.2) becomes
$d(q^{2})=\sum_{k=0}^{\infty}z^{k}f_{k}(x)=\sum_{k=0}^{\infty}z^{k}\Big{(}\sum_{n=0}^{k}x^{n}f_{kn}(0)+x^{k+1}B_{k}(x)\Big{)}.$
(35)
Omitting all the intermediate tedious derivations (which, nevertheless, are
quite obvious), these double sums can be equivalently present as the sum of
the three independent terms as follows:
$d(q^{2})=z\sum_{k=0}^{\infty}z^{k}\sum_{m=0}^{\infty}\Phi_{km}(0)+a\sum_{k=0}^{\infty}a^{k}\sum_{m=0}^{\infty}A_{km}(x)+d^{PT}(q^{2}),$
(36)
where the constant $a=xz=\Delta^{2}/M^{2}$ and the dimensionless functions
$A_{km}(x)$ are regular functions of $x$: otherwise they remain arbitrary.
$d^{PT}(q^{2})$ denotes the terms which do not depend on the mass gap
$\Delta^{2}$ at all, i.e., it is nothing but the Lorentz structure of the PT
gluon propagator (2.14), indeed. The summation over $m$ explicitly shows that
all iterations invoke each NP IR singularity labeled by $k$ in the first term
of the expansion (4.5). Thus it is the general NL formal expansion in powers
of the mass gap (this is explicitly seen from appendix A).
Going back to the gluon momentum variable $q^{2}$, one obtains
$d(q^{2};\Delta^{2})=d^{TNP}(q^{2};\Delta^{2})+d^{PT}(q^{2})=d^{INP}(q^{2};\Delta^{2})+d^{MPT}(q^{2};\Delta^{2})+d^{PT}(q^{2}),$
(37)
where the superscripts ”INP” and ”MPT” stand for the intrinsically NP and
mixed PT parts of the TNP term, respectively (for reasons see discussion
below). In other words, in the general NL iteration solution the TNP part
itself is a sum of the two independent terms, i.e.,
$d^{TNP}(q^{2};\Delta^{2})=d^{INP}(q^{2};\Delta^{2})+d^{MPT}(q^{2};\Delta^{2})$.
Their explicit expressions are
$d^{INP}(q^{2};\Delta^{2})=\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}\sum_{k=0}^{\infty}\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}^{k}\Phi_{k}=\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}\sum_{k=0}^{\infty}\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}^{k}\sum_{m=0}^{\infty}\Phi_{km}$ (38)
and
$d^{MPT}(q^{2};\Delta^{2})=\Bigl{(}{\Delta^{2}\over
M^{2}}\Bigr{)}\sum_{k=0}^{\infty}\Bigl{(}{\Delta^{2}\over
M^{2}}\Bigr{)}^{k}A_{k}(q^{2})=\Bigl{(}{\Delta^{2}\over
M^{2}}\Bigr{)}\sum_{k=0}^{\infty}\Bigl{(}{\Delta^{2}\over
M^{2}}\Bigr{)}^{k}\sum_{m=0}^{\infty}A_{km}(q^{2}).$ (39)
Here and everywhere below all the quantities depend on the parameters of the
theory, namely $\Delta^{2}=\Delta^{2}(\lambda,\alpha,\xi,g^{2})$ and
$A_{k}(q^{2})=\sum_{m=0}^{\infty}A_{km}(q^{2};\lambda,\alpha,\xi,g^{2})$. At
the same time, $\Phi_{km}$ depends in addition on the parameter $a$ as well,
i.e., $\Phi_{km}=\Phi_{km}(\lambda,\alpha,\xi,g^{2},a)$.
### IV.1 The exact structure of the NL iteration solution
The full gluon propagator (2.11) thus becomes the sum of the three independent
terms, namely
$D_{\mu\nu}(q;\Delta^{2})=D^{TNP}_{\mu\nu}(q;\Delta^{2})+D^{PT}_{\mu\nu}(q)=D^{INP}_{\mu\nu}(q;\Delta^{2})+D^{MPT}_{\mu\nu}(q;\Delta^{2})+D^{PT}_{\mu\nu}(q),$
(40)
where
$D^{INP}_{\mu\nu}(q;\Delta^{2})=iT_{\mu\nu}(q)d^{INP}(q^{2};\Delta^{2}){1\over
q^{2}}=iT_{\mu\nu}(q){\Delta^{2}\over(q^{2})^{2}}L(q^{2};\Delta^{2})$ (41)
with
$L(q^{2};\Delta^{2})=\sum_{k=0}^{\infty}\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}^{k}\Phi_{k}=\sum_{k=0}^{\infty}\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}^{k}\sum_{m=0}^{\infty}\Phi_{km},$ (42)
while
$D^{MPT}_{\mu\nu}(q;\Delta^{2})=iT_{\mu\nu}(q)d^{MPT}(q^{2};\Delta^{2}){1\over
q^{2}}$ (43)
with $d^{MPT}(q^{2};\Delta^{2})$ given in Eq. (4.13) and
$D^{PT}_{\mu\nu}(q)=i\Bigr{[}T_{\mu\nu}(q)d^{PT}(q^{2})+\xi
L_{\mu\nu}(q)\Bigl{]}{1\over q^{2}}$ (44)
with $d^{PT}(q^{2})$ given in Eq. (2.14). For the direct NL iteration
procedure see appendix A, as mentioned above.
Let us emphasize that the general problem of convergence of formal (but
regularized) series, which appear in these relations, is irrelevant here. In
other words, it does not make any sense to discuss the convergence of such
kind of series before the renormalization program is performed (which will
allow one to see whether or not the mass gap survives it at all). The problem
how to remove the UV overlapping divergences 9 and usual overall ones 1 ; 2 ;
10 ; 11 is a standard one, i.e., it is not our problem, anyway (let us remind
that the mass gap does not survive in the PT $q^{2}\rightarrow\infty$ limit).
Our problem will be how to deal with severe infrared (IR) ($q^{2}\rightarrow
0$) singularities due to their novelty and genuine (intrinsic) NP character
(in this limit the mass gap dominates the structure of the full gluon
propagator). Fortunately, there already exists a well-elaborated mathematical
formalism for this purpose, namely the distribution theory (DT) 12 , into
which the dimensional regularization method (DRM) 13 should be correctly
implemented (see also Refs. 14 ; 15 ).
The INP part of the full gluon propagator is characterized by the presence of
severe power-type (or, equivalently, NP) IR singularities $(q^{2})^{-2-k},\
k=0,1,2,3,...$. So these IR singularities are defined as more singular than
the power-type IR singularity of the free gluon propagator $(q^{2})^{-1}$,
which thus can be defined as the PT IR singularity. The INP part of the full
gluon propagator (4.10), apart from the structure $(\Delta^{2}/q^{4})$, is
nothing but the corresponding Laurent expansion (explicitly shown in Eq.
(4.11)) in integer powers of $q^{2}$ accompanied by the corresponding powers
of the mass gap squared and multiplied by the $q^{2}$-independent factors, the
so-called residues
$\Phi_{k}(\lambda,\alpha,\xi,g^{2},a)=\sum_{m=0}^{\infty}\Phi_{km}(\lambda,\alpha,\xi,g^{2},a)$.
The sum over $m$ indicates that an infinite number of iterations (all
iterations) of the above-mentioned corresponding regularized skeleton loop
integrals invokes each severe IR singularity labeled by $k$. It is worth
emphasizing that the Laurent expansion (4.11) cannot be summed up into the
some known function, since its residues are, in general, arbitrary. However,
this arbitrariness is not a problem. The functional dependence, which has been
established exactly, is all that matters (this will be explicitly shown in the
subsequent paper). Let us note that the expansions (4.10)-(4.11) have been
independently obtained in Ref. 14 in a rather different way.
The MPT part of the full gluon propagator (4.12), which has the power-type PT
IR singularity only, remains undetermined, but depends on the mass gap (that
is why we call this term as the mixed PT contribution, but it vanishes in the
formal PT $\Delta^{2}=0$ limit). This is the price we have paid to fix exactly
the functional dependence of the INP part of the full gluon propagator. With
respect to the character of the IR singularity it should be combined with the
PT gluon propagator, leading to the so-called general PT (GPT) term, namely
$D^{GPT}_{\mu\nu}(q;\Delta^{2})=D^{MPT}_{\mu\nu}(q;\Delta^{2})+D^{PT}_{\mu\nu}(q)=i\Bigr{[}T_{\mu\nu}(q)d^{GPT}(q^{2};\Delta^{2})+\xi
L_{\mu\nu}(q)\Bigl{]}{1\over q^{2}},$ (45)
where $d^{GPT}(q^{2};\Delta^{2})=d^{MPT}(q^{2};\Delta^{2})+d^{PT}(q^{2})$ is
regular at small $q^{2}$, while $d^{MPT}(q^{2};\Delta^{2}=0)=0$ and hence
$d^{GPT}(q^{2};\Delta^{2}=0)=d^{PT}(q^{2})$. Thus both terms MPT and PT
present the PT-type contributions to the full gluon propagator (4.6). It is
worth reminding that all the three terms, which appear in the right-hand-side
of Eq. (4.9) are valid in the whole energy/momentum range, i.e., they are not
asymptotics. At the same time, we have achieved the separation between the
terms responsible for the NP (dominating in the IR ($q^{2}\rightarrow 0$)) and
the nontrivial PT (dominating in the UV ($q^{2}\rightarrow\infty$)) dynamics
in the true QCD vacuum. The structure of this solution shows clearly that the
deep IR region interesting for confinement and other NP effects is dominated
by the mass gap. In the formal PT $\Delta^{2}=0$ limit, the nontrivial PT
dynamics is all that matters.
## V INP gluon propagator
In accordance with our prescription, one should subtract all the types of the
PT contributions in order to get the relevant gluon propagator for the truly
NP QCD. As it follows from discussion above, in the case of the NL iteration
solution, we should subtract the two terms. Doing so in Eq. (4.9), on account
of Eq. (4.14), one finally obtains
$D_{\mu\nu}(q;\Delta^{2})\rightarrow
D^{INP}_{\mu\nu}(q;\Delta^{2})=D_{\mu\nu}(q;\Delta^{2})-D^{GPT}_{\mu\nu}(q;\Delta^{2}),$
(46)
and hence $d(q^{2})\rightarrow d^{INP}(q^{2})$ as well, so that
$D^{INP}_{\mu\nu}(q;\Delta^{2})=iT_{\mu\nu}(q){\Delta^{2}\over(q^{2})^{2}}L(q^{2};\Delta^{2})=iT_{\mu\nu}(q){\Delta^{2}\over(q^{2})^{2}}\sum_{k=0}^{\infty}\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}^{k}\Phi_{k},$ (47)
where $\Delta^{2}=\Delta^{2}(\lambda,\alpha,\xi,g^{2})$ and
$\Phi_{k}=\Phi_{k}(\lambda,\alpha,\xi,g^{2})=\sum_{m=0}^{\infty}\Phi_{km}(\lambda,\alpha,\xi,g^{2})$.
In this connection, let us note that after the subtraction (5.1) is completed
we can put the intermediate parameter $a=1$, to equate thus the auxiliary
fixed mass to the mass gap itself, i.e., put $M^{2}=\Delta^{2}$, not losing
generality. In the deep IR region ($q^{2}\rightarrow 0$) the mass gap is only
one that’s really matters. All other masses introduced from a technical point
of view in order to clarify the derivations play only auxiliary role.
It is important to emphasize that the INP gluon propagator (5.2) is uniquely
defined because there exists a special regularization expansion for severe
(i.e., NP) IR singularities, while for the PT IR singularity such kind of
expansion does not exist at all (see Refs. 12 ; 14 ; 15 and references
therein). This just determines the principal difference between the NP and PT
IR singularities. It is also exactly defined because of its two features. The
first one is that the INP gluon propagator depends only on the transversal
degrees of freedom of gauge bosons. The second one is that in the formal PT
$\Delta^{2}=0$ limit the INP gluon propagator vanishes. Thus, one can conclude
that the presence of severe IR singularities only is the first necessary
condition, while the regular dependence on the mass gap and transversality is
only second sufficient condition for the unique and exact separation of the
INP gluon propagator from the PT gluon propagator. At the same time, the TNP
gluon propagator is not uniquely defined, since it contains the MPT part, see
Eq. (4.9). In other words, the INP gluon propagator is free of all the types
of the PT contributions (”contaminations”). Just it should replace the full
gluon propagator in order to calculate the physical observables, processes,
etc. from first principles in low-energy QCD after the corresponding
renormalization program is performed.
The INP gluon propagator satisfies its own equation of motion. For the sake of
completeness, let us begin with the SD equation for the TNP gluon propagator 4
, namely
$\displaystyle D^{TNP}_{\mu\nu}(q;\Delta^{2})$ $\displaystyle=$ $\displaystyle
D^{0}_{\mu\rho}(q)iT_{\rho\sigma}(q)[-q^{2}\Pi^{s}(q^{2};D^{PT})+q^{2}\Pi^{s}(q^{2};D)+\Delta^{2}]D^{PT}_{\sigma\nu}(q)$
(48) $\displaystyle+$ $\displaystyle
D^{0}_{\mu\rho}(q)iT_{\rho\sigma}(q)[q^{2}\Pi^{s}(q^{2};D)+\Delta^{2}]D^{TNP}_{\sigma\nu}(q;\Delta^{2})$
with
$D^{TNP}_{\mu\nu}(q)=i\left\\{T_{\mu\nu}(q)d^{TNP}(q^{2})+\xi
L_{\mu\nu}(q)\right\\}{1\over q^{2}}.$ (49)
Here and below we omit the dependence on the mass gap in the propagators and
their Lorentz structures, for simplicity. On account of this decomposition,
the ”solution” of the previous equation is
$d^{TNP}(q^{2})={\Pi^{s}(q^{2};D^{PT})-\Pi^{s}(q^{2};D)-(\Delta^{2}/q^{2})\over[1+\Pi^{s}(q^{2};D)+(\Delta^{2}/q^{2})][1+\Pi^{s}(q^{2};D^{PT})]}.$
(50)
This expression coincides with the definition of
$d^{TNP}(q^{2})=d(q^{2})-d^{PT}(q^{2})$ on account of the explicit expressions
(2.14) and (2.15), as it should be.
From Eq. (4.9) it follows that
$D^{TNP}_{\mu\nu}(q)=D^{INP}_{\mu\nu}(q)+D^{MPT}_{\mu\nu}(q),$ (51)
and substituting it into Eq. (5.3), one obtains the SD equation for the INP
gluon propagator, namely
$\displaystyle D^{INP}_{\mu\nu}(q)=$ $\displaystyle-$ $\displaystyle
D^{MPT}_{\mu\nu}(q)+D^{0}_{\mu\rho}(q)iT_{\rho\sigma}(q)[q^{2}\Pi^{s}(q^{2};D)+\Delta^{2}]D^{MPT}_{\sigma\nu}(q)$
(52) $\displaystyle+$ $\displaystyle
D^{0}_{\mu\rho}(q)iT_{\rho\sigma}(q)[-q^{2}\Pi^{s}(q^{2};D^{PT})+q^{2}\Pi^{s}(q^{2};D)+\Delta^{2}]D^{PT}_{\sigma\nu}(q)$
$\displaystyle+$ $\displaystyle
D^{0}_{\mu\rho}(q)iT_{\rho\sigma}(q)[q^{2}\Pi^{s}(q^{2};D)+\Delta^{2}]D^{INP}_{\sigma\nu}(q).$
Using the decompositions (2.2), (4.12) and (4.13), it can be simplified to
$\displaystyle q^{2}D^{INP}_{\mu\nu}(q)=$ $\displaystyle-$ $\displaystyle
iT_{\mu\nu}(q)\Bigl{(}1+\Pi^{s}(q^{2};D)+(\Delta^{2}/q^{2})\Bigr{)}d^{MPT}(q)$
(53) $\displaystyle-$ $\displaystyle
iT_{\mu\nu}(q)\Bigl{(}-\Pi^{s}(q^{2};D^{PT})+\Pi^{s}(q^{2};D)+(\Delta^{2}/q^{2})\Bigr{)}d^{PT}(q)$
$\displaystyle-$ $\displaystyle
T_{\mu\sigma}(q)T_{\rho\sigma}(q)[q^{2}\Pi^{s}(q^{2};D)+\Delta^{2}]D^{INP}_{\sigma\nu}(q),$
where $d^{MPT}(q)$ and $d^{PT}(q)$ are given in Eqs. (4.8) and (2.14),
respectively. This equation is of no practical use due to its complicated
structure. Fortunately, we already have the explicit expression for the INP
gluon propagator (4.10)-(4.11) or, equivalently, (5.2). It is only one to be
used in order to derive renormalized gluon propagator with the correct
confinement properties.
However, from Eq. (5.8) it follows an important observation that like the TNP
SD equation (5.3) this equation cannot be reduced to the free gluon
propagator, when the interaction is to be switched off (i.e., setting formally
$\Pi^{s}(q^{2};D^{PT})=\Pi^{s}(q^{2};D)=\Delta^{2}=0$). Evidently, to the same
conclusion one comes from the explicit expressions (4.8) and (5.5), on account
of the relation
$d^{INP}(q^{2};\Delta^{2})=d^{TNP}(q^{2};\Delta^{2})-d^{MPT}(q^{2};\Delta^{2})$,
which follows from Eq. (4.6). So in INP QCD the gluon propagator is always
”dressed” as well, and thus this theory has no free gluon propagator in its
formalism. As it has been argued in our initial work 4 , it makes it possible
to suppress the emission and absorbtion of the colored dressed gluons at large
distances by the renormalization of the mass gap. Both the suppression of the
dressed gluons and the absence of the free gluons are necessary for the
explanation of gluon confinement by INP QCD (see our next paper). On the other
hand, the full gluon propagator (2.19) which satisfies Eq. (2.15) is reduced
to the free gluon propagator when the interaction is switched off. There is no
mechanism to suppress the emission and absorbtion of the free gluons at large
distances 4 . That is why the full gluon propagator (2.19) is not confining,
while the INP one (5.2) can be.
The subtraction (5.1) seems to be necessary, indeed. It makes the relevant
gluon propagator (5.2) transversal and excludes the free gluons from the
theory at the same time.
## VI Conclusions
The structure of the full gluon propagator in the presence of the regularized
mass gap has been firmly established. We have shown explicitly that in its
presence at least two independent and different typed of formal solutions for
the regularized full gluon propagator exist. No
truncations/approximations/assumptions are made in order to show the existence
of these general types of solutions. Also, our approach, in general, and the
above-mentioned solutions, in particular, is gauge-invariant, since no special
gauge has been chosen. Let us emphasize that before the renormalization
program is performed the gauge invariance should be understood in this sense
only.
In the presence of the mass gap the gluons may acquire an effective gluon
masses, depending on the gauge choice (the so-called massive solution (3.6)),
but a gauge-fixing parameter remains arbitrary, i.e., a gauge is not fixed by
hand (see remarks above). Its relation to the solution of the color
confinement problem is not clear, even after the renormalization program is
performed.
The general NL iteration solution (4.9)-(4.13) for the full gluon propagator
depends explicitly on the mass gap. It is always severely singular in the
$q^{2}\rightarrow 0$ limit, so the gluons remain massless, and this does not
depend on the gauge choice. However, we argued that only the INP gluon
propagator (5.2) is to be used for the numerical calculations of physical
observables, processes, etc. in low-energy QCD from first principles.
It is worth emphasizing that there exists only one general restriction on the
behavior of $\Pi^{s}(q^{2};D)$, which enters the corresponding gluon SD
equation (2.15), in the explicit presence of the mass gap within our approach,
namely
$q^{2}\Pi^{s}(q^{2};D)\rightarrow 0,\quad q^{2}\rightarrow 0,$ (54)
at any $D$. It stems from the second of the exact decompositions (2.5), since
the subtracted gluon self-energy in this limit (or more precisely at
$q^{2}\rightarrow\mu^{2}$) should go to zero. Otherwise the invariant function
$\Pi^{s}(q^{2};D)$ remains arbitrary (but it is logarithmic divergent at
infinity). Both general types of formal solutions the massive solution and the
NL iteration one satisfy it. The existence of some other solution(s) for the
full gluon propagator, satisfying the general condition (6.1), should not be
excluded $a\ priori$. Let us remind that the gluon SD equation (2.15) is
highly NL, so the number of independent solutions is not fixed. Any concrete
solution obtained by lattice QCD or by the analytical approach based on the SD
system of equations is a particular case of the general types (finite or
singular at zero gluon momentum) of the formal solutions established here.
They are subject to the different truncations/approximations/assumptions and
the concrete gauge choice imposed on the invariant function
$\Pi^{s}(q^{2};D)$, which, in general, remains arbitrary but satisfying the
above-mentioned general constraint (6.1) within our approach (see, for example
recent papers 16 ; 17 ; 18 ; 19 ; 20 ; 21 ; 22 and references therein. Let us
also point out Refs. 23 ; 24 ; 25 ; 26 as well, where the gluon propagator is
finite and contains the mass scale parameter. However, it, apparently, cannot
be interpreted as gluon effective mass).
The INP solution (5.2) is interesting for confinement, but the two important
problems remain to solve. The first problem is how to perform the
renormalization program for the regularized mass gap
$\Delta^{2}\equiv\Delta^{2}(\lambda,\alpha,\xi,g^{2})$, and to see whether the
mass gap survives it or not (it has been already discussed in our previous
work 4 ). The second problem is how to treat correctly severe IR singularities
$(q^{2})^{-2-k},\ k=0,1,2,3,...$ inevitably present in this solution (see a
few brief remarks above in section IV). Both problems will be addressed and
solved in our subsequent paper.
###### Acknowledgements.
Support by HAS-JINR grant (P. Levai) is to be acknowledged. The author is
grateful to P. Forgács, J. Nyiri, C. Wilkin, T. Biró, M. Faber, Á. Lukács, M.
Vasúth and especially to A.V. Kouzushin for useful discussions, remarks and
help.
## Appendix A Direct NL iteration procedure
In order to find a formal solution for the regularized full gluon propagator
(2.11), on account of its effective charge (2.16), let us rewrite the latter
one in the form of the corresponding transcendental (i.e., not algebraic)
equation, namely
$d(q^{2})=1-\Bigl{[}\Pi^{s}(q^{2};d)+{\Delta^{2}\over
q^{2}}c(d)\Bigr{]}d(q^{2})=1-P(q^{2};d)d(q^{2}),$ (55)
where Eq. (2.4) has been already used, and instead of $D$ an equivalent
dependence on $d$ is introduced. It is suitable for the formal NL iteration
procedure. For future purposes, it is convenient to introduce short-hand
notations as follows:
$\displaystyle c(d=d^{(0)}+d^{(1)}+d^{(2)}+...+d^{(m)}+...)$ $\displaystyle=$
$\displaystyle c_{m}\equiv c_{m}(\lambda,\alpha,\xi,g^{2}),$
$\displaystyle\Pi^{s}(q^{2};d=d^{(0)}+d^{(1)}+d^{(2)}+...+d^{(m)}+...)$
$\displaystyle=$ $\displaystyle\Pi^{s}_{m}(q^{2}),$ (56)
and
$P_{m}(q^{2})=\Bigl{[}\Pi^{s}_{m}(q^{2})+{\Delta^{2}\over
q^{2}}c_{m}\Bigr{]},\ m=0,1,2,3,...\ .$ (57)
Via the corresponding subscript $m$ it is explicitly seen which iteration for
the gluon form factor $d$ is actually done in $c(d)$, $\Pi^{s}(q^{2};d)$ and
$P(q^{2};d)$. Let us also point out that all the invariant functions
$\Pi^{s}_{m}(q^{2})$ can be expand in a formal Taylor series near the finite
subtraction point $\alpha$. If it were possible to express the full gluon form
factor $d(q^{2})$ in terms of these quantities then it would be the formal
solution for the full gluon propagator. In fact, this is nothing but the
skeleton loops expansion, since the regularized skeleton loop integrals,
contributing to the gluon self-energy as mentioned above, have to be iterated.
This is the so-called general NL iteration solution. This formal expansion is
not a PT series. The magnitude of the coupling constant squared and the
dependence of the regularized skeleton loop integrals on it is completely
arbitrary.
It is instructive to describe the general iteration procedure in some details.
Evidently, $d^{(0)}=1$, and this corresponds to the approximation of the full
gluon propagator by its free counterpart. Doing the first iteration in Eq.
(A1), one thus obtains
$d(q^{2})=1-P_{0}(q^{2})+...=1+d^{(1)}(q^{2})+...,$ (58)
where obviously
$d^{(1)}(q^{2})=-P_{0}(q^{2}).$ (59)
Carrying out the second iteration, one gets
$d(q^{2})=1-P_{1}(q^{2})[1+d^{(1)}(q^{2})]+...=1+d^{(1)}(q^{2})+d^{(2)}(q^{2})+...,$
(60)
where
$d^{(2)}(q^{2})=-d^{(1)}(q^{2})-P_{1}(q^{2})[1-P_{0}(q^{2})].$ (61)
Doing the third iteration, one further obtains
$d(q^{2})=1-P_{2}(q^{2})[1+d^{(1)}(q^{2})+d^{(2)}(q^{2})]+...=1+d^{(1)}(q^{2})+d^{(2)}(q^{2})+d^{(3)}(q^{2})+...,$
(62)
where
$d^{(3)}(q^{2})=-d^{(1)}(q^{2})-d^{(2)}(q^{2})-P_{2}(q^{2})[1-P_{1}(q^{2})(1-P_{0}(q^{2}))],$
(63)
and so on for the next iterations.
Thus up to the third iteration, one finally arrives at
$d(q^{2})=\sum_{m=0}^{\infty}d^{(m)}(q^{2})=1-\Bigl{[}\Pi^{s}_{2}(q^{2})+{\Delta^{2}\over
q^{2}}c_{2}\Bigr{]}\Bigl{[}1-\Bigl{[}\Pi^{s}_{1}(q^{2})+{\Delta^{2}\over
q^{2}}c_{1}\Bigr{]}\Bigl{[}1-\Pi^{s}_{0}(q^{2})-{\Delta^{2}\over
q^{2}}c_{0}\Bigr{]}\Bigr{]}+...\ .$ (64)
We restrict ourselves by the iterated gluon form factor up to the third term,
since this already allows to show explicitly some general features of the NL
iteration solution.
### A.1 Splitting/shifting procedure
Doing some tedious algebra, the previous expression (A10) can be rewritten as
follows:
$\displaystyle d(q^{2})$ $\displaystyle=$
$\displaystyle\Bigl{[}1-\Pi^{s}_{2}(q^{2})+\Pi^{s}_{1}(q^{2})\Pi^{s}_{2}(q^{2})-\Pi^{s}_{0}(q^{2})\Pi^{s}_{1}(q^{2})\Pi^{s}_{2}(q^{2})+...\Bigr{]}$
(65) $\displaystyle+$ $\displaystyle{\Delta^{2}\over
q^{2}}\Bigl{[}\Pi^{s}_{2}(q^{2})c_{1}+\Pi^{s}_{1}(q^{2})c_{2}-\Pi^{s}_{0}(q^{2})\Pi^{s}_{1}(q^{2})c_{2}-\Pi^{s}_{0}(q^{2})\Pi^{s}_{2}(q^{2})c_{1}-\Pi^{s}_{1}(q^{2})\Pi^{s}_{2}(q^{2})c_{0}+...\Bigr{]}$
$\displaystyle-$ $\displaystyle{\Delta^{4}\over
q^{4}}\Bigl{[}\Pi^{s}_{0}(q^{2})c_{1}c_{2}+\Pi^{s}_{1}(q^{2})c_{0}c_{2}+\Pi^{s}_{2}(q^{2})c_{0}c_{1}+...\Bigr{]}$
$\displaystyle-$ $\displaystyle{\Delta^{2}\over
q^{2}}\Bigl{[}c_{2}-{\Delta^{2}\over q^{2}}c_{1}c_{2}+{\Delta^{4}\over
q^{4}}c_{0}c_{1}c_{2}+...\Bigr{]}.$
This formal expansion contains three different types of terms. The first type
are the terms which contain only different combinations of
$\Pi^{s}_{m}(q^{2})$ (they are not multiplied by inverse powers of $q^{2}$);
the third type of terms contains only different combinations of
$(\Delta^{2}/q^{2})$. The second type of terms contains the so-called mixed
terms, containing the first and third types of terms in different
combinations. The two last types of terms are multiplied by the corresponding
powers of $1/q^{2}$. Such structure of terms will be present in each iteration
term for the full gluon form factor. However, any of the mixed terms can be
split exactly into the first and third types of terms. For this purpose the
formal Taylor expansions for $\Pi^{s}_{m}(q^{2})$ around the finite
subtraction point $\alpha$ should be used. Thus an exact IR structure of the
full gluon form factor (which just is our primary goal to establish) is
determined not only by the third type of terms. It gains contributions from
the mixed terms as well, but without changing its functional dependence (see
remarks below). To demonstrate this in some detail, it is convenient to
express the previous expansion (A11) in terms of dimensionless variables and
parameters introduced in section IV, namely
$z={\Delta^{2}\over q^{2}},\quad x={q^{2}\over M^{2}},\quad
a=zx={\Delta^{2}\over M^{2}},\quad\alpha={\mu^{2}\over M^{2}},$ (66)
where $M^{2}$ is some fixed mass squared, and $\mu^{2}$ is the fixed point
close to $q^{2}=0$ (to be not mixed up with the tensor index). Also, in the
formal PT $\Delta^{2}=0$ limit $a=0$ as well, since $M^{2}$ is fixed. On
account of the relations (A12), the expansion (A11) becomes
$\displaystyle d(x)$ $\displaystyle=$
$\displaystyle\Bigl{[}1-\Pi^{s}_{2}(x)+\Pi^{s}_{1}(x)\Pi^{s}_{2}(x)-\Pi^{s}_{0}(x)\Pi^{s}_{1}(x)\Pi^{s}_{2}(x)+...\Bigr{]}$
(67) $\displaystyle+$ $\displaystyle
z\Bigl{[}\Pi^{s}_{2}(x)c_{1}+\Pi^{s}_{1}(x)c_{2}-\Pi^{s}_{0}(x)\Pi^{s}_{1}(x)c_{2}-\Pi^{s}_{0}(x)\Pi^{s}_{2}(x)c_{1}-\Pi^{s}_{1}(x)\Pi^{s}_{2}(x)c_{0}+...\Bigr{]}$
$\displaystyle-$ $\displaystyle
z^{2}\Bigl{[}\Pi^{s}_{0}(x)c_{1}c_{2}+\Pi^{s}_{1}(x)c_{0}c_{2}+\Pi^{s}_{2}(x)c_{0}c_{1}+...\Bigr{]}$
$\displaystyle-$ $\displaystyle z\Bigl{[}c_{2}-\Bigl{(}{a\over
x}\Bigr{)}c_{1}c_{2}+\Bigl{(}{a\over
x}\Bigr{)}^{2}c_{0}c_{1}c_{2}+...\Bigr{]}.$
Taking into account the above-mentioned formal Taylor expansions
$\Pi^{s}_{m}(x)=\sum_{n=0}^{\infty}(x-\alpha)^{n}\Pi^{(n)}_{m}(\alpha)=\sum_{n=0}^{\infty}\Bigl{[}\sum_{k=0}^{n}p_{nk}x^{k}\alpha^{n-k}\Bigr{]}\Pi^{(n)}_{m}(\alpha),$
(68)
for example, the mixed term $z\Pi^{s}_{2}(x)c_{1}$ can be then exactly
split/decomposed as follows:
$c_{1}z\Pi^{s}_{2}(x)=c_{1}z\sum_{n=0}^{\infty}\Bigl{[}\sum_{k=0}^{n}p_{nk}x^{k}\alpha^{n-k}\Bigr{]}\Pi^{(n)}_{2}(\alpha)=zP_{1}(\alpha)+P_{0}(\alpha)+O_{2}(x).$
(69)
Here and below the dependence on all other possible parameters is not shown,
for simplicity. The dimensionless function $O_{2}(x)$ is of the order $x$ at
small $x$; otherwise it remains arbitrary. The first term now is to be shifted
to the third type of terms, while the remaining terms are to be shifted to the
first type of terms. All other mixed terms of similar structure should be
treated absolutely in the same way.
The mixed term $z^{2}\Pi^{s}_{0}(x)c_{1}c_{2}$ can be split as
$c_{1}c_{2}z^{2}\Pi^{s}_{0}(x)=c_{1}c_{2}z^{2}\sum_{n=0}^{\infty}\Bigl{[}\sum_{k=0}^{n}p_{nk}x^{k}\alpha^{n-k}\Bigr{]}\Pi^{(n)}_{0}(\alpha)=z^{2}P_{2}(\alpha)+zN_{1}(\alpha)+N_{0}(\alpha)+O_{0}(x),$
(70)
where the dimensionless function $O_{0}(x)$ is of the order $x$ at small $x$;
otherwise it remains arbitrary. Again the first two terms should be shifted to
the third type of terms, while the last two terms should be shifted to the
first type of terms.
Similarly to the formal Taylor expansion (A14), we can write
$\Pi^{s}_{m}(x)\Pi^{s}_{m^{\prime}}(x)=\Pi^{s}_{mm^{\prime}}(x)=\sum_{n=0}^{\infty}(x-\alpha)^{n}\Pi^{(n)}_{mm^{\prime}}(\alpha)=\sum_{n=0}^{\infty}\Bigl{[}\sum_{k=0}^{n}p_{nk}x^{k}\alpha^{n-k}\Bigr{]}\Pi^{(n)}_{mm^{\prime}}(\alpha).$
(71)
Then, for example the mixed term $z\Pi^{s}_{0}(x)\Pi^{s}_{1}(qx)c_{2}$ can be
split as
$c_{2}z\Pi^{s}_{0}(x)\Pi^{s}_{1}(x)=c_{2}z\Bigr{)}\Pi_{01}(x)=c_{2}z\sum_{n=0}^{\infty}\Bigl{[}\sum_{k=0}^{n}p_{nk}x^{k}\alpha^{n-k}\Bigr{]}\Pi^{(n)}_{01}(\alpha)=zM_{1}(\alpha)+M_{0}(\alpha)+O_{01}(x),$
(72)
where the dimensionless function $O_{01}(x)$ is of the order $x$ at small $x$;
otherwise it remains arbitrary. Again the first term should be shifted to the
third type of terms, while other two terms are to be shifted to the first type
of terms.
Completing this exact splitting/shifting procedure in the expansion (A13), and
restoring the explicit dependence on the dimensional variable and parameters
(A12), one can equivalently present the initial expansion (A11) as follows:
$d(q^{2})=\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}B_{1}(\lambda,\alpha,\xi,g^{2},a)+\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}^{2}B_{2}(\lambda,\alpha,\xi,g^{2},a)+\Bigl{(}{\Delta^{2}\over
q^{2}}\Bigr{)}^{3}B_{3}(\lambda,\alpha,\xi,g^{2},a)+...+d_{3}(q^{2};\Delta^{2})+...\
,$ (73)
since the coefficients of the above-used expansions depend, in general, on the
same set of parameters: $\lambda,\alpha,\xi,g^{2},a$, etc. The invariant
function $d_{3}(q^{2};\Delta^{2})$ is dimensionless, and it is free of the
power-type IR singularities; otherwise it remains arbitrary. In the formal PT
$\Delta^{2}=0$ limit it survives, and is to be reduced to the sum of the first
type of terms in the expansion (A11). In other words, it is a sum of
$d^{MPT}(q^{2})$ and $d^{PT}(q^{2})$ up to third order, which have been
defined in section IV. The generalization to the next iterations is almost
obvious, and one finally obtains expansions (4.9)-(4.13) for the full gluon
propagator.
Concluding, let us underline that the splitting/shifting procedure does not
change the structure of the NL iteration solution at small $q^{2}$. It only
changes the coefficients at inverse powers of $q^{2}$ in the corresponding
expansion. In other words, it makes it possible to rearrange the terms in the
initial expansion (A11) in order to get it in the final form (A19). Also, in
the $q^{2}\rightarrow 0$ limit, it is legitimate to suppress the subtracted
gluon self-energy in comparison with the mass gap term in the initial Eq.
(A1). Nevertheless, as a result of the splitting/shifting procedure, which
becomes almost trivial in this case, one will obtain the same expansion (A19)
with only different residues, as just mentioned above. It is worth emphasizing
that residues remain completely arbitrary (undetermined) in any case.
## References
* (1) W. Marciano, H. Pagels, Phys. Rep. C 36 (1978) 137.
* (2) M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory
(AW, Advanced Book Program, 1995).
* (3) A. Jaffe, E. Witten, Yang-Mills Existence and Mass Gap,
$http://www.claymath.org/prize-problems/,\ http://www.arthurjaffe.com$ .
* (4) V. Gogokhia, Int. J. Theor. Phys. (2009) DOI: 10.1007/s10773-009-0101-3; arXiv:0806.0247 [hep-th, hep-ph].
* (5) J.M. Cornwall, Phys. Rev. D 26 (1982) 1453.
* (6) V.A. Rubakov, Classical Gauge Fields (Editorial YRSS, Moscaw, 1999).
* (7) Quark Matter 2005, Edited by T. Csorgo, G. David, P. Levai, G. Papp (ELSEVIER, Amsterdam-…-St. Louis, 2005).
* (8) M. Gyulassy, L. McLerran, arXiv:nucl-th/0405013.
* (9) M. Baker, Ch. Lee, Phys. Rev. D 15 (1977) 2201.
* (10) C. Itzykson, J.-B. Zuber, Quantum Field Theory (Mc Graw-Hill Book Company, 1984).
* (11) T. Muta, Foundations of QCD (Word Scientific, 1987).
* (12) I.M. Gel’fand, G.E. Shilov, Generalized Functions, Vol. I (Academic Press, New York, 1968).
* (13) G. ’t Hooft, M. Veltman, Nucl. Phys. B 44 (1972) 189.
* (14) V. Gogohia, Phys. Lett. B 584 (2004) 225.
* (15) V. Gogohia, Phys. Lett. B 618 (2005) 103.
* (16) V.G. Bornyakov, V.K. Mitrjushkin, M. Müller-Preussker, arXiv:0812.2761 [hep-lat].
* (17) I.L. Bogolubsky, E.-M. Igenfritz, M. Müller-Preussker, A. Sternbeck, arXiv:0901.0736 [hep-lat].
* (18) A. Cucchieri, T. Mendes, arXiv:0904.4033 [hep-lat].
* (19) A. Cucchieri, T. Mendes, Phys. Rev. Lett., 100 (2008) 241601, arXiv:0712.3517 [hep-lat].
* (20) A.C. Aguilar, D. Binosi, J. Papavassiliou, Phys. Rev. D 78 (2008) 025010, arXiv:0802.1870 [hep-ph].
* (21) R. Alkofer, L. von Smekal, Phys. Rep. 353 (2001) 281.
* (22) C.S. Fischer, A. Maas, J.H. Pawlowski, arXiv:0810.1987 [hep-ph].
* (23) S.P. Sorella, arXiv:0905.1010 [hep-th].
* (24) D. Dudal, J.A. Gracey, S.P. Sorella, N. Vandersickel, H. Verschelde, Phys. Rev. D 78 (2008) 065047, arXiv:0806.4348 [hep-th].
* (25) D. Zwanziger, arXiv:0904.2380 [hep-th].
* (26) K.-I. Kondo, arXiv:0907.3249 [hep-th].
|
arxiv-papers
| 2009-04-15T09:21:43 |
2024-09-04T02:49:01.888627
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. Gogokhia",
"submitter": "V. Gogokhia",
"url": "https://arxiv.org/abs/0904.2266"
}
|
0904.2422
|
# Higher derivatives estimate for the 3D Navier-Stokes equation
Alexis Vasseur Department of Mathematics, University of Texas
Abstract: In this article, a non linear family of spaces, based on the energy
dissipation, is introduced. This family bridges an energy space (containing
weak solutions to Navier-Stokes equation) to a critical space (invariant
through the canonical scaling of the Navier-Stokes equation). This family is
used to get uniform estimates on higher derivatives to solutions to the 3D
Navier-Stokes equations. Those estimates are uniform, up to the possible
blowing-up time. The proof uses blow-up techniques. Estimates can be obtained
by this means thanks to the galilean invariance of the transport part of the
equation.
Keywords: Navier-Stokes equation, fluid mechanics, blow-up techniques.
Mathematics Subject Classification: 76D05, 35Q30.
## 1 Introduction
In this paper, we investigate estimates of higher derivatives of solutions to
the incompressible Navier-Stokes equations in dimension 3, namely:
$\begin{array}[]{l}\displaystyle{\partial_{t}u+\mathrm{div}(u\otimes u)+\nabla
P-\Delta u=0\qquad t\in(0,\infty),\ x\in\mathbb{R}^{3},}\\\\[8.53581pt]
\displaystyle{\mathrm{div}u=0.}\end{array}$ (1)
The initial value problem is endowed with the conditions:
$\displaystyle u(0,\cdot)=u^{0}\in L^{2}(\mathbb{R}^{3}).$
The existence of weak solutions for this problem was proved long ago by Leray
[7] and Hopf [5]. For this, Leray introduces a notion of weak solution. He
shows that for any initial value with finite energy $u^{0}\in
L^{2}(\mathbb{R}^{3})$ there exists a function $u\in
L^{\infty}(0,\infty;L^{2}(\mathbb{R}^{3}))\cap
L^{2}(0,\infty;\dot{H}^{1}(\mathbb{R}^{3}))$ verifying (1) in the sense of
distribution. From that time on, much effort has been made to establish
results on the uniqueness and regularity of weak solutions. However those two
questions remain yet mostly open. Especially it is not known until now if such
a weak solution can develop singularities in finite time, even considering
smooth initial data. We present our main result on a laps of time $(0,T)$
where the solution is indeed smooth (with possible blow-ups both at $t=0$ and
$t=T$). We will carefully show, however, that the estimates do not depend on
the blow-up time $T$, but only on $\|u^{0}\|_{L^{2}}$ and $\inf(t,1)$. The aim
of this paper is to show the following theorem.
###### Theorem 1
For any $t_{0}>0$, any $\Omega$ bounded subset of
$(t_{0},\infty)\times\mathbb{R}^{3}$, any integer $n\geq 1$, any $\gamma>0$,
and any $p\geq 0$ such that
$\frac{4}{p}>n+1,$ (2)
there exists a constant $C$, such that the following property holds.
For any smooth solution $u$ of (1) on $(0,T)$ (with possible blow-up at 0 and
$T$), we have
$\|\nabla^{n}u\|_{L^{p}(\Omega\cap[(0,T)\times\mathbb{R}^{3}])}\leq
C\left(\|u^{0}\|^{2(1+\gamma)/p}_{L^{2}(\mathbb{R}^{3})}+1\right).$
Note that the constant $C$ does not depend on the solution $u$ nor on the
blowing-up time $T$.
Note that for $n\geq 3$ we consider $L^{p}$ spaces with $p<1$. Those spaces
are not complete. For this reason the result cannot be easily extend to
general weak solutions after the possible blow-up time. However, up to $d=2$,
the result can be proven in this context. For this reason, along the proof, we
will always consider suitable weak solutions, following [2]. That is,
solutions verifying in addition to (1) the generalized energy inequality in
the sense of distribution:
$\partial_{t}\frac{|u|^{2}}{2}+\mathrm{div}\left(u\frac{|u|^{2}}{2}\right)+\mathrm{div}(uP)+|\nabla
u|^{2}-\Delta\frac{|u|^{2}}{2}\leq 0\qquad t\in(0,\infty),\
x\in\mathbb{R}^{3}.$ (3)
Moreover, by interpolation, the result of Theorem 1 can be extended to the
whole real derivative coefficients, $1<d\leq 2$, for
$\|\Delta^{d/2}u\|_{L^{p}}$ with
$\frac{4}{p}>d+1.$
Our result can be seen as a kind of anti-Sobolev result. Indeed, as we will
see later, $\|\nabla u\|^{2}_{L^{2}}$ is used as a pivot quantity to control
higher derivatives on the solution. The result for $d=2$ was obtained in a
slightly better space, with completely different techniques by Lions [9]. He
shows that $\nabla^{2}u$ can be bounded in the Lorentz space $L^{4/3,\infty}$.
In a standard way, using the energy inequality and interpolation, we get
estimates on $\Delta^{d/2}u\in L^{p}((0,\infty)\times\mathbb{R}^{3})$ for
$\frac{5}{p}=d+\frac{3}{2},\qquad 0\leq d\leq 1.$ (4)
The Serrin-Prodi conditions (see [14],[4], [16]) ensure the regularity for
solutions such that $\Delta^{d/2}u\in L^{p}((0,\infty)\times\mathbb{R}^{3})$
for
$\frac{5}{p}=d+1,\qquad 0\leq d<\infty.$ (5)
Those two families of spaces are given by an affine relation on $d$ with
respect to $1/p$ with slope $5$. Notice that the family of spaces present in
Theorem 1 has a different slope. Imagine, that we were able to extend this
result along the same line with $d<1$. For $d=0$, we would obtain almost $u\in
L^{4}((0,\infty)\times\mathbb{R}^{3})$, which would imply that the energy
inequality (3) is an equality (see [17]). Notice also that the line of this
new family of spaces crosses the line of the critical spaces (5) at $d=-1$,
$1/p=0$. This point corresponds (at least formally) to the Tataru and Koch
result on regularity of solutions small in
$L^{\infty}(0,\infty;BMO^{-1}(\mathbb{R}^{3}))$ (see [6]). However, at this
time, due to the “anti-Sobolev” feature of the proof, obtaining results for
$d<1$ seems out of reach.
To see where lie the difficulties, let us focus on the result on the third
derivatives. Consider the gradient of the Navier-Stokes equations (1).
$\partial_{t}\nabla u-\Delta\nabla u=-\nabla u\cdot\nabla
u-\nabla^{2}P-(u\cdot\nabla)\nabla u.$
Note that the two first right-hand side terms lie in
$L^{1}((0,\infty)\times\mathbb{R}^{3})$ (for the pressure term, see [9]).
Parabolic regularity are not complete in $L^{1}$. This justify the fact that
we miss the limit case $L^{1}$. But, surprisingly, the worst term is the
transport one $(u\cdot\nabla)\nabla u$. To control it in $L^{1}$ using the
control on $D^{2}u$ in $L^{4/3,\infty}$ of Lions [9], we would need $u\in
L^{4,1}$, which is not known. To overcome this difficulty, we will consider
the solution in another frame, locally, by following the flow.
The idea of the proof comes from the result of partial regularity obtained by
Caffarelli, Kohn and Nirenberg [2]. This paper extended the analysis about the
possible singular points set, initialized by Scheffer in a series of paper
[10, 11, 12, 13]. The main remark in [2] is that the dissipation of entropy
$\mathcal{D}(u)=\int_{0}^{\infty}\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,dx\,dt$
(6)
has a scaling, through the standard invariance of the equation, which is far
more powerful that any other quantities from the energy scale (4). Let us be
more specific. The standard invariance of the equation gives that for any
$(t_{0},x_{0})\in\mathbb{R}^{+}\times\mathbb{R}^{3}$ and $\varepsilon>0$, if
$u$ is a suitable solution of the Navier-Stokes equations (1) (3), then
$u_{\varepsilon}(t,x)=\varepsilon u(t_{0}+\varepsilon^{2}t,x_{0}+\varepsilon
x)$ (7)
is also solution to (1) (3). The dissipation of energy of this quantity is
then given by
$\mathcal{D}(u_{\varepsilon})=\varepsilon^{-1}\mathcal{D}(u).$
This power of $\varepsilon$ made possible in [2] to show that the Hausdorff
dimension of the set of blow-up points is at most 1. This was a great
improvement of the result obtained by Scheffer who gives 5/3 as an upper bound
for the Hausdorff dimension of this set. We can notice that it is what we get
considering the quantity of the energy scale (4) with $d=0,p=10/3$:
$\mathcal{F}(u)=\int_{0}^{\infty}\int_{\mathbb{R}^{3}}|u|^{10/3}\,dx\,dt.$
Indeed:
$\mathcal{F}(u_{\varepsilon})=\varepsilon^{-5/3}\mathcal{F}(u).$
The idea of this paper is to give a quantitative version of the result of [2],
in the sense, of getting control of norms of the solution which have the same
nonlinear scaling that $\mathcal{D}$. Indeed, for any norm of the non linear
scaling (2), we have (in the limit case)
$\|\nabla^{n}u_{\varepsilon}\|^{p}_{L^{p}}=\varepsilon^{-1}\|\nabla^{n}u\|^{p}_{L^{p}}.$
The paper is organized as follows. In the next section, we give some
preliminaries and fix some notations. We introduce the local frame following
the flow in the third section. The fourth section is dedicated to a local
result providing a universal control of the higher derivatives of $u$ from a
local control of the dissipation of the energy $\|\nabla u\|^{2}_{L^{2}}$ and
a corresponding quantity on the pressure (see Proposition 10). Ideally, we
would like to consider a quantity on the pressure which has the same nonlinear
scaling as $\mathcal{D}(u)$. The corresponding quantity is
$\|\nabla^{2}P\|_{L^{1}}$. Unfortunately, we need a slightly better
integrability in time for the local study. This is the reason why we miss the
limit case $L^{p,\infty}$ with
$\frac{4}{p}=n+1.$
This is also the reason why we need to work with fractional Laplacian for the
pressure: $\|\Delta^{-s}\nabla^{2}P\|_{L^{p}}$ with $0<s<1/2$. In the last
section, we show how this local study leads to our main theorem.
## 2 Preliminaries and notations
Let us denote $Q_{r}=(-r^{2},0)\times B_{r}$ where $B_{r}=B(0,r)$, the ball in
$\mathbb{R}^{3}$ of radius $r$ and centered at 0.
For $F\in L^{p}(\mathbb{R}^{+}\times\mathbb{R}^{3})$, we define the Maximal
function in $x$ only by
$MF(t,x)=\sup_{r>0}\frac{1}{r^{3}}\int_{B_{r}}|F(t,x+y)|\,dy.$
We recall that for any $1<p<\infty$, there exists $C_{p}$ such that for any
$F\in L^{p}(\mathbb{R}^{+}\times\mathbb{R}^{3})$
$\|MF\|_{L^{p}(\mathbb{R}^{+}\times\mathbb{R}^{3})}\leq
C_{p}\|F\|_{L^{p}(\mathbb{R}^{+}\times\mathbb{R}^{3})}.$
Moreover, there exists a constant $C$ such that for any $F\in
L^{1}(\mathbb{R}^{+};\mathcal{H}(\mathbb{R}^{3}))$, (where $\mathcal{H}$
stands for the Hardy space), then
$\|MF\|_{L^{1}(\mathbb{R}^{+}\times\mathbb{R}^{3})}\leq
C\|F\|_{L^{1}(\mathbb{R}^{+};\mathcal{H}(\mathbb{R}^{3}))}.$
We begin with an interpolation lemma. It is a straightforward consequence of a
result in [1]. We state it here for further reference.
###### Lemma 2
For any function F such that $(-\Delta)^{d_{1}/2}F$ lies in
$L^{p_{1}}(0,\infty;L^{q_{1}}(\mathbb{R}^{3}))$ and $(-\Delta)^{d_{2}/2}F\in
L^{p_{2}}(0,\infty;L^{q_{2}}(\mathbb{R}^{3}))$ with
$d_{1},d_{2}\in\mathbb{R},\qquad 1\leq p_{1},p_{2}\leq\infty,\qquad
1<q_{1},q_{2}<\infty,$
we have $(-\Delta)^{d/2}F\in L^{p}(0,\infty;L^{q}(\mathbb{R}^{3}))$ with
$\displaystyle\|(-\Delta)^{d/2}F\|_{L^{p}(0,\infty;L^{q}(\mathbb{R}^{3}))}$
$\displaystyle\qquad\leq\|(-\Delta)^{d_{1}/2}F\|^{\theta}_{L^{p_{1}}(0,\infty;L^{q_{1}}(\mathbb{R}^{3}))}\|(-\Delta)^{d_{2}/2}F\|^{1-\theta}_{L^{p_{2}}(0,\infty;L^{q_{2}}(\mathbb{R}^{3}))},$
for any $d,p,q$ such that
$\displaystyle\frac{1}{q}=\frac{\theta}{q_{1}}+\frac{1-\theta}{q_{2}},$
$\displaystyle\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{2}},$
$\displaystyle d=\theta d_{1}+(1-\theta)d_{2},$
where $0<\theta<1$.
Proof. Exercise 31 page 168 in [1] shows that for any $0<t<\infty$, we have
$\|(-\Delta)^{d/2}F(t)\|_{L^{p}(\mathbb{R}^{3})}\leq\|(-\Delta)^{d_{1}/2}F(t)\|^{\theta}_{L^{p_{1}}(\mathbb{R}^{3})}\|(-\Delta)^{d_{2}/2}F(t)\|^{1-\theta}_{L^{p_{2}}(\mathbb{R}^{3})}.$
Interpolation in the time variable gives the result. In the second lemma we
show that we can control a local $L^{1}$ norm on a function $f$ by its mean
value and some local control on the maximal function of $(-\Delta)^{-s}\nabla
f$, $0<s<1/2$. This extends the fact that we can control the local $L^{1}$
norm by the mean value and a local $L^{p}$ norm of the gradient. But due to
the nonlocal feature of the fractional Laplacian, we need to consider the
maximal function to recapture all the information needed.
###### Lemma 3
Let $0<s<1/2$, $q\geq 1$, $p\geq 1$. For any $\phi\in
C^{\infty}(\mathbb{R}^{3})$, $\phi\geq 0$, compactly supported in $B_{1}$ with
$\int_{\mathbb{R}^{3}}\phi(x)\,dx=1$, there exists $C>0$ such that, for any
function $f\in L^{q}(\mathbb{R}^{3})$ with $(-\Delta)^{-s}\nabla f\in
L^{p}(\mathbb{R}^{3})$ and $|\int f\phi\,dx|$ bounded, we have $f\in
L^{1}(B_{1})$ and
$\|f\|_{L^{1}(B_{1})}\leq
C\left(\left|\int_{\mathbb{R}^{3}}f(x)\phi(x)\,dx\right|+\|M((-\Delta)^{-s}\nabla
f)\|_{L^{p}(B_{1})}\right).$
Proof. Let us denote $g=(-\Delta)^{-s}\nabla f$. Since $f\in
L^{q}(\mathbb{R}^{3})$, we have
$f=-(-\Delta)^{s-1}\mathrm{div}g.$
So, for any $x\in B_{1}$
$f(x)=C_{s}\int_{\mathbb{R}^{3}}\frac{g(y)}{|x-y|^{2(1+s)}}\cdot\frac{(x-y)}{|x-y|}\,dy,$
and
$\displaystyle f(x)-\int_{\mathbb{R}^{3}}\phi(z)\,f(z)\,dz$
$\displaystyle\qquad=C_{s}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\phi(z)g(y)\left(\frac{(x-y)/|x-y|}{|x-y|^{2(1+s)}}-\frac{(z-y)/|z-y|}{|y-z|^{2(1+s)}}\right)\,dy\,dz.$
Note that, for $k\geq 2$, $y\in B_{2^{k}}\setminus B_{2^{k-1}}$, $x\in B_{1}$,
$z\in B_{1}$, we have
$\left|\frac{(x-y)/|x-y|}{|x-y|^{2(1+s)}}-\frac{(z-y)/|z-y|}{|y-z|^{2(1+s)}}\right|\leq\frac{C}{2^{k(3+2s)}}.$
Moreover
$\displaystyle\int_{B_{1}}\int_{B_{1}}\int_{B_{2}}\phi(z)|g(y)|\left|\frac{(x-y)/|x-y|}{|x-y|^{2(1+s)}}-\frac{(z-y)/|z-y|}{|y-z|^{2(1+s)}}\right|\,dy\,dz\,dx$
$\displaystyle\qquad\leq\int_{B_{3}}\int_{B_{1}}\int_{B_{2}}\frac{\phi(z)|g(y)|}{|x|^{2(1+s)}}\,dy\,dz\,dx+\int_{B_{1}}\int_{B_{3}}\int_{B_{2}}\frac{\sup|\phi||g(y)|}{|z|^{2(1+s)}}\,dy\,dz\,dx$
$\displaystyle\qquad\leq 2C_{s}\|g\|_{L^{1}(B_{1})}\leq
2C_{s}\|Mg\|_{L^{1}(B_{1})},$
since $2(1+s)<3$. Hence
$\displaystyle\qquad\qquad\left\|f-\int\phi(z)f(z)\,dz\right\|_{L^{1}(B_{1})}$
$\displaystyle\leq\int_{B_{1}}\int_{B_{1}}\int_{B_{2}}\phi(z)|g(y)|\left|\frac{(x-y)/|x-y|}{|x-y|^{2(1+s)}}-\frac{(z-y)/|z-y|}{|y-z|^{2(1+s)}}\right|\,dy\,dz\,dx$
$\displaystyle\qquad+\sum_{k=2}^{\infty}\int_{B_{1}}\int_{B_{1}}\int_{(B_{2^{k}}\setminus
B_{2^{k-1}})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\phi(z)|g(y)|\left|\frac{(x-y)/|x-y|}{|x-y|^{2(1+s)}}-\frac{(z-y)/|z-y|}{|y-z|^{2(1+s)}}\right|$
$\displaystyle\leq
2C_{s}\|Mg\|_{L^{1}(B_{1})}+C\sum_{k=2}^{\infty}\int_{B_{2^{k}}}\frac{|g(y)|}{2^{k(3+2s)}}\,dy$
$\displaystyle\leq
2C_{s}\|Mg\|_{L^{1}(B_{1})}+8C\sum_{k=2}^{\infty}2^{-2sk}\frac{1}{|B_{2^{k+1}}|}\int_{B_{1}}\int_{B_{2^{k+1}}}|g(y+u)|\,dy\,du$
$\displaystyle\leq
2C_{s}\|Mg\|_{L^{1}(B_{1})}+C\|Mg\|_{L^{1}(B_{1})}\sum_{k=2}^{\infty}[2^{-2s}]^{k}$
$\displaystyle\leq C_{s}\|Mg\|_{L^{1}(B_{1})},$
whenever $0<s<1/2$.
We give now very standard results of parabolic regularity. There are not even
optimal, but enough for our study.
###### Lemma 4
For any $1<p<\infty$, $t_{0}>0$, there exists a constant $C$ such that the
following is true. Let $f,g\in L^{p}((-t_{0},0)\times\mathbb{R}^{3})$ be
compactly supported in $B_{1}$. Then there exists a unique $u\in
L^{p}(-t_{0},0;W^{1,p}(\mathbb{R}^{3}))$ solution to
$\displaystyle\partial_{t}u-\Delta u=g+\mathrm{div}f,\qquad-t_{0}\leq t\leq
0,\ \ x\in\mathbb{R}^{3},$ $\displaystyle u(-t_{0},x)=0,\qquad
x\in\mathbb{R}^{3}.$
Moreover,
$\|u\|_{L^{p}(-t_{0},0;W^{1,p}(B_{1}))}\leq
C(\|f\|_{L^{p}((-t_{0},0)\times\mathbb{R}^{3})}+\|g\|_{L^{p}((-t_{0},0)\times\mathbb{R}^{3})}).$
(8)
If $g\in L^{1}(-t_{0},0;L^{\infty}(\mathbb{R}^{3}))$ and $f\in
L^{1}(-t_{0},0;W^{1,\infty}(\mathbb{R}^{3}))$, then
$\|u\|_{L^{\infty}(-t_{0},0)\times\mathbb{R}^{3})}\leq
C(\|g\|_{L^{1}(-t_{0},0;L^{\infty}(\mathbb{R}^{3}))}+\|f\|_{L^{1}(-t_{0},0;W^{1,\infty}(\mathbb{R}^{3}))}).$
Proof. We get the solution using the Green function:
$u(t,x)=\int_{-t_{0}}^{t}\frac{1}{4\pi(t-s)^{3/2}}\int_{\mathbb{R}^{3}}e^{-\frac{|x-y|^{2}}{4(t-s)}}(g(s,y)+\mathrm{div}f(s,y))\,dy\,ds.$
From this formulation, using that $z^{n}e^{-z^{2}}$ are bounded functions, we
find that
$|u(t,x)|\leq C\frac{\|f\|_{L^{1}((-t_{0},0)\times
B_{1})}+\|g\|_{L^{1}((-t_{0},0)\times B_{1})}}{|x|^{3}},\qquad\mathrm{for}\
|x|>2,-t_{0}\leq t<0.$ (9)
Standard Solonnikov’s parabolic regularization result gives (8) (see for
instance [15]). Finally, if $g\in L^{1}(-t_{0},0;L^{\infty}(\mathbb{R}^{3}))$
and $f\in L^{1}(-t_{0},0;W^{1,\infty}(\mathbb{R}^{3}))$, then the function
$v(t,x)=\int_{0}^{t}(\|g(s)\|_{L^{\infty}}+\|\mathrm{div}f(s)\|_{L^{\infty}})\,ds$
is a supersolution thanks to (9). The global bound follows.
The last lemma of this section is a standard decomposition of the pressure
term as a close range part and a long range part.
###### Lemma 5
Let $\overline{B}$ and $\underline{B}$ be two balls such that
$\overline{B}\subset\subset\underline{B}.$
Then for any $1<p<\infty$, there exists a constant $C>0$ and a family of
constants $\\{C_{d,q}\ \setminus\ d,q\ \ \mathrm{integers}\\}$ (depending only
on $p$, $\underline{B}$ and $\overline{B}$) such that for any $R\in
L^{1}(\underline{B})$ and $A\in[L^{p}(\underline{B})]^{N\times N}$ symmetric
matrix, verifying
$-\Delta R=\mathrm{div}\mathrm{div}A,\qquad\mathrm{in}\ \ \underline{B},$
we have a decomposition
$R=R_{1}+R_{2},$
with, for any integer $q\geq 0$, $d\geq 0$:
$\displaystyle\|R_{1}\|_{L^{p}(\overline{B})}\leq
C\|A\|_{L^{p}(\underline{B})},$
$\displaystyle\|\nabla^{d}R_{2}\|_{L^{\infty}(\overline{B})}\leq
C_{d,q}\left(\|A\|_{L^{1}(\underline{B})}+\|R\|_{W^{-q,1}(\underline{B})}\right).$
Moreover, if $A$ is Lipschitzian, then we can choose $R_{1}$ such that
$\|R_{1}\|_{L^{\infty}(\overline{B})}\leq C\left(\|\nabla
A\|_{L^{\infty}(\underline{B})}+\|A\|_{L^{\infty}(\underline{B})}\right).$
Proof.
Let $B^{*}$ be a a ball such that
$\overline{B}\subset\subset B^{*}\subset\subset\underline{B},$
with a distance between $\overline{B}$ and ${B^{*}}^{c}$ bigger that $D/2$,
where $D$ is the distance between $\overline{B}$ and $\underline{B}^{c}$.
Consider a smooth nonnegative cut-off function $\psi$, $0\leq\psi\leq 1$ such
that
$\displaystyle\psi(x)$ $\displaystyle=$ $\displaystyle 1\qquad\mathrm{in}\
B^{*},$ $\displaystyle=$ $\displaystyle 0\qquad\mathrm{in}\
\underline{B}^{c}.$
Then the function $\psi R$ (defined in $\mathbb{R}^{3}$) is solution in
$\mathbb{R}^{3}$ to
$\displaystyle-\Delta(\psi R)$ $\displaystyle=$
$\displaystyle\mathrm{div}\mathrm{div}(\psi A)$
$\displaystyle+R\Delta\psi+A:\nabla^{2}\psi$
$\displaystyle-2\mathrm{div}\\{\nabla\psi\cdot A+R\nabla\psi\\}.$
We denote
$\displaystyle R_{1}=(-\Delta)^{-1}\mathrm{div}\mathrm{div}(\psi A),$
$\displaystyle
R_{2}=(-\Delta)^{-1}\left(R\Delta\psi+A:\nabla^{2}\psi-2\mathrm{div}\\{\nabla\psi\cdot
A+R\nabla\psi\\}\right).$
We have, on $\overline{B}$, $R=R_{1}+R_{2}$. The operator
$(-\Delta)^{-1}\mathrm{div}\mathrm{div}$ is a Riesz operator, so there exists
a constant (depending only on $p$ and $\psi$) such that
$\displaystyle\|R_{1}\|_{L^{p}(\mathbb{R}^{3})}\leq C\|\psi
A\|_{L^{p}(\mathbb{R}^{3})}\leq C\|A\|_{L^{p}(\underline{B})},$
$\displaystyle\|R_{1}\|_{C^{\alpha}(\mathbb{R}^{3})}\leq C\|\psi
A\|_{C^{\alpha}(\mathbb{R}^{3})}\leq C\left(\|\nabla
A\|_{L^{\infty}(\underline{B})}+\|A\|_{L^{\infty}(\underline{B})}\right).$
Using the fact that $\nabla\psi$ and $\nabla^{2}\psi$ vanishes on
$B^{*}\cup\underline{B}^{c}$, we have for any $x\in\overline{B}$:
$\displaystyle|\nabla^{d}R_{2}(x)|=\left|\int_{\mathbb{R}^{3}}\nabla^{d}\left(\frac{1}{|x-y|}\right)\left(R\Delta\psi+A:\nabla^{2}\psi\right)(y)\,dy\right.$
$\displaystyle\qquad\qquad\qquad\left.+2\int_{\mathbb{R}^{3}}\nabla^{d+1}\left(\frac{1}{|x-y|}\right)\\{\nabla\psi\cdot
A+R\nabla\psi\\}(y)\,dy\right|$
$\displaystyle\qquad\leq\|\nabla^{2}\psi\|_{L^{\infty}}\|A\|_{L^{1}(\underline{B})}\sup_{|x-y|\geq
D/2}\left|\nabla^{d}\left(\frac{1}{|x-y|}\right)\right|$
$\displaystyle\qquad\qquad+2\|\nabla\psi\|_{L^{\infty}}\|A\|_{L^{1}(\underline{B})}\sup_{|x-y|\geq
D/2}\left|\nabla^{d+1}\left(\frac{1}{|x-y|}\right)\right|$
$\displaystyle\qquad\qquad+\|R\|_{W^{-q,1}(\underline{B})}\sup_{|x-y|\geq
D/2}\left|\nabla^{q}\left[\nabla^{d}\left(\frac{1}{|x-y|}\right)\Delta\psi\right]\right|$
$\displaystyle\qquad\qquad+2\|R\|_{W^{-q,1}(\underline{B})}\sup_{|x-y|\geq
D/2}\left|\nabla^{q}\left[\nabla^{d+1}\left(\frac{1}{|x-y|}\right)\nabla\psi\right]\right|$
$\displaystyle\qquad\leq
C_{d}\left[\left(\frac{2}{D}\right)^{d+2}+\left(\frac{2}{D}\right)^{d+1}\right]\|A\|_{L^{1}(\underline{B})}$
$\displaystyle\qquad\qquad+C_{d,q}\left[\left(\frac{2}{D}\right)^{d+1}+\left(\frac{2}{D}\right)^{q+d+2}\right]\|R\|_{W^{-q,1}(\underline{B})}.$
## 3 Blow-up method along the trajectories
Our result relies on a local study, which was the keystone of the partial
regularity result of [2]. (see [8] for an other proof). We use, here, the
version of [18]. This version is better for our purpose because it requires a
bound on the pressure only in $L^{p}$ in time for any $p>1$.
###### Proposition 6
[18] For any $p>1$, there exists $\eta>0$, such that the following property
holds. For any $u$, suitable weak solution to the Navier-Stokes equation (1),
(3), in $Q_{1}$, such that
$\displaystyle\sup_{-1<t<0}\\!\\!\left(\int_{B_{1}}\\!\\!|u(t,x)|^{2}\,dx\right)+\\!\int_{Q_{1}}\\!\\!|\nabla
u|^{2}\,dx\,dt\\!+\\!\int_{-1}^{0}\\!\\!\left(\int_{B_{1}}\\!\\!|P|\,dx\right)^{p}\\!dt\leq\eta,$
(10)
we have
$\sup_{(t,x)\in Q_{1/2}}|u(t,x)|\leq 1.$
As explained in the introduction, the proof of Theorem 1 relies on this local
control. From there we can get control on higher derivatives of $u$. We first
show the following lemma. It introduces the pivot quantity. Note that the
ideal pivot quantity would be $\|\nabla
u\|^{2}_{L^{2}(L^{2})}+\|\nabla^{2}P\|_{L^{1}(L^{1})}$. This is because this
quantity scales as $1/\varepsilon$ through the canonical scaling. However, to
use Proposition 6 locally, we need a better integrability in time on the
pressure. For this reason, we add the quantity on the pressure involving the
fractional Laplacian. We get a better integrability in time on the pressure,
at the cost of a slightly worst rate of change in $\varepsilon$ through the
canonical scaling. Finally, due to the nonlocal character of the fractional
Laplacian, the maximal function is used in order to recapture all the local
information needed (see Lemma 3).
###### Lemma 7
For any $0<\delta<1$, there exists $\gamma>0$ and a constant $C>0$ such that
for any $u$ solution to (1) (3), with $u^{0}\in L^{2}(\mathbb{R}^{3})$, we
have
$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\left(|M((-\Delta)^{-\delta/2}\nabla^{2}P)|^{1+\gamma}+|\nabla^{2}P|+|\nabla
u|^{2}\right)\,dx\,dt$ $\displaystyle\qquad\qquad\leq
C\left(\|u^{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|u^{0}\|^{2(1+\gamma)}_{L^{2}(\mathbb{R}^{3})}\right).$
Moreover, $\gamma$ converges to 0 when $\delta$ converges to 0.
Proof. Integrating in $x$ the energy equation (3) gives that
$\int_{0}^{\infty}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\,dx\,dt\leq\|u^{0}\|^{2}_{L^{2}(\mathbb{R}^{3})},$ (11)
together with
$\|u\|^{2}_{L^{\infty}(0,\infty;L^{2}(\mathbb{R}^{3}))}\leq\|u^{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}.$
By Sobolev imbedding and interpolation, this gives in particular that
$\|u\|^{2}_{L^{4}(0,\infty;L^{3}(\mathbb{R}^{3}))}\leq
C\|u^{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}.$ (12)
For the pressure, we have $\nabla^{2}P\in L^{1}(\mathcal{H})$ (see Lions [9]).
Indeed,
$\displaystyle\nabla^{2}P=(\nabla^{2}\Delta^{-1})\sum_{ij}\partial_{i}u_{j}\partial_{j}u_{i}$
$\displaystyle\qquad=(\nabla^{2}\Delta^{-1})\sum_{i}(\partial_{i}u)\cdot\nabla
u_{i}.$
For any $i$, we have $\mathrm{rot}(\nabla u_{i})=0$ and $\mathrm{div}\
\partial_{i}u=0$. Hence, from the div-rot lemma (see Coifman, Lions, Meyer and
Semmes [3]), we have
$\|\sum_{i}\partial_{i}u\cdot\nabla u_{i}\|_{L^{1}(\mathcal{H})}\leq\|\nabla
u\|^{2}_{L^{2}}.$
But $\nabla^{2}\Delta^{-1}$ is a Riesz operator (in $x$ only) which is bounded
from $\mathcal{H}$ to $\mathcal{H}$. Hence:
$\|\nabla^{2}P\|_{L^{1}(\mathbb{R}^{+}\times\mathbb{R}^{3})}\leq
C\|\nabla^{2}P\|_{L^{1}(\mathbb{R}^{+};\mathcal{H}(\mathbb{R}^{3}))}\leq
C\|\nabla u\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R}^{3})}.$ (13)
By Sobolev imbedding, for any $0<s<1$, we have
$\|(-\Delta)^{-s/2}\nabla^{2}P\|_{L^{1}(0,\infty;L^{p}(\mathbb{R}^{3}))}\leq
C\|u^{0}\|^{2}_{L^{2}}$ (14)
for
$\frac{1}{p}=1-\frac{s}{3}.$
we have also
$(-\Delta)^{-1/2}\nabla^{2}P=\sum_{ij}[(-\Delta)^{-3/2}\nabla^{2}\partial_{i}](\partial_{j}u_{i}u_{j}).$
The operators $(-\Delta)^{-3/2}\nabla^{2}\partial_{i}$ are Riesz operators so,
together with (11) (12), we have
$\|(-\Delta)^{-1/2}\nabla^{2}P\|_{L^{4/3}(0,\infty;L^{6/5}(\mathbb{R}^{3}))}\leq
C\|u^{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}.$ (15)
By interpolation with (14), using Lemma 2 with $\theta=1/(1+4s)$, we find
$\|M[(-\Delta)^{-\delta/2}\nabla^{2}P]\|_{L^{1+\gamma}((0,\infty)\times\mathbb{R}^{3})}\leq
C\|u^{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}$
with
$\delta=\frac{5s}{1+4s},\qquad\qquad\gamma=\frac{s}{1+3s}.$
Note that $\gamma$ converges to 0 when $\delta$ goes to 0. This, together with
(13) and (11), gives the result.
Let us fix from now on a smooth cut-off function $0\leq\phi\leq 1$ compactly
supported in $B_{1}$ and such that
$\int_{\mathbb{R}^{3}}\phi(x)\,dx=1.$ (16)
For any $\varepsilon>0$, we define
$u_{\varepsilon}(t,x)=\int_{\mathbb{R}^{3}}\phi(y)u(t,x+\varepsilon y)\,dy.$
(17)
Note that $u_{\varepsilon}\in L^{\infty}(0,\infty;C^{\infty}(\mathbb{R}^{3}))$
and $\mathrm{div}u_{\varepsilon}=0$. We define the flow:
$\begin{array}[]{l}\displaystyle{\frac{\partial X}{\partial
s}=u_{\varepsilon}(s,X(s,t,x))}\\\\[8.5359pt]
\displaystyle{X(t,t,x)=x.}\end{array}$ (18)
Consider, for any $0<\delta<1$ and $\eta^{*}>0$:
$\Omega^{\delta}_{\varepsilon}=\left\\{(t,x)\in(4\varepsilon^{2},\infty)\times\mathbb{R}^{3}\
|\
\frac{1}{\varepsilon}\int_{t-4\varepsilon^{2}}^{t}\int_{B_{2\varepsilon}}\\!\\!\\!\\!F^{\delta}(s,X(s,t,x)+y)\,ds\,dy\leq\eta^{*}\varepsilon^{\delta}\right\\},$
where
$F^{\delta}(t,x)=|M((-\Delta)^{-\delta/2}\nabla^{2}P)|^{1+\gamma}+|\nabla
u|^{2}+|\nabla^{2}P|,$
and $\gamma$ is defined in Lemma 7. We then have the following lemma.
###### Lemma 8
There exists a constant $C$ such that for any $0<\varepsilon<1$, $0<\delta<1$,
and $\eta^{*}>0$ we have
$|[\Omega^{\delta}_{\varepsilon}]^{c}|\leq
C\left(\frac{\|u^{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|u^{0}\|^{2(1+\gamma)}_{L^{2}(\mathbb{R}^{3})}}{\eta^{*}}\right)\varepsilon^{4-\delta}.$
Proof. Define for $t>4\varepsilon^{2}$
$F^{\delta}_{\varepsilon}(t,x)=\frac{1}{(2\varepsilon)^{5}}\int_{t-4\varepsilon^{2}}^{t}\int_{B_{2\varepsilon}}F^{\delta}(s,X(s,t,x)+y)\,ds\,dy.$
(19)
We have
$\displaystyle\qquad\qquad\int_{4\varepsilon^{2}}^{\infty}\int_{\mathbb{R}^{3}}F^{\delta}_{\varepsilon}(t,x)\,dx\,dt$
$\displaystyle=\int_{4\varepsilon^{2}}^{\infty}\int_{\mathbb{R}^{3}}\frac{1}{(2\varepsilon)^{5}}\int_{-4\varepsilon^{2}}^{0}\int_{B_{2\varepsilon}}F^{\delta}(t+s,X(t+s,t,x)+y)\,ds\,dy\,dx\,dt$
$\displaystyle=\frac{1}{(2\varepsilon)^{5}}\int_{B_{2\varepsilon}}\int_{-4\varepsilon^{2}}^{0}\int_{4\varepsilon^{2}}^{\infty}\int_{\mathbb{R}^{3}}F^{\delta}(t+s,X(t+s,t,x)+y)\,dx\,dt\,ds\,dy$
$\displaystyle=\frac{1}{(2\varepsilon)^{5}}\int_{B_{2\varepsilon}}\int_{-4\varepsilon^{2}}^{0}\int_{4\varepsilon^{2}}^{\infty}\int_{\mathbb{R}^{3}}F^{\delta}(t+s,z+y)\,dz\,dt\,ds\,dy$
$\displaystyle\leq\left(\frac{1}{(2\varepsilon)^{5}}\int_{B_{2\varepsilon}}\int_{-4\varepsilon^{2}}^{0}\,ds\,dy\right)\int_{0}^{\infty}\int_{\mathbb{R}^{3}}F^{\delta}(\underline{t},\underline{z})\,d\underline{z}\,d\underline{t}$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\left(|M((-\Delta)^{-\delta/2}\nabla^{2}P)|^{1+\gamma}+|\nabla
u|^{2}+|\nabla^{2}P|\right)\,dx\,dt.$
In the second equality, we have used Fubini, in the third we have used the
fact that $X$ is an incompressible flow. In the fourth equality we did the
change of variable in $(t,z)$
$\underline{t}=t+s\qquad\underline{z}=y+z.$
We then find, thanks to Tchebychev inequality,
$\left|\left\\{F^{\delta}_{\varepsilon}(t,x)\geq\frac{\eta^{*}\varepsilon^{\delta}}{2(2\varepsilon)^{4}}\right\\}\right|\leq
2^{5}\frac{\int_{0}^{\infty}\int_{\mathbb{R}^{3}}F^{\delta}_{\varepsilon}(t,x)\,dx\,dt}{\eta^{*}}\varepsilon^{4-\delta}.$
We conclude thanks to Lemma 7.
We fix $\delta>0$. For any fixed $(t,x)\in\Omega^{\delta}_{\varepsilon}$ with
$t\geq 4\varepsilon^{2}$, we define $v_{\varepsilon},P_{\varepsilon}$,
(depending on this fixed point $(t,x)$) as functions of two local new
variables $(s,y)\in Q_{2}$:
$\displaystyle v_{\varepsilon}(s,y)=\varepsilon
u(t+\varepsilon^{2}s,X(t+\varepsilon^{2}s,t,x)+\varepsilon y)$
$\displaystyle\qquad\qquad\qquad-\varepsilon
u_{\varepsilon}(t+\varepsilon^{2}s,X(t+\varepsilon^{2}s,t,x)),$ (20)
$\displaystyle
P_{\varepsilon}(s,y)=\varepsilon^{2}P(t+\varepsilon^{2}s,X(t+\varepsilon^{2}s,t,x)+\varepsilon
y)$ $\displaystyle\qquad\qquad\qquad+\varepsilon
y\partial_{s}[u_{\varepsilon}(t+\varepsilon^{2}s,X(t+\varepsilon^{2}s,t,x))].$
(21)
We have the following proposition.
###### Proposition 9
The function $(v_{\varepsilon},P_{\varepsilon})$ is solution to (1) (3) for
$(s,y)\in(-4,0)\times\mathbb{R}^{3}$. It verifies:
$\displaystyle\int_{\mathbb{R}^{3}}\phi(y)v_{\varepsilon}(s,y)\,dy=0,\qquad
s\geq-4,$ (22) $\displaystyle\int_{-4}^{0}\int_{B_{2}}|\nabla
v_{\varepsilon}|^{2}\,dy\,ds\leq\eta^{*},$ (23)
$\displaystyle\int_{-4}^{0}\int_{B_{2}}|\nabla^{2}P_{\varepsilon}|\,dy\,ds\leq\eta^{*},$
(24)
$\displaystyle\int_{-4}^{0}\int_{B_{2}}|M[(-\Delta)^{-\delta/2}\nabla^{2}P_{\varepsilon}]|^{1+\gamma}\,dy\,ds\leq\eta^{*}.$
(25)
Proof. The fact that $(v_{\varepsilon},P_{\varepsilon})$ is solution to (1)
(3) and verifies (22) comes from its definition (20), (21), (16) and (17). We
have
$\begin{array}[]{l}\displaystyle{\qquad\qquad\int_{Q_{2}}(|\nabla
v_{\varepsilon}|^{2}+|\nabla^{2}P_{\varepsilon}|)\,dy\,ds+\int_{Q_{2}}|M[(-\Delta)^{-\delta/2}\nabla^{2}P_{\varepsilon}]|^{1+\gamma}\,dy\,ds}\\\\[8.5359pt]
\displaystyle{=\int_{Q_{2}}\left(\varepsilon^{4}(|\nabla
u|^{2}+|\nabla^{2}P|)+\varepsilon^{(4-\delta)(1+\gamma)}|M[(-\Delta)^{-\delta/2}\nabla^{2}P]|^{1+\gamma}\right)}\\\\[8.5359pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\displaystyle{(t+\varepsilon^{2}s,X(t+\varepsilon^{2}s,t,x)+\varepsilon
y)\,dy\,ds}\\\\[8.5359pt]
\displaystyle{\leq\frac{1}{\varepsilon^{1+\delta}}\int_{t-4\varepsilon^{2}}^{t}\int_{B_{2\varepsilon}}(|\nabla
u|^{2}+|\nabla^{2}P|+M[(-\Delta)^{-\delta/2}\nabla^{2}P]^{1+\gamma})}\\\\[8.5359pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\displaystyle{(s,X(s,t,x)+y)\,ds\,dy}\\\\[8.5359pt]
\leq\eta^{*}.\end{array}$ (26)
In the first equality, we used the definition of $v_{\varepsilon}$ and
$P_{\varepsilon}$, in the second, we used the change of variable
$(t+\varepsilon^{2}s,\varepsilon y)\to(s,y)$ (together with the fact that
$\delta<4$ and $\gamma\geq 0$), and the last inequality comes from the fact
that $(s,y)$ lies in $\Omega^{\delta}_{\varepsilon}$. Our aim is to apply
proposition 6 to $v_{\varepsilon}$. It will be a consequence of the following
section.
## 4 Local study
This section is dedicated to the following Proposition.
###### Proposition 10
For any $\gamma>0$ and any $0<\delta<1$, there exists a constant
$\overline{\eta}<1$, and a sequence of constants $\\{C_{n}\\}$ such that for
any solution $(u,P)$ of (1) (3) in $Q_{2}$ verifying
$\displaystyle\int_{\mathbb{R}^{3}}\phi(y)u(t,x)\,dx=0,\qquad t\geq-4,$ (27)
$\displaystyle\int_{-4}^{0}\int_{B_{2}}|\nabla
u|^{2}\,dx\,dt\leq\overline{\eta},$ (28)
$\displaystyle\int_{-4}^{0}\int_{B_{2}}|\nabla^{2}P|\,dx\,dt\leq\overline{\eta},$
(29)
$\displaystyle\int_{-4}^{0}\int_{B_{2}}|M[(-\Delta)^{-\delta/2}\nabla^{2}P]|^{1+\gamma}\,dx\,dt\leq\overline{\eta},$
(30)
the velocity $u$ is infinitely differentiable in $x$ at $(0,0)$ and
$|\nabla^{n}u(0,0)|\leq C_{n}.$
Proof. We want to apply Proposition 6. Then, by a bootstrapping argument we
will get uniform controls on higher derivatives. For this, we first need a
control of $u$ in $L^{\infty}(L^{2})$ and a control on $P$ in
$L^{\gamma+1}(L^{1})$. The equation is on $\nabla P$ (not the pressure
itself). Therefore, changing $P$ by $P-\int_{B_{2}}\phi P\,dx$ we can assume
without loss of generality that
$\int_{\mathbb{R}^{3}}\phi(x)P(t,x)\,dx=0,\qquad-4<t<0.$
To get a control in $L^{1+\gamma}(L^{1})$ on the pressure it is then enough to
control $\nabla P$.
Step 1: Control on $u$ in $L^{\infty}(L^{3/2})$ in $Q_{3/2}$. Thanks to
Hypothesis (27), there exists a constant $C$, depending only on $\phi$, such
that for any $-4<t<0$
$\|u(t)\|_{L^{6}(B_{2})}\leq C\|\nabla u(t)\|_{L^{2}(B_{2})}.$ (31)
So
$\|(u\cdot\nabla)u\|_{L^{1}(-4,0;L^{3/2}(B_{2}))}\leq C\|\nabla
u\|^{2}_{L^{2}(Q_{2})}\leq C\overline{\eta}.$
We need the same control on $\nabla P$. First, multiplying (1) by $\phi(x)$,
integrating in $x$, and using Hypothesis (27), we find for any $-4<t<0$
$\int\phi(x)(u\cdot\nabla)u\,dx+\int\phi(x)\nabla P\,dx-\int\Delta\phi
u\,dx=0.$ (32)
So
$\left\|\int\phi(x)\nabla P\,dx\right\|_{L^{1}(-4,0)}\leq C\left(\|\nabla
u\|^{2}_{L^{2}(Q_{2})}+\|u\|_{L^{2}(-4,0;L^{6}(B_{2}))}\right)\leq
C\sqrt{\overline{\eta}}.$
But, as for $u$,
$\left\|\nabla P-\int\phi\nabla P\,dx\right\|_{L^{1}(-4,0;L^{3/2}(B_{2}))}\leq
C\|\nabla^{2}P\|_{L^{1}(Q_{2})}.$
So, finally
$\||(u\cdot\nabla)u|+|\nabla P|\|_{L^{1}(-4,0;L^{3/2}(B_{2}))}\leq
C\sqrt{\overline{\eta}}.$ (33)
Note that
$\displaystyle\frac{3}{2}\frac{u}{|u|^{1/2}}\partial_{t}u=\frac{3}{2}\frac{1}{|u|^{1/2}}\partial_{t}\frac{|u|^{2}}{2}$
$\displaystyle\qquad\qquad=\frac{3}{2}|u|^{1/2}\partial_{t}|u|=\partial_{t}|u|^{3/2},$
$\displaystyle\frac{3}{2}\frac{u}{|u|^{1/2}}\Delta
u=\frac{3}{2}\mathrm{div}\left(\frac{u}{|u|^{1/2}}\nabla
u\right)-\frac{3}{2}\frac{|\nabla
u|^{2}}{|u|^{1/2}}+\frac{3}{4}\frac{|\nabla|u||^{2}}{|u|^{1/2}}$
$\displaystyle\qquad\qquad\leq\Delta|u|^{3/2},$
since $|\nabla u|\geq|\nabla|u||$.
We consider $\psi_{1}\in C^{\infty}(\mathbb{R}^{4})$ a nonnegative function
compactly supported in $Q_{2}$ with $\psi_{1}=1$ in $Q_{3/2}$ and
$|\nabla_{t,x}\psi_{1}|+|\nabla_{t,x}^{2}\psi_{1}|\leq C.$
Multiplying (1) by $(3/2)\psi_{1}(t,x)u/|u|^{1/2}$ and integrating in $x$
gives
$\displaystyle\qquad\frac{d}{dt}\int\psi_{1}(t,x)|u|^{3/2}\,dx$
$\displaystyle\leq\int(|\partial_{t}\psi_{1}|+|\Delta\psi_{1}|)|u|^{3/2}\,dx$
$\displaystyle\qquad\qquad+\frac{3}{2}\|\psi_{1}^{1/3}|u|^{1/2}\|_{L^{3}(\mathbb{R}^{3})}\|\psi_{1}^{2/3}((u\cdot\nabla)u+\nabla
P)\|_{L^{3/2}(B_{2})}$
$\displaystyle\leq\int(|\partial_{t}\psi_{1}|+|\Delta\psi_{1}|)|u|^{3/2}\,dx$
$\displaystyle\qquad\qquad+\frac{3}{2}\left(\int\psi_{1}(t,x)|u|^{3/2}\,dx\right)^{1/3}\|((u\cdot\nabla)u+\nabla
P)\|_{L^{3/2}(B_{2})}$
$\displaystyle\leq\alpha(t)\left(1+\int\psi_{1}(t,x)|u|^{3/2}\,dx\right),$
with
$\alpha(t)=\int(|\partial_{t}\psi_{1}|+|\Delta\psi_{1}|)|u|^{3/2}\,dx+\frac{3}{2}\|((u\cdot\nabla)u+\nabla
P)\|_{L^{3/2}(B_{2})}.$
Thanks to (31) and (33)
$\|\alpha\|_{L^{1}(-4,0)}\leq C\sqrt{\overline{\eta}}.$
Denoting $Y(t)=1+\int\psi_{1}(t,x)|u|^{3/2}\,dx$, we have
$\dot{Y}\leq\alpha Y,\qquad Y(-4)=1.$
Gronwall’s lemma gives that for any $-4<t<0$ we have
$Y(t)\leq exp\left(\int_{-4}^{t}\alpha(s)\,ds\right).$
Hence, for $\overline{\eta}$ small enough:
$\|u\|_{L^{\infty}(-(3/2)^{2},0;L^{3/2}(B_{3/2}))}\leq
C{\overline{\eta}}^{1/3}.$ (34)
Step 2: Control on $u$ in $L^{\infty}(L^{2})$ in $Q_{1}$.
We consider $\psi_{2}\in C^{\infty}(\mathbb{R}^{4})$ a nonnegative function
compactly supported in $Q_{3/2}$ with $\psi_{2}=1$ in $Q_{1}$ and
$|\nabla_{t,x}\psi_{2}|+|\nabla_{t,x}^{2}\psi_{2}|\leq C.$
Multiplying inequality (3) by $\psi_{2}$ and integrating in $x$ gives
$\displaystyle\qquad\qquad\frac{d}{dt}\left(\int\psi_{2}\frac{|u|^{2}}{2}\,dx\right)$
$\displaystyle\leq\int
u\cdot\nabla\psi_{2}\left(\frac{|u|^{2}}{2}+P\right)\,dx+\int(\partial_{t}\psi_{2}+\Delta\psi_{2})\frac{|u|^{2}}{2}\,dx.$
equalities (31) together with (33) and Sobolev imbedding gives
$\||u|^{2}+P\|_{L^{1}(-(3/2)^{2},0;L^{3}(B_{3/2}))}\leq
C{\overline{\eta}}^{1/2}.$
Together with (34), this gives that
$\|u\|_{L^{\infty}(-1,0;L^{2}(B_{1}))}\leq C{\overline{\eta}}^{1/4}.$ (35)
Step 3. $L^{\infty}$ bound in $Q_{1/2}$. We need now to get better
integrability in time on the pressure.
From (32) and (35), we get
$\left\|\int\phi(x)\nabla P\,dx\right\|_{L^{2}(-1,0)}\leq
C\sqrt{\overline{\eta}}.$
With Lemma 3 and (30), this gives for $\gamma<1$
$\|\nabla P\|_{L^{1+\gamma}(-1,0;L^{1}(B_{1}))}\leq C\sqrt{\overline{\eta}}.$
Together with (35), (28), and Proposition 6, this shows that for
$\overline{\eta}$ small enough, we have
$|u|\leq 1\qquad\mathrm{in}\ \ Q_{1/2}.$
Step 4: Obtaining more regularity. We now obtain higher derivative estimates
by a standard bootstrapping method. We give the details carefully to ensure
that the bounds obtained are universal, that is, do not depend on the actual
solution $u$.
For $n\geq 1$ we define $r_{n}=2^{-n-3}$, $\overline{B}_{n}=B_{r_{n}}$ and
$\overline{Q}_{n}=Q_{r_{n}}$. We denote also $\overline{\psi}_{n}$ such that
$0\leq\overline{\psi}_{n}\leq 1$, $\overline{\psi}_{n}\in
C^{\infty}(\mathbb{R}^{4})$,
$\displaystyle\overline{\psi}_{n}(t,x)$ $\displaystyle=$ $\displaystyle
1\qquad(t,x)\in\overline{Q}_{n},$ $\displaystyle=$ $\displaystyle
0\qquad(t,x)\in\overline{Q}_{n-1}^{c}.$
For every $n$ we have
$\partial_{t}\nabla^{n}u+\mathrm{div}A_{n}+\nabla R_{n}-\Delta\nabla^{n}u=0,$
(36)
with
$A_{n}=\nabla^{n}(u\otimes u),\qquad R_{n}=\nabla^{n}P.$
So we have
$\|A_{n}\|_{L^{p}(\overline{Q}_{n-1})}\leq
C_{n}\|u\|^{2}_{L^{2p}(-r^{2}_{n-1},0;W^{n,2p}(\overline{B}_{n-1}))}$ (37)
and thanks to Lemma 5, we can split $R_{n}$ as
$R_{n}=R_{1,n}+R_{2,n},$
with
$\displaystyle\|R_{1,n}\|_{L^{p}(\overline{Q}_{n-1})}\leq
C_{n}\|A_{n}\|_{L^{p}(\overline{Q}_{n-2})},$ (38)
$\displaystyle\|R_{2,n}\|_{L^{1}(-r^{2}_{n-1},0;W^{2,\infty}(\overline{B}_{n-1}))}\leq
C_{n}\left(\|A_{n}\|_{L^{p}(\overline{Q}_{n-2})}+\|\nabla
P\|_{L^{1}(\overline{Q}_{n-2})}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\leq
C_{n}\left(\|A_{n}\|_{L^{p}(\overline{Q}_{n-2})}+1\right).$ (39)
Moreover we have:
$\displaystyle\partial_{t}(\overline{\psi}_{n}\nabla^{n}u)-\Delta(\overline{\psi}_{n}\nabla^{n}u)$
$\displaystyle\qquad=-\mathrm{div}(A_{n}\overline{\psi}_{n})+\nabla\overline{\psi}_{n}A_{n}$
$\displaystyle\qquad\qquad-\nabla(\overline{\psi}_{n}R_{n})+(\nabla\overline{\psi}_{n})R_{n}$
$\displaystyle\qquad\qquad+\Delta\overline{\psi}_{n}\nabla^{n}u-2\mathrm{div}(\nabla\overline{\psi}_{n}\nabla^{n}u)$
$\displaystyle\qquad\qquad+(\partial_{t}\overline{\psi}_{n})\nabla^{n}u.$
Note that $\overline{\psi}_{n}\nabla^{n}u=0$ on $\partial\overline{Q}_{n-1}$.
So
$\overline{\psi}_{n}\nabla^{n}u=V_{1,n}+V_{2,n}$ (40)
with
$\displaystyle\partial_{t}V_{1,n}-\Delta
V_{1,n}=-\mathrm{div}(A_{n}\overline{\psi}_{n})+\nabla\overline{\psi}_{n}A_{n}$
$\displaystyle\qquad-\nabla(\overline{\psi}_{n}R_{1,n})+(\nabla\overline{\psi}_{n})R_{1,n}$
$\displaystyle\qquad+\Delta\overline{\psi}_{n}\nabla^{n}u-2\mathrm{div}(\nabla\overline{\psi}_{n}\nabla^{n}u)$
$\displaystyle\qquad+(\partial_{t}\overline{\psi}_{n})\nabla^{n}u$
$\displaystyle\qquad\qquad\qquad=F_{n},$ $\displaystyle
V_{1,n}=0\qquad\mathrm{for}\ t=-r_{n-1}^{2},$
and
$\displaystyle\partial_{t}V_{2,n}-\Delta
V_{2,n}=-\nabla(\overline{\psi}_{n}R_{2,n})+R_{2,n}(\nabla\overline{\psi}_{n}),$
$\displaystyle V_{2,n}=0\qquad\mathrm{for}\ t=-r_{n-1}^{2}.$
Thanks to (37) and (38), we have
$\|F_{n}\|_{L^{p}(-r^{2}_{n-1},0;W^{-1,p}(\overline{B}_{n-1}))}\leq
C_{n}\left(1+\|u\|^{2}_{L^{2p}(-r^{2}_{n-2},0;W^{n,2p}(\overline{B}_{n-2}))}\right).$
So, from Lemma 4,
$\displaystyle\|V_{1,n}\|_{L^{p}(-r^{2}_{n-1},0;W^{1,p}(\overline{B}_{n-1}))}\leq
C\|F_{n}\|_{L^{p}(-r^{2}_{n-1},0;W^{-1,p}(\mathbb{R}^{3}))},$
$\displaystyle\|V_{2,n}\|_{L^{\infty}(-r^{2}_{n-1},0;W^{1,\infty}(\overline{B}_{n-1}))}\leq
C\|\overline{\psi}_{n}\nabla
R_{2,n}\|_{L^{1}(-r^{2}_{n-1};W^{1,\infty}(\mathbb{R}^{3}))}$
$\displaystyle\qquad\qquad\qquad\qquad+C\|R_{2,n}(\nabla\overline{\psi}_{n})\|_{L^{1}(-r^{2}_{n-1}W^{1,\infty}(\mathbb{R}^{3}))}$
$\displaystyle\qquad\qquad\leq
C_{n}\left(1+\|u\|^{2}_{L^{2p}(-r^{2}_{n-2},0;W^{n,2p}(\overline{B}_{n-2}))}\right),$
where we have used (37) and (39) in the last line.
Hence, from (40) and using that $\overline{\psi}_{n}=1$ on $\overline{Q}_{n}$,
we have for any $1<p<\infty$
$\|\nabla^{n}u\|_{L^{p}(-r^{2}_{n},0;W^{1,p}(\overline{B}_{n}))}\leq
C_{n}\left(1+\|u\|^{2}_{L^{2p}(-r^{2}_{n-2},0;W^{n,2p}(\overline{B}_{n-2}))}\right).$
By induction we find that for any $n\geq 1$, and any $1\leq p<\infty$, there
exists a constant $C_{n,p}$ such that
$\|u\|_{L^{2^{-n}p}(-r^{2}_{n},0;W^{n,2^{-n}p}(\overline{B}_{n}))}\leq
C_{n,p}.$
This is true for any $p$, so for $n$ fixed, taking $p$ big enough and using
Sobolev imbedding, we show that for any $1\leq q<\infty$, there exists a
constant $C_{n,q}$ such that
$\|u\|_{L^{q}(-r^{2}_{n+1},0;W^{n,\infty}(\overline{B}_{n+1}))}\leq C_{n,q}.$
As (37), we get that
$\|A_{n}\|_{L^{1}(-r^{2}_{n+3},0;W^{2,\infty}(\overline{B}_{n+3}))}\leq
C_{n}.$
Thanks to Lemma 5, we get
$\displaystyle\|R_{1,n}\|_{L^{1}(-r^{2}_{n+4},0;W^{1,\infty}(\overline{B}_{n+4}))}\leq
C_{n},$
$\displaystyle\|R_{2,n}\|_{L^{1}(-r^{2}_{n+4},0;W^{1,\infty}(\overline{B}_{n+4}))}\leq
C_{n}.$
Hence
$\|\partial_{t}\nabla^{n}u\|_{L^{1}(-r^{2}_{n+4},0;L^{\infty}(\overline{B}_{n+4}))}\leq
C_{n},$
and finally
$\|\nabla^{n}u\|_{L^{\infty}(\overline{Q}_{n+4})}\leq C_{n}.$
## 5 From local to global
Let us fix $\delta>0$. We take $\eta^{*}\leq\overline{\eta}$ and consider any
$\varepsilon>0$ such that $4\varepsilon^{2}\leq t_{0}$. Then from Proposition
10 and Proposition 9, for any
$(t,x)\in\Omega^{\delta}_{\varepsilon}\cap\\{t\geq t_{0}\\}$, we have
$|\nabla^{n}_{y}v_{\varepsilon}(0,0)|\leq C_{n},$
where $v_{\varepsilon}$ is defined by (20). But for any $n\geq 1$, we have
$\nabla^{n}_{y}v_{\varepsilon}(0,0)=\varepsilon^{n+1}\nabla^{n}u(t,x).$
Hence
$\left|\left\\{(t,x)\in\Omega\setminus|\nabla^{n}u(t,x)|\geq\frac{C_{n}}{\varepsilon^{n+1}}\right\\}\right|\leq|[\Omega^{\delta}_{\varepsilon}]^{c}|.$
And thanks to Lemma 8, This measure is smaller than
$\frac{C}{\eta^{*}}\left(\|u^{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}+\|u^{0}\|_{L^{2}(\mathbb{R}^{3})}^{2(\gamma+1)}\right)\varepsilon^{4-\delta}.$
We denote
$R=\left(1+\frac{4}{t_{0}}\right)^{\frac{n+1}{2}}.$
For $k\geq 1$, we use our estimate with $\varepsilon^{n+1}=R^{-k}$ to get
$\left|\left\\{(t,x)\in\Omega\setminus\frac{|\nabla^{n}u(t,x)|}{C_{n}}\geq
R^{k}\right\\}\right|\leq\frac{C\left(1+\|u^{0}\|_{L^{2}(\mathbb{R}^{3})}^{2(\gamma+1)}\right)}{R^{k\frac{4-\delta}{n+1}}}.$
So, for $p<\frac{4-\delta}{n+1}$
$\displaystyle\left\|\frac{\nabla^{n}u}{C_{n}}\right\|^{p}_{L^{p}(\Omega)}\leq\left|\left\\{(t,x)\in\Omega\setminus\frac{|\nabla^{n}u(t,x)|}{C_{n}}\leq
R\right\\}\right|R^{p}$
$\displaystyle\qquad\qquad+\sum_{k=1}^{\infty}R^{(k+1)p}\left|\left\\{(t,x)\in\Omega\setminus\frac{|\nabla^{n}u(t,x)|}{C_{n}}\geq
R^{k}\right\\}\right|$
$\displaystyle\qquad\leq|\Omega|R^{p}+CR^{p}\left(1+\|u^{0}\|_{L^{2}(\mathbb{R}^{3})}^{2(\gamma+1)}\right)\sum_{k=1}^{\infty}R^{k\left(p-\frac{4-\delta}{n+1}\right)}$
$\displaystyle\leq|\Omega|R^{p}+\frac{CR^{p}}{1-R^{p-\frac{4-\delta}{n+1}}}\left(1+\|u^{0}\|_{L^{2}(\mathbb{R}^{3})}^{2(\gamma+1)}\right).$
The results holds for any $\delta>0$ which ends the proof of Theorem 1.
Acknowledgment: This work was partially supported by NSF Grant DMS-0607053. We
thank Prof. Caffarelli for many insightful discussions and advices.
## References
* [1] J. Bergh and J. Löfström. Interpolation spaces. An introduction. Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223.
* [2] L. Caffarelli, R. Kohn, and L. Nirenberg. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math., 35(6):771–831, 1982.
* [3] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes. Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9), 72(3):247–286, 1993.
* [4] E. B. Fabes, B. F. Jones, and N. M. Rivière. The initial value problem for the Navier-Stokes equations with data in $L^{p}$. Arch. Rational Mech. Anal., 45:222–240, 1972.
* [5] E. Hopf. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr., 4:213–231, 1951.
* [6] H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. Adv. Math., 157(1):22–35, 2001.
* [7] J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta. Math., 63:183–248, 1934.
* [8] F. Lin. A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math., 51(3):241–257, 1998.
* [9] P.-L. Lions. Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications.
* [10] V. Scheffer. Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math., 66(2):535–552, 1976.
* [11] V. Scheffer. Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys., 55(2):97–112, 1977.
* [12] V. Scheffer. The Navier-Stokes equations in space dimension four. Comm. Math. Phys., 61(1):41–68, 1978.
* [13] V. Scheffer. The Navier-Stokes equations on a bounded domain. Comm. Math. Phys., 73(1):1–42, 1980.
* [14] J. Serrin. The initial value problem for the Navier-Stokes equations. In Nonlinear Problems (Proc. Sympos., Madison, Wis., pages 69–98. Univ. of Wisconsin Press, Madison, Wis., 1963.
* [15] V. A. Solonnikov. A priori estimates for solutions of second-order equations of parabolic type. Trudy Mat. Inst. Steklov., 70:133–212, 1964.
* [16] M. Struwe. On partial regularity results for the Navier-Stokes equations. Comm. Pure Appl. Math., 41(4):437–458, 1988.
* [17] R. Temam. Navier-Stokes equations. AMS Chelsea Publishing, Providence, RI, 2001. Theory and numerical analysis, Reprint of the 1984 edition.
* [18] A. Vasseur. A new proof of partial regularity of solutions to Navier-Stokes equations. NoDEA Nonlinear Differential Equations Appl., 14(5-6):753–785, 2007\.
|
arxiv-papers
| 2009-04-16T03:25:12 |
2024-09-04T02:49:01.898036
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexis F Vasseur",
"submitter": "Alexis Vasseur",
"url": "https://arxiv.org/abs/0904.2422"
}
|
0904.2513
|
# Significant foreground unrelated non-acoustic anisotropy on the one degree
scale in WMAP 5-year observations
Bi-Zhu Jiang11affiliation: Physics Department and Center for Astrophysics,
Tsinghua University, Beijing 100084, China. 22affiliation: Department of
Physics, University of Alabama, Huntsville, AL 35899. , Richard
Lieu22affiliation: Department of Physics, University of Alabama, Huntsville,
AL 35899. , Shuang-Nan Zhang11affiliation: Physics Department and Center for
Astrophysics, Tsinghua University, Beijing 100084, China. 22affiliation:
Department of Physics, University of Alabama, Huntsville, AL 35899.
33affiliation: Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing, China , and Bart
Wakker44affiliation: Department of Astronomy, University of Wisconsin, 475 N.
Charter St, Madison, WI 53706, USA
###### Abstract
The spectral variation of the cosmic microwave background (CMB) as observed by
WMAP was tested using foreground reduced WMAP5 data, by producing subtraction
maps at the 1∘ angular resolution between the two cosmological bands of V and
W, for masked sky areas that avoid the Galactic disk. The resulting $V-W$ map
revealed a non-acoustic signal over and above the WMAP5 pixel noise, with two
main properties. Firstly, it possesses quadrupole power at the $\approx$ 1
$\mu K$ level which may be attributed to foreground residuals. Second, it
fluctuates also at all values of $\ell>$ 2, especially on the $1^{\circ}$
scale ($200\lesssim\ell\lesssim 300$). The behavior is random and symmetrical
about zero temperature with a r.m.s. amplitude of $\approx$ 7 $\mu K$, or 10 %
of the maximum CMB anisotropy, which would require a ‘cosmic conspiracy’ among
the foreground components if it is a consequence of their existences. Both
anomalies must be properly diagnosed and corrected if ‘precision cosmology’ is
the claim. The second anomaly is, however, more interesting because it opens
the question on whether the CMB anisotropy genuinely represents primordial
density seeds.
## 1 Introduction
Studies of the cosmic microwave background (CMB, Penzias and Wilson (1965)),
the afterglow radiation of the Big Bang, are currently in a period of
renaissance after the breakthrough discovery of anisotropy by the COBE mission
(Smoot et al (1992)). Confirmed with much improved resolution and statistics
by WMAP (Hinshaw et al (2009)), the phenomenon provides vital information on
the primordial ‘seeds’ of structure formation. The anisotropy is attributed to
frequency shift of CMB light induced by these ‘seed’ density perturbations,
which has the unique property that it leads to changes in the temperature of
the black body spectrum and not the shape of it. The CMB has maximum
anisotropy power at the 1∘ scale, or harmonic number $\ell\approx$ 220, with
lower amplitude secondary and tertiary peaks at higher $\ell$.
The $\Lambda$CDM cosmological model (Spergel et al(2007)) explains the entire
power spectrum remarkably using six parameters, by attributing the peaks to
acoustic oscillations of baryon and dark matter fluids, as long wavelength
modes of density contrast enter the horizon and undergo causal physical
evolution. CMB light emitted from within an overdense region of the
oscillation are redshifted by a constant fractional amount, resulting in a
cold spot, which is a lowering by $\delta T$ of the black body temperature
$T$, and is frequency independent, i.e. $\delta T/T=\delta\nu/\nu=$ constant.
The opposite effect of blueshift applies to underdense regions, leading to hot
spots. Therefore, if the anisotropy is genuinely due to acoustic oscillations,
the inferred change in $T$ at a given region should be the same for all the
‘clean’ frequency passbands of the WMAP mission. Since a corresponding
variation of the CMB flux $B(\nu,T)$ at any given frequency $\nu$ is $\delta
B=(\partial B/\partial T)\delta T$ if the cause is solely $\delta T$ with no
accompanying distortion of the functional form of $B$ itself, the expected
$\delta B$ at constant $\delta T$ is then the ‘dipole spectrum’ $\partial
B/\partial T$ which is well measured by COBE-FIRAS (Mather et al(1994)).
Moreover, the WMAP data are calibrated w.r.t. this dipole response.
A noteworthy point about the acoustic peaks is that one needs to employ the
technique of cross correlation to reduce the noise contamination at high
$\ell$, especially the harmonics of the second and higher acoustic peaks.
Specifically one computes the all-sky cross power spectrum
$C_{\ell}^{ij}=\frac{1}{2\ell+1}\sum_{m}~{}a_{\ell m}^{i}a_{\ell m}^{j*},$ (1)
where the indices $i$ and $j$ denote independent data streams with
uncorrelated noise that arise from a pair of maps at different frequency bands
(or same band but taken at different times), and $a_{\ell m}^{i}=\delta
T_{\ell m}^{i}$ is the apparent CMB temperature anisotropy for the spherical
harmonics $(\ell,m)$ as recorded by observation $i$. Since the use of multiple
passbands is crucial to the accurate profiling of the acoustic oscillations,
it is important that we do compare them with care, down to the level of
measurement uncertainties. Only a priori statistically consistent maps should
be cross correlated, in the sense that any real discrepancies between such
maps may carry vital information about new physical processes that their cross
power spectrum does not reveal. In one previous attempt to address this point
(see Figure 9 of Bennett et al(2003a)) WMAP1 data downgraded to an angular
resolution commensurate with COBE were used to produce a difference
(subtraction) map between the two missions. When displayed side by side with
the map of the expected noise for each resulting pixel, the two maps did
appear consistent. Nevertheless, this powerful method of probing the CMB
anisotropy does, in the context of the specific datasets used by Bennett et
al(2003a), suffer from one setback: it is limited by the sensitivity and
resolution of COBE.
In another test of a similar kind, we observe that each amplitude $a_{\ell
m}^{i}$ can further be factorized as $a_{\ell m}^{i}=a_{\ell m}b_{\ell}^{i}$,
where the array $b_{\ell}^{i}$ accounts for the smoothing effects of both the
beam and the finite sky map pixel size, and $a_{\ell m}=\delta T_{\ell m}$ is
the true amplitude of the CMB anisotropy. The results (see Figure 13 of
Hinshaw et al (2007)) indicate agreement of the variance $C_{\ell}^{ij}$,
hence $\delta T_{\ell}$, within the margin of a few percent for $\ell\lesssim$
400 among the many cross power spectra formed by the various possible
combinations of pairs of all-sky maps. This offers more ground for optimism,
but to be definitive the remaining discrepancy needs to be demonstrably
attributed to noise, instrumental systematics, or foreground emission.
The purpose of our investigation is to perform further, more revealing
comparisons than the two past ones described above, initially by focussing
upon the angular scale of the first acoustic peak, which is $\sim$ 1∘. Our
analysis will be done in both real (angular) and harmonic domains, because
while most of the effort have hitherto been pursued in the latter, the former
is the domain in which the raw data were acquired and organized.
## 2 The all sky difference map between the WMAP5 V and W bands
We adopted the Healpix111See http://healpix.jpl.nasa.gov. pixelization scheme
to ensure that all pixels across the sky have the same area (or solid angle).
Firstly the W band data is smoothed to the V band resolution. Then the whole
sky map is downgraded to $\approx$ 1∘ diameter (corresponding to nside of 64
in the parametrization of the WMAP database), which is not only commensurate
with the scale of global maximum $\delta T$ power, but also large enough to
prevent data over-sampling due to the use of too high a resolution, as the
size is comfortably bigger than the beam width of the WMAP V band (61 GHz)
larger than that of the W band.
The resulting $\delta T$ values for the two cosmological passbands of V and W,
span $\approx$ 35,000 clean (i.e. ext-masked222ext is short for external
temperature analysis. and foreground subtracted333For foreground subtracted
WMAP5 maps see
http://lambda.gsfc.nasa.gov/product/map/current/m_products.cfm.) pixels, from
which a $V-W$ difference map at this $\approx$ 1∘ resolution was made. After
removing the monopole and dipole residuals (the latter aligned with the
original COBE dipole), this map is displayed in Figure 1 along with the
corresponding pixel noise map for reference; the latter represents the
expected appearance of the $V-W$ map if the CMB anisotropy is genuinely
acoustic in nature, so that the map would consist only of null pixels should
the WMAP5 instruments that acquired them be completely noise free. When
comparing the real data map of Figure 1a with the simulated map of Figure 1b,
the former appears visibly noisier on the resolution scale $\approx$ 1∘;
moreover, the Leo and Aquarius (i.e. the first and third) sky quadrants
contain more cold pixels than the other half of the sky, indicative of the
existence of a quadrupole residual.
The extra signals revealed by the $V-W$ subtraction map are elucidated further
in respect of their aforementioned properties by examining the statistical
distribution of the pixel values across the four sky quadrants. As shown in
Figure 2, the distribution of the 1∘ anisotropy is considerably wider than
that expected from the WMAP5 pixel noise for all the quadrants, by $\approx$
10 $\mu K$, which is $\sim$ 10 % of the $\approx$ 75 $\mu K$ power in the
first acoustic peak, and is therefore very significant. A detailed
confirmation by Gaussian curve fitting is given in Table 1.
The $V-W$ quadrupole is more subtle, and is evident in the residual plots at
the bottom of each graph in Figure 2, from which a slight skewness of the data
to the right is apparent in quadrants 1 and 3 (the quadrants of the CMB
dipole), with 2 and 4 exhibiting the opposite behavior. For this reason, the
effect does not manifest itself as shifts in the Gaussian mean value $\mu$ of
Table 1. Rather, the high statistical significance of both the quadrupole and
the degree-scale signals, with the former having a magnitude of $\approx$ 1
$\mu K$, are established by computing the cross power spectra of the
temperature difference maps, Figure 4. This was performed at the resolution of
nside$=$ 64 using the PolSpice software444Available from
http://www.planck.fr/article141.html.. From Figure 4 also, the presence of
excess non-acoustic anisotropy at all harmonics $\ell>2$, including the
cosmologically important $\theta\approx 1^{\circ}$ angular scale, appears
robust. At the $1^{\circ}$ scale ($200\lesssim\ell\lesssim 300$), the r.m.s.
is about 7 $\mu K$, or 10 % of the maximum CMB anisotropy. Lastly, the $V-W$
quadrupole may be displayed in isolation by arranging the data of the
subtracted map as a multipole expansion
$\delta T(\theta,\phi)=\sum_{\ell,m}a_{\ell m}Y_{\ell m}(\theta,\phi),$ (2)
and evaluating at $\ell=2$ the amplitude
$\delta T_{\ell}(\theta,\phi)=\sum_{m}a_{\ell m}Y_{\ell m}(\theta,\phi),$ (3)
(note $\delta T_{\ell}(\theta,\phi)$ is always a real number if the original
data $\delta T(\theta,\phi)$ are real). The ensuing whole sky map is in Figure
3, and the coordinates of the axes are in Table 2.
## 3 Interpretation of results
The WMAP5 $V-W$ map reveals two principal anomalies to be explained: (a) the
quadrupole at $\ell=2$, with an amplitude of $\approx 1\mu K$, and (b) the
higher harmonic signals, especially the $\approx 8~{}\mu K$ anisotropy at
$\ell\gtrsim$ 200 (Figure 4). Similar findings are also made by others, like
the noticeable hemispherical power asymmetry in the WMAP1 analysis of Eriksen
et al (2004) and confirmed in the WMAP5 data by Hoftuft et al (2009), or the
large scale distribution investigated by Diego et al (2009). Also because both
(a) and (b) are not small effects, claims to precision cosmology are
overstatements until they are properly accounted for and the cosmological
model accordingly adjusted.
Concerning (a), unlike the dipole, there is no previous known CMB quadrupole
of sufficient amplitude to justify its dismissal as a cross band calibration
residual. In fact, our reported amplitude of $1~{}\mu$K is about 7 % of the
211 $\mu$K2 WMAP5 anisotropy in the unsubtracted maps of the individual bands
at $\ell=$ 2, which is far larger than the calibration uncertainty of
$\approx$ 0.5 % (Hinshaw et al (2009)) for each band.
It will probably be more rewarding to search for remaining foreground
contamination not yet removed by the standard data filtering and correction
procedures of the WMAP5 team (Bennett et al(2003b), Gold et al(2009)). Thermal
dust emission might have a power law spectrum with an index too close to that
of the Rayleigh-Jeans tail in the V and W bands for an appreciable V - W
signal, although this is an interesting scenario worthy of further study
(Diego et al 2009). We consider here another possibility, viz. free-free
emission from High Velocity Clouds (HVCs, Wakker et al (2009) and references
therein). The clouds are moving at velocities sufficiently large for any
H$\alpha$ emission from them to be outside the range555Example of a HVC missed
by WHAM is Hill et al 2009, a cloud of unit emission measure. A notable
exception (counter example) would be the HVC K-complex (Haffner et al 2001),
with an emission measure of 1.1 units, that happens to fall inside the
velocity window of WHAM. of the WHAM survey, the database employed to estimate
the free-free contribution to the WMAP foreground. HVC parameters for the
larger and brighter clouds can reach: $n_{e}\approx$ 0.2 $cm^{-3}$ and column
density $\approx$ 3 $\times$ 1019 cm-2 (Wakker et al 2008). This corresponds
to an emission measure of two units, or 6 $\times$ 1018 cm-5, or $\approx$ 0.6
$\mu K$ of V-W temperature excess (Finkbeiner D.P. (2003)), on par with the 1
$\mu K$ of our observed quadrupole. Moreover, as can be seen from the all-sky
map of $N_{{\rm HI}}$ and an estimate of the V-W excess in Figure 5 when they
are compared with Figures 3 and 4, the strength and distribution of HVCs do
appear to be responsible for a non-negligible fraction of the observed anomaly
on very large scales. Further work in this area is clearly necessary, and will
be pursued in a future, separate paper.
We now turn to (b), the effect that occurs on the much smaller and
cosmologically most significant angular scale of 1∘. Calibration issues are
again immediately excluded here, since the 8 $\mu K$ anomalous amplitude is on
par with the pixel noise of WMAP5 for the scale in question (Table 1).
Moreover, because the subtracted $V-W$ dipole and the (unsubtracted) $V-W$
quadrupole, the latter being (a), are both relatively feeble phenomena, of
amplitudes $\approx$ 0.2 and 1 $\mu K$ respectively as compared to the 7 $\mu
K$ amplitude of (b), the prospect of smaller scale fluctuations having been
enhanced by a larger scale one can be ruled out here. CMB spectral distortion
during the recombination era, or subsequently from the Sunyaev-Zeld́ovich (SZ)
scattering, or from other foreground re-processing that were not properly
compensated by the data cleaning procedure of WMAP5, could all be responsible
for the observed anomaly. Although the first two interactions (Sunyaev and
Chluba (2008), Birkinshaw and Gull (1983)) exert much smaller influences than
7 $\mu K$ (bearing in mind that the degree of SZ needs to be averaged over the
scale of the whole sky), the foreground could potentially play a relevant role
in a similar way as it did at very low $\ell$. Thus, in respect of free-free
emission by HVCs alone, until a full survey at high angular resolution is
performed one cannot be certain that the emission measure from these clouds is
too weak to account for our (b) anomaly. However, the action of the foreground
is systematic in that it does not lead to random and symmetric temperature
excursions (about zero) between two frequencies of V and W. More precisely,
because the sources or sinks involved have a characteristic spectrum that
differs from black body in a specific way, any widening in Figure 2 of the
data distribution w.r.t. the expected simulated gaussian ought to be highly
asymmetric. This obviously contradicts our findings, i.e. we note from Figure
2 that the widening of the data histogram is highly symmetric. As a result,
the symptoms do not point to the foreground as responsible cause.
## 4 Conclusion
We performed a new way of testing the black body nature of the CMB degree
scale anisotropy, by comparing the all-sky distribution of temperature
difference between the WMAP5 cosmological bands of V and W, with their
expected pixel noise behavior taken fully into consideration by means of
simulated data. In this way a non acoustic signal is found in the ext-masked
$V-W$ map at the $\approx$ 1∘ resolution of nside $=$ 64, with the following
two properties. It has a quadrupole amplitude $\approx$ 1 $\mu$K (Figures 2,
3, and 4) which may in part be attributed to unsubstracted foreground
emission. It also has excess anisotropy (or fluctuation) on all scales $\ell>$
2, including and especially the scales of $200\lesssim\ell\lesssim 300$ where
most of the acoustic power resides, and about which the anomaly we reported is
in the form of a symmetric random excursion about zero temperature with a
r.m.s. $\approx$ 8 $\mu K$ (Figures 2 and 4, Table 1) which is $\approx$ 10 %
of the maximum acoustic amplitude found at $\ell\approx$ 220\. This type of
excursion frustrates attempts to explain the effect as foreground residuals,
i.e. it opens the question of whether the WMAP anisotropy on the 1∘ scale is
genuinely related to the seeds of structure formation.
In any case, it is clear that both anomalies have sufficiently large
magnitudes to warrant their diagnoses through future, further investigations,
if the status of precision cosmology is to be reinstated.
Figure 1: The ext-masked and point sources subtracted WMAP5 $V-W$ map, viz.
the difference map between the CMB anisotropy as measured in the V band and
the W band, for the real data after the removal of residual monopole and
dipole components (top), and simulated pixel noise that reflect precisely the
observational condition (bottom). Both maps are plotted in Galactic
coordinates with the Galactic center $(l,b)=(0,0)$ in the middle and Galactic
longitude $l$ increasing to the left. To avoid the problems of beam size
variation from one band to the next, the W band data is smoothed to the V band
resolution, then the pixels were downgraded to the common resolution of
nside$=$ 64 using the foreground-reduced WMAP5 data (see section 2); this
resolution under-samples the data in both bands. The color scale is coded
within a symmetrical range: those pixels with values beyond $\pm 40~{}\mu$K
are displayed in the same (extreme) color; most of such pixels are around the
masked regions. The existence of additional non- black body signal in the real
data can readily be seen from this comparison, as the simulated map is
noticeably quieter.
Figure 2: The data points show quadrant sky occurrence frequency distribution
of the difference in the degree-scale (nside$=64$) anisotropy between the
WMAP5 V and W bands, while the errors in the data are due to the WMAP5 pixel
noise for the same ext-masked quadrant sky area, i.e. they are the statistical
fluctuations in the various parts of the solid line, which gives the mean
histogram of this noise. The orientation of each quadrant follows the same
convention as the sky maps of Figure 1, with the 1st and 3rd quadrants marking
the COBE dipole. Figure 3: $V-W$ quadrupole of the nside$=64$ WMAP5
temperature difference maps, after ext-masking and point source subtraction.
The mathematical procedure of extracting each multipole $\ell$ is given in
eqs. (3) and (4) of the text, and the software used to do these computations
was from anafast of Healpix.
Figure 4: Real and simulated (noise) power spectra of the WMAP5 $V-W$ map. These are V-W cross power spectra computed by cross correlating the first three years of observations with the last two. The errors in the real data of the first two graphs represent the pixel noise power of the last graph, i.e. 4c is the average of 1,000 simulated realizations of the V-W WMAP5 pixel noise. Thus, if the noise power at harmonic $\ell$ is $(\delta T_{\ell})^{2}$ from 4c, the upper error bar in 4a and 4b will extend from $T_{l}^{2}$ to $(T_{\ell}+\delta T_{\ell})^{2}$ where $T_{\ell}$ is the observed V-W anisotropy of each real data point (given by the intersection of the error bars with the zig-zag line) in 4a and 4b. The rising trend ($\sim l^{2}$) of all three curves towards higher $l$ simply reflects the relatively larger pixel noise for smaller angular areas. For $l>$ 200 the real data of 4a and 4b rapidly become noise dominated. Figure 5: Upper map shows 21 cm data of HVCs with HI column density ($N_{{\rm HI}}$) larger than 7 $\times$ 1018 cm-2 (i.e. the greyscale shows $N_{{\rm HI}}$ with the outer contour at 7 $\times$ 1018 cm-2). Complex C is the cloud in the region $l=$ 90∘ – 130∘, $b=$ 40∘ – 60∘. Complex A is around $l=$ 150∘, $b=$ 30∘ – 45∘. The Magellanic Stream (MS) and Bridge is at $l=$ 280∘ – 310∘, $b<$ -30∘. The Leading Arm of the MS, plus some other bright HVCs are at $l=$ 240∘ – 300∘, $b=$ 10∘ – 30∘. Lower map gives our estimated V-W temperature excess due to HVCs. Note that because the dynamic range of conversion from $N_{{\rm HI}}$ to this excess (via free-free emission measure $EM$ of $N_{{\rm HII}}$) is not linear (e.g. Putman et al 2003, Hill et al 2009). Our approach is to assign 0.5 and 1.0 unit of $EM$, or 0.15 and 0.3 $\mu$K of V-W excess, to every direction with $N_{{\rm HI}}\geq$ 2 $\times$ 1019 cm-2 and 5 $\times$ 1019 cm-2 respectively. V - W | $\mu(\mu$K) | error ($\mu$K) | $\sigma$ ($\mu$K) | error ($\mu$K)
---|---|---|---|---
| WMAP5 | -0.23 | 0.15 | 16.23 | 0.13
Quadrant 1 | Simulation | 0.00 | 0.13 | 14.70 | 0.12
| Difference $\Delta$ | -0.23 | 0.20 | 6.88 | 0.40
| WMAP5 | 0.24 | 0.12 | 14.47 | 0.10
Quadrant 2 | Simulation | -0.04 | 0.12 | 12.10 | 0.10
| Difference $\Delta$ | 0.28 | 0.17 | 7.94 | 0.24
| WMAP5 | -0.11 | 0.16 | 16.22 | 0.13
Quadrant 3 | Simulation | 0.03 | 0.15 | 14.70 | 0.12
| Difference $\Delta$ | -0.14 | 0.22 | 6.86 | 0.40
| WMAP5 | 0.40 | 0.13 | 14.80 | 0.10
Quadrant 4 | Simulation | -0.01 | 0.13 | 12.26 | 0.10
| Difference $\Delta$ | 0.41 | 0.18 | 8.30 | 0.23
Table 1: Parameters for the gaussian curves that fitted the WMAP5 data and the
pixel noise histograms (the latter are the solid lines) of Figure 2. Each
parameter uncertainty is set by the $\chi^{2}_{{\rm min}}+1$ criterion, which
represents the usual 68 % (or unit standard deviation) confidence interval for
one interesting parameter, when the error bars shown in Figure 2 are employed
for fitting both the real and pixel noise data. The difference in the width
$\sigma$ between the two models, which gives the distribution width of the
additional random signal, is given by
$(\Delta\sigma)^{2}=\sigma_{r}^{2}-\sigma_{s}^{2}$. The smaller simulated
gaussian widths for quadrants 2 and 4 (relative to 1 and 3) is due to the
higher exposure times there (which contain the heavily scanned ecliptic poles)
leading to lower pixel noise. V-W quadrupole location $(l,b)$
---
hot | $(-132.1^{\circ},-14.4^{\circ})$,$(48.0^{\circ},14.4^{\circ})$
cold | $(-81.5^{\circ},68.0^{\circ})$,$(98.5^{\circ},-68.0^{\circ})$
Table 2: Orientation of the quadrupole in the WMAP5 V-W map of Figure 3.
We are grateful to the referee for very valuable suggestions towards the
improvement of this paper. Lyman Page, Priscilla Frisch, Gary Zank, and Barry
Welsh are also acknowledged for helpful discussions. Some of the results were
obtained by means of the HEALPix package (G$\acute{o}$rski et al (2005)).
## References
* Bennett et al (2003a) Bennett, C.L., et al. 2003a, ApJS, 148, 1
* Bennett et al (2003b) Bennett, C.L., et al. 2003b, ApJS, 148, 97
* Birkinshaw and Gull (1983) Birkinshaw, M. and Gull, S.F. 1983, Nature, 302, 315
* Diego et al (2009) Diego, J.M., Cruz, M., Gonz$\acute{a}$lez-Nuevo, J., Maris, M., Ascasibar, Y., Burigana, C., preprint(arXiv:0901.4344 [astro-ph]), MNRAS submitted
* Eriksen et al (2004) Eriksen, H.K., Hansen, F.K., Banday, A.J., G$\acute{o}$rski, K.M., Lilje, P.B. 2004, ApJ, 605, 14
* Gold et al (2009) Gold, B., et al. 2009, ApJS, 180, 265
* G$\acute{o}$rski et al (2005) G$\acute{o}$rski, K.M., Hivon, E., Banday, A.J., Wandelt, B.D., Hansen, F.K., Reinecke, M., and Bartelmann, M. 2005, ApJ, 622, 759-771
* Finkbeiner D.P. (2003) Finkbeiner, D.P. 2003, ApJS, 146, 407
* Haffner et al (2001) Haffner, L.M., Reynolds, R.J., Tufte, S.L., 2001, ApJ, 556, L33
* Hill et al (2009) Hill, A.S., Haffner, L.M., Reynolds, R.J. 2009, ApJ, 703, 1832
* Hinshaw et al (2007) Hinshaw, G., et al. 2007, ApJS, 170, 288
* Hinshaw et al (2009) Hinshaw, G., et al. 2009, ApJS, 180, 225
* Hoftuft et al (2009) Hoftuft, J., Eriksen, H.K., Banday, A.J., G$\acute{o}$rski, K.M., Hansen, F.K., Lijie, P.B. 2009, ApJ, 699, 2
* Hou et al (2009) Hou, Z., Banday, A.J., G$\acute{o}$rski, K.M. 2009, MNRAS, 396, 3
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* Sunyaev and Chluba (2008) Sunyaev, R.A. and Chluba, J. 2008, ASPC 395, 35S
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|
arxiv-papers
| 2009-04-16T15:07:23 |
2024-09-04T02:49:01.908253
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bi-Zhu Jiang, Richard Lieu, Shuang-Nan Zhang, and Bart Wakker",
"submitter": "Bizhu Jiang",
"url": "https://arxiv.org/abs/0904.2513"
}
|
0904.2549
|
# Planetary nebulae and the chemical evolution of the Magellanic Clouds
W. J. Maciel 11affiliation: Instituto de Astronomia, Geofísica e Ciências
Atmosféricas, Universidade de São Paulo, Brazil. R. D. D. Costa
11affiliationmark: and T. E. P. Idiart11affiliationmark: W. J. Maciel, R. D.
D. Costa and T. E. P. Idiart: Instituto de Astronomia, Geofísica e Ciências
Atmosféricas, Universidade de São Paulo - Rua do Matão 1226, CEP 05508-900,
São Paulo SP, Brazil (maciel@astro.iag.usp.br, roberto@astro.iag.usp.br,
thais@astro.iag.usp.br.)
###### Abstract
The determination of accurate chemical abundances of planetary nebulae (PN) in
different galaxies allows us to obtain important constraints of chemical
evolution models for these systems. We have a long term program to derive
abundances in the galaxies of the Local Group, particularly the Large and
Small Magellanic Clouds. In this work, we present our new results on these
objects and discuss their implications in view of recent abundance
determinations the literature. In particular, we obtain distance-independent
correlations involving He, N, O, Ne, S, and Ar, and compare the results with
data from our own Galaxy and other galaxies in the Local Group. As a result of
our observational program, we have a large database of PN in the Galaxy and
the Magellanic Clouds, so that we can obtain reliable constraints to the
nucleosynthesis processes in the progenitor stars in galaxies of different
metallicities.
ISM: planetary nebulae: general galaxies: Magellanic Clouds galaxies:
abundances
## 0.1 Introduction
The study of the chemical evolution of the galaxies in the Local Group,
particularly the Milky Way and the Magellanic Clouds, can be significantly
improved by the consideration of the chemical abundances of planetary nebulae
(PN) (see for example Maciel et al. 2006a, Richer & McCall 2006, Buzzoni et
al. 2006, and Ciardullo 2006). These objects are produced by low and
intermediate mass stars, with main sequence masses roughly between 0.8 and
8$\,M_{\odot}$, and present a reasonably large age and metallicity spread. As
a conclusion, they provide important constraints to the chemical evolution
models applied to these systems, and can also be used to test nucleosynthetic
processes in the PN progenitor stars. In particular, the PN abundances in the
nearby Magellanic Clouds can be derived with a high acuracy, comparable to the
objects in the Milky Way, so that they can be especially useful in the study
of the chemical evolution of these galaxies. In this work, we present some
recent results on the determination of chemical abundances from PN in the
Large and Small Magellanic Clouds derived by our group, and compare these
results with recent data from our own Galaxy and other galaxies in the Local
Group. We also take advantage of the inclusion of similar determinations from
the recent literature, so that the database of PN in the Magellanic Clouds is
considerably increased, allowing a better determination of observational
constraints of the nucleosynthetic processes ocurring in the progenitor stars.
Preliminary results of this work have been presented by Maciel, Costa & Idiart
(2006a, 2008).
## 0.2 The Sample
We have considered a sample of PN both in the LMC and SMC on the basis of
observations secured at the 1.6m LNA telescope located in southeast Brazil and
the ESO 1.5m telescope in La Silla, Chile. Details of the observations and the
resulting abundances can be found in the following references: de Freitas
Pacheco et al. (1993a, 1993b), Costa et al. (2000), and Idiart et al. (2007).
In these papers, abundances of He, N, O, S, Ne and Ar have been determined for
23 nebulae in the LMC and 46 objects in the SMC. The abundances presented in
Idiart et al. (2007) were based on average fluxes obtained by taking into
account some recent results from the literature, so that there may be some
differences compared with our originally derived values. For details the
reader is referred to the discussion in that paper.
In order to increase the PN database in the Magellanic Clouds, we have also
taken into account the samples by Stasińska et al. (1998), which included
abundances of He, N, O, and Ne for 61 nebulae in the SMC and 139 objects in
the LMC, and Leisy & Dennefeld (2006), containing 37 objects in the SMC and
120 nebulae in the LMC. In Stasińska et al. (1998), a collection was obtained
of photometric and spectroscopic data of PN in five different galaxies,
including the Magellanic Clouds. Although the original sources of the data are
rather heterogeneous, the plasma diagnostics and determination of the chemical
abundances were processed in the same way, so that the degree of homogeneity
of the data was considerably increased. The Leisy & Dennefeld (2006) sample is
a more homogeneous one, in which a larger fraction of the observations were
made by the authors themselves, and all abundances were re-derived in an
homogeneous way, as in Stasińska et al. (1998). As we will show in the next
section, the similarity of the methods in the abundance determinations
warrants comparable abundances, so that a larger sample was obtained.
## 0.3 Results and discussion
### 0.3.1 Average abundances
Average abundances of all elements in the SMC and LMC according to the three
samples considered are shown in Table 1. Helium abundances are given as He/H
by number as usual, while for the heavier elements the quantity given is
$\epsilon({\rm X})=\log{\rm X/H}+12$. Although the samples considered here are
probably the largest ones with carefullly derived abundances in the Magellanic
Clouds, they cannot be considered as complete. The total number of PN in these
systems is not known, but recent estimates point to about 130 and 980 objects
for the SMC and LMC, respectively (cf. Jacoby 2006 and Shaw 2006). Therefore,
incompleteness effects may still affect the results presented in this paper.
The He abundances show a good agreement in all samples within the average
uncertainties. The IAG and Leisy samples show a slightly higer He abundance in
the LMC compared to the SMC, but the differences between these objects are in
all cases smaller than the estimated uncertainties.
The O/H abundances, which are in general the best determined of all heavy
elements considered here, also show a good agreement among the samples.
Moreover, in all cases the LMC is richer than the SMC, as expected, and the
average metallicity difference is in the range 0.3 to 0.5 dex, which is
consistent with the metallicities given by Stanghellini (2008), namely
$Z=0.004$ and $Z=0.008$ for the SMC and LMC, respectively.
The Ar/H and Ne/H ratios show a similar behaviour as O/H, noticing that the
IAG data do not include Ne abundances for the LMC, and that Stasińska et al.
(1998) do not list Ar/H abundances for both galaxies. The sulfur abundances
seem to be less reliable, as can be seen from the large standard deviations
obtained in the IAG and Leisy & Dennefeld samples. Moreover, the estimated
average S/H ratio in the SMC is slightly larger than in the LMC according to
the IAG data, contrary to our expectations, while in the Leisy & Dennefeld
sample the LMC abundance is larger by only 0.27 dex in comparison with the
SMC. In fact, these characteristics of the sulfur abundances in Magellanic
Cloud PN can be observed in previous analyses, as for example in the summary
by Kwok (2000, Table 19.1, p. 202), where the S/H ratios in the SMC and LMC
are indistinguishable within the given uncertainties. Clearly, the
determination of S/H abundances in the Magellanic Cloud PN - and galactic
nebulae as well - is apparently affected by some additional effects, as
compared to the previous elements. In the following we will give further
evidences on the problem of sulfur abundances in planetary nebulae.
From Table 1, it can be seen that the nitrogen abundances also follow the same
pattern as O/H, Ar/H, and Ne/H, even though the N/H ratio is affected by the
dredge-up episodes occuring in the PN progenitor stars. This is further
discussed in Section 5, but from the results shown in the last column of Table
1, it is suggested that the average nitrogen contamination from the PN
progenitor stars is small. Average N/H abundances of of Magellanic Cloud PN
are given by Stanghellini (2008), where an effort was made to take into
account objects of different morphologies. The average N/H abundances in the
whole sample are $1.48\times 10^{-4}$ and $0.29\times 10^{-4}$ for the LMC and
SMC, respectively, which correspond to $\epsilon({\rm N})=8.17$ and 7.46, in
the notation of Table 1. These results correctly indicate that the LMC is
richer in N than the SMC, as also reported in Table 1, and the absolute value
of the SMC abundances given by Stanghellini (2008) is very similar to the
results of the 3 samples considered here, but the average abundance for the
LMC nebulae is much higher than our results. In fact, the N/H abundances of
the LMC given in Stanghellini (2008) are close to the Milky Way values, which
is paradoxical, as the LMC has a much lower metallcity than the Galaxy. Part
of the discrepancy may be caused by the fact that the sample used in that
paper includes a larger proportion of bipolar nebulae, which are ejected by
higher mass progenitor stars, which produce a larger nitrogen contamination
than the lower mass objects. This problem needs further clarification.
-2cm-2cm
Table 1: Average abundances of PN in the Magellanic Clouds.
| He | O | S | Ar | N | Ne
---|---|---|---|---|---|---
IAG/USP | | | | | |
SMC | $0.097\pm 0.035$ | $7.89\pm 0.44$ | $6.98\pm 0.58$ | $5.59\pm 0.36$ | $7.35\pm 0.49$ | $7.14\pm 0.42$
LMC | $0.119\pm 0.023$ | $8.40\pm 0.20$ | $6.72\pm 0.31$ | $6.01\pm 0.25$ | $7.69\pm 0.50$ | —
Stasińska | | | | | |
SMC | $0.094\pm 0.025$ | $7.74\pm 0.50$ | — | — | $7.46\pm 0.37$ | $7.10\pm 0.40$
LMC | $0.090\pm 0.032$ | $8.10\pm 0.31$ | — | — | $7.76\pm 0.45$ | $7.44\pm 0.41$
Leisy | | | | | |
SMC | $0.093\pm 0.025$ | $8.01\pm 0.29$ | $6.86\pm 0.67$ | $5.57\pm 0.27$ | $7.39\pm 0.47$ | $7.14\pm 0.36$
LMC | $0.105\pm 0.035$ | $8.26\pm 0.35$ | $7.13\pm 0.67$ | $5.99\pm 0.26$ | $7.77\pm 0.57$ | $7.46\pm 0.48$
Orion | $0.098\pm 0.004$ | $8.55\pm 0.07$ | $7.02\pm 0.10$ | $6.52\pm 0.18$ | $7.78\pm 0.12$ | $7.82\pm 0.16$
Sun | $0.092\pm 0.009$ | $8.80\pm 0.11$ | $7.26\pm 0.08$ | $6.48\pm 0.11$ | $7.97\pm 0.07$ | $8.08\pm 0.01$
30 Dor | $0.087\pm 0.001$ | $8.33\pm 0.02$ | $6.84\pm 0.10$ | $6.09\pm 0.10$ | $7.05\pm 0.08$ | $7.65\pm 0.06$
NGC 346 | — | $8.15$ | $6.40$ | $5.82$ | $6.81$ | $7.32$
### 0.3.2 Abundances of individual nebulae
Figure 1: Abundances of O/H (solid dots), N/H (empty triangles) and Ne/H
(empty circles) from the sample by Stasińska et al. (1998) as a function of
data from the IAG/USP group for the SMC. Figure 2: The same as Fig. 1 for the
LMC. No Ne/H data is available in the IAG sample for this galaxy. Figure 3:
Abundances of O/H (solid dots), S/H (stars), Ar/H (crosses), N/H (empty
triangles), and Ne/H (empty circles) from the sample by Leisy & Dennefeld
(2006) as a function of data from the IAG/USP group for the SMC. Figure 4: The
same as Fig. 3 for the LMC. No Ne/H data is available in the IAG sample.
In order to illustrate the internal agreement of the three PN samples
considered in this work, we present in Figs. 1 and 2 the abundances of O/H
(solid dots), N/H (empty triangles) and Ne/H (empty circles) as derived by
Stasińska et al. (1998) as a function of the IAG/USP data, for the SMC and LMC
nebulae, respectively. The same comparisons are shown in Figs. 3 and 4 taking
into account the data by Leisy & Dennefeld (2006), in which case we also
include abundances of S/H (stars) and Ar/H data (crosses). An average error
bar is included at the lower right corner of the figures. The agreement of
both samples with our own data is generally very good, within the average
uncertainties of the abundance data, which are about 0.1 to 0.2 dex for the
best derived abundances, and of 0.2 to 0.3 for the less accurate element
ratios. Some scatter is to be expected, especially taking into account that
the abundances of several nebulae are flagged as uncertain (:) by Leisy &
Dennefeld (2006). The main discrepancies between the IAG data and the results
by Stasińska et al. (1998) occur for a few objects in the SMC, for which our
O/H and Ne/H abundances differ by an amount larger than the average
uncertainty (cf. Fig. 1), while for the LMC a small group of nebulae have
higher N/H abundances as derived by Stasińska et al. (1998) (cf. Fig. 2).
Concerning the Leisy & Dennefeld (2006) sample, the main discrepancies are
restricted to some S/H data, as can be seen from Figs. 3 and 4. The origin of
these discrepancies is not clear, but it should be stressed that the vast
majority of the objects in common in the 3 samples have similar results, as
illustrated in Figs. 1 to 4.
### 0.3.3 Metallicity differences: the Galaxy and the Magellanic Clouds
The PN abundances of the heavy elements O, S, Ne, and Ar as given in Table 1
are expected to reflect the average metallicities of the Magellanic Clouds,
which are a few dex lower than in the Milky Way, as these elements are not
produced by the PN progenitor stars. This can be confirmed by comparing the PN
abundances with the abundances in the Orion Nebula, which can be taken as
representative of the present heavy element abundances in the Galaxy. From the
compilation by Stasińska (2004), we obtain the abundances given at the end of
Table 1. For comparison purposes, the average solar abundances from the same
source are also included. For the Orion Nebula and the Sun the uncertainties
given are not the intrinsic uncertainties of the data, but the dispersion of
the measurements in the recent literature as considered by Stasińska (2004).
It can be seen that the Orion Nebula abundances are higher than the Magellanic
Clouds PN by about 0.2 to 0.5 dex for the LMC and 0.5 to 0.8 dex for the SMC
in the case of oxygen. The average for the Orion Nebula, $\epsilon_{ON}({\rm
O})\simeq 8.55$, is also essentially the same as in the galactic PN,
$\epsilon_{PN}({\rm O})\simeq 8.65$ (cf. Maciel et al. 2006a). For S, Ne, and
Ar a similar comparison is obtained, although the S abundance in the LMC is
actually somewhat higher than in the Orion Nebula according to the data by
Leisy & Dennefeld (2006). The difference in the abundances is also smaller in
the case of nitrogen, which is a clear evidence of the N enhancement in the PN
progenitor stars.
### 0.3.4 The metallicity distribution
Figure 5: The O/H abundance distribution of the Magellanic Clouds from the
IAG/USP data. Figure 6: The same as Fig. 5 for the data by Stasińska et al.
(1998). Figure 7: The same as Fig. 5 for the data by Leisy & Dennefeld (2006).
The available data on PN in the Magellanic Clouds can be used to infer the
metallicity distribution in these systems, on the basis of the derived
abundances of O, S, Ne, and Ar. A comparison of the distributions in different
systems can be used to infer their average metallicities, with consequences on
the star formation rates. As an example, Figs. 5. 6 and 7 show the O/H
distribution in the Magellanic Clouds according to the three samples
considered in this work. The metallicity difference between the SMC and the
LMC can be clearly observed in all samples, amounting to about 0.4 to 0.5 dex
in average. The difference is especially well defined in our sample, as shown
in Fig. 5. In a comparison with the Milky Way, the galactic disk nebulae
extend to a higher metallicity, up to $\epsilon({\rm O})\simeq 9.2$, while the
LMC objects reach $\epsilon({\rm O})\simeq 8.8$ and the lowest metallicities
in the SMC are about $\epsilon({\rm O})\simeq 7.0$. Concerning the remaining
elements that are not affected by the evolution of the PN progenitor stars,
the Ar/H abundance distribution has a similar pattern, while the S/H data is
less clear, as already mentioned. We will discuss this element in more detail
in the next section. For Ne/H we have no IAG data for the LMC, but the larger
Leisy & Dennefeld (2006) sample clearly confirms the 0.4 to 0.5 dex difference
between the LMC and the SMC.
The metallicity distribution of the PN as shown in Figs. 5, 6, and 7 can also
be compared with galactic data, both for disk and bulge PN. Cuisinier et al.
(2000) considered a sample of 30 bulge nebulae and a compilation containing
about 200 disk PN, and concluded that both O/H distributions are similar,
peaking around 8.7–8.8 dex, and extending form $\epsilon({\rm O})\simeq 8.0$
to $\epsilon({\rm O})\simeq 9.2$. More recently, Escudero et al. (2004)
obtained a similar distribution using a bulge sample twice as large, which
extended to about 7.5 dex (see also Costa et al. 2008). According to Figs.
5–7, the O/H distributions are displaced relative to the Milky Way by
approximately 0.4 and 0.7 dex towards shorter metallicities for the LMC and
SMC, respectively, in good agreement with the results discussed in Section
4.1.
### 0.3.5 Abundance correlations: elements not produced by the progenitor
stars
Photoionized nebulae, comprising both PN and HII regions, are extremely useful
to study chemical abundances in different systems. While HII regions reflect
the present chemical composition of star-forming systems, PN is helpful to
trace the time evolution of the abundances, especially when an effort is made
to establish their age distribution (see for example Maciel et al. 2006b). The
elements S, Ar and Ne are probably not produced by the PN progenitor stars, as
they are manufactured in the late evolutionary stages of massive stars.
Therefore, S, Ar, and Ne abundances as measured in PN should reflect the
interstellar composition at the time the progenitor stars were formed. Since
in the interstellar medium of star-forming galaxies such as the Magellanic
Clouds the production of O and Ne is believed to be dominated by type II
supernovae, we may conclude that the original O and Ne abundances are not
significantly modified by the stellar progenitors of bright PN.
The variation of the ratios S/H, Ar/H and Ne/H with O/H usually show a good
positive correlation for all studied systems in the Local Group, with similar
slopes close to unity. The main differences lie in the average metallicity of
the different galaxies, which can be inferred from the observed metallicity
range, as we have seen in the previous section.
Fig. 8 shows the Ne/H ratio as a function of O/H for the SMC, while Fig. 9
corresponds to the LMC. In these figures we include the combined samples
mentioned in Section 2 as follows: IAG/USP data (filled circles), Stasińska et
al. (1998) (empty circles), and Leisy & Dennefeld (2006) (crosses). Average
error bars are included at the lower right corner of the figures. It can be
seen that the correlation is very good, with a slope in the range 0.8–0.9 in
both cases. The Ne/H $\times$ O/H relation is probably the best example
provided by PN regarding the nucleosynthesis in massive stars. This
correlation is very well defined, as shown in Figs. 8 and 9, and is
essentially the same as derived from HII regions in different star forming
galaxies of the Local Group, including the Milky Way, and in emission line
galaxies as well, as clearly shown by Richer & McCall (2006) and Richer (2006,
see also Henry et al. 2006).
The Ar/H data shows a similar correlation with O/H, as can be seen from Figs.
10 and 11, but the correlation is poorer, which may be due to the fact that
the samples are smaller, since Stasińska et al. (1998) do not present argon
data. Again, the main discrepancy lies in the S/H data, as can be seen from
Figs. 12 and 13. Although most objects define a positive correlation, which is
especially true for the LMC, the dispersion is much larger in the S/H data
compared to the previous elements, again suggesting that a problem remains in
the interpretation of the S/H abundances in planetary nebulae. In particular,
both the IAG/USP and Leisy & Dennefeld data suggest a scattering diagram on
the S/H $\times$ O/H plane for the SMC, with an average abundance around
$\epsilon({\rm S/H})=\log({\rm S/H})+12\simeq 7.0$. A weaker correlation
involving sulfur is to be expected, since the diagnostic lines for this
element are weaker than e.g. for oxygen or neon. However, the real situation
may be more complex, so that a more detailed discussion is appropriate.
Figure 8: Distance-independent correlation of Ne/H $\times$ O/H for the SMC.
Filled circles: IAG/USP data; empty circles: Stasińska et al. (1998); crosses:
Leisy & Dennefeld 2006). Figure 9: The same as Fig. 8, for the LMC. No IAG/USP
data is available for this object. Figure 10: Distance-independent correlation
of Ar/H $\times$ O/H for the SMC. Symbols are as in Fig. 8. Figure 11: The
same as Fig. 10, for the LMC. Figure 12: Distance-independent correlation of
S/H $\times$ O/H for the SMC. Symbols are as in Fig. 8. Figure 13: The same as
Fig. 12, for the LMC. Figure 14: Comparison of the Spitzer results by Bernard-
Salas et al. (2008) and the IAG/USP sample. circles: SMC, Ne/H data;
triangles: SMC, S/H abundances; crosses: LMC, S/H data.
A hint on the problem of the sulfur abundances in PN can be obtained by
comparing our S/H abundances with the recent determinations by Bernard-Salas
et al. (2008), who have presented Ne/H and S/H abundances for 25 PN in the
Magellanic Clouds using Spitzer data. These results have been obtained on the
basis of high-resolution spectroscopic observations in the infrared, and are
in principle more accurate compared with the abundances of our present sample,
since the uncertainties in the electron temperatures do not affect the
infrared lines, the interstellar extinction effects are smaller, and the use
of the often uncertain ionization correction factors is greatly reduced (cf.
Bernard-Salas et al. 2008). A comparison of the Ne/H and S/H abundances from
this source and those by the IAG/USP group is shown in Fig. 14, where the
adopted uncertainties are also shown. There are eleven objects in common,
which is a small but representative sample. In the figure, the circles refer
to Ne/H and the triangles for S/H for the SMC, while the crosses are S/H data
for PN in the LMC.
It can be seen that the Ne/H abundances show a very good agreement with the
infrared data, while for S/H there is a tedency for our values to be larger
than those by Bernard-Salas et al. (2008). Although the differences are not
very large except for a few nebulae, it may be suggested that the S/H data
presented here should be considered as upper limits. Inspecting Figs. 12 and
13, that would be expected especially for those nebulae having lower oxygen
abundances, which would explain the scatter diagram observed in Fig. 12. In
Bernard-Salas et al. (2008), a similar comparison of the Spitzer S/H
abundances with data by Leisy & Dennefeld (2006) was presented, and it was
shown that the latter are also systematically larger than the infrared
results. This was interpreted by Bernard-Salas et al. (2008) as the ionization
correction factors used by Leisy & Dennefeld (2006) overestimated the
contribution of the S+3 ion to the total sulfur abundances. In fact, several
of the S/H values for Magellanic Cloud PN in Leisy & Dennefeld (2006) are
flagged as upper limits. While commenting on the large dispersion of their
$\log{\rm S/H}\ \times\log{\rm O/H}$ plot, the authors stress that the sulfur
abundances are affected by several problems, such as the lack of [SIV] or
[SIII] lines, blending with oxygen lines, and innacuracies in the adopted
electron temperatures. By considering only the nebulae for which the sulfur
data is more reliable, Leisy & Dennefeld (2006) obtain a somewhat reduced
dispersion on the $\log{\rm S/H}\ \times\log{\rm O/H}$ plane, but it is still
concluded that the sulfur abundances are not good metallicity indicators for
Magellanic Cloud planetary nebulae.
A discussion of the sulfur abundance problem in PN was recently given by Henry
et al. (2004, 2006). These authors identified a so-called “sulfur anomaly”, or
the lack of agreement of the S/H ratio in PN with corresponding data from HII
regions and other objects. From an analysis of the abundances in Milky Way
planetary nebulae, HII regions and blue compact galaxies, it was suggested
that the origin of the “sulfur anomaly” is probably linked to the presence of
S+3 ions, which would affect the total sulfur abundances, at least in some
nebulae. According to this view, the abundances of at least some of the
galactic PN are underestimated, in the sense that the measured S/H ratio is
lower than expected on the basis of the derived O/H abundances. If this
explanation is valid for Magellanic Cloud PN, it would probably affect those
objects with higher O/H ratios, so it is an alternative to the previous
suggestion based on the comparison of optical abundances with infrared data.
However, other factors may play a role, such as the weakness of the sulfur
lines, the assumptions leading to the ionization correction factors, etc., so
that this problem deserves further investigation.
### 0.3.6 Abundance correlations: elements produced by the progenitor stars
Considering now the elements that are produced during the evolution of the PN
progenitor stars, namely, He and N, Figs. 15 and 16 show the derived
correlations of N/H and O/H for the SMC and LMC, respectively. As expected, a
positive correlation is observed, which is especially evident in the case of
the LMC, but the dispersion of the data is larger than in the case of Ne and
Ar. This is due to the fact that the PN display both the original N present at
the formation of the star and the contamination that is dredged up at the AGB
branch of the stellar evolution. In other words, the N/H ratio measured in PN
shows some contamination, or enrichment, in comparison with the original
abundances in the progenitor star.
Figure 15: Distance-independent correlation of N/H $\times$ O/H for the SMC.
Symbols are as in Fig. 8. Figure 16: The same as Fig. 15, for the LMC. Figure
17: Distance-independent correlation of N/H $\times$ He/H for the SMC. Symbols
are as in Fig. 8. Figure 18: The same as Fig. 17, for the LMC. Figure 19:
Distance-independent correlation of N/O $\times$ He/H for the SMC. Symbols are
as in Fig. 8. Figure 20: The same as Fig. 19, for the LMC. Figure 21:
Distance-independent correlation of N/O $\times$ O/H for the SMC. Symbols are
as in Fig. 8. Figure 22: The same as Fig. 21, for the LMC.
An estimate of the nitrogen enrichment from the PN progenitor stars can be
made by comparing the average N/H abundances of Table 1 with those of HII
regions. The Orion value given in the table is similar to the PN abundances
for the 3 samples considered, but HII regions in the lower metallicity
Magellanic Clouds have accordingly lower nitrogen abundances. As an example,
for 30 Doradus, the brightest HII region in the LMC, Peimbert (2003) estimates
$\epsilon({\rm N})=7.05$ based on echelle spectrophotometry, assuming no
temperature fluctuations ($t^{2}=0.00$). Comparing this result with the data
of Table 1, an average enrichment of about 0.6–0.7 dex is obtained for the N/H
ratio. Concerning HII regions in the SMC, Relaño et al. (2002) estimate
$\epsilon({\rm N})=6.81$ for NGC 346 on the basis of photoionization models,
which implies an enrichment of 0.5–0.6 dex for the PN samples listed in Table
1. These enrichment factors may be affected by the chemical evolution of the
host galaxy, which includes the average increase of the metallicity as the
galaxy evolves, but it is interesting that similar factors are obtained both
for the LMC and SMC. The quoted values for 30 Dor and NGC 346 are included at
the bottom of Table 1, as representative of HII in the Magellanic Clouds.
Figs. 17 and 18 show the N/H abundances as a function of the He/H ratio, while
Figs. 19 and 20 are the corresponding plots for N/O as a function of He/H. As
pointed out in the literature (cf. Kwok 2000), these ratios present
enhancements relative to the average interstellar values. The dispersion is
again large, but a positive correlation can also be observed, as expected,
since the same processes that increase the nitrogen abundances in PN also
affect the He/H ratio. A plot similar to Figs. 19 and 20 was presented by Shaw
(2006), in an effort to separate PN of different morphologies. In the LMC,
some objects in the sample by Stasińska et al. (1998) have very low He
abundances while the N/O ratio is normal, suggesting that neutral helium may
be present in these objects. As pointed out by Maciel et al. (2006a), the N/O
$\times$ He/H ratios in the Magellanic Clouds support the correlation observed
in the Milky Way, but the N/O ratio is comparatively lower. The O/H ratio
corresponding to the SMC is also lower, which can be interpreted as an
evidence that the lower metallicity environment in the SMC leads to a smaller
fraction of Type I PN, which are formed by the more massive stars in the
Intermediate Mass Star bracket (cf. Stanghellini et al. 2003).
Finally, Figs. 21 and 22 show the N/O ratios as a function of the O/H
abundances. The conversion of oxygen into nitrogen by the ON cycling in the PN
progenitor stars has been suggested in the literature as an explanation for
the anticorrelation between N/O and O/H in planetary nebulae (cf. Costa et al.
2000, Stasińska et al. 1998, Perinotto et al. 2006). This relation is
approximately valid on the basis of PN data in several galaxies of the Local
Group, as discussed by Richer & McCall (2006). From Figs. 21 and 22, we
conclude that the Magellanic Cloud data support such anticorrelation,
particularly in the case of the SMC. As discussed by Maciel et al. (2006a) the
Milky Way data define a mild anticorrelation, especially in the case of
$\epsilon({\rm O})=\log({\rm O/H})+12>8.0$, which is better defined by the
SMC/LMC.
Acknowledgements. This work was partly supported by FAPESP and CNPq.
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|
arxiv-papers
| 2009-04-16T17:42:07 |
2024-09-04T02:49:01.915346
|
{
"license": "Public Domain",
"authors": "W. J. Maciel, R. D. D. Costa, T. E. P. Idiart",
"submitter": "Walter J. Maciel",
"url": "https://arxiv.org/abs/0904.2549"
}
|
0904.2718
|
11institutetext: Osservatorio Astronomico di Padova, INAF, vicolo
dell’Osservatorio 5, I-35122 Padova, Italy
11email: marco.gullieuszik@oapd.inaf.it,enrico.held@oapd.inaf.it
22institutetext: European Southern Observatory, Casilla 19001, Santiago 19,
Chile
22email: isaviane@eso.org 33institutetext: Joint Astronomy Centre, 660 N.
A’ohoku Place, University Park, Hilo, HI 96720, USA
33email: l.rizzi@jach.hawaii.edu
# New constraints on the chemical evolution of the dwarf spheroidal galaxy Leo
I from VLT spectroscopy ††thanks: Based on data collected at the European
Southern Observatory, Paranal, Chile, Proposals No. 69.D-0455 and 71.D-0219
M. Gullieuszik 11 E. V. Held 11 I. Saviane 22 L. Rizzi 33
(Received …; accepted …)
We present the spectroscopy of red giant stars in the dwarf spheroidal galaxy
Leo I, aimed at further constraining its chemical enrichment history.
Intermediate-resolution spectroscopy in the Ca ii triplet spectral region was
obtained for 54 stars in Leo I using FORS2 at the ESO Very Large Telescope.
The equivalent widths of Ca ii triplet lines were used to derive the
metallicities of the target stars on the [Fe/H] scale of Carretta & Gratton,
as well as on a scale tied to the global metal abundance, [M/H]. The
metallicity distribution function for red giant branch (RGB) stars in Leo I is
confirmed to be very narrow, with mean value [M/H]$\simeq-1.2$ and dispersion
$\sigma_{\rm[M/H]}\simeq 0.2$. By evaluating all contributions to the
measurement error, we provide a constraint to the intrinsic metallicity
dispersion, $\sigma_{\rm[M/H],0}=0.08$. We find a few metal-poor stars (whose
metallicity values depend on the adopted extrapolation of the existing
calibrations), but in no case are stars more metal-poor than [Fe/H] $=-2.6$.
Our measurements provide a hint of a shallow metallicity gradient of $-0.27$
dex Kpc-1 among Leo I red giants. The gradient disappears if our data are
combined with previous spectroscopic datasets in the literature, so that any
firm conclusions about its presence must await new data, particularly in the
outer regions. By combining the metallicities of the target stars with their
photometric data, we provide age estimates and an age-metallicity relation for
a subset of red giant stars in Leo I. Our age estimates indicate a rapid
initial enrichment, a slowly rising metal abundance –consistent with the
narrowness of the metallicity distribution– and an increase of $\sim 0.2$ dex
in the last few Gyr. The estimated ages also suggest a radial age gradient in
the RGB stellar populations, which agrees with the conclusions of a parallel
study of asymptotic giant branch stars in Leo I from near-infrared photometry.
Together, these studies provide the first evidence of stellar population
gradients in Leo I.
###### Key Words.:
Galaxies: dwarf spheroidal – Galaxies: individual (Leo I) – Stars: abundances
– Local Group – Galaxies: stellar content
††offprints: M. Gullieuszik Figure 1: Digitized Sky Survey image of the Leo I
field, centred at $10^{h}08^{m}28\aas@@fstack{s}1$, $+12\degr 18\arcmin
23\arcsec$ (J2000). Starred symbols mark the 4 metal-poor member stars, the
circles are other member stars, squares represent interlopers.
## 1 Introduction
A first-order estimate of the distribution of stellar metallicities in
resolved galaxies can be obtained from photometry of red giant branch (RGB)
stars. However, young metal-rich stars have the same colours as older metal-
poor stars, a phenomenon known as “age-metallicity degeneracy”. Although
improved photometric metallicity estimates can be obtained by combining
optical and near-infrared photometry, with reduced degeneracy effects
(Gullieuszik et al. 2007, 2008), the problem is never entirely overcome, and
age and metallicity inextricably contribute to the observed colour.
More direct measurements of the stellar metallicity distribution function
(MDF) are obtained from spectroscopy. The most accurate determinations come
from high-resolution abundance analysis which provides information on the
relative abundances of chemical elements, and therefore the strongest
constraints to the galaxy evolution models (e.g. Tolstoy et al. 2003;
Lanfranchi & Matteucci 2007). High-resolution spectroscopy, however, is
limited to the brightest stars in nearby dwarf galaxies, and becomes
unfeasible for more distant galaxies even within the Local Group and using
10m-class telescopes. A viable alternative for obtaining metallicities for a
large number of stars (as needed to derive statistically significant stellar
MDFs) is using low- or intermediate-resolution spectroscopy. The infrared Ca
triplet (CaT) method, originally devised to measure metallicities of stars in
Galactic globular clusters (Armandroff & Zinn 1988; Armandroff & Da Costa
1991; Rutledge et al. 1997b), has now been applied quite extensively to RGB
stars in dwarf galaxies. Using this method, values of [Fe/H] for dwarf
galaxies have been obtained for Fornax (Tolstoy et al. 2001; Pont et al. 2004;
Battaglia et al. 2006), Carina (Koch et al. 2006), Sculptor (Tolstoy et al.
2001), Leo II (Bosler et al. 2007; Koch et al. 2007a), NGC 6822 (Tolstoy et
al. 2001) and the LMC (Cole et al. 2005).
In this paper we present new CaT spectroscopy for the dwarf spheroidal (dSph)
galaxy Leo I. Leo I is known to have formed the bulk of its stars at an
intermediate epoch (e.g., Gallart et al. 1999). Along with Leo II, Leo I is
one the most distant dSph satellites of the Milky Way, for which the influence
of tidal interaction with our Galaxy on evolution must have been more limited
than for nearby galaxies such as Carina or Fornax. As such, deriving its
chemical enrichment history is of foremost importance for our knowledge of the
evolution of dSph galaxies. Gallart et al. (1999) derived the metallicity and
reconstructed the star-formation history (SFH) of Leo I from HST/WFPC2 data –
they found a metallicity ranging from [Fe/H] $=-1.4$ to [Fe/H] $=-2.3$. Using
the same data set, Dolphin (2002) found higher metallicity, ranging from
[Fe/H] $=-0.8$ to [Fe/H] $=-1.2$. From the colour of RGB stars, Held et al.
(2000) derived a mean metallicity [Fe/H] $\sim-1.6$ on the Zinn & West (1984)
scale.
The metallicity distribution of RGB stars in Leo I has recently been
investigated by Bosler et al. (2007) and Koch et al. (2007b). Bosler et al.
(2007) used the CaT method to analyse Keck-LRIS spectra of 102 RGB stars, and
found ${\rm[Fe/H]}=-1.34$ on the [Fe/H] scale of Carretta & Gratton (1997)
(hereafter, CG97). The authors also proposed a new calibration based on Ca
abundance, yielding a mean metallicity $\text{[Ca/H]}=-1.34$ ($\sigma=0.21$).
Also using measurements of the Ca ii triplet lines, Koch et al. (2007b) found
a mean metallicity [Fe/H] $=-1.31$ on the CG97 scale for 58 red giants. In
both studies, the MDF is well described by a Gaussian function with a
1$\sigma$ width of 0.25 dex, and a full range in [Fe/H] of approximately 1
dex.
Leo I is an interesting target also because, to date, it is one of the few
Local Group dwarf galaxies showing scarce evidence of a population gradient.
Held et al. (2000) found that the old horizontal branch stars of Leo I are
radially distributed as the intermediate-age helium-burning stars; Koch et al.
(2007b) found no significant metallicity radial gradient. Different
conclusions were found in our companion paper, based on near-infrared
photometry (Held et al. 2009), showing that intermediate-age asymptotic giant
branch (AGB) stars are more concentrated in the central region than old RGB
stars.
A new, independent data set of metallicities for Leo I stars, also based on
the CaT method, was obtained by us at the ESO VLT using high signal-to-noise
FORS2 spectra. The new sample has negligible overlap with the previous data
sets, thus effectively increasing the number of Leo I stars with direct
metallicity measurements. Using the new data, this paper provides an
independent determination of the MDF of Leo I and further constraints on its
evolution, based on an analysis of metallicity and age gradients and the age-
metallicity relation.
## 2 Observations and reduction
### 2.1 Target selection
Our targets were selected from the colour-magnitude diagram (CMD) of Leo I.
For the central region of the galaxy, we relied upon $B$,$V$-band photometry
from Held et al. (2000), obtained with the EMMI instrument at the NTT
telescope at the ESO La Silla Observatory. For the stars in the outer regions,
we used the $BV$ photometry originally obtained for a study of RR Lyrae
variable stars in Leo I (Held et al. 2001), based on observations carried out
with the Wide Field Imager at the 2.2m ESO-MPI telescope.
The spectroscopic targets were selected among the brightest RGB stars of Leo
I, down to 1 mag below the RGB tip. We avoided any colour constraints that
might bias the age/metallicity distribution. After mask design (for which we
were guided only by geometric constraints) we were left with 61 targets in 4
masks.
The identifiers, coordinates, and $BV$ photometry of the stars in our final
sample (excluding a few targets with too low S/N ratio to allow any
measurements) are listed in Table New constraints on the chemical evolution of
the dwarf spheroidal galaxy Leo I from VLT spectroscopy ††thanks: Based on
data collected at the European Southern Observatory, Paranal, Chile, Proposals
No. 69.D-0455 and 71.D-0219, and the targets are shown in Fig. 1. The location
of the target stars in the CMD of Leo I is shown in Fig. 2.
Figure 2: Target stars in the colour-magnitude diagram of Leo I. The filled
squares represent red giant stars in Leo I, the diamonds are the 4 metal-poor
red giant members, while crosses mark non-members.
### 2.2 Observations
Table 2: Log of the observations. Field | Night | $t_{\text{exp}}$ (sec)
---|---|---
NGC 4590 | 6 | May | 2002 | $15+60$
NGC 5927 | 6 | May | 2002 | $60+300$
NGC 6171 | 6 | May | 2002 | $10+60$
NCG 5904 | 3 | May | 2003 | $60+300$
NGC 6397 | 5 | May | 2003 | $60+300$
NGC 6528 | 5 | May | 2003 | $60+300$
NGC 4372 | 24 | May | 2003 | $60+300$
NGC 6752 | 24 | May | 2003 | $60+300$
Leo I field U | 4 | May | 2003 | $2\times 2870$
Leo I field R | 20,21 | Dec | 2003 | $2\times 2870$
Leo I field D | 22 | May | 2003 | $2\times 2870$
Leo I field L | 16,25 | Jun | 2003 | $2\times 2870$
The observations were carried out in service mode in two runs between May 2002
and December 2003 using FORS2, the multi-mode optical instrument mounted on
the Cassegrain focus of the Yepun (VLT-UT4) 8.2m telescope at the ESO Paranal
Observatory. We used FORS2 in MXU mode with the 1028z+29 grism and the
OG590+32 order-blocking filter. With this setup, the spectral coverage is
approximately 7700 Å to 9500 Å, with a dispersion 0.85 Å pixel-1. The selected
targets were observed with 4 masks using $0\aas@@fstack{\prime\prime}80$
slits. For each Leo I mask, two spectra were taken. The observing log and
exposure times are given in Table 2. We observed also RGB stars in 8 Galactic
globular clusters (GCs) in a wide range of metallicity with the same
instrumental setup in order to calibrate our measurements onto a known
metallicity scale. Two different exposure times were used (long and short
exposure) to prevent saturation of the brightest RGB stars. The photometry of
the GC stars was taken from the data compilation of Rutledge et al. (1997b).
Figure 3: Examples of normalised, background-subtracted spectra of RGB stars
in Leo I (stars #17, #19, #54, from top to bottom). The spectra are shown on a
rest-frame wavelength scale and vertically shifted for clarity.
### 2.3 Data Reduction
The basic reduction of multi-object spectra was performed using standard
procedures in IRAF 111 IRAF is distributed by the National Optical Astronomy
Observatory, which is operated by the Association of Universities for Research
in Astronomy (AURA) under cooperative agreement with the National Science
Foundation.. Bias and flat-field corrections were applied to all images using
the ccdproc task. Due to the long exposure times, the scientific frames for
Leo I contain a large number of cosmic ray hits. These were effectively
cleaned using the IRAF program lacos (van Dokkum 2001) on the bias-subtracted,
flat-fielded images. The multi-object spectra were extracted with the apall
task and wavelength calibrated using HeNeAr lamp exposures taken at the end of
each night.
The two spectra taken for each Leo I target were combined to increase the S/N
ratio. However, we also retained the two individual spectra to estimate the
uncertainties in the wavelength calibration and line strengths. Finally, the
continuum was normalised in the region between 8400 Å and 8800 Å, by excluding
in the process the Ca ii and other relatively strong absorption lines. Typical
sky-subtracted spectra of three Leo I stars with different metallicities are
shown in Fig. 3.
The average signal-to-noise ratio per pixel was calculated from the rms of the
combined spectra in two wavelength windows free from strong spectral features,
8580–8620 Å and 8710–8750 Å. The values were checked against those measured on
the raw spectra, and found to be consistent within a few percent. The S/N
ratio, listed in Table New constraints on the chemical evolution of the dwarf
spheroidal galaxy Leo I from VLT spectroscopy ††thanks: Based on data
collected at the European Southern Observatory, Paranal, Chile, Proposals No.
69.D-0455 and 71.D-0219, is $\gtrsim 20$ for all stars in our sample, with a
mean value of $\sim 50$.
## 3 Radial velocities and membership
Figure 4: Heliocentric radial velocities of stars in the Leo I field. The
distribution is fitted with a Gaussian centred at 271 km s-1and with
dispersion 13.7 km s-1. The vertical dashed lines represents the $3\sigma$
limits used to select members of Leo I.
Radial velocities were measured for target stars to establish their
membership. Only the two reddest lines were used to this purpose, since the
bluest line of the Ca triplet is weak and, for the systemic velocity of Leo I,
overlapping with a strong sky line. The line wavelengths were obtained from
the central values $\lambda_{m}$ of the fitted profile (see Eq. 1 below), and
compared with the laboratory air wavelengths 8542.09 Å and 8662.14 Å. A radial
velocity was measured for each individual spectrum of each star by combining
the measurements of the two lines $\lambda_{8542}$ and $\lambda_{8662}$. Then,
the radial velocity was calculated as the mean of the two values independently
measured from the single spectra. The results, corrected to heliocentric
velocities using the rvcorr task, are given in Table New constraints on the
chemical evolution of the dwarf spheroidal galaxy Leo I from VLT spectroscopy
††thanks: Based on data collected at the European Southern Observatory,
Paranal, Chile, Proposals No. 69.D-0455 and 71.D-0219. The distribution of
radial velocities, shown in Fig. 4, is well fitted by a Gaussian function
centred at 271 km s-1 with a dispersion 13.7 km s-1. All but three stars have
heliocentric radial velocities within $3\sigma$ of the peak, and therefore are
considered members of Leo I (members have ID $<100$ in Table New constraints
on the chemical evolution of the dwarf spheroidal galaxy Leo I from VLT
spectroscopy ††thanks: Based on data collected at the European Southern
Observatory, Paranal, Chile, Proposals No. 69.D-0455 and 71.D-0219).
Radial velocity errors (given in Col. 7 of Table New constraints on the
chemical evolution of the dwarf spheroidal galaxy Leo I from VLT spectroscopy
††thanks: Based on data collected at the European Southern Observatory,
Paranal, Chile, Proposals No. 69.D-0455 and 71.D-0219) were estimated by
calculating the differences between radial velocities measured on the two
individual spectra. The distribution of the differences is approximately
Gaussian with a mean $-6.4$ km s-1 and dispersion 10.4 km s-1. The mean
velocity difference provides a good estimate of the mask-to-mask systematic
errors, which include wavelength calibration errors and the centring errors of
the targets on the slitlets, while the standard deviation represents a
combination of random errors. We assume
$\sigma_{v}=\sigma_{v_{1}-v_{2}}/\sqrt{2}=7.4$ km s-1 as a good approximation
to the standard error of the radial velocity measured on the combined
spectrum. Therefore our systemic radial velocity of Leo I is 271 $\pm\,6.4$
(systematic) $\pm\,7.4$ (random) km s-1. Our error estimates are consistent
with the 0.85 Å pixel-1 spectral resolution provided by our instrumental
setup, which corresponds to a resolution in radial velocity of
$\leavevmode\nobreak\ \sim 30$ km s-1pixel-1 in the CaT wavelength range. Koch
et al. (2007b) measured 284.2 km s-1 with a velocity dispersion of 9.9 km s-1,
while $282.6\pm 9.8$ km s-1 was the value measured by Bosler et al. (2007).
Most recently, Mateo et al. (2008) obtained a mean heliocentric velocity
$282.9\pm 0.5$ km s-1 and a dispersion $9.2\pm 0.4$ km s-1 from echelle
spectroscopy of 328 Leo I members. The systematic difference of $\sim 10$ km
s-1 between our mean velocity estimate and previous results is consistent with
pointing errors. In the CaT wavelength range, a velocity of 10 km s-1
corresponds to a shift of $\sim 1/3$ pixel on the detector (see above), or
$0\aas@@fstack{\prime\prime}08$ on the sky.
## 4 Equivalent widths and metallicity
### 4.1 Equivalent width measurements
Figure 5: Upper panel: correlation between two independent EW measurements on
individual spectra of Leo I stars. The dashed lines show the $3\sigma$
interval. Lower panel: histogram of the differences between the two
measurements, fitted by a Gaussian profile with $\sigma_{\Delta\Sigma W}=0.44$
Å.
We measured the equivalent widths (EWs) of CaT lines in the spectra of target
stars in Leo I and the calibrating GCs as follows. We first normalised the
spectra over a wavelength interval encompassing, for each line, the side bands
defined by Armandroff & Da Costa (1991). Equivalent widths were then measured
for the two stronger CaT lines in the co-added spectra by fitting a model
profile over the line central bandpasses as defined by the same authors.
Following Cole et al. (2004), the fitted model is the sum of a Gaussian and a
Lorentzian profiles with a common line centre $\lambda_{m}$,
$F(\lambda)=1-A_{G}\exp\left[-\frac{(\lambda-\lambda_{m})^{2}}{2\sigma^{2}}\right]\\\
-A_{L}\left[\frac{(\lambda-\lambda_{m})^{2}}{\Gamma^{2}}+1\right]^{-1}$ (1)
with the best-fit parameters determined using a Levenberg-Marquardt least-
squares algorithm (coded in an idl procedure by C. Markwardt222
http://cow.physics.wisc.edu/$\sim$craigm/idl/idl.html). The Ca ii line
strength was then defined as the unweighted sum of the two equivalent widths,
$\Sigma W=EW_{8542}+EW_{8662}$ (2)
The sum equivalent widths are given in Table New constraints on the chemical
evolution of the dwarf spheroidal galaxy Leo I from VLT spectroscopy ††thanks:
Based on data collected at the European Southern Observatory, Paranal, Chile,
Proposals No. 69.D-0455 and 71.D-0219 for the red giants in Leo I and in Table
4 for the RGB stars in the template GCs.
To estimate the equivalent width measurement errors, we also measured the CaT
line strengths independently on the individual spectra of each star in Leo I.
A comparison of the two $\Sigma W$ measurements is shown in Fig. 5, where they
appear to be well correlated. The histogram of the differences $\Delta_{\Sigma
W}=(\Sigma W_{2}-\Sigma W_{1})$ (shown in Fig. 5, lower panel) is well fitted
by a Gaussian function centred at $\langle\Delta_{\Sigma W}\rangle=0.03$ and
with a standard deviation $\sigma_{\Delta\Sigma W}=0.44$ Å. Since the two
individual spectra have comparable $S/N$ ratio, we adopt $\sigma_{\Sigma
W}=\sigma_{\Delta\Sigma W}/\sqrt{2}=0.31$ Å as our error estimate for $\Sigma
W$ measured on the combined spectrum. For comparison, the half-range
$\epsilon\,_{\Sigma W}=\left|\Delta_{\Sigma W}/2\right|$ is listed for each
Leo I star in Table New constraints on the chemical evolution of the dwarf
spheroidal galaxy Leo I from VLT spectroscopy ††thanks: Based on data
collected at the European Southern Observatory, Paranal, Chile, Proposals No.
69.D-0455 and 71.D-0219.
Table 4: Observed stars in template globular clusters.
Cluster | ID | $V$ | $B-V$ | $\Sigma W$(Å)
---|---|---|---|---
M 5 | II-45 | 14.75 | 0.82 | 4.62
| II-50 | 13.92 | 0.96 | 5.12
| II-51 | 14.05 | 0.96 | 4.90
| II-80 | 14.31 | 0.91 | 4.89
| II-74 | 13.82 | 1.01 | 5.05
| I-2 | 13.87 | 1.02 | 5.24
| I-50 | 13.91 | 0.97 | 5.06
| I-61 | 13.37 | 1.17 | 5.40
| I-68 | 12.37 | 1.52 | 6.25
| I-71 | 13.01 | 1.29 | 5.57
NGC 4372 | 20 | 12.88 | 1.57 | 3.78
| 14 | 14.29 | 1.29 | 2.88
| 13 | 12.72 | 1.73 | 4.13
| 10 | 13.82 | 1.29 | 3.00
| 95 | 14.48 | 1.30 | 2.72
| 91 | 14.45 | 1.33 | 2.81
| 89 | 14.49 | 1.29 | 2.83
| 141 | 12.93 | 1.65 | 4.01
| 74 | 14.17 | 1.40 | 3.09
| 76 | 14.18 | 1.30 | 3.16
| 77 | 14.19 | 1.28 | 3.07
NGC 6171 | 62 | 13.97 | 1.62 | 5.83
| 100 | 14.21 | 1.40 | 5.50
| I | 13.89 | 1.46 | 5.66
| F | 13.39 | 1.70 | 6.21
NGC 6397 | 328 | 12.07 | 0.93 | 3.19
| 326 | 12.78 | 0.89 | 2.89
| 337 | 12.58 | 0.90 | 2.83
| 343 | 11.42 | 1.13 | 3.34
| 361 | 11.67 | 1.08 | 3.18
NGC 6528 | R2-8 | 15.79 | 1.89 | 6.80
| R1-42 | 16.46 | 1.62 | 6.34
| R2-41 | 16.30 | 1.64 | 6.44
NGC 6752 | 4 | 13.70 | 0.85 | 3.86
| 8 | 11.96 | 1.05 | 4.99
| 28 | 13.17 | 0.89 | 4.18
| 29 | 11.79 | 1.17 | 4.98
| 30 | 12.15 | 1.12 | 4.83
NGC 5927 | 133 | 14.75 | 1.97 | 6.45
| 372 | 14.66 | 2.11 | 6.28
| 335 | 14.44 | 1.94 | 6.63
| 190 | 14.29 | 2.02 | 6.71
| 65 | 14.64 | 1.92 | 6.58
NGC 4590 | 144 | 12.80 | 1.29 | 3.72
| 239 | 14.19 | 0.87 | 2.88
| II72 | 15.03 | 0.85 | 2.60
| 30 | 14.15 | 0.87 | 2.64
| 74 | 14.59 | 0.84 | 2.36
| 119 | 13.62 | 0.95 | 2.90
Notes. The IDs, magnitudes, and colours of stars are those given in the
original photometry papers quoted by Rutledge et al. (1997a).
Figure 6: The sum of the equivalent widths of the two most reliable CaT lines
plotted as a function of the magnitude difference from the HB level. The
calibration globular clusters are represented by open symbols (coded with
different colours in the electronic edition of the journal). The metallicity
of the clusters increases from the bottom to the top (see Table 5). The
straight lines are best fits to the EWs for each cluster, assuming a common
best-fit slope. Filled squares refer to Leo I RGB stars.
### 4.2 Reduced EW
According to the CaT method (Armandroff & Da Costa 1991), the gravity and
$T_{\rm eff}$ dependence of CaT lines is accounted for by introducing a linear
correction to the line strengths which depends on the star’s luminosity, that
is
$W^{\prime}=\Sigma W+\beta(V-V_{\text{HB}})$ (3)
where $\beta$ is a constant and $(V-V_{\text{HB}})$ is the difference between
the magnitude of the star and the horizontal branch (HB) in the $V$ band. In
globular clusters, this reduced equivalent width $W^{\prime}$ was found to be
well correlated with metallicity (Armandroff & Zinn 1988; Armandroff & Da
Costa 1991; Rutledge et al. 1997b). This provides the empirical basis for the
validity of the CaT method. Applied to composite stellar populations, the
method is less straightforward and has been widely discussed in the recent
literature. We will return to this point later on.
In Fig. 6, we plot the sum of equivalent widths $\Sigma W$ versus $V-V_{{\rm
HB}}$ for all stars with good S/N spectra, both in Leo I and the template GCs.
The magnitude of the HB (of old stars) in Leo I, $V_{{\rm HB}}=22.60$, is from
Held et al. (2001), while for the template GCs, $V_{\text{HB}}$ was taken from
Rutledge et al. (1997a) (listed in Table 5 together with clusters’
metallicities). For all the globular clusters, our CaT line strengths define
clean, well separated linear sequences generally consistent with a constant
slope and having different, metallicity-dependent zero points. In this
diagram, the Leo I stars show quite a large dispersion, although most of them
are located between the sequences of NGC 6397 and M 5. This spread in CaT line
strengths is real, being larger than the typical measurement error
$\sigma_{\Sigma W}\approx 0.3$ Å. By assuming a common slope and lumping
together the data for all globular clusters, we derived a slope
$\beta=0.627\pm 0.021$. This value can only be compared with previous results
that use the same definition of $\Sigma W$. This is the case for Tolstoy et
al. (2001), who found $\beta=0.64\pm 0.02$, in agreement with our result.
### 4.3 Metallicity calibration
Table 5: Parameters for the calibration Galactic globular clusters. Cluster | [Fe/H]ZW | [Fe/H]CG | [M/H] | $V_{\text{HB}}$
---|---|---|---|---
NGC 6528 | $-0.23$ | $-0.10$ | $-0.03$ | 17.10
NGC 5927 | $-0.31$ | $-0.46$ | $-0.37$ | 16.60
NGC 6171 (M107) | $-0.99$ | $-0.87$ | $-0.70$ | 15.70
NGC 5904 (M 5) | $-1.40$ | $-1.11$ | $-0.90$ | 15.06
NGC 6752 | $-1.54$ | $-1.42$ | $-1.21$ | 13.70
NGC 6397 | $-1.91$ | $-1.82$ | $-1.65$ | 12.87
NGC 4372 | $-2.08$ | $-1.94$ | $-1.74$ | 15.30
NGC 4590 (M68) | $-2.09$ | $-1.99$ | $-1.81$ | 15.68
Table 6: Metallicity calibrations. Type | Calibration
---|---
quadratic | [Fe/H]${}^{\text{Z}W}$ = | $0.088\,{W^{\prime}}^{2}$ | $-0.184\,W^{\prime}$ | $-2.079$
| [Fe/H]${}^{\text{C}G}$ = | $0.072\,{W^{\prime}}^{2}$ | $-0.076\,W^{\prime}$ | $-2.122$
| [M/H] = | $0.051\,{W^{\prime}}^{2}$ | $+0.056\,W^{\prime}$ | $-2.125$
linear | [Fe/H]${}^{\text{Z}W}$ = | | $0.359\,W^{\prime}$ | $-2.845$
| [Fe/H]${}^{\text{C}G}$ = | | $0.391\,W^{\prime}$ | $-2.806$
| [M/H] = | | $0.395\,W^{\prime}$ | $-2.628$
Figure 7: The metallicity of the reference globular clusters against the
reduced EW of CaT lines, for three adopted metallicity scales. The solid lines
are quadratic fits, while the dashed straight lines are linear fits obtained
for clusters with [Fe/H]$<-0.6$.
Using our reduced CaT equivalent widths $W^{\prime}$ and the published
metallicities for the Galactic globular clusters, we re-determined the
calibration relations between $W^{\prime}$ and metallicity on 3 different
abundance scales: the ${\rm[Fe/H]}$ scales of Zinn & West (1984) and Carretta
& Gratton (1997) and the global metallicity ${\rm[M/H]}$, as defined by
Salaris et al. (1993). The metallicities of the GCs (Table 5) were taken from
Ferraro et al. (1999), except for the metal-rich cluster NGC 6528, for which
the more recent results of Zoccali et al. (2004) were adopted
(${\rm[Fe/H]}=-0.1$, ${\rm[}\alpha\rm{/Fe]}=0.1$). The global metallicity of
NGC 6528 (${\rm[M/H]}=-0.03$) was calculated using the relation from Salaris
et al. (1993):
$\begin{split}{\rm[M/H]}=&{\rm[Fe/H]}+\log(0.683f_{\alpha}+0.362)\\\ \log
f_{\alpha}=&{\rm[}\alpha\rm{/Fe]}\end{split}$ (4)
Figure 7 shows the $W^{\prime}$-metallicity relations for the three scales,
along with quadratic fits to the whole dataset and linear fits to the metal-
poor and intermediate globular clusters. The quadratic relations provide a
better fit to the GC metallicities over the whole metallicity range of
template GCs. The curvature is driven by the data for two most metal-rich
globular clusters, consistently with a fall in sensitivity of the Ca ii index
at high metallicity. Previous studies which included metal-rich GCs also found
quadratic relations (Armandroff & Da Costa 1991; Da Costa & Armandroff 1995;
Carretta et al. 2001; Bosler et al. 2007). Linear relations have been proposed
by other studies (most recently, Cole et al. 2004; Koch et al. 2007b; Carrera
et al. 2007) using metal-rich open clusters to constrain the metal-rich end of
the $W^{\prime}$ – [Fe/H] relation. A full discussion of the behaviour of CaT
line strengths against metallicity is beyond the scope of this paper, and will
be presented in a future paper along with a large dataset of calibrating
globular clusters. For our data, a linear relation indeed provides a good fit
for stars less metal-rich than the template cluster NGC 6171 ([M/H] $=-0.70$)
(Fig. 7). In the case of a metal-poor system such as Leo I, the linear and
quadratic relations give similar results except for the most metal-poor stars.
Our calibration is presently quite uncertain near the metal-poor end, being
based on one globular cluster (NGC 4590). For this cluster, Pritzl et al.
(2005) give a lower metallicity ([Fe/H] $\sim-2.3$) than that adopted in Table
5, yielding a better agreement with our linear calibration.
The main source of error on [Fe/H] (or [M/H]) is the uncertainty on the
measured equivalent width $\Sigma W$, since other sources of error, such as
photometric errors for stars on the upper RGB of Leo I, the error on the HB
level, or even the uncertainties associated to the fit parameters of the
calibration relations, are negligible compared to the $\Sigma W$ measurement
errors. A metallicity uncertainty can be computed for each star by error
propagation using the values of $\epsilon\,_{\Sigma W}$ in Table New
constraints on the chemical evolution of the dwarf spheroidal galaxy Leo I
from VLT spectroscopy ††thanks: Based on data collected at the European
Southern Observatory, Paranal, Chile, Proposals No. 69.D-0455 and 71.D-0219
and the calibrations in Table 6. However, a more meaningful metallicity
uncertainty is obtained using $\sigma_{W^{\prime}}$ as our estimate of the
measurement error. For the quadratic [M/H] calibration in Table 6,
$\sigma_{W^{\prime}}=0.31$ Å implies a [M/H] error $\sigma_{\rm err}\simeq
0.14$ dex for stars with [M/H] $=-1.20$.
The sources of uncertainty related to the CaT method itself are more difficult
to quantify and predict. All traditional calibrations refer to Galactic
globular clusters, which are simple and nearly coeval old stellar populations,
and the applicability of these calibrations to complex stellar populations is
not obvious. In our case, Leo I stars are on average several Gyr younger than
those in GCs. At a given luminosity, a star in Leo I has a different mass from
a GC star with the same metallicity and $V-V_{\text{HB}}$. However, recent
studies have shown that the CaT method can be used for complex stellar
populations younger that those in globular clusters (Cole et al. 2004; Pont et
al. 2004; Battaglia et al. 2008). Battaglia et al. (2008) have compared a
linear metallicity calibration that uses the CaT of RGB stars in two dSph
(Sculptor and Fornax) with spectroscopic [Fe/H] values obtained from their
high-resolution studies. The metallicities are in good agreement, although
with some residual trends of about 0.1–0.2 dex, in the range
$-2.5<\text{[Fe/H]}<-0.8$. These studies conclude that for ages older than 2.5
Gyr, the CaT line strengths are little affected by age, and suggest that the
overall uncertainty related to age effects is $<0.2$ dex.
### 4.4 A new metallicity scale?
Figure 8: Metallicity distribution of Leo I stars using a linear (left panels)
or quadratic (right panels) calibration relation. In the upper panels we plot
the MDF obtained with the [M/H] calibration. The metallicity distributions on
the [Fe/H] scale of Carretta & Gratton (1997) are shown in the lower panels,
together with previous results from Bosler et al. (2007) and Koch et al.
(2007b).
Most CaT metallicity measurements in nearby galaxies use a [Fe/H] scale based
on observations of Galactic globular clusters (e.g. Pont et al. 2004; Cole et
al. 2004; Battaglia et al. 2006; Koch et al. 2006). However, the relative
abundances of $\alpha$-elements (including Ca) in Local Group dwarfs are on
average lower than in the Milky Way halo stars and GCs (Shetrone et al. 2001,
2003; Tolstoy et al. 2003; Geisler et al. 2005; Pritzl et al. 2005). To
overcome this problem, Bosler et al. (2007) proposed a new calibration of CaT
lines against the [Ca/H] abundance, based on high-resolution spectroscopy of
Galactic star clusters. In their hypothesis, the [Ca/H] calibration is less
affected by the difference in [Ca/Fe] abundance ratios between red giant stars
in globular clusters and dwarf spheroidal galaxies. However, the strength of
CaT lines is also determined by other parameters (gravity and $T_{\rm eff}$)
in addition to Ca abundance. A comparison of Ca abundances derived from CaT
lines with the results of high-resolution abundance measurements for stars in
two dSph galaxies (Battaglia et al. 2008) shows that, while the CaT lines
trace both Ca and Fe, their dependence on Fe abundance is stronger. Similarly,
[Ca/H] ratios derived from CaT lines for stars in Leo II dSph and globular
clusters (Shetrone et al. 2009) systematically differ from those obtained from
mid-resolution synthetic spectra, with a residual trend that is a function of
metallicity. Since the effective temperature of red giants in globular
clusters is driven by their global metallicity [M/H] (Salaris et al. 1993), an
empirical metallicity ranking based on CaT and a global metallicity [M/H]
scale, proposed here for the first time, appears to be the most empirically
sound. In fact, [M/H] (or, equivalently, $Z$) takes into account the
abundances of both the $\alpha$-elements and Fe. Our data are therefore
calibrated using the [M/H] calibration in addition to the common [Fe/H]
scales.
### 4.5 A concluding remark
We conclude this section with a consideration that ought to be kept in mind
throughout all the following discussion. While we give the metallicity of the
stars in three flavours ([Fe/H]ZW, [Fe/H]CG, and [M/H]), this does not imply
that we are determining the three metallicity parameters at the same time. The
only observable quantity is the reduced equivalent width $W^{\prime}$. The
calibration of $W^{\prime}$ in terms of metallicity relies on the assumption
that $W^{\prime}$ is correlated with metallicity, i.e. a star in Leo I has the
same metallicity as a star in a GC with the same $W^{\prime}$. The key
questions are: what are the real drivers that determine the strength of the
CaT lines? Do two stars with the same iron-to-hydrogen ratio but different
$\alpha$-elements composition have the same $W^{\prime}$? Some of these
effects have been discussed by Battaglia et al. (2008), and are further
addressed by a large observational program by our group whose results will be
presented in future papers.
## 5 The metallicity of Leo I stars
### 5.1 The observed metallicity distribution
Table 7: Mean metallicity and standard deviation of red giants in Leo I. Scale | fit | mean | $\sigma$
---|---|---|---
${\rm[Fe/H]}^{\rm ZW}$ | linear | $-1.53$ | $0.17$
| quadratic | $-1.55$ | $0.21$
${\rm[Fe/H]}^{\rm CG}$ | linear | $-1.37$ | $0.18$
| quadratic | $-1.41$ | $0.21$
${\rm[M/H]}$ | linear | $-1.18$ | $0.19$
| quadratic | $-1.22$ | $0.20$
In Fig. 8 we show the metallicity distribution of Leo I red giant stars as
derived from our data using both the [Fe/H] metallicity scale of Carretta &
Gratton (1997) and the [M/H] scale. The parameters of the distribution (mean
and standard deviation, excluding the 4 most metal-poor stars in Fig. 8) are
given in Table 7 for both scales, along with the results on the Zinn & West
(1984) scale for ease of comparison with previous literature. The
metallicities of individual Leo I stars are listed in Table New constraints on
the chemical evolution of the dwarf spheroidal galaxy Leo I from VLT
spectroscopy ††thanks: Based on data collected at the European Southern
Observatory, Paranal, Chile, Proposals No. 69.D-0455 and 71.D-0219. Using the
CG97 scale, the distribution is centred at [Fe/H] $\simeq-1.4$ with a standard
deviation $\sigma_{{\rm[Fe/H]}}\simeq 0.2$. For the [M/H] calibration, the
average is [M/H] $\simeq-1.2$ with the same scatter.
The results obtained from the linear and quadratic calibrations are very
similar in all cases, as expected since most of the Leo I stars have
metallicities lower than ${\rm[Fe/H]}=-1.0$. The choice of the linear or
quadratic relation only affects the metallicity of 4 metal-poor stars, having
[M/H]$\lesssim-2$. In this range, the calibration is extrapolated beyond the
most metal-poor globular cluster, which makes the metallicity of the 4 stars
quite uncertain and dependent on the adopted calibration. In the case of the
CG97 calibration, which yields the lowest extrapolated values, the 4 stars
have $-2.6<{\rm[Fe/H]}<-2.2$. Visual inspection of the targets on a Leo I
image indicates normal star-like profiles (i.e. no blends). We therefore
conclude that, while a few stars may have low metallicity, there is so far no
evidence of extremely metal-poor stars in Leo I. Spectral synthesis methods
(see, e.g., Kirby et al. 2008) will be used in a future paper to obtain more
secure metallicity estimates for these metal-poor stars from a different
spectral interval.
Our determination of the metallicity of Leo I agrees well with the results of
Bosler et al. (2007) (${\rm[Fe/H]}=-1.34$) and Koch et al. (2007b) ([Fe/H]
$=-1.31$), in particular when a linear calibration is used as in the previous
papers. On the other hand, our [M/H] values are in better agreement with the
[Ca/H] results of Bosler et al. (2007). Using 9 stars in common with Bosler et
al. (2007) and 5 stars in common with Koch et al. (2007b), we compared the
metallicities star-by-star. The mean differences are $\Delta{\rm[Fe/H]}^{\rm
CG}=-0.04\pm 0.14$ (rms of the sample) and $\Delta{\rm[Fe/H]}^{\rm
CG}=-0.17\pm 0.11$, respectively, in agreement with the shifts between the
mean values of the MDFs. The rms values are consistent with our measurement
error (see next section).
### 5.2 Intrinsic metallicity dispersion and clues on the evolution of Leo I
The observed MDF in Fig. 8 is the convolution of the intrinsic metallicity
distribution of stars in Leo I and the measurement errors. The real abundance
spread can be estimated by adopting a Gaussian model for the intrinsic MDF,
which yields $\sigma_{\text{OBS}}^{2}=\sigma_{0}^{2}+\sigma_{\text{err}}^{2}$,
where $\sigma_{\text{OBS}}$ is the observed metallicity dispersion,
$\sigma_{0}$ is the intrinsic dispersion, and $\sigma_{\text{err}}$ is the
measurement error.
If we adopt a quadratic [M/H] calibration and a typical measurement error
$\sigma_{\rm err}\simeq 0.14$ dex, the measurement scatter largely contributes
to the observed metallicity dispersion. The observed scatter implies an
intrinsic metallicity dispersion $\sigma_{{\rm[Fe/H]},0}=0.14$ for the CG97
scale and $\sigma_{{\rm[M/H]},0}=0.08$ for [M/H]. The intrinsic abundance
dispersion of Leo I stars is therefore very small, even smaller than
previously thought, and this happens in spite of the relatively wide range of
ages of the stellar populations. This is an important constraint to the
chemical evolution across the life of the galaxy.
Figure 9: The MDF of Leo I RGB stars on the [M/H] metallicity scale, compared
with a simple model with a low effective yield (dashed line) and a model with
a prompt initial enrichment (solid line).
In order to model the metallicity distribution and the chemical evolution of
Leo I, detailed models have to be put forth (such as those of Lanfranchi &
Matteucci 2007) properly taking into account the chemical and dynamical
evolution of the galaxy. However, some order-of-magnitude physical insight on
the evolution of Leo Ican already be obtained using basic considerations. The
metallicity distribution of RGB stars in Leo I is compared in Fig. 9 with a
simple closed-box model with a low effective yield consistent with a
continuous loss of gas (e.g., Pagel 1997). In order to reproduce the peak of
the observed MDF, we have to adopt an effective yield $y=0.025Z_{\odot}$, and
$y=0.040Z_{\odot}$, for the distributions based on the [Fe/H] and [M/H]
metallicity scales, respectively. This is clearly much lower than the value
found in the solar vicinity ($y=1.2Z_{\odot}$; e.g. Portinari et al. 2004), in
a way consistent with the loss of metals driven by a galactic wind (Hartwick
1976; Pagel 1997). Still, even allowing for a gas outflow, the number of
metal-poor stars is largely overestimated by the simple model, as shown in
Fig. 9. The fit is considerably improved by assuming a prompt early enrichment
with an initial metallicity $Z_{0}=0.02\,Z_{\odot}$ (${\rm[Fe/H]}=-1.7$)
(continuous line in Fig. 9). Although very simplistic, this conclusion agrees
with the finding that the metal-poor tail of the MDF in 4 Local Group dwarf
spheroidal galaxies (Helmi et al. 2006) is significantly different from that
of the Galactic halo, lacking stars below [Fe/H] $=-3$. What this “toy model”
tells us is that the narrow MDF of the Leo I stars can be understood as a
combination of fast enrichment from an initial generation of stars, and
subsequent loss of metals through outflows. This situation is common among
Local Group dwarfs, but the MDF of Leo I is the narrowest observed to date
(cf. Tolstoy et al. 2001; Pont et al. 2004; Koch et al. 2006, 2007a).
### 5.3 Radial metallicity gradients
Figure 10: Metallicities of Leo I stars on the [Fe/H] scale of CG97, plotted
against the elliptical radius (see text). Filled dots: data in this paper;
open squares: data from Bosler et al. (2007); open triangles: Koch et al.
(2007b); circles with error bars: our data, binned in 1$\aas@@fstack{\prime}$5
bins. The error bars of the binned data represent the abundance scatter
($1\sigma$) in each bin. The crosses are the 4 metal-poor stars in our sample.
The typical errors of each study are shown in the upper right corner of the
plot. The solid line is a fit to our (binned) data, while the dashed line
represents a fit to all available spectroscopic data.
Radial variations in the stellar populations are common in the dwarf
spheroidals of the Local Group, where the younger and more metal-rich
populations are often concentrated toward the galaxy centre (Harbeck et al.
2001; Saviane et al. 2001; Pont et al. 2004; Tolstoy et al. 2004; Koch et al.
2006). Leo I remains one of the few dSph’s showing little evidence of a
population gradient (Held et al. 2000; Koch et al. 2007b). In particular, the
spectroscopic investigation of Koch et al. (2007b), extending to quite large
radial distances, did not detect a significant metallicity gradient.
Our new spectroscopic sample of red giants allows us to further search for
radial variations in the metallicity of Leo I stars. As the radial coordinate,
we have adopted the semi-major axis $r$ of ellipses passing through the
projected sky position of each star. The ellipses have the centre at
$10^{h}08^{m}28\aas@@fstack{s}1$, $+12\degr 18\arcmin 23\arcsec$ (J2000) and a
fixed position angle and ellipticity ($\text{PA}=79^{\circ}$, $\epsilon=0.21$;
Irwin & Hatzidimitriou 1995).
In Fig. 10, the metallicities of Leo I stars in our sample (see Table New
constraints on the chemical evolution of the dwarf spheroidal galaxy Leo I
from VLT spectroscopy ††thanks: Based on data collected at the European
Southern Observatory, Paranal, Chile, Proposals No. 69.D-0455 and 71.D-0219)
are plotted against the distance from the centre. To directly compare our data
with results from previous studies, we used the metallicity obtained from the
calibration in terms of [Fe/H]${}^{\text{CG}}$. Data from Bosler et al. (2007)
and Koch et al. (2007b) were shifted to account for the small differences in
mean metallicity (of the order 0.1 dex or less) between the MDF’s (Fig. 8). A
linear relation was fitted both to our data alone (solid line) and to all
available metallicity measurements in the literature (dashed line), excluding
stars with [Fe/H] $<-2$.
The fit to our new FORS2 data yields a radial gradient of $-0.02$ dex
arcmin-1, or $-0.27$ dex Kpc-1. In our sample, stars more metal-rich than
[Fe/H] $=-1.3$ are only found in the central region of Leo I, with
$a<5\aas@@fstack{\prime}5$. In contrast, the fit to the merged spectroscopic
sample suggests a radially constant metallicity, in agreement with the
conclusions of Koch et al. (2007b). To quantify the gradient, we have used a
Kolmogorov-Smirnov test to compare the metallicity distributions of stars with
$1\aas@@fstack{\prime}2<a<5\arcmin$ (inner sample) and
$5\arcmin<a<8\aas@@fstack{\prime}2$ (outer sample), considering only the
radial interval covered by our data. The two metallicity distributions are
similar in shape, with the MDF in the inner region peaked at higher
metallicity. The hypothesis that the inner and outer sample are drawn from the
same parent population can be rejected at a 90% level using our data, and only
at a non-significant 62% level using all spectroscopic data.
We conclude that, while our data provide a hint of a weak radial metallicity
gradient in Leo I, the statistical significance of this result is at present
low. More stars need to be observed, particularly in the outer region of the
galaxy, before definite conclusions can be drawn.
## 6 The age of Leo I stars
### 6.1 The age-metallicity relation
Figure 11: The age-metallicity relation of Leo I RGB stars in our sample, on
the [M/H] scale. The error bars in age represent the first and third quartile
of the confidence intervals obtained through Monte Carlo realisations (see
text for details). For the metallicity errors the representative value
discussed in Sect. 4.3 is adopted. Also shown are the metallicities from high-
resolution spectroscopy of two Leo I RGB stars from Tolstoy et al. (2003) (big
filled circles). The side histograms are the marginal distributions in
metallicity and age. The solid line in the top panel represents the SFH
derived by Dolphin (2002) from HST photometry, normalised to the total number
of stars in our sample.
With the stellar metallicities of Leo I stars known from spectroscopy, ages
could be estimated by comparing stars’ locations in the CMD with a grid of
theoretical isochrones (the models of Pietrinferni et al. 2004, were used to
this purpose). Absolute magnitudes and dereddened colours were computed
adopting a colour excess $E_{B-V}=0.04$, a total-to-differential extinction
ratio $R_{V}=A_{V}/E_{B-V}=3.1$, and a true distance modulus $(m-M)_{0}=22.04$
(Held et al. 2009).
The stellar ages were interpolated in two steps. First, we used a set of
theoretical isochrones of fixed age and different metallicities to find, for a
star of given age and luminosity, a metallicity-colour relation and (from the
known colour) an interpolated metallicity. This step was repeated for all
model ages, yielding for each data point in the CMD a set of theoretical age-
metallicity pairs each consistent with the star’s magnitude and colour. This
age-metallicity look-up table (spanning the full range from old, metal-poor
stars to young, metal-rich stars) allowed us to compute an interpolated age
for each star from its spectroscopic metallicity. We chose the [M/H] scale for
the input value, as the most directly related to the mass fraction of metals
($Z$) used in stellar models. For a number of stars, ages could not be derived
because the observed colour and/or magnitude were outside the range covered by
the isochrones.
The method was checked against Galactic globular clusters with ages given by
the literature. In particular, a small correction was applied to the isochrone
colours so as to yield a correct age for NGC 5904 (M 5), assumed to be 12 Gyr
(Sandquist et al. 1996) and the closest in metallicity to Leo I among the
clusters listed in Table 5. Thus, our ages for Leo I stars are essentially
referred to M 5, which is in our view the most correct approach given the
considerable uncertainties in the isochrone colours.
The resulting ages are listed in Table New constraints on the chemical
evolution of the dwarf spheroidal galaxy Leo I from VLT spectroscopy ††thanks:
Based on data collected at the European Southern Observatory, Paranal, Chile,
Proposals No. 69.D-0455 and 71.D-0219 along with their uncertainties,
estimated as follows. For each star, we performed a set of 100 Monte Carlo
experiments by randomly varying the input quantities in intervals consistent
with their uncertainties. We adopted a standard error 0.02 mag in the $V$
magnitude, 0.05 mag in $(B-V)$ colour, and 0.15 dex in [M/H]. The latter was
chosen conservatively large to account for the inherent uncertainties in the
metallicity scale. The median and quartiles of the age distributions of
randomly generated “stars” corresponding to each observed stars are listed in
Table New constraints on the chemical evolution of the dwarf spheroidal galaxy
Leo I from VLT spectroscopy ††thanks: Based on data collected at the European
Southern Observatory, Paranal, Chile, Proposals No. 69.D-0455 and 71.D-0219.
In general, the uncertainties are of the order 50%, which reasonably reflects
the large uncertainties in the process.
The age-metallicity relation derived from our data is shown in Fig. 11. The
chemical evolution of Leo I seems to be very slow, in accord with the
narrowness of the MDF. If the 4 metal-poor stars are excluded, there is a
trend for stars younger than 5 Gyr to be on average more metal-rich by about
0.2–0.3 dex. Similar conclusions were drawn, from a different data set, by
Bosler et al. (2004). The scatter in the age-metallicity relation appears to
be smaller than observed in other galaxies (Battaglia et al. 2006; Tolstoy et
al. 2003, and references therein). Our data are in agreement with the results
from high-resolution spectroscopy for 2 stars (Tolstoy et al. 2003).
We can use our age determinations also to obtain a SFH of Leo I. This can be
done since our target selection was designed to avoid any bias in age and/or
metallicity (see Sect. 2); the age distribution of our target stars is then
proportional to the SFH. Our age measures are in agreement with the SFHs
derived by HST photometry. In the upper panel of Fig. 11 our age distribution
is compared with the Dolphin (2002) SFH, showing only small differences that
can be explained by statistical fluctuations.
### 6.2 Radial distribution of stellar ages
To complete our analysis of the population gradients, we investigated the
possible presence of a radial variation in the age of Leo I stars. Figure 12
shows the ages plotted against the elliptical distance. While old stars are
found at all radii, young stars appear to be concentrated at small distances
from the centre. To quantify this finding, we have plotted in Fig. 12 the
cumulative distributions of two subsamples of stars in the inner
($r<3\aas@@fstack{\prime}9$) and outer ($r>3\aas@@fstack{\prime}9$) region,
respectively. This limit was chosen to have the same number of stars in each
subsample. A Kolmogorov-Smirnov test indicates that the null hypothesis that
the distributions are drawn from the same parent population can be rejected at
$>99.9$% confidence level. Similarly, we have plotted the cumulative
distributions of two subsamples of stars with age $<4.6$ Gyr and $>4.6$ Gyr.
Also in this case, the probability of the null hypothesis can be rejected at a
level $>99.9$%.
This suggestion of an age gradient among RGB stars is strengthened by the
detection of a radial gradient in the fraction of upper-AGB stars in Leo I,
which points to a concentration of intermediate-age populations towards the
galaxy centre (Held et al. 2009).
Figure 12: Ages of the Leo I stars in our sample as a function of the
elliptical radius (central panel). The upper panel shows the cumulative radial
distributions of stars with ages smaller and larger (dotted line) than 4.6
Gyr. The cumulative age distributions of stars in the inner
($r<3\aas@@fstack{\prime}9$) and outer ($r>3\aas@@fstack{\prime}9$, dotted
line) region are shown in the right panel (the different age and radial
intervals are colour-coded in the electronic version of the journal). Younger
stars appear to be more concentrated towards the centre of Leo I.
## 7 Summary and conclusions
We have presented spectroscopic measurements of RGB stars in the Leo I dSph
from observations carried out with the FORS2 spectrograph at the ESO VLT. We
derived radial velocities for 57 stars with good S/N ratio, 54 of which have
been found to be Leo I members. Among these, 14 stars are in common with
previous spectroscopic studies.
We measured the metallicities of RGB stars in Leo I from the equivalent widths
of Ca ii triplet lines, using the [Fe/H] metallicity scales of Carretta &
Gratton (1997) and Zinn & West (1984). In addition, we derived a new
calibration tied to the [M/H] ranking of Galactic globular clusters, which
accounts for the abundance of both Fe-group and $\alpha$ elements.
The metallicity distribution (MDF) of Leo I stars is symmetric and very
narrow. If we adopt a quadratic calibration of Ca ii line strengths against
[Fe/H], the mean metallicity is ${\rm[Fe/H]}=-1.41$ with a measured dispersion
$0.21$ dex on the Carretta & Gratton (1997) scale, in agreement with previous
spectroscopic studies. The new [M/H] calibration yields a mean value [M/H]
$=-1.22$ with a dispersion $0.20$ dex. By subtracting the measurement errors,
we estimated a very low intrinsic metallicity dispersion,
$\sigma_{\rm[M/H]}=0.08$, which represents a constraint for modelling the
chemical evolution of this isolated dwarf galaxy. As pointed out by previous
studies, this narrow MDF is inconsistent with a simple “closed-box” chemical
evolution model, even adopting a very low effective yield to account for
galactic outflows expelling the metals produced by SNe winds. A prompt initial
chemical enrichment may explain the very small number of extremely metal poor
stars (we find only 4 stars with [Fe/H] $<-2$). Together, the two effects can
explain the small abundance dispersion of Leo I stars, which gives the
narrowest observed MDF among Local Group dwarf galaxies. However, detailed
chemical evolution models (e.g., Lanfranchi & Matteucci 2007) are needed to
gain a complete picture of the evolution of Leo I.
Our data for RGB stars also provide an indication of a weak radial metallicity
gradient in Leo I, of $-0.27$ dex Kpc-1. In fact, all of our stars with
[M/H]$>-1.3$ are found in the inner region ($r\lesssim 5^{\prime}$). However,
by combining our observations with previous spectroscopic datasets in the
literature, the radial variation becomes insignificant. More observations in
the outskirt of Leo I with a quality comparable to those presented here, are
required to definitively establish the presence of an abundance gradient.
The metallicities of the RGB stars in our Leo I sample have been combined with
existing photometric data to yield age estimates and an age-metallicity
relation. Our age determinations are consistent with the SFH derived by
Gallart et al. (1999) and Dolphin (2002) from HST photometry. The age-
metallicity relation of Leo I red giants is quite flat, again suggesting a
rapid initial enrichment. An increase in metal abundance by $\sim 0.2-0.3$ dex
in the last 5 Gyr is possibly related to the main star-formation episode at
intermediate ages. Since Leo I only hosts a minor old ($>10$ Gyr) stellar
component, the chemical history of the galaxy is not well constrained at early
epochs. Its most metal-poor stars must have formed out of a medium pre-
enriched by a lost generation of stars, either before or after the galaxy had
started assembling.
We have provided the first evidence of a radial variation in the ages of red
giants in Leo I. Despite the uncertainties in age determination, our direct
measurement of a radial variation of stellar ages seems quite convincing, with
a Kormogorov-Smirnov test confirming, at a high level of statistical
significance, that stars in the inner part of Leo I are on average younger
than those in the outer regions. This result agrees with the conclusions of a
parallel study of intermediate-age AGB stars in Leo I from near-infrared
photometry (Held et al. 2009). In the emerging scenario, the first generation
of Leo I stars uniformly formed throughout this isolated dwarf spheroidal
galaxy. The bulk of intermediate-age stars originated from an interstellar
medium, poorly enriched by previous stellar generations mainly because of the
effects of stellar winds. Younger stellar populations preferentially formed in
the central regions, from gas somewhat enriched as seen from the age-
metallicity relation in the last few Gyr. In this framework, our results on
the radial distribution of Leo I stellar populations are not in contrast with
previous results which found no gradients. The lack of detection of an age
gradient by Held et al. (2000) can be explained considering that the mean age
of the red-clump stars used by Held et al. (2000) as tracers of intermediate-
age populations, is $\sim 5$ Gyr, which is older than that of upper AGB stars
used by Held et al. (2009). As shown in Fig. 12, there are no clear radial
variations in the age distribution of Leo I stars with ages greater than $\sim
5$ Gyr.
###### Acknowledgements.
We thank A. Koch for providing us with unpublished data. M.G. wishes to thank
the European Southern Observatory at Santiago, Chile for partial funding
through DGDF and for hospitality during a visit in which this paper was
partially written. This research was partially funded by PRIN MIUR 2007
“Galactic astroarchaeology: the local route to cosmology” (P.I. F. Matteucci).
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Table 1: Spectroscopic sample in the Leo I field.
ID | $\alpha$ (J2000) | $\delta$ (J2000) | $B-V$ | $V$ | $v$ | $\Delta v$ | S/N | other
---|---|---|---|---|---|---|---|---
| | | | | km s-1 | km s-1 | |
1 | 10:08:28.43 | +12:15:54.1 | 1.39 | 19.62 | $268.2$ | 1.5 | 57 |
2 | 10:08:16.47 | +12:15:59.5 | 1.48 | 19.59 | $264.5$ | 3.2 | 46 |
3 | 10:08:36.41 | +12:16:02.2 | 1.46 | 19.89 | $261.9$ | 5.7 | 37 |
4 | 10:08:17.58 | +12:16:15.7 | 1.61 | 19.55 | $267.0$ | 0.9 | 42 |
5 | 10:08:19.45 | +12:16:34.1 | 1.10 | 20.28 | $268.6$ | 3.2 | 32 | B8391
6 | 10:08:40.46 | +12:16:38.0 | 1.35 | 19.78 | $273.5$ | 0.4 | 36 | B25113
7 | 10:08:31.00 | +12:16:40.6 | 1.32 | 19.85 | $282.4$ | 3.1 | 58 | B18214
8 | 10:08:03.62 | +12:16:46.4 | 1.50 | 19.74 | $301.0$ | 7.4 | 59 |
9 | 10:08:32.56 | +12:16:54.9 | 1.19 | 20.45 | $290.7$ | 1.5 | 33 |
10 | 10:08:08.79 | +12:17:05.9 | 1.07 | 20.00 | $304.3$ | 2.2 | 90 |
11 | 10:08:10.61 | +12:17:08.0 | 1.20 | 20.05 | $262.2$ | 6.8 | 62 | B4173
12 | 10:08:49.42 | +12:17:14.8 | 1.18 | 20.44 | $281.2$ | 2.5 | 18 |
13 | 10:08:34.90 | +12:17:17.8 | 1.16 | 20.01 | $287.8$ | 6.8 | 38 |
14 | 10:08:39.66 | +12:17:19.9 | 1.37 | 19.68 | $258.4$ | 3.7 | 44 | K195
15 | 10:08:57.38 | +12:17:20.8 | 1.18 | 20.44 | $280.3$ | 6.6 | 34 | K833
16 | 10:08:55.47 | +12:17:21.9 | 1.26 | 20.14 | $274.2$ | 10.9 | 43 |
17 | 10:08:11.82 | +12:17:29.4 | 1.05 | 20.31 | $260.0$ | 0.6 | 51 |
18 | 10:08:29.89 | +12:17:31.7 | 1.30 | 19.67 | $262.8$ | 4.8 | 43 |
19 | 10:08:07.52 | +12:17:34.6 | 1.25 | 20.16 | $269.6$ | 7.4 | 52 |
20 | 10:08:14.21 | +12:17:36.1 | 1.11 | 19.90 | $247.6$ | 32.2 | 67 | B5496
21 | 10:08:15.40 | +12:17:38.3 | 1.43 | 19.48 | $266.1$ | 3.0 | 37 |
22 | 10:08:46.36 | +12:17:41.3 | 1.33 | 20.32 | $261.9$ | 2.9 | 30 |
23 | 10:08:20.50 | +12:17:45.0 | 1.61 | 19.64 | $263.7$ | 0.3 | 50 |
24 | 10:08:06.29 | +12:17:45.3 | 1.28 | 20.16 | $269.9$ | 7.1 | 49 | B3135
25 | 10:08:50.70 | +12:17:46.9 | 1.15 | 20.31 | $270.5$ | 6.7 | 35 | K677
26 | 10:07:55.77 | +12:17:55.2 | 1.14 | 20.48 | $277.3$ | 6.0 | 47 |
27 | 10:08:01.84 | +12:17:56.6 | 1.43 | 19.82 | $275.4$ | 5.4 | 61 | B2488
28 | 10:08:45.25 | +12:17:57.8 | 1.28 | 20.08 | $248.9$ | 5.5 | 52 |
29 | 10:08:38.58 | +12:18:21.0 | 0.94 | 19.71 | $274.5$ | 8.3 | 41 |
30 | 10:08:56.58 | +12:18:22.4 | 1.13 | 20.22 | $271.1$ | 0.3 | 30 |
31 | 10:08:34.85 | +12:18:22.6 | 1.24 | 19.98 | $256.2$ | 0.1 | 47 |
32 | 10:08:15.96 | +12:18:25.3 | 1.17 | 20.25 | $282.9$ | 3.0 | 34 |
33 | 10:07:57.28 | +12:18:26.1 | 1.40 | 19.74 | $284.3$ | 7.5 | 69 |
34 | 10:08:44.09 | +12:18:29.0 | 1.13 | 19.66 | $278.8$ | 9.5 | 42 |
35 | 10:08:47.87 | +12:18:29.9 | 1.46 | 19.70 | $248.8$ | 8.0 | 39 |
36 | 10:08:36.01 | +12:18:32.6 | 1.30 | 19.58 | $254.0$ | 0.6 | 45 |
37 | 10:07:59.36 | +12:18:35.7 | 1.60 | 19.75 | $273.9$ | 6.3 | 54 |
38 | 10:08:58.70 | +12:18:37.1 | 1.44 | 19.85 | $257.2$ | 8.2 | 46 |
39 | 10:08:41.92 | +12:18:39.7 | 1.16 | 20.22 | $250.4$ | 7.8 | 31 | B25820
40 | 10:08:51.73 | +12:18:45.7 | 1.28 | 20.23 | $278.9$ | 3.0 | 49 |
41 | 10:08:20.21 | +12:18:46.6 | 1.41 | 19.62 | $258.0$ | 0.1 | 27 |
42 | 10:08:33.75 | +12:18:47.0 | 1.26 | 19.23 | $275.4$ | 0.0 | 50 |
43 | 10:08:24.22 | +12:18:53.3 | 1.38 | 19.93 | $259.0$ | 2.4 | 42 |
44 | 10:08:40.60 | +12:19:02.8 | 1.25 | 19.66 | $276.9$ | 2.7 | 48 |
45 | 10:08:30.67 | +12:19:30.0 | 1.00 | 20.34 | $259.4$ | 4.6 | 39 |
46 | 10:08:21.15 | +12:19:43.6 | 1.52 | 19.72 | $270.8$ | 1.0 | 35 |
47 | 10:08:39.42 | +12:20:05.9 | 1.16 | 19.74 | $271.2$ | 3.8 | 51 |
48 | 10:08:37.29 | +12:20:12.2 | 1.24 | 19.88 | $270.7$ | 1.3 | 43 | K351
49 | 10:08:13.35 | +12:20:13.8 | 1.27 | 19.64 | $292.2$ | 2.9 | 55 |
50 | 10:08:22.17 | +12:20:14.9 | 1.38 | 19.15 | $289.8$ | 0.9 | 59 |
51 | 10:08:28.41 | +12:20:29.6 | 1.51 | 19.61 | $267.6$ | 2.1 | 64 | K137
52 | 10:08:14.96 | +12:20:43.9 | 1.47 | 19.58 | $266.6$ | 2.9 | 50 |
53 | 10:08:19.17 | +12:20:48.9 | 1.38 | 19.87 | $255.3$ | 2.0 | 38 | B8203
54 | 10:08:18.24 | +12:20:54.6 | 1.27 | 20.02 | $268.0$ | 0.7 | 31 |
101 | 10:07:51.18 | +12:17:36.6 | 1.52 | 19.56 | $49.6$ | 8.5 | 72 |
102 | 10:08:05.23 | +12:18:13.5 | 1.05 | 19.96 | $-27.9$ | 5.5 | 52 |
103 | 10:08:53.43 | +12:18:27.4 | 1.36 | 19.88 | $90.4$ | 8.7 | 55 |
Notes. $v$ is the heliocentric radial velocity and $\Delta v$ the absolute
semi-difference in radial velocity of the individual spectra. The last column
gives the identification of stars in common with Bosler et al. (2007) and Koch
et al. (2007b).
Table 3: Measurements of metallicity and age for stars in Leo I.
Notes. $\epsilon\,_{\Sigma W}$ is the absolute semi-difference of the
equivalent widths measured on the individual spectra. The listed metallicity
values were calculated from $W^{\prime}$ using the quadratic calibration. The
last two columns give the lower and upper confidence intervals of our age
estimates (see text for details).
|
arxiv-papers
| 2009-04-17T15:00:10 |
2024-09-04T02:49:01.928270
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Gullieuszik, E. V. Held, I. Saviane and L. Rizzi",
"submitter": "Marco Gullieuszik",
"url": "https://arxiv.org/abs/0904.2718"
}
|
0904.2769
|
†† Copyright ©2009 By Author
Non Homogeneous Poisson Process Model based Optimal Modular Software Testing
using Fault Tolerance
Amit K Awasthi and Sanjay Chaudhary
Pranveer Singh Institute of Technology,
NH-2, Kanpur-Agra Highway, Kanpur, UP, India
###### Abstract
In software development process we come across various modules. Which raise
the idea of priority of the different modules of a software so that important
modules are tested on preference. This approach is desirable because it is not
possible to test each module regressively due to time and cost constraints.
This paper discusses on some parameters, required to prioritize several
modules of a software and provides measure of optimal time and cost for
testing based on non homogeneous Poisson process.
Keywords: Non Homogeneous Poisson Process, Optimal Test Policy, Software Life
Cycle Length, Testing Time, Module Test Prioritization, Fault Tolerance.
## 1 Introduction
Whenever a software is developed a question about its reliability comes in
front. We need some tool to be sure that software is working properly. That
is, there is a need of software testing, to find out any faults that might
exist, before releasing the product. For this purpose, software product is
tested carefully but regressive testing is not feasible always, as it can be
very expensive in form of cost and time both. That s why, a modular testing is
a suggestive approach so that the Testing Authority can test the software’s
important modules preferably and may save time and cost.
It is impractical to test the software till all the bugs are removed, the
tester should also be aware of the optimal testing time and cost required to
test the modules. We also allow a bit of faults in the accepted range instead
of making it 100% error free. For this reason, this paper attempts to provide
an optimal boundary values for time and cost considering the actual percentage
of faults obtained in testing. A project manager should be familiar with the
points where it should stop testing and go for release or rejection.
A lot of work has been done in the area of optimal software testing. McDaid
and Wilson (2001) gave three plans to settle on the problem of decision - How
long to test software? by introducing the optimal time measure [2]. Musa and
Ackerman used the concept of reliability to make the decision [3]. Ehrlich,
Prasanna, Stampfel and Wu also tried to find out the cost of a stop test
decision [4]. But one of the most suitable models for the problem of
determining optimal cost and time is proposed by Goel and Okumoto [5]. They
gave a non homogeneous Poisson process based model to determine the optimal
cost and time for software [6][7]. Praveen et al. enhanced their work by
proposing a cumulative priority based elucidation to find out optimal software
testing period [8].
In this paper, we consider the new idea of modular approach to test software.
We suggest here to assign a weight on each modules depending on various
parameters. Hierarchies of the modules also plays an imporatant role in
decision as preceder module will always affact their dependent modules. We
enhanced previous ideas by adding this hierachical module concept.
The next section briefly explains background and related work. Section 3
provides the module prioritization schema based on various factors and our
approach to test the software to determine that the software is OK for release
or not. Section 4 brings an example where this approach is applied. Last
section concludes finally.
## 2 Background and Related Work
### 2.1 Non homogeneous poisson process
A Poisson process is one of the most significant random processes in
probability theory. It is widely used to model random points in time and space
such as the times of radioactive emissions, the arrival times of customers at
a service center and the positions of flaws in a piece of material. Several
important probability distributions arise naturally from the Poisson process.
The Poisson process is a collection of random variables where $N(t)$ is the
number of events that have occurred up to time $t$ (starting from time 0) [8].
The number of events between time $a$ and time $b$ is given as $N(b).N(a)$ and
has a Poisson distribution. A Non-Homogeneous process is a process with rate
parameter $\lambda(t)$ such that the rate parameter of the process is a
function of time e.g. the arrival rate of vehicles in a traffic light signal.
### 2.2 Related work by Goel and Okumoto
Faults present in the system causes software failure at random times. Let
$N(t)$ (where $t>0$) be the cumulative number of failures at time $t$ (either
CPU time or calendar time). According to Goel and Okumoto [5], Let $m(t)$ be
the expected number of faults detected by time $t$ can be shown as 1:
$m(t)=a(1-e^{-bt})$ (1)
where, $m(\infty)=a$ so that a represents the expected number of software
failures to be eventually encountered and b is the detection rate for an
individual fault.
According to Goel and Okumoto, the operational performance of a system is to a
large extent dependent on testing time. Longer testing phase leads to enhanced
performance. Also, cost of fixing a default during operation is generally much
more than during testing. However, the time spent in testing delays the
product release, which leads to additional costs. The objective is to
determine optimal release time to minimize cost by reducing testing time. Goel
and Okumoto gave the parameters $c_{1},c_{2},c_{3},t$ and $T$ which are as
follows:
$c_{1}$ = cost of fixing a fault during testing
$c_{2}$ = cost of fixing a fault during operation $(c_{2}>c_{1})$
$c_{3}$ = cost of testing per unit time
$t$ = software life cycle length
$T$ = software release time (same as testing time)
Since $m(t)$ represents the expected number of faults during $(0,t)$ the
expected costs of fixing faults during the testing and operational phases are
$c_{1}m(T)$ and $c_{2}(m(t)-m(T))$ respectively. Further, the testing cost
during a time period $T$ is $c_{3}(T)$. If there is a cost associated with
delay in meeting a delivery plan, such a cost could be included in $c_{3}$.
Combining the above costs, the total expected cost is given by (2).
$C(T)=c_{1}m(T)+c_{2}[m(t)-m(T)]+c_{3}(T)$ (2)
This policy minimizes the average cost and depends on the ratio of $a*b$ and
$C_{r}=c_{3}/(c_{2}-c_{1})$ (3)
Two cases arise, $ab>C_{r}$ and $ab\leq C_{r}$
Case I : If $ab>C_{r}$, the optimal policy is to take
$T^{*}=min(T_{0},t)$ (4)
where $T_{0}=1/bln(ab/C_{r})$
Case II : If $ab<=C_{r}$, then $T=0$. If the cost of testing or cost of delay
in release are very high, the solution favors no testing at all i.e.
$T^{*}=0$.
On the other hand, if the cost of fixing a fault after release is very high as
compared to the usefulness of the system, the solution will tend to favor not
using the system i.e. $T^{*}=t$.
### 2.3 Related work by Praveen et al.
This paper suggests prioritizing the software modules into 5 categories namely
very high, high, medium, low and very low. Then they calculate optimal cost
and time similar to Goal and Okumoto work. To find out maximum allowable cost
and time stringency concept is used here. Stringency is the maximum allowable
deviation from the optimum which is decided by the organization.
Then they advise to start testing the software to calculate the actual time
and actual cost for each priority category. The deviation from optimal testing
time and optimal cost can be calculated from (5) and (6).
$\alpha={(T_{a}-T^{*})\over T^{*}}$ (5)
Where,
$\alpha$ = deviation from optimal time
$T_{a}$ = actual testing time
$T^{*}$ = optimal testing time calculated from (4), and
$\beta={(C_{a}-C_{0})\over C_{0}}$ (6)
Where,
$\beta$ = deviation from optimal cost
$C_{a}$ = actual testing cost
$C_{0}$ = optimal testing cost calculated from (2)
Limiting factor $\delta$ is given by (7)
$\delta=\alpha+\beta$ (7)
Afterwards they cumulatively calculate the limiting factor to determine
whether further software testing is required.
### 2.4 Related work by Ohba
The above discussed models view the software as single unit, regardless of the
structural or functional relationship among software subsystems (modules).
Based on the concept of redundancy, recovery block techinique [15] and
N-version program techinique [14] s-independently produce multiple versions of
the software to perform the same function.
Most software reliability models assume s-independence of faults. However,
Ohba [16] argues that faults are s-dependent because of the logical or
functional dependency within a program. Ohba observed an S-shaped software
reliability growth curve, as opposed to the exponential growth curve for the
s-independence models. The model is characterized by:
$m(t)=n.[1-(1+\phi.t).exp(-\phi.t)]$ (8)
Unlike most software reliability models that use execution time, the S-shaped
model is generally observed when calendar time is used.
### 2.5 Musa-Okumoto
Musa & Okumoto [17] proposed a logarithmic Poisson execution-time model where
the observed number of failures by time $t$ is NHPP. This model adds a decay
parameter, and is characterized by:
$m(t)=(1/\theta).\log(\lambda.\theta.t+1)$ (9)
## 3 Proposed Approach
### 3.1 Components Priority
To ensure that the component prioritization is uniform and effective, it is
imperative to introduce a schema [13]. The following parameters may be helpful
to decide the priority of the components.
Production Time This is the amount of work carried out by an employee on the
project. This parameter keeps the track of total person hours for a module.
Module priority will increase as Production time increases.
Decision density High complexity may result in bad understandability and more
errors. Complex procedures also need more time to develop and test. Therefore,
excessive complexity should be avoided. Too complex procedures should be
simplified by rewriting or splitting into several procedures. Complexity is
often positively correlated to code size. A big program or function is likely
to be complex as well. These are not equal, however. A procedure with
relatively few lines of code might be far more complex than a long one. We
recommend the combined use of lines of code and complexity metrics to detect
complex code. The total cyclomatic complexity for a module is calculated as
follows.
$TCC=Sum(CC)-Count(CC)+1$ (10)
Cyclomatic complexity is usually higher in longer procedures. How much
decision is there actually, compared to lines of code? This is where you need
decision density (also called cyclomatic density).
$DD=CC/LLOC$ (11)
where LLOC id logical lines of codes. This parameter shows the average
decision density of the code lines within the modules.
Programming Path This parameter suggest that what environment for coding is
used. Costs associated with technology required for the component. What are
the importance of current technology for this component. How much experts are
available for such technologies.
Size of Components How much code had done?
Skill of fault reporters/resolvers Source of origin of fault suggested is how
much reliable. Errors are reported technically or just by inexperience of
user. Actually in our model, we consider that faults are collected using some
bug tracking system which is open to customer too.
Weight priority This includes the ranking given by developers, managers and
customer based on the requirements and previous experiences. It also includes
risk factors.
Code reusability If an earlier source code can be used in the current work
with little or no modifications then we call it code reusability. This lessens
the requirements of testing the code again as it has already been tested
earlier.
Coupling It is the measure of connectedness of one module to another. It is
given as-
$C=1-\left(k\over{(d_{i}+ac_{i}+d_{o}+bc_{o}+g_{d}+cg_{c}+w+r)}\right)$ (12)
Where $C$ = Coupling
$d_{i}$ = number of input data parameters
$c_{i}$ = number of input control parameters
$d_{o}$ = number of output data parameters
$c_{o}$ = number of output control parameters
$g_{d}$ = number of global variables used as data
$g_{c}$ = number of global variables used as control
$w$ = number of modules called (fan-out)
$r$ = number of modules calling the module under consideration (fan-in)
the values of $k$ and $a,b$ and $c$ may be adjusted as more experimental
verification occurs [11].
Layout appropriateness For a specific layout (i.e., a specific GUI design),
cost can be assigned to each sequence of actions according to the following
relationship:
$\textrm{cost}=\Sigma[\textrm{frequency of transition}(k)\times\textrm{cost of
transition}(k)]$ (13)
where $k$ is a specific transition from one layout entity to the next as a
specific task is accomplished. Layout appropriateness is defined as
$LA=100\times[(\textrm{cost of LA}-\textrm{optimal layout})/(\textrm{cost of
proposed layout})]$ (14)
where $LA=100$ for an optimal layout.
Maintenance $M_{T}$ = the number of modules in the current release $F_{c}$ =
the number of modules in the current release that have been changed $F_{a}$ =
the number of modules in the current release that have been added $F_{d}$ =
the number of modules from the preceding release that were deleted in the
current release
The software maturity index is computed in the following manner:
$SMI=[M_{T}-(F_{a}+F_{c}+F_{d})]/M_{T}$ (15)
As $SMI$ approaches 1.0, the product begins to stabilize. SMI may also be used
as parameter for planning software maintenance activities.
The parameters are not limited as above. Some other parameters may also be
used. Even fuzzy parametes may also included.
### 3.2 Weight Parameter for Each Component
In our system these parameters are based on neural networks. Assume that
$w_{1,ij},~{}(i=1,2,3,...,p;j=1,2,3,...,q;)$ are the weight between $i$-th
unit on sensory layer and $j$-th unit on association layer. And,
$w_{2,jk},~{}(j=1,2,3,...,q;k=1,2,3,...,r;)$ are the weight between $j$-th
unit on association layer and $k$-th unit on response layer. $x_{i}$ represent
the normalized input variables to the $i$-th unit on sensory layer and $y_{k}$
represent the output values. We apply normalized values of fault level, fault
reporter, etc to input values $x_{i}$. Cosider the logistic activation
function, sigmod function
$f(x)={1\over{1+e^{-\theta x}}}$ (16)
Then the input-out rules of each unit on each layer are
$h_{j}=f(\sum_{i=1}^{p}{w_{1,ij}x_{i}})$ (17)
$y_{k}=f(\sum_{j=1}^{q}{w_{2,jk}h_{ji}})$ (18)
We apply the multi-layered neural networks by propagation in order to learn
the interaction among software components [18]. Now as the error in $y_{k}$
may be given as
$\epsilon_{k}=\frac{1}{2}\sum_{k=1}^{r}{(y_{k}-d_{k})^{2}})$ (19)
where $d_{k}$ are the target input values for the output values. We consider
the estimation and prediction model so that the property of interation among
software components accumulates on the connection weight of neural networks.
Finally, we may obtain the total weight parameter $p_{k}$ which represents the
level of importance for each component
$p_{k}=\frac{y_{k}}{\sum_{k=1}^{r}{y_{k}}}$ (20)
### 3.3 Our Extension to Goel and Okumoto Scheme
In Goel-Okumoto method, $m(t)$ represents the faults during $(0,t)$, the
expected costs of fixing faults during the testing and operational phases are
$c_{1}m(T)$ and $c_{2}(m(t)-m(T))$ respectively. Further, the testing cost
during a time period $T$ is $c_{3}(T)$. If there is a cost associated with
delay in meeting a delivery plan, such a cost could be included in $c_{3}$.
Here we assume that software developement is in muti-version environemt.
During the developement phase of current version some, fault appears in
previous version. It is clear that cost to repair that fault goes to previous
version’s cost, which we could not include here. But fault appearing in
previous version is nearly equivalent to finding fault is current version. The
cost for this could not be same as $c_{1}$. We assume this newly associated
cost as $c_{4}$. Now if $n(t)$ represents the faults in previous version
during $(0,t)$, the expected costs of fixing faults during the testing and
operational phases is $c_{4}n(T)$. Thus, total expected cost is now
$C(T)=c_{1}m(T)+c_{2}[m(t)-m(T)-n(T)]+c_{3}(T)+c_{4}n(T)$ (21)
### 3.4 Component Importance basis Testing
Now, we decide level of priority on the basis of parameter $p_{k}$. In order
to resolve tie cases manual decision may be prefered. If some dependent module
should be given much more prefernce if its parent module is not tested. After
prioritzing the modules, try to find optimum cost and time parameters in very
similar way to Goel’s Model.
Let $T$ and $C$ be the total time and cost available to release the software.
Our aim is to the test all the modules within $T$ and $C$. But if we are not
able to do this then at least the components with very high priority must be
tested. We set the fault tolerance = 0 for the first time testing of all the
components of a particular category (e.g. Very High) and find out actual time
and cost for testing.
If optimal cost and time parameters $C^{*}$, $T^{*}$ are determined, then we
can compute a expected cost as limiting factor $\delta=f(T,T^{*},C,C^{*})$.
i.e.
$\delta=p\frac{(C-C^{*})}{C^{*}}+(1-p)\frac{(T-T^{*})}{T^{*}}$ (22)
where $p$ is odds in in favour of cost.
## References
* [1] Onoma, K., W.T. Tsai, M. Poonawala and H. Suganuma, Regression Testing in an Industrial Environment, Comm. ACM, vol. 41, no. 5, pp. 81-86, May 1988.
* [2] McDaid, Kevin and Wilson Simon P., Deciding How Long to Test Software, The Statistician, Royal Statistical Society, Part 2, 50, pp. 117-134, 2001.
* [3] Musa, J.D. and Ackerman A.F., Quantifying Software Validation: When to Stop Testing, IEEE Software, vol.6, Issue 3, pp. 19-27, May 1989.
* [4] Ehrlich W., Prasanna b., Stampfel J. and Wu J., Determining the Cost of a Stop-Test Decision IEEE Software, vol. 10, Issue 2, pp. 33-42, March 1993.
* [5] Goel, A.L. and Okumoto K., When to stop testing and start using software, Proc. of ACM, pp. 131-137, 1981.
* [6] Goel, A.L. and Okumoto K., A Time Dependent Error Detection Rate Model for Software Performance Assessment with Applications, Proc. National Computer Conference, RADC-TR-80-179, May 1980.
* [7] Goel A.L. and Okumoto K., A Time Dependent Error Detection Rate Model for Software Reliability and Other Performance Measures, IEEE Transactions on Reliability, vol. R-28, no. 3, pp. 206-211, August 1979.
* [8] Praveen R Srivastava, Deepak Pareek, Kailash Sati, Dinesh C Pujari and G Raghurama, Non Homogenous Poisson Process Based Cumulative Priority Model for Determining Optimal Software Testing Period, ACM SIGSOFT Software Engineering Notes, vol. 33, no. 2, March 2008.
* [9] Jones Capers, Applied Software Measurement, McGraw-Hill, New Your, NY, 1991.
* [10] Praveen R Srivastava, Krishan Kumar and G. Raghurama, Test Case Prioritization Based on Requirements and Risk Factors, ACM SIGSOFT Software Engineering Notes, vol. 33, no. 4, July 2008.
* [11] R. S. pressman, Software Engineering: A Practitioner s Ap-proach, McGraw hill, 6th Edition. 2005.
* [12] Praveen R Srivastava, Model for Optimizing Software Testing Period using Non Homogenous Poisson Process based on Cumulative Test Case Prioritization, IEEE TENCON, Hyderabad, India, 18-21 Nov., 2008.
* [13] Praveen R Srivastava, Deepak Pareek, Component Prioritization Schema for Achieving Maximum Time and Cost Benefits from Software Testing, IEEE Region 10 Colloquium and the Third ICIIS, Kharagpur, INDIA December 8-10 2008.
* [14] A. Avizienis, ”The N-Vesrsion approach to fault tolerant software”, IEEE Tran. Software Engineering, Vol SE-11, pp 1411-1423, 1985.
* [15] H. Hecht, ”Fault tolerance software”, IEEE Trans. Reliability, Vol. R-28, pp. 227-232, 1979.
* [16] M. Ohba, Software reliability analysis models , ZBM J. Research and Development, vol 28, num 4, pp 428-443, 1984.
* [17] J.D. Musa, A. Iannino, K. Okumoto, Software Reliability: Measurement, Prediction, Application, 1987; McGraw-Hill.
* [18] E. D., Karnin, A simple procedure for pruning back propagation trained neural networks, IEEE Trans. Neural Networks, 1, pp. 239–242, 1990.
|
arxiv-papers
| 2009-04-17T19:40:05 |
2024-09-04T02:49:01.938932
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Amit K Awasthi and Sanjay Chaudhary",
"submitter": "Amit K Awasthi",
"url": "https://arxiv.org/abs/0904.2769"
}
|
0904.2770
|
ITM Probe: analyzing information flow in protein networks
Aleksandar Stojmirović and Yi-Kuo Yu***to whom correspondence should be
addressed
National Center for Biotechnology Information
National Library of Medicine
National Institutes of Health
Bethesda, MD 20894
United States
#### Summary:
Founded upon diffusion with damping, ITM Probe is an application for modelling
information flow in protein interaction networks without prior restriction to
the sub-network of interest. Given a context consisting of desired origins and
destinations of information, ITM Probe returns the set of most relevant
proteins with weights and a graphical representation of the corresponding sub-
network. With a click, the user may send the resulting protein list for
enrichment analysis to facilitate hypothesis formation or confirmation.
#### Availability:
ITM Probe web service and documentation can be found at
www.ncbi.nlm.nih.gov/CBBresearch/qmbp/mn/itm_probe
#### Contact:
yyu@ncbi.nlm.nih.gov
## 1 Introduction
Protein interaction networks are presently under intensive research (Bader et
al., 2008). Recently, a number of authors have applied the concept of random
walk (with truncation) to extract biologically relevant information from
protein interaction networks (Nabieva et al., 2005; Tu et al., 2006; Suthram
et al., 2008). These approaches, however, do not model information
loss/leakage that naturally occurs in all networks. For example, in cellular
networks, proteases constantly degrade proteins, diminishing the strength of
information propagation. We have recently developed a mathematical framework
to model information flow in interaction networks with a novel ingredient,
damping/aging of information (Stojmirović and Yu, 2007). Implementing the
theory, we have constructed a web application ITM Probe, which also contains a
new model of information propagation: information channel.
ITM Probe models information flow in a protein interaction network through
discrete random walks. Unlike classical random walks, our model allows the
walker a certain probability to _dissipate_ or _damp_ (that is, to leave the
network) at each step. Each walk, simulating a possible information path,
terminates either by dissipation or by reaching a boundary node.
We distinguish two types of boundary nodes: _sources_ (emitting information)
and _sinks_ (absorbing information). ITM Probe offers three models: absorbing,
emitting and channel. For any network node, the corresponding weight returned
by the emitting model is the expected number of visits to that node by a
random walk originating at given source(s). The absorbing model, on the other
hand, returns the likelihood of a random walk starting at that node to
terminate at sink(s). The channel model combines the emitting and absorbing
models: it contains both sources and sinks as boundary and reports the
expected numbers of visits to any network node from random walks originating
at sources and terminating at sinks.
Each selection of boundary nodes and dissipation rates provides the biological
_context_ for the information transmission modelled. Small dissipation allows
random walks to explore the nodes farther away from their origin while large
dissipation evaporates quickly most walks. For the channel model, dissipation
controls how much a random walk can deviate from the shortest path from
sources to sinks. We call the set of most significant nodes, in terms of the
weights returned, an _Information Transduction Module_ (ITM).
## 2 Usage
Both the absorbing and emitting models navigate neighborhoods of selected
nodes and illuminate the protein complexes associated with them. However, the
absorbing model can reveal relatively distant ‘leaf’ nodes linked to a sink by
a nearly unique path, while the emitting model favors highly connected
clusters. The channel model is suited for discovery of potential pathways
linking proteins of interest or biological functions associated with them.
Using multiple sources may reveal the potential points of crosstalk between
information channels, while a solution of multiple sinks chosen according to a
set of competing hypotheses may suggest the most biologically plausible
pathways among many possible ones.
Every model of ITM Probe requires an interaction graph, the boundary nodes
(sources and/or sinks) and the damping factors as input. The damping factors
may be specified directly or by setting the desired average path-length
(emitting/channel model) or the average likelihood of absorption at sinks
(absorbing model).
Although our mathematical framework can be applied to any directed graph, our
web service presently supports only the yeast (Saccharomyces cerevisiae)
physical interaction networks derived from the BioGRID (Stark et al., 2006)
database. We offer three yeast networks: Full, Reduced and Directed. The Full
network consists of all interactions from the BioGRID as an undirected graph,
while the Reduced consists only of those interactions that are from low-
throughput experiments (that is, from publications reporting less than $300$
interactions) or are reported by at least two independent publications. The
Directed network is derived from Reduced by turning all interactions labelled
as ‘Biochemical activity’ into directed links (bait $\to$ prey).
To assist in silico investigations on the impact of knocking out certain
genes, ITM Probe allows users to specify nodes to exclude from the network.
Furthermore, it is known (Steffen et al., 2002) that proteins with a large
number of non-specific interaction partners might overtake the true signaling
proteins in the information flow modeling. Therefore, ITM Probe by default
excludes from the yeast networks the proteins that may provide undesirable
shortcuts, such as cytoskeleton proteins, histones and chaperones. The user
may choose to lengthen or shorten this list.
### Output and analysis
ITM Probe outputs a list of the top ranking nodes together with an image of
the sub-network consisting of these nodes (Fig. 1). Images are produced using
the Graphviz suite (Gansner and North, 2000). Each protein listed is linked to
its full description in several external databases. The number of nodes to be
listed can be specified directly by the user or determined automatically from
the model results through a criterion such as participation ratio (Stojmirović
and Yu, 2007) or the cutoff value. The resulting weights for all nodes can be
downloaded in the CSV format for further analysis.
Figure 1: An example ITM from running the ITM Probe channel model.
Each ITM image can be rendered and saved in multiple formats (SVG, PNG, JPEG,
EPS and PDF). For each rendering, the users can choose which aspects of
results to display, the color map and the scale for presentation (linear or
logarithmic). When multiple boundary points are specified, it is possible to
obtain an overview of all of their contributions simultaneously by selecting
the color mixture scheme (Fig. 1). In this case, each source (channel/emitting
model) or sink (absorbing model) is assigned a basic CMY (cyan, magenta or
yellow) color and the coloring of each displayed node is a result of mixing
the colors corresponding to its source- or sink- specific values for each of
the boundary points.
While it is possible to specify any proteins in the network as sources and
sinks, not every context produces biologically meaningful results. To
facilitate biological interpretation of the users’ results, we have locally
implemented a Gene Ontology (GO) (Ashburner et al., 2000) enrichment tool
based on GO::TermFinder of Boyle et al. (2004). It compares a given input list
of proteins to the lists annotated with GO terms and finds those GO terms that
statistically best explain the input list. Every ITM Probe results page
contains a query form allowing the user to specify the number of the top
ranking proteins to consider for GO term enrichment analysis.
### Example
Histone acetyltransferases remodel chromatin by acetylating histone octamers
and hence may play an important role in transcription activation (Sterner and
Berger, 2000). To explore the interface between them and the RNA Polymerase II
core in yeast, we choose three histone acetyltransferases (Hat1p, Gcn5p,
Elp3p) as sources and a catalytic subunit Rpo21p of RNA Polymerase II as a
sink for the channel model (Fig. 1). From the color mixing image it appears
that Elp3p and Gcn5p interact with Rpo21p through a wide channel of proteins,
while Hat1p seems to be remote from Rpo21p. This prompts the hypothesis that
Hat1p is not directly involved in transcription activation. Enrichment
analysis, using the 16 nodes (shown in magenta color in Fig. 1) mostly visited
from Hat1p, shows that Hat1p and these nodes participate mainly in DNA
replication and only indirectly in transcription regulation, thus reinforcing
the hypothesis. Similar analysis on the nodes associated with Elp3p indicates
the interaction is almost exclusively through the elongator complex. The nodes
associated with Gcn5p are less specific, indicating a more generic interface,
but are all involved mRNA transcription.
## 3 Outlook
We plan to include interaction networks from additional organisms, once their
coverage/quality becomes comparable to those from yeast. In principle, the
analysis from ITM Probe can be integrated with existing partial knowledge to
form a broad picture of possible communication paths in cellular processes.
The concept of context-specific analysis may find applications beyond
biological networks.
## Acknowledgments
This work was supported by the Intramural Research Program of the National
Library of Medicine at National Institutes of Health. ITM Probe implementation
relies on a variety of open source projects, which we acknowledge on our
website.
## References
* Ashburner et al. (2000) Ashburner, M. et al. (2000). Gene ontology: tool for the unification of biology. the gene ontology consortium. Nat Genet, 25, 25–29.
* Bader et al. (2008) Bader, S. et al. (2008). Interaction networks for systems biology. FEBS Lett, 582(8), 1220–4.
* Boyle et al. (2004) Boyle, E. I. et al. (2004). GO::TermFinder–open source software for accessing gene ontology information and finding significantly enriched gene ontology terms associated with a list of genes. Bioinformatics, 20, 3710–3715.
* Gansner and North (2000) Gansner, E. R. and North, S. C. (2000). An open graph visualization system and its applications to software engineering. Software — Practice and Experience, 30(11), 1203–1233.
* Nabieva et al. (2005) Nabieva, E. et al. (2005). Whole-proteome prediction of protein function via graph-theoretic analysis of interaction maps. Bioinformatics, 21 Suppl 1, 302–310.
* Stark et al. (2006) Stark, C. et al. (2006). BioGRID: a general repository for interaction datasets. Nucleic Acids Res, 34(Database issue), D535–9.
* Steffen et al. (2002) Steffen, M. et al. (2002). Automated modelling of signal transduction networks. BMC Bioinformatics, 3, 34.
* Sterner and Berger (2000) Sterner, D. E. and Berger, S. L. (2000). Acetylation of histones and transcription-related factors. Microbiol Mol Biol Rev, 64(2), 435–459.
* Stojmirović and Yu (2007) Stojmirović, A. and Yu, Y.-K. (2007). Information flow in interaction networks. J Comput Biol, 14(8), 1115–43.
* Suthram et al. (2008) Suthram, S. et al. (2008). eQED: an efficient method for interpreting eQTL associations using protein networks. Mol. Syst. Biol., 4, 162.
* Tu et al. (2006) Tu, Z. et al. (2006). An integrative approach for causal gene identification and gene regulatory pathway inference. Bioinformatics, 22, e489–496.
|
arxiv-papers
| 2009-04-17T19:52:32 |
2024-09-04T02:49:01.948573
|
{
"license": "Public Domain",
"authors": "Aleksandar Stojmirovi\\'c and Yi-Kuo Yu",
"submitter": "Aleksandar Stojmirovi\\'c",
"url": "https://arxiv.org/abs/0904.2770"
}
|
0904.2809
|
# Bound of Entanglement of Assistance and Monogamy Constraints
Zong-Guo Li Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
Shao-Ming Fei Department of Mathematics, Capital Normal University, Beijing
100037, China Institut für Angewandte Mathematik, Universität Bonn, 53115,
Germany Sergio Albeverio Institut für Angewandte Mathematik, Universität
Bonn, 53115, Germany W. M. Liu Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100080, China
###### Abstract
We investigate the entanglement of assistance which quantifies capabilities of
producing pure bipartite entangled states from a pure tripartite state. The
lower bound and upper bound of entanglement of assistance are obtained. In the
light of the upper bound, monogamy constraints are proved for arbitrary
n-qubit states.
###### pacs:
03.67.Mn, 03.65.Ud, 03.65.Yz
## I Introduction
In quantum information theory, entanglement is a vital resource for some
practical applications such as quantum cryptography, quantum teleportation and
quantum computation bennett ; nielsen . During the last decade, this inspired
a great deal of effort for detecting and quantifying the entanglement wootters
; chen ; mintert0 ; mintert1 ; gao ; li1 ; ou ; mintert2 ; li2 . On the other
hand, the creation and distribution of entanglement is also of central
interest in quantum information processing. More specially the distribution of
bipartite entanglement is a key ingredient for performing certain quantum-
information processing tasks such as teleportation.
One of the methods for generating bipartite entanglement is the entanglement
of assistance that is defined in Refs. cohen ; dp . It quantifies the
entanglement which could be created by reducing a multipartite entangled state
to an entangled state with fewer parties (e.g. bipartite) via measurements.
Such producing of entanglement, also called “assisted entanglement”, is a
special case of the _localizable entanglement_ localizable , which is
especially important for quantum communication, where quantum repeaters are
needed to establish bipartite entanglement over a long length scale hj1 . For
a pure $2\otimes 2\otimes n$ state, the analytical formula of entanglement of
assistance has been derived by Laustsen _et al._ laustsen , whereas the
calculation of entanglement of assistance is not easy for a general pure
tripartite state gour .
In this paper, we explore the entanglement of assistance for a general pure
tripartite state in terms of I-concurrence rungta . We obtain a lower bound of
entanglement of assistance, which is also the lower bound of a tripartite
entanglement measure, the entanglement of collaboration. This may help to
characterize the localizable entanglement. Furthermore, an upper bound is also
obtained. Deducing from the upper bound of entanglement of assistance, we find
a proper form of entanglement monogamy inequality for arbitrary N-qubit
states, which is analogous to the monogamy constraints for concurrence
proposed by Coffman _et al._ coffman and proven by Osborne _et al._ tobias
for the general case.
The paper is organized as follows: In Sec. II, we derive a lower bound and
upper bound of entanglement of assistance for pure tripartite states. In Sec.
III, monogamy constraints are proved in terms of this upper bound. Finally in
Sec. IV we conclude with a discussion of our results.
## II Bound of entanglement of Assistance
We consider a pure ($d_{1}\times d_{2}\times N$) tripartite state shared by
three parties referred to as Alice, Bob and Charlie, who performs a
measurement on his party to yield a known bipartite entangled state shared by
Alice and Bob. Charlie’s aim is to maximize the entanglement of the state
between Alice and Bob. This maximum average entanglement that he can create is
called entanglement of assistance, which was originally defined in terms of
entropy of entanglement dp ; cohen . In this paper, we define entanglement of
assistance in terms of the entanglement measure I-concurrence:
$\displaystyle
E_{a}(|\psi\rangle_{ABC})\\!\equiv\\!E_{a}(\rho_{AB})\\!\equiv\\!\textrm{max}\sum_{i}p_{i}C(|\phi_{i}\rangle_{AB}),$
which is maximized over all possible pure-state decompositions of
$\rho_{AB}=\textrm{Tr}_{C}[|\psi\rangle_{ABC}\langle\psi|]=\sum_{i}p_{i}|\phi_{i}\rangle_{AB}\langle\phi_{i}|$.
By applying the method in Ref. mintert0 , we can obtain the lower bound of
entanglement of assistance for pure tripartite states.
For any given pure-state decomposition of $\rho_{AB}$,
$\rho_{AB}=\sum_{i}p_{i}|\phi_{i}\rangle_{AB}\langle\phi_{i}|$, we have
$\displaystyle E_{a}(|\psi\rangle_{ABC})\\!$ $\displaystyle=$
$\displaystyle\\!\textrm{max}\sum_{i}p_{i}C(|\phi_{i}\rangle_{AB})$ (1)
$\displaystyle=$
$\displaystyle\\!\textrm{max}\sum_{i}p_{i}\sqrt{\sum_{mn}|\langle\phi_{i}|S_{mn}|\phi_{i}^{*}\rangle|^{2}}$
$\displaystyle\geq$
$\displaystyle\\!\textrm{max}\sqrt{\sum_{mn}(\sum_{i}p_{i}|\langle\phi_{i}|S_{mn}|\phi_{i}^{*}\rangle|)^{2}},$
where $S_{mn}=L_{m}\otimes L_{n}$, $L_{m},m=1,...,d_{1}(d_{1}-1)/2$,
$L_{n},n=1,...,d_{2}(d_{2}-1)/2$ are the generators of group $SO(d_{1})$ and
$SO(d_{2})$ respectively. The inequality holds according to the Minkowski
inequality
$[\sum\limits_{i=1}(\sum\limits_{k}x_{i}^{k})^{p}]^{1/p}\leq\sum_{k}[\sum\limits_{i=1}(x_{i}^{k})^{p}]^{1/p},\text{
}p>1$. Consider the eigenvalue decomposition of $\rho_{AB}$, $\rho_{AB}=\Psi
M\Psi^{\dagger}$, where $M$ is a diagonal matrix whose diagonal elements are
the eigenvalues of $\rho$, and $\Psi$ is a unitary matrix whose columns are
the eigenvectors of $\rho$. Taking into account the relation $\Phi
W^{1/2}=\Psi M^{1/2}U$, where $U$ is a right-unitary matrix, we can rewrite
inequality (1) as
$\displaystyle\\!\\!E_{a}(\rho_{AB})\\!$ $\displaystyle\geq$
$\displaystyle\\!\textrm{max}\sqrt{\sum_{mn}(\sum_{i}|\Phi^{T}W^{\frac{1}{2}}S_{mn}W^{\frac{1}{2}}\Phi|_{ii})^{2}}$
$\displaystyle=$
$\displaystyle\\!\textrm{max}\sqrt{\sum_{mn}(\sum_{i}|U^{T}M^{\frac{1}{2}}\Psi^{T}S_{mn}\Psi
M^{\frac{1}{2}}U|_{ii})^{2}}.$
In terms of the Cauchy-Schwarz inequality
$(\sum_{i}x_{i}^{2})^{\frac{1}{2}}(\sum_{i}y_{i}^{2})^{\frac{1}{2}}\geq\sum_{i}x_{i}y_{i}$,
the inequality
$\displaystyle
E_{a}(\rho_{AB})\\!\geq\\!\textrm{max}\sum_{i}\left|U^{T}\left(\sum_{mn}z_{mn}A_{mn}\right)U\right|_{ii}$
(2)
is implied for any $z_{mn}=y_{mn}exp(i\theta_{mn})$ with $y_{mn}\geq 0$ and
$\sum_{mn}y_{mn}^{2}=1$, where $A_{mn}=M^{\frac{1}{2}}\Psi^{T}S_{mn}\Psi
M^{\frac{1}{2}}$. Since $\sum_{mn}z_{mn}A^{mn}$ is a symmetric matrix, we can
always find a unitary matrix $U$ such that
$\sum_{i}|U^{T}(\sum_{mn}z_{mn}A_{mn})U|_{ii}=\|\sum_{mn}z_{mn}A^{mn}\|$ as
shown in Ref. horn , where $\|\cdot\|$ stands for the trace norm defined by
$\|G\|=\textrm{Tr}(GG^{\dagger})^{1/2}$. For an arbitrary unitary matrix $V$,
we have
$\displaystyle\sum_{i}|V^{T}(\sum_{mn}z_{mn}A_{mn})V|_{ii}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\sum_{i}|V^{T}(U^{-1})^{T}U^{T}(\sum_{mn}z_{mn}A_{mn})UU^{-1}V|_{ii}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\sum_{i}|V^{T}(U^{-1})^{T}Diag(\lambda_{1},\lambda_{2}\cdots)U^{-1}V|_{ii}$
$\displaystyle\\!\\!\leq\\!\\!$
$\displaystyle\sum_{ij}|(U^{-1}V)_{ij}|^{2}\lambda_{i}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\sum_{i}\lambda_{i},$
where $\lambda_{i}(z)$s, dependent on the choice of the $y$ and $\theta$, are
the singular values of the matrix $\mathcal{T}=\sum_{mn}z_{mn}A^{mn}$, i.e.,
the square roots of the eigenvalues of the positive Hermitian matrix
$\mathcal{T}\mathcal{T}^{\dagger}$. Therefore the maximum of Eq. (2) is given
by
$\underset{z\in\mathbf{C}}{max}\left(\sum_{i}\lambda_{i}(z)\right)=\underset{z\in\mathbf{C}}{max}\|\sum_{mn}z_{mn}A^{mn}\|$.
Hence, we arrive at the lower bound of entanglement of assistance for a pure
tripartite state as following:
$\displaystyle
E_{a}(\rho_{AB})\\!\geq\\!\underset{z\in\mathbf{C}}{\textrm{max}}\|\sum_{mn}z_{mn}A^{mn}\|.$
(3)
Furthermore the entanglement of collaboration gour2 ; gour3 quantifies the
maximum amount of entanglement that can be generated between two parties from
a tripartite state with collaborations composed of local operations and
classical communication among the three parties. It has been shown by Gour
_et. al._ gour2 that, for tripartite states, the entanglement of
collaboration is greater than or equal to entanglement of assistance in terms
of a given entanglement measure. Therefore our lower bound is also the one for
entanglement of collaboration, which can be tightened by numerical
optimization. Our bound may help to characterize localizable entanglement. For
a pure $2\times 2\times N$ state, this lower bound is consistent with the
result of Ref. laustsen .
We can also obtain the upper bound of entanglement of assistance. From the
definition of entanglement of assistance, we have
$\displaystyle[E_{a}(\rho_{AB})]^{2}$ $\displaystyle=$
$\displaystyle[\textrm{max}\sum_{i}p_{i}C(|\phi_{i}\rangle_{AB})]^{2}$
$\displaystyle\\!\leq\\!$
$\displaystyle\textrm{max}\sum_{i}[\sqrt{p_{i}}C(|\phi_{i}\rangle_{AB})]^{2}\sum_{i}(\sqrt{p_{i}})^{2}$
$\displaystyle\\!=\\!$
$\displaystyle\textrm{max}\sum_{i}2p_{i}[1-\textrm{Tr}(\rho_{i}^{A})^{2}]$
$\displaystyle\\!\leq\\!$ $\displaystyle 2(1-\textrm{Tr}\rho_{A}^{2}),$
where $\rho_{i}^{A}=\textrm{Tr}_{B}|\phi_{i}\rangle_{AB}\langle\phi_{i}|$. The
first inequality holds according to the Cauchy-Schwarz inequality tj ; the
last one, which has also been proved in Ref. vicente , holds due to the convex
property of $\textrm{Tr}\rho_{A}^{2}$.
Define the upper bound as the tangle of assistance
$\tau_{a}(\rho_{AB})\equiv\textrm{max}\sum_{i}p_{i}[C(|\phi_{i}\rangle_{AB})]^{2}$.
Similar to the entanglement of assistance that satisfies the monogamy
constraints for n-qubit pure state gour1 ; monogamy , we show below that the
tangle of assistance also exhibits monogamy constraints for arbitrary n-qubit
states.
## III Monogamy inequality
Consider a pure tripartite state $|\Psi\rangle_{ABC}$. The tangle of
assistance is defined by
$\displaystyle\tau_{a}(|\Psi\rangle_{ABC})$ $\displaystyle=$
$\displaystyle\underset{\\{p_{x},|\psi_{x}\rangle\\}}{\textrm{max}}\sum_{x}p_{x}[C(|\psi_{x}\rangle)]^{2}$
$\displaystyle=$
$\displaystyle\underset{\\{p_{x},|\psi_{x}\rangle\\}}{\textrm{max}}\sum_{x}p_{x}S_{2}[\textrm{Tr}_{B}(|\psi_{x}\rangle\langle\psi_{x}|)],$
where the linear entropy $S_{2}[\rho]=2[1-\textrm{Tr}(\rho)^{2}]$, and the
maximum runs over all pure-state decompositions $\\{p_{x},|\psi_{x}\rangle\\}$
of
$\rho_{AB}=\textrm{Tr}_{C}(|\Psi\rangle_{ABC}\langle\Psi|)=\sum_{x}p_{x}|\psi_{x}\rangle\langle\psi_{x}|$.
In the case of pure state $\rho_{AB}$, the tangle of assistance is the square
of concurrence of this state.
###### Theorem 1
For an arbitrary n-qubit state, the tangle of assistance satisfies,
$\displaystyle\tau_{a}(\rho_{A_{1}A_{2}})+\tau_{a}(\rho_{A_{1}A_{3}})+\cdots+\tau_{a}(\rho_{A_{1}A_{n}})$
(4) $\displaystyle\geq$ $\displaystyle\tau_{a}(\rho_{A_{1}(A_{2}A_{3}\cdots
A_{n})}),$
where $\tau_{a}(\rho_{A_{1}(A_{2}A_{3}\cdots A_{n})})$ denotes the tangle of
assistance in the bipartite partition $A_{1}|A_{2}A_{3}\cdots A_{n}$.
Proof: First of all, we prove the following inequality
$\tau_{a}(\rho_{AB})+\tau_{a}(\rho_{AC})\geq\tau_{a}(\rho_{A(BC)}),$ (5)
for arbitrary tripartite states $\rho_{ABC}$ in $2\times 2\times 2^{n-2}$
system.
We first prove Eq. (5) for pure states. In this case, due to the local-unitary
invariance of $\tau_{a}(\rho_{AC})$, we can rotate the basis of subsystem $C$
into the local Schmidt basis $|V_{k}\rangle$, $k=1,\cdots,4$, given by the
eigenvectors of $\rho_{C}=Tr_{AB}(\rho_{ABC})$. In this way we can regard the
$2^{n-2}$-dimensional qudit $C$ as an effective four-dimensional qudit.
Therefore, we simply need to prove Eq. (5) for a $2\times 2\times 4$ pure
state $ABC$.
For pure states of a tripartite system $ABC$ of two qubits $A$ and $B$ and a
four-level system $C$, we have
$\displaystyle\tau_{a}(\rho_{A(BC)})-\tau_{a}(\rho_{AC})$ $\displaystyle=$
$\displaystyle
S_{2}(\rho_{A})-\underset{\\{p_{j},|\phi_{j}\rangle\\}}{\textrm{max}}\sum_{j}p_{j}S_{2}[\textrm{Tr}_{C}(|\phi_{j}\rangle\langle\phi_{j}|)],$
where $\sum_{j}p_{j}|\phi_{j}\rangle\langle\phi_{j}|=\rho_{AC}$. It can be
shown that any pure-state decomposition of $\rho_{AC}$ can be realized by
positive-operator-valued measures (POVMs) $\\{M_{x}\\}$ performed by Bob, the
rank of which is 1 (for more details see gour ; lp ). Therefore, we get the
the following expression
$\displaystyle\tau_{a}(\rho_{AC})=\underset{\\{M_{x}\\}}{\textrm{max}}\sum_{x}p_{x}S_{2}(\rho_{x}),$
(6)
where the maximum runs over all rank-1 POVMs on Bob’s system,
$p_{x}=\textrm{Tr}(I_{A}\otimes M_{x}\rho_{AB})$ is the probability of outcome
$x$, and $\rho_{x}=\textrm{Tr}_{B}(I_{A}\otimes M_{x}\rho_{AB})/p_{x}$ is the
posterior state in Alice’s subsystem. For convenience, we take the definition
$\displaystyle
I(\rho_{AB}):=S_{2}(\rho_{A})-\underset{\\{M_{x}\\}}{\textrm{max}}\sum_{x}p_{x}S_{2}(\rho_{x}).$
By comparing $I(\rho_{AB})$ with Eq. (5) for pure tripartite states, we see
that it is sufficient to prove the inequality
$\displaystyle I(\rho_{AB})\leq\tau_{a}(\rho_{AB}),$
for all two-qubit states $\rho_{AB}$.
We first derive a computable formula for $I(\rho_{AB})$. Any bipartite quantum
state $\rho_{AB}$ may be written as
$\displaystyle\rho_{AB}=\Lambda\otimes I_{B}(|V_{B^{\prime}B}\rangle\langle
V_{B^{\prime}B}|),$ (7)
where $V_{B^{\prime}B}$ is the symmetric two-qubit purification of the reduced
density operator $\rho_{B}$ on an auxiliary qubit system $B^{\prime}$ and
$\Lambda$ is a qubit channel from $B^{\prime}$ to $A$. Deducing from Eq. (6)
we have
$\displaystyle\rho_{x}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\textrm{Tr}_{B}(I_{A}\otimes M_{x}\rho_{AB})/p_{x}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\textrm{Tr}_{B}[(I_{A}\otimes
M_{x})(\Lambda\otimes I_{B})|V_{B^{\prime}B}\rangle\langle
V_{B^{\prime}B}|)]/p_{x}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\Lambda[\textrm{Tr}_{B}(I_{A}\otimes
M_{x}|V_{B^{\prime}B}\rangle\langle V_{B^{\prime}B}|)]/p_{x}.$
Since the rank of $M_{x}$ is 1, $\textrm{Tr}_{B}(I_{A}\otimes
M_{x}|V_{B^{\prime}B}\rangle\langle V_{B^{\prime}B}|)]$ is a pure state.
Moreover, all pure-state decompositons of
$\rho_{B}^{\prime}=\textrm{Tr}_{B}(|V_{B^{\prime}B}\rangle\langle
V_{B^{\prime}B}|)=\rho_{B}$ can be realized by the rank-1 POVM measurements
$\\{M_{x}\\}$ operating on subsystem $B$ of $|V_{B^{\prime}B}\rangle\langle
V_{B^{\prime}B}|$. Hence $I(\rho_{AB})$ satisfies
$I(\rho_{AB})=S_{2}[\Lambda(\rho_{B})]-\underset{\\{p_{x},|\psi_{x}\rangle\\}}{\textrm{max}}\sum_{x}p_{x}S_{2}[\Lambda(|\psi_{x}\rangle)],$
(8)
where the maximum runs over all pure-state decompositions
$\\{p_{x},|\psi_{x}\rangle\\}$ of $\rho_{B}$ such that
$\sum_{x}p_{x}|\psi_{x}\rangle\langle\psi_{x}|=\rho_{B}$.
The action of a qubit channel $\Lambda$ on a single-qubit state
$\rho=(I+\mathbf{r}\cdot\boldsymbol{\upsigma})/2$, where
$\boldsymbol{\upsigma}$ is the vector of Pauli operators, may be written as
$\Lambda(\rho)=[I+(\mathbf{L}\mathbf{r}+\mathbf{l})\cdot\boldsymbol{\upsigma}]/2$,
where $\mathbf{L}$ is a $3\times 3$ real matrix and $\mathbf{l}$ is a three-
dimensional vector. In this Pauli basis, the possible pure-state
decompositions of $\rho_{B}$ are represented by all possible sets of
probabilities $\\{p_{j}\\}$ and unit vectors $\\{\mathbf{r}_{j}\\}$ such that
$\sum_{j}p_{j}\mathbf{r}_{j}=\mathbf{r}_{B}$, where
$(I+\mathbf{r}_{B}\cdot\boldsymbol{\upsigma})/2=\rho_{B}$. In terms of the
Block representation of one-qubit states, the linear entropy $S_{2}$ is given
by $S_{2}[(I+\mathbf{r}\cdot\boldsymbol{\upsigma})/2]=1-|\mathbf{r}|^{2}$. In
this way we get the following equation
$S_{2}[\Lambda(I+\mathbf{r}\cdot\boldsymbol{\upsigma})/2]=1-(\mathbf{L}\mathbf{r}+\mathbf{l})^{T}(\mathbf{L}\mathbf{r}+\mathbf{l})$.
Substituting $\mathbf{r}_{j}=\mathbf{r}_{B}+\mathbf{x}_{j}$, one can easily
check that Eq. (8) reduces to the following one whose value is determined by
$\\{p_{j},\mathbf{x}_{j}\\}$ subject to the conditions
$\sum_{j}p_{j}\mathbf{x}_{j}=0$ and $|\mathbf{r}_{B}+\mathbf{x}_{j}|=1$,
$\displaystyle I(\rho_{AB})$ (9) $\displaystyle\\!=\\!$ $\displaystyle
S_{2}[\Lambda(\rho_{B})]-\underset{\\{p_{j},\mathbf{x}_{j}\\}}{\textrm{max}}\sum_{j}p_{j}S_{2}[\Lambda(\frac{I+(\mathbf{r}_{B}+\mathbf{x}_{j})\cdot\boldsymbol{\upsigma}}{2})]$
$\displaystyle\\!=\\!$ $\displaystyle
1-(\mathbf{L}\mathbf{r}_{B}+\mathbf{l})^{T}(\mathbf{L}\mathbf{r}_{B}+\mathbf{l})$
$\displaystyle\\!-\\!$
$\displaystyle\underset{\\{p_{j},\mathbf{x}_{j}\\}}{\textrm{max}}\sum_{j}p_{j}\Big{\\{}1-[\mathbf{L}(\mathbf{r}_{B}+\mathbf{x}_{j})+\mathbf{l}]^{T}[\mathbf{L}(\mathbf{r}_{B}+\mathbf{x}_{j})+\mathbf{l}]\Big{\\}}$
$\displaystyle=$
$\displaystyle\underset{\\{p_{j},\mathbf{x}_{j}\\}}{\textrm{min}}\sum_{j}p_{j}(\mathbf{x}^{T}_{j}\mathbf{L}^{T}\mathbf{L}\mathbf{x}_{j}).$
Without loss of generality, we assume that $\mathbf{L}^{T}\mathbf{L}$ is
diagonal with diagonal elements $\lambda_{x}\leq\lambda_{y}\leq\lambda_{z}$.
The constrains $|\mathbf{r}_{B}+\mathbf{x}_{j}|=1$ lead to the identities
$(\mathbf{x}^{x}_{j})^{2}=1-|\mathbf{r}_{B}|^{2}-2\mathbf{r}_{B}^{T}\mathbf{x}_{j}-(\mathbf{x}^{y}_{j})^{2}-(\mathbf{x}^{z}_{j})^{2}$.
Substituting this into Eq. (9), we get
$I(\rho_{AB})=\lambda_{x}(1-|\mathbf{r}_{B}|^{2})+\underset{\\{p_{j},\mathbf{x}_{j}\\}}{\textrm{min}}\sum_{j}p_{j}[(\lambda_{y}-\lambda_{x})(\mathbf{x}^{y}_{j})^{2}+(\lambda_{z}-\lambda_{x})(\mathbf{x}^{z}_{j})^{2}]$.
This expression is obviously minimized by choosing
$\mathbf{x}^{z}_{j}=\mathbf{x}^{y}_{j}=0$ for all $j$. Then from the condition
$|\mathbf{r}_{B}+\mathbf{x}_{j}|=1$, $\mathbf{x}^{x}_{j}$ have two solutions.
The ensemble of two states corresponding to such two solutions can reach the
minimum $\lambda_{x}(1-|\mathbf{r}_{B}|^{2})$.
As $S_{2}(\rho_{B})=(1-|\mathbf{r}_{B}|^{2})$, we obtain the following
computable expression: $I(\rho_{AB})=\lambda_{min}S_{2}(\rho_{B})$. Note that
a local filtering operation of the form $\rho^{\prime}_{AB}=\frac{(I\otimes
B)\rho_{AB}(I\otimes B^{\dagger})}{\textrm{Tr}[(I\otimes
B^{\dagger}B)\rho_{AB}]}$ leaves $\mathbf{L}$ invariant and transforms
$S_{2}(\rho_{B^{\prime}})=\frac{\textrm{det}(B)^{2}}{\textrm{Tr}[(I\otimes
B^{\dagger}B)\rho_{AB}]^{2}}S_{2}(\rho_{B})$ frank .
If the local filtering operator $B$ is invertible, we can get the conclusion
that there does not exist a pure-state decomposition
$\\{q_{j},|\psi_{j}\rangle\\}$ of $\rho^{\prime}_{AB}$ such that
$\tau_{a}(\rho^{\prime}_{AB})>\frac{\textrm{det}(B)^{2}}{\textrm{Tr}[(I\otimes
B^{\dagger}B)\rho_{AB}]}\tau_{a}(\rho_{AB})$ by the contradiction. For the
case that the operator $B$ is not invertible, such pure-state decomposition
also doesn’t exist. Furthermore, there exists exactly an optimal pure-state
decomposition $\\{p_{i},|\phi_{i}\rangle\\}$ of the state $\rho_{AB}$ for
$\tau_{a}(\rho_{AB})$ such that $\sum_{i}p_{i}C[\frac{(I\otimes
B)(|\phi_{i}\rangle\langle\phi_{i}|I\otimes
B^{\dagger})}{\textrm{Tr}[(I\otimes
B^{\dagger}B)\rho_{AB}]}]^{2}=\frac{\textrm{det}(B)^{2}}{\textrm{Tr}[(I\otimes
B^{\dagger}B)\rho_{AB}]^{2}}\tau_{a}(\rho_{AB})$. Therefore, the tangle of
assistance
$\tau_{a}(\rho^{\prime}_{AB})=\frac{\textrm{det}(B)^{2}}{\textrm{Tr}[(I\otimes
B^{\dagger}B)\rho_{AB}]^{2}}\tau_{a}(\rho_{AB})$. Since
$I(\rho^{\prime}_{AB})=\frac{\textrm{det}(B)^{2}}{\textrm{Tr}[(I\otimes
B^{\dagger}B)\rho_{AB}]^{2}}\lambda_{min}S_{2}(\rho_{B})$, it transforms
exactly in the same way as the tangle of assistance
$\tau_{a}(\rho^{\prime}_{AB})$ does. As there always exists a filtering
operation for which $\rho_{B}^{\prime}\propto I$, we can assume, without loss
of generality, that $S_{2}(\rho_{B})=1$.
So let us consider $\rho_{AB}$ with
$\rho_{B}=\textrm{Tr}_{A}(\rho_{AB})=\frac{1}{2}I$. In terms of Pauli
operators, we can rewrite the pure state as follows:
$\displaystyle\frac{(I\otimes B)|V_{B^{\prime}B}\rangle\langle
V_{B^{\prime}B}|(I\otimes B^{\dagger})}{\textrm{Tr}[(I\otimes
B^{\dagger}B)|V_{B^{\prime}B}\rangle\langle V_{B^{\prime}B}|]}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{1}{4}[I+\sum_{i}m_{i}I\otimes\sigma_{i}+\sum_{i}n_{i}\sigma_{i}\otimes
I+\sum_{ij}O_{ij}\sigma_{i}\otimes\sigma_{j}],$
where $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$ are $\sigma_{x}$,
$\sigma_{y}$ and $\sigma_{z}$ respectively. Then we get the conclusion from
its purity and unity reduced density, that $m_{i}=n_{i}=0$ for all i and the
$3\times 3$ real matrix $O$ is orthogonal. Thus we have
$\rho_{AB}=\frac{1}{4}\Lambda\otimes
I_{B}[I+\sum_{ij}O_{ij}\sigma_{i}\otimes\sigma_{j}]=\frac{1}{4}[I+\sum_{i}l_{i}\sigma_{i}\otimes
I+\sum_{ij}(LO)_{ij}\sigma_{i}\otimes\sigma_{j}]$. As unitary operator $U_{1}$
satisfies the equation
$U_{1}\sigma_{i}U_{1}^{\dagger}=\sum_{j}P_{ij}\sigma_{j}$, where $P$ is a real
orthogonal $3\times 3$ matrix, we can always find local unitary operators, in
terms of the theorem of singular value decomposition, so that $U_{1}\otimes
U_{2}\rho_{AB}U_{1}^{\dagger}\otimes
U_{2}^{\dagger}=\frac{1}{4}[I+\sum_{i}(lP)_{i}\sigma_{i}\otimes
I+\sum_{ij}(QLOP)_{ij}\sigma_{i}\otimes\sigma_{j}]=\frac{1}{4}[I+\sum_{i}l^{\prime}_{i}\sigma_{i}\otimes
I+\sum_{i}(L^{\prime})_{ii}\sigma_{i}\otimes\sigma_{i}]$, where $Q$ and $P$
are real orthogonal matrix and $L^{\prime}$ is a diagonal matrix with its
diagonal elements the singular values of $L$. Because of the local-unitary
invariance of $\tau_{a}(\rho_{AB})$ and $I(\rho_{AB})$, without loss of
generality, we assume that
$\rho_{AB}=\frac{1}{4}[I+\sum_{i}t_{i}\sigma_{i}\otimes
I+\sum_{i}(R)_{ii}\sigma_{i}\otimes\sigma_{i}]$, where $R$ is a diagonal
matrix with its diagonal elements the singular values of $L$. Due to the
positivity of
$\displaystyle\rho_{AB}=\\!\\!\\!\frac{1}{4}\left(\\!\\!\\!\begin{array}[]{cccc}1+R_{3}+t_{3}\\!\\!\\!&\\!\\!\\!0\\!\\!\\!&\\!\\!\\!t_{1}-it_{2}\\!\\!\\!&\\!\\!\\!R_{1}-R_{2}\\\
0\\!\\!\\!&\\!\\!\\!1-R_{3}+t_{3}\\!\\!\\!&\\!\\!\\!R_{1}+R_{2}\\!\\!\\!&\\!\\!\\!t_{1}-it_{2}\\\
t_{1}+it_{2}\\!\\!\\!&\\!\\!\\!R_{1}+R_{2}\\!\\!\\!&\\!\\!\\!1-R_{3}-t_{3}\\!\\!\\!&\\!\\!\\!0\\\
R_{1}-R_{2}\\!\\!\\!&\\!\\!\\!t_{1}+it_{2}\\!\\!\\!&\\!\\!\\!0\\!\\!\\!&\\!\\!\\!1+R_{3}-t_{3}\\\
\end{array}\\!\\!\right)\\!\\!,$
the inequality $1-t_{1}^{2}-t_{2}^{2}-t_{3}^{2}\geq R_{3}^{2}$ must hold.
Therefore we obtain
$\displaystyle\\!\\!\\!\tau_{a}(\rho_{AB})\geq[C_{a}(\rho_{AB})]^{2}$
$\displaystyle\geq$
$\displaystyle\\!\\!\\!\textrm{Tr}[\sigma_{y}\otimes\sigma_{y}\rho^{*}_{AB}\sigma_{y}\otimes\sigma_{y}\rho_{AB}]$
$\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{16}\left[4+4(R_{1}^{2}+R_{2}^{2}+R_{3}^{2})-4(t_{1}^{2}+t_{2}^{2}+t_{3}^{2})\right]$
$\displaystyle\geq$
$\displaystyle\\!\\!\\!\frac{1}{4}[R_{1}^{2}+R_{2}^{2}+2R_{3}^{2}]$
$\displaystyle\geq$
$\displaystyle\\!\\!\\!\lambda_{min}(\mathbf{L}^{T}\mathbf{L}).$
This inequalities imply that $I(\rho_{AB})\leq\tau_{a}(\rho_{AB})$ for all
two-qubit states $\rho_{AB}$, which then proves Eq. (5) for pure states.
Now we extend Eq. (5) to mixed state case. Consider the maximizing pure-state
decomposition $\\{p_{x},|\psi_{x}\rangle\\}$ for $\tau_{a}(\rho_{A(BC)})$. By
applying the inequality Eq. (5) and taking into account the concavity of
$\tau_{a}$, we have
$\displaystyle\tau_{a}(\rho_{A(BC)})$ $\displaystyle=$
$\displaystyle\sum_{x}p_{x}\tau_{a}(\rho^{x}_{A(BC)})$ $\displaystyle\leq$
$\displaystyle\sum_{x}p_{x}[\tau_{a}(\rho^{x}_{AB})+\tau_{a}(\rho^{x}_{AC})]$
$\displaystyle\leq$ $\displaystyle\tau_{a}(\rho_{AB})+\tau_{a}(\rho_{AC}),$
where $\rho^{x}_{A(BC)}=|\psi_{x}\rangle\langle\psi_{x}|$.
Let $C=C_{1}C_{2}$ be a $2\times 2^{n-3}$ system and apply Eq. (5), then we
get
$\displaystyle\tau_{a}(\rho_{A(BC)})\\!\\!\\!$
$\displaystyle\\!\\!\\!\leq\tau_{a}(\rho_{AB})+\tau_{a}(\rho_{AC})$
$\displaystyle\\!\\!\\!\leq\tau_{a}(\rho_{AB})+\tau_{a}(\rho_{AC_{1}})+\tau_{a}(\rho_{AC_{2}}).$
Successively applying Eq. (5) to partitions of $C$, we obtain the inequality
Eq. (4) by induction. $\blacksquare$
In fact, Eq. (4) turns out to be an equality for product states under
partition $A|BC_{1}\cdots C_{n}$. For the generalized GHZ states, Eq. (4) is a
strictly inequality.
## IV Discussion
In summary, as an important quantity in quantum computation, the entanglement
of assistance has been investigated in terms of I-concurrence for pure
tripartite states. We have obtained a lower bound of entanglement of
assistance, which is also the lower bound of the tripartite entanglement
measure, the entanglement of collaboration. In stead of great difficulty
involved in computing the entanglement of collaboration, the lower bound Eq.
(3) can be calculated in a numerical optimization to make a good estimation of
entanglement of collaboration. Moreover, an upper bound is also obtained. In
the light of the upper bound of entanglement of assistance, we find a proper
form of entanglement monogamy inequality for arbitrary N-qubit states.
This work was supported by NSFC under grants Nos. 60525417, 10740420252,
10874235, 10875081, 10675086, the NKBRSFC under grants Nos. 2006CB921400,
2009CB930704, KZ200810028013 and NKBRPC(2004CB318000).
## References
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|
arxiv-papers
| 2009-04-18T02:44:46 |
2024-09-04T02:49:01.970436
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zong-Guo Li, Shao-Ming Fei, Sergio Albeverio and W. M. Liu",
"submitter": "ZongGuo Li",
"url": "https://arxiv.org/abs/0904.2809"
}
|
0904.2824
|
# On the Grothendieck groups of toric stacks
Zheng Hua
## 1 Introduction
In this note, we prove that the Grothendieck group of a smooth complete toric
Deligne-Mumford stack is torsion free. This statement holds when the generic
point is stacky. We also construct an example of open toric stack with torsion
in K-theory. This is a part of the author’s Ph.D thesis. A similar result has
been proved by Goldin, Harada, Holm, Kimura and Knutson in [GHHKK] using
symplectic methods.
## 2 Grothendieck groups of reduced stacks
Let $N$ be a free abelian group of rank $d$ and
$N_{\mathbb{R}}=N\otimes{\mathbb{R}}$. Given a complete simplicial fan
$\Sigma$ in $N_{\mathbb{R}}$, one chooses an integral element $v_{i}$ in each
of the one-dimensional cones of $\Sigma$. This defines a stacky fan
$\bf{\Sigma}$ in the sense of [BCS]. We denote the corresponding toric
Deligne-Mumford stack by ${\mathcal{X}}_{\bf\Sigma}$. Recall the Grothendieck
group $K_{0}({\mathcal{X}}_{\bf\Sigma})$ is defined to be the free abelian
group generated by all formal combinations of coherent sheaves on
${\mathcal{X}}_{\bf\Sigma}$ modding out by the short exact sequences. Each
element $v_{i}$ corresponds to a toric invariant divisor $E_{i}$. This divisor
$E_{i}$ determines an invertible sheaf ${\mathcal{O}}(E_{i})$. We denote its
equivalent class in $K_{0}({\mathcal{X}}_{\bf\Sigma})$ by $R_{i}$. The ring
structure of $K_{0}({\mathcal{X}}_{\bf\Sigma})$ is given by tensor product of
coherent sheaves. K-theory of smooth toric stacks has been studied in [BH]. In
particular they computed $K_{0}({\mathcal{X}}_{\bf\Sigma})$ explicitly by
writing out its generators and relations.
###### Theorem 2.1.
[BH] Let $B$ be the quotient of the Laurent polynomial ring
${\mathbb{Z}}[x_{1},x_{1}^{-1},\ldots,x_{n},x_{n}^{-1}]$ by the ideal
generated by the relations
* •
$\prod_{i=1}^{n}x_{i}^{\langle m,v_{i}\rangle}=1$ for any dual vector $m\in
M=Hom(N,{\mathbb{Z}})$,
* •
$\prod_{i\in S}(1-x_{i})=0$ for any set $S\subseteq[1,\ldots,n]$ such that
$\\{v_{i}|i\in S\\}$ are not contained in any cone of $\Sigma$.
Then the map from $B$ to $K_{0}({\mathcal{X}}_{\bf\Sigma})$ which sends
$x_{i}$ to $R_{i}$ is an isomorphism of rings.
Our main result is the following.
###### Theorem 2.2.
The Grothendieck group $K_{0}({\mathcal{X}}_{\bf\Sigma})$ of a complete smooth
toric Deligne-Mumford stack ${\mathcal{X}}_{\bf\Sigma}$ is a free
${\mathbb{Z}}$ module.
###### Proof.
We denote the Laurent polynomial ring
${\mathbb{Z}}[x_{1},x_{1}^{-1},\ldots,x_{n},x_{n}^{-1}]$ by $R$. Let $A=R/I$,
where $I$ is generated by $\prod_{i\in S}(1-x_{i})=0$ for any set
$S\subseteq[1,\ldots,n]$ such that $\\{v_{i}|i\in S\\}$ are not contained in
any cone of $\Sigma$. And $B=A/J$, where $J$ is generated by $n$ Laurent
polynomials ${h_{j}=\prod_{i=1}^{n}x_{i}^{\langle m_{j},v_{i}\rangle}-1}$
where $m_{j}$ is an integral basis of $M$.
First we want to replace $h_{j}$ by
$g_{j}=\prod_{<m_{j},v_{i}>>0}x_{i}^{<m_{j},v_{i}>}-\prod_{<m_{j},v_{i}><0}x_{i}^{-<m_{j},v_{i}>}$.
They generate the same ideal $J$ but this collection avoids negative powers.
To prove $B$ is a free ${\mathbb{Z}}$ module we need to show that the
multiplication map $B\rightarrow pB$ is an injection for any prime $p$. Let
$K(g_{1},\ldots,g_{d})$ and $K(g_{1},\ldots,g_{d},p)$ be the Koszul complexes
for sequences ${g_{1},\ldots,g_{d}}$ and ${g_{1},\ldots,g_{d},p}$ of elements
of the ring $A$. These two Koszul complexes are related by the following
lemma.
###### Lemma 2.3.
[E] Let $\phi:K(g_{1},\ldots,g_{d})\rightarrow K(g_{1},\ldots,g_{d})$ be the
map of multiplication by $p$. Then $K(g_{1},\ldots,g_{d},p)$ equals
$Cone(\phi)[-1]$. Here Cone means mapping cone of complexes.
###### Proof.
See corollary 17.11. of [E]. ∎
According to this lemma, we get a long exact sequence of cohomology groups:
$\begin{CD}\ldots
@>{}>{}>H^{i}(K(g_{1},\ldots,g_{d},p))@>{}>{}>H^{i}(K(g_{1},\ldots,g_{d}))\\\
@>{\phi}>{}>H^{i}(K(g_{1},\ldots,g_{d}))@>{}>{}>H^{i+1}(K(g_{1},\ldots,g_{d},p))@>{}>{}>\ldots\end{CD}$
(2.1)
We will show that all the cohomology groups of $K(g_{1},\ldots,g_{d})$ and
$K(g_{1},\ldots,g_{d},p)$ vanish except at one position. More precisely, the
only non vanishing piece of (2.1) is:
$\begin{CD}0@>{}>{}>H^{n}(K(g_{1},\ldots,g_{d},p))\cong
B@>{p}>{}>H^{n}(K(g_{1},\ldots,g_{d}))\cong B\\\
@>{}>{}>H^{n+1}(K(g_{1},\ldots,g_{d},p))\cong B/pB@>{}>{}>0\end{CD}$
To prove this we need a result about Cohen-Macaulay properties of Stanley-
Reisner rings.
###### Theorem 2.4.
Let $A^{\prime}={\mathbb{Z}}[x_{1},\ldots,x_{n}]/I$. Ring $A^{\prime}$ is
Cohen-Macaulay.
###### Proof.
If we make a change of variables $x_{i}$ to $1-x_{i}$, then we see that
$A^{\prime}$ is nothing but the Stanley-Reisner ring associated to supporting
polytope of $\Sigma$. It is a general fact that the Stanley-Reisner ring of
polytopes are CM over any field(See Chapter 5 of [BrHe]). Furthermore one can
show it is actually CM over ${\mathbb{Z}}$(See Exercise 5.1.25 of [BrHe]). We
will sketch the solution of this exercise in the following remark. ∎
###### Remark 2.5.
Consider the flat morphism ${\mathbb{Z}}\to A^{\prime}$. For any maximal ideal
$\mathfrak{q}\subset A^{\prime}$, we have $\mathfrak{q}\cap{\mathbb{Z}}=(p)$.
In order to show $A^{\prime}$ is CM it suffices to check it for each fiber,
i.e. $A^{\prime}_{\mathfrak{q}}/pA^{\prime}_{\mathfrak{q}}$ is CM for all the
maximal ideal $\mathfrak{q}$. If $(p)$ is not $(0)$ then
$A^{\prime}_{\mathfrak{q}}/pA^{\prime}_{\mathfrak{q}}=(A^{\prime}\otimes{\mathbb{Z}}/p{\mathbb{Z}})_{\mathfrak{q}}$.
It is CM because Stanley-Reisner ring over the field is CM. So we just need to
show that for any maximal ideal $\mathfrak{q}$, the restriction
$\mathfrak{q}\cap{\mathbb{Z}}$ is not $(0)$. Suppose this is the case, we will
have an inclusion ${\mathbb{Z}}\to A^{\prime}/\mathfrak{q}$. However, since we
assume $\mathfrak{q}\cap{\mathbb{Z}}=(0)$, the field $A^{\prime}/\mathfrak{q}$
must have characteristic zero. But this contradicts the fact that $A^{\prime}$
is finitely generated over ${\mathbb{Z}}$ because ${\mathbb{Q}}$ is not
finitely generated over ${\mathbb{Z}}$.
###### Corollary 2.6.
The ring $A$ is Cohen-Macaulay.
###### Proof.
Because $A$ is a localization of $A^{\prime}$ and being CM ring is a local
property, $A$ is CM by Theorem 2.4. ∎
###### Remark 2.7.
It follows from the general theory of Stanley Reisner ring (Theorem $5.1.16$
of [BrHe]) that $A^{\prime}$ has Krull dimension $d+1$.
###### Lemma 2.8.
[E] Suppose $M$ is a finitely generated module over ring $A$ and
$I=(x_{1},\ldots,x_{n})\subset A$ is a proper ideal. If $depth(I)=r$ then
$H^{i}(M\bigotimes K(x_{1},\ldots,x_{n}))=0$ for $i<r$, while
$H^{r}(M\bigotimes K(x_{1},\ldots,x_{n}))=M/IM$.
###### Lemma 2.9.
The quotient $A/J$ is a finitely generated abelian group.
###### Proof.
Let ${\bf k}$ be any field and $f$ be an arbitrary map from $A/J$ to ${\bf
k}$. Maximal ideals of $A/J$ are in one to one correspondence with such map
$f$. We want to solve for $(x_{1},\ldots,x_{n})$ that satisfy equations in
ideal $I$ and $J$ in the field ${\bf k}$. Recall elements of ideal $I$ are in
form of $\prod_{i\in S}(1-x_{i})$ for any subset $S\subseteq[1,\ldots,n]$ such
that one dimensional rays $v_{i},i\in S$ are not contained in any cone of
$\Sigma$. So $x_{i}$ equals 1 outside some cone $\sigma$. Then equations in
$J$ reduce to $\prod_{v_{i}\in\sigma}x_{i}^{\langle m,v_{i}\rangle}$=1. We can
choose the dual vector $m$ such that $\langle m,v_{i}\rangle=0$ for all but
one $i$. Say $\langle m,v_{i}\rangle=d_{i}$. The number $d_{i}$ only depends
on the fan but not on the field ${\bf k}$. This implies that $1-x_{i}^{d_{i}}$
maps to 0 for any map $f$ from $A/J$ to ${\bf k}$, i.e. $1-x_{i}^{d_{i}}$ is
contained in any maximal ideal of $A/J$. Because $A/J$ is a finitely generated
${\mathbb{Z}}$ algebra the Jacobson radical coincides with nilradical. So
$(1-x_{i}^{d_{i}})^{N}$=0 for any $i$. We can pick a large enough integer $N$
uniformly for any $x_{i}$ such that there exists a ${\mathbb{Z}}$ basis
consisting of monomials with powers of each $x_{i}$ between 0 and $Nd_{i}$.
This proves the statement of the lemma. ∎
By theorem 2.4, remark 2.7 and lemma 2.9 we can prove:
###### Corollary 2.10.
The ideal $J=(g_{1},\ldots,g_{d})$ has depth $d$.
###### Proof.
Because $A$ is CM, by the definition of CM rings $depth(J)=codim(J)$. The
quotient $A/J$ is finitely generated over ${\mathbb{Z}}$, therefore, of Krull
dimension one. By remark 2.7 $codim(J)=d$ and $depth(J)=d$. ∎
This corollary above together with lemma 2.8 imply the Koszul complex
$K(g_{1},\ldots,g_{d})$ has only one nonzero cohomology
$H^{d}(K(g_{1},\ldots,g_{d}))=B=A/J$. On the other hand, the lemma 2.9 imples
$B/pB$ is a finite dimensional vector space over ${\mathbb{Z}}/p$. By similar
argument with the corollary above we get $depth(J,p)=d+1$. Then
$H^{i}(K(g_{1},\ldots,g_{d},p))=B/pB$ when $i=d+1$ and zero otherwise. Now by
applying the long exact sequence (2.1) we prove the multiplication map by $p$
is an injection. This finish the proof of theorem 2.2. ∎
###### Remark 2.11.
The proof of theorem 2.2 can be generalized to the non complete toric stacks
satsifying a condition called “shellability”. This is a combinatorial
condition on the underlying simplicial complex of the toric stack(See [BrHe]
for details of this definition). It is proved in [BrHe] that Stanley-Reisner
rings of shellable simplicial complexes are Cohen-Macaulay. However, we will
see in Chapter 4 that Grothendieck groups of open toric stacks are not
necessarily free.
## 3 Grothendieck groups of non-reduced stacks
Now we remove the assumption that $N$ is a free abelian group. Then the
corresponding toric stack will have nontrivial stabilizer at the generic
point. We will generalize theorem 2.2 to this setting. Recall the derived Gale
dual of the homomorphism $\beta:{\mathbb{Z}}^{n}\to N$ is the homomorphism
$\beta^{\vee}:({\mathbb{Z}}^{n})^{\vee}\to DG(\beta)$. When $N$ is torsion
free, $DG(\beta)$ is the Picard group. The general definition of $DG(\beta)$
involves a projective resolution of $N$. We refer to [BCS] for details.
Theorem 2.1 can be generalized to the case when $N$ has torsion. Notice the
ring ${\mathbb{Z}}[x_{1},x_{1}^{-1},\ldots,x_{n},x_{n}^{-1}]/J$ is the
representation ring of the algebraic group $Hom(DG(\beta),{\mathbb{C}}^{*})$
when $N$ is torsion free. If $N$ has torsion then
$Hom(DG(\beta),{\mathbb{C}}^{*})$ maps to $({\mathbb{C}}^{*})^{n}$ with finite
kernel. After replacing
${\mathbb{Z}}[x_{1},x_{1}^{-1},\ldots,x_{n},x_{n}^{-1}]/J$ by the
representation ring of $Hom(DG(\beta),{\mathbb{C}}^{*})$ we can generalize
Theorem 2.1 to non reduced case(See section $6$ of [BH] for more details).
###### Theorem 3.1.
Let $N$ be a finitely generated abelian group and $\bf\Sigma$ is a stacky fan
in $N$. The Grothendieck Group $K_{0}({\mathcal{X}}_{\bf\Sigma})$ is a free
${\mathbb{Z}}$ module.
###### Proof.
Let’s denote the $N_{free}$ for the quotient $N/torsion(N)$ and
${\mathcal{X}}_{red}$ for the reduced stack associated to $N_{free}$. Recall
the Grothendieck group $K_{0}({\mathcal{X}}_{\bf\Sigma})$ is the quotient of
representation ring of $Hom(DG(\beta),{\mathbb{C}}^{*})$ by the ideal $I$
generated by Stanley-Reisner relations. Let’s denote the Gale dual group of
the reduced stack ${\mathcal{X}}_{red}$ by $DG(\beta_{red})$. The quotient map
$N\to N_{free}$ induces an inclusion on Gale dual groups $DG(\beta_{red})\to
DG(\beta)$, whose cokernel is isomorphic to $torsion(N)$. Now we see the
Grothendieck groups $K_{0}({\mathcal{X}}_{\bf\Sigma})$ and
$K_{0}({\mathcal{X}}_{red})$ are isomorphic to the group rings
${\mathbb{Z}}[DG(\beta)]$ and ${\mathbb{Z}}[DG(\beta_{red})]$. If we fix a
lifting from $torsion(N)$ to $DG(\beta)$, then we get a coset decomposition
$DG(\beta)=\sqcup_{y\in torsion(N)}(yDG(\beta_{red}))$. This induce a coset
decomposition of the group ring ${\mathbb{Z}}[DG(\beta)]$ such that each coset
is isomorphic with ${\mathbb{Z}}[DG(\beta_{red})]$. Since
${\mathbb{Z}}[DG(\beta_{red})]$ is torsion free by theorem 2.2, we prove the
theorem. ∎
## 4 Grothendieck groups of non complete stacks
Theorem 2.1 holds for non complete toric stacks too. But our proof for
freeness of K-theory relies on the shellability of the underlying simplicial
complex of the toric stack. There are many non complete toric stacks whose
underlying simplicial complexes are _not_ shellable. For example, we can take
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. Denote its four toric invariant
divisors by $E_{1},E_{2},E_{3}$ and $E_{4}$. Let point $P$(resp. $Q$) be the
intersection of $E_{1}$ and $E_{2}$(resp. $E_{3}$ and $E_{4}$). Simplicial
complex of ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\backslash\\{P,Q\\}$ is not
shellable.
Actually, there are examples of non complete toric stacks such that their
Grothendieck groups have torsions. The following example is due to Lev
Borisov.
###### Example 4.1.
Let’s take a dimension five weighted projective stack
${\mathbb{P}}(1,1,1,1,2,2)$. Denote its toric invariant divisors by
$E_{1},E_{2},\ldots,E_{6}$, where $E_{1},\ldots,E_{4}$ have weights one and
$E_{5},E_{6}$ have weights two. Let ${\mathcal{X}}$ be the substack
${\mathbb{P}}(1,1,1,1,2,2)\backslash\\{(E_{1}\cap E_{2}\cap E_{3}\cap
E_{4})\cup(E_{5}\cap E_{6})\\}$. By theorem 2.1
$K_{0}({\mathcal{X}})=\frac{{\mathbb{Z}}[t,t^{-1}]}{\langle(1-t)^{4},(1-t^{2})^{2}\rangle}$
It is easy to check that $t(1-t)^{2}$ is a torsion element.
## References
* [BCS] L.A. Borisov, L. Chen, G.G. Smith, _The orbifold Chow ring of toric Deligne-Mumford stacks._ J. Amer. Math. Soc. 18 (2005), no. 1, 193–215.
* [BH] L.A. Borisov, R.P. Horja, _On the $K$-theory of smooth toric DM stacks._ Snowbird lectures on string geometry, 21–42, Contemp. Math., 401, Amer. Math. Soc., Providence, RI, 2006.
* [BrHe] W. Bruns, J. Herzog, _Cohen-Macaulay rings_. Cambridge Studies in advanced mathematics 39. Cambridge Univ. Press, Cambridge, 1993.
* [E] D. Eisenbud, _Commutative Algebra with a View Toward Algebraic Geometry_ , GTM 150.
* [GHHKK] R. Goldin, M. Harada, T. Holm, T. Kimura, A. Knutson. _MSRI talk on workshop in Combinatorial, Enumerative and Toric Geometry by Tara Holm_.
Department of Mathematics, University of Wisconsin-Madison,
Madison, WI, 53706, U.S.
hua@math.wisc.edu
|
arxiv-papers
| 2009-04-18T06:34:00 |
2024-09-04T02:49:01.976660
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zheng Hua",
"submitter": "Zheng Hua",
"url": "https://arxiv.org/abs/0904.2824"
}
|
0904.2837
|
# ASYMPTOTIC PROPERTIES OF RANDOM MATRICES OF LONG-RANGE PERCOLATION MODEL
S.Ayadi LMV - Laboratoire de Mathématiques de Versailles, Université de
Versailles Saint-Quentin-en-Yvelines, 78035 Versailles (FRANCE). E-mail:
ayadi@math.uvsq.fr
Abstract: We study the spectral properties of matrices of long-range
percolation model. These are $N\times N$ random real symmetric matrices
$H=\\{H(i,j)\\}_{i,j}$ whose elements are independent random variables taking
zero value with probability $1-\psi\left((i-j)/b\right)$,
$b\in\mathbb{R}^{+}$, where $\psi$ is an even positive function with
$\psi(t)\leq{1}$ and vanishing at infinity. We study the resolvent
$G(z)=(H-z)^{-1},\ Imz\neq{0}$ in the limit $N,b\rightarrow\infty$,
$b=O(N^{\alpha}),\ 1/3<\alpha<1$ and obtain the explicit expression
$T(z_{1},z_{2})$ for the leading term of the correlation function of the
normalized trace of resolvent $g_{N,b}(z)=N^{-1}TrG(z)$. We show that in the
scaling limit of local correlations, this term leads to the expression
$(Nb)^{-1}T(\lambda+r_{1}/N+i0,\lambda+r_{2}/N-i0)=b^{-1}\sqrt{N}|r_{1}-r_{2}|^{-3/2}(1+o(1))$
found earlier by other authors for band random matrix ensembles. This shows
that the ratio $b^{2}/N$ is the correct scale for the eigenvalue density
correlation function and that the ensemble we study and that of band random
matrices belong to the same class of spectral universality.
AMS Subject Classifications: 15A52, 45B85, 60F99.
Key Words: random matrices, asymptotic properties, percolation model.
running title: Asymptotic properties for percolation model.
## 1 Introduction
Random matrices play an important role in various fields of mathemathics and
physics. The eigenvalue distribution of large matrices was initially
considered by E.Wigner to model the statistical properties of the energy
spectrum of heavy nuclei (see e.g. the collection of early papers [27]).
Further investigations have led to numerous applications of random matrices of
infinite dimensions in such branches of theoretical physics as statistical
mechanics of disordered spin systems, solid state physics, quantum chaos
theory, quantum field theory and others (see monographs and reviews [3, 9, 12,
14]). In mathematics, the spectral theory of random matrices has revealed deep
links with the orthogonal polynomials theory, integrable systems,
representation theory, combinatorics, free probability theory, and others [4,
11, 28, 30].
The first result of the spectral theory of large random matrices concerns the
eigenvalue distribution of the Wigner ensemble $A_{N}$ of $N\times N$ real
symmetric matrices of the form
$A_{N}(i,j)=\frac{1}{\sqrt{N}}a(i,j),\quad|i|,|j|\leq{n},$ (1.1)
where $N=2n+1$ and $\\{a(i,j);\ -n\leq{i}\leq{j}\leq{n}\\}$ are independent
and identically distributed random variables defined on the same probability
space $(\Omega,\mathfrak{F},{\bf P})$ such that
${\bf E}\\{a(i,j)\\}=0,\quad{\bf E}\\{a(i,j)^{2}\\}=v^{2}(1+\delta_{ij}),$
(1.2)
where
$\delta_{ij}=\left\\{\begin{array}[]{lll}0&\textrm{if}&i\neq{j},\\\
1&\textrm{if}&i=j\end{array}\right.$
is the Kronecker symbol and ${\bf E}\\{\cdot\\}$ is the mathematical
expectation with respect to ${\bf P}$.
Denoting by $\lambda^{(n)}_{-n}\leq{\ldots}\leq{\lambda^{(n)}_{n}}$ the
eigenvalues of $A_{N}$, the normalized eigenvalue counting function is defined
by
$\sigma_{n}(\lambda,A_{N})=N^{-1}\sharp\\{\lambda^{(n)}_{j}\leq{\lambda}\\}.$
(1.3)
E.Wigner [31] proved that if $a(i,j)$ has all order finite moments, the
eigenvalue counting measure $d\sigma_{n}(\lambda,A_{N})$ converges weakly in
average as $n\rightarrow\infty$ to a distribution $d\sigma_{sc}(\lambda)$,
where the nondecreasing function $\sigma_{sc}(\lambda)$ is differentiable and
its derivative $\rho_{sc}$ is given by
$\rho_{sc}{(\lambda)}=\sigma_{sc}^{{}^{\prime}}(\lambda)=\frac{1}{2\pi
v^{2}}\left\\{\begin{array}[]{lll}\sqrt{4v^{2}-\lambda^{2}}&\textrm{if}&|\lambda|\leq{2\sqrt{v^{2}}},\\\
0&\textrm{if}&|\lambda|>2v.\end{array}\right.$ (1.4)
This limiting distribution (1.4) is known as the Wigner distribution, or the
semicircle law. A proof of the Wigner’s result based on the resolvent
technique is given in [26, 22, 23].
Important generalizations of the Wigner’s ensemble are given by the band and
dilute random matrix ensembles [20]. In the band random matrices model, the
matrix elements take zero value outside the band of width $b_{n}$ along the
principal diagonal, for some positive sequence $(b_{n})_{n\geq{0}}$ of real
numbers. This ensemble can be obtained from $A_{N}$ (1.1) by multiplying each
$a(i,j)$ by $I_{(-1/2,1/2)}\left((i-j)/b_{n}\right)$, where
$I_{B}(t)=\left\\{\begin{array}[]{lll}1&\textrm{if}&t\in{B},\\\
0&\textrm{if}&t\in\mathbb{R}\setminus B\end{array}\right.$
is the indicator function of the interval $B$. The ensemble of dilute random
matrices can be obtained from $A_{N}$ (1.1) by multiplying $a(i,j)$ by
independent Bernoulli random variables of parameter $p_{n}/N$. Assuming that
$b(n)=o(n)$ for large $n$, the semicircle law is observed for both ensembles,
in the limit $b_{n}\rightarrow\infty$ (see [25]) and $p_{n}\rightarrow\infty$
as $n\rightarrow\infty$ (see [20]).
The crucial observation made numerically [7] and then supported in the
theoretical physics (see [13, 29]) is that the ratio $b^{2}/n$ is the critical
one for the corresponding transition in spectral properties of band random
matrices. In [16], it was proved that the ratio
$\tilde{\alpha}=\lim_{n\rightarrow\infty}b^{2}/n$ naturally arises when one
considers the leading term of this correlation function on the local scale.
This can be regarded as the support of the conjecture that the local
properties of spectra of band random matrices depend on $\tilde{\alpha}$.
Let us describe our results in more details. We are interested in a
generalization of the both ensembles mentioned above. Roughly speaking, we
consider the band random matrices with a random width. To proceed, we consider
the ensemble $\\{H_{n,b}\\}$ of random $N\times N$ matrices, $N=2n+1$ whose
entries $H_{n,b}$ is obtained as follows: we multiply each matrix element
$a(i,j)$ by some Bernoulli random variable $d_{b}(i,j)$ with parameter
$\psi\left((i-j)/b\right)$.
The family $\\{d_{b}(i,j);\ |i|,|j|\leq{n}\\}$ can be regarded as the
adjacency matrix of the family of random graphs $\\{\Gamma_{n}\\}$ with
$N=2n+1$ vertices $(i,j)$ such that the average number of edges attached to
one vertex is $b_{n}$. Hence, each edge $e(i,j)$ of the graph is present with
probability $\psi\left((i-j)/b\right)$ and not present with probability
$1-\psi\left((i-j)/b\right)$. Below are some well known examples:
1. $-$
In theoretical physics, the ensemble $\\{\Gamma_{n}\\}$ with
$\psi(t)=e^{-|t|^{s}}$ is referred to as the Long-Range Percolation Model (see
for example [8] and references therein). Our ensemble can be regarded as a
modification of the adjacency matrices of $\\{\Gamma_{n}\\}$. To our best
knowledge, the spectral properties of this model has not been studied yet.
2. $-$
It is easy to see that if one takes $b_{n}=N$ and $\psi\equiv 1$, then one
recovers the Wigner ensemble $(1.1)$.
3. $-$
If one considers $\psi(t)=I_{(-1/2,1/2)}(t)$, one gets the band random matrix
ensemble [25].
In present paper, we consider the resolvent
$G_{n,b}=\left(H_{n,b}-zI\right)^{-1}$ and study the asymptotic expansion of
the correlation function
$C_{n}(z_{1},z_{2})={\bf E}\\{g_{n,b}(z_{1})g_{n,b}(z_{2})\\}-{\bf
E}\\{g_{n,b}(z_{1})\\}{\bf E}\\{g_{n,b}(z_{2})\\},$
where we denoted $g_{n,b}=N^{-1}\mathrm{Tr}G_{n,b}(z)$. Keeping $z_{l}$ far
from the real axis, we consider the leading term $T(z_{1},z_{2})$ of this
expansion and find explicit expression for it. This term
$T(r_{1}+i0,r_{2}-i0)$ regarded on the local scale $r_{1}-r_{2}=r/N$ exhibits
different behavior depending on the rate of decay of the profile function
$\psi(t)$.
Our main conclusion is that if $\psi(t)\sim|t|^{-1-\nu}$ as
$t\rightarrow\infty$, then the value $\nu=2$ separates two major cases. If
$\nu\in(1,2)$, then the limit of $T(r)$ depend on $\nu$. If
$\nu\in(2,+\infty)$, then
$\frac{1}{Nb}T(r)=-const\cdot\frac{\sqrt{N}}{b}\cdot\frac{1}{|r|^{3/2}}(1+o(1)).$
This asymptotic expression coincides with the result obtained in [16] for band
random matrices. Then one can conclude that the ensemble under consideration
and the band random matrix ensemble belong to the same universality class.
The outline of this paper is as follows. In section 2, we define the random
matrix ensemble $H_{n,b}$ of long-range percolation model, we state our main
results and describe the scheme of their proofs. In section 3, we study the
correlation function $C_{n,b}(z_{1},z_{2})$ and obtain the main relation for
it. In section 4, we show that ${\bf Var}\\{g_{n,b}(z)\\}$ is bounded by
$(Nb)^{-1}$ and find the leading term $T(z_{1},z_{2})$ of the correlation
function under the moment condition that $\sup_{i,j}{\bf
E}|a(i,j)|^{14}<\infty$. In section 5, we prove the auxiliary facts used in
section 4. Expressions derived in section 4 are analyzed in section 6, where
the asymptotic behavior of $T(z_{1},z_{2})$ is studied and the issue of the
universal bihaviour is discussed.
## 2 The ensemble, main results and technical tools
### 2.1 The ensemble and main results
Let us consider a family of independent real random variables ${\cal
A}_{n}=\\{a(i,j);\ |i|,|j|\leq{n}\\}$ satisfying (1.1). Let $\psi(t)$,
$t\in\mathbf{R}$, be a real continuous even function such that:
$0\leq{\psi{(t)}}\leq{1},\quad\int_{\mathbb{R}}\psi{(t)}dt=1.$ (2.1)
Given real $b>0$, we introduce a family of independent Bernoulli random
variables ${\cal D}_{b}=\\{d_{b}(i,j);\ |i|,|j|\leq{n}\\}$ with the law
$d_{b}(i,j)=\left\\{\begin{array}[]{lll}1&\textrm{with
probability}&\psi\left((i-j)/b\right)\\\ 0&\textrm{with
probability}&1-\psi\left((i-j)/b\right).\end{array}\right.$ (2.2)
This family is independent of the family ${\cal A}_{n}$. We assume that ${\cal
A}_{n}$ and ${\cal D}_{n}$ are defined on the same probability space
$(\Omega,\mathfrak{F},{\bf P})$ and we denote by ${\bf E}\\{\cdot\\}$ the
mathematical expectation with respect to ${\bf P}$.
We define a real symmetric $N\times N$ random matrix $H_{n,b}$ by equality:
$H_{n,b}(i,j)=\frac{1}{\sqrt{b}}a(i,j)d_{b}(i,j),\quad
i\leq{j},\quad|i|,|j|\leq{n},$ (2.3)
where $b\leq{N}$, $N=2n+1$. Here and below the family $\\{H_{n,b}\\}$ is
referred to as the ensemble of random matrices of long-range percolation
model. In what follows, we will need the existence of several absolute moments
of $a(i,j)$ that we denote by
$\mu_{l}=\sup_{|i|,|j|\leq{n}}{\bf E}\\{|a(i,j)|^{l}\\},$ (2.4)
where the upper bound for $l$ will be specified later.
We consider the resolvent
$G_{n,b}(z)=(H_{n,b}-z)^{-1},\quad\mathrm{Im}z\neq{0}.$
Its normalized trace $g_{n,b}(z)$ coincides with the Stieltjes transform of
the normalized eigenvalue counting function $\sigma_{n,b}(\lambda;H_{n,b})$
(1.3):
$g_{n,b}(z)=\frac{1}{N}\mathrm{Tr}G_{n,b}(z)=\int(\lambda-z)^{-1}d\sigma_{n,b}(\lambda,H_{n,b}),\
\mathrm{Im}z\neq{0}.$ (2.5)
In [1], we have proved that if $\mu_{3}<\infty$ (2.4) and $1\ll b\ll N$, then
$\lim_{n,b\rightarrow\infty}{\bf E}\\{g_{n,b}(z)\\}=w(z)$
for $z\in\Lambda_{\eta}$, where
$\Lambda_{\eta}=\\{z\in\mathbf{C}:\
\eta\leq{|\mathrm{Im}z|}\\},\quad\eta=2v+1$ (2.6)
and the limiting function $w(z)$ verifies equation
$w(z)=\frac{1}{-z-v^{2}w(z)}$ (2.7)
with $v$ is determined by (1.2). Equation (2.7) has a unique solution in the
class of functions such that $\mathrm{Im}w(z)\mathrm{Im}z>0$,
$\mathrm{Im}z\neq{0}$. This solution $w(z)$ is the Stieltjes transform of the
semi-circle distribution (1.4). This result shows that the semi-circle law is
valid for random matrices of long-range percolation model. As a by-product of
proof, we have shown that
${\bf Var}\\{g_{n,b}(z)\\}=o(1),\ \quad z\in\Lambda_{\eta},\quad\hbox{ as
}\quad n,b\rightarrow\infty$ (2.8)
and that the convergence $g_{n,b}(z)\rightarrow{w(z)}$ holds in probability.
In this paper, we improve the result (2.8) in two stages. On the first one we
show that ${\bf Var}\\{g_{n,b}(z)\\}=O\left((Nb)^{-1}\right)$ in the limit
$n,b\rightarrow\infty$ such that
$b=O\left(n^{\alpha}\right),\quad 1/3<\alpha<1$ (2.9)
and this gives the convergence $g_{n,b}(z)\rightarrow{w(z)}$ with probability
1. Next, we find the explicit form of the leading term of the correlation
function
$C_{n,b}(z_{1},z_{2})={\bf E}\\{g_{n,b}(z_{1})g_{n,b}(z_{2})\\}-{\bf
E}\\{g_{n,b}(z_{1})\\}{\bf E}\\{g_{n,b}(z_{2})\\}.$
We now formulate the main result of the paper, where we denote
$w_{1}=w(z_{1})$ and $w_{2}=w(z_{2})$ are given by (2.7).
###### Theorem 2.1.
Let ${\cal A}_{n}$ be such that, in addition to (1.2), the following
properties are verified :
${\bf E}\\{a(i,j)^{3}\\}={\bf E}\\{a(i,j)^{5}\\}=0,\quad{\bf
E}\\{a(i,j)^{2m}\\}=V_{2m}(1+\delta_{ij})^{m},\quad m=2,3$ (2.10)
for all $i\leq{j}$, $\mu_{14}<\infty$ (2.4) and
$\int_{\mathbb{R}}\sqrt{\psi(t)}dt<\infty.$
Then in the limit $n,b\rightarrow\infty$ (2.9) and for
$z_{l}\in\Lambda_{\eta}$ (2.6), $l=1,2$, equality
$C_{n,b}(z_{1},z_{2})=\frac{1}{Nb}T(z_{1},z_{2})+o\left(\frac{1}{Nb}\right)$
(2.11)
holds with $T$ is given by the formula
$T(z_{1},z_{2})=Q(z_{1},z_{2})+\frac{2\Delta
w_{1}^{3}w_{2}^{3}}{(1-v^{2}w_{1}^{2})(1-v^{2}w_{2}^{2})}$ (2.12)
with
$Q(z_{1},z_{2})=\frac{v^{2}w_{2}^{2}w_{1}^{2}}{\pi(1-v^{2}w_{1}^{2})(1-v^{2}w_{2}^{2})}\int_{\mathbb{R}}\frac{\tilde{\psi}(p)}{[1-v^{2}w_{1}w_{2}\tilde{\psi}(p)]^{2}}dp,$
(2.13)
where $\tilde{\psi}(p)$ is the Fourier transform of $\psi$
$\tilde{\psi}(p)=\int_{\mathbb{R}}\psi(t)e^{ipt}dt$
and
$\Delta=V_{4}\int_{\mathbb{R}}\psi(t)dt-3v^{4}\int_{\mathbb{R}}\psi^{2}(t)dt.$
(2.14)
Under this conditions, Theorem 2.1 and relation (2.12) remain true with
$\Delta$ replaced by
$\lim_{n,b\rightarrow\infty}\sup_{|i|\leq{n}}\left(b\sum_{|j|\leq{n}}{\bf
E}\\{H(i,j)^{4}\\}-3{\bf E}\\{H(i,j)^{2}\\}^{2}\right)$
$=\lim_{n,b\rightarrow\infty}\sup_{|i|\leq{n}}\left(\frac{1}{b}\sum_{|j|\leq{n}}(1+\delta_{ij})^{2}\left[V_{4}\psi(\frac{i-j}{b})-3v^{4}\psi(\frac{i-j}{b})^{2}\right]\right).$
We would like to note that the form of (2.12) generalizes the expressions
obtained in [16] and [19]. Namely, the term $Q(z_{1},z_{2})$ is derived for
the case when the entries of random matrices $H$ are gaussian random
variables. The ensemble we consider is very similar to the band random
matrices, but it represents a different model. The form of the last term is
exactly the same as the one obtained in [19] for the Wigner random matrices.
This shows that this term " forgets " the band-like structure of our matrices.
All our computations and formulas are valid in the case of band random
matrices $H_{n,b}(i,j)=b^{-1/2}a(i,j)[\psi\left((i-j)/b\right)]^{1/2}$ with
not necessarily gaussian $a(i,j)$. Therefore Theorem 2.1 generalizes the
results of paper [16]. In the case of band random matrices, one obtains the
same expressions (2.11) and (2.12) with $\Delta$ (2.14) replaced by
$\Delta_{band}=(V_{4}-3v^{4})\int\psi^{2}(t)dt$, provided $a(i,j)$ are the
same as in Theorem 2.1.
The results of Theorem 2.1 are used to study the universality properties of
eigenvalue distribution. We do this in Section 6.
### 2.2 Cumulant expansions and resolvent identities
We prove Theorem 2.1 and Theorem 2.2 by using the method proposed in papers
[19, 20] and further developed in a series of works [1, 16]. The basic tools
of this method are given by the resolvent identities combined with the
cumulant expansions technique.
#### 2.2.1 The cumulant expansions formula
Let us consider a family $\\{X_{t}:\ t=1,\ldots,m\\}$ of independent real
random variables defined on the same probability space such that ${\bf
E}\\{|X_{t}|^{q+2}\\}<\infty$ for some $q\in\mathbb{N}$ and $t=1,\ldots,m$.
Then for any complex-valued function $F(u_{1},\ldots,u_{m})$ of the class
$\mathcal{C}_{\infty}(\mathbb{R}^{m})$ and for all $j$, one has
${\bf E}\\{X_{t}F(X_{1},\ldots,X_{m})\\}=\sum_{r=0}^{q}\frac{K_{r+1}}{r!}{\bf
E}\left\\{\frac{\partial^{r}F(X_{1},\ldots,X_{m})}{(\partial{X_{t}})^{r}}\right\\}+\epsilon_{q}(X_{t}),$
(2.15)
where $K_{r}=Cum_{r}(X_{t})$ is the r-th cumulant of $X_{t}$ and the remainder
$\epsilon_{q}(X_{t})$ can be estimated by inequality
$|\epsilon_{q}(X_{t})|\leq{C_{q}\sup_{U\in\mathbb{R}^{m}}\left|\frac{\partial^{q+1}F(U)}{\partial{u_{t}^{q+1}}}\right|{\bf
E}\\{|X_{t}|^{q+2}\\}},$ (2.16)
where $C_{q}$ is a constant. Relations (2.15) and (2.16) can be proved by
multiple using of the Taylor’s formula (see [1, 19] for the proofs).
###### Remark 2.1.
The cumulants $K_{r}$ can be expressed in terms of the moments
$\breve{\mu}_{r}={\bf E}(X_{t}^{r})$ of $X_{t}$.
Indeed, let $f_{t}$ be a complex-valued function of one real variable such
that
$f_{t}(x)=F(X_{1},\ldots,X_{t-1},x,X_{t+1},\ldots,X_{n})$
and $f_{t}^{(r)}$ is its r-th derivative.
* $\bullet$
If $q=1$ and ${\bf E}\\{X_{t}\\}=0$, then
$K_{1}=\breve{\mu}_{1}=0,\quad K_{2}=\breve{\mu}_{2}$ (2.17)
and the remainder $\epsilon_{1}(X_{t})$ is given by:
$\epsilon_{1}(X_{t})=\frac{1}{2}{\bf
E}\left\\{X_{t}^{3}f_{t}^{(2)}(x_{0})\right\\}-K_{2}{\bf
E}\left\\{X_{t}f_{t}^{(2)}(x_{1})\right\\}.$ (2.18)
* $\bullet$
If $q=3$ and ${\bf E}\\{X_{t}\\}={\bf E}\\{X_{t}^{3}\\}=0$, then
$K_{1}=K_{3}=0,\quad K_{2}=\breve{\mu}_{2},\quad
K_{4}=\breve{\mu}_{4}-3\breve{\mu}^{2}_{2}$ (2.19)
and the remainder $\epsilon_{3}(X_{t})$ is given by:
$\displaystyle\epsilon_{3}(X_{t})=$ $\displaystyle\frac{1}{4!}{\bf
E}\left\\{X_{t}^{5}f_{t}^{(4)}(x_{0})\right\\}-\frac{K_{2}}{3!}{\bf
E}\left\\{X_{t}^{3}f_{t}^{(4)}(x_{1})\right\\}$
$\displaystyle-\frac{K_{4}}{3!}{\bf
E}\left\\{X_{t}f_{t}^{(4)}(x_{2})\right\\}.$ (2.20)
* $\bullet$
If $q=5$ and ${\bf E}\\{X_{t}\\}={\bf E}\\{X_{t}^{3}\\}={\bf
E}\\{X_{t}^{5}\\}=0$, then the cumulants $K_{r}$, $r=1,\ldots,4$ are given by
(2.19),
$K_{5}=0,\quad
K_{6}=\breve{\mu}_{6}-15\breve{\mu}_{4}\breve{\mu}_{2}+30\breve{\mu}^{3}_{2}$
(2.21)
and the remainder $\epsilon_{5}(X_{t})$ is given by:
$\displaystyle\epsilon_{5}(X_{t})=$ $\displaystyle\frac{1}{6!}{\bf
E}\left\\{X_{t}^{7}f_{t}^{(6)}(x_{0})\right\\}-\frac{K_{2}}{5!}{\bf
E}\left\\{X_{t}^{5}f_{t}^{(6)}(x_{1})\right\\}$
$\displaystyle-\frac{K_{4}}{(3!)^{2}}{\bf
E}\left\\{X_{t}^{3}f_{t}^{(6)}(x_{2})\right\\}-\frac{K_{6}}{5!}{\bf
E}\left\\{X_{t}f_{t}^{(6)}(x_{3})\right\\},$ (2.22)
where for each $\nu=0,\ldots,3$, $x_{\nu}$ is a real random variable that
depends on $X_{t}$ and such that $|x_{\nu}|\leq{|X_{t}|}$. In what follows, we
denote $f_{t}^{(r)}(x_{\nu})=[\partial^{r}{F}/\partial{X_{t}^{r}}]^{(\nu)}$.
#### 2.2.2 Resolvent identities
For any two real symmetric $n\times n$ matrices $h$ and $\tilde{h}$ and any
non-real $z$ the resolvent identity
$(h-zI)^{-1}=(\tilde{h}-zI)^{-1}-(h-zI)^{-1}(h-\tilde{h})(\tilde{h}-zI)^{-1}$
(2.23)
is valid. Regarding (2.23) with, $\tilde{h}=0$ and denoting
$G=\left(h-zI\right)^{-1}$, we get equality
$G(i,j)=\zeta\delta_{ij}-\zeta\sum_{s=1}^{n}G(i,s)h(s,j),\quad\zeta={-z^{-1}},$
(2.24)
where $h(i,j),\ i,j=1,\ldots,n$ are the entries of the matrix $h$, $G(i,j)$
are the entries of the resolvent $G$ and $\delta$ denotes the Kronecker
symbol.
Using (2.23) we derive for $G=\left(h-zI\right)^{-1}$, $|\mathrm{Im}z|\neq{0}$
equality
$\frac{\partial{G(s,t)}}{\partial{h(j,k)}}=-\frac{1}{1+\delta_{jk}}\left[G(s,j)G(k,t)+G(s,k)G(j,t)\right].$
(2.25)
We will also need two more formulas based on (2.25); these are expressions for
$\partial^{2}{G(i,j)}/\partial{h(j,i)^{2}}$ and
$\partial^{3}{G(i,j)}/\partial^{3}{h(j,i)}$. We present them later.
#### 2.2.3 The scheme of the proof of Theorem 2.1
In this subsection we present a schema of computation of the leading terms of
$C_{n,b}(z_{1},z_{2})$ (cf. (2.11)).
Let us denote $g_{l}=g_{n,b}(z_{l})$, $l=1,2$ (everywhere below, we omit the
subscripts $n$,$b$ when no confusion can arise). For a given a random
variable, we denote $\xi^{0}=\xi-{\bf E}\xi$. Then using identity
${\bf E}\\{\xi^{0}g^{0}\\}={\bf E}\\{\xi^{0}g\\},$ (2.26)
we rewrite $C_{12}=C_{n,b}(z_{1},z_{2})$ as
$C_{12}={\bf E}\\{g^{0}_{1}g_{2}\\}=\frac{1}{N}\sum_{|i|\leq{n}}R_{12}(i)$
with $R_{12}(i)={\bf E}\\{g^{0}_{1}G_{2}(i,i)\\}$. Applying the resolvent
identity (2.23) to $G_{2}(i,i)$, we obtain equality
$R_{12}(i)=-\zeta_{2}\sum_{|p|\leq{n}}{\bf E}\\{g^{0}_{1}G_{2}(i,p)H(p,i)\\}.$
(2.27)
To compute ${\bf E}\\{g^{0}_{1}G_{2}(i,p)H(p,i)\\}$, we use the cumulants
expansion method (2.15), and get
$\displaystyle{\bf E}\\{g^{0}_{1}G_{2}(i,p)H(p,i)\\}=$ $\displaystyle
K_{2}{\bf
E}\left\\{\frac{\partial\left(g^{0}_{1}G_{2}(i,p)\right)}{\partial{H(p,i)}}\right\\}$
$\displaystyle+\frac{K_{4}}{6}{\bf
E}\left\\{\frac{\partial^{3}\left(g^{0}_{1}G_{2}(i,p)\right)}{\partial{H(p,i)^{3}}}\right\\}+\tau_{ip},$
(2.28)
where $K_{r}$ is the r-th cumulant of $H(p,i)$ and $\tau_{ip}$ vanishes.
Substituting this equality in (2.27) and using (2.25), we obtain that
$\displaystyle\frac{\partial\\{g^{0}_{1}G_{2}(i,p)\\}}{\partial{H(p,i)}}=$
$\displaystyle g^{0}_{1}\frac{\partial
G_{2}(i,p)}{\partial{H(p,i)}}+G_{2}(i,p)\frac{1}{N}\sum_{|s|\leq{n}}\frac{\partial
G_{1}(s,s)}{\partial{H(p,i)}}$ $\displaystyle=$
$\displaystyle-\frac{1}{1+\delta_{pi}}g^{0}_{1}[G_{2}(i,p)^{2}+G_{2}(i,i)G_{2}(p,p)]$
$\displaystyle-\frac{1}{1+\delta_{pi}}\left\\{\frac{2}{N}G^{2}_{1}(i,p)G_{2}(i,p)\right\\},$
(2.29)
where we used (2.25) in the form
$\frac{\partial\\{g^{0}_{1}G_{2}(i,p)\\}}{\partial{H(p,i)}}=\left\\{\frac{\partial\left(g^{0}_{1}G_{2}(i,p)\right)}{\partial{h(p,i)}}|_{h=H}\right\\}.$
We get relation
$\displaystyle R_{12}(i)=$ $\displaystyle\zeta_{2}v^{2}{\bf
E}\left\\{g^{0}_{1}G_{2}(i,i)\sum_{|p|\leq{n}}G_{2}(p,p)\frac{1}{b}\psi\left(\frac{p-i}{b}\right)\right\\}$
$\displaystyle+\frac{\zeta_{2}v^{2}}{b}\sum_{|p|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,p)^{2}\\}\psi\left(\frac{p-i}{b}\right)$
$\displaystyle+\frac{2\zeta_{2}v^{2}}{Nb}\sum_{|p|\leq{n}}{\bf
E}\\{G_{1}^{2}(i,p)G_{2}(i,p)\\}\psi\left(\frac{p-i}{b}\right)$
$\displaystyle-\frac{\zeta_{2}}{6}\sum_{|p|\leq{n}}K_{4}{\bf
E}\left\\{\frac{\partial^{3}(g^{0}_{1}G_{2}(i,p))}{\partial{H(p,i)^{3}}}\right\\}+\Phi_{n,b}(i),$
(2.30)
where $\sup_{|i|\leq{n}}|\Phi_{n,b}(i)|$ vanishes as $n,b\rightarrow\infty$
(2.9) (see subsection 3.2 for more details). Also we have taken into account
that (cf. (2.19))
$K_{2}(p,i)=K_{2}\left(H_{n,b}(p,i)\right)=\frac{1}{b}{\bf
E}\\{a(p,i)^{2}d_{n,b}(p,i)^{2}\\}=\frac{v^{2}}{b}\psi\left(\frac{p-i}{b}\right)(1+\delta_{pi}).$
Let us return to relation (2.30). We observe that the first term of the right-
hand side (RHS) can be expressed in terms of $R_{12}$. This gives the
possibility to obtain an equation of $R_{12}$. The second term vanishes in the
limit $n,b\rightarrow\infty$ (we give later the explicit formulation). The
third term represents the leading term of the correlations function (which
provides the first expression of (2.12)). The fourth term gives the
contribution of the order $O((Nb)^{-1})$ to (2.11) (which provides the second
expression of the leading term (2.12)). The last term $\Phi_{n,b}(i)$ gives
the contribution of the order $o((Nb)^{-1})$ to (2.11) (see Lemma 3.2).
## 3 Correlation function of the resolvent
In this section we give the main relation of the correlation function
$C_{n,b}(z_{1},z_{2})$. In what follows, we will need two elementary
inequalities
$|G(i,p)|\leq{||G||}\leq{\frac{1}{|\mathrm{Im}z|}},$ (3.1)
and
$\sum_{|p|\leq{n}}|G(i,p)|^{2}=||G\vec{e}_{i}||^{2}\leq{\frac{1}{|\mathrm{Im}z|^{2}}},\quad|i|\leq{n}$
(3.2)
that hold for the resolvent of any real symmetric matrix. Here and below we
consider $||e||_{2}^{2}=\sum_{i}|e(i)|^{2}$ and denote by
$||G||=\sup_{||e||_{2}=1}||Ge||_{2}$ the corresponding operator norm.
### 3.1 Derivation of relations for $R_{12}(i)$
Let us consider the average ${\bf E}\\{g^{0}_{1}G_{2}(i,p)H(p,i)\\}$. For each
pair $(i,p)$, $g^{0}_{1}G_{2}(i,p)$ is a smooth function of $H(p,i)$. Its
derivatives are bounded because of equation (2.25) and (3.1). In particular
$|D^{6}_{pi}\\{\hat{g}^{0}_{1}\hat{G}_{2}(i,p)\\}|\leq{C\left(|\mathrm{Im}z_{1}|^{-1}+|\mathrm{Im}z_{2}|^{-1}\right)^{8}},$
where $C$ is an absolute constant. Here and thereafter we use the notation
$D_{pi}$ for $\partial/\partial{H(p,i)}$.
According to the definition of $H$ and the condition $\mu_{7}<\infty$ (2.4),
the seven absolute moment of $H(p,i)$ is of order $1/(b^{7/2})$. Then we can
apply (2.15) with $q=5$ to ${\bf E}\\{g^{0}_{1}G_{2}(i,p)H(p,i)\\}$ and using
(2.23), we get relation
* $\bullet$
if $p<i$
$\displaystyle{\bf E}\\{g^{0}_{1}G_{2}(i,p)H(p,i)\\}=$ $\displaystyle
K_{2}\left(H(p,i)\right){\bf
E}\left\\{D^{1}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\\}$
$\displaystyle+\frac{K_{4}\left(H(p,i)\right)}{6}{\bf
E}\left\\{D^{3}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\\}$
$\displaystyle+\frac{K_{6}\left(H(p,i)\right)}{120}{\bf
E}\left\\{D^{5}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\\}+\tilde{\epsilon}_{pi}$
(3.3)
with
$\displaystyle\tilde{\epsilon}_{pi}=$ $\displaystyle\frac{1}{6!}{\bf
E}\left\\{H(p,i)^{7}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(0)}\right\\}$
$\displaystyle-\frac{K_{2}\left(H(p,i)\right)}{5!}{\bf
E}\left\\{H(p,i)^{5}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(1)}\right\\}$
$\displaystyle-\frac{K_{4}\left(H(p,i)\right)}{(3!)^{2}}{\bf
E}\left\\{H(p,i)^{3}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(2)}\right\\}$
$\displaystyle-\frac{K_{6}\left(H(p,i)\right)}{5!}{\bf
E}\left\\{H(p,i)[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(3)}\right\\},$ (3.4)
where the cumulants are given by (cf. (2.19)-(2.21))
$K_{2}\left(H(p,i)\right)=\frac{v^{2}}{b}\psi(\frac{p-i}{b})(1+\delta_{pi}),\quad
K_{4}\left(H(p,i)\right)=\frac{\Delta_{pi}}{b^{2}}(1+\delta_{pi})^{2}$ (3.5)
with
$\Delta_{pi}=V_{4}\psi\left((p-i)/b\right)-3v^{4}\psi\left((p-i)/b\right)^{2}$
and
$K_{6}\left(H(p,i)\right)=\frac{\theta_{pi}}{b^{3}}(1+\delta_{pi})^{3},$ (3.6)
with
$\theta_{pi}=V_{6}\psi\left((p-i)/b\right)-15V_{4}v^{2}\psi\left((p-i)/b\right)^{2}+30v^{6}\psi\left((p-i)/b\right)^{3}$.
In (3.4), we have denoted for each pair $(p,i)$
$[g^{0}_{1}G_{2}(i,p)]^{(\nu)}=\\{g^{(\nu)}\\}^{0}_{pi}(z_{1})G^{(\nu)}_{pi}(i,p;z_{2}),\quad\nu=0,\ldots,3$
and $G^{(\nu)}_{pi}(z_{l})=(H^{(\nu)}_{pi}-z_{l})^{-1}$, $l=1,2$ with real
symmetric
$H^{(\nu)}_{pi}(r,s)=\left\\{\begin{array}[]{lll}H(r,s)&\textrm{if}&(r,s)\neq(p,i);\\\
H^{(\nu)}(p,i)&\textrm{if}&(r,s)=(p,i),\end{array}\right.$
where $|H^{(\nu)}(p,i)|\leq{|H(p,i)|}$, $\nu=0,\ldots,3$ (see subsection 2.2.1
for more detail).
* $\bullet$
If $i<p$, then using equality $H(p,i)=H(i,p)$, we get
$\displaystyle{\bf E}\\{g^{0}_{1}G_{2}(i,p)H(i,p)\\}=$ $\displaystyle
K_{2}\left(H(i,p)\right){\bf
E}\left\\{D^{1}_{ip}\left(g^{0}_{1}G_{2}(i,p)\right)\right\\}$
$\displaystyle+\frac{K_{4}\left(H(i,p)\right)}{6}{\bf
E}\left\\{D^{3}_{ip}\left(g^{0}_{1}G_{2}(i,p)\right)\right\\}$
$\displaystyle+\frac{K_{6}\left(H(i,p)\right)}{120}{\bf
E}\left\\{D^{5}_{ip}\left(g^{0}_{1}G_{2}(i,p)\right)\right\\}+\tilde{\tilde{\epsilon}}_{ip},$
(3.7)
where $\tilde{\tilde{\epsilon}}_{ip}$ is given by (3.4) with replaced $D_{pi}$
by $D_{ip}$ and $K_{r}$ are the cumulants of $H(i,p)$ as in (3.5)-(3.6).
* $\bullet$
If $p=i$, then
$\displaystyle{\bf E}\\{g^{0}_{1}G_{2}(i,i)H(i,i)\\}=$ $\displaystyle
K_{2}\left(H(i,i)\right){\bf
E}\left\\{D^{1}_{ii}\left(g^{0}_{1}G_{2}(i,i)\right)\right\\}$
$\displaystyle+\frac{K_{4}\left(H(i,i)\right)}{6}{\bf
E}\left\\{D^{3}_{ii}\left(g^{0}_{1}G_{2}(i,i)\right)\right\\}$
$\displaystyle+\frac{K_{6}\left(H(i,i)\right)}{120}{\bf
E}\left\\{D^{5}_{ii}\left(g^{0}_{1}G_{2}(i,i)\right)\right\\}+\tilde{\tilde{\tilde{\epsilon}}}_{ii},$
(3.8)
where $\tilde{\tilde{\tilde{\epsilon}}}_{ii}$ is given by (3.4) with replaced
$D_{pi}$ by $D_{ii}$ and $K_{r}$ are the cumulants of $H(i,i)$ as in
(3.5)-(3.6).
Substituting (3.3), (3.7) and (3.8) into (2.27) and using (2.29), we obtain
equality
$\displaystyle R_{12}(i)=$ $\displaystyle\zeta_{2}v^{2}{\bf
E}\left\\{g^{0}_{1}G_{2}(i,i)\sum_{|p|\leq{n}}G_{2}(p,p)\frac{1}{b}\psi\left(\frac{p-i}{b}\right)\right\\}$
$\displaystyle+\frac{\zeta_{2}v^{2}}{b}\sum_{|p|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,p)^{2}\\}\psi\left(\frac{p-i}{b}\right)$
$\displaystyle+\frac{2\zeta_{2}v^{2}}{Nb}\sum_{|p|\leq{n}}{\bf
E}\\{G_{1}^{2}(i,p)G_{2}(i,p)\\}\psi\left(\frac{p-i}{b}\right)$
$\displaystyle-\frac{\zeta_{2}}{6}\sum_{|p|\leq{n}}\frac{\Delta_{pi}}{b^{2}}{\bf
E}\left\\{D^{3}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\\}-\frac{\zeta_{2}\Delta_{ii}}{2b^{2}}{\bf
E}\left\\{D^{3}_{ii}\left(g^{0}_{1}G_{2}(i,i)\right)\right\\}$
$\displaystyle-\frac{\zeta_{2}}{120}\sum_{|p|\leq{n}}\frac{\theta_{pi}}{b^{3}}(1+\delta_{pi})^{3}{\bf
E}\left\\{D^{5}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\\}+\epsilon_{i}$
(3.9)
with
$\displaystyle\epsilon_{i}=$
$\displaystyle-\zeta_{2}\sum_{|p|\leq{n}}\frac{1}{6!}{\bf
E}\left\\{H(p,i)^{7}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(0)}\right\\}$
$\displaystyle+\zeta_{2}\sum_{|p|\leq{n}}\frac{K_{2}}{5!}{\bf
E}\left\\{H(p,i)^{5}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(1)}\right\\}$
$\displaystyle+\zeta_{2}\sum_{|p|\leq{n}}\frac{K_{4}}{(3!)^{2}}{\bf
E}\left\\{H(p,i)^{3}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(2)}\right\\}$
$\displaystyle+\zeta_{2}\sum_{|p|\leq{n}}\frac{K_{6}}{5!}{\bf
E}\left\\{H(p,i)[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(3)}\right\\},$ (3.10)
where $K_{r}$ are the cumulants of $H(p,i)$ as in (3.5)-(3.6).
### 3.2 Main relation for $R_{12}(i)$
To give the complete description of $R_{12}$, we use the notation
$U(p,i)=\frac{1}{b}\psi\left(\frac{p-i}{b}\right),\quad
U_{G}(i)=\sum_{|p|\leq{n}}G(p,p)U(p,i)$
and introduce the identity
${\bf E}\\{\xi g\\}={\bf E}\\{\xi\\}{\bf E}\\{g\\}+{\bf E}\\{\xi g^{0}\\}.$
(3.11)
Then, we rewrite the first term of the RHS of (3.9) in the form
$\displaystyle\zeta_{2}v^{2}{\bf E}$
$\displaystyle\left\\{g^{0}_{1}G_{2}(i,i)\sum_{|p|\leq{n}}G_{2}(p,p)U(p,i)\right\\}$
$\displaystyle=\zeta_{2}v^{2}R_{12}(i){\bf
E}\\{U_{G_{2}}(i)\\}+\zeta_{2}v^{2}{\bf
E}\\{g^{0}_{1}G_{2}(i,i)U^{0}_{G_{2}}(i)\\}.$
Now computing the partial derivatives with the help of (2.25), we obtain the
following relation for $R_{12}$
$\displaystyle R_{12}(i)=$ $\displaystyle\zeta_{2}v^{2}R_{12}(i){\bf
E}\\{U_{G_{2}}(i)\\}+\zeta_{2}v^{2}{\bf
E}\\{g^{0}_{1}G_{2}(i,i)U^{0}_{G_{2}}(i)\\}$
$\displaystyle+\frac{2\zeta_{2}v^{2}}{N}\sum_{|p|\leq{n}}F_{12}(i,p)U(p,i)+\frac{1}{Nb}\Upsilon_{12}(i)+\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}$
(3.12)
with $F_{12}(i,p)={\bf E}\\{G_{1}^{2}(i,p)G_{2}(i,p)\\}$,
$\Upsilon_{12}(i)=\frac{\zeta_{2}}{b}\sum_{|p|\leq{n}}\Delta_{pi}{\bf
E}\left\\{[G_{1}^{2}(i,i)G_{1}(p,p)+G_{1}^{2}(p,p)G_{1}(i,i)]G_{2}(i,i)G_{2}(p,p)\right\\},$
(3.13)
the terms $Y_{r}(i)$, $r=1,\ldots,7$ are given by relations
$\displaystyle Y_{1}(i)=$
$\displaystyle\frac{\zeta_{2}}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,i)^{2}G_{2}(p,p)^{2}\\}\Delta_{pi},$ $\displaystyle
Y_{2}(i)=$ $\displaystyle\zeta_{2}v^{2}{\bf
E}\left\\{g^{0}_{1}\sum_{|p|\leq{n}}G_{2}(i,p)^{2}U(p,i)\right\\},$
$\displaystyle Y_{3}(i)=$
$\displaystyle\frac{\zeta_{2}}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}\left\\{g^{0}_{1}G_{2}(i,p)^{4}+6g^{0}_{1}G_{2}(i,p)^{2}G_{2}(i,i)G_{2}(p,p)\right\\}\Delta_{pi},$
$\displaystyle Y_{4}(i)=$
$\displaystyle\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\leq{n}}{\bf
E}\left\\{G_{1}^{2}(i,p)G_{2}(i,p)^{3}+3G_{1}^{2}(i,p)G_{2}(i,p)G_{2}(i,i)G_{2}(p,p)\right\\}\Delta_{pi},$
$\displaystyle Y_{5}(i)=$
$\displaystyle\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\leq{n}}{\bf
E}\left\\{G_{1}^{2}(i,p)G_{1}(i,p)G_{2}(i,p)^{2}+G_{1}^{2}(i,p)G_{1}(i,p)G_{2}(i,i)G_{2}(p,p)\right\\}\Delta_{pi}$
$\displaystyle+\frac{\zeta_{2}}{Nb^{2}}\sum_{|p|\leq{n}}{\bf
E}\left\\{G_{1}^{2}(i,i)G_{1}(p,p)G_{2}(i,p)^{2}+G_{1}^{2}(p,p)G_{1}(i,i)G_{2}(i,p)^{2}\right\\}\Delta_{pi}$
$\displaystyle+\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\leq{n}}{\bf
E}\left\\{G_{1}^{2}(i,p)G_{1}(i,p)^{2}G_{2}(i,p)+G_{1}^{2}(i,i)G_{1}(p,p)G_{1}(i,p)G_{2}(i,p)\right\\}\Delta_{pi}$
$\displaystyle+\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\leq{n}}{\bf
E}\left\\{G_{1}^{2}(i,p)G_{1}(p,p)G_{1}(i,i)G_{2}(i,p)\right\\}\Delta_{pi}$
$\displaystyle+\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\leq{n}}{\bf
E}\left\\{G_{1}^{2}(p,p)G_{1}(i,i)G_{1}(i,p)G_{2}(i,p)\right\\}\Delta_{pi},$
$\displaystyle Y_{6}(i)=$ $\displaystyle-\frac{3\zeta_{2}}{b^{2}}\left({\bf
E}\\{g_{1}^{0}G_{2}(i,i)^{4}\\}+\frac{1}{N}{\bf
E}\\{G^{2}_{1}(i,i)G_{2}(i,i)^{3}\\}\right)\Delta_{ii}$
$\displaystyle-\frac{3\zeta_{2}}{Nb^{2}}{\bf
E}\left\\{G^{2}_{1}(i,i)G_{1}(i,i)G_{2}(i,i)[G_{1}(i,i)+G_{2}(i,i)]\right\\}\Delta_{ii},$
$\displaystyle Y_{7}(i)=$
$\displaystyle-\frac{\zeta_{2}}{120}\sum_{|p|\leq{n}}\frac{\theta_{pi}}{b^{3}}(1+\delta_{pi})^{3}{\bf
E}\left\\{D^{5}_{pi}(g^{0}_{1}G_{2}(i,p))\right\\}$
and $\epsilon_{i}$ given by (3.10). The first and the second terms of the RHS
of (3.12) is expressed in terms of $R_{12}$ and this finally gives a closed
relation for $R_{12}$. The third and forth terms of the RHS of (3.12) give a
non-zero contribution to $R_{12}$ that provide the expression of the leading
term $T(z_{1},z_{2})$ (2.12). We will compute this contribution later (see
subsection 4.3). The two last terms of (3.12) contributes with $o((Nb)^{-1})$
to (2.11). We formalize this proposition in the following two statements.
###### Lemma 3.1.
Under conditions of Theorem 2.1, the estimate
$\max_{r=1,2}\left\\{\sup_{|i|\leq{n}}|Y_{r}(i)|\right\\}=O\left(b^{-2}n^{-1}+b^{-2}[{\bf
Var}\\{g_{1}\\}]^{1/2}\right).$ (3.14)
is true in the limit $n,b\rightarrow\infty$ (2.9).
We postpone the proof of Lemma 3.1 to the next section.
###### Lemma 3.2.
Under conditions of Theorem 2.1, the estimate
$\max_{r=3,4,5,6,7}\left\\{\sup_{|i|\leq{n}}|Y_{r}(i)|\right\\}=O\left(b^{-2}n^{-1}+b^{-2}[{\bf
Var}\\{g_{1}\\}]^{1/2}\right)$ (3.15)
and
$\sup_{|i|\leq{n}}|\epsilon_{i}|=O\left(b^{-2}n^{-1}+b^{-2}[{\bf
Var}\\{g_{1}\\}]^{1/2}\right)$ (3.16)
are true in the limit $n,b\rightarrow\infty$ (2.9).
Proof of Lemma 3.2. We start with (3.15). Inequality (3.1) and (3.2) implies
that if $z_{l}\in\Lambda_{\eta}$, then
$|Y_{3}(i)|\leq{\frac{7[V_{4}+3v^{4}]}{\eta^{3}b^{2}}\sum_{|p|\leq{n}}{\bf
E}|g_{1}^{0}G_{2}(i,p)^{2}|}=O\left(\frac{1}{b^{2}}\\{{\bf
Var}\\{g_{1}\\}\\}^{1/2}\right).$
To estimate $Y_{r}$, $r=4,5$, we use (3.1), (3.2) and inequality
$\displaystyle\sum_{|i|\leq{n}}{\bf E}|G^{m}_{1}(i,p)G_{2}(i,p)|$
$\displaystyle\leq{{\bf
E}\left(\sum_{|i|\leq{n}}|G^{m}_{1}(i,p)|^{2}\right)^{1/2}\left(\sum_{|i|\leq{n}}|G_{2}(i,p)|^{2}\right)^{1/2}}$
$\displaystyle\leq{\frac{1}{\eta^{m+1}}}$ (3.17)
with $m=1,2$. Then we get that
$|Y_{4}(i)|\leq{8[V_{4}+v^{4}]/(\eta^{6}Nb^{2})}$. Using (3.1), (3.2) and
(3.17) with $m=1,2$, we obtain that the terms $\sup_{i}|Y_{r}(i)|$, $r=5,6$
are all of the order indicated in (3.15).
Let us estimate $Y_{7}$. Let us accept for the moment that
${\bf E}|D^{5}_{pi}\\{g^{0}_{1}G_{2}(i,p)\\}|=O\left(N^{-1}+[{\bf
Var}\\{g_{1}\\}]^{1/2}\right),\ \hbox{ as }\quad n,p\rightarrow\infty$ (3.18)
holds. Using this estimate and relation (2.1), we obtain that
$\sum_{|p|\leq{n}}\left|\frac{\theta_{pi}}{b}\right|\leq{c\sum_{|p|\leq{n}}\frac{1}{b}\psi\left(\frac{p-i}{b}\right)}=O(1)$
and that
$\sup_{|i|\leq{n}}|Y_{7}(i)|=O\left(b^{-2}n^{-1}+b^{-2}\\{{\bf
Var}\\{g_{1}\\}\\}^{1/2}\right)$
where $c$ is a constant.
Now let use prove (3.18). Using (2.25) and (3.1), we get for
$z_{1}\in\Lambda_{\eta}$
$D_{pi}\\{g^{0}_{1}\\}=\frac{1}{N}\sum_{|t|\leq{n}}D_{pi}\\{G_{1}(t,t)\\}=-\frac{2}{N}G^{2}_{1}(i,p)=O\left(\frac{1}{N}\right).$
It is easy to show that
$D^{r}_{pi}\\{g^{0}_{1}\\}=O\left(\frac{1}{N}\right),\quad r=1,2,\ldots,\
z\in\Lambda_{\eta}.$ (3.19)
Then (3.18) follows from (3.19) and (3.1). Estimate (3.15) is proved.
To proceed with estimates of $\epsilon_{i}$ (3.16), we use the following
simple statement, proved in the previous work [1].
###### Lemma 3.3.
(see [1]) If $z_{l}\in\Lambda_{\eta},\ l=1,2$, under conditions of Theorem
2.1, the estimates
${\bf Var}([g_{n,b}(z_{l})]^{(\nu)})=O\left({\bf
Var}\\{g_{n,b}(z_{l})\\}+b^{-1}N^{-2}\right),\quad\nu=0,\ldots,3$ (3.20)
and
$D^{6}_{pi}\left\\{g^{0}_{1}G_{2}(i,p)\right\\}=O\left(N^{-1}+|g_{1}^{0}|\right)$
(3.21)
are true in the limit $n,b\longrightarrow\infty$ (2.9).
Now regarding the first term of (3.10) and using (3.20) and (3.21), we obtain
inequality
$\displaystyle\sum_{|p|\leq{n}}{\bf
E}|H(p,i)^{7}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(0)}|\leq{c_{1}\sum_{|p|\leq{n}}{\bf
E}\left\\{\frac{|H(p,i)|^{7}}{N}+|H(p,i)|^{7}|[g^{0}_{1}]^{(0)}|\right\\}}$
$\displaystyle\leq{c_{1}\sum_{|p|\leq{n}}\frac{\hat{\mu}_{7}}{Nb^{7/2}}\psi\left(\frac{p-i}{b}\right)+c_{1}\sum_{|p|\leq{n}}\frac{(\hat{\mu}_{14})^{1/2}}{b^{7/2}}\left(\psi\left(\frac{p-i}{b}\right)\right)^{1/2}\left({\bf
Var}\\{[g_{1}]^{(0)}\\}\right)^{1/2}}$
$\displaystyle=O\left(N^{-1}b^{-2}+b^{-2}[{\bf Var}\\{g_{1}\\}]^{1/2}\right),$
(3.22)
where $c$ is a constant.
Regarding the last term of the right-hand side of (3.10) and using (3.20) and
(3.21), we obtain inequality
$\displaystyle\sum_{|p|\leq{n}}K_{6}{\bf
E}|H(p,i)[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(3)}|$
$\displaystyle\leq{\frac{c_{2}}{b^{2}}\left(\frac{1}{b}\sum_{|p|\leq{n}}\psi\left(\frac{p-i}{b}\right)\right)\left(\frac{\hat{\mu}_{1}\psi(\frac{p-i}{b})}{Nb^{1/2}}+\frac{\hat{\mu}^{1/2}_{2}\left(\psi(\frac{p-i}{b})\right)^{1/2}}{b^{1/2}}\left({\bf
Var}\\{[g_{1}]^{(3)}\\}\right)^{1/2}\right)}$
$\displaystyle=O\left(N^{-1}b^{-2}+b^{-2}[{\bf Var}\\{g_{1}\\}]^{1/2}\right),$
(3.23)
where $c_{2}$ is a constant.
Repeating previous computations of (3.23), we obtain that
$\displaystyle\sum_{|p|\leq{n}}K_{4}{\bf
E}|H(p,i)^{3}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(2)}|+\sum_{|p|\leq{n}}K_{2}{\bf
E}|H(p,i)^{5}D^{6}_{pi}[g^{0}_{1}G_{2}(i,p)]^{(1)}|$
$\displaystyle=O\left(N^{-1}b^{-2}+b^{-2}[{\bf Var}\\{g_{1}\\}]^{1/2}\right).$
(3.24)
Then (3.16) follows from the estimates given by relations (3.22), (3.23) and
(3.24). Lemma 3.2 is proved. $\hfill\blacksquare$
Let us come back to relation (3.12). Using equality
$\sum_{|p|\leq{n}}{\bf E}\\{g^{0}_{1}G_{2}(i,i)G^{0}_{2}(p,p)\\}U(p,i)={\bf
E}\\{g^{0}_{1}U_{G_{2}}^{0}(i)G_{2}^{0}(i,i)\\}+U_{R_{12}}(i){\bf
E}\\{G_{2}(i,i)\\},$
we obtain the following relation
$\displaystyle R_{12}(i)=$ $\displaystyle\zeta_{2}v^{2}R_{12}(i)U_{{\bf
E}(G_{2})}(i)+\zeta_{2}v^{2}U_{R_{12}}(i){\bf E}\\{G_{2}(i,i)\\}$
$\displaystyle+\frac{2\zeta_{2}v^{2}}{N}\sum_{|p|\leq{n}}F_{12}(i,p)U(p,i)+\frac{1}{Nb}\Upsilon_{12}(i)$
$\displaystyle+\tau(i)+\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i},$ (3.25)
where $F_{12}(i,p)$ is the same as in (3.12), $\Upsilon_{12}$ is given by
(3.13) and
$\tau(i)=\zeta_{2}v^{2}{\bf E}\\{g^{0}_{1}U_{G_{2}}^{0}(i)G_{2}^{0}(i,i)\\}.$
(3.26)
Relation (3.25) is the main equality used for the proof of Theorem 2.1. We use
(3.25) twice : at the first stage we estimate the variance ${\bf
Var}\\{g_{n,b}(z)\\}$ and at the second one we obtain explicit expressions for
the leading term of $C_{n,b}(z_{1},z_{2})$. This will be done this in the next
section.
## 4 Variance and leading term of $C_{n,b}(z_{1},z_{2})$
In this section we give the estimate of the variance and the proof of Theorem
2.1, postponing some technical results to the next section.
### 4.1 Estimate of the variance
Let us define an auxiliary variable
$q_{2}(i)=\frac{\zeta_{2}}{1-\zeta_{2}v^{2}U_{g_{2}}(i)},$ (4.1)
where $g_{2}(i)={\bf E}G_{2}(i,i)$. Then we can rewrite (3.25) in the form
$\displaystyle R_{12}(i)=$ $\displaystyle
v^{2}q_{2}(i)U_{R_{12}}(i)g_{2}(i)+\frac{1}{Nb}\left(2v^{2}q_{2}(i)b[F_{12}U](i,i)+q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i)\right)$
$\displaystyle+q_{2}(i)\zeta^{-1}_{2}\left(\tau(i)+\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}\right)$
(4.2)
with
$[F_{12}U](i,i)=\sum_{|p|\leq{n}}F_{12}(i,p)U(p,i),$
where $F_{12}(i,p)$ is the same as in (3.12) and $\Upsilon_{12}$ is given by
(3.13).
Now let us estimate the terms of the RHS of (4.2).Taking into account that
$U(p,i)\leq{b^{-1}}$ and using inequalities (3.17) with $m=2$, it is easy to
see that if $z_{l}\in\Lambda_{\eta}$, then
$\frac{1}{N}|[F_{12}U](i,i)|\leq{\frac{1}{\eta^{3}Nb}}=O(\frac{1}{Nb}).$ (4.3)
Let us estimate $\Upsilon_{12}$ (3.13). Using (3.1) and inequality
$|\Delta_{pi}|\leq{[V_{4}+3v^{4}]\psi((p-i)/b)}$, we obtain that
$|q_{2}(i)|\leq{\frac{1}{|\mathrm{Im}z_{2}|}},\quad z_{2}\in\Lambda_{\eta}$
(4.4)
and that
$|q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i)|\leq{\frac{2[V_{4}+3v^{4}]}{\eta^{6}}\sum_{|p|\leq{n}}\frac{1}{b}\psi\left(\frac{p-i}{b}\right)}=O(1).$
(4.5)
To estimate the term $\tau$ (3.26), we use the following statement.
###### Lemma 4.1.
Under the conditions of Theorem 2.1, the estimate
$\sup_{|i|,|s|\leq{n}}|{\bf
E}g^{0}(z)G^{0}(i,i)U^{0}_{G}(s)|=O\left(n^{-1}b^{-2}+b^{-2}[{\bf
Var}\\{g(z)\\}]^{1/2}\right)$ (4.6)
is true in the limit $n,b\rightarrow\infty$ (2.9).
We prove Lemma 4.1 in section 5.
It follows from results of Lemmas 3.1, 3.2 and relation (4.6), that if
$z_{j}\in\Lambda_{\eta}$, then
$\sup_{|i|\leq{n}}\left|q_{2}(i)\zeta^{-1}_{2}\left(\tau(i)+\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}\right)\right|=O\left(N^{-1}b^{-2}+b^{-2}\\{{\bf
Var}\\{g_{1}\\}\\}^{1/2}\right).$ (4.7)
Let us denote $r_{12}=\sup_{i}|R_{12}(i)|$. Regarding estimates (4.3), (4.5)
and (4.7), we derive from (4.2) inequality
$r_{12}\leq{\frac{v^{2}}{\eta^{2}}r_{12}+\frac{A}{bN}+\frac{1}{b^{2}}\sqrt{r_{12}}}$
for some constant A. Since $r_{12}$ is bounded for all
$z_{l}\in\Lambda_{\eta}$, then $r_{12}=O((Nb)^{-1}+b^{-4})$. Using condition
(2.9) and taking $z=z_{1}=\overline{z_{2}}$, one obtains that
${\bf Var}\\{g_{n,b}(z)\\}=O\left(\frac{1}{Nb}\right).$ (4.8)
Substituting (4.8) into (4.7), we obtain that
$\sup_{|i|\leq{n}}\left|q_{2}(i)\zeta^{-1}_{2}\left(\tau(i)+\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}\right)\right|=o\left(\frac{1}{Nb}\right)$
(4.9)
in the limit $n,b\rightarrow\infty$ (2.9) and for all
$z_{l}\in\Lambda_{\eta}$, $l=1,2$. This proves (2.11).
### 4.2 Leading term of the correlation function
Assuming that (4.9) is true, we rewrite (4.2) in the form
$R_{12}(i)=v^{2}q_{2}(i)g_{2}(i)U_{R_{12}}(i)+\frac{1}{Nb}f_{12}(i)+\Gamma(i)$
(4.10)
with
$f_{12}(i)=2v^{2}q_{2}(i)b[F_{12}U](i,i)+q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i),$
(4.11)
where $F_{12}(i,p)$ is the same as in (3.25) and $\Upsilon_{12}$ is given by
(3.13). We have denoted the vanishing terms by
$\Gamma(i)=q_{2}(i)\zeta^{-1}_{2}\left(\tau(i)+\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}\right).$
(4.12)
To obtain an explicit expression for the leading term of
$C_{n,b}(z_{1},z_{2})$, it is necessary to study in detail the variables
$F_{12}$ and $\Upsilon_{12}$. Let us formulate the corresponding statements
and the auxiliary relations needed. Given a positive integer $L$, set
$B_{L}\equiv B_{L}(n,b)=\left\\{i\in\mathbb{Z};\ |i|\leq{n-bL}\right\\}.$
(4.13)
###### Lemma 4.2.
If $z\in\Lambda_{\eta}$, then for arbitrary positive $\epsilon$ and large
enough values of $n$ and $b$ (2.9) there exists a positive integer
$L=L(\epsilon)$ such that relations
$\sup_{i\in{B_{L}}}\left|b[F_{12}U](i,i)-\frac{w_{2}w_{1}^{2}}{2\pi(1-v^{2}w_{1}^{2})}\int_{\mathbb{R}}\frac{\tilde{\psi}(p)}{[1-v^{2}w_{1}w_{2}\tilde{\psi}(p)]^{2}}dp\right|\leq{\epsilon}$
(4.14)
and
$\sup_{i\in{B_{L}}}\left|q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i)-\frac{2\Delta
w^{3}_{1}w^{3}_{2}}{1-v^{2}w^{2}_{1}}\right|\leq{\epsilon}$ (4.15)
hold for enough $n$ and $b$ satisfying (2.9) with $\Delta$ is given by (2.14).
The proof of Lemma 4.2 is based on the following statement formulated for the
product $G_{1}G_{2}$.
###### Lemma 4.3.
Given positive $\epsilon$, there exists a positive integer $L=L(\epsilon)$
such that relations
$\sup_{i\in{B_{L}}}\left|{\bf
E}\\{G^{2}_{1}(i,i)\\}-\frac{w^{2}_{1}}{1-v^{2}w^{2}_{1}}\right|\leq{\epsilon},$
(4.16) $\sup_{i\in{B_{L}}}\left|b\sum_{|s|\leq{n}}{\bf
E}\\{G_{1}(i,s)G_{2}(i,s)\\}U^{k}(s,i)-\frac{1}{2\pi}\int_{\mathbb{R}}\frac{w_{1}w_{2}\tilde{\psi}^{k}(p)}{1-v^{2}w_{1}w_{2}\tilde{\psi}(p)}dp\right|\leq{\epsilon}$
(4.17)
and
$\sup_{i\in{B_{L}}}\left|\sum_{|s|\leq{n}}{\bf
E}\\{G_{1}(i,s)G_{2}(i,s)\\}-\frac{w_{1}w_{2}}{1-v^{2}w_{1}w_{2}}\right|\leq{\epsilon}$
(4.18)
hold for enough $n$ and $b$ satisfying (2.9) for all $k\in\mathbb{N}$, all
$z_{j}\in\Lambda_{\eta}$, $j=1,2$.
We postpone the proof of Lemma 4.3 to the next section.
### 4.3 Proof of Lemma 4.2 and Theorem 2.1
#### 4.3.1 Proof of Lemma 4.2
We start with (4.14). Let us consider the average $F_{12}(i,s)={\bf
E}\\{G^{2}_{1}(i,s)G_{2}(i,s)\\}$. Applying to $G_{2}(i,s)$ the resolvent
identity (2.23), we obtain equality
$F_{12}(i,s)=\zeta_{2}\delta_{is}{\bf
E}\\{G^{2}_{1}(i,i)\\}-\zeta_{2}\sum_{|p|\leq{n}}{\bf
E}\\{G^{2}_{1}(i,s)G_{2}(i,p)H(p,s)\\}.$
Applying formula (2.15) to ${\bf E}\\{G^{2}_{1}(i,s)G_{2}(i,p)H(p,s)\\}$ with
$q=3$ and taking into account (2.25), we get relation
$\displaystyle F_{12}(i,s)=$ $\displaystyle\zeta_{2}\delta_{is}{\bf
E}\\{G^{2}_{1}(i,i)\\}+\zeta_{2}v^{2}[t_{12}U](i,s){\bf
E}\\{G^{2}_{1}(s,s)\\}$
$\displaystyle+\zeta_{2}v^{2}[F_{12}U](i,s)g_{1}(s)+\zeta_{2}v^{2}F_{12}(i,s)U_{g_{2}}(s)$
$\displaystyle+\sum_{r=1}^{5}\beta_{r}(i,s),$ (4.19)
where we denoted $g_{l}(s)={\bf E}\\{G_{l}(s,s)\\}$, $l=1,2$,
$t_{12}(i,s)={\bf E}\\{G_{1}(i,s)G_{2}(i,s)\\}$ and the terms $\beta_{l}$,
$l=1,\ldots,5$ are given by:
$\displaystyle\beta_{1}(i,s)=$
$\displaystyle\zeta_{2}v^{2}\sum_{|p|\leq{n}}{\bf
E}\\{G^{2}_{1}(p,s)G_{1}(i,s)G_{2}(i,p)\\}U(p,s)$
$\displaystyle+\zeta_{2}v^{2}\sum_{|p|\leq{n}}{\bf
E}\\{G^{2}_{1}(i,s)G_{1}(p,s)G_{2}(i,p)\\}U(p,s)$
$\displaystyle+\zeta_{2}v^{2}\sum_{|p|\leq{n}}{\bf
E}\\{G^{2}_{1}(i,s)G_{2}(p,s)G_{2}(i,p)\\}U(p,s),$
$\displaystyle\beta_{2}(i,s)=$
$\displaystyle\zeta_{2}v^{2}\sum_{|p|\leq{n}}{\bf
E}\\{G_{1}(i,p)G_{2}(i,p)(G^{2}_{1}(s,s))^{0}\\}U(p,s)$
$\displaystyle+\zeta_{2}v^{2}\sum_{|p|\leq{n}}{\bf
E}\\{G^{2}_{1}(i,p)G_{2}(i,p)G^{0}_{1}(s,s)\\}U(p,s),$
$\displaystyle\beta_{3}(i,s)=$ $\displaystyle\zeta_{2}v^{2}{\bf
E}\left\\{G^{2}_{1}(i,s)G_{2}(i,s)U^{0}_{G_{2}}(s)\right\\},$
$\displaystyle\beta_{4}(i,s)=$
$\displaystyle-\frac{\zeta_{2}}{6}\sum_{|p|\leq{n}}K_{4}{\bf
E}\left\\{D^{3}_{ps}\left(G^{2}(i,s)G_{2}(i,p)\right)\right\\},$
and
$\displaystyle\beta_{5}(i,s)=$
$\displaystyle-\frac{\zeta_{2}}{4!}\sum_{|p|\leq{n}}{\bf
E}\left\\{H(p,s)^{5}[D^{4}_{ps}(G^{2}(i,s)G_{2}(i,p))]^{(0)}\right\\}$
$\displaystyle+\frac{\zeta_{2}}{3!}\sum_{|p|\leq{n}}K_{2}{\bf
E}\left\\{H(p,s)^{3}[D^{4}_{ps}(G^{2}(i,s)G_{2}(i,p))]^{(1)}\right\\}$
$\displaystyle+\frac{\zeta_{2}}{3!}\sum_{|p|\leq{n}}K_{4}{\bf
E}\left\\{H(p,s)[D^{4}_{ps}(G^{2}(i,s)G_{2}(i,p))]^{(2)}\right\\}$
with $K_{r}$ are the cumulants of $H(p,s)$ as in (3.5)-(3.6).
Let us accept for the moment that
$\max_{j=1,\ldots,5}\left\\{\sup_{|i|,|s|\leq{n}}|\beta_{r}(i,s)|\right\\}=O\left(b^{-1}\right),\quad\hbox{
as }\quad n,b\rightarrow\infty$ (4.20)
holds for enough $n$ and $b$ satisfying (2.9). Using them and the definition
of $q_{2}(s)$ (4.1), we rewrite (4.19) in the form
$F_{12}(i,s)=v^{2}g_{1}(s)q_{2}(s)[F_{12}U](i,s)+R_{1}(i,s)+R_{2}(i,s)+\beta(i,s),$
(4.21)
where we denoted
$R_{1}(i,s)=q_{2}(i){\bf E}\\{G^{2}_{1}(i,i)\\}\delta_{is},$ (4.22)
$R_{2}(i,s)=v^{2}q_{2}(s)[t_{12}U](i,s){\bf E}\\{G^{2}_{1}(s,s)\\}$ (4.23)
and the vanishing term
$\beta(i,s)=\frac{q_{2}(s)}{\zeta_{2}}\sum_{r=1}^{5}\beta_{r}(i,s).$
We define the linear operator $W$ that acts on the space of $N\times{N}$
matrices $F$ according to the formula
$[WF](i,s)=v^{2}g_{1}(s)q_{2}(s)\sum_{|p|\leq{n}}F(i,p)U(p,s).$
It is easy to see that if $z_{l}\in\Lambda_{\eta}$, then the estimates (3.1)
and (4.4) imply that $|R_{1}|\leq{\eta^{-3}}$ and
$|R_{2}|\leq{v^{2}\eta^{-5}}$ and that
$||W||_{(1,1)}\leq{\frac{v^{2}}{\eta^{2}}}<\frac{1}{2},$ (4.24)
where the norm of $N\times{N}$ matrix $A$ is determined as
$||A||_{(1,1)}=\sup_{i,s}|A(i,s)|$. This estimate verified by the direct
computation of the norm $||WA||_{(1,1)}$ with $||A||_{(1,1)}=1$. Then (4.21)
can be rewritten as
$F_{12}(i,s)=\sum_{m=0}^{\infty}\left[W^{m}\left(R_{1}+R_{2}+\beta\right)\right](i,s).$
(4.25)
The next steps of the proof of (4.14) are very elementary. To do this, we
start with the following statements, proved in the previous work [1].
###### Lemma 4.4.
(see [1]) Given positive $\epsilon$, there exists a positive integer
$L=L(\epsilon)$ such that relations
$\sup_{i\in{B_{L}}}|{\bf E}\\{G(i,i;z)\\}-w(z)|\leq{\epsilon},\quad
z\in\Lambda_{\eta}$ (4.26)
and
$\sup_{i\in{B_{L}}}|q(i;z)-w(z)|\leq{2\epsilon}\quad z\in\Lambda_{\eta}$
(4.27)
hold for enough $n$ and $b$ satisfying (2.9), where $w$ and $q$ are given by
(2.7) and (4.1).
Now let us return to relation (4.25). We consider the first $M$ terms of the
infinite series and use the decay of the matrix elements
$U(i,s)=U^{(b)}(i,s)$. If one considers (4.22) and (4.23) with $i$ and $s$
taken far enough from the endpoints -$n$, $n$, then the variables $g_{1}(j)$,
$q_{2}(k)$ enter into the finite series with $j$ and $k$ also far from the
endpoints. Then one can use relations (4.26) and (4.27) and replace $g_{1}$
and $q_{2}$ by the constant values $w_{1}$ and $w_{2}$, respectively. This
substitution leads to simplified expressions with error terms that vanish as
$n,b\rightarrow\infty$. The second step is similar. It is to show that we can
use Lemma 4.3 and replace the terms $R_{1}$ and $R_{2}$ of the finite series
of (4.22)and (4.23) by corresponding expressions given by formulas (4.16) and
(4.17).
Let us start to perform this program. Taking into account the estimate of
$\beta$ (4.20) and using bounded-ness of the terms $R_{1}$ and $R_{2}$, we can
deduce from (4.25) equality
$b\sum_{|s|\leq{n}}F_{12}(i,s)U(s,i)=b\sum_{m=0}^{M}\left[W^{m}(R_{1}+R_{2}).U\right](i,i)+\kappa_{1}(i,i),$
(4.28)
where $M>0$ is such that given $\epsilon>0$ and $|\kappa_{1}(i,i)|<\epsilon$
for large enough $b$ and $N$. Now let us find such $h>0$ that the following
holds
$\sup_{h\leq{|t|}}\psi(t)<\epsilon\ \hbox{ and }\
\int_{h\leq{|t|}}\psi(t)dt\leq{\epsilon}.$
We determine the matrix
$\hat{U}(i,p)=\left\\{\begin{array}[]{lll}U(i,p)&\textrm{if}&|i-p|\leq{bh};\\\
0&\textrm{if}&|i-p|>bh\end{array}\right.$
and denote by $\hat{W}$ the corresponding linear operator
$[\hat{W}F](i,s)=v^{2}g_{1}(s)q_{2}(s)\sum_{|p|\leq{n}}F_{12}(i,p)\hat{U}(p,s).$
Certainly , $\hat{W}$ admits the same estimate as $W$ (4.24). Given
$\epsilon>0$ and $L>0$ the large number. Let us denote by $Q$ the first
natural greater than $(M+k)h$. Then one can write that
$b\sum_{m=0}^{M}\left[W^{m}(R_{1}+R_{2}).U\right](i,i)=b\sum_{m=0}^{M}\left[\hat{W}^{m}(R_{1}+R_{2})\hat{U}\right](i,i)+\kappa_{2}(i,i),$
(4.29)
where
$\sup_{i\in{B_{L+Q}}}|\kappa_{2}(i,i)|\leq{\epsilon},\quad\hbox{ as }\
n,b\longrightarrow{\infty}.$ (4.30)
The proof of (4.30) uses elementary computations. Indeed, $\kappa_{2}(i,i)$ is
represented as the sum of $M+1$ terms of the form
$b\sum_{|s_{r}|\leq{n}}^{*}\nu^{2m}g_{1}(s_{1})q_{2}(s_{1})\ldots,g_{1}(s_{m})q_{2}(s_{m})[R_{1}+R_{2}](i,s_{m+1})$
$\times U(s_{m+1},s_{m})\ldots U(s_{1},i),$
where the sum is taken over the values of $s_{j}$ such that
$|s_{j}-s_{j+1}|>bh$ at least for one of the numbers $j\leq{m}$.
Now remembering the a priori bounds for $R_{1}$ (4.22) and $R_{2}$ (4.23), one
obtains the following estimate of $\kappa_{2}$:
$\displaystyle\sup_{|i|\leq{n}}|\kappa_{2}(i,i)|\leq$
$\displaystyle{\sum_{m=0}^{M}\frac{v^{2m}}{\eta^{2m+3}}\sum_{|s_{r}|\leq{n}}^{*}bU(i,s_{1})\ldots
U(s_{m},s_{m+1})}$
$\displaystyle+\sum_{m=0}^{M}\frac{v^{2m+2}}{\eta^{2m+5}}\sum_{|s_{r}|\leq{n}}^{*}bU(i,s_{1})\ldots
U(s_{m},s_{m+1}).$ (4.31)
Assuming that $|s_{j}-s_{j+1}|>bh$ and using inequality
$\displaystyle\sum_{|s_{i}|\leq{n}}U(i,s_{1})\ldots U(s_{j-1},s_{j})$
$\displaystyle\leq{\sum_{s_{i}\in\mathbb{Z}}U(i,s_{1})\ldots
U(s_{j-1},s_{j})}$ (4.32)
$\displaystyle\leq{\left[\int_{-\infty}^{+\infty}\psi(t)dt+\frac{\psi(0)}{b}\right]^{j}}$
$\displaystyle\leq{(1+1/b)^{j}},$ (4.33)
one sees that for large enough $b$ and $n$,
$\sum_{|s_{j}|\leq{n}}U^{j}(i,s_{j})\epsilon
U^{m-j}(s_{j+1},s_{m+1})\leq{\epsilon}.$
Let us also mention here that given $\epsilon>0$, one has large enough $n$ and
$b$ that
$\sup_{i\in B_{L+Q}}|\sum_{|s|\leq{n}}U^{j}(i,s)-1|\leq{\epsilon},$ (4.34)
where $j\leq{M}$. This follows from elementary computations related with the
differences
$P_{b}=\frac{1}{b}\sum_{t\in\mathbb{Z}}\psi\left(\frac{t}{b}\right)-\int_{\mathbb{R}}\psi(s)ds$
(4.35)
and
$T_{n,b}(i)\equiv
T(i)=\frac{1}{b}\sum_{|t|\leq{n}}\psi\left(\frac{t-i}{b}\right)-\frac{1}{b}\sum_{t\in\mathbb{Z}}\psi\left(\frac{t}{b}\right).$
(4.36)
that vanish in the limit $1\ll b\ll n$ (see previous work [1] for more
details).
This reasoning when slightly modified is used to estimate the second term in
the RHS of (4.31). Now one can write that
$\sup_{|i|\leq{n}}|\kappa_{2}(i,i)|\leq{2\epsilon\sum_{m=0}^{M}m\left[\frac{v^{2}}{\eta^{2}}\right]^{m}}\leq{\epsilon}.$
Regarding the RHS of (4.29) with $i\in B_{L+Q}$, one observes that the
summations run over such values of $s_{r}$ that $|i-s_{1}|\leq{bh}$,
$|s_{r}-s_{r+1}|\leq{bh}$, and thus $s_{j}\in B_{L}$ for all $j\leq{k+m-1}$.
This means that we can apply relations (4.26) and (4.27) to the RHS of (4.29)
and to replace $g_{1}$ by $w_{1}$, $q_{2}$ by $w_{2}$. From (4.28), it follows
that
$\displaystyle b[F_{12}U](i,i)=$
$\displaystyle\sum_{m=0}^{M}[v^{2}w_{1}w_{2}]^{m}\
b\sum_{|s_{m+1}|\leq{n}}\left(R_{1}(i,s_{m+1})+R_{2}(i,s_{m+1})\right)\hat{U}^{m+1}(s_{m+1},i)$
$\displaystyle+\kappa_{3}(i,i)$
with
$\sup_{i\in B_{L+Q}}|\kappa_{3}(i,i)|\leq{4\epsilon}.$
Finally, applying Lemma 4.3 to the expressions involved in $R_{l}$ and taking
into account that
$\sup_{i\in
B_{L+Q}}|bU^{m+1}(i,i)-\frac{1}{2\pi}\int_{\mathbb{R}}\tilde{\psi}^{m+1}(p)dp|\leq{\epsilon},$
(4.37)
we obtain equality
$\displaystyle b[F_{12}U](i,i)=$
$\displaystyle\frac{1}{2\pi}\frac{w^{2}_{1}w_{2}}{1-v^{2}w^{2}_{1}}\sum_{m=0}^{M}[v^{2}w_{1}w_{2}]^{m}\int_{\mathbb{R}}\tilde{\psi}^{m+1}(p)dp$
$\displaystyle+\frac{1}{2\pi}\frac{w^{2}_{1}w_{2}}{1-v^{2}w^{2}_{1}}\sum_{m=0}^{M}[v^{2}w_{1}w_{2}]^{m}v^{2}\int_{\mathbb{R}}\frac{w_{1}w_{2}\tilde{\psi}^{m+1}(p)}{1-v^{2}w_{1}w_{2}\tilde{\psi}(p)}dp+\kappa_{4}(i,i)$
(4.38)
with
$\sup_{i\in B_{L+Q}}|\kappa_{4}(i,i)|\leq{\epsilon}.$
Passing back in (4.38) to the infinite series and simplifying them, we arrive
at the expression standing in the RHS of (4.14). Relation (4.14) is proved.
Now let us prove (4.20). Inequality $U(p,s)\leq{b^{-1}}$, (3.1) and (3.17)
imply that if $z_{l}\in\Lambda_{\eta}$, the estimate
$\max_{r=1,2}\left\\{\sup_{|i|,|s|\leq{n}}|\beta_{r}(i,s)|\right\\}=O(b^{-1})$
(4.39)
holds for enough $n$ and $b$ satisfying (2.9). To estimate $\beta_{3}$, we use
the following estimate of the diagonal elements of the resolvent $G$, proved
in the previous work [1].
###### Lemma 4.5.
(see [1]) If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the
estimate
$\sup_{|s|\leq{n}}{\bf E}\\{|U^{0}_{G}(s;z)|^{2}\\}=O(b^{-2})$ (4.40)
holds for enough $n$ and $b$ satisfying (2.9).
Then inequality (3.1) and estimate (4.40), imply that
$\sup_{|i|,|s|\leq{n}}|\beta_{3}(i,s)|=O(b^{-1}),\quad
z_{1},z_{2}\in\Lambda_{\eta}\quad\hbox{ as }\ n,b\rightarrow\infty.$ (4.41)
Using inequality
$|K_{4}\left(H(p,s)\right)|\leq{\frac{4|\Delta_{ps}|}{b^{2}}}\leq{\frac{4[V_{4}+3v^{4}]}{b^{2}}\psi\left(\frac{p-s}{b}\right)}$
(4.42)
and relations (3.1) and (2.25), we obtain that
$|{\bf
E}\left\\{D^{3}_{ps}\left(G^{2}(i,s)G_{2}(i,p)\right)\right\\}|=O(1),\quad\hbox{
as }\ n,b\rightarrow\infty$
and conclude that
$\sup_{|i|,|s|\leq{n}}|\beta_{4}(i,s)|=O(b^{-1}),\quad
z_{1},z_{2}\in\Lambda_{\eta}\quad\hbox{ as }\ n,b\rightarrow\infty.$ (4.43)
Regarding the term $\beta_{5}$ and using similar arguments as those to the
proof of (3.16) (see (3.22)-(3.23)), we conclude that
$\sup_{|i|,|s|\leq{n}}|\beta_{5}(i,s)|=O(b^{-1}),\quad
z_{1},z_{2}\in\Lambda_{\eta}\quad\hbox{ as }\ n,b\rightarrow\infty.$ (4.44)
Now (4.20) follows from (4.39), (4.41), (4.43) and (4.44).
To complete the proof of Lemma 4.2, let us prove (4.15). To do this we use the
following simple statement, proved in the previous work [1].
###### Lemma 4.6.
(see [1]) If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the
estimate
$\sup_{|s|\leq{n}}{\bf E}\\{|G(s,s;z)^{0}|^{2}\\}=O(b^{-1})$ (4.45)
holds for enough $n$ and $b$ satisfying (2.9).
We introduce the variable $M_{12}(i)=q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i)$
with $\Upsilon_{12}$ is given by (3.13). Using identity (3.11) and estimate
(4.45), we obtain that
$\displaystyle M_{12}(i)=$ $\displaystyle q_{2}(i)g_{2}(i){\bf
E}\\{G^{2}_{1}(i,i)\\}\sum_{|p|\leq{n}}\frac{\Delta_{pi}}{b}g_{1}(p)g_{2}(p)$
$\displaystyle+q_{2}(i)g_{1}(i)g_{2}(i)\sum_{|p|\leq{n}}\frac{\Delta_{pi}}{b}{\bf
E}\\{G^{2}_{1}(p,p)\\}g_{2}(p)+o(1),\quad\hbox{ as }\ n,b\rightarrow\infty.$
(4.46)
If one considers (4.46) with $i$ taken far enough from the endpoints $-n$,
$n$, then one can use relation (4.26) and (4.27) and replace $g_{1}$, $g_{2}$
and $q_{2}$ by the constant values $w_{1}$ and $w_{2}$. This substitution
leads to simplified expressions with error terms that vanish as
$n,b\rightarrow\infty$. To finish the proof, we use relation (2.1) and Lemma
4.3 and replace the terms $\sum_{p}\Delta_{pi}/b$ and $G^{2}_{1}$ of $M_{12}$
by the corresponding expressions given by relations (2.14) and (4.16). This
proves (4.15). Lemma 4.2 is proved. $\hfill\blacksquare$
#### 4.3.2 Proof of Theorem 2.1
Let us return to relation (4.10). We introduce the linear operator
$W^{(g_{2},q_{2})}$ acting on vectors $e\in\mathbb{C}^{N}$ with components
$e(i)$ as follows;
$\\{W^{(g_{2},q_{2})}(e)\\}(i)=v^{2}g_{2}(i)q_{2}(i)\sum_{|p|\leq{n}}e(p)U(p,i).$
(4.47)
As a matter of fact, we can rewrite (4.10) in the following form:
$[I-W^{(g_{2},q_{2})}](R_{12})(i)=\frac{1}{Nb}f_{12}(i)+\Gamma(i),$ (4.48)
where $f_{12}$ and $\Gamma$ are given by (4.11) and (4.12). It is easy to see
that if $z\in\Lambda_{\eta}$, then inequalities (3.1) and (4.4) imply that
$||W^{(g_{2},q_{2})}||_{1}\leq{\frac{v^{2}}{(2v+1)^{2}}}<{\frac{1}{2}},$
where $||W^{(g_{2},q_{2})}||_{1}=\sup_{|V|_{1}=1}|W^{(g_{2},q_{2})}(V)|_{1}$
and $|V|_{1}=\sup_{i}|V(i)|$. Then (4.48) can be rewritten in the form
$R_{12}(i)=\frac{1}{Nb}\sum_{m=0}^{\infty}\left([W^{(g_{2},q_{2})}]^{m}\vec{f}_{12}\right)(i)+o\left(\frac{1}{Nb}\right).$
Regarding the trace
$\frac{1}{N}\sum_{|i|\leq{n}}R_{12}(i)=\frac{1}{N}\sum_{i\in
B_{L}}R_{12}(i)+\frac{2bL}{N}O\left(\frac{1}{Nb}\right)=\frac{1}{N}\sum_{i\in
B_{L}}R_{12}(i)+o\left(\frac{1}{Nb}\right)$
and repeating the same arguments of the proof of (4.14) presented above, we
can write that
$R_{12}(i)=\frac{1}{Nb}\sum_{m=0}^{M}\sum_{|t|\leq{n}}f_{12}(t)(v^{2}w^{2}_{2}U)^{m}(t,i)+\frac{1}{Nb}\Delta^{(2)}(i)$
with $\sup_{i\in B_{L}}|\Delta^{(2)}(i)|=o(1)$. Finally, observing that
$f_{12}(t)$ asymptotically does not depend on $t$ (see Lemma 4.2), we arrive
with the help of (4.34), at the expression (2.12). Theorem 2.1 is proved.
$\hfill\blacksquare$
## 5 Proof of auxiliary statement
The main goal of this section is to prove Lemmas 3.1, 4.1 and 4.3.
### 5.1 Proof of Lemma 3.1
#### 5.1.1 Estimate of the term $Y_{1}$ (3.12)
Here we have to use the resolvent identity (2.23) and the cumulants expansion
formula (2.15) twice. However, the computations are based on the same
inequalities as those of the proofs of Lemma 3.2. Regarding
$Y_{1}(i)=\zeta_{2}b^{-2}\sum_{p}{\bf
E}\\{g^{0}_{1}G_{2}(i,i)^{2}G_{2}(p,p)^{2}\\}\Delta_{pi}$, we apply to
$G_{2}(i,i)$ the resolvent identity (2.23). Then we get relation
$\displaystyle Y_{1}(i)=$
$\displaystyle{\zeta^{2}_{2}}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}\\}\Delta_{pi}$
$\displaystyle-\frac{\zeta^{2}_{2}}{b^{2}}\sum_{|s|,|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2}H(s,i)\\}\Delta_{pi}.$
Applying the formula (2.15) with $q=3$ to ${\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2}H(s,i)\\}$, we obtain that
$\displaystyle Y_{1}(i)=$
$\displaystyle\frac{\zeta^{2}_{2}}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}(g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2})\Delta_{pi}$
$\displaystyle+\frac{\zeta^{2}_{2}v^{2}}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)^{2}G_{2}(p,p)^{2}U_{G_{2}}(i)\\}\Delta_{pi}+\sum_{r=1}^{3}Q_{r}(i)$
(5.1)
with
$\displaystyle Q_{1}(i)=$
$\displaystyle\frac{2\zeta^{2}_{2}v^{2}}{Nb^{2}}\sum_{|s|,|p|\leq{n}}{\bf
E}\\{G_{1}^{2}(i,s)G_{2}(i,s)G_{2}(i,i)G_{2}(p,p)^{2}\\}U(s,i)\Delta_{pi}$
$\displaystyle+\frac{3\zeta^{2}_{2}v^{2}}{b^{2}}\sum_{|s|,|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}G_{2}(i,s)^{2}\\}U(s,i)\Delta_{pi}$
$\displaystyle+\frac{4\zeta^{2}_{2}v^{2}}{b^{2}}\sum_{|s|,|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)G_{2}(i,s)G_{2}(i,p)G_{2}(p,s)\\}U(s,i)\Delta_{pi},$
$\displaystyle Q_{2}(i)=$
$\displaystyle-\frac{\zeta^{2}_{2}}{b^{2}}\sum_{|s|,|p|\leq{n}}\frac{K_{4}}{6}{\bf
E}\left\\{D_{si}^{3}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2})\right\\}\Delta_{pi}$
and
$\displaystyle Q_{3}(i)=$
$\displaystyle-\frac{\zeta^{2}_{2}}{b^{2}4!}\sum_{|s|,|p|\leq{n}}{\bf
E}\left\\{H(s,i)^{5}[D_{si}^{4}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2})]^{(0)}\right\\}\Delta_{pi}$
$\displaystyle+\frac{\zeta^{2}_{2}}{b^{2}3!}\sum_{|s|,|p|\leq{n}}K_{2}{\bf
E}\left\\{H(s,i)^{3}[D_{si}^{4}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2})]^{(1)}\right\\}\Delta_{pi}$
$\displaystyle+\frac{\zeta^{2}_{2}}{b^{2}3!}\sum_{|s|,|p|\leq{n}}K_{4}{\bf
E}\left\\{H(s,i)[D_{si}^{4}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2})]^{(2)}\right\\}\Delta_{pi},$
where $K_{r}$, $r=2,4$ are the cumulants of $H(s,i)$ as in (3.5). Applying to
the second term of the RHS of (5.1) identity (3.11) and using the definition
of $q_{2}(i)$ (4.1), we obtain that
$\displaystyle Y_{1}(i)=$
$\displaystyle\frac{\zeta_{2}q_{2}(i)}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}\\}\Delta_{pi}$
$\displaystyle+\frac{\zeta_{2}v^{2}q_{2}(i)}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)^{2}G_{2}(p,p)^{2}U^{0}_{G_{2}}(i)\\}\Delta_{pi}$
$\displaystyle+\frac{q_{2}(i)}{\zeta_{2}}\sum_{r=1}^{3}Q_{r}(i).$ (5.2)
Regarding the first term of the RHS of this equality, we apply the resolvent
identity (2.23) to $G_{2}(i,i)$. Repeating the usual computations based on the
formula (2.15) (with $q=3$) and relation (2.25), we obtain that
$\frac{\zeta_{2}q_{2}(i)}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}\\}\Delta_{pi}=\frac{\zeta_{2}}{b^{2}}q^{2}_{2}(i)\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(p,p)^{2}\\}\Delta_{pi}$
$+\frac{\zeta_{2}v^{2}}{b^{2}}q^{2}_{2}(i)\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}U^{0}_{G_{2}}(i)\\}\Delta_{pi}+\frac{q_{2}(i)}{\zeta_{2}}\sum_{r=1}^{3}\breve{Q}_{r}(i)$
(5.3)
with
$\displaystyle\breve{Q}_{1}(i)=$
$\displaystyle\frac{2\zeta^{2}_{2}v^{2}q_{2}(i)}{Nb^{2}}\sum_{|s|,|p|\leq{n}}{\bf
E}\\{G_{1}^{2}(i,s)G_{2}(i,s)G_{2}(p,p)^{2}\\}U(s,i)\Delta_{pi}$
$\displaystyle+\frac{\zeta^{2}_{2}v^{2}q_{2}(i)}{b^{2}}\sum_{|s|,|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(p,p)^{2}G_{2}(i,s)^{2}\\}U(s,i)\Delta_{pi}$
$\displaystyle+\frac{4\zeta^{2}_{2}v^{2}q_{2}(i)}{b^{2}}\sum_{|s|,|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(p,p)G_{2}(i,s)G_{2}(i,p)G_{2}(p,s)\\}U(s,i)\Delta_{pi},$
$\displaystyle\breve{Q}_{2}(i)=$
$\displaystyle-\frac{\zeta^{2}_{2}q_{2}(i)}{b^{2}}\sum_{|s|,|p|\leq{n}}\frac{K_{4}}{6}{\bf
E}\left\\{D_{si}^{3}(g^{0}_{1}G_{2}(i,s)G_{2}(p,p)^{2})\right\\}\Delta_{pi}$
and
$\displaystyle\breve{Q}_{3}(i)=$
$\displaystyle-\frac{\zeta^{2}_{2}}{b^{2}4!}\sum_{|s|,|p|\leq{n}}{\bf
E}\left\\{H(s,i)^{5}[D_{si}^{4}(g^{0}_{1}G_{2}(i,s)G_{2}(p,p)^{2})]^{(0)}\right\\}\Delta_{pi}$
$\displaystyle+\frac{\zeta^{2}_{2}}{b^{2}3!}\sum_{|s|,|p|\leq{n}}K_{2}{\bf
E}\left\\{H(s,i)^{3}[D_{si}^{4}(g^{0}_{1}G_{2}(i,s)G_{2}(p,p)^{2})]^{(1)}\right\\}\Delta_{pi}$
$\displaystyle+\frac{\zeta^{2}_{2}}{b^{2}3!}\sum_{|s|,|p|\leq{n}}K_{4}{\bf
E}\left\\{H(s,i)[D_{si}^{4}(g^{0}_{1}G_{2}(i,s)G_{2}(p,p)^{2})]^{(2)}\right\\}\Delta_{pi},$
where $K_{r}$, $r=2,4$ are the cumulants of $H(s,i)$ as in (3.5).
Substituting (5.3) into (5.2), we obtain that
$\displaystyle Y_{1}(i)=$
$\displaystyle\frac{\zeta_{2}}{b^{2}}q^{2}_{2}(i)\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(p,p)^{2}\\}\Delta_{pi}$
$\displaystyle+\frac{\zeta_{2}v^{2}}{b^{2}}q^{2}_{2}(i)\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}U^{0}_{G_{2}}(i)\\}\Delta_{pi}$
$\displaystyle+\frac{\zeta_{2}v^{2}q_{2}(i)}{b^{2}}\sum_{|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,i)^{2}G_{2}(p,p)^{2}U^{0}_{G_{2}}(i)\\}\Delta_{pi}$
$\displaystyle+\frac{q_{2}(i)}{\zeta_{2}}\sum_{r=1}^{3}Q_{r}(i)+\frac{q_{2}(i)}{\zeta_{2}}\sum_{r=1}^{3}\breve{Q}_{r}(i).$
(5.4)
Now let us estimate each term of the RHS of this equality. If one assumes for
a while that
$\sup_{|p|\leq{n}}|{\bf
E}\\{g_{1}^{0}G_{2}(p,p)^{2}\\}|=O\left(N^{-1}b^{-1}+b^{-1}[{\bf
Var}\\{g_{1}\\}]^{1/2}\right)$ (5.5)
holds for enough $n$ and $b$ satisfying (2.9). Then this estimate and
relations (4.4), (4.42) and (4.40) imply that the fist, the second and the
third terms of the RHS of (5.4) are of the order indicated in the RHS of
(3.14).
Inequality (3.1), (3.2), (3.17) (with $m=1$ and $m=2$) and (4.4) imply that
the term $q_{2}(i)\zeta^{-1}_{2}[Q_{1}(i)+\breve{Q}_{1}(i)]$ is of the order
indicated in the RHS of (3.14). Using similar arguments as those of the proof
of (3.16) (see (3.22)-(3.24)) and the following estimates (cf. (3.20)-(3.21))
$D_{si}^{r}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2})=O\left(N^{-1}+|g_{1}^{0}|\right),\quad
r=3,4$
and
${\bf Var}\\{[g_{n,b}(z_{l})]^{(\nu)}\\}=O\left({\bf
Var}\\{g_{n,b}(z_{l})\\}+b^{-1}N^{-2}\right),\quad\nu=0,1,2,$
we obtain that the terms $Q_{r}$, $r=2,3$ are of the order indicated in the
RHS of (3.14). We conclude that the terms $\breve{Q}_{r}$, $r=2,3$ and
$\sup_{i}|Y_{1}(i)|$ are of the order indicated in the RHS of (3.14).
Now let us prove (5.5). Let us apply the resolvent identity (2.23) to
$G_{2}(p,p)$. Repeating the usual computations based on the formula (2.15)
(with $q=3$) and relation (2.25), we obtain that
$\displaystyle{\bf E}\\{g_{1}^{0}G_{2}(p,p)^{2}\\}=$ $\displaystyle
q_{2}(p){\bf E}\\{g_{1}^{0}G_{2}(p,p)\\}+q_{2}(p){\bf
E}\\{g_{1}^{0}G_{2}(p,p)^{2}U^{0}_{G_{2}}(p)\\}$
$\displaystyle+3q_{2}(p)v^{2}\sum_{|s|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(p,s)^{2}G_{2}(p,p)\\}U(s,p)$
$\displaystyle+\frac{2v^{2}}{N}q_{2}(p)\sum_{|s|\leq{n}}{\bf
E}\\{G^{2}_{1}(s,p)G_{2}(s,p)G_{2}(p,p)\\}U(s,p)$
$\displaystyle-\frac{q_{2}(p)}{6}\sum_{|s|\leq{n}}K_{4}{\bf
E}\left\\{D_{sp}^{3}(g^{0}_{1}G_{2}(p,p)G_{2}(p,s))\right\\}-q_{2}(p)\tilde{Q}(i)$
(5.6)
with
$\displaystyle\tilde{Q}(i)=$ $\displaystyle-\frac{1}{4!}\sum_{|s|\leq{n}}{\bf
E}\left\\{H(s,p)^{5}[D_{sp}^{4}(g^{0}_{1}G_{2}(p,p)G_{2}(p,s))]^{(0)}\right\\}$
$\displaystyle+\frac{1}{3!}\sum_{|s|\leq{n}}K_{2}{\bf
E}\left\\{H(s,p)^{3}[D_{sp}^{4}(g^{0}_{1}G_{2}(p,p)G_{2}(p,s))]^{(1)}\right\\}$
$\displaystyle+\frac{1}{3!}\sum_{|s|\leq{n}}K_{4}{\bf
E}\left\\{H(s,p)[D_{sp}^{4}(g^{0}_{1}G_{2}(p,p)G_{2}(p,s))]^{(2)}\right\\},$
where $K_{r}$, $r=2,4$ are the cumulants of $H(s,p)$ as in (3.5).
Let us estimate each term of the RHS of (5.6). It is easy to show that the
estimate of the first term of the RHS of (5.6) follows from the following
statement, proved in the previous work [1].
###### Lemma 5.1.
(see [1]) If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the
estimate
$\sup_{|p|\leq{n}}|{\bf
E}\\{g_{1}^{0}G_{2}(p,p)\\}|=O\left(b^{-1}n^{-1}+b^{-1}[{\bf
Var}\\{g_{1}\\}]^{1/2}\right)$ (5.7)
holds in the limit $n,b\rightarrow\infty$ (2.9).
Then (5.5) follows from this Lemma.and the estimate (4.40) and the similar
arguments used in the estimates of the terms $Q_{r}$, $r=1,2,3$ in (5.4).
Estimate (5.5) is proved.
#### 5.1.2 Estimate of $Y_{2}$ (3.12)
We rewrite $Y_{2}$ in the form $Y_{2}(i)=\zeta_{2}v^{2}\sum_{s}{\bf
E}\\{M(i,s)\\}U(s,i)$, where we denoted
${\bf E}\\{M(i,s)\\}={\bf E}\\{g^{0}_{1}G_{2}(i,s)^{2}\\}.$
To proceed with estimate of $Y_{2}$, we use the resolvent identity (2.23) and
the cumulants expansion formula (2.15) twice. However, the computations are
based on the results of Lemma 4.5, 4.6 and 5.1. Therefore we just indicate the
main lines of the proof and do not go into the details. Applying to
$G_{2}(i,s)$ the resolvent identity (2.23), we get equality
${\bf E}M(i,s)=\zeta_{2}\delta_{is}{\bf
E}\\{g^{0}_{1}G_{2}(i,i)\\}-\zeta_{2}\sum_{|t|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)H(t,s)\\}.$ (5.8)
Regarding the first term of the RHS of this equality and using relation (5.7),
it is easy to see that the term
$\sum_{|s|\leq{n}}\zeta_{2}\delta_{is}{\bf
E}\\{g^{0}_{1}G_{2}(i,i)\\}U(s,i)=\zeta_{2}\frac{\psi(0)}{b}{\bf
E}\\{g^{0}_{1}G_{2}(i,i)\\}$
is the value of order indicated in (3.14). Let us consider the second term of
(5.8). Applying formula (2.15) with $q=5$ to ${\bf
E}\\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)H(t,s)\\}$ and taking account relations
(2.25) and (3.11), we obtain that
$-\zeta_{2}\sum_{|t|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)H(t,s)\\}=\sum_{l=1}^{7}\Theta_{l}(i,s),$
(5.9)
where
$\displaystyle\Theta_{1}(i,s)=$ $\displaystyle v^{2}\zeta_{2}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)^{2}\\}{\bf E}U_{G_{2}}(s),$
$\displaystyle\Theta_{2}(i,s)=$ $\displaystyle v^{2}\zeta_{2}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)^{2}U^{0}_{G_{2}}(s)\\},$
$\displaystyle\Theta_{3}(i,s)=$
$\displaystyle\frac{2v^{2}\zeta_{2}}{N}\sum_{|t|\leq{n}}{\bf
E}\\{G^{2}_{1}(s,t)U(t,s)G_{2}(i,s)G_{2}(i,t)\\},$
$\displaystyle\Theta_{4}(i,s)=$ $\displaystyle
v^{2}\zeta_{2}\sum_{|t|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,s)\\}U(t,s),$
$\displaystyle\Theta_{5}(i,s)=$ $\displaystyle
2v^{2}\zeta_{2}\sum_{|t|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)G_{2}(t,s)G_{2}(i,t)\\}U(t,s),$
$\displaystyle\Theta_{6}(i,s)=$
$\displaystyle-\zeta_{2}\sum_{|t|\leq{n}}\frac{K_{4}\left(H(t,s)\right)}{6}{\bf
E}\\{D^{3}_{ts}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))\\}$
and
$\Theta_{7}(i,s)=-\zeta_{2}\sum_{|t|\leq{n}}\frac{K_{6}\left(H(t,s)\right)}{5!}{\bf
E}\\{D^{5}_{ts}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))\\}+\tilde{\Theta}_{7}(i,s)$
with
$\displaystyle\tilde{\Theta}_{7}(i,s)=$
$\displaystyle-\frac{\zeta_{2}}{6!}\sum_{|t|\leq{n}}{\bf
E}\left\\{H(t,s)^{7}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))]^{(0)}\right\\}$
$\displaystyle+\frac{\zeta_{2}}{5!}\sum_{|t|\leq{n}}K_{2}\left(H(t,s)\right){\bf
E}\left\\{H(t,s)^{5}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))]^{(1)}\right\\}$
$\displaystyle+\frac{\zeta_{2}}{(3!)^{2}}\sum_{|t|\leq{n}}K_{4}\left(H(t,s)\right){\bf
E}\left\\{H(t,s)^{3}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))]^{(2)}\right\\}$
$\displaystyle+\frac{\zeta_{2}}{5!}\sum_{|t|\leq{n}}K_{6}\left(H(t,s)\right){\bf
E}\left\\{H(t,s)[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))]^{(3)}\right\\},$
where $K_{r}\left(H(t,s)\right)$, $r=2,4,6$ are the cumulants of $H(t,s)$ as
in (3.5)-(3.6).
The term $\Theta_{1}$ is of the form $v^{2}\zeta_{2}{\bf E}\\{M(i,s)\\}{\bf
E}U_{G_{2}}(s)$ and can be put to the left hand side of (5.8). The terms
$\Theta_{2}$ and $\Theta_{3}$ are of the order indicated in the RHS of (3.14).
This can be shown with the help of the estimate (4.40) and inequality (eg.
[1])
$\displaystyle\left|\sum_{|s|,|t|\leq{n}}G^{2}_{1}(s,t)G_{2}(i,s)G_{2}(i,t)\right|$
$\displaystyle\leq{||G_{1}^{2}||\left(\sum_{|s|\leq{n}}|G_{2}(i,s)|^{2}\right)^{1/2}\left(\sum_{|t|\leq{n}}|G_{2}(i,t)|^{2}\right)^{1/2}}\leq{\frac{1}{\eta^{4}}}.$
(5.10)
Regarding $\Theta_{4}$, we apply the resolvent identity (2.23) to the factor
$G_{2}(s,s)$. Repeating the usual computations based on the formula (2.15)
with $q=5$ and taking into account relations (2.25) and (3.11), we obtain that
$\Theta_{4}(i,s)=v^{2}\zeta^{2}_{2}\sum_{|t|\leq{n}}{\bf
E}\\{M(i,t)\\}U(t,s)+v^{2}\zeta_{2}\Theta_{4}(i,s){\bf
E}U_{G_{2}}(s)+\sum_{l=1}^{8}\Omega_{l}(i,s),$ (5.11)
where
$\displaystyle\Omega_{1}(i,s)=$ $\displaystyle
v^{4}\zeta^{2}_{2}\sum_{|t|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,s)U^{0}_{G_{2}}(s)\\}U(t,s),$
$\displaystyle\Omega_{2}(i,s)=$ $\displaystyle
v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,t)^{2}G_{2}(s,p)^{2}\\}U(p,s)U(t,s),$
$\displaystyle\Omega_{3}(i,s)=$
$\displaystyle\frac{2v^{4}\zeta^{2}_{2}}{N}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{G^{2}_{1}(p,s)G_{2}(i,t)^{2}G_{2}(s,p)\\}U(p,s)U(t,s),$
$\displaystyle\Omega_{4}(i,s)=$ $\displaystyle
2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,t)G_{2}(i,s)G_{2}(p,t)G_{2}(s,p)\\}U(p,s)U(t,s),$
$\displaystyle\Omega_{5}(i,s)=$ $\displaystyle
2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,t)G_{2}(i,p)G_{2}(s,t)G_{2}(s,p)\\}U(p,s)U(t,s),$
$\displaystyle\Omega_{6}(i,s)=$
$\displaystyle-\zeta^{2}_{2}v^{2}\sum_{|t|,|p|\leq{n}}\frac{K_{4}\left(H(p,s)\right)}{6}{\bf
E}\\{D^{3}_{ps}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))\\}U(t,s),$
$\displaystyle\Omega_{7}(i,s)=$
$\displaystyle-\zeta^{2}_{2}v^{2}\sum_{|t|,|p|\leq{n}}\frac{K_{6}\left(H(p,s)\right)}{5!}{\bf
E}\\{D^{5}_{ps}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))\\}U(t,s)$
and
$\displaystyle\Omega_{8}($ $\displaystyle i,s)$ $\displaystyle=$
$\displaystyle-\frac{\zeta^{2}_{2}v^{2}}{6!}\sum_{|t|,|p|\leq{n}}{\bf
E}\left\\{H(p,s)^{7}[D_{ps}^{6}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))]^{(0)}\right\\}U(t,s)$
$\displaystyle+\frac{\zeta^{2}_{2}v^{2}}{5!}\sum_{|t|,|p|\leq{n}}K_{2}\left(H(p,s)\right){\bf
E}\left\\{H(p,s)^{5}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))]^{(1)}\right\\}U(t,s)$
$\displaystyle+\frac{\zeta^{2}_{2}v^{2}}{(3!)^{2}}\sum_{|t|,|p|\leq{n}}K_{4}\left(H(p,s)\right){\bf
E}\left\\{H(p,s)^{3}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))]^{(2)}\right\\}U(t,s)$
$\displaystyle+\frac{\zeta^{2}_{2}v^{2}}{5!}\sum_{|t|,|p|\leq{n}}K_{6}\left(H(p,s)\right){\bf
E}\left\\{H(p,s)[D_{ts}^{6}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))]^{(3)}\right\\}U(t,s)$
with $K_{r}\left(H(p,s)\right)$, $r=2,4,6$ are the cumulants of $H(p,s)$ as in
(3.5)-(3.6).
The terms $\Omega_{l}$, $l=1,\ldots,5$ are of the order indicated in the RHS
of (3.14). This can be shown with the help of the estimate (4.40) and the
inequalities (3.1), (3.2), (3.17) and (5.10). The term $\Omega_{6}$ contains
$272$ terms that are of the order indicated in the RHS of (3.14). This can be
checked by direct computations with the use of (3.17) and (5.10). Using
similar argument as those of the proofs of (3.15) and (3.16) (see
(3.22)-(3.24)), and the following estimate (cf. (3.18))
${\bf
E}|D^{5}_{pi}\\{g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p)\\}|=O\left(N^{-1}+[{\bf
Var}\\{g_{1}\\}]^{1/2}\right),\ \hbox{ as }\quad n,p\rightarrow\infty,$ (5.12)
we conclude that the terms $\Omega_{7}$ and $\Omega_{8}$ are of the order
indicated in the RHS of (3.14). Then, the relation (5.11) is of the form that
leads to the estimates needed for $\sum_{s}{\bf E}\\{M(i,s)\\}U(s,i)$.
Regarding $\Theta_{5}(i,s)$, we apply the resolvent identity (2.23) to the
factor $G_{2}(t,s)$. Repeating the usual computations based on the formula
(2.15) with $q=5$ and taking into account relations (2.25) and (3.11), we
obtain that
$\Theta_{5}(i,s)=2v^{2}\zeta^{2}_{2}\frac{\psi(0)}{b}{\bf
E}M(i,s)+v^{2}\zeta_{2}\Theta_{5}(i,s){\bf
E}U_{G_{2}}(s)+\sum_{l=1}^{9}\Omega^{{}^{\prime}}_{l}(i,s),$ (5.13)
where
$\displaystyle\Omega^{{}^{\prime}}_{1}(i,s)=$ $\displaystyle
2v^{4}\zeta^{2}_{2}\sum_{|t|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,s)U^{0}_{G_{2}}(s)\\}U(t,s),$
$\displaystyle\Omega^{{}^{\prime}}_{2}(i,s)=$ $\displaystyle
2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{g_{1}^{0}G_{2}(i,p)G_{2}(s,s)G_{2}(i,t)G_{2}(t,p)\\}U(p,s)U(t,s),$
$\displaystyle\Omega^{{}^{\prime}}_{3}(i,s)=$
$\displaystyle\frac{4v^{4}\zeta^{2}_{2}}{N}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{G^{2}_{1}(p,s)G_{2}(i,s)G_{2}(i,t)G_{2}(t,p)\\}U(p,s)U(t,s),$
$\displaystyle\Omega^{{}^{\prime}}_{4}(i,s)=$ $\displaystyle
4v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)G_{2}(p,s)G_{2}(i,t)G_{2}(t,p)\\}U(p,s)U(t,s),$
$\displaystyle\Omega^{{}^{\prime}}_{5}(i,s)=$ $\displaystyle
2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)G_{2}(i,p)G_{2}(t,p)G_{2}(s,t)\\}U(p,s)U(t,s),$
$\displaystyle\Omega^{{}^{\prime}}_{6}(i,s)=$ $\displaystyle
2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\leq{n}}{\bf
E}\\{g^{0}_{1}G_{2}(i,s)^{2}G_{2}(p,t)^{2}\\}U(p,s)U(t,s),$
$\displaystyle\Omega^{{}^{\prime}}_{7}(i,s)=$
$\displaystyle-2\zeta^{2}_{2}v^{2}\sum_{|t|,|p|\leq{n}}\frac{K_{4}\left(H(p,s)\right)}{6}{\bf
E}\\{D^{3}_{ps}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))\\}U(t,s),$
$\displaystyle\Omega^{{}^{\prime}}_{8}(i,s)=$
$\displaystyle-2\zeta^{2}_{2}v^{2}\sum_{|t|,|p|\leq{n}}\frac{K_{6}\left(H(p,s)\right)}{5!}{\bf
E}\\{D^{5}_{ps}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))\\}U(t,s)$
and
$\displaystyle\Omega^{{}^{\prime}}$ ${}_{9}(i,s)$ $\displaystyle=$
$\displaystyle-\frac{2\zeta^{2}_{2}v^{2}}{6!}\sum_{|t|,|p|\leq{n}}{\bf
E}\left\\{H(p,s)^{7}[D_{ps}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))]^{(0)}\right\\}U(t,s)$
$\displaystyle+\frac{2\zeta^{2}_{2}v^{2}}{5!}\sum_{|t|,|p|\leq{n}}K_{2}\left(H(p,s)\right){\bf
E}\left\\{H(p,s)^{5}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))]^{(1)}\right\\}U(t,s)$
$\displaystyle+\frac{2\zeta^{2}_{2}v^{2}}{(3!)^{2}}\sum_{|t|,|p|\leq{n}}K_{4}\left(H(p,s)\right){\bf
E}\left\\{H(p,s)^{3}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))]^{(2)}\right\\}U(t,s)$
$\displaystyle+\frac{2\zeta^{2}_{2}v^{2}}{5!}\sum_{|t|,|p|\leq{n}}K_{6}\left(H(p,s)\right){\bf
E}\left\\{H(p,s)[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))]^{(3)}\right\\}U(t,s)$
with $K_{r}\left(H(p,s)\right)$, $r=2,4,6$ are the cumulants of $H(p,s)$ as in
(3.5)-(3.6).
The terms $\sum_{s}\Omega^{{}^{\prime}}_{l}(i,s)U(s,i)$, $l=1,\ldots,6$ are of
the order indicated in the RHS of (3.14). This can be shown with the help of
the estimate (4.40) and the inequalities (3.1), (3.2), (3.17) and (5.10). The
term $\Omega^{{}^{\prime}}_{7}$ contains $356$ terms that are of the order
indicated in the RHS of (3.14). This can be checked by direct computations
with the use of (3.17) and (5.10). Using similar argument as those of the
proofs of (3.15) and (3.16) (see (3.22)-(3.24)), and the following estimate
(cf. (3.18))
${\bf
E}|D^{5}_{pi}\\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p)\\}|=O\left(N^{-1}+[{\bf
Var}\\{g_{1}\\}]^{1/2}\right),\ \hbox{ as }\quad n,p\rightarrow\infty,$ (5.14)
we conclude that the terms $\Omega^{{}^{\prime}}_{8}$ and
$\Omega^{{}^{\prime}}_{9}$ are of the order indicated in the RHS of (3.14).
Then, the form of (5.13) is also such that, being substituted into (5.9) and
then into (5.8), it leads to the needed estimates.
The term $\Theta_{6}(i,s)$ contains $67$ terms. These terms can be gathered
into three groups. In each group, the terms are estimated by the same values
with the help of the same computations.
We give estimates for the typical cases. Using (3.1), (3.2) and (3.17) (with
$m=1$), we get for the terms of the first group:
$\displaystyle\left|\frac{\zeta_{2}}{N}\sum_{|t|\leq{n}}K_{4}\left(H(t,s)\right){\bf
E}\\{G_{1}^{2}(t,t)G_{1}(s,s)G_{1}(t,s)G_{2}(i,s)G_{2}(i,t)\\}\cdot\frac{1}{(1+\delta_{ts})^{3}}\right|$
$\displaystyle\leq{\frac{V_{4}+3v^{4}}{\eta^{5}Nb^{2}}\sum_{|t|\leq{n}}{\bf
E}|G_{1}(t,s)G_{2}(i,t)|}\leq{\frac{V_{4}+3v^{4}}{\eta^{7}Nb^{2}}}.$
For the terms of the second group, we obtain estimates
$\displaystyle\left|\zeta_{2}\sum_{|t|\leq{n}}K_{4}\left(H(t,s)\right){\bf
E}\\{g_{1}^{0}G_{2}(s,s)^{2}G_{2}(i,t)^{2}\\}\cdot\frac{1}{(1+\delta_{ts})^{3}}\right|$
$\displaystyle\leq{\frac{V_{4}+3v^{4}}{\eta^{3}b^{2}}{\bf
E}|g_{1}^{0}|\sum_{|t|\leq{n}}|G_{2}(i,t)^{2}|}\leq{\frac{[V_{4}+3v^{4}]\sqrt{{\bf
Var}\\{g_{1}\\}}}{\eta^{5}b^{2}}}.$
Finally, for the terms of the third group, we get inequalities
$\displaystyle\left|\sum_{|s|\leq{n}}\frac{\zeta_{2}}{N}\sum_{|t|\leq{n}}K_{4}\left(H(t,s)\right){\bf
E}\\{G_{1}^{2}(s,s)G_{1}(t,t)G_{2}(t,t)G_{2}(i,s)^{2}\\}U(s,i)\cdot\frac{1}{(1+\delta_{ts})^{3}}\right|$
$\displaystyle\leq{\frac{V_{4}+3v^{4}}{\eta^{5}Nb}\sum_{|s|\leq{n}}{\bf
E}|G_{2}(i,s)^{2}|\sum_{|t|\leq{n}}U(t,s)U(s,i)}=O\left(\frac{1}{Nb^{2}}\right).$
Gathering all the estimates of $67$ terms, we obtain that
$\left|\sum_{|s|\leq{n}}\Theta_{6}(i,s)U(s,i)\right|=O\left(\frac{1}{Nb^{2}}+\frac{\sqrt{{\bf
Var}\\{g_{1}\\}}}{b^{2}}\right).$
Using similar argument as those of the proofs of (3.15) and (3.16) (see
(3.22)-(3.24)), we conclude that $\Theta_{7}$ and $\sup_{i}|Y_{2}(i)|$ are of
the order indicated in (3.14). Estimate (3.14) is proved and so Lemma 3.1 is
proved.$\hfill\blacksquare$
### 5.2 Proof of Lemma 4.1
Let us consider the variable
$K(i,s)={\bf E}\\{RG^{0}(i,i)\\}={\bf E}\\{R^{0}G(i,i)\\},$
where we denoted $R=g^{0}U^{0}_{G}(s)$. Applying to $G_{2}(i,i)$ the resolvent
identity (2.23) and taking account formula (2.15) with $q=3$ and relation
(2.25), we obtain that
${\bf E}\\{R^{0}G(i,i)\\}=\zeta v^{2}{\bf
E}\\{R^{0}G(i,i)U_{G}(i)\\}+\sum_{a=1}^{5}l_{a}(i,s)$ (5.15)
with
$\displaystyle l_{1}(i,s)=$ $\displaystyle\zeta v^{2}\sum_{|p|\leq{n}}{\bf
E}\\{R^{0}G(i,p)^{2}\\}U(p,i),$ $\displaystyle l_{2}(i,s)=$ $\displaystyle
2\zeta v^{2}\sum_{|p|,|t|\leq{n}}{\bf
E}\\{g^{0}G(t,p)G(t,i)G(i,p)\\}U(t,s)U(p,i),$ $\displaystyle l_{3}(i,s)=$
$\displaystyle\frac{2\zeta v^{2}}{N}\sum_{|p|,|t|\leq{n}}{\bf
E}\\{G(p,t)G(i,t)U^{0}_{G}(s)G(i,p)\\}U(p,i),$ $\displaystyle l_{4}(i,s)=$
$\displaystyle-\frac{\zeta}{6}\sum_{|p|\leq{n}}K_{4}{\bf
E}\\{D^{3}_{pi}(R^{0}G(i,p))\\}$
and
$\displaystyle l_{5}(i,s)=$
$\displaystyle-\frac{\zeta}{4!}\sum_{|p|\leq{n}}{\bf
E}\left\\{H(p,i)^{5}[D_{pi}^{4}(R^{0}G(i,p))]^{(0)}\right\\}$
$\displaystyle+\frac{\zeta}{3!}\sum_{|p|\leq{n}}K_{2}{\bf
E}\left\\{H(p,i)^{3}[D_{pi}^{4}(R^{0}G(i,p))]^{(1)}\right\\}$
$\displaystyle+\frac{\zeta}{3!}\sum_{|p|\leq{n}}K_{4}{\bf
E}\left\\{H(p,i)[D_{pi}^{4}(R^{0}G(i,p))]^{(2)}\right\\},$
where $K_{r}$, $r=2,4$ are the cumulants of $H(p,i)$ as in (3.5). Let us use
the identity
${\bf E}R^{0}XY={\bf E}RX^{0}{\bf E}Y+{\bf E}RY^{0}{\bf E}X+{\bf
E}RX^{0}Y^{0}-{\bf E}R{\bf E}X^{0}Y^{0},$
and rewrite (5.15) in the form
${\bf
E}\\{R^{0}G(i,i)\\}=K(i,s)=v^{2}q(i)g(i)\sum_{|t|\leq{n}}K(t,s)U(t,i)+\Pi(i,s)$
(5.16)
with
$\displaystyle\Pi(i,s)=$ $\displaystyle v^{2}q(i)\left[{\bf
E}\\{RU^{0}_{G}(i)G^{0}(i,i)\\}-{\bf E}\\{g^{0}U^{0}_{G}(s)\\}{\bf
E}\\{G^{0}(i,i)U^{0}_{G}(i)\\}\right]$
$\displaystyle+\frac{q(i)}{\zeta}\sum_{a=1}^{5}l_{a}(i,s),$ (5.17)
where $g(i)={\bf E}\\{G(i,i)\\}$ and $q$ is given by (4.1). Now we rewrite
(5.16) in the form of a vector equality
$\vec{K}(.,s)=[I-W^{(q,g)}]^{-1}\vec{\Pi}(.,s),$
where we denote by $W^{(q,g)}$ the linear operator acting on a vector $e$ with
components $e(i)$ as
$[W^{(q,g)}e](i)=v^{2}q(i)g(i)\sum_{|t|\leq{n}}e(t)U(t,i)$
and vectors $[\vec{\Pi}(.,s)](i)=\Pi(i,s)$. It is easy to see that if
$z\in\Lambda_{\eta}$, then $||W^{(q,g)}||\leq{\frac{1}{2}}$. Thus, to prove
relation (4.6), it is sufficient to show that
$\sup_{|i|,|s|\leq{n}}|\Pi(i,s)|=O\left(\frac{1}{Nb^{2}}+\frac{1}{b^{2}}\left({\bf
Var}\\{g\\}\right)^{1/2}\right).$ (5.18)
Let us prove (5.18). Taking into account inequality (3.1), (3.2), (5.10) and
estimate (4.40), we obtain that
$|l_{a}(i,s)|\leq{\frac{c}{b^{2}}\left({\bf Var}\\{g\\}\right)^{1/2}}\quad\
a=1,2$ (5.19)
and
$|l_{3}(i,s)|\leq{\frac{c}{Nb^{2}}},$ (5.20)
where c is a constant. Using similar arguments as those of the proof of (3.16)
(see (3.22)-(3.24)) and the following estimates (cf. (3.20)-(3.21))
$D_{pi}^{r}(R^{0}G(i,p))=O\left(N^{-1}+|g_{1}^{0}|\right),\quad r=3,4$
and
${\bf Var}\\{[g_{n,b}(z)]^{(\nu)}\\}=O\left({\bf
Var}\\{g_{n,b}(z)\\}+b^{-1}N^{-2}\right),\quad\nu=0,1,2,$
we obtain that the terms $l_{a}$, $a=4,5$ are of the order indicated in the
RHS of (4.6). Finally, we derive inequality
$\displaystyle|\Pi(i,s)|\leq$ $\displaystyle{c\left({\bf
Var}\\{g\\}\right)^{1/2}\left(\left({\bf
E}|U^{0}_{G}(i)|^{4}\right)^{1/2}+\frac{1}{b^{2}}\left({\bf
E}|U^{0}_{G}(i)|^{2}\right)^{1/2}\right)}$
$\displaystyle+c\left(\frac{1}{Nb^{2}}+\frac{1}{b^{2}}\left({\bf
Var}\\{g\\}\right)^{1/2}\right),$ (5.21)
where $c$ is a constant. Then (5.18), (5.21) and Lemma 4.1 follow from (4.40)
and the following estimate.
###### Lemma 5.2.
If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the estimate
$\sup_{|s|\leq{n}}{\bf E}\\{|U^{0}_{G}(s;z)|^{4}\\}=O(b^{-4})$ (5.22)
holds in the limit $n,b\rightarrow\infty$.
Proof of Lemma 5.2. Let us consider variable
${\bf
E}\\{U^{0}_{G_{1}}(x_{1})U^{0}_{G_{2}}(x_{2})U^{0}_{G_{3}}(x_{3})U^{0}_{G_{4}}(x_{4})\\}={\bf
E}[U^{0}_{G_{1}}(x_{1})U^{0}_{G_{2}}(x_{2})U^{0}_{G_{3}}(x_{3})]^{0}U_{G_{4}}(x_{4}).$
Set $T=U^{0}_{G_{1}}U^{0}_{G_{2}}U^{0}_{G_{3}}$ and
$M(x_{1},x_{2},x_{3},t)={\bf E}T^{0}G_{4}(t,t)$. We apply to $G_{4}(t,t)$ the
resolvent identity $(3.2)$ and obtain
${\bf E}T^{0}G_{4}(t,t)=-\zeta_{4}\sum_{|s|\leq{n}}{\bf
E}\\{T^{0}G_{4}(t,s)H(s,t)\\}.$
Applying (2.15) to ${\bf E}\\{T^{0}G_{4}(t,s)H(s,t)\\}$ with $q=3$ and taking
into account (2.25), we get relation
$\displaystyle{\bf E}T^{0}G_{4}(t,t)$ $\displaystyle=\zeta_{4}v^{2}{\bf
E}\\{T^{0}G_{4}(t,t)U_{G_{4}}(t)\\}+\zeta_{4}v^{2}{\bf
E}\left\\{T^{0}\sum_{|s|\leq{n}}G_{4}(t,s)^{2}U(s,t)\right\\}$
$\displaystyle+2\zeta_{4}v^{2}\sum_{(i,j,k)}{\bf
E}\left\\{U^{0}_{G_{i}}(x_{i})U^{0}_{G_{j}}(x_{j})\sum_{|y|,|s|\leq{n}}G_{k}(y,s)G_{k}(t,y)G_{4}(t,s)U(y,x_{k})U(s,t)\right\\}$
$\displaystyle+\zeta_{4}\Gamma_{1}(t)+\zeta_{4}\Gamma_{2}(t)$ (5.23)
with
$\Gamma_{1}(t)=-\sum_{|s|\leq{n}}\frac{K_{4}}{3!}{\bf
E}\left\\{D^{3}_{st}(T^{0}G_{4}(t,s))\right\\}$ (5.24)
and
$\displaystyle\Gamma_{2}(t)=$ $\displaystyle-\frac{1}{4!}\sum_{|s|\leq{n}}{\bf
E}\left\\{H(s,t)^{5}[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(0)}\right\\}$
$\displaystyle+\sum_{|s|\leq{n}}\frac{K_{2}}{3!}{\bf
E}\left\\{H(s,t)^{3}[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(1)}\right\\}$
$\displaystyle+\sum_{|s|\leq{n}}\frac{K_{4}}{3!}{\bf
E}\left\\{H(s,t)[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(2)}\right\\},$
(5.25)
where $K_{r}$, $r=2,4$ are the cumulants of $H(s,t)$ as in (3.5). In (5.23),
we introduce the notation
$\sum_{(i,j,k)}\xi(x_{i},x_{j},x_{k})=\xi(x_{1},x_{2},x_{3})+\xi(x_{1},x_{3},x_{2})+\xi(x_{2},x_{3},x_{1}).$
Applying to the first term of the RHS of (5.23) relation (3.11) and using
$q_{4}(t)$ (4.1), we obtain that
$\displaystyle{\bf E}T^{0}G_{4}(t,t)$ $\displaystyle=q_{4}(t)v^{2}{\bf
E}\\{T^{0}G_{4}(t,t)U^{0}_{G_{4}}(t)\\}+q_{4}(t)v^{2}{\bf
E}\left\\{T^{0}\sum_{|s|\leq{n}}G_{4}(t,s)^{2}U(s,t)\right\\}$
$\displaystyle+2q_{4}(t)v^{2}\sum_{(i,j,k)}{\bf
E}\left\\{U^{0}_{G_{i}}(x_{i})U^{0}_{G_{j}}(x_{j})\sum_{|y|,|s|\leq{n}}G_{k}(y,s)G_{k}(t,y)G_{4}(t,s)U(y,x_{k})U(s,t)\right\\}$
$\displaystyle+q_{4}(t)\left(\Gamma_{1}(t)+\Gamma_{2}(t)\right).$
Now gathering relation given by (2.1), (3.1), (3.2), (5.10), (4.4) and
$\sup_{|t|\leq{n}}{\bf E}|T^{0}U^{0}_{G_{4}}(t)|\leq{{\bf
E}|T|\sup_{|t|\leq{n}}{\bf E}|U^{0}_{G_{4}}(t)|}+\sup_{|t|\leq{n}}{\bf
E}|TU^{0}_{G_{4}}(t)|$
imply the following inequality
$\displaystyle|\sum_{|t|\leq{n}}M(x_{1},x_{2},x_{3},t)U(t,x_{4})|\leq$
$\displaystyle{\frac{v^{2}}{\eta^{2}}\sup_{|t|\leq{n}}{\bf
E}|TU^{0}_{G_{4}}(t)|+\frac{v^{2}}{\eta^{2}}{\bf E}|T|\sup_{|t|\leq{n}}{\bf
E}|U^{0}_{G_{4}}(t)|}$ $\displaystyle+\frac{2v^{2}}{\eta^{3}b}{\bf
E}|T|+\frac{6v^{2}}{\eta^{4}b^{2}}{\bf
E}|U^{0}_{G_{i}}(x_{i})U^{0}_{G_{j}}(x_{j})|$
$\displaystyle+\frac{1}{\eta}\sup_{|t|\leq{n}}|\Gamma_{1}(t)+\Gamma_{2}(t)|.$
(5.26)
Henceforth, for sake of clarity, we consider
$G=G_{1}=G_{3}=\bar{G}_{2}=\bar{G}_{4}$ and $x=x_{r}$, $r=1,\ldots,4$, then we
get $T=\left(U^{0}_{G}(x)\right)^{2}U^{0}_{\bar{G}}(x)$ and
${\bf E}|T|\leq{\left({\bf E}|U^{0}_{G}|^{4}\right)^{1/2}\left({\bf
E}|U^{0}_{G}|^{2}\right)^{1/2}}.$ (5.27)
Let us assume for the moment that
$\sup_{|t|\leq{n}}|\Gamma_{1}(t)+\Gamma_{2}(t)|=O\left(b^{-4}+b^{-2}\sqrt{W}\right),\quad
z\in\Lambda{\eta}$ (5.28)
with $W=\sup_{x}{\bf E}|U^{0}_{G}(x)|^{4}$. Now returning to (5.26) and
gathering estimates given by relations (4.40), (5.27) and (5.28) imply the
following estimate
$W\leq{A_{1}b^{-2}\sqrt{W}+A_{2}b^{-4}},$
where $A_{1}$, $A_{2}$ are some constants. This proves (5.22).
To complete the proof of Lemma 5.2, let us prove (5.28). To do this, we use
the following statement.
###### Lemma 5.3.
If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the estimates
$D^{r}_{st}\left(U^{0}_{G}(x)\right)=O(b^{-1}),\ \quad r=1,\ldots,4,$ (5.29)
$D^{r}_{st}\left(T^{0}\bar{G}(t,s)\right)=O\left(b^{-3}+b^{-2}|U^{0}_{G}(x)|+b^{-1}|U^{0}_{G}(x)|^{2}+|U^{0}_{G}(x)|^{3}\right),\quad
r=3,4$ (5.30)
and
${\bf E}|[U^{0}_{G}(x)]^{(\nu)}|^{2r}=O\left(b^{-3r}+{\bf
E}|U^{0}_{G}(x)|^{2r}\right),\ r=1,2$ (5.31)
hold for all $\nu=0,1,2$, all $|x|\leq{n}$ and large enough $n$ and $b$
satisfying (2.9).
We prove this Lemma at the end of this subsection.
Let us return to the proof of (5.28). Regarding the variable $\Gamma_{1}$
(5.24) and using (4.40), (4.42) and (5.30), one gets with the help of (5.27)
that
$\sum_{|s|\leq{n}}\frac{2[V_{4}+3v^{4}]}{3b}{\bf
E}|D^{3}_{st}(T^{0}\bar{G}(t,s))|U(s,t)=O\left(b^{-4}+b^{-2}\sqrt{W}\right).$
(5.32)
Now let us estimate $\Gamma_{2}$ (5.25). Regarding the first term of the RHS
of (5.25) and using (4.40), (5.30) and (5.31), we obtain inequality
$\displaystyle\sum_{|s|\leq{n}}{\bf
E}|H(s,t)^{5}[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(0)}|$
$\displaystyle\leq{c\sum_{|s|\leq{n}}{\bf
E}\left\\{\frac{|H(s,t)|^{5}}{b^{3}}+\frac{|H(s,t)|^{5}}{b^{2}}|[U_{G}^{0}(x)]^{(0)}|+\frac{|H(s,t)|^{5}}{b}|[U_{G}^{0}(x)]^{(0)}|^{2}\right\\}}$
$\displaystyle+c\sum_{|s|\leq{n}}{\bf
E}\left\\{|H(s,t)|^{5}|[U_{G}^{0}(x)]^{(0)}|^{3}\right\\}$
$\displaystyle\leq{c\sum_{|s|\leq{n}}\left[\frac{\mu_{5}}{b^{11/2}}\psi\left(\frac{s-t}{b}\right)+\frac{\mu_{10}^{1/2}}{b^{9/2}}\left({\bf
E}|[U_{G}^{0}(x)]^{(0)}|^{2}\right)^{1/2}\psi\left(\frac{s-t}{b}\right)^{1/2}\right]}$
$\displaystyle+c\sum_{|s|\leq{n}}\left[\frac{\mu_{10}^{1/2}}{b^{7/2}}\left({\bf
E}|[U_{G}^{0}(x)]^{(0)}|^{4}\right)^{1/2}\psi\left(\frac{s-t}{b}\right)^{1/2}\right]$
$\displaystyle=O\left(\frac{1}{b^{9/2}}+\frac{1}{b^{5/2}}\sqrt{W}\right).$
(5.33)
Repeating the arguments used to prove (5.33), it is easy to show that the term
$\sum_{|s|\leq{n}}\frac{K_{2}}{3!}{\bf
E}\left\\{H(s,t)^{3}[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(1)}\right\\}+\sum_{|s|\leq{n}}\frac{K_{4}}{3!}{\bf
E}\left\\{H(s,t)[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(2)}\right\\}$
is of the order indicated in the RHS of (5.28) and that
$\sup_{|t|\leq{n}}|\Gamma_{2}(t)|=O\left(b^{-4}+b^{-2}\sqrt{W}\right).$ (5.34)
Then the estimate (5.28) follows from (5.32) and (5.34). Lemma 5.2 is proved.
Proof of Lemma 5.3. We prove Lemma 5.2 with $r=1$ because the general case
does not differ from this one. We start with the proof of (5.29). Using
(2.25), we obtain that
$D^{1}_{st}\left(U^{0}_{G}(x)\right)=-2\sum_{|k|\leq{n}}G(k,s)G(k,t)U(k,x).$
Then estimate (5.29) (with $r=1$) follows from this relation and inequality
$|U(k,x)|\leq{b^{-1}}$ and (3.17) (with $m=1$). The general case does not
differ from this one, so the estimate (5.29) is proved.
Let us prove (5.30). Remembering that $T=[U^{0}_{G}(x)]^{2}U^{0}_{\bar{G}}(x)$
and using (2.25) and (5.29), we obtain that
$D^{1}_{st}\\{T^{0}\\}=O(b^{-1}|U^{0}_{G}(x)|^{2}),$
$D^{2}_{st}\\{T^{0}\\}=O\left(b^{-2}|U^{0}_{G}(x)|+b^{-1}|U^{0}_{G}(x)|^{2}\right),$
$D^{3}_{st}\\{T^{0}\\}=O\left(b^{-3}+b^{-2}|U^{0}_{G}(x)|+b^{-1}|U^{0}_{G}(x)|^{2}\right).$
Now it is easy to show that (5.30) is true.
Finally, we prove (5.31) with $r=1$ because the general case does not doffer
from this one. To simplify computation, we use the notation: for each pair
$(s,t)$ and $\nu=0,1,2$, let $H^{(\nu)}_{st}=H^{(\nu)}=\hat{H}$ be the matrix
defined by
$\hat{H}(r,i)=\left\\{\begin{array}[]{lll}H(r,i),&\textrm{if}&(r,i)\neq(s,t);\\\
\hat{H}(s,t),&\textrm{if}&(r,i)=(s,t)\end{array}\right.$
with $|\hat{H}(s,t)|\leq{|H(s,t)|}$ and its resolvent by
$G^{(\nu)}_{sp}(z)=\hat{G}(z)$. Then the resolvent identity (2.23) imply that
$\displaystyle U_{\hat{G}}(x)=$ $\displaystyle
U_{G}(x)-\frac{1}{b}\sum_{|k|,|r|,|i|\leq{n}}\hat{G}(k,r)\\{\hat{H}-H\\}(r,i)G(i,k)\psi\left(\frac{x-k}{b}\right)$
$\displaystyle=$ $\displaystyle
U_{G}(x)-\frac{1}{b}\sum_{|k|\leq{n}}B(k,s,t)\psi\left(\frac{x-k}{b}\right)$
with $B(k,s,t)=\hat{G}(k,s)[\hat{H}(s,t)-H(s,t)]G(t,k)$. Then inequality (3.1)
implies that
$\displaystyle{\bf E}|U^{0}_{\hat{G}}(x)|^{2}$ $\displaystyle\leq{2{\bf
E}|U^{0}_{G}(x)|^{2}+\frac{2}{b^{2}}{\bf
E}\left|\sum_{|k|\leq{n}}B^{0}(k,s,t)\psi\left(\frac{x-k}{b}\right)\right|^{2}}$
$\displaystyle\leq{2{\bf E}|U^{0}_{G}(x)|^{2}+\frac{8}{\eta^{4}b^{2}}{\bf
E}\left(|\hat{H}(s,t)|+{\bf E}|H(s,t)|\right)^{2}}$ $\displaystyle\leq{2{\bf
E}|U^{0}_{G}(x)|^{2}+\frac{8}{\eta^{4}b^{3}}\left[{\bf
E}|a(s,p)|^{2}\psi\left(\frac{s-p}{b}\right)+3\left({\bf
E}|a(s,p)|\right)^{2}\psi\left(\frac{s-p}{b}\right)^{2}\right]}.$
This proves (5.31). Lemma 5.3 is proved. $\hfill\blacksquare$
### 5.3 Proof of Lemma 4.3.
We prove relation (4.17) with $k=1$ because the general case does not differ
from this one. To derive relations for the average value of the variable
$t_{12}(i,s)={\bf E}G_{1}(i,s)G_{2}(i,s)$, we use identity (2.23) and relation
(2.15) (with $q=3$) and repeat the proof of relation (4.14). Simple
computations lead to
$\displaystyle t_{12}(i,s)=$
$\displaystyle\zeta_{2}g_{1}(i)\delta_{is}+\zeta_{2}v^{2}t_{12}(i,s)U_{g_{2}}(s)$
$\displaystyle+\zeta_{2}v^{2}\sum_{|p|\leq{n}}t_{12}(i,p)g_{1}(s)U(p,s)+\sum_{j=1}^{6}\gamma_{j}(i,s),$
(5.35)
with
$\displaystyle\gamma_{1}(i,s)=$
$\displaystyle\zeta_{2}v^{2}\sum_{|p|\leq{n}}{\bf
E}\\{G_{1}(i,s)G_{2}(i,p)G_{1}(p,s)\\}U(p,s),$ $\displaystyle\gamma_{2}(i,s)=$
$\displaystyle\zeta_{2}v^{2}\sum_{|p|\leq{n}}{\bf
E}\left\\{G_{1}(i,s)G_{2}(i,p)G_{2}(p,s)\right\\}U(p,s)$
$\displaystyle\gamma_{3}(i,s)=$ $\displaystyle\zeta_{2}v^{2}{\bf
E}\\{G_{1}(i,s)G_{2}(i,s)U^{0}_{G_{1}}(s)\\},$ $\displaystyle\gamma_{4}(i,s)=$
$\displaystyle\zeta_{2}v^{2}{\bf
E}\left\\{G^{0}_{1}(s,s)\sum_{|p|\leq{n}}G_{1}(i,p)G_{2}(i,p)U(p,s)\right\\},$
$\displaystyle\gamma_{5}(i,s)=$
$\displaystyle-\frac{\zeta_{2}}{6}\sum_{|p|\leq{n}}K_{4}{\bf
E}\left\\{D^{3}_{ps}(G_{1}(i,s)G_{2}(i,p))\right\\}$
and
$\displaystyle\gamma_{6}(i,s)=$
$\displaystyle-\frac{\zeta_{2}}{4!}\sum_{|p|\leq{n}}{\bf
E}\left\\{H(p,s)^{5}[D^{4}_{ps}\left(G_{1}(i,s)G_{2}(i,p)\right)]^{(0)}\right\\}$
$\displaystyle+\frac{\zeta_{2}}{3!}\sum_{|p|\leq{n}}K_{2}{\bf
E}\left\\{H(p,s)^{3}[D^{4}_{ps}\left(G_{1}(i,s)G_{2}(i,p)\right)]^{(1)}\right\\}$
$\displaystyle+\frac{\zeta_{2}}{3!}\sum_{|p|\leq{n}}K_{4}{\bf
E}\left\\{H(p,s)[D^{4}_{ps}\left(G_{1}(i,s)G_{2}(i,p)\right)]^{(2)}\right\\},$
where $K_{r}$, $r=2,4$ are the cumulants of $H(p,s)$ as in (3.5). Using
(3.17), it is easy to show that
$\sup_{|i|,|s|\leq{n}}|\gamma_{1}(i,s)|=o(b^{-1}),\quad\sup_{|i|\leq{n}}|\sum_{|s|\leq{n}}\gamma_{1}(i,s)|=o(b^{-1}).$
The same is valid for $\gamma_{2}$. Similar estimates for $\gamma_{3}$,
$\gamma_{4}$, $\gamma_{5}$ and $\gamma_{6}$ follow from relations (4.40),
(4.45) and simple arguments as those to the proof of (4.20) (see
(4.42)-(4.44)). Thus, (5.35) implies that
$t_{12}(i,s)=g_{1}(i)q_{2}(i)\delta_{is}+v^{2}g_{1}(s)q_{2}(s)\\{t_{12}U\\}(i,s)+\Delta(i,s),$
(5.36)
where
$\sup_{|i|,|s|\leq{n}}|\Delta(i,s)|=o(1)\quad\hbox{ and
}\quad\sup_{|i|\leq{n}}|\sum_{|s|\leq{n}}\Delta(i,s)|=o(1)$ (5.37)
in the limit $n,b\rightarrow\infty$. We rewrite relation (5.36) in the matrix
form (cf. (4.25))
$t_{12}=\\{I-W^{(g_{1},q_{2})}\\}^{-1}(Diag(g_{1}q_{2})+\Delta)=\sum_{m=0}^{+\infty}\\{W^{(g_{1},q_{2})}\\}^{m}(Diag(g_{1}q_{2})+\Delta).$
(5.38)
Now we can apply to (5.38) the same arguments as in the proof of (4.14).
Replacing $g_{1}$ and $q_{2}$ by $w_{1}$ and $w_{2}$, respectively, we derive
from (5.37) that for $i\in B_{L+Q}$,
$t_{12}(i,s)=\sum_{m=0}^{M}v^{2m}(w_{1}w_{2})^{m+1}[U^{m}](i,s)+o(1),\quad
n,b\rightarrow\infty.$ (5.39)
Multiplying both sides of (5.39) by $U(s,i)$ and summing over $s$, we obtain
the relation
$\sum_{|s|\leq{n}}t_{12}(i,s)U(s,i)=\sum_{m=0}^{M}v^{2m}(w_{1}w_{2})^{m+1}[U^{m+1}](i,i)+o(1),\
N,b\rightarrow\infty.$
Now convergence (4.37) implies the relation that leads, with $M$ replaced by
$\infty$, to (4.17).
To prove (4.18), let us sum (5.39) over $s$. The second part of (5.37) tells
us that the terms $\Delta$ remain small when summed over $s$. Thus we can
write relations
$\sum_{|s|\leq{n}}t_{12}(i,s)=\sum_{m=0}^{M}(v^{2}w_{1}w_{2})^{m+1}\sum_{|s|\leq{n}}[U^{m}](i,s)+o(1),\
N,b\rightarrow\infty.$ (5.40)
Taking into account estimates for terms (4.35)-(4.36) (see previous work [1]
for more details), it is easy to observe that convergence (4.34) together with
(5.40) imply (4.18). Finally, we prove (4.16). To derive relations for the
average value of variable $t_{11}(i,s)={\bf E}G_{1}(i,s)G_{1}(i,s)$, we repeat
the proof of (4.18) and replace $G_{2}$ by $G_{1}$. Then one obtains (4.16).
Lemma 4.3 is proved. $\hfill\blacksquare$
## 6 Asymptotic properties of $T(z_{1},z_{2})$
The asymptotic expression for $T(z_{1},z_{2})$ regarded in the limit
$z_{1}=\lambda_{1}+i0$, $z_{2}=\lambda_{2}+i0$ supplies one with the
information about the local properties of eigenvalue distribution provided
that $\lambda_{1}-\lambda_{2}=O(N^{-1})$. Indeed, according to (2.5), the
formal definition of the eigenvalue density
$\rho_{n,b}(\lambda)=\sigma^{{}^{\prime}}_{n,b}(\lambda)$ is
$\rho_{n,b}(\lambda)=\frac{1}{2i}[g_{n,b}(\lambda+i0)-g_{n,b}(\lambda-i0)].$
We consider the density-density correlation function of $\rho_{n,b}$
$R_{n,b}(\lambda_{1},\lambda_{2})=-\frac{1}{4}\sum_{\delta_{1},\delta_{2}=-1,1}\delta_{1}\delta_{2}C_{N,b}(\lambda_{1}+i\delta_{1}0,\lambda_{2}+i\delta_{2}0).$
In general, even if $R_{n,b}$ can be rigorously determined, it is difficult to
carry out direct study of it. Taking into account relation $(2.13)$, one can
simpler-expression
$\Xi_{n,b}(\lambda_{1},\lambda_{2})=-\frac{1}{4Nb}\sum_{\delta_{1},\delta_{2}=-1,+1}\delta_{1}\delta_{2}T(\lambda_{1}+i\delta_{1}0,\lambda_{1}+i\delta_{1}0)$
(6.1)
and assume that it corresponds to the leading term to
$R_{n,b}(\lambda_{1},\lambda_{2})$ in the limit $n,b\rightarrow\infty$.
It should be noted that for Wigner random matrices this approach is justified
by the study of the simultaneous limiting transition $N\rightarrow\infty$,
$\mathrm{Im}z_{j}\rightarrow 0$ in the studies of $C_{N}(z_{1},z_{2})$ [5, 6,
10, 18].
###### Theorem 6.1.
Let $T(z_{1},z_{2})$ is given by (2.12). Assume that function $\hat{\psi}(p)$
is such that there exist positive constants $c_{1}$, $\delta$ and $v>1$ that
$\hat{\psi}(p)=\hat{\psi}(0)-c_{1}|p|^{\nu}+o(|p|^{\nu})$ (6.2)
for all $p$ such that $|p|\leq{\delta}$, $\delta\rightarrow 0$. Then
$\Xi_{n,b}(\lambda_{1},\lambda_{2})=\frac{1}{Nb}\frac{c_{2}}{|\lambda_{1}-\lambda_{2}|^{2-1/v}}(1+o(1))$
(6.3)
for $\lambda_{j}$, $j=1,2$ satisfying
$\lambda_{1},\lambda_{2}\rightarrow\lambda\in(-2v,2v).$ (6.4)
We see from (2.12) that there are two terms in $T(z_{1},z_{2})$. The first was
found in [16] for band random matrices, the second coincides with that found
in [19] for the ensemble of Wigner random matrices. The proof of (6.3)
consists of two parts already done in [16] and [19]. For completeness, we
reproduce here these computations.
Proof of Theorem 6.1. Let us start with the term of (6.1) that correspond to
$\delta_{1}\delta_{2}=-1$. It follows from (2.7) that
$\frac{1-v^{2}w_{1}w_{2}}{w_{1}w_{2}}=\frac{z_{1}-z_{2}}{w_{1}-w_{2}}.$ (6.5)
The above identity yields relations
$\epsilon|w(\lambda+i\epsilon)|^{2}=\mathrm{Im}w(\lambda+i\epsilon)(1-v^{2}|w(\lambda+i\epsilon)|^{2})\quad\hbox{
and }\quad|w(\lambda+i0)|^{2}=v^{-2}$
for $\lambda$ such that $\mathrm{Im}{w}(\lambda+i0)>0$. Combining these
relations with (1.4) for the real and imaginary parts of
$w(\lambda+i0)=\tau(\lambda)+i\rho(\lambda)$, we obtain that
$v^{2}\tau^{2}=\frac{\lambda^{2}}{4v^{2}}\quad\hbox{ and }\quad
v^{2}\rho^{2}=1-\frac{\lambda^{2}}{4v^{2}}$ (6.6)
(here and below we omit the variable $\lambda$). This implies the existence of
the limits $w(z_{1})=\overline{w(z_{2})}$ for (6.4). One can easily deduce
from (6.5) that in the limit (6.4)
$\frac{1-v^{2}w(z_{1})w(z_{2})}{w(z_{1})w(z_{2})}=\frac{\lambda_{1}-\lambda_{2}}{2i\rho}=o(1).$
(6.7)
Also we have that
$(1-v^{2}w^{2}_{1})(1-v^{2}w^{2}_{2})=2-2v^{2}(\tau^{2}-\rho^{2})=4v^{2}\rho^{2}.$
(6.8)
Now let us consider the leading term of the correlation function. Rewrite
(2.12) as
$\displaystyle T(z_{1},z_{2})=$ $\displaystyle
Q(z_{1},z_{2})+Q^{{}^{\prime}}(z_{1},z_{2})+\frac{2v^{2}Q(z_{1},z_{2})}{(1-v^{2}w^{2}_{1})(1-v^{2}w^{2}_{2})}$
$\displaystyle=$
$\displaystyle\frac{2v^{2}S(z_{1},z_{2})}{(1-v^{2}w^{2}_{1})(1-v^{2}w^{2}_{2})}+Q^{{}^{\prime}}(z_{1},z_{2})$
with
$S(z_{1},z_{2})=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{w^{2}_{1}w^{2}_{2}\hat{\psi}(p)}{(1-v^{2}w_{1}w_{2}\hat{\psi}(p))^{2}}dp$
(6.9)
and
$Q^{{}^{\prime}}(z_{1},z_{2})=\frac{2\Delta
v^{4}w_{1}^{3}w_{2}^{3}}{(1-v^{2}w_{1}^{2})(1-v^{2}w_{2}^{2})},$ (6.10)
where $\Delta$ is given by (2.14). It is easy to observe that relations (6.6)
and $|w(\lambda+i0)|^{2}=|w(\lambda-i0)|^{2}=1/v^{2}$ imply that (cf. [19])
$Q^{{}^{\prime}}(\lambda_{1}+i0,\lambda_{2}-i0)+Q^{{}^{\prime}}(\lambda_{1}-i0,\lambda_{2}+i0)=\frac{\Delta}{v^{4}\rho^{2}}.$
(6.11)
Now let us consider $S(z_{1},z_{2})$ (6.9) and let us write
$S(z_{1},z_{2})=\frac{1}{2\pi}\left\\{\int_{-\delta}^{\delta}+\int_{\mathbf{R}\setminus(-\delta,\delta)}\right\\}\frac{w^{2}_{1}w^{2}_{2}\hat{\psi}(p)}{(1-v^{2}w_{1}w_{2}\hat{\psi}(p))^{2}}dp=I_{1}+I_{2}.$
Relation (6.5) and (6.7) imply equality
$[1-v^{2}w_{1}w_{2}\hat{\psi}(p)]^{2}=[\hat{\Psi}(p)-1]^{2}(1+o(1)).$ (6.12)
Since $\psi(t)$ is monotone, then
$\liminf_{p\in\mathbf{R}\setminus(-\delta,\delta)}[\Psi(p)-1]^{2}>0.$
This means that $I_{2}<\infty$ in the limit (6.4). Relations (6.2), (6.7) and
(6.12) imply in the limit (6.4) and that if we take
$\delta|\lambda_{1}-\lambda_{2}|^{-1/\nu}\rightarrow\infty$, we obtain
asymptotically (cf. [16])
$I_{1}(\lambda_{1}+i0,\lambda_{2}-i0)+I_{1}(\lambda_{1}-i0,\lambda_{2}+i0)=4B_{v}(c_{1})\frac{(2v\rho)^{2-1/\nu}}{|\lambda_{1}-\lambda_{2}|^{2-1/\nu}},$
(6.13)
where
$B_{v}(c_{1})=\frac{1}{2\pi
c^{1/\nu}_{1}}\left[\int_{0}^{\infty}\frac{ds}{1+s^{2\nu}}-2\int_{0}^{\infty}\frac{ds}{(1+s^{2\nu})^{2}}\right]$
(6.14)
and $c_{1}$ is as in (6.2). To prove (6.3), it remains to consider the sum
$I_{1}(\lambda_{1}+i0,\lambda_{2}-i0)+I_{2}(\lambda_{1}-i0,\lambda_{2}+i0).$
It is easy to observe that relations of the form (6.8) imply the bounded ness
of this sum in the limit (6.4).
Now gathering relations (6.8), (6.11) and (6.13), we derive that
$\Xi_{n,b}(\lambda_{1},\lambda_{2})=\frac{1}{Nb}\frac{B_{\nu}(c_{1})}{(2v\rho)^{1/\nu}}\frac{1}{|\lambda_{1}-\lambda_{2}|^{2-1/\nu}}(1+o(1)).$
(6.15)
This proves (6.3). $\hfill\blacksquare$
Let us discuss two consequences of Theorem 6.1.
* $\bullet$
If $\nu=2$ and $c_{1}=\int t^{2}\psi(t)<\infty$. Regarding the RHS of (2.11)
in the limit (6.4) with $\lambda_{j}=\lambda+\frac{r_{j}}{N}$, $j=1,2$, we
obtain the asymptotic relation (see [16])
$\displaystyle\Xi(\lambda_{1},\lambda_{2})$
$\displaystyle=-\frac{B_{2}(c_{1})}{2\sqrt{2}(v^{2}\rho)^{1/2}}\frac{\sqrt{N}}{b}\frac{1}{|r_{1}-r_{2}|^{3/2}}(1+o(1))$
$\displaystyle=-C\frac{\sqrt{N}}{b}\frac{1}{|r_{1}-r_{2}|^{3/2}}(1+o(1)),\quad
C>0.$ (6.16)
* $\bullet$
If $\Psi(t)=O(|t|^{-1-\nu})$ with $1<\nu<2$, we obtain the asymptotic relation
(see [16])
$\Xi(\lambda_{1},\lambda_{2})=\frac{B_{\nu}(c_{1})}{(2v^{2}\rho)^{1/\nu}}\frac{N^{1-1/\nu}}{b}\frac{1}{|r_{1}-r_{2}|^{2-1/\nu}}(1+o(1))$
(6.17)
and conclude that the expression for (6.1) is proportional to
$\frac{N^{1-1/\nu}}{b}\frac{1}{|r_{1}-r_{2}|^{2-1/\nu}}.$
The form of asymptotic expressions (6.16) and (6.17) coincides with the
expressions determined by Khorunzhy and Kirsch (see [16]) for the spectral
correlation function of band random matrices [16].
The first conclusion is that the leading terms of the ensemble we study (see
(2.3)) and the ensemble of band random matrices are different but in the local
scale, the form (6.16) and (6.17) is the same. More precisely, the tow
ensembles mentioned above belong to the same class of spectral universality.
Our main conclusion is that the limiting expression for
$\Xi_{n,b}(\lambda_{1},\lambda_{2})$ exhibits different behavior depending on
the rate of decay of $\psi(t)$ at infinity. In both cases (see (6.16) and
(6.17)) the exponents do not depend on the particular form of the function
$\psi(t)$. Moreover, in the first case the exponents do not depend on $\psi$
at all. This can be regarded as a kind of spectral universality for the random
matrix ensembles $\\{H_{n,b}\\}$ (2.3). One can deduce that these
characteristics also do not depend on the probability distribution of the
random variables $a(i,j)$ (1.1).
## References
* [1] S. Ayadi: Semicircle Law For Random Matrices Of Long-Range Percolation Model. Arxiv PR/0806.4497v1, to appear in Random Operators and Stochastic Eqs. N4, Volume 16, (2009).
* [2] S. Ayadi: Asymptotic properties of random matrices of long-range percolation model. (submited in ROSE)
* [3] D. Bessis, C. Itzykson, J. B. Zuber. Quantum field theory thechniques in graphical enumeration. Adv. Appl. Math. 1, 109-157 (1980)
* [4] P. Bleher and A. Its. Semiclassical asymptotics of orthogonal polynomials, Rieman-Hilbert problem, and universality in the matrix model. Annals of Mathematics, 150, 185-266 (1999)
* [5] A. Boutet de Monvel, Khorunzhy: Asymptotic distribution of smoothed eigenvalue density: I. Gaussian random matrices, Random Oper. Stoch. Eqs. 7, 1-22 (1999) II. Wigner random matrices, Random Oper. Stoch. Eqs. 7, 149-167 (1999)
* [6] E. Brézin, A. Zee: Universality of the correlations between eigenvalues of large random matrices.Nucl. Phys. B 402 no. 3, 613-627 (1993); Ambjorn J, Jurkiewicz J, Makeenko Yu M.:Multiloop correlators for two-dimensional quantum gravity. Phys. lett. B 251 (4), 517-524 (1990)
* [7] G. Casati, L. Molinari, F. Izrailev. Scaling properties of band random matrices. Phys. Rev. Lett. 64 1851 (1990)
* [8] D. Coppersmith, D. Gamarnik, M. I. Sviridenko: The diametre of long-range percolation graph. In Mathematics and Computer Science II. Trends Math., Birkhauser, Basel, 147-159 (2002)
* [9] A. Crisanti, G. Paladin, A. Vulpiani. Products of Random Matrices in Statistical Physics. Berlin: Springer, (1993)
* [10] F. J. Dyson: Statistical theory of the energy levels of complex systems (III).J.Math. Phys 3, 166-175 (1962)
* [11] P. A. Deift, A. Its, X. Zhou. A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math 146, 149-235 (1997)
* [12] T. Guhr, A. Müller-Groeling, H. A. Weidenmüller. Random -matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189-425 (1998)
* [13] Y. V. Fyodorov, A. D. Mirlin. Scaling properties of localization in random band matrices. A $\sigma$ model approach. Phys. Rev. Lett. 67, 2405 (1991)
* [14] F. Haake. Quantum Signatures of Chaos. Berlin: Springer, (1991)
* [15] S. Janson, T. Luczak, A. Rucinski. Random Graphs. John Wiles and Sons, Inc. New York. (2002)
* [16] A. Khorunzhy, W. Kirsch: On Asymptotic Expansions and Scales of Spectral Universality in Band Random Matrix Ensembles.Commun. Math. Phys. 231, 223-255 (2002)
* [17] O. Khorunzhiy, W. Kirsch, P. Müller. Lifshitz tails for spectra of Erdős-Rènyi random graphs. Ann. Appl. Probab. Volume 16, Number 1, 295-309, (2006)
* [18] A. Khorunzhy: On smoothed density of states for Wigner random matrices. Rand. Oper. Stoch. Eqs. 5, 147-162 (1997)
* [19] A. Khorunzhy, B. Khoruzhenko, L. Pastur: Asymptotic properties of large random matrices with independent entries.J.Math. Phys. 37 , 5033-5060 (1996)
* [20] A. Khorunzhy, B. Khoruzhenko, L. Pastur, M. Shcherbina. Large-n limit in statistical mechanics and the spectral theory of disordered systems. In Phase Transitions and Critical Phenomena, Vol.15, edg C.Domb and J.L.Lebowitz.Academic Press, London, pp. 73-239, (1992)
* [21] A. Khorunzhy, L. Pastur: On the eigenvalue distribution of the deformed Wigner ensemble of random matrices. Adv. Soviet. Math. 19, 97-107 (1994)
* [22] V. Marchenko, L. Pastur. Math. USSR-sb 1, 457 (1967)
* [23] V. Marchenko, L. Pastur: Eigenvalue distribution of some class of random matrices. Matem. Sbornik. 72, 507 (1972)
* [24] M. L. Mehta: Random matrices, 2nd ed. Academic, New York, (1991)
* [25] S. A. Molchanov, L. Pastur, A. Khorunzhy: Eigenvalue distribution for band random matrices in the limit of their infinite rank. Teor. Matem. Fizika 99, (1992)
* [26] L. A. Pastur. Theor. Math.Phys. 10, 67 (1972)
* [27] C. Porter: Statistical Theories of Spectra: Fluctuations. New York: Acad. Press, (1965)
* [28] A. B. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207, 697-733 (1999)
* [29] P. Sylvestrov. Summing graphs for random band matrices. Phys. Rev. E 55, 6419-6432 (1997)
* [30] D. Voiculescu, K. J. Dykema, A. Nica. Free Random Variables, A noncommutative probability approch to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. CRM Monograph Series, 1. Providence, RI: AMS, 1992
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|
arxiv-papers
| 2009-04-18T09:26:39 |
2024-09-04T02:49:01.982666
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Slim Ayadi",
"submitter": "Ayadi Slim",
"url": "https://arxiv.org/abs/0904.2837"
}
|
0904.2839
|
# On the classification of unstable $H^{\ast}V-A$-modules
Dorra BOURGUIBA 111 pris en charge par l’unité de recherche 00/UR/15-05.
Faculté de Sciences–Mathématiques, Université de Tunis, TN-1060 Tunis,
Tunisie. e-mail: dorra.bourguiba@fst.rnu.tn
###### Abstract
In this work, we begin studying the classification, up to isomorphism, of
unstable $\mathrm{H}^{\ast}V-A$-modules $E$ such that
$\mathbb{F}_{2}\otimes_{\mathrm{H}^{\ast}V}E$ is isomorphic to a given
unstable $A$-module $M$. In fact this classification depends on the structure
of $M$ as unstable $A$-module. In this paper, we are interested in the case
$M$ a nil-closed unstable $A$-module and the case $M$ is isomorphic to
$\sum^{n}\mathbb{F}_{2}$. We also study, for $V=\mathbb{Z}/2\mathbb{Z}$, the
case $M$ is the Brown-Gitler module $\mathrm{J}(2)$.
## 1 Introduction
Let $V$ be an elementary abelian 2-group of rank $d$, that is a group
isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{d},\;d\in\mathbb{N}$, $BV$ be a
classifying space for the group $V$ and
$\mathrm{H}^{\ast}V=H^{\ast}(BV;\mathbb{F}_{2})$. We recall that
$\mathrm{H}^{\ast}V$ is an $\mathbb{F}_{2}$-polynomial algebra
$\mathbb{F}_{2}[t_{1},\ldots,t_{d}]$ on $d$ generators $t_{i},1\leq i\leq d$,
of degree one.
Let $A$ be the mod.2 Steenrod algebra and $\mathcal{U}$ the category of
unstable $A$-modules. We recall that $\mathrm{H}^{\ast}V-\mathcal{U}$ is the
category whose objects are unstable $\mathrm{H}^{\ast}V-A$-modules and
morphisms are $\mathrm{H}^{\ast}V$-linear and $A$-linear maps of degree zero.
For example, the mod.2 equivariant cohomology of a $V$-CW-complex, which is
the cohomology of the Borel construction, is an unstable
$\mathrm{H}^{\ast}V-A$-module.
Let $E$ be an unstable $\mathrm{H}^{*}V-A$-module, we denote by $\overline{E}$
the unstable $A$-module
$\mathbb{F}_{2}\otimes_{\mathrm{H}^{\ast}V}E=E/\widetilde{\mathrm{H}^{*}}V.E$,
where $\widetilde{\mathrm{H}^{*}}V$ denotes the augmentation ideal of
$\mathrm{H}^{*}V$ .
We have the following problem:
$\mathcal{(P)}$ : Let $M$ be an unstable A-module.
Classify, up to isomorphism, unstable $\mathrm{H}^{\ast}V-A$-modules
such that $\overline{E}\cong M$ (as unstable $A$-modules).
It is clear that, for every subgroup $W$ of $V$, the unstable
$\mathrm{H}^{\ast}V-A$-module:
$\mathrm{H}^{\ast}W\otimes M$
is a solution for the problem $\mathcal{(P)}$.
For $W=0$, a solution of $\mathcal{(P)}$ is given by the unstable
$\mathrm{H}^{\ast}V-A$-module $M$ which is trivial as an
$\mathrm{H}^{\ast}V$-module.
For $W=V$, a solution of $\mathcal{(P)}$ is given by the unstable
$\mathrm{H}^{\ast}V-A$-module $\mathrm{H}^{\ast}V\otimes M$ which is free as
an $\mathrm{H}^{\ast}V$-module.
If $V=\mathbb{Z}/2\mathbb{Z}$ and $M=\Sigma N$ a suspension of an unstable
$A$-module $N$, then we have, at least, the following two solutions of the
problem $\mathcal{(P)}$ which are free as
$H^{*}(\mathbb{Z}/2\mathbb{Z})$-modules:
1. 1.
$\Sigma(H^{*}(\mathbb{Z}/2\mathbb{Z})\otimes N)$.
2. 2.
$((H^{*}(\mathbb{Z}/2\mathbb{Z})^{\geq 1})\otimes N$.
These two solutions are different as unstable $A$-modules (here
$H^{*}(\mathbb{Z}/2\mathbb{Z})^{\geq 1}$ is the sub-algebra of
$H^{*}(\mathbb{Z}/2\mathbb{Z})$ of elements of degree bigger than or equal to
one). This shows that the solutions of the problem $\mathcal{(P)}$ i.e. the
classification, up to isomorphism, of unstable $\mathrm{H}^{\ast}V-A$-modules
such that $\overline{E}\cong M$ (as unstable $A$-modules), depends on the
structure of $E$ as an $\mathrm{H}^{\ast}V$-module and on the structure of $M$
as unstable $A$-module.
In this paper we will discuss the solutions of $\mathcal{(P)}$ if $M$ is a
nil-closed unstable $A$-module and $E$ is free as an $H^{*}V$-module and the
solutions of $\mathcal{(P)}$ if $M$ is isomorphic to $\sum^{n}\mathbb{F}_{2}$
or to $\mathrm{J}(2)$ and $E$ is free as an $H^{*}V$-module .
We begin by proving the following result (which is solution of $(\mathcal{P})$
when $M$ is a nil-closed unstable $A$-module ).
###### Theorem 1.1.
Let $E$ be unstable $\mathrm{H}^{\ast}V-A$-module which is free as an
$\mathrm{H}^{\ast}V$-module. If $\overline{E}$ is a nil-closed unstable
$A$-module, then there exists two reduced $\mathcal{U}$-injectives
$I_{0},\;I_{1}$ and an $\mathrm{H}^{\ast}V-A$-linear map
$\varphi:\mathrm{H}^{\ast}V\otimes I_{0}\rightarrow\mathrm{H}^{\ast}V\otimes
I_{1}$ such that:
1. 1.
$E\cong ker\varphi$
2. 2.
$\overline{E}\cong ker\overline{\varphi}$
The proof of this result is based on the classification of
$\mathrm{H}^{\ast}V-\mathcal{U}$-injectives and on some properties of the
injective hull in the category $\mathrm{H}^{\ast}V-\mathcal{U}$.
Our work is naturally motivated by topology as shown in the study of homotopy
fixed points of a $\mathbb{Z}/2$-action (see [L1]). Let $X$ be a space
equipped with an action of $\mathbb{Z}/2$ and $X^{\mathrm{h}\mathbb{Z}/2}$
denote the space of homotopy fixed points of this action. The problem of
determining the $\bmod{.\hskip 2.0pt2}$ cohomology of
$X^{\mathrm{h}\mathbb{Z}/2}$ (we ignore deliberately the questions of
$2$-completion) involves two steps:
* –
determining the $\bmod{.\hskip 2.0pt2}$ equivariant cohomology
$\mathrm{H}^{*}_{\mathbb{Z}/2}X$;
* –
determining $\mathrm{Fix}_{\mathbb{Z}/2}\hskip
2.0pt\mathrm{H}^{*}_{\mathbb{Z}/2}X$ (for the definition of the functor
$\mathrm{Fix}_{\mathbb{Z}/2}$ see section 2).
For the first step, see for example [DL], the main information one has about
the $\mathbb{Z}/2$-space $X$ is that the Serre spectral sequence, for
$\bmod{.\hskip 2.0pt2}$ cohomology, associated to the fibration
$X\rightarrow X_{\mathrm{h}\mathbb{Z}/2}\rightarrow\mathrm{B}\mathbb{Z}/2$
collapses ($X_{\mathrm{h}\mathbb{Z}/2}$ denotes the Borel construction
$\mathrm{E}\mathbb{Z}/2\times_{\mathbb{Z}/2}X$). This collapsing implies that
$\mathrm{H}^{*}_{\mathbb{Z}/2}X$ is $\mathrm{H}$-free and that
$\overline{\mathrm{H}^{*}_{\mathbb{Z}/2}X}$ is canonically isomorphic to
$\mathrm{H}^{*}X$. This gives clearly a topological application of problem
$(\mathcal{P})$.
We then prove the following results (related to the case $\overline{E}$ is
$\sum^{n}\mathbb{F}_{2}$ and $\mathrm{J}(2)$).
###### Theorem 1.2.
Let $E$ be unstable $\mathrm{H}^{\ast}V-A$-module which is free as an
$\mathrm{H}^{\ast}V$-module. If $\overline{E}$ is isomorphic to
$\sum^{n}\mathbb{F}_{2}$, then there exists an element $u$ in
$\mathrm{H}^{\ast}V$ such that:
1. 1.
$u=\displaystyle\prod_{i}\theta_{i}^{\alpha_{i}}$, where
$\theta_{i}\in(\mathrm{H}^{1}V)\setminus\\{0\\}$ and $\alpha_{i}\in\mathbb{N}$
2. 2.
$E\cong\sum^{d}u\mathrm{H}^{\ast}V$ with $d+\displaystyle\sum_{i}\alpha_{i}=n$
###### Proposition 1.3.
Let $E$ be an $\mathrm{H}-A$-module which is $\mathrm{H}$-free and such that
$\overline{E}$ is isomorphic to $\mathrm{J}(2)$ then:
$E\cong\mathrm{H}\otimes\mathrm{J}(2)$
or
$E$ is the sub-$\mathrm{H}-A$-module of $\mathrm{H}\oplus\sum\mathrm{H}$
generated by $(t,\Sigma 1)$ and $(t^{2},0)$.
The proofs of these two results are based on Smith theory, some properties of
the functor $\mathrm{F}ix$ and on a result of J.P. Serre.
The paper is structured as follows. In section 2, we introduce the definitions
of reduced and nil-closed unstable $A$-modules. We give the classification of
injective modules in the category $\mathcal{U}$ and in the category
$H^{*}V-\mathcal{U}$. We also recall the algebraic Smith theory. In section 3,
we establish some properties of $E$ when $\overline{E}$ is a reduced unstable
$A$-module. The results will be useful in section 4, where we give the
solutions of the problem ($\mathcal{P}$) when $E$ is free as an
$\mathrm{H}^{\ast}V$-module and $\overline{E}$ is nil-closed. In section 5, we
give some topological applications. In section 6, we give the solutions of the
problem ($\mathcal{P}$) when $E$ is free as an $\mathrm{H}^{\ast}V$-module and
$\overline{E}$ is isomorphic to $\sum^{n}\mathbb{F}_{2}$, we also give a
topological application. In section 7, we solve the problem ($\mathcal{P}$)
when $\overline{E}$ is the Brown-Gitler module $\mathrm{J}(2)$ and $V$ is
$\mathbb{Z}/2\mathbb{Z}$.
Acknowledgements. I would like to thank Professor Jean Lannes and Professor
Said Zarati for several useful discussions. I am grateful to the referee for
his suggestions.
## 2 Preliminaries on the categories $\mathcal{U}$ and
$\mathrm{H}^{\ast}V-\mathcal{U}$
In this section, we will fix some notations, recall some definitions and
results about the categories $\mathcal{U}$ and
$\mathrm{H}^{\ast}V-\mathcal{U}$.
### 2.1 Nilpotent unstable $A$-modules
Let $N$ be an unstable $A$-module. We denote by $Sq_{0}$ the
$\mathbb{Z}/2\mathbb{Z}$-linear map:
$Sq_{0}:N\rightarrow N,\;x\mapsto Sq_{0}(x)=Sq^{\mid x\mid}x.$
An unstable $A$-module $N$ is called nilpotent if:
$\forall\;x\in N,\;\exists\;n\in\mathbb{N};\;Sq_{0}^{n}x=0.$
For example, finite unstable $A$-modules and suspension of unstable
$A$-modules are nilpotent. Let
$Tor^{\mathrm{H}^{\ast}V}_{1}(\mathbb{F}_{2},N)$ be the first derived functor
of the functor
$\mathbb{F}_{2}\otimes_{\mathrm{H}^{\ast}V}-\;:\mathrm{H}^{\ast}V-\mathcal{U}\rightarrow\mathcal{U}$,
we have the following useful result.
###### Proposition 2.1.1.
([S] page 150) Let $N$ be an unstable $\mathrm{H}^{\ast}V-A$-module, then the
unstable $A$-module $Tor^{\mathrm{H}^{\ast}V}_{1}(\mathbb{F}_{2},N)$ is
nilpotent.
### 2.2 Reduced unstable $A$-modules
An unstable $A$-module $M$ is called reduced if the
$\mathbb{Z}/2\mathbb{Z}$-linear map:
$Sq_{0}:M\rightarrow M,\;x\mapsto Sq_{0}(x)=Sq^{\mid x\mid}x,$
is an injection.
Another characterization of reduced unstable $A$-module in terms of nilpotent
modules is the following.
###### Lemma 2.2.1.
([LZ1]) An unstable $A$-module is reduced if it does not contain a non-trivial
nilpotent module.
In particular, any $A$-linear map from a nilpotent $A$-module to a reduced one
is trivial.
### 2.3 Nil-closed unstable $A$-modules
Let $M$ be an unstable $A$-module. We denote by $Sq_{1}$ the
$\mathbb{Z}/2\mathbb{Z}$-linear map:
$Sq_{1}:N\rightarrow N,\;x\mapsto Sq_{1}(x)=Sq^{\mid x\mid-1}x.$
###### Definition 2.3.1.
([EP]) An unstable $A$-module $M$ is called nil-closed if:
1. 1.
$M$ is reduced.
2. 2.
$Ker(Sq_{1})=Im(Sq_{0})$.
We have the following two characterizations of unstable nil-closed
$A$-modules.
###### Lemma 2.3.2.
([LZ1]) Let $M$ be an unstable $A$-module and $\mathcal{E}(M)$ be its
injective hull. The unstable $A$-module $M$ is nil-closed if and only if $M$
and the quotient $\mathcal{E}(M)/M$ are reduced.
Let $Ext^{s}_{\mathcal{U}}(-,M)$ be the s-th derived functor of the functor
$\mathrm{H}om_{\mathcal{U}}(-,M)$.
###### Lemma 2.3.3.
([LZ1]) An unstable $A$-module $M$ is nil-closed if and only if
$Ext^{s}_{\mathcal{U}}(N,M)=0$ for any nilpotent unstable $A$-module $N$ and
$s=0,1$.
### 2.4 Injectives in the category $\mathcal{U}$
Let $I$ be an unstable $A$-module, $I$ is called an injective in the category
$\mathcal{U}$ or $\mathcal{U}$-injective for short, if the functor
$\mathrm{H}om_{\mathcal{U}}(-,I)$ is exact.
The classification of $\mathcal{U}$-injectives (see [LZ1], [LS]) is the
following.
Let $\mathrm{J}(n),\;n\in\mathbb{N}$, be the $n$-th Brown- Gitler module,
characterized up to isomorphism, by the functorial bijection on the unstable
$A$-module M:
$\mathrm{H}om_{\mathcal{U}}(M,\mathrm{J}(n))\cong\mathrm{H}om_{\mathbb{F}_{2}}(M^{n},\mathbb{F}_{2})$
Clearly $\mathrm{J}(n)$ is an $\mathcal{U}$-injective and it is a finite
module.
Let $\mathcal{L}$ be a set of representatives for $\mathcal{U}$-isomorphism
classes of indecomposable direct factors of
$\mathrm{H}^{\ast}(\mathbb{Z}/2\mathbb{Z})^{m},\;m\in\mathbb{N}$ (each class
is represented in $\mathcal{L}$ only once).
We have:
###### Theorem 2.4.1.
Let $I$ be an $\mathcal{U}$-injective module. Then there exists a set of
cardinals $a_{L,n}\;,(L,n)\in\mathcal{L}\times\mathbb{N}$, such that
$I\cong\displaystyle\bigoplus_{(L,n)}(L\otimes\mathrm{J}(n))^{\oplus a_{L,n}}$
.
Conversely, any unstable $A$-module of that form is $\mathcal{U}$-injective.
Let’s remark that $\mathrm{H}^{\ast}V$ is an $\mathcal{U}$-injective.
### 2.5 The injectives of the category $\mathrm{H}^{\ast}V-\mathcal{U}$
The classification of injectives of the category
$\mathrm{H}^{\ast}V-\mathcal{U}\;\;(\mathrm{H}^{\ast}V-\mathcal{U}$-injectives
for short) is given by Lannes-Zarati [LZ2] as follows.
Let $\mathrm{J}_{V}(n),\;n\in\mathbb{N}$, be the unstable
$\mathrm{H}^{\ast}V-A$-module characterized, up to isomorphism, by the
functorial bijection on the unstable $\mathrm{H}^{\ast}V-A$-module M:
$\mathrm{H}om_{\mathrm{H}^{\ast}V-\mathcal{U}}(M,\mathrm{J}_{V}(n))\cong\mathrm{H}om_{\mathbb{F}_{2}}(M^{n},\mathbb{F}_{2})$
Clearly $\mathrm{J}_{V}(n)$ is an $\mathrm{H}^{\ast}V-\mathcal{U}$-injective.
Let $\mathcal{W}$ be the set of subgroups of $V$ and let
$(W,n)\in\mathcal{W}\times\mathbb{N}$, we write
$E(V,W,n)=\mathrm{H}^{\ast}V\otimes_{\mathrm{H}^{\ast}V/W}\mathrm{J}_{V/W}(n)$
(in this formula $\mathrm{H}^{\ast}V$ is an $\mathrm{H}^{\ast}V/W$-module via
the map induced in mod.2 cohomology by the canonical projection $V\rightarrow
V/W$).
###### Theorem 2.5.1.
([LZ2]) If I is an injective of the category of
$\mathrm{H}^{\ast}V-\mathcal{U}$, then
$I\cong\displaystyle\bigoplus_{(L,W,n)\in\mathcal{L}\times\mathcal{W}\times\mathbb{N}}(E(V,W,n)\otimes_{\mathbb{F}_{2}}L)^{\oplus_{a_{L,W,n}}}$.
Conversely, each $\mathrm{H}^{\ast}V-A$-module of this form is an
$\mathrm{H}^{\ast}V-\mathcal{U}$-injective.
Clearly $\mathrm{H}^{\ast}V$ is an $\mathrm{H}^{\ast}V-\mathcal{U}$-injective.
### 2.6 Algebraic Smith theory
#### 2.6.1 The functors $\mathrm{F}ix$
We introduce the functors $\mathrm{F}ix$ ([L1], [LZ2]). We denote by
$\mathrm{F}ix_{V}:\mathrm{H}^{\ast}V-\mathcal{U}\rightarrow\mathcal{U}$
the left adjoint of the functor
$\mathrm{H}^{\ast}V\otimes-:\mathcal{U}\rightarrow\mathrm{H}^{\ast}V-\mathcal{U}$
We have the functorial bijection:
$\displaystyle\mathrm{H}om_{\mathrm{H}^{\ast}V-\mathcal{U}}(N,\;H^{\ast}V\otimes
P)\cong\mathrm{H}om_{\mathcal{U}}(\mathrm{F}ix_{V}N,\;P)$
for every unstable $\mathrm{H}^{\ast}V-A$-module $N$ and every unstable
$A$-module $P$.
The functor $\mathrm{F}ix_{V}$ has the following properties.
2.6.1.1. The functor $\mathrm{F}ix_{V}$ is an exact functor.
2.6.1.2. Let $N$ be an unstable $\mathrm{H}^{\ast}V-A$-module and
$\mathcal{E}(N)$ be its injective hull. Then, the module
$\mathrm{F}ix_{V}\mathcal{E}(N)$ is the injective hull of $\mathrm{F}ix_{V}N$.
#### 2.6.2
Let $N$ be an unstable $\mathrm{H}^{\ast}V-A$-module, we denote by
$\eta_{{}_{V}}:\;N\rightarrow\mathrm{H}^{\ast}V\otimes\mathrm{F}ix_{{}_{V}}N$
the adjoint of the identity of $Fix_{{}_{V}}N$. We denote by
$\mathrm{c}_{V}=\displaystyle\prod_{u\in\mathrm{H}^{1}V-\\{0\\}}u$ the top
Dickson invariant, we have the following result (see [LZ2] corollary 2.3).
###### Proposition 2.6.1.
Let $N$ be an unstable $\mathrm{H}^{\ast}V-A$-module. The localization of the
map $\eta_{{}_{V}}$
$\eta_{{}_{V}}[\mathrm{c}_{V}^{-1}]:N[\mathrm{c}_{V}^{-1}]\rightarrow\mathrm{H}^{\ast}V[\mathrm{c}_{v}^{-1}]\otimes\mathrm{F}ix_{V}N$
is an injection.
This shows in particular, that if $N$ is torsion-free then the map
$\eta_{{}_{V}}$ is an injection.
The proposition 2.6.1 can be reformulated as follows.
###### Proposition 2.6.2.
Let $N$ be an unstable $\mathrm{H}^{\ast}V-A$-module. If $N$ is torsion-free
then its injective hull in $\mathrm{H}^{\ast}V-\mathcal{U}$ is free as an
$H^{*}V$-module and is isomorphic to
$\displaystyle\bigoplus_{(L,n)\in\mathcal{L}\times\mathbb{N}}(\mathrm{H}^{\ast}V\otimes\mathrm{J}(n))\otimes
L$
###### Proof.
Since the module is torsion-free then the map
$\eta_{{}_{V}}:\;N\rightarrow\mathrm{H}^{\ast}V\otimes\mathrm{F}ix_{{}_{V}}N$
adjoint of the identity of $\mathrm{F}ix_{{}_{V}}N$ is an injection. So $N$ is
a sub-$\mathrm{H}^{\ast}V-A$-module of
$\mathrm{H}^{\ast}V\otimes\mathrm{F}ix_{{}_{V}}N$. By 2.6.1.1 and 2.6.1.2, we
have that the injective hull of $N$ is isomorphic to
$\mathrm{H}^{\ast}V\otimes I$, where $I$ is an $\mathcal{U}$-injective. ∎
###### Remark 2.6.3.
As a consequence of proposition 2.6.2, we have that if $E$ is an unstable
$\mathrm{H}^{\ast}V-A$-module which is free as an $\mathrm{H}^{\ast}V$-module
then its injective hull (in the category $\mathrm{H}^{\ast}V-\mathcal{U}$) is
also free as an $\mathrm{H}^{\ast}V$-module.
###### Proposition 2.6.4.
[LZ2]. Let $N$ be an unstable $\mathrm{H}^{\ast}V-A$-module which is of finite
type as an $\mathrm{H}^{\ast}V$-module. The localization of the map
$\eta_{{}_{V}}$
$\eta_{{}_{V}}[\mathrm{c}_{V}^{-1}]:N[\mathrm{c}_{V}^{-1}]\rightarrow\mathrm{H}^{\ast}V[\mathrm{c}_{V}^{-1}]\otimes\mathrm{F}ix_{V}N$
is an isomorphism.
In particular, the previous result shows that:
1. 1.
If $N$ is free as an $\mathrm{H}^{\ast}V$-module, then the map $\eta_{V}$ is
an injection.
2. 2.
The isomorphism of the proposition proves that
$dim\overline{E}=dim\mathrm{F}ix_{V}E$ where $dim$ is the total dimension (see
[LZ2]).
## 3 Some properties of $E$ when $\overline{E}$ is reduced
In this section we will prove some algebraic results which will be useful for
section 4. In fact, we will analyze the relation between an unstable
$\mathrm{H}^{\ast}V-A$-module $E$ and its (associated) unstable $A$-module
$\overline{E}$. For this, we will begin by giving some technical results.
### 3.1 Technical results
###### Lemma 3.1.1.
Let $P$ and $Q$ be unstable $\mathrm{H}^{\ast}V-A$-modules, free as
$\mathrm{H}^{\ast}V$-modules and $f:P\rightarrow Q$ an
$\mathrm{H}^{\ast}V-A$-linear map. If the induced map
$\overline{f}:\overline{P}\rightarrow\overline{Q}$ is an injection then $f$ is
also an injection.
###### Proof.
Let’s denote by $Imf$ the image of $f$, by $\widetilde{f}:P\rightarrow Imf$
the natural surjection and by $i:Imf\hookrightarrow Q$ the inclusion of $Imf$
in $Q$. Since $\overline{f}$ is an injection so the induced map
$\overline{(\widetilde{f})}$ is an isomorphism of unstable $A$-modules and
then the induced map $\overline{i}$ is an injection. This shows that
$\overline{Imf}$ is the image of $\overline{f}$. Since the module $Imf$ is a
sub-$\mathrm{H}^{\ast}V$-module of the $\mathrm{H}^{\ast}V$-free module $Q$
and $\overline{i}:\overline{Imf}\hookrightarrow\overline{Q}$ is an injection,
so $Imf$ is free as an $\mathrm{H}^{\ast}V$-module. In particular, we have
that $Tor_{1}^{\mathrm{H}^{\ast}V}(\mathbb{F}_{2},Imf)$=0 (see for example
[R]). Let’s denote by $N$ the kernel of the map $\widetilde{f}$, so we have
the following short exact sequence in $\mathrm{H}^{\ast}V-\mathcal{U}$:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.62497pt\raise 9.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 6.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 68.62497pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.62497pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
88.6562pt\raise 12.61111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
9.61111pt\hbox{$\scriptstyle{\widetilde{f}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 106.43399pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
106.43399pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Imf\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
145.21411pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
145.21411pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces\,.$
By applying the functor ($\mathbb{F}_{2}\otimes_{\mathrm{H}^{\ast}V}-$) to the
previous sequence, we prove that $\overline{N}$ is trivial (since the map
$\overline{(\widetilde{f})}$ is an isomorphism and $Imf$ is free as an
$\mathrm{H}^{\ast}V-A$-module). Hence the module $N$ is trivial and the map
$f$ is an injection.
∎
The converse of this lemma is not true in general, but we have the following
result:
###### Lemma 3.1.2.
Let $P$ and $Q$ be unstable $\mathrm{H}^{\ast}V-A$-modules, free as
$\mathrm{H}^{\ast}V$-modules and $f:P\rightarrow Q$ an
$\mathrm{H}^{\ast}V-A$-linear map which is an injection. If $\overline{P}$ is
a reduced unstable $A$-module, then the induced map
$\overline{f}:\overline{P}\rightarrow\overline{Q}$ is an injection.
###### Proof.
We denote by $C$ the quotient of $Q$ by $P$, we have the following short exact
sequence in $\mathrm{H}^{\ast}V-\mathcal{U}$:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.21873pt\raise 12.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
9.1111pt\hbox{$\scriptstyle{f}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
67.30902pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.30902pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
90.21457pt\raise 9.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 6.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 105.21457pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.21457pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
143.07706pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
143.07706pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces\,.$
By applying the functor ($\mathbb{F}_{2}\otimes_{\mathrm{H}^{\ast}V}-$) to the
previous sequence, we obtain an exact sequence in $\mathcal{U}$:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Tor^{\mathrm{H}^{\ast}V}_{1}(\mathbb{F}_{2},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
69.02911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
69.02911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
86.52911pt\raise 11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
8.83888pt\hbox{$\scriptstyle{\overline{f}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 104.02911pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
104.02911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
124.02911pt\raise 9.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 6.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 139.02911pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
139.02911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
174.02911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
174.02911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces\,.$
Since $\overline{P}$ is reduced as unstable $A$-module and
$Tor_{1}^{{\mathrm{H}^{\ast}V}}(\mathbb{F}_{2},C)$ is nilpotent (see
proposition 2.1.1), then the map $\overline{f}$ is an injection.
∎
### 3.2 Statement of some properties of $E$ when $\overline{E}$ is reduced
The first result of this paragraph concerns the relation between the injective
hull of $E$ and the induced module $\overline{E}$.
###### Theorem 3.2.1.
Let $E$ be an unstable $\mathrm{H}^{\ast}V-A$-module which is free as an
$H^{*}V$-module and let $\mathcal{E}(E)$ be its injective hull (in the
category $\mathrm{H}^{\ast}V-\mathcal{U}$). We suppose that $\overline{E}$ is
reduced and let $I$ be its injective hull in the category $\mathcal{U}$.
Then $\mathcal{E}(E)$ is isomorphic, as an unstable
$\mathrm{H}^{*}V-A$-module, to $\mathrm{H}^{*}V\otimes I$.
###### Proof.
Since $E$ is free as an $\mathrm{H}^{*}V$-module, then $\mathcal{E}(E)$ is
isomorphic, in the category $\mathrm{H}^{*}V-\mathcal{U}$, to
$\mathrm{H}^{*}V\otimes J$, where $J$ is an $\mathcal{U}$-injective (see
proposition 2.6.2).
Let’s denote by $i$ the inclusion of $E$ in $\mathcal{E}(E)$, we have, by
lemma 3.1.2, that the induced map $\overline{i}$ is an injection. We will
prove, by using the definition, that $J$ is the injective hull of
$\overline{E}$, in the category $\mathcal{U}$. Let $P$ be a sub-$A$-module of
$J$ such that the $A$-module $(\overline{i})^{-1}(P)$ is trivial, we have to
show that the unstable $A$-module $P$ is trivial.
Since $(\overline{i})^{-1}(P)$ is trivial then the composition:
$\pi\circ\overline{i}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.0pt\raise 11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise
8.83888pt\hbox{$\scriptstyle{\overline{i}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
49.01184pt\raise 10.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
7.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 66.00693pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
66.00693pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{J/P}$}}}}}}}\ignorespaces}}}}\ignorespaces$
is an injection. By lemma 3.1.1, the following composition
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathrm{H}^{*}V\otimes
J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}V\otimes(J/P)}$
is an injection, which proves that the unstable $\mathrm{H}^{*}V-A$-module
$i^{-1}(\mathrm{H}^{*}V\otimes P)$ is trivial. Since $\mathrm{H}^{*}V\otimes
J$ is the injective hull of $E$ so the unstable $\mathrm{H}^{*}V-A$-module
$\mathrm{H}^{*}V\otimes P$ is trivial. ∎
###### Corollary 3.2.2.
Let $E$ be an unstable $\mathrm{H}^{\ast}V-A$-module such that:
1. 1.
$E$ is free as an $\mathrm{H}^{*}V$-module.
2. 2.
$\overline{E}$ is reduced as unstable $A$-module.
Then $E$ is reduced as unstable $A$-module.
###### Proof.
We have, by theorem 3.2.1, that the injective hull of $E$ is
$\mathrm{H}^{*}V\otimes I$, where $I$ is the injective hull of $\overline{E}$
in $\mathcal{U}$. Since $\overline{E}$ is reduced, then $I$ is a reduced
$\mathcal{U}$-injective. This shows that $E$ is reduced as an unstable
$A$-module because its injective hull (in the category
$\mathrm{H}^{\ast}V-\mathcal{U}$) is $\mathrm{H}^{*}V\otimes I$ which is
reduced as unstable $A$-module. ∎
###### Remark 3.2.3.
In the previous result the condition (1): $E$ is free as an
$\mathrm{H}^{*}V$-module is necessary. In fact, the finite
$\mathrm{H}-A$-module $\mathrm{J}_{\mathbb{Z}/2\mathbb{Z}}(1)$ is not free as
an $\mathrm{H}$-module and not reduced as an unstable $A$-module, however
$\overline{\mathrm{J}_{\mathbb{Z}/2\mathbb{Z}}(1)}=\mathbb{F}_{2}$ is a
reduced unstable $A$-module. Observe that
$\mathrm{J}_{\mathbb{Z}/2\mathbb{Z}}(1)$ is isomorphic, as unstable
$A$-module, to $\mathbb{F}_{2}\oplus\sum\mathbb{F}_{2}$, the structure of
$\mathrm{H}$-module is given by: $t.\iota=\Sigma\iota$, where $\iota$ is the
generator of $\mathbb{F}_{2}$ and $t$ the generator of $\mathrm{H}$.
Observe that the converse of corollary 3.2.2 is false. In fact, the
$\mathrm{H}-A$-module $E=\mathrm{H}^{\geq 1}$ is reduced as unstable
$A$-module however the unstable $A$-module
$\overline{E}\cong\sum\mathbb{F}_{2}$ is not reduced.
## 4 Description of $E$ when $\overline{E}$ is nil-closed
The main result of this paragraph concerns the relation between the two first
terms of a (minimal) injective resolution of $E$ and $\overline{E}$.
###### Theorem 4.1.
Let $E$ be an unstable $\mathrm{H}^{\ast}V-A$-module which is free as an
$\mathrm{H}^{*}V$-module. We suppose that:
1. 1.
$\overline{E}$ is nil-closed.
2. 2.
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{....}$
is the beginning of a (minimal) $\mathcal{U}$\- injective resolution of
$\overline{E}$.
Then there exists an $\mathrm{H}^{\ast}V-A$-linear map
$\varphi:\mathrm{H}^{*}V\otimes I_{0}\rightarrow\mathrm{H}^{*}V\otimes I_{1}$
such that:
1. 1.
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}V\otimes
I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathrm{H}^{*}V\otimes
I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{....}$
is the beginning of a (minimal) injective resolution of $E$ (in the category
$\mathrm{H}^{*}V-\mathcal{U}$).
2. 2.
$\overline{\varphi}=i_{1}$
###### Proof.
The unstable $A$-module $\overline{E}$ is nil-closed so is reduced, we have
then, by theorem 3.2.1, that the injective hull of $E$ is
$\mathrm{H}^{*}V\otimes I_{0}$. We denote by $C_{0}$ the quotient of
$\mathrm{H}^{\ast}V\otimes I_{0}$ by $E$. We have the following short exact
sequence in $\mathrm{H}^{\ast}V-\mathcal{U}$:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
51.25252pt\raise 15.64444pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
12.64444pt\hbox{$\scriptstyle{i_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 67.45831pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.45831pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathrm{H}^{*}V\otimes
I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
92.45828pt\raise 9.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 6.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 107.45828pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
107.45828pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
145.32077pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
145.32077pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces\,.$
Since the induced map $\overline{i_{0}}$ is an injection (see lemma 3.1.2),
then the unstable $A$-module
$Tor_{1}^{\mathrm{H}^{\ast}V}(\mathbb{F}_{2},C_{0})$ is trivial; this shows
that the module $C_{0}$ is free as an $\mathrm{H}^{*}V$-module (see for
example [NS], proposition A.1.5).
We verify that the $\mathcal{U}$-injective hull of $\overline{C_{0}}$ is
$I_{1}$ and that $C_{0}$ is reduced since $\overline{C_{0}}$ is reduced (see
corollary 3.2.2). This implies, by theorem 3.2.1, that the
$\mathrm{H}^{*}V-\mathcal{U}$-injective hull of $C_{0}$ is isomorphic to
$\mathrm{H}^{*}V\otimes I_{1}$. ∎
###### Remark 4.2.
let $M$ be a nil-closed unstable $A$-module and
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{0}}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{....}$
be the beginning of a (minimal) $\mathcal{U}$-injective resolution of $M$. We
denote by
$(\mathrm{H}om_{\mathrm{H}^{\ast}V-\mathcal{U}}(\mathrm{H}^{\ast}V\otimes
I_{0},\;\mathrm{H}^{\ast}V\otimes I_{1}))_{i_{1}}$
the set of $\mathrm{H}^{\ast}V-A$-linear map
$\varphi:\mathrm{H}^{\ast}V\otimes I_{0}\rightarrow\mathrm{H}^{\ast}V\otimes
I_{1}$ such that $\overline{\varphi}=i_{1}$.
Using Lannes T-functor (see [L1]) we have:
$(\mathrm{H}om_{\mathrm{H}^{\ast}V-\mathcal{U}}(\mathrm{H}^{\ast}V\otimes
I_{0},\;\mathrm{H}^{\ast}V\otimes
I_{1}))_{i_{1}}\cong(\mathrm{H}om_{\mathcal{U}}(T_{V}I_{0},\;I_{1}))_{i_{1}}$
where $(\mathrm{H}om_{\mathcal{U}}(T_{V}I_{0},\;I_{1}))_{i_{1}}$ is the set of
$A$-linear map $\psi:T_{V}I_{0}\rightarrow I_{1}$ such that $\psi\circ
i=i_{1}$, where $i:I_{0}\hookrightarrow T_{V}I_{0}$ denotes the natural
inclusion.
The kernel of any element
$\psi\in(\mathrm{H}om_{\mathcal{U}}(T_{V}I_{0},\;I_{1}))_{i_{1}}$, which is
free as an $\mathrm{H}^{*}V$-module, is an unstable
$\mathrm{H}^{\ast}V-A$-module such that $\overline{ker\psi}\cong M$.
###### Remark 4.3.
If $\overline{E}$ is an $\mathcal{U}$-injective then the only unstable free
$\mathrm{H}^{\ast}V-A$-module, up to isomorphism, solution of the problem
$(\mathcal{P})$ is $\mathrm{H}^{\ast}V\otimes\overline{E}$.
Let $n$ be an even integer. The unstable free $\mathrm{H}-A$-modules, up to
isomorphism, solution of the problem $(\mathcal{P})$ when $M$ is
$\mathrm{H}^{*}BSO(n)$ are $\mathrm{H}^{*}BO(n)$ and
$\mathrm{H}\otimes\mathrm{H}^{*}BSO(n)$. We verify that these two
$\mathrm{H}-A$-modules are not isomorphic in the category
$\mathrm{H}-\mathcal{U}$ (since it does not exist an $A$-linear section of the
projection $\mathrm{H}^{*}BO(n)\rightarrow\mathrm{H}^{*}BSO(n)$).
## 5 Applications
### 5.1
Our first application concerns the determination of the $\bmod{.\hskip
2.0pt2}$ cohomology of the mapping space $\mathbf{hom}\hskip
1.0pt(\mathrm{B}\hskip 1.0pt(\mathbb{Z}/2^{n}),Y)$ whose domain is a
classifying space for the group $\mathbb{Z}/2^{n}$ and whose range is a space
$Y$ such that $\mathrm{H}^{*}Y$ is concentrated in even degrees.
We will just recall some facts, ignoring the p-completion problems. For
further details see [DL].
One proceeds by induction on the integer $n$. Let us set
$\hskip 24.0ptX=\mathbf{hom}\hskip 1.0pt(\mathrm{E}\hskip
1.0pt(\mathbb{Z}/2^{n})/(\mathbb{Z}/2^{n-1}),Y)\hskip 24.0pt.$
The space $X$ has the homotopy type of $\mathbf{hom}\hskip
1.0pt(\mathrm{B}\hskip 1.0pt(\mathbb{Z}/2^{n-1}),Y)$ and is equipped of an
action $\mathbb{Z}/2$ such that one has a homotopy equivalence
$\hskip 24.0pt\mathbf{hom}\hskip 1.0pt(\mathrm{B}\hskip
1.0pt(\mathbb{Z}/2^{n}),Y)\cong X^{\mathrm{h}\hskip 1.0pt\mathbb{Z}/2}\hskip
24.0pt,$
$X^{\mathrm{h}\hskip 1.0pt\mathbb{Z}/2}$ denoting the homotopy fixed point
space: $\mathbf{hom}_{\mathbb{Z}/2}\hskip 1.0pt(\mathrm{E}\hskip
1.0pt\mathbb{Z}/2,X)$. Using $\mathrm{Fix}_{\mathbb{Z}/2}$-theory [L1], one
gets:
$\hskip 24.0pt\mathrm{H}^{*}\mathbf{hom}\hskip 1.0pt(\mathrm{B}\hskip
1.0pt(\mathbb{Z}/2^{n}),Y)\cong\mathrm{Fix}_{\mathbb{Z}/2}\hskip
2.0pt\mathrm{H}^{*}_{\mathbb{Z}/2}\hskip 1.0ptX\hskip 24.0pt.$
Since the computation of the functor $\mathrm{Fix}_{\mathbb{Z}/2}$ on an
unstable $\mathrm{H}-\mathrm{A}$-module is not difficult in general, the
determination of the $\bmod{.\hskip 2.0pt2}$ cohomology of the mapping space
$\mathbf{hom}\hskip 1.0pt(\mathrm{B}\hskip 1.0pt(\mathbb{Z}/2^{n}),Y)$ is
reduced to the determination of the unstable $\mathrm{H}-\mathrm{A}$-module
$\mathrm{H}^{*}_{\mathbb{Z}/2}\hskip 1.0ptX$. As we are going to explain, this
last point is closely related to problem $(\mathcal{P})$.
One knows by induction on $n$ that the $\bmod{.\hskip 2.0pt2}$ cohomology of
the space $X$ as the one of the space $Y$ is concentrated in even degrees and
one checks that the action of $\mathbb{Z}/2$ on $\mathrm{H}^{*}(Y;\mathbb{Z})$
is trivial. These two facts imply that the Serre spectral sequence, for
$\bmod{.\hskip 2.0pt2}$ cohomology, associated to the fibration
$X\rightarrow X_{\mathrm{h}\mathbb{Z}/2}\rightarrow\mathrm{B}\mathbb{Z}/2$
collapses ($X_{\mathrm{h}\mathbb{Z}/2}$ denotes the Borel construction
$\mathrm{E}\mathbb{Z}/2\times_{\mathbb{Z}/2}X$). This collapsing implies in
turn that $\mathrm{H}^{*}_{\mathbb{Z}/2}X$ is $\mathrm{H}$-free and that
$\overline{\mathrm{H}^{*}_{\mathbb{Z}/2}X}$ is isomorphic to
$\mathrm{H}^{*}X$. So the determination of $\mathrm{H}^{*}\mathbf{hom}\hskip
1.0pt(\mathrm{B}\hskip 1.0pt(\mathbb{Z}/2^{n}),Y)$ is indeed reduced to the
resolution of a problem $(\mathcal{P})$.
We conclude this subsection by a concrete example (we follow [De], section 6);
we take $n=2$ and $Y=\mathrm{BSU}(2)$. Using
$\mathrm{T}_{\mathbb{Z}/2}$-computations one sees that $X$ has the homotopy
type of $\mathrm{BSU}(2)\coprod\mathrm{BSU}(2)$; one checks also that the
$\mathbb{Z}/2$-action preserves the connected components. The
$(\mathcal{P})$-problem asociated to the determination of the unstable
$\mathrm{H}-\mathrm{A}$-module $\mathrm{H}^{*}_{\mathbb{Z}/2}\hskip 1.0ptX$ is
the following one:
Find the unstable $\mathrm{H}-\mathrm{A}$-modules $E$ such that
* –
$E$ is $\mathrm{H}$-free;
* –
the unstable $\mathrm{A}$-module $\overline{E}$ is isomorphic to
$\mathrm{H}^{*}\mathrm{BSU}(2)$.
Using the fact that the injective hull, in the category
$\mathrm{H}-\mathcal{U}$, of $E$ is $\mathrm{H}\otimes\mathrm{H}$ (see theorem
3.2), one checks that one has two possibilities:
* –
$E\cong\mathrm{H}\otimes\mathrm{H}^{*}\mathrm{BSU}(2)$;
* –
$E\cong\mathrm{H}\otimes_{\mathrm{H}^{*}\mathrm{BU}(1)}\mathrm{H}^{*}\mathrm{BU}(2)$
(the structures of unstable $\mathrm{H}^{*}\mathrm{BU}(1)-\mathrm{A}$-modules
on $\mathrm{H}=\mathrm{H}^{*}\mathrm{BO}(1)$ and
$\mathrm{H}^{*}\mathrm{BU}(2)$ are respectively induced by the inclusion of
$\mathrm{O}(1)$ in $\mathrm{U}(1)$ and the determinant homomorphism from
$\mathrm{U}(2)$ to $\mathrm{U}(1)$).
### 5.2
The theorem 4.1 can be illustrated, topologically, as follows:
###### Proposition 5.2.1.
Let $X$ be a CW-complex on which acts an elementary abelian group 2-group $V$.
Suppose that:
1. 1.
$\mathrm{H}^{\ast}X$ is nil-closed
2. 2.
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{\ast}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{....}$
is the beginning of a (minimal) $\mathcal{U}$-injective resolution of
$\mathrm{H}^{\ast}X$
3. 3.
$\mathrm{H}^{\ast}_{V}X$ is free as an $\mathrm{H}^{*}V$-module.
Then there exists an $\mathrm{H}^{\ast}V-A$-linear map
$\varphi:\;\mathrm{H}^{*}V\otimes I_{0}\rightarrow\mathrm{H}^{*}V\otimes
I_{1}$ such that:
1. 1.
$\mathrm{H}^{\ast}_{V}X\cong Ker(\varphi)$.
2. 2.
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{\ast}_{V}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}V\otimes
I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathrm{H}^{*}V\otimes
I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{....}$
is the beginning of a (minimal) injective resolution of
$\mathrm{H}^{\ast}_{V}X$ (in the category $\mathrm{H}^{*}V-\mathcal{U}$).
3. 3.
$\overline{\varphi}=\alpha:I_{0}\rightarrow I_{1}$.
In particular, we have:
###### Corollary 5.2.2.
Let $X$ be a CW-complex on which acts an elementary abelian group 2-group $V$.
Suppose that:
1. 1.
$\mathrm{H}^{\ast}X$ is a reduced $\mathcal{U}$-injective,
2. 2.
$\mathrm{H}^{\ast}_{V}X$ is free as an $\mathrm{H}^{*}V$-module.
Then $\mathrm{H}^{\ast}_{V}X\cong\mathrm{H}^{\ast}V\otimes H^{*}X$.
## 6 Description of $E$ when $\overline{E}$ is isomorphic to
$\sum^{n}\mathbb{F}_{2}$
In this section, we prove the following result.
###### Theorem 6.1.
Let $E$ be unstable $\mathrm{H}^{\ast}V-A$-module which is free as an
$\mathrm{H}^{\ast}V$-module. If $\overline{E}$ is isomorphic to
$\sum^{n}\mathbb{F}_{2}$, then there exists an element $u$ in
$\mathrm{H}^{\ast}V$ such that:
1. 1.
$u=\displaystyle\prod_{i}\theta_{i}^{\alpha_{i}}$, where
$\theta_{i}\in(\mathrm{H}^{1}V)\setminus\\{0\\}$ and $\alpha_{i}\in\mathbb{N}$
2. 2.
$E\cong\sum^{d}u\mathrm{H}^{\ast}V$ with
$d+\displaystyle\sum_{i}\alpha_{i}=n$.
###### Proof.
Let $N$ be an unstable $A$-module, we denote by $\mathrm{d}imN$ the total
dimension of $N$ that is $\mathrm{d}im\;N=\sum_{i}\mathrm{d}im\;N^{i}$. We
have the equality
$\mathrm{d}im\;\overline{E}=1=\mathrm{d}im\;\mathrm{F}ix_{{}_{V}}E$ (see
[LZ3]), so we deduce that $\mathrm{F}ix_{{}_{V}}E=\sum^{l}\mathbb{F}_{2}$,
where $l\in\mathbb{N}$. Let
$\eta_{{}_{V}}:\;E\rightarrow\mathrm{H}^{\ast}V\otimes\mathrm{F}ix_{{}_{V}}E$
be the adjoint of the identity of $\mathrm{F}ix_{{}_{V}}E$ (see [LZ2]). Since
the map $\eta_{V}$ is an injection, then the module $E$ is a
sub-$\mathrm{H}^{\ast}V-A$-module of $\sum^{l}\mathrm{H}^{\ast}V$. Let’s write
$E=\sum^{l}E^{\prime}$, where $E^{\prime}$ is
sub-$\mathrm{H}^{\ast}V-A$-module of $\mathrm{H}^{\ast}V$ . By a result of
J-P. Serre (see [Se]), there exists $N$ such that:
$\mathrm{c}_{V}^{N}\mathrm{H}^{\ast}V\subset
E^{\prime}\subset\mathrm{H}^{\ast}V$. Since $E^{\prime}$ is free as an
$\mathrm{H}^{\ast}V$-module and of dimension one, then there exists
$u\in\mathrm{\widetilde{H}}^{\ast}V$ such that
$E^{\prime}=u\mathrm{H}^{\ast}V$. The inclusion
$\mathrm{c}_{V}^{N}\mathrm{H}^{\ast}V\subset u\mathrm{H}^{\ast}V$ proves that
$u=\displaystyle\prod_{i}\theta_{i}^{\alpha_{i}}$, where
$\theta_{i}\in(\mathrm{H}^{1}V)\setminus\\{0\\}$ and
$\alpha_{i}\in\mathbb{N}$. ∎
###### Remark 6.2.
We remark that by the previous result, we can determinate $E$ when
$\overline{E}$ is isomorphic to $\mathbb{F}_{2}\oplus\sum^{n}\mathbb{F}_{2}$.
In this case, we verify that
$E\cong\mathrm{H}^{\ast}V\oplus\sum^{d}u\mathrm{H}^{\ast}V$, where
$u=\displaystyle\prod_{i}\theta_{i}^{\alpha_{i}}$,
$\theta_{i}\in\mathrm{H}^{\ast}V\setminus\\{0\\}$, $\alpha_{i}\in\mathbb{N}$
and $d+\displaystyle\sum_{i}\alpha_{i}=n$. In fact, since the
$\mathrm{H}^{\ast}V-\mathcal{U}$-injective module $\mathrm{H}^{\ast}V$ is a
sub-$\mathrm{H}^{\ast}V$-module of $E$, then $E\cong\mathrm{H}^{\ast}V\oplus
E^{\prime}$, where $E^{\prime}$ is an unstable $\mathrm{H}^{\ast}V-A$-module,
free as an $\mathrm{H}^{\ast}V$-module and such that
$\overline{E^{\prime}}\cong\sum^{n}\mathbb{F}_{2}$. The result holds from
theorem 6.1.
6.3 Example
We give an example showing how to realize topologically the cases of theorem
6.1 and remark 6.2.
Let $\rho:V\rightarrow\mathrm{O}(d)$ be a group homomorphism. $\rho$ gives
both an action of $V$ on $\mathrm{D}^{d}$, $\mathrm{S}^{d-1}$ and a
$d$-dimensional orthogonal bundle whose mod.2 Euler class is denoted by
$e(\rho)$.
The long exact sequence of the pair ($\mathrm{D}^{d},\;\mathrm{S}^{d-1}$) and
the Thom isomorphism give the long (Gysin) exact sequence (see for example
[Hu]):
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*-1}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{V}^{*-1}\mathrm{S}^{d-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma^{-d}\mathrm{H}^{*}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\smile
e(\rho)}$$\textstyle{\mathrm{H}^{*}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}_{V}\mathrm{S}^{d-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
The decomposition $\rho\cong\displaystyle\oplus_{i=1}^{d}\;\rho_{i}$ of the
representation $\rho$ into orthogonal representations of dimension 1 gives
$e(\rho)=\prod_{i}e(\rho_{i}).$ We have now two cases.
\- If none of the representations $\rho_{i}$ is trivial then $e(\rho)$ is non
zero and $\mathrm{H}^{\ast}_{V}(\mathrm{D}^{d},\mathrm{S}^{d-1})$ is
isomorphic to $e(\rho)\mathrm{H}^{*}V$ as an
$\mathrm{H}^{*}V-\mathrm{A}$-module. This illustrates theorem 6.1.
\- Otherwise, let’s write $\rho=\sigma\oplus\tau$, $\sigma$ (resp. $\tau$)
being the direct sum of the non trivial (resp. trivial) representations
$\rho_{i}$. Then
$\mathrm{H}^{*}_{V}\mathrm{S}^{d-1}\cong\mathrm{H}^{*}V\oplus\Sigma^{\mathrm{d}im\tau}\;e(\sigma)\mathrm{H}^{*}V$
and $\mathrm{H}^{\ast}_{V}(\mathrm{S}^{d-1})$ is an illustration of the remark
6.2.
## 7 Determination of $E$ when $V$ is $\mathbb{Z}/2\mathbb{Z}$ and
$\overline{E}$ is $\mathrm{J}(2)$
In this section, we assume that $V$ is $\mathbb{Z}/2\mathbb{Z}$ and
$\overline{E}$ is the Brown-Gitler module $\mathrm{J}(2)$.
We denote by $\mathrm{H}=\mathbb{F}_{2}[t]$ the cohomology of
$\mathbb{Z}/2\mathbb{Z}$, where $t$ is an element of $\mathrm{H}$ of degree
one. We have the following result.
###### Proposition 7.1.
Let $E$ be an $\mathrm{H}-A$-module which is $\mathrm{H}$-free and such that
$\overline{E}$ is isomorphic to $\mathrm{J}(2)$ then:
$E\cong\mathrm{H}\otimes\mathrm{J}(2)$
or
$E$ is the sub-$\mathrm{H}-A$-module of $\mathrm{H}\oplus\sum\mathrm{H}$
generated by $(t,\Sigma 1)$ and $(t^{2},0)$.
###### Proof.
This proof uses the Smith theory (see [DW], [LZ2] theorem 2.1) which gives us
an exact sequence (*) in $\mathrm{H}-\mathcal{U}$:
$(*)\;\;\;\;\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
8.18748pt\hbox{$\scriptstyle{\eta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 67.45831pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.45831pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}\otimes\mathrm{F}ixE\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
90.41663pt\raise 9.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 6.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 105.41663pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.41663pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
143.27911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
143.27911pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces$
where $C$ the quotient of $\mathrm{H}\otimes\mathrm{F}ixE$ is finite and also
$\mathrm{F}ixE$ is finite.
If the module $C$ is trivial then $E$ is isomorphic to
$\mathrm{H}\otimes\mathrm{J}(2)$.
When $C$ is a non trivial module. By applying the functor
$\mathbb{F}_{2}\otimes_{\mathrm{H}}-$ to the exact sequence (*), we obtain:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sum\tau
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{E}=\mathrm{J}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{F}ixE\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
where $\tau C$ is the trivial part of $C$ (see [BHZ]).
Let’s denote by $Q$ the quotient of $\overline{E}$ by $\sum\tau C$. By
properties of the module $\mathrm{J}(2)$, we have that $\sum\tau
C=\sum^{2}\mathbb{F}_{2}$ and $Q=\sum\mathbb{F}_{2}$. The exact sequence:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sum\mathbb{F}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{FixE\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
gives that $FixE\cong\sum\mathbb{F}_{2}\oplus\overline{C}$. One checks that
the module $\overline{C}$ is either isomorphic to $\mathbb{F}_{2}$ or
$\sum\mathbb{F}_{2}$. If $\overline{C}=\sum\mathbb{F}_{2}$ then
$\mathrm{F}ixE\cong\sum\mathbb{F}_{2}\oplus\sum\mathbb{F}_{2}$ as an unstable
$A$-module, which implies that the module $E$ is a suspension which is
impossible because $\overline{E}=\mathrm{J}(2)$ is not a suspension. We
conclude that $\overline{C}=\mathbb{F}_{2}$. Since $\tau C=\sum\mathbb{F}_{2}$
then we get $C$ is isomorphic to $\mathrm{H}^{\leq 1}$, where
$\mathrm{H}^{\leq 1}$ denotes the sub-$\mathrm{H}-A$-module of $\mathrm{H}$
consisting of elements of degree less or equal than 1. We have the following
exact sequence in $\mathrm{H}-\mathcal{U}$:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.45831pt\raise 9.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 6.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 67.45831pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.45831pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}\oplus\sum\mathrm{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
92.94647pt\raise 11.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise
8.18748pt\hbox{$\scriptstyle{\varphi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 110.23605pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
110.23605pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathrm{H}^{\leq
1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
147.73605pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
6.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
147.73605pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
The module $E$, we are searching for, is the kernel of $\varphi$ and we check
that it is the sub-$\mathrm{H}-\mathrm{A}$-module of
$\mathrm{H}\oplus\sum\mathrm{H}$ generated by the elements $(t,\Sigma 1)$ and
$(t^{2},0)$. ∎
###### Remark 7.2.
Let be $\mathbb{Z}/2\mathbb{Z}$ act on a real projective space
$\mathbb{R}\mathrm{P}^{2}$; let $x_{0}$ be a fixed point of this action (the
set of fixed point is not empty for example by an argument of Lefschetz
number). We have:
\- The Serre spectral sequence collapses to give that:
$\mathrm{H}^{\ast}_{V}(\mathbb{R}\mathrm{P}^{2},x_{0})$ is $\mathrm{H}$-free
and $\overline{\mathrm{H}^{\ast}_{V}(\mathbb{R}\mathrm{P}^{2},x_{0})}$ is
isomorphic to $\mathrm{J}(2)$.
\- In [DW], Dwyer and Wilkerson have shown that
$\mathrm{H}^{\ast}_{V}\mathbb{R}\mathrm{P}^{2}=\mathbb{F}_{2}[t,y]/(f)$ where
$y$ restricts to $x$ and $f=y^{i}(y+t)^{j}$ for $i+j=3$. It is easy to check
that this computation agrees with theorem 7.1.
## References
* [BHZ] D.Bourguiba, S.Hammouda, S.Zarati: Profondeur et cohomologie équivariante, African Diaspora Mathematics Research, Special Issue Vol 4 Number 3, 11-21.
* [De] F.X.Dehon Cobordisme complexe des espaces profinis et foncteur $\mathrm{T}$ de Lannes, Mémoires de la Société Mathématique de France 98, SMF 2004.
* [DL] F.X.Dehon, J.Lannes: Sur les espaces fonctionnels dont la source est le classifiant d’un groupe de Lie compact, commutatif I.H.E.S. 89 (1999) 127-177.
* [DW] W.G.Dwyer, C.W.Wilkerson: Smith theory revisited, Annals of Mathematics, 127(1988) 191-198.
* [EP] M.J.Errockh, C.Peterson: Injective resolutions of unstable modules, Journal of Pure and Applied Algebra 97(1994) 37-50.
* [Hu] D.Husemoller: Fibre bundles, McGraw-Hill, series in higher mathematics, 1966.
* [L1] J.Lannes: Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire, Publ. I.H.E.S. 75 (1992) 135-224.
* [LS] J.Lannes, L.Shwartz: Sur la structure des $A$-modules instables injectifs, Topology 28 (1989) 153-169.
* [LZ1] J.Lannes, S.Zarati: Sur les $\mathcal{U}$-injectifs, Ann. Scient. Ec. Norm. Sup. 19 (1986) 1-31.
* [LZ2] J.Lannes, S.Zarati: Théorie de Smith algébrique et classification des $\mathrm{H}^{\ast}V-\mathcal{U}$-injectifs, Bull. Soc. Math. France 123 (1995) 189-223.
* [LZ3] J.Lannes, S.Zarati: Tor et Ext-dimensions des $\mathrm{H}^{*}V-\mathrm{A}$-modules instables qui sont de type fini comme $\mathrm{H}^{*}V$-modules, Progress in Mathematics, Birkhäuser Verlag, vol 136 (1996) 241-253.
* [NS] M.D.Neusel, L.Smith: Invariant theory of finite groups, volume 94 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002.
* [R] J.Rotman: An introduction to homological algebra, Academic Press, 1979.
* [S] L.Schwartz: Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, University of Chicago Press, 1984.
* [Se] J-P.Serre: Sur la dimension cohomologique des groupes profinis, Topology 3. (1965), 413-420.
|
arxiv-papers
| 2009-04-18T10:19:11 |
2024-09-04T02:49:01.998662
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dorra Bourguiba",
"submitter": "Dorra Bourguiba",
"url": "https://arxiv.org/abs/0904.2839"
}
|
0904.2990
|
# Probing high-density behavior of symmetry energy from pion emission in
heavy-ion collisions
Zhao-Qing Fenga111Corresponding author. Tel. +86 931 4969215.
_E-mail address:_ fengzhq@impcas.ac.cn, Gen-Ming Jinb
a _Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
People’s Republic of China_
Abstract
Within the framework of the improved isospin dependent quantum molecular
dynamics (ImIQMD) model, the emission of pion in heavy-ion collisions in the
region 1 A GeV as a probe of nuclear symmetry energy at supra-saturation
densities is investigated systematically, in which the pion is considered to
be mainly produced by the decay of resonances $\triangle$(1232) and N*(1440).
The total pion multiplicities and the $\pi^{-}/\pi^{+}$ yields are calculated
for selected Skyrme parameters SkP, SLy6, Ska and SIII, and also for the cases
of different stiffness of symmetry energy with the parameter SLy6. Preliminary
results compared with the measured data by the FOPI collaboration favor a hard
symmetry energy of the potential term proportional to
$(\rho/\rho_{0})^{\gamma_{s}}$ with $\gamma_{s}=2$.
_PACS_ : 25.75.-q, 13.75.Gx, 25.80.Ls
_Keywords:_ ImIQMD model; pion emission; Skyrme parameters; symmetry energy
Heavy-ion collisions induced by radioactive beam at intermediate energies play
a significant role to extract the information of nuclear equation of state
(EoS) of isospin asymmetric nuclear matter under extreme conditions. Besides
nucleonic observables such as rapidity distribution and flow of free nucleons
and light clusters (such as deuteron, triton and alpha etc.), also mesons
emitted from the reaction zone can be probes of the hot and dense nuclear
matter. The energy per nucleon in the isospin asymmetric nuclear matter is
usually expressed as
$E(\rho,\delta)=E(\rho,\delta=0)+E_{\textrm{sym}}(\rho)\delta^{2}+\textsc{O}(\delta^{2})$
in terms of baryon density $\rho=\rho_{n}+\rho_{p}$, relative neutron excess
$\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$, energy per nucleon in a
symmetric nuclear matter $E(\rho,\delta=0)$ and bulk nuclear symmetry energy
$E_{\textrm{sym}}=\frac{1}{2}\frac{\partial^{2}E(\rho,\delta)}{\partial\delta^{2}}\mid_{\delta=0}$.
In general, two different forms have been predicted by some microscopical or
phenomenological many-body approaches. One is the symmetry energy increases
monotonically with density, and the other is the symmetry energy increases
initially up to a supra-saturation density and then decreases at higher
densities. Based on recent analysis of experimental data associated with
transport models, a symmetry energy of the form $E_{\textrm{sym}}(\rho)\approx
31.6(\rho/\rho_{0})^{\gamma}$ MeV with $\gamma=0.69-1.05$ was extracted for
densities between 0.1$\rho_{0}$ and 1.2$\rho_{0}$ [1, 2]. The symmetry energy
at supra-saturation densities can be investigated by analyzing isospin
sensitive observables in theoretically, such as the neutron/proton ratio of
emitted nucleons, $\pi^{-}/\pi^{+}$, $\Sigma^{-}/\Sigma^{+}$ and $K^{0}/K^{+}$
[2]. Recently, a very soft symmetry energy at supra-saturation densities was
pointed out by fitting the FOPI data [3] using IBUU04 model [4]. With the
establishment of high-energy radioactive beam facilities in the world, such as
the CSR (IMP in Lanzhou, China), FAIR (GSI in Darmstadt, Germany), RIKEN
(Japan), SPIRAL2 (GANIL in Caen, France) and FRIB (MSU, USA) [2], the high-
density behavior of the symmetry energy can be studied more detail
experimentally in the near future. The emission of pion in heavy-ion
collisions in the region 1 A GeV is especially sensitive as a probe of
symmetry energy at supra-saturation densities. Further investigations of the
pion emissions in the 1 A GeV region are still necessary by improving
transport models or developing some new approaches. The ImIQMD model has been
successfully applied to treat heavy-ion fusion reactions near Coulomb barrier
[5, 6, 7]. Recently, Zhang _et al_ analyzed the neutron-proton spectral double
ratios to extract the symmetry energy per nucleon at sub-saturation density
with a similar model [8]. To investigate the pion emission, we further include
the inelastic channels in nucleon-nucleon collisions.
In the ImIQMD model, the time evolutions of the baryons and pions in the
system under the self-consistently generated mean-field are governed by
Hamilton’s equations of motion, which read as
$\displaystyle\dot{\mathbf{p}}_{i}=-\frac{\partial
H}{\partial\mathbf{r}_{i}},\quad\dot{\mathbf{r}}_{i}=\frac{\partial
H}{\partial\mathbf{p}_{i}}.$ (1)
Here we omit the shell correction part in the Hamiltonian $H$ as described in
Ref. [6]. The Hamiltonian of baryons consists of the relativistic energy, the
effective interaction potential and the momentum dependent part as follows:
$H_{B}=\sum_{i}\sqrt{\textbf{p}_{i}^{2}+m_{i}^{2}}+U_{int}+U_{mom}.$ (2)
Here the $\textbf{p}_{i}$ and $m_{i}$ represent the momentum and the mass of
the baryons.
The effective interaction potential is composed of the Coulomb interaction and
the local interaction
$U_{int}=U_{Coul}+U_{loc}.$ (3)
The Coulomb interaction potential is written as
$U_{Coul}=\frac{1}{2}\sum_{i,j,j\neq
i}\frac{e_{i}e_{j}}{r_{ij}}erf(r_{ij}/\sqrt{4L})$ (4)
where the $e_{j}$ is the charged number including protons and charged
resonances. The $r_{ij}=|\mathbf{r}_{i}-\mathbf{r}_{j}|$ is the relative
distance of two charged particles.
The local interaction potential is derived directly from the Skyrme energy-
density functional and expressed as
$U_{loc}=\int V_{loc}(\rho(\mathbf{r}))d\mathbf{r}.$ (5)
The local potential energy-density functional reads
$\displaystyle
V_{loc}(\rho)=\frac{\alpha}{2}\frac{\rho^{2}}{\rho_{0}}+\frac{\beta}{1+\gamma}\frac{\rho^{1+\gamma}}{\rho_{0}^{\gamma}}+\frac{g_{sur}}{2\rho_{0}}(\nabla\rho)^{2}+\frac{g_{sur}^{iso}}{2\rho_{0}}[\nabla(\rho_{n}-\rho_{p})]^{2}+$
$\displaystyle\left(a_{sym}\frac{\rho^{2}}{\rho_{0}}+b_{sym}\frac{\rho^{1+\gamma}}{\rho_{0}^{\gamma}}+c_{sym}\frac{\rho^{8/3}}{\rho_{0}^{5/3}}\right)\delta^{2}+g_{\tau}\rho^{8/3}/\rho_{0}^{5/3},$
(6)
where the $\rho_{n}$, $\rho_{p}$ and $\rho=\rho_{n}+\rho_{p}$ are the neutron,
proton and total densities, respectively, and the
$\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$ is the isospin asymmetry. The
coefficients $\alpha$, $\beta$, $\gamma$, $g_{sur}$, $g_{sur}^{iso}$,
$g_{\tau}$ are related to the Skyrme parameters $t_{0},t_{1},t_{2},t_{3}$ and
$x_{0},x_{1},x_{2},x_{3}$ [6]. The parameters of the potential part in the
symmetry energy term are also derived directly from Skyrme energy-density
parameters as
$\displaystyle a_{sym}=-\frac{1}{8}(2x_{0}+1)t_{0}\rho_{0},\quad
b_{sym}=-\frac{1}{48}(2x_{3}+1)t_{3}\rho_{0}^{\gamma},$ $\displaystyle
c_{sym}=-\frac{1}{24}\left(\frac{3}{2}\pi^{2}\right)^{2/3}\rho_{0}^{5/3}[3t_{1}x_{1}-t_{2}(5x_{2}+4)].$
(7)
The momentum dependent term in the Hamiltonian is the same of the form in Ref.
[9] and expressed as
$U_{mom}=\frac{\delta}{2}\sum_{i,j,j\neq
i}\frac{\rho_{ij}}{\rho_{0}}[\ln(\epsilon(\textbf{p}_{i}-\textbf{p}_{j})^{2}+1)]^{2},$
(8)
with
$\rho_{ij}=\frac{1}{(4\pi
L)^{3/2}}\exp\left[-\frac{(\textbf{r}_{i}-\textbf{r}_{j})^{2}}{4L}\right],$
(9)
which does not distinguish between protons and neutrons. Here the $L$ denotes
the square of the pocket-wave width, which is dependent on the mass number of
the nucleus. The parameters $\delta$ and $\epsilon$ were determined by fitting
the real part of the proton-nucleus optical potential as a function of
incident energy.
In Table 1 we list the ImIQMD parameters related to several typical Skyrme
forces after including the momentum dependent interaction. The parameters
$\alpha$, $\beta$ and $\gamma$ are redetermined in order to reproduce the
binding energy ($E_{B}$=-16 MeV) of symmetric nuclear matter at saturation
density $\rho_{0}$ and to satisfy the relation $\frac{\partial
E/A}{\partial\rho}\mid_{\rho=\rho_{0}}$=0 for a given incompressibility.
Combined Eq.(7) with the kinetic energy part, the symmetry energy per nucleon
in the ImIQMD model is given by
$E_{sym}(\rho)=\frac{1}{3}\frac{\hbar^{2}}{2m}\left(\frac{3}{2}\pi^{2}\rho\right)^{2/3}+a_{sym}\frac{\rho}{\rho_{0}}+b_{sym}\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}+c_{sym}\left(\frac{\rho}{\rho_{0}}\right)^{5/3}.$
(10)
More clearly compared with other transport models, the symmetry energy can be
expressed as
$E_{sym}(\rho)=\frac{1}{3}\frac{\hbar^{2}}{2m}\left(\frac{3}{2}\pi^{2}\rho\right)^{2/3}+\frac{1}{2}C_{sym}\left(\frac{\rho}{\rho_{0}}\right)^{\gamma_{s}}.$
(11)
The value $\gamma_{s}=1$ is used in IQMD model [10, 11]. In Fig. 1 we show a
comparison of the energy per nucleon in symmetric nuclear matter with and
without the momentum dependent potentials in the left panel and the nuclear
symmetry energy in the right panel for different cases of Skyrme forces SkP,
Sly6, Ska and SIII from Eq. (10), $\gamma_{s}$=0.5 (soft) and 2 (hard) with
$C_{sym}$=32 MeV in Eq. (11), and also compared with the form
$E_{sym}=31.6(\rho/\rho_{0})^{\mu}$ MeV ($\mu$=0.5 and $\mu$=2) [1].
Analogously to baryons, the Hamiltonian of pions is represented as
$H_{\pi}=\sum_{i=1}^{N_{\pi}}\left(\sqrt{\textbf{p}_{i}^{2}+m_{\pi}^{2}}+V_{i}^{Coul}\right),$
(12)
where the $\textbf{p}_{i}$ and $m_{\pi}$ represent the momentum and the mass
of the pions. The Coulomb interaction is given by
$V_{i}^{Coul}=\sum_{j=1}^{N_{B}}\frac{e_{i}e_{j}}{r_{ij}},$ (13)
where the $N_{\pi}$ and $N_{B}$ is the total number of pions and baryons
including charged resonances. Thus, the pion propagation in the whole stage is
guided essentially by the Coulomb force. The in-medium pion potential in the
mean field is not considered in the model. However, the inclusion of the pion
optical potential based on the perturbation expansion of the $\Delta$-hole
model gives negligible influence on the transverse momentum distribution [12].
The pion is created by the decay of the resonances $\triangle$(1232) and
N*(1440) which are produced in inelastic NN scattering. The cross section of
direct pion production is very small in the considered energies and not
included in the model [13]. The reaction channels are given as follows:
$\displaystyle NN\leftrightarrow N\triangle,$ $\displaystyle NN\leftrightarrow
NN^{\ast},$ $\displaystyle NN\leftrightarrow\triangle\triangle,$
$\displaystyle\Delta\leftrightarrow N\pi,$ $\displaystyle
N^{\ast}\leftrightarrow N\pi.$ (14)
The cross sections of each channel to produce resonances are parameterized by
fitting the data calculated with the one-boson exchange model [14]. In the 1 A
GeV region, there are mostly $\Delta$ resonances which disintegrate into a
$\pi$ and a nucleon, however, the $N^{\ast}$ yet gives considerable
contribution to the high energetic pion yield. The energy and momentum
dependent decay width is used in the calculation [15].
Pion meson in heavy-ion collisions is mainly produced at supra-saturation
densities of compressed nuclear matter larger than the normal density
$\rho_{0}$. The production of pions is influenced by the $\triangle$(1232) and
the Fermi motion of baryons in the vicinity of the threshold energies. The
$\pi^{-}$/$\pi^{+}$ ratio is a sensitive probe to extract the high-density
behavior of the symmetry energy per energy. Shown in Fig. 2 is a comparison of
the measured total pion multiplicity and $\pi^{-}$/$\pi^{+}$ yields by the
FOPI collaboration in central 197Au+197Au collisions [3] and the results
calculated by IQMD model [10] as well as by the ImIQMD model for Skyrme
parameters SkP, SLy6, Ska and SIII, which correspond to different modulus of
incompressibility as listed in table 1. The total multiplicity of pion is
mainly determined by the cross sections of the channels $NN\leftrightarrow
N\triangle$. The ImIQMD model with four Skyrme parameters predicts rather well
the total yields at higher incident energies, but slightly overestimates the
values near threshold energies, which may be influenced by the in-medium cross
sections. In this work, we use the in-vacuum cross sections of nucleon-nucleon
elastic and inelastic collisions. Reasonable consideration of the in-medium
inelastic collisions in producing $\Delta$ and $N^{\ast}$ is still an open
problem in transport models, which have been performed in Giessen-BUU model
[16]. Using the isobar model, one gets the ratio $\pi^{-}$/$\pi^{+}$=1.95 for
pions from the $\Delta$ resonance, and $\pi^{-}$/$\pi^{+}$=1.7 from the
$N^{\ast}$ for the system 197Au+197Au [17]. These relations are globally
valid, i.e. independent of the pion energy. On the other hand, the statistical
model predicts that the $\pi^{-}$/$\pi^{+}$ ratio is sensitive to the
difference in the chemical potentials of neutrons and protons by the relation
$\pi^{-}/\pi^{+}\propto\exp[2(\mu_{n}-\mu_{p})/T]=\exp[8\delta
E_{sym}(\rho)/T]$, where the $T$ is nuclear temperature [18]. The observed
energy dependence of the $\pi^{-}$/$\pi^{+}$ ratio is due to the re-scattering
and absorption process of pions and nucleons in the mean field of the
compressed nuclear matter. We use the free absorption cross sections in
collisions of pions and nucleons by fitting the experimental data. The branch
ratio of the charged $\pi$ and $\pi^{0}$ is determined by the Clebsch-Gordan
coefficients with the decay of the resonances $\triangle$(1232) and N*(1440).
The $\pi^{-}/\pi^{+}$ ratio is sensitive to the stiffness of the symmetry
energy at the lower incident energies. The ImIQMD model can predict the
decrease trend of the $\pi^{-}/\pi^{+}$ ratio with incident energy. While the
ImIQMD model with different Skyrme parameters gives the same excitation
functions of the total pion multiplicity owing to the same cross sections in
the production of pions and resonances for each case, the $\pi^{-}/\pi^{+}$
yields is different resulting from the symmetry energy.
The compressed nuclear matter with central density about two times of the
normal density is formed in heavy-ion collisions in the 1 A GeV region. To
extract more information of symmetry energy in heavy-ion collisions from the
pion production, in Fig. 3 we calculated the time evolution of average central
density from low to high incident energies and the excitation functions of the
$\pi^{-}$/$\pi^{+}$ ratios with the force SLy6, but different stiffness of the
symmetry energy which corresponds to hard ($\gamma_{s}$=2), linear
($\gamma_{s}$=1), soft ($\gamma_{s}$=0.5) and supersoft (SIII)), and also
compared with IQMD results [10] as well as the FOPI data [3]. The ImIQMD model
gives larger values of $\pi^{-}$/$\pi^{+}$ than the ones calculated by IQMD,
which mainly results from the cross section of the channel
$N\pi\rightarrow\Delta$ and the larger coefficient $C_{sym}$. We considered
the pion absorption process according to the Breit-Wigner formula with the
cross section given in Ref. [13]. Our calculations show that a stiff symmetry
energy is close to experimental data. The results does not support a very soft
symmetry energy at high-density from analyzing the same experimental data
reported in Ref. [4]. Situation is different in IBUU04 model, each nucleon in
the evolution is enforced by the symmetry potential associated with isospin
and momentum. Inversely, a transport model reported in Ref. [19] also
predicted the larger ratios for stiffer symmetry energy from the analysis of
the $\pi^{-}$/$\pi^{+}$ and $K^{0}/K^{+}$ yields. The influence of the
symmetry energy on pion production in heavy-ion collisions is also studied
from the distribution of transverse momentum of the total charged pions and
the ratio $\pi^{-}$/$\pi^{+}$ for the cases of stiff and soft symmetry
energies as shown in Fig. 4. The $\pi^{-}$ mesons are mostly produced from
neutron-neutron collisions, and for a stiff symmetry energy, a wider high-
density zone is formed in the calculation of the ImIQMD model. The larger
$\pi^{-}$/$\pi^{+}$ ratio is also clear in the momentum distribution and the
larger errors at the higher transverse momentum are resulted from the limited
simulation events.
The final $\pi^{-}$/$\pi^{+}$ ratio with different stiffness of the symmetry
energy is shown in Fig. 5 as a function of N/Z of the systems in the reactions
40Ca+40Ca, 96Ru+96Ru, 96Zr+96Zr and 197Au+197Au, and also plotted the ratios
of N/Z and (N/Z)2 as a function of N/Z at incident energy 0.4A GeV and 1.5A
GeV, respectively. The FOPI data [3] and the results calculated by IQMD model
[10] are also given for a comparison. Experimental data and calculations show
that an increase trend of the $\pi^{-}$/$\pi^{+}$ ratio in realistic heavy ion
collisions than that predicted by the isobar model is found at near threshold
energy 0.4A GeV, especially for the larger N/Z systems. The ratio decreases
with the incident energy and the value is located between the lines of (N/Z)2
and N/Z at incident energy 1.5A GeV. The phenomena can be explained from the
fact that the symmetry energy enhances the N/Z ratio in the high-density
region at lower incident energy. The decrease of the $\pi^{-}$/$\pi^{+}$ ratio
with the incident energy is mainly owing to the production of pions from
secondary nucleon-nucleon collisions, such as a neutron converts a proton by
producing $\pi^{-}$. Subsequent collisions of the energetic proton can convert
again to neutron by producing $\pi^{+}$. One can see that the stiff symmetry
energy is also close to the experimental data. Recently, a moderately soft
symmetry energy with $\gamma_{s}\simeq 0.9\pm 0.3$ was extracted from the
analysis of neutron-proton elliptic flow of the FOPI/LAND data for the
reaction 197Au+197Au using the UrQMD model [20]. Further experimental works
associated transport models should be performed in more details to get
reliable information of the high-density trend of the symmetry energy in
heavy-ion collisions.
In summary, the pion production in heavy-ion collisions in the region 1 A GeV
is investigated systematically by using the ImIQMD model. The total
multiplicity of produced pion and the $\pi^{-}/\pi^{+}$ ratio in central
collisions are calculated for the selected Skyrme parameters SkP, SLy6, Ska,
SIII which correspond to different modulus of incompressibility of symmetric
nuclear matter and different cases of the stiffness of symmetry energy, and
compared them with the experimental data by the FOPI collaborations as well as
IQMD results. The $\pi^{-}/\pi^{+}$ excitation functions for the reaction
197Au+197Au and the dependence of the $\pi^{-}/\pi^{+}$ ratio on N/Z of
reaction systems at energy 0.4A GeV are compared with the force SLy6, but
different stiffness of the symmetry energy. Calculations show that a stiffer
symmetry energy of the potential term with $\gamma_{s}=2$ is close to the
experimental data.
Acknowledgements
This work was supported by the National Natural Science Foundation of China
under Grant Nos. 10805061 and 10775061, the special foundation of the
president fund, the west doctoral project of Chinese Academy of Sciences, and
major state basic research development program under Grant No. 2007CB815000.
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* [7] N. Wang, Z.X. Li, X.Z Wu, et al., Phys. Rev. C 69 (2004) 034608.
* [8] Y.X. Zhang, P. Danielewicz, M. Famiano, et al., Phys. Lett. B 664 (2008) 145.
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* [16] A.B. Larionov, W. Cassing, S. Leupold, U. Mosel, Nucl. Phys. A 696 (2001) 747; A.B. Larionov, U. Mosel, Nucl. Phys. A 728 (2003) 135\.
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* [20] W. Trautmann, M. Chartier, Y. Leifels, el al., arXiv:0907.2822.
Table 1: ImIQMD parameters and properties of symmetric nuclear matter for
Skyrme effective interactions after the inclusion of the momentum dependent
interaction with parameters $\delta$=1.57 MeV and $\epsilon$=500 c2/GeV2
Parameters SkM* Ska SIII SVI SkP RATP SLy6 $\alpha$ (MeV) -325.1 -179.3 -128.1
-123.0 -357.7 -250.3 -296.7 $\beta$ (MeV) 238.3 71.9 42.2 51.6 286.3 149.6
199.3 $\gamma$ 1.14 1.35 2.14 2.14 1.15 1.19 1.14 $g_{sur}$(MeV fm2) 21.8 26.5
18.3 14.1 19.5 25.6 22.9 $g_{sur}^{iso}$(MeV fm2) -5.5 -7.9 -4.9 -3.0 -11.3
0.0 -2.7 $g_{\tau}$ (MeV) 5.9 13.9 6.4 1.1 0.0 11.0 9.9 $C_{sym}$ (MeV) 30.1
33.0 28.2 27.0 30.9 29.3 32.0 $a_{sym}$ (MeV) 62.4 29.8 38.9 42.9 94.0 79.3
130.6 $b_{sym}$ (MeV) -38.3 -5.9 -18.4 -22.0 -63.5 -58.2 -123.7 $c_{sym}$
(MeV) -6.4 -3.0 -3.8 -5.5 -13.0 -4.1 12.8 $\rho_{\infty}$ (fm-3) 0.16 0.155
0.145 0.144 0.162 0.16 0.16 $m_{\infty}^{\ast}/m$ 0.639 0.51 0.62 0.73 0.77
0.56 0.57 $K_{\infty}$ (MeV) 215 262 353 366 200 239 230
Figure 1: The density dependence of the energy per nucleon in symmetric
nuclear matter at temperature T=0 MeV with and without the momentum dependent
potentials (left panel) and comparison of the density dependence of the
nuclear symmetry energy for different Skyrme forces SkP, Sly6, Ska and SIII,
and the symmetry energy $E_{sym}=31.6(\rho/\rho_{0})^{\gamma}$ MeV (the two
cases $\gamma$=0.5 and $\gamma$=2) taken in Refs. [1, 2] (right panel). Figure
2: Comparison of calculated pion multiplicity and $\pi^{-}$/$\pi^{+}$ ratios
in central 197Au+197Au collisions with different Skyrme parameters, and
compared with IQMD results [10] as well as the FOPI data [3]. Figure 3:
Evolution of average central density at different incident energies (left
panel) and the excitation functions of the $\pi^{-}$/$\pi^{+}$ ratios at
different stiffness of the symmetry energy (hard, linear, soft and supersoft),
and compared with IQMD results [10] as well as the FOPI data [3] (right
panel). Figure 4: Distributions of transverse momentum of final $\pi^{-}$ and
$\pi^{+}$ and the ratio $\pi^{-}$/$\pi^{+}$ for the cases of stiff and soft
symmetry energies in the reaction 197Au+197Au at incident energy $E_{lab}=$
0.4A GeV. Figure 5: The $\pi^{-}$/$\pi^{+}$ yields as a function of the
neutron over proton N/Z of reaction systems for head on collisions at incident
energy $E_{lab}=$ 0.4A GeV and 1.5A GeV, respectively.
|
arxiv-papers
| 2009-04-20T09:41:03 |
2024-09-04T02:49:02.015655
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhao-Qing Feng, Gen-Ming Jin",
"submitter": "Zhaoqing Feng",
"url": "https://arxiv.org/abs/0904.2990"
}
|
0904.2994
|
# Influence of entrance channels on formation of superheavy nuclei in massive
fusion reactions
Zhao-Qing Fenga111Corresponding author. Tel. +86 931 4969215.
_E-mail address:_ fengzhq@impcas.ac.cn, Jun-Qing Lia, Gen-Ming Jina
a _Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China_
Abstract
Within the framework of the dinuclear system (DNS) model, the production cross
sections of superheavy nuclei Hs (Z=108) and Z=112 combined with different
reaction systems are analyzed systematically. It is found that the mass
asymmetries and the reaction Q values of the combinations play a very
important role on the formation cross sections of the evaporation residues.
Both methods by solving the master equations along the mass asymmetry degree
of freedom (1D) and along the proton and the neutron degrees of freedom (2D)
are compared each other and with the available experimental results.
_PACS:_ 25.70.Jj, 24.10.-i, 25.60.Pj
_Keywords:_ DNS model; production cross sections; mass asymmetries; reaction Q
values
The synthesis of heavy or superheavy nuclei (SHN) is a very important subject
in nuclear physics motivated with respect to the island of stability which is
predicted theoretically, and has obtained much experimental research with
fusion-evaporation reactions [1, 2]. Combinations with a doubly magic nucleus
or nearly magic nucleus are usually chosen owing to the larger reaction $Q$
values. Six new elements with Z=107-112 were synthesized in cold fusion
reactions for the first time and investigated at GSI (Darmstadt, Germany) with
the heavy-ion accelerator UNILAC and the SHIP separator [1, 3]. Recently,
experiments on the synthesis of element 113 in the 70Zn+209Bi reaction have
been performed successfully at RIKEN (Tokyo, Japan) [4]. However, it is
difficulty to produce heavier SHN in the cold fusion reactions because of the
smaller production cross sections that are lower than 1 pb for $Z>113$. The
superheavy elements Z=113-116, 118 were synthesized at FLNR in Dubna (Russia)
with the double magic nucleus 48Ca bombarding actinide nuclei [5, 6, 7]. New
heavy isotopes 259Db and 265Bh have also been synthesized at HIRFL in Lanzhou
(China) [8]. Further experimental works are necessary in order to testify the
new synthesized SHN. A better understanding of the formation of SHN in the
massive fusion reactions is still a challenge for theory.
In this letter, we focus on the influence of the entrance mass asymmetry and
the reaction Q value of projectile-target combinations on the production cross
sections of superheavy residues. In the DNS model, the evaporation residue
cross section is expressed as a sum over partial waves with angular momentum
$J$ at the centre-of-mass energy $E_{c.m.}$ [9, 10, 11],
$\sigma_{ER}(E_{c.m.})=\frac{\pi\hbar^{2}}{2\mu
E_{c.m.}}\sum_{J=0}^{J_{max}}(2J+1)T(E_{c.m.},J)P_{CN}(E_{c.m.},J)W_{sur}(E_{c.m.},J).$
(1)
Here, $T(E_{c.m.},J)$ is the transmission probability of the two colliding
nuclei overcoming the Coulomb potential barrier in the entrance channel to
form the DNS. The $P_{CN}$ is the probability that the system will evolve from
a touching configuration into the compound nucleus in competition with quasi-
fission of the DNS and fission of the heavy fragment. The last term is the
survival probability of the formed compound nucleus, which can be estimated
with the statistical evaporation model by considering the competition between
neutron evaporation and fission [9]. We take the maximal angular momentum as
$J_{max}=30$ since the fission barrier of the heavy nucleus disappears at high
spin [12].
In order to describe the fusion dynamics as a diffusion process along proton
and neutron degrees of freedom, the fusion probability is obtained by solving
a set of master equations numerically in the potential energy surface of the
DNS. The time evolution of the distribution probability function
$P(Z_{1},N_{1},E_{1},t)$ for fragment 1 with proton number $Z_{1}$ and neutron
number $N_{1}$ with excitation energy $E_{1}$ is described by the following
master equations [13],
$\displaystyle\frac{dP(Z_{1},N_{1},E_{1},t)}{dt}=\sum_{Z_{1}^{\prime}}W_{Z_{1},N_{1};Z_{1}^{\prime},N_{1}}(t)\left[d_{Z_{1},N_{1}}P(Z_{1}^{\prime},N_{1},E_{1}^{\prime},t)-d_{Z_{1}^{\prime},N_{1}}P(Z_{1},N_{1},E_{1},t)\right]+$
$\displaystyle\sum_{N_{1}^{\prime}}W_{Z_{1},N_{1};Z_{1},N_{1}^{\prime}}(t)\left[d_{Z_{1},N_{1}}P(Z_{1},N_{1}^{\prime},E_{1}^{\prime},t)-d_{Z_{1},N_{1}^{\prime}}P(Z_{1},N_{1},E_{1},t)\right]-$
$\displaystyle\left[\Lambda^{qf}(\Theta(t))+\Lambda^{fis}(\Theta(t))\right]P(Z_{1},N_{1},E_{1},t).$
(2)
Here $W_{Z_{1},N_{1};Z_{1}^{\prime},N_{1}}$
($W_{Z_{1},N_{1};Z_{1},N_{1}^{\prime}}$) is the mean transition probability
from the channel $(Z_{1},N_{1},E_{1})$ to
$(Z_{1}^{\prime},N_{1},E_{1}^{\prime})$ (or $(Z_{1},N_{1},E_{1})$ to
$(Z_{1},N_{1}^{\prime},E_{1}^{\prime})$) , and $d_{Z_{1},N_{1}}$ denotes the
microscopic dimension corresponding to the macroscopic state
$(Z_{1},N_{1},E_{1})$. The sum is taken over all possible proton and neutron
numbers that fragment $Z_{1}^{\prime},N_{1}^{\prime}$ may take, but only one
nucleon transfer is considered in the model with $Z_{1}^{\prime}=Z_{1}\pm 1$
and $N_{1}^{\prime}=N_{1}\pm 1$. The excitation energy $E_{1}$ is determined
by the dissipation energy from the relative motion and the potential energy
surface of the DNS. The motion of nucleons in the interacting potential is
governed by the single-particle Hamiltonian [9, 10]. The evolution of the DNS
along the variable R leads to the quasi-fission of the DNS. The quasi-fission
rate $\Lambda^{qf}$ and the fission rate $\Lambda^{fis}$ can be estimated with
the one-dimensional Kramers formula [10, 11].
In the relaxation process of the relative motion, the DNS will be excited by
the dissipation of the relative kinetic energy. The local excitation energy is
determined by the excitation energy of the composite system and the potential
energy surface of the DNS. The potential energy surface (PES) of the DNS is
given by
$\displaystyle
U(Z_{1},N_{1},Z_{2},N_{2};J,\textbf{R};\beta_{1},\beta_{2},\theta_{1},\theta_{2})=B(Z_{1},N_{1})+B(Z_{2},N_{2})-\left[B(Z,N)+V^{CN}_{rot}(J)\right]+$
$\displaystyle
V(Z_{1},N_{1},Z_{2},N_{2};J,\textbf{R};\beta_{1},\beta_{2},\theta_{1},\theta_{2})$
(3)
with $Z_{1}+Z_{2}=Z$ and $N_{1}+N_{2}=N$. Here $B(Z_{i},N_{i})(i=1,2)$ and
$B(Z,N)$ are the negative binding energies of the fragment $(Z_{i},N_{i})$ and
the compound nucleus $(Z,N)$, respectively, in which the shell and the pairing
corrections are included reasonably. The $V^{CN}_{rot}$ is the rotation energy
of the compound nucleus. The $\beta_{i}$ represent the quadrupole deformations
of the two fragments. The $\theta_{i}$ denote the angles between the collision
orientations and the symmetry axes of deformed nuclei. The interaction
potential between fragment $(Z_{1},N_{1})$ and $(Z_{2},N_{2})$ includes the
nuclear, Coulomb and centrifugal parts, the details are given in Ref. [10]. In
the calculation, the distance R between the centers of the two fragments is
chosen to be the value which gives the minimum of the interaction potential,
in which the DNS is considered to be formed. So the PES depends on the proton
and neutron numbers of the fragment. In Fig.1 we give the potential energy
surface in the reaction 30Si+252Cf as functions of the protons and neutrons of
the fragments in the left panel. The incident point is shown by the solid
circle and the minimum way in the PES is added by the thick line. The driving
potential as a function of the mass asymmetry that was calculated in Ref. [9,
10] is also given in the right panel and compared with the minimum way in the
left panel. The driving potential at the incident point in 1D PES is located
at the maximum value, so there is no the inner fusion barrier for the system,
which results in a too large fusion probability. Therefore, we solve the
master equations within the 2D PES to get the fusion probability for the
systems with larger mass asymmetries.
The formation probability of the compound nucleus at the Coulomb barrier $B$
(here a barrier distribution $f(B)$ is considered) and for angular momentum
$J$ is given by[9, 10]
$P_{CN}(E_{c.m.},J,B)=\sum_{Z_{1}=1}^{Z_{BG}}\sum_{N_{1}=1}^{N_{BG}}P(Z_{1},N_{1},E_{1},\tau_{int}(E_{c.m.},J,B)).$
(4)
We obtain the fusion probability as
$P_{CN}(E_{c.m.},J)=\int f(B)P_{CN}(E_{c.m.},J,B)dB,$ (5)
where the barrier distribution function is taken in asymmetric Gaussian form.
The survival probability of the excited compound nucleus cooled by the neutron
evaporation in competition with fission is expressed as follows:
$W_{sur}(E_{CN}^{\ast},x,J)=P(E_{CN}^{\ast},x,J)\prod\limits_{i=1}^{x}\left(\frac{\Gamma_{n}(E_{i}^{\ast},J)}{\Gamma_{n}(E_{i}^{\ast},J)+\Gamma_{f}(E_{i}^{\ast},J)}\right)_{i},$
(6)
where the $E_{CN}^{\ast},J$ are the excitation energy and the spin of the
compound nucleus, respectively. The $E_{i}^{\ast}$ is the excitation energy
before evaporating the $i$th neutron, which has the relation
$E_{i+1}^{\ast}=E_{i}^{\ast}-B_{i}^{n}-2T_{i},$ (7)
with the initial condition $E_{1}^{\ast}=E_{CN}^{\ast}$. The energy
$B_{i}^{n}$ is the separation energy of the $i$th neutron. The nuclear
temperature $T_{i}$ is given as $E_{i}^{\ast}=aT_{i}^{2}-T_{i}$ with the level
density parameter $a$. $P(E_{CN}^{\ast},x,J)$ is the realization probability
of emitting $x$ neutrons. The widths of neutron evaporation and fission are
calculated using the statistical model. The details can be found in Refs. [9,
11].
With this procedure introduced above, we calculated the evaporation residue
excitation functions using the 1D and 2D master equations in the reaction
48Ca+238U as shown in Fig.2 represented by dashed and solid lines,
respectively, and compared them with the experimental data performed in Dubna
[14] and at GSI [15]. The GSI results show that the formation cross sections
in the 3n channel at the same excitation energy with 35 MeV have a slight
decrease, which are in a good agreement with our 1D calculations. In the whole
range, the 2D calculations give smaller cross sections than 1D master
equations owing to the decrease of the fusion probability. For the considered
system, the value of the PES at the incident point is located at the line of
the minimum way. So the 1D master equations can give reasonable results.
However, for the systems with larger mass asymmetries and larger quadrupole
deformation parameters, e.g. 16O+238U, 22Ne+244Pu, etc, the 1D master
equations give too large fusion probabilities.
The synthesis of heavy or superheavy nuclei through fusing two stable nuclei
is inhibited by the so-called quasi-fission process. The entrance channel
combinations of projectile and target will influence the fusion dynamics. The
suppression of the evaporation residue cross sections for less fissile
compound systems such as 216Ra and 220Th when reactions are involved in
projectiles heavier than 12C and 16O was observed in Refs. [16]. The wider
width of the mass distributions for the fission-like fragments was also
reported in Ref. [17]. In Fig.3 we calculated the transmission and fusion
probabilities using the 2D master equations for the reactions 34S+238U,
64Fe+208Pb and 136Xe+136Xe which lead to the same compound nucleus 272Hs
formation. The larger transmission probabilities were found in the reactions
64Fe+208Pb and 136Xe+136Xe owing to the larger Q values (absolute values).
Smaller mass asymmetries of the two systems result in a decrease of the fusion
probabilities. The evaporation residue excitation functions in 1n-5n channels
are shown in Fig.4. The competition of the capture and the fusion process of
the three systems leads to different trends of the evaporation channels. The
3n and 4n channels in the reaction 34S+238U, 1n and 2n channels in the
reaction 64Fe+208Pb are favorable to produce the isotopes 269,268Hs and
271,270Hs. Although the system 136Xe+136Xe consists of two magic nuclei, the
higher inner fusion barrier decreases the fusion probabilities and enhances
the quasi-fission rate of the DNS, hence leads to the smaller cross sections
of the Hs isotopes. The upper limit cross sections for evaporation residues
$\sigma_{(1-3)n}\leq$4 pb were observed in a recent experiment [18], which are
much lower than the ones predicted by the fusion by diffusion model [19]. In
the DNS model, the larger mass asymmetry favors the nucleon transfer from the
light projectile to heavy target, and therefore enhances the fusion
probability of two colliding nuclei.
The superheavy element Z=112 was synthesized at GSI with the new isotope
277112 in cold fusion reaction 70Zn+208Pb [20] and also fabricated with more
neutron-rich isotopes 282,283112 in 48Ca induced reaction 48Ca+238U. We
analyzed the combinations 30Si+252Cf, 36S+250Cm, 40Ar+244Pu and 48Ca+238U
which lead to the production of new isotopes of the element Z=112 between the
cold fusion reactions and the 48Ca induced reactions as shown in Fig.5. The
2n, 3n and 4n channels in the reaction 30Si+252Cf, and the 4n channel in the
reaction 36S+250Cm have larger cross section to produce new isotopes due to
the larger fusion probabilities of the two colliding nuclei.
In summary, we systematically analyzed the entrance channel effects of
synthesizing SHN using the DNS model. The systems with larger entrance mass
asymmetry and larger reaction Q value can enhance the capture and fusion
probabilities of two colliding nuclei. Calculations were carried out for the
reactions 34S+238U, 64Fe+208Pb and 136Xe+136Xe which lead to the same compound
nucleus formation. The 2n, 3n and 4n channels in the reaction 30Si+252Cf, and
the 4n channel in the reaction 36S+250Cm are favorable to synthesize new
isotopes of the element Z=112 at the stated excitation energies.
Acknowledgements
We would like to thank Prof. Werner Scheid for carefully reading the
manuscript. This work was supported by the National Natural Science Foundation
of China under Grant No. 10805061, the special foundation of the president
fellowship, the west doctoral project of Chinese Academy of Sciences, and
major state basic research development program under Grant No. 2007CB815000.
## References
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* [2] Yu.Ts. Oganessian, J. Phys. G 34 (2007) R165; Nucl. Phys. A 787 (2007) 343c.
* [3] G. Münzenberg, J. Phys. G 25 (1999) 717.
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* [7] Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Phys. Rev. C 74 (2006) 044602.
* [8] Z.G. Gan, Z. Qin, H.M. Fan, et al., Eur. Phys. J. A 10 (2001) 21; Z.G. Gan, J.S. Guo, X.L. Wu, et al., Eur. Phys. J. A 20 (2004) 385.
* [9] Z.Q. Feng, G.M. Jin, F. Fu, J.Q. Li, Nucl. Phys. A 771 (2006) 50.
* [10] Z.Q. Feng, G.M. Jin, J.Q. Li, W. Scheid, Phys. Rev. C 76 (2007) 044606.
* [11] Z.Q. Feng, G.M. Jin, J.Q. Li, W. Scheid, Nucl. Phys. A 816 (2009) 33.
* [12] P. Reiter, T.L. Khoo, T. Lauritsen, et al., Phys. Rev. Lett. 84 (2000) 3542.
* [13] M.H. Huang, Z.G. Gan, Z.Q. Feng, et al., Chin. Phys. Lett. 25 (2008) 1243.
* [14] Yu.Ts. Oganessian, V.K. Utyonkov, Yu.V. Lobanov, et al., Phys. Rev. C 70 (2004) 064609.
* [15] S. Hofmann, D. Ackermann, S. Antalic, et al., Eur. Phys. J. A 32 (2007) 251.
* [16] A.C. Berriman, D.J. Hinde, M. Dasgupta, et al., Nature 413 (2001) 144; D.J. Hinde, M. Dasgupta, A. Mukherjee, Phys. Rev. Lett. 89 (2002) 282701.
* [17] R.G. Thomas, D.J. Hinde, D. Duniec, et al., Phys. Rev. C 77 (2008) 034610.
* [18] Yu.Ts. Oganessian, S.N. Dmitriev, A.V. Yeremin, et al., Phys. Rev. C 79 (2009) 024608.
* [19] W.J. Swiatecki, K. Siwek-Wilczynska, J. Wilczynski, Int. J. Mod. Phys. E 13 (2004) 261.
* [20] S. Hofmann, V. Ninov, F.P. Heßberger, et al., Z. Phys. A 350 (1995) 277.
Figure 1: The potential energy surface of the DNS in the reaction 30Si+252Cf
as functions of the protons and neutrons of the fragments (left panel) and the
mass asymmetry coordinate (right panel). Figure 2: Comparison of the
calculated evaporation residue excitation functions using the 1D and 2D master
equations with the available experimental data in the reaction 48Ca+238U.
Figure 3: Calculated transmission and fusion probabilities as functions of the
excitation energies of the compound nucleus for the reactions 34S+238U,
64Fe+208Pb and 136Xe+136Xe. Figure 4: Comparison of the calculated evaporation
residue cross sections in 1n-5n channels using the 2D master equations for the
reactions 34S+238U, 64Fe+208Pb and 136Xe+136Xe. Figure 5: The same as in
Fig.4, but for the reactions 30Si+252Cf, 36S+250Cm, 40Ar+244Pu and 48Ca+238U
leading to the formation of the element Z=112.
|
arxiv-papers
| 2009-04-20T09:58:42 |
2024-09-04T02:49:02.021746
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhao-Qing Feng, Jun-Qing Li, Gen-Ming Jin",
"submitter": "Zhaoqing Feng",
"url": "https://arxiv.org/abs/0904.2994"
}
|
0904.2996
|
# Pion Production in Heavy-ion Collisions in the 1 A GeV region 111Supported
by the National Natural Science Foundation of China under Grant No. 10805061,
the special foundation of the president fellowship, the west doctoral project
of Chinese Academy of Sciences, and major state basic research development
program 2007CB815000.
FENG Zhao-Qing1,2222Tel: 0931-4969215, 13893620698; Email:
fengzhq@impcas.ac.cn, JIN Gen-Ming1,2
###### Abstract
Within the framework of the improved isospin dependent quantum molecular
dynamics (ImIQMD) model, the pion emission in heavy-ion collisions in the
region 1 A GeV is investigated systematically, in which the pion is considered
to be mainly produced by the decay of resonances $\triangle$(1232) and
N*(1440). The in-medium dependence and Coulomb effects of the pion production
are included in the calculation. Total pion multiplicity and $\pi^{-}/\pi^{+}$
yields are calculated for the reaction 197Au+197Au in central collisions for
selected Skyrme parameters SkP, SLy6, Ska, SIII and compared them with the
measured data by the FOPI collaboration.
1 _Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China_
2 _Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion
Accelerator of Lanzhou,
Lanzhou 730000, China_
_PACS_ : 25.75.-q, 13.75.Gx, 25.80.Ls
Heavy-ion collisions at intermediate energies play a significant role to
extract the information of the nuclear equation of state (EoS) under extreme
conditions, i.e., at high densities and high temperature. Besides nucleonic
observables such as rapidity distribution and flow, also mesons emitted from
the reaction zone can be probes of the hot and dense nuclear matter, that are
also the interest physics at the Cooling Storage Ring (CSR) energies in
Lanzhou.${}^{\cite[cite]{[\@@bibref{}{Zh08}{}{}]}}$ The emission of pion in
heavy-ion collisions in the region 1 A GeV is especially sensitive as probes
of isospin asymmetric EoS at supra-saturation
densities.${}^{\cite[cite]{[\@@bibref{}{Li08}{}{}]}}$ Spectra of the pion
emission in heavy-ion collisions have been measured by the Kaos and FOPI
collaborations and analyzed systematically by the present theoretical
transport models.${}^{\cite[cite]{[\@@bibref{}{Re07,Mu95}{}{}]}}$ A comparison
of the various transport approaches was made in Ref. [5]. The present
theoretical models overpredict the total pion multiplicity if using free
nucleon-nucleon (NN) cross sections below 2 A GeV region compared with
experimental data. Further investigations of the pion emissions in the 1 A GeV
region are still necessary by improving transport models or developing some
new approaches. The improved isospin-dependent quantum molecular dynamics
model has been successfully applied to treat fusion dynamics and reaction
mechanism of two colliding nuclei near Coulomb
barrier.${}^{\cite[cite]{[\@@bibref{}{Fe05,Fe08,Wa04}{}{}]}}$ To investigate
the pion emission, we further include the inelastic channels in nucleon-
nucleon collisions in the ImIQMD model.
In the ImIQMD model, the time evolutions of the baryons and pions in the
system under the self-consistently generated mean-field are governed by
Hamilton’s equations of motion, which read as
$\displaystyle\dot{\mathbf{p}}_{i}=-\frac{\partial
H}{\partial\mathbf{r}_{i}},\quad\dot{\mathbf{r}}_{i}=\frac{\partial
H}{\partial\mathbf{p}_{i}}.$ (1)
Here we omit the shell correction part in the Hamiltonian $H$ as described in
Ref. [7]. The Hamiltonian of baryons consists of the relativistic energy, the
effective interaction potential and the momentum dependent part as follows:
$H_{B}=\sum_{i}\sqrt{\textbf{p}_{i}^{2}+m_{i}^{2}}+U_{int}+U_{mom}.$ (2)
Here the $\textbf{p}_{i}$ and $m_{i}$ represent the momentum and the mass of
the baryons.
The effective interaction potential is composed of the Coulomb interaction and
the local interaction
$U_{int}=U_{Coul}+U_{loc}.$ (3)
The Coulomb interaction potential is written as
$U_{Coul}=\frac{1}{2}\sum_{i,j,j\neq
i}\frac{e_{i}e_{j}}{r_{ij}}erf(r_{ij}/\sqrt{4L})$ (4)
where the $e_{j}$ is the charged number including protons and charged
resonances. The $r_{ij}=|\mathbf{r}_{i}-\mathbf{r}_{j}|$ is the relative
distance of two charged particles. The local interaction potential is derived
directly from the Skyrme energy-density functional and expressed as
$U_{loc}=\int V_{loc}(\rho(\mathbf{r}))d\mathbf{r}.$ (5)
The local potential energy-density functional reads [7]
$V_{loc}(\rho)=\frac{\alpha}{2}\frac{\rho^{2}}{\rho_{0}}+\frac{\beta}{1+\gamma}\frac{\rho^{1+\gamma}}{\rho_{0}^{\gamma}}+\frac{g_{sur}}{2\rho_{0}}(\nabla\rho)^{2}+\frac{g_{sur}^{iso}}{2\rho_{0}}[\nabla(\rho_{n}-\rho_{p})]^{2}+\frac{C_{sym}}{2\rho_{0}}\rho^{2}\delta^{2}+g_{\tau}\rho^{8/3}/\rho_{0}^{5/3},$
(6)
where the $\rho$ is the baryon density and the
$\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$ is the isospin asymmetry with
the proton density $\rho_{p}$ and the neutron density $\rho_{n}$. The momentum
dependent part in the Hamiltonian is expressed as
$U_{mom}=\frac{\delta}{2}\sum_{i,j,j\neq
i}\frac{\rho_{ij}}{\rho_{0}}[\ln(\epsilon(\textbf{p}_{i}-\textbf{p}_{j})^{2}+1)]^{2},$
(7)
with
$\rho_{ij}=\frac{1}{(4\pi
L)^{3/2}}\exp\left[-\frac{(\textbf{r}_{i}-\textbf{r}_{j})^{2}}{4L}\right].$
(8)
Here the $L$ denotes the square of the pocket-wave width, which is dependent
on the size of the nucleus.
In Table 1 we list the ImIQMD parameters related to several typical Skyrme
forces after including the momentum dependent interaction. The parameters
$\alpha$, $\beta$, $\gamma$, $g_{\tau}$, $g_{sur}$, $g_{sur}^{iso}$, $\delta$
and $\epsilon$ are related to the Skyrme parameters $t_{0},t_{1},t_{2},t_{3}$
and $x_{0},x_{1},x_{2},x_{3}$, and determined in order to reproduce the
binding energy ($E_{B}$=-16 MeV) of symmetric nuclear matter at saturation
density for a given incompressibility as well as the correct momentum
dependence of the real part of the proton-nucleus optical potential. In the
following calculation we take the Skyrme parameter SLy6, which can give the
good properties from finite nucleus to neutron star [9].
Analogously to baryons, the Hamiltonian of pions is represented as
$H_{\pi}=\sum_{i=1}^{N_{\pi}}\left(\sqrt{\textbf{p}_{i}^{2}+m_{\pi}^{2}}+V_{i}^{Coul}\right),$
(9)
where the $\textbf{p}_{i}$ and $m_{\pi}$ represent the momentum and the mass
of the pions. The Coulomb interaction is given by
$V_{i}^{Coul}=\sum_{j=1}^{N_{B}}\frac{e_{i}e_{j}}{r_{ij}},$ (10)
where the $N_{\pi}$ and $N_{B}$ is the total number of pions and baryons
including charged resonances. Thus, the pion propagation in the whole stage is
guided essentially by the Coulomb effect. The in-medium pion potential in the
mean field is not considered in the model. However, the inclusion of the pion
optical potential based on the perturbation expansion of the $\Delta$-hole
model gives negligible influence on the transverse momentum
distribution.${}^{\cite[cite]{[\@@bibref{}{Fu97}{}{}]}}$
The pion is created by the decay of the resonances $\triangle$(1232) and
N*(1440) which are produced in inelastic NN scattering. The direct pion
production cross section is very small in the considered energies and not
included in the model.${}^{\cite[cite]{[\@@bibref{}{Ba01}{}{}]}}$ The reaction
channels are given as follows:
$\displaystyle NN\leftrightarrow N\triangle,$ $\displaystyle NN\leftrightarrow
NN^{\ast},$ $\displaystyle NN\leftrightarrow\triangle\triangle,$
$\displaystyle\Delta\leftrightarrow N\pi,$ $\displaystyle
N^{\ast}\leftrightarrow N\pi.$ (11)
The cross section of each channel to produce resonances are taken the values
calculated with the one-boson exchange
model.${}^{\cite[cite]{[\@@bibref{}{Hu94}{}{}]}}$ Transport models
overpredicted the total pion production with the free cross section. In the
ImIQMD model, we use the free elastic cross section and the in-medium
inelastic cross section which is given by
$\sigma^{inelastic}_{medium}=(\frac{\mu_{BB}^{\ast}}{\mu_{BB}})^{2}\sigma^{inelastic}_{free}$
with the free baryon-baryon (BB) inelastic cross section
$\sigma^{inelastic}_{free}$ and the reduced effective mass $\mu_{BB}^{\ast}$
(free mass $\mu_{BB}$). The experimental data of total elastic and inelastic
cross sections${}^{\cite[cite]{[\@@bibref{}{Ca93}{}{}]}}$ are parameterized in
the ImIQMD model as shown in Fig.1.
In the 1 A GeV region, there are mostly $\Delta$ resonances which disintegrate
into a $\pi$ and a nucleon, however, the $N^{\ast}$ yet gives considerable
contribution to the high energetic pion yield. The energy and momentum
dependent decay width is used in the ImIQMD model and expressed as
$\Gamma(|\textbf{p}|)=\frac{a_{1}|\textbf{p}|^{3}}{(1+a_{2}|\textbf{p}|^{2})(a_{3}+|\textbf{p}|^{2})}\Gamma_{0},$
(12)
which originates from the p-wave resonances. The p is the momentum of the
created pion (in GeV/c) in the resonance rest frame. The values $a_{1}=$22.48
(17.22), $a_{2}=$39.69 and $a_{3}=$0.04 (0.09) are used for the $\Delta$
($N^{\ast}$) with bare decay width $\Gamma_{0}=$0.12 GeV (0.2
GeV).${}^{\cite[cite]{[\@@bibref{}{Hu94}{}{}]}}$ In Fig.2 we show a comparison
of the time evolution of the $\pi$, $\Delta$ and $N^{\ast}$ production in the
reaction 197Au+197Au for head on collisions at 1 A GeV for two cases of the
bare decay and energy dependent decay widths. Both methods almost give the
same yield of the pion production. In the following calculation, we use the
energy and momentum dependent decay width. In Fig.3 we give the multiplicity
of produced pion as a function of the impact parameter for the same system at
1 A GeV energy. The numbers of produced $\pi^{-}$, $\pi^{0}$and $\pi^{+}$ are
reduced with increasing the impact parameter because of the decrease of the
participants of the ’fire ball’ formed in the heavy-ion collisions.
The emission of the produced pion is sensitive to the incident energy owing to
the size of the compressed nuclear matter. We calculated the transverse
momentum distribution of $\pi^{-}$, $\pi^{0}$and $\pi^{+}$ in central
197Au+197Au collisions at different incident energies as shown in Fig.4. The
larger and wider distributions were found at the higher incident energies due
to the larger participant numbers of the collision nucleons. The high energy
pions originate from the early phase and the decay of the N*(1440) resonance
also plays a significant role.${}^{\cite[cite]{[\@@bibref{}{Ma98}{}{}]}}$ In
Fig.5 we compare the total pion number and the $\pi^{-}$/$\pi^{+}$ ratio with
the FOPI data in central 197Au+197Au
collisions${}^{\cite[cite]{[\@@bibref{}{Re07}{}{}]}}$ for the Skyrme
parameters SkP, SLy6, Ska and SIII which correspond to the different
incompressibility modulus as listed in table 1. The calculated value of the
total pion number is related to the incompressibility modulus $K_{\infty}$ and
the effective mass in nuclear medium. Over the whole domain, the force SLy6 is
nice and can reproduce the experimental data. But the parameter slight
overpredicts the total pion multiplicity at lower incident energies and
underestimates the value at higher incident energies if using the above in-
medium inelastic cross section. The in-medium elastic and inelastic cross
sections are still open problems in transport model calculations, which should
be calculated by microscopic many-body models and then parameterized to add
into transport models. The $\pi^{-}$/$\pi^{+}$ ratio is interest for
extracting the high density behavior of the symmetry energy per
nucleon.${}^{\cite[cite]{[\@@bibref{}{Li08}{}{}]}}$ Using the isobar model,
one gets the ratio $\pi^{-}$/$\pi^{+}$=1.95 for pions from the $\Delta$
resonance, and $\pi^{-}$/$\pi^{+}$=1.7 from the $N^{\ast}$ for the system
197Au+197Au.${}^{\cite[cite]{[\@@bibref{}{St86}{}{}]}}$ These relations are
globally valid, i.e. independent of the pion energy. The observed energy
dependence of the $\pi^{-}$/$\pi^{+}$ ratio is due to the influence of the
Coulomb force and the symmetry energy interaction. The $\pi^{-}/\pi^{+}$ ratio
is sensitive to the stiffness of the symmetry energy at the lower incident
energies. Recently, a soft nuclear symmetry energy at supra-saturation
densities was pointed out by fitting the FOPI data with the IBUU04
model.${}^{\cite[cite]{[\@@bibref{}{Xi09}{}{}]}}$ In the ImIQMD model, we only
consider the linear dependence of the symmetry energy term on the baryon
density as shown in Eq.(6). The inclusion of the density-dependent symmetry
energy in the ImIQMD model is in progress.
In summary, the pion production in heavy-ion collisions in the region 1 A GeV
for the reaction 197Au+197 is investigated systematically by using the ImIQMD
model. The distribution of the transverse momentum is calculated at different
incident energies. The total number of produced pion and the $\pi^{-}/\pi^{+}$
ratio are calculated in central collisions for selected Skyrme parameters SkP,
SLy6, Ska, SIII and compared them with the FOPI data. Deviations from the
simple isobar model originate from the Coulomb and the symmetry interactions.
The $\pi^{-}/\pi^{+}$ ratio is sensitive to the stiffness of the symmetry
energy at the lower incident energies that may be further investigated at the
CSR energies.
We would like to thank Prof. Lie-Wen Chen, Prof. Wei Zuo and Dr. Gao-Chan Yong
for fruitful discussions.
## References
* [1] Zhan W L, Xia J W, Zhao H W et al (HIRFL-CSR Group) 2008 _Nucl. Phys. A_ 805 533c
* [2] Li B A, Chen L W, Ko C M 2008 _Phys. Rep._ 464 113
* [3] Reisdorf W, Stockmeier M, Andronic A et al (FOPI collaboration) 2007 _Nucl. Phys. A_ 781 459
* [4] C. Müntz et al (KaoS collaboration) 1995 _Z. Phys. A_ 352 175
* [5] Kolomeitsev E E, Hartnack C, Barz H W et al 2005 _J. Phys. G_ 31 S741
* [6] Feng Z Q, Zhang F S, Jin G M, Huang X 2005 _Nucl. Phys. A_ 750 232
* [7] Feng Z Q, Jin G M, Zhang F S 2008 _Nucl. Phys. A_ 802 91
* [8] Wang N, Li Z X, Wu X Z et al 2004 _Phys. Rev. C_ 69 034608\.
* [9] Chabanat E, Bonche P, Haensel P et al 1997 _Nucl. Phys. A_ 627 710
* [10] Fuchs C, Sehn L, Lehmann E et al 1997 _Phys. Rev. C_ 55 411\.
* [11] Li B A, Sustich A T, Zhang B, Ko C M 2001 _Int. J Mod. Phys. E_ 10 1
* [12] Catherine L-L, François L 1993 _Rev. Mod. Phys._ 65 47
* [13] Huber S, Aichelin J 1994 _Nucl. Phys. A_ 573 587
* [14] Maheswari V S, Fuchs C, Faessler A et al 1998 _Nucl. Phys. A_ 628 669
* [15] Stock R 1986 _Phys. Rep._ 135 259
* [16] Xiao Z G, Li B A, Chen L W et al 2009 _Phys. Rev. Lett._ 102 062502
Table 1: ImIQMD parameters and properties of symmetric nuclear matter for
Skyrme effective interactions after the inclusion of the momentum dependent
interaction with parameters $\delta$=1.57 MeV and $\epsilon$=500 c2/GeV2
Parameters SkM* Ska SIII SVI SkP RATP SLy6 $\alpha$ (MeV) -325.1 -179.3 -128.1
-123.0 -357.7 -250.3 -296.7 $\beta$ (MeV) 238.3 71.9 42.2 51.6 286.3 149.6
199.3 $\gamma$ 1.14 1.35 2.14 2.14 1.15 1.19 1.14 $g_{sur}$(MeV fm2) 21.8 26.5
18.3 14.1 19.5 25.6 22.9 $g_{sur}^{iso}$(MeV $fm^{2}$) -5.5 -7.9 -4.9 -3.0
-11.3 0.0 -2.7 $g_{\tau}$ (MeV) 5.9 13.9 6.4 1.1 0.0 11.0 9.9 $C_{sym}$ (MeV)
30.1 33.0 28.2 27.0 30.9 29.3 32.0 $\rho_{\infty}$ (fm-3) 0.16 0.155 0.145
0.144 0.162 0.16 0.16 $m_{\infty}^{\ast}/m$ 0.639 0.51 0.62 0.73 0.77 0.56
0.57 $K_{\infty}$ (MeV) 215 262 353 366 200 239 230
Figure 1: Comparison of nucleon-nucleon cross sections parameterized in ImIQMD
and the experimental data.${}^{\cite[cite]{[\@@bibref{}{Ca93}{}{}]}}$ Figure
2: Production of pion, delta and N* for head-on collisions in the reaction
197Au+197Au at 1 A GeV as functions of evolution time with the energy
dependent decay width (left panel) and fixed width (right panel). Figure 3:
Final multiplicities of $\pi^{-}$, $\pi^{0}$ and $\pi^{+}$ as a function of
impact parameter for head-on collisions in the reaction 197Au+197Au at 1 A
GeV. Figure 4: Final transverse momentum distribution in central 197Au+197Au
collisions at different incident energies. Figure 5: Calculated excitation
functions of the total pion multiplicity (left panel) and the ratio
$\pi^{-}$/$\pi^{+}$ (right panel) in central 197Au+197Au collisions and
compared with the FOPI data.${}^{\cite[cite]{[\@@bibref{}{Re07}{}{}]}}$
|
arxiv-papers
| 2009-04-20T10:10:25 |
2024-09-04T02:49:02.026331
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhao-Qing Feng, Gen-Ming Jin",
"submitter": "Zhaoqing Feng",
"url": "https://arxiv.org/abs/0904.2996"
}
|
0904.3171
|
# Spectral theory for a mathematical model of the weak interaction: The decay
of the intermediate vector bosons $\textbf{{W}}^{\pm}$. I.
J.-M. Barbaroux Centre de Physique Théorique, Luminy Case 907, 13288
Marseille Cedex 9, France and Département de Mathématiques, Université du Sud
Toulon-Var, 83957 La Garde Cedex, France
barbarou@univ-tln.fr and J.-C. Guillot Centre de Mathématiques Appliquées,
UMR 7641, École Polytechnique - C.N.R.S, 91128 Palaiseau Cedex, France
Jean-Claude.Guillot@polytechnique.edu
###### Abstract.
We consider a Hamiltonian with cutoffs describing the weak decay of spin $1$
massive bosons into the full family of leptons. The Hamiltonian is a self-
adjoint operator in an appropriate Fock space with a unique ground state. We
prove a Mourre estimate and a limiting absorption principle above the ground
state energy and below the first threshold for a sufficiently small coupling
constant. As a corollary, we prove absence of eigenvalues and absolute
continuity of the energy spectrum in the same spectral interval.
## 1\. Introduction
In this article, we consider a mathematical model of the weak interaction as
patterned according to the Standard Model in Quantum Field Theory (see [18,
31]). We choose the example of the weak decay of the intermediate vector
bosons $W^{\pm}$ into the full family of leptons.
The mathematical framework involves fermionic Fock spaces for the leptons and
bosonic Fock spaces for the vector bosons. The interaction is described in
terms of annihilation and creation operators together with kernels which are
square integrable with respect to momenta. The total Hamiltonian, which is the
sum of the free energy of the particles and antiparticles and of the
interaction, is a self-adjoint operator in the Fock space for the leptons and
the vector bosons and it has an unique ground state in the Fock space for a
sufficiently small coupling constant.
The weak interaction is one of the four fundamental interactions known up to
now. But the weak interaction is the only one which does not generate bound
states. As it is well known it is not the case for the strong, electromagnetic
and gravitational interactions. Thus we are expecting that the spectrum of the
Hamiltonian associated with every model of weak decays is absolutely
continuous above the energy of the ground state and this article is a first
step towards a proof of such a statement. Moreover a scattering theory has to
be established for every such Hamiltonian.
In this paper we establish a Mourre estimate and a limiting absorption
principle for any spectral interval above the energy of the ground state and
below the mass of the electron for a small coupling constant.
Our study of the spectral analysis of the total Hamiltonian is based on the
conjugate operator method with a self-adjoint conjugate operator. The methods
used in this article are taken largely from [4] and [13] and are based on [3]
and [25]. Some of the results of this article has been announced in [8].
For other applications of the conjugate operator method see [1, 5, 6, 9, 10,
11, 12, 14, 15, 17, 21, 26].
For related results about models in Quantum Field Theory see [7] and [28] in
the case of the Quantum Electrodynamics and [2] in the case of the weak
interaction.
The paper is organized as follows. In section 2, we give a precise definition
of the model we consider. In section 3, we state our main results and in the
following sections, together with the appendix, detailed proofs of the results
are given.
Acknowledgments. One of us (J.-C. G) wishes to thank Laurent Amour and Benoît
Grébert for helpful discussions. The authors also thank Walter Aschbacher for
valuable remarks. The work was done partially while J.M.-B. was visiting the
Institute for Mathematical Sciences, National University of Singapore in 2008.
The visit was supported by the Institute.
## 2\. The model
The weak decay of the intermediate bosons $W^{+}$ and $W^{-}$ involves the
full family of leptons together with the bosons themselves, according to the
Standard Model (see [18, Formula (4.139)] and [31]).
The full family of leptons involves the electron $e^{-}$ and the positron
$e^{+}$, together with the associated neutrino $\nu_{e}$ and antineutrino
$\bar{\nu}_{e}$, the muons $\mu^{-}$ and $\mu^{+}$ together with the
associated neutrino $\nu_{\mu}$ and antineutrino $\bar{\nu}_{\mu}$ and the tau
leptons $\tau^{-}$ and $\tau^{+}$ together with the associated neutrino
$\nu_{\tau}$ and antineutrino $\bar{\nu}_{\tau}$.
It follows from the Standard Model that neutrinos and antineutrinos are
massless particles. Neutrinos are left-handed, i.e., neutrinos have helicity
$-1/2$ and antineutrinos are right handed, i.e., antineutrinos have helicity
$+1/2$.
In what follows, the mathematical model for the weak decay of the vector
bosons $W^{+}$ and $W^{-}$ that we propose is based on the Standard Model, but
we adopt a slightly more general point of view because we suppose that
neutrinos and antineutrinos are both massless particles with helicity $\pm
1/2$. We recover the physical situation as a particular case. We could also
consider a model with massive neutrinos and antineutrinos built upon the
Standard Model with neutrino mixing [27].
Let us sketch how we define a mathematical model for the weak decay of the
vector bosons $W^{\pm}$ into the full family of leptons.
The energy of the free leptons and bosons is a self-adjoint operator in the
corresponding Fock space (see below) and the main problem is associated with
the interaction between the bosons and the leptons. Let us consider only the
interaction between the bosons and the electrons, the positrons and the
corresponding neutrinos and antineutrinos. Other cases are strictly similar.
In the Schrödinger representation the interaction is given by (see [18, p159,
(4.139)] and [31, p308, (21.3.20)])
(2.1)
$I=\int\mathrm{d}^{3}\\!x\,\overline{\Psi_{e}}(x)\gamma^{\alpha}(1-\gamma_{5})\Psi_{\nu_{e}}(x)W_{\alpha}(x)+\int\mathrm{d}^{3}\\!x\,\overline{\Psi_{\nu_{e}}}(x)\gamma^{\alpha}(1-\gamma_{5})\Psi_{e}(x)W_{\alpha}(x)^{*}\
,$
where $\gamma^{\alpha}$, $\alpha=0,1,2,3$ and $\gamma_{5}$ are the Dirac
matrices and $\Psi_{.}(x)$ and $\overline{\Psi_{.}}(x)$ are the Dirac fields
for $e_{-}$, $e_{+}$, $\nu_{e}$ and $\bar{\nu}_{e}$.
We have
$\begin{split}&\Psi_{e}(x)=\big{(}\frac{1}{2\pi}\big{)}^{\frac{3}{2}}\sum_{s=\pm\frac{1}{2}}\int\mathrm{d}^{3}\\!p\,(b_{e,+}(p,s)\frac{u(p,s)}{\sqrt{p_{0}}}\mathrm{e}^{ip.x}+b_{e,-}^{*}(p,s)\frac{v(p,s)}{\sqrt{p_{0}}}\mathrm{e}^{-ip.x})\
,\\\ &\overline{\Psi_{e}}(x)=\Psi_{e}(x)^{\dagger}\gamma^{0}\ .\end{split}$
Here $p_{0}=(|p|^{2}+m_{e}^{2})^{\frac{1}{2}}$ where $m_{e}>0$ is the mass of
the electron and $u(p,s)$ and $v(p,s)$ are the normalized solutions to the
Dirac equation (see [18, Appendix]).
The operators $b_{e,+}(p,s)$ and $b_{e,+}^{*}(p,s)$ (respectively
$b_{e,-}(p,s)$ and $b_{e,-}^{*}(p,s)$) are the annihilation and creation
operators for the electrons (respectively the positrons) satisfying the
anticommutation relations (see below).
Similarly we define $\Psi_{\nu_{e}}(x)$ and $\overline{\Psi_{\nu_{e}}}(x)$ by
substituting the operators $c_{\nu_{e},\pm}(p,s)$ and
$c_{\nu_{e},\pm}^{*}(p,s)$ for $b_{e,\pm}(p,s)$ and $b_{e,\pm}^{*}(p,s)$ with
$p_{0}=|p|$. The operators $c_{\nu_{e},+}(p,s)$ and $c_{\nu_{e},+}^{*}(p,s)$
(respectively $c_{\nu_{e},-}(p,s)$ and $c_{\nu_{e},-}^{*}(p,s)$) are the
annihilation and creation operators for the neutrinos associated with the
electrons (respectively the antineutrinos).
For the $W_{\alpha}$ fields we have (see [30, §5.3]).
$W_{\alpha}(x)=\big{(}\frac{1}{2\pi}\big{)}^{\frac{3}{2}}\sum_{\lambda=-1,0,1}\int\frac{\mathrm{d}^{3}\\!k}{\sqrt{2k_{0}}}(\epsilon_{\alpha}(k,\lambda)a_{+}(k,\lambda)\mathrm{e}^{ik.x}+\epsilon_{\alpha}^{*}(k,\lambda)a_{-}^{*}(k,\lambda)\mathrm{e}^{-ik.x})\
.$
Here $k_{0}=(|k|^{2}+m_{W}^{2})^{\frac{1}{2}}$ where $m_{W}>0$ is the mass of
the bosons $W^{\pm}$. $W^{+}$ is the antiparticule of $W^{-}$. The operators
$a_{+}(k,\lambda)$ and $a_{+}^{*}(k,\lambda)$ (respectively $a_{-}(k,\lambda)$
and $a_{-}^{*}(k,\lambda)$) are the annihilation and creation operators for
the bosons $W^{-}$ (respectively $W^{+}$) satisfying the canonical commutation
relations. The vectors $\epsilon_{\alpha}(k,\lambda)$ are the polarizations of
the massive spin 1 bosons $W^{\pm}$ (see [30, Section 5.2]).
The interaction (2.1) is a formal operator and, in order to get a well defined
operator in the Fock space, one way is to adapt what Glimm and Jaffe have done
in the case of the Yukawa Hamiltonian (see [16]). For that sake, we have to
introduce a spatial cutoff $g(x)$ such that $g\in L^{1}({\mathbb{R}}^{3})$,
together with momentum cutoffs $\chi(p)$ and $\rho(k)$ for the Dirac fields
and the $W_{\mu}$ fields respectively.
Thus when one develops the interaction $I$ with respect to products of
creation and annihilation operators, one gets a finite sum of terms associated
with kernels of the form
$\chi(p_{1})\,\chi(p_{2})\,\rho(k)\,\hat{g}(p_{1}+p_{2}-k)\ ,$
where $\hat{g}$ is the Fourier transform of $g$. These kernels are square
integrable.
In what follows, we consider a model involving terms of the above form but
with more general square integrable kernels.
We follow the convention described in [30, section 4.1] that we quote: “The
state-vector will be taken to be symmetric under interchange of any bosons
with each other, or any bosons with any fermions, and antisymmetric with
respect to interchange of any two fermions with each other, in all cases,
wether the particles are of the same species or not”. Thus, as it follows from
section 4.2 of [30], fermionic creation and annihilation operators of
different species of leptons will always anticommute.
Concerning our notations, from now on, $\ell\in\\{1,2,3\\}$ denotes each
species of leptons. $\ell=1$ denotes the electron $e^{-}$ the positron $e^{+}$
and the neutrinos $\nu_{e}$, $\bar{\nu}_{e}$. $\ell=2$ denotes the muons
$\mu^{-}$, $\mu^{+}$ and the neutrinos $\nu_{\mu}$ and $\bar{\nu}_{\mu}$, and
$\ell=3$ denotes the tau-leptons and the neutrinos $\nu_{\tau}$ and
$\bar{\nu}_{\tau}$.
Let $\xi_{1}=(p_{1},\ s_{1})$ be the quantum variables of a massive lepton,
where $p_{1}\in{\mathbb{R}}^{3}$ and $s_{1}\in\\{-1/2,\ 1/2\\}$ is the spin
polarization of particles and antiparticles. Let $\xi_{2}=(p_{2},\ s_{2})$ be
the quantum variables of a massless lepton where $p_{2}\in{\mathbb{R}}^{3}$
and $s_{2}\in\\{-1/2,\ 1/2\\}$ is the helicity of particles and antiparticles
and, finally, let $\xi_{3}=(k,\ \lambda)$ be the quantum variables of the spin
$1$ bosons $W^{+}$ and $W^{-}$ where $k\in{\mathbb{R}}^{3}$ and
$\lambda\in\\{-1,\ 0,\ 1\\}$ is the polarization of the vector bosons (see
[30, section 5]). We set $\Sigma_{1}={\mathbb{R}}^{3}\times\\{-1/2,\ 1/2\\}$
for the leptons and $\Sigma_{2}={\mathbb{R}}^{3}\times\\{-1,\ 0,\ 1\\}$ for
the bosons. Thus $L^{2}(\Sigma_{1})$ is the Hilbert space of each lepton and
$L^{2}(\Sigma_{2})$ is the Hilbert space of each boson. The scalar product in
$L^{2}(\Sigma_{j})$, $j=1,2$ is defined by
(2.2) $(f,\ g)=\int_{\Sigma_{j}}\overline{f(\xi)}g(\xi)\mathrm{d}\xi,\quad
j=1,2\ .$
Here
$\int_{\Sigma_{1}}\mathrm{d}\xi=\sum_{s=+\frac{1}{2},-\frac{1}{2}}\int\mathrm{d}p\quad\mbox{and}\quad\int_{\Sigma_{2}}\mathrm{d}\xi=\sum_{\lambda=0,1,-1}\int\mathrm{d}k,\quad(p,k\in{\mathbb{R}}^{3})\
.$
The Hilbert space for the weak decay of the vector bosons $W^{+}$ and $W^{-}$
is the Fock space for leptons and bosons that we now describe.
Let ${\mathfrak{S}}$ be any separable Hilbert space. Let
$\otimes_{a}^{n}{\mathfrak{S}}$ (resp. $\otimes_{s}^{n}{\mathfrak{S}}$) denote
the antisymmetric (resp. symmetric) $n$-th tensor power of ${\mathfrak{S}}$.
The fermionic (resp. bosonic) Fock space over ${\mathfrak{S}}$, denoted by
${\mathfrak{F}}_{a}({\mathfrak{S}})$ (resp.
${\mathfrak{F}}_{s}({\mathfrak{S}}))$, is the direct sum
(2.3)
${\mathfrak{F}}_{a}({\mathfrak{S}})=\bigoplus_{n=0}^{\infty}\bigotimes_{a}^{n}{\mathfrak{S}}\quad(\mbox{resp.
}{\mathfrak{F}}_{s}({\mathfrak{S}})=\bigoplus_{n=0}^{\infty}\bigotimes_{s}^{n}{\mathfrak{S}})\
,$
where
$\otimes_{a}^{0}{\mathfrak{S}}=\otimes_{s}^{0}{\mathfrak{S}}\equiv{\mathbb{C}}$.
The state $\Omega=(1,0,0,\ldots,0,\ldots)$ denotes the vacuum state in
${\mathfrak{F}}_{a}({\mathfrak{S}})$ and in
${\mathfrak{F}}_{s}({\mathfrak{S}})$.
For every $\ell$, ${\mathfrak{F}}_{\ell}$ is the fermionic Fock space for the
corresponding species of leptons including the massive particle and
antiparticle together with the associated neutrino and antineutrino, i.e.,
(2.4)
${\mathfrak{F}}_{\ell}=\bigotimes^{4}{\mathfrak{F}}_{a}(L^{2}(\Sigma_{1}))\,\quad\ell=1,2,3\
.$
We have
(2.5) ${\mathfrak{F}}_{\ell}=\bigoplus_{q_{\ell}\geq 0,\bar{q}_{\ell}\geq
0,r_{\ell}\geq 0,\bar{r}_{\ell}\geq
0}{\mathfrak{F}}_{\ell}^{(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})}\
,$
with
(2.6)
${\mathfrak{F}}_{\ell}^{(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})}=(\otimes_{a}^{q_{\ell}}L^{2}(\Sigma_{1}))\otimes(\otimes_{a}^{\bar{q}_{\ell}}L^{2}(\Sigma_{1}))\otimes(\otimes_{a}^{r_{\ell}}L^{2}(\Sigma_{1}))\otimes(\otimes_{a}^{\bar{r}_{\ell}}L^{2}(\Sigma_{1}))\
.$
Here $q_{\ell}$ (resp. $\bar{q}_{\ell}$) is the number of massive particle
(resp. antiparticles) and $r_{\ell}$ (resp. $\bar{r}_{\ell}$) is the number of
neutrinos (resp. antineutrinos). The vector $\Omega_{\ell}$ is the associated
vacuum state. The fermionic Fock space denoted by ${\mathfrak{F}}_{L}$ for the
leptons is then
(2.7) ${\mathfrak{F}}_{L}=\otimes_{\ell=1}^{3}{\mathfrak{F}}_{\ell}\ ,$
and $\Omega_{L}=\otimes_{\ell=1}^{3}\Omega_{\ell}$ is the vacuum state.
The bosonic Fock space for the vector bosons $W^{+}$ and $W^{-}$, denoted by
${\mathfrak{F}}_{W}$, is then
(2.8)
${\mathfrak{F}}_{W}={\mathfrak{F}}_{s}(L^{2}(\Sigma_{2}))\otimes{\mathfrak{F}}_{s}(L^{2}(\Sigma_{2}))\simeq{\mathfrak{F}}_{s}(L^{2}(\Sigma_{2})\oplus
L^{2}(\Sigma_{2}))\ .$
We have
${\mathfrak{F}}_{W}=\bigoplus_{t\geq 0,\bar{t}\geq
0}{\mathfrak{F}}_{W}^{(t,\bar{t})}\ ,$
where
${\mathfrak{F}}_{W}^{(t,\bar{t})}=(\otimes_{s}^{t}L^{2}(\Sigma_{2}))\otimes(\otimes_{s}^{\bar{t}}L^{2}(\Sigma_{2}))$.
Here $t$ (resp. $\bar{t}$) is the number of bosons $W^{-}$ (resp. $W^{+}$).
The vector $\Omega_{W}$ is the corresponding vacuum.
The Fock space for the weak decay of the vector bosons $W^{+}$ and $W^{-}$,
denoted by ${\mathfrak{F}}$, is thus
${\mathfrak{F}}={\mathfrak{F}}_{L}\otimes{\mathfrak{F}}_{W}$
and $\Omega=\Omega_{L}\otimes\Omega_{W}$ is the vacuum state.
For every $\ell\in\\{1,2,3\\}$ let ${\mathfrak{D}}_{\ell}$ denote the set of
smooth vectors $\psi_{\ell}\in{\mathfrak{F}}_{\ell}$ for which
$\psi_{\ell}^{{(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})}}$ has a
compact support and
$\psi_{\ell}^{(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})}=0$ for all
but finitely many ${(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})}$. Let
${\mathfrak{D}}_{L}=\widehat{\bigotimes}_{\ell=1}^{3}{\mathfrak{D}}_{\ell}\ .$
Here $\hat{\otimes}$ is the algebraic tensor product.
Let ${\mathfrak{D}}_{W}$ denote the set of smooth vectors
$\phi\in{\mathfrak{F}}_{W}$ for which $\phi^{(t,\bar{t})}$ has a compact
support and $\phi^{(t,\bar{t})}=0$ for all but finitely many $(t,\bar{t})$.
Let
${\mathfrak{D}}={\mathfrak{D}}_{L}\hat{\otimes}\,{\mathfrak{D}}_{W}\ .$
The set ${\mathfrak{D}}$ is dense in ${\mathfrak{F}}$.
Let $A_{\ell}$ be a self-adjoint operator in ${\mathfrak{F}}_{\ell}$ such that
${\mathfrak{D}}_{\ell}$ is a core for $A_{\ell}$. Its extension to
${\mathfrak{F}}_{L}$ is, by definition, the closure in ${\mathfrak{F}}_{L}$ of
the operator $A_{1}\otimes{\mathbf{1}}_{2}\otimes{\mathbf{1}}_{3}$ with domain
${\mathfrak{D}}_{L}$ when $\ell=1$, of the operator ${\mathbf{1}}_{1}\otimes
A_{2}\otimes{\mathbf{1}}_{3}$ with domain ${\mathfrak{D}}_{L}$ when $\ell=2$,
and of the operator ${\mathbf{1}}_{1}\otimes{\mathbf{1}}_{2}\otimes A_{3}$
with domain ${\mathfrak{D}}_{L}$ when $\ell=3$. Here ${\mathbf{1}}_{\ell}$ is
the operator identity on ${\mathfrak{F}}_{\ell}$.
The extension of $A_{\ell}$ to ${\mathfrak{F}}_{L}$ is a self-adjoint operator
for which ${\mathfrak{D}}_{L}$ is a core and it can be extended to
${\mathfrak{F}}$. The extension of $A_{\ell}$ to ${\mathfrak{F}}$ is, by
definition, the closure in ${\mathfrak{F}}$ of the operator
$\tilde{A}_{\ell}\otimes{\mathbf{1}}_{W}$ with domain ${\mathfrak{D}}$, where
$\tilde{A}_{\ell}$ is the extension of $A_{\ell}$ to ${\mathfrak{F}}_{L}$. The
extension of $A_{\ell}$ to ${\mathfrak{F}}$ is a self-adjoint operator for
which ${\mathfrak{D}}$ is a core.
Let $B$ be a self-adjoint operator in ${\mathfrak{F}}_{W}$ for which
${\mathfrak{D}}_{W}$ is a core. The extension of the self-adjoint operator
$A_{\ell}\otimes B$ is, by definition, the closure in ${\mathfrak{F}}$ of the
operator $A_{1}\otimes{\mathbf{1}}_{2}\otimes{\mathbf{1}}_{3}\otimes B$ with
domain ${\mathfrak{D}}$ when $\ell=1$, of the operator
${\mathbf{1}}_{1}\otimes A_{2}\otimes{\mathbf{1}}_{3}\otimes B$ with domain
${\mathfrak{D}}$ when $\ell=2$, and of the operator
${\mathbf{1}}_{1}\otimes{\mathbf{1}}_{2}\otimes A_{3}\otimes B$ with domain
${\mathfrak{D}}$ when $\ell=3$. The extension of $A_{\ell}\otimes B$ to
${\mathfrak{F}}$ is a self-adjoint operator for which ${\mathfrak{D}}$ is a
core.
We now define the creation and annihilation operators. For each $\ell=1,2,3$,
$b_{\ell,\epsilon}(\xi_{1})$ (resp. $b^{*}_{\ell,\epsilon}(\xi_{1})$) is the
annihilation (resp. creation) operator for the corresponding species of
massive particle when $\epsilon=+$ and for the corresponding species of
massive antiparticle when $\epsilon=-$. Similarly, for each $\ell=1,2,3$,
$c_{\ell,\epsilon}(\xi_{2})$ (resp. $c^{*}_{\ell,\epsilon}(\xi_{2})$) is the
annihilation (resp. creation) operator for the corresponding species of
neutrino when $\epsilon=+$ and for the corresponding species of antineutrino
when $\epsilon=-$. The operator ${a_{\epsilon}}(\xi_{3})$ (resp.
${a^{*}_{\epsilon}}(\xi_{3})$) is the annihilation (resp. creation) operator
for the boson $W^{-}$ when $\epsilon=+$ and for the boson $W^{+}$ when
$\epsilon=-$.
Let $\Psi\in{\mathfrak{D}}$ be such that
$\Psi=\left(\Psi^{(Q)}\right)_{Q}\ ,$
with
$Q=\Big{(}{(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})}_{\ell=1,2,3},\,(t,\bar{t})\Big{)}$,
and
$\Psi^{(Q)}=\left(\otimes_{\ell=1}^{3}\Psi^{(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})}\right)\otimes\varphi^{(t,\bar{t})}\
,$
where
$(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell},t,\bar{t})\in{\mathbb{N}}^{6}$.
Here,
$(\Psi^{(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})})_{q_{\ell}\geq
0,\bar{q}_{\ell}\geq 0,r_{\ell}\geq 0,\bar{r}_{\ell}\geq
0}\in{\mathfrak{D}}_{\ell}$, and $(\varphi^{(t,\bar{t})})_{t\geq 0,\bar{t}\geq
0}\in{\mathfrak{D}}_{W}$.
Let
$\begin{split}Q_{\ell,+}&=\Big{(}(q_{\ell^{\prime}},\bar{q}_{\ell^{\prime}},r_{\ell^{\prime}},\bar{r}_{\ell^{\prime}})_{\ell^{\prime}<\ell},\,(q_{\ell}+1,\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell}),\,(q_{\ell^{\prime}},\bar{q}_{\ell^{\prime}},r_{\ell^{\prime}},\bar{r}_{\ell^{\prime}})_{\ell^{\prime}>\ell},\,(t,\bar{t})\Big{)}\
,\\\
Q_{\ell,-}&=\Big{(}(q_{\ell^{\prime}},\bar{q}_{\ell^{\prime}},r_{\ell^{\prime}},\bar{r}_{\ell^{\prime}})_{\ell^{\prime}<\ell},\,(q_{\ell},\bar{q}_{\ell}+1,r_{\ell},\bar{r}_{\ell}),\,(q_{\ell^{\prime}},\bar{q}_{\ell^{\prime}},r_{\ell^{\prime}},\bar{r}_{\ell^{\prime}})_{\ell^{\prime}>\ell},\,(t,\bar{t})\Big{)}\
,\\\
\tilde{Q}_{\ell,+}&=\Big{(}(q_{\ell^{\prime}},\bar{q}_{\ell^{\prime}},r_{\ell^{\prime}},\bar{r}_{\ell^{\prime}})_{\ell^{\prime}<\ell},\,(q_{\ell},\bar{q}_{\ell},r_{\ell}+1,\bar{r}_{\ell}),\,(q_{\ell^{\prime}},\bar{q}_{\ell^{\prime}},r_{\ell^{\prime}},\bar{r}_{\ell^{\prime}})_{\ell^{\prime}>\ell},\,(t,\bar{t})\Big{)}\
,\\\
\tilde{Q}_{\ell,-}&=\Big{(}(q_{\ell^{\prime}},\bar{q}_{\ell^{\prime}},r_{\ell^{\prime}},\bar{r}_{\ell^{\prime}})_{\ell^{\prime}<\ell},\,(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell}+1),\,(q_{\ell^{\prime}},\bar{q}_{\ell^{\prime}},r_{\ell^{\prime}},\bar{r}_{\ell^{\prime}})_{\ell^{\prime}>\ell},\,(t,\bar{t})\Big{)}\
,\end{split}$
and
$\begin{split}Q_{b,+}&=\Big{(}(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})_{\ell=1,2,3},\,(t+1,\bar{t})\Big{)}\
,\\\
Q_{b,-}&=\Big{(}(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})_{\ell=1,2,3},\,(t,\bar{t}+1)\Big{)}\
.\end{split}$
We define
$\begin{split}&(b_{\ell,+}(\xi_{1})\Psi)^{(Q)}(\,.\,;\,\xi_{1}^{(1)},\xi_{1}^{(2)},\ldots,\xi_{1}^{(q_{\ell})};\,.\,)\\\
&=\sqrt{q_{\ell}+1}\,\Pi_{\ell^{\prime}<\ell}\
(-1)^{q_{\ell^{\prime}}+\bar{q}_{\ell^{\prime}}}\Psi^{(Q_{\ell,+})}(\,.\,;\xi_{1},\xi_{1}^{(1)},\xi_{1}^{(2)},\ldots,\xi_{1}^{(q_{\ell})};\,.\,)\\\
&(b_{\ell,-}(\xi_{1})\Psi)^{(Q)}(\,.\,;\,\xi_{1}^{(1)},\xi_{1}^{(2)},\ldots,\xi_{1}^{(\bar{q}_{\ell})};\,.\,)\\\
&=\sqrt{\bar{q}_{\ell}+1}\,(-1)^{q_{\ell}}\Pi_{\ell^{\prime}<\ell}\
(-1)^{q_{\ell^{\prime}}+\bar{q}_{\ell^{\prime}}}\Psi^{(Q_{\ell,-})}(\,.\,;\xi_{1},\xi_{1}^{(1)},\xi_{1}^{(2)},\ldots,\xi_{1}^{(\bar{q}_{\ell})};\,.\,)\
,\end{split}$
$\begin{split}&(c_{\ell,+}(\xi_{2})\Psi)^{(Q)}(\,.\,;\,\xi_{2}^{(1)},\xi_{2}^{(2)},\ldots,\xi_{2}^{(r_{\ell})};\,.\,)\\\
&=\sqrt{r_{\ell}+1}\,(-1)^{q_{\ell}+\bar{q}_{\ell}}\Pi_{\ell^{\prime}<\ell}\
(-1)^{q_{\ell^{\prime}}+\bar{q}_{\ell^{\prime}}+r_{\ell^{\prime}}+\bar{r}_{\ell^{\prime}}}\Psi^{(\tilde{Q}_{\ell,+})}(\,.\,;\xi_{2},\xi_{2}^{(1)},\xi_{2}^{(2)},\ldots,\xi_{2}^{(r_{\ell})};\,.\,)\\\
&(c_{\ell,-}(\xi_{2})\Psi)^{(Q)}(\,.\,;\,\xi_{2}^{(1)},\xi_{2}^{(2)},\ldots,\xi_{2}^{(\bar{r}_{\ell})};\,.\,)\\\
&=\sqrt{\bar{r}_{\ell}+1}\,(-1)^{q_{\ell}+\bar{q}_{\ell}+r_{\ell}}\Pi_{\ell^{\prime}<\ell}\
(-1)^{q_{\ell^{\prime}}+\bar{q}_{\ell^{\prime}}+r_{\ell^{\prime}}+\bar{r}_{\ell^{\prime}}}\Psi^{(\tilde{Q}_{\ell,-})}(\,.\,;\xi_{2},\xi_{2}^{(1)},\xi_{2}^{(2)},\ldots,\xi_{2}^{(\bar{r}_{\ell})};\,.\,)\
,\end{split}$
and
$\begin{split}&(a_{+}(\xi_{3})\Psi)^{(Q)}(\,.\,;\,\xi_{3}^{(1)},\xi_{3}^{(2)},\ldots,\xi_{3}^{(t)};\,.\,)\\\
&=\sqrt{t+1}\Psi^{(Q_{b,+})}(\,.\,;\,\xi_{3},\xi_{3}^{(1)},\xi_{3}^{(2)},\ldots,\xi_{3}^{(t)};\,.\,)\
,\\\
&(a_{-}(\xi_{3})\Psi)^{(Q)}(\,.\,;\,\xi_{3}^{(1)},\xi_{3}^{(2)},\ldots,\xi_{3}^{(\bar{t})};\,.\,)\\\
&=\sqrt{\bar{t}+1}\Psi^{(Q_{b,-})}(\,.\,;\,\xi_{3},\xi_{3}^{(1)},\xi_{3}^{(2)},\ldots,\xi_{3}^{(\bar{t})};\,.\,)\
.\end{split}$
As usual, $b^{*}_{\ell,\epsilon}(\xi_{1})$ (resp.
$c^{*}_{\ell,\epsilon}(\xi_{2})$) is the formal adjoint of
$b_{\ell,\epsilon}(\xi_{1})$ (resp. $c_{\ell,\epsilon}(\xi_{2})$). For
example, we have
$\begin{split}&(b^{*}_{\ell,\epsilon}(\xi_{1})\Psi)^{(Q_{\ell,+})}(\,.\,;\xi_{1}^{(1)},\xi_{1}^{(2)},\ldots,\xi_{1}^{(q_{\ell})},\xi_{1}^{(q_{\ell}+1)};\,.\,)\\\
&=\frac{1}{\sqrt{q_{\ell}+1}}\prod_{\ell^{\prime}<\ell}(-1)^{q_{\ell^{\prime}}+\bar{q}_{\ell^{\prime}}}\\\
&\sum_{i=1}^{q_{\ell}+1}(-1)^{i+1}\delta(\xi_{1}-\xi_{1}^{(i)})\Psi^{(Q)}(\,.\,;\xi_{1}^{(1)},\xi_{1}^{(2)},\ldots,\widehat{\xi_{1}^{(i)}},\ldots,\xi_{1}^{(q_{\ell}+1)};\,.\,)\
,\end{split}$
where $\widehat{.}$ denotes that the $i$-th variable has to be omitted, and
$\delta(\xi_{1}-\xi_{1}^{(i)})=\delta_{s_{1}s_{1}^{(i)}}\delta(p_{1}-p_{1}^{(i)})$.
The operator $a^{*}_{\epsilon}(\xi_{3})$ is the formal adjoint of
$a_{\epsilon}(\xi_{3})$ and we have
$\begin{split}&(a^{*}_{+}(\xi_{3})\Psi)^{(Q_{b,+})}(\,.\,;\xi_{3}^{(1)},\xi_{3}^{(2)},\ldots,\xi_{3}^{(t+1)};.)\\\
&=\frac{1}{\sqrt{t+1}}\sum_{i=1}^{t+1}\delta(\xi_{3}-\xi_{3}^{(i)})\Psi^{(Q)}(\,.\,;\xi_{3}^{(1)},\ldots,\widehat{\xi_{3}^{(i)}},\ldots,\xi_{3}^{(t+1)};\,.\,)\end{split}$
where
$\delta(\xi_{3}-\xi_{3}^{(i)})=\delta_{\lambda\lambda^{(i)}}\delta(k-k^{(i)})$.
The following canonical anticommutation and commutation relations hold.
$\begin{split}&\\{b_{\ell,\epsilon}(\xi_{1}),b^{*}_{\ell^{\prime},\epsilon^{\prime}}(\xi_{1}^{\prime})\\}=\delta_{\ell\ell^{\prime}}\delta_{\epsilon\epsilon^{\prime}}\delta(\xi_{1}-\xi_{1}^{\prime})\
,\\\
&\\{c_{\ell,\epsilon}(\xi_{2}),c^{*}_{\ell^{\prime},\epsilon^{\prime}}(\xi_{2}^{\prime})\\}=\delta_{\ell\ell^{\prime}}\delta_{\epsilon\epsilon^{\prime}}\delta(\xi_{2}-\xi_{2}^{\prime})\
,\\\
&[a_{\epsilon}(\xi_{3}),a^{*}_{\epsilon^{\prime}}(\xi_{3}^{\prime})]=\delta_{\epsilon\epsilon^{\prime}}\delta(\xi_{3}-\xi_{3}^{\prime})\
,\\\
&\\{b_{\ell,\epsilon}(\xi_{1}),b_{\ell^{\prime},\epsilon^{\prime}}(\xi_{1}^{\prime})\\}=\\{c_{\ell,\epsilon}(\xi_{2}),c_{\ell^{\prime},\epsilon^{\prime}}(\xi_{2}^{\prime})\\}=0\
,\\\ &[a_{\epsilon}(\xi_{3}),a_{\epsilon^{\prime}}(\xi_{3}^{\prime})]=0\ ,\\\
&\\{b_{\ell,\epsilon}(\xi_{1}),c_{\ell^{\prime},\epsilon^{\prime}}(\xi_{2})\\}=\\{b_{\ell,\epsilon}(\xi_{1}),c^{*}_{\ell^{\prime},\epsilon^{\prime}}(\xi_{2})\\}=0\
,\\\
&[b_{\ell,\epsilon}(\xi_{1}),a_{\epsilon^{\prime}}(\xi_{3})]=[b_{\ell,\epsilon}(\xi_{1}),a^{*}_{\epsilon^{\prime}}(\xi_{3})]=[c_{\ell,\epsilon}(\xi_{2}),a_{\epsilon^{\prime}}(\xi_{3})]=[c_{\ell,\epsilon}(\xi_{2}),a^{*}_{\epsilon^{\prime}}(\xi_{3})]=0\
.\end{split}$
Here, $\\{b,b^{\prime}\\}=bb^{\prime}+b^{\prime}b$,
$[a,a^{\prime}]=aa^{\prime}-a^{\prime}a$.
We recall that the following operators, with $\varphi\in L^{2}(\Sigma_{1})$,
$\begin{split}&b_{\ell,\epsilon}(\varphi)=\int_{\Sigma_{1}}b_{\ell,\epsilon}(\xi)\overline{\varphi(\xi)}\mathrm{d}\xi,\quad
c_{\ell,\epsilon}(\varphi)=\int_{\Sigma_{1}}c_{\ell,\epsilon}(\xi)\overline{\varphi(\xi)}\mathrm{d}\xi\
,\\\
&b^{*}_{\ell,\epsilon}(\varphi)=\int_{\Sigma_{1}}b^{*}_{\ell,\epsilon}(\xi){\varphi(\xi)}\mathrm{d}\xi,\quad
c^{*}_{\ell,\epsilon}(\varphi)=\int_{\Sigma_{1}}c^{*}_{\ell,\epsilon}(\xi){\varphi(\xi)}\mathrm{d}\xi\end{split}$
are bounded operators in ${\mathfrak{F}}$ such that
(2.9)
$\|b^{\sharp}_{\ell,\epsilon}(\varphi)\|=\|c^{\sharp}_{\ell,\epsilon}(\varphi)\|=\|\varphi\|_{L^{2}}\
,$
where $b^{\sharp}$ (resp. $c^{\sharp}$) is b (resp. $c$) or $b^{*}$ (resp.
$c^{*}$).
The operators $b^{\sharp}_{\ell,\epsilon}(\varphi)$ and
$c^{\sharp}_{\ell,\epsilon}(\varphi)$ satisfy similar anticommutaion relations
(see e.g. [29]).
The free Hamiltonian $H_{0}$ is given by
$\begin{split}H_{0}&=H_{0}^{(1)}+H_{0}^{(2)}+H_{0}^{(3)}\\\
&=\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int
w_{\ell}^{(1)}(\xi_{1})b^{*}_{\ell,\epsilon}(\xi_{1})b_{\ell,\epsilon}(\xi_{1})\mathrm{d}\xi_{1}+\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int
w_{\ell}^{(2)}(\xi_{2})c^{*}_{\ell,\epsilon}(\xi_{2})c_{\ell,\epsilon}(\xi_{2})\mathrm{d}\xi_{2}\\\
&+\sum_{\epsilon=\pm}\int
w^{(3)}(\xi_{3})a^{*}_{\epsilon}(\xi_{3})a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\
,\end{split}$
where
$\begin{split}&w_{\ell}^{(1)}(\xi_{1})=(|p_{1}|^{2}+m_{\ell}^{2})^{\frac{1}{2}},\quad\mbox{with}\
0<m_{1}<m_{2}<m_{3}\ ,\\\ &w_{\ell}^{(2)}(\xi_{2})=|p_{2}|\ ,\\\
&w^{(3)}(\xi_{3})=(|k|^{2}+m^{2}_{W})^{\frac{1}{2}}\ ,\end{split}$
where $m_{W}$ is the mass of the bosons $W^{+}$ and $W^{-}$ such that
$m_{W}>m_{3}$.
The spectrum of $H_{0}$ is $[0,\,\infty)$ and $0$ is a simple eigenvalue with
$\Omega$ as eigenvector. The set of thresholds of $H_{0}$, denoted by $T$, is
given by
$T=\\{p\,m_{1}+q\,m_{2}+r\,m_{3}+s\,m_{W};(p,\,q,\,r,\,s)\in{\mathbb{N}}^{4}\mbox{
and }p+q+r+s\geq 1\\}\ ,$
and each set $[t,\infty)$, $t\in T$, is a branch of absolutely continuous
spectrum for $H_{0}$.
The interaction, denoted by $H_{I}$, is given by
(2.10) $H_{I}=\sum_{\alpha=1}^{2}H_{I}^{(\alpha)}\ ,$
where
(2.11)
$\begin{split}H_{I}^{(1)}=&\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int
G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})b^{*}_{\ell,\epsilon}(\xi_{1})c^{*}_{\ell,\epsilon^{\prime}}(\xi_{2})a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\\\
&+\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int\overline{G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})}a^{*}_{\epsilon}(\xi_{3})c_{\ell,\epsilon^{\prime}}(\xi_{2})b_{\ell,\epsilon}(\xi_{1})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\
,\end{split}$ (2.12)
$\begin{split}H_{I}^{(2)}=&\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int
G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})b^{*}_{\ell,\epsilon}(\xi_{1})c^{*}_{\ell,\epsilon^{\prime}}(\xi_{2})a^{*}_{\epsilon}(\xi_{3})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\\\
&+\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int\overline{G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})}a_{\epsilon}(\xi_{3})c_{\ell,\epsilon^{\prime}}(\xi_{2})b_{\ell,\epsilon}(\xi_{1})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\
.\end{split}$
The kernels $G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(.,.,.)$, $\alpha=1,2$,
are supposed to be functions.
The total Hamiltonian is then
(2.13) $H=H_{0}+gH_{I},\quad g>0\ ,$
where $g$ is a coupling constant.
The operator $H_{I}^{(1)}$ describes the decay of the bosons $W^{+}$ and
$W^{-}$ into leptons. Because of $H_{I}^{(2)}$ the bare vacuum will not be an
eigenvector of the total Hamiltonian for every $g>0$ as we expect from the
physics.
Every kernel $G_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})$,
computed in theoretical physics, contains a $\delta$-distribution because of
the conservation of the momentum (see [18] [30, section 4.4]). In what
follows, we approximate the singular kernels by square integrable functions.
Thus, from now on, the kernels
$G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$ are supposed to satisfy the
following hypothesis .
###### Hypothesis 2.1.
For $\alpha=1,2$, $\ell=1,2,3$, $\epsilon,\epsilon^{\prime}=\pm$, we assume
(2.14)
$G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})\in
L^{2}(\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2})\ .$
###### Remark 2.2.
A similar model can be written down for the weak decay of pions $\pi^{-}$ and
$\pi^{+}$ (see [18, section 6.2]).
###### Remark 2.3.
The total Hamiltonian is more general than the one involved in the theory of
weak interaction because, in the Standard Model, neutrinos have helicity
$-1/2$ and antineutrinos have helicity $1/2$.
In the physical case, the Fock space, denoted by ${\mathfrak{F}}^{\prime}$, is
isomorphic to ${\mathfrak{F}}_{L}^{\prime}\otimes{\mathfrak{F}}_{W}$, with
${\mathfrak{F}}_{L}^{\prime}=\bigotimes_{\ell=1}^{3}{\mathfrak{F}}_{\ell}^{\prime}\
,$
and
${\mathfrak{F}}_{\ell}^{\prime}=(\otimes_{a}^{2}L^{2}(\Sigma_{1}))\otimes(\otimes_{a}^{2}L^{2}({\mathbb{R}}^{3}))\
.$
The free Hamiltonian, now denoted by $H_{0}^{\prime}$, is then given by
$\begin{split}H_{0}^{\prime}=&\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int
w_{\ell}^{(1)}(\xi_{1})b^{*}_{\ell,\epsilon}(\xi_{1})b_{\ell,\epsilon}(\xi_{1})\mathrm{d}\xi_{1}+\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int_{{\mathbb{R}}^{3}}|p_{2}|c^{*}_{\ell,\epsilon}(p_{2})c_{\ell,\epsilon}(p_{2})\mathrm{d}p_{2}\\\
&+\sum_{\epsilon=\pm}\int
w^{(3)}(\xi_{3})a^{*}_{\epsilon}(\xi_{3})a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\
,\end{split}$
and the interaction, now denoted by $H_{I}^{\prime}$, is the one obtained from
$H_{I}$ by supposing that $G^{(\alpha)}(\xi_{1},(p_{2},s_{2}),\xi_{3})=0$ if
$s_{2}=\epsilon\frac{1}{2}$. The total Hamiltonian, denoted by $H^{\prime}$,
is then given by $H^{\prime}=H_{0}^{\prime}+g\,H_{I}^{\prime}$. The results
obtained in this paper for $H$ hold true for $H^{\prime}$ with obvious
modifications.
Under Hypothesis 2.1 a well defined operator on ${\mathfrak{D}}$ corresponds
to the formal interaction $H_{I}$ as it follows.
The formal operator
$\int
G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})b^{*}_{\ell,\epsilon}(\xi_{1})c^{*}_{\ell,\epsilon^{\prime}}(\xi_{2})a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}$
is defined as a quadratic form on
$({\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W})\times({\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W})$
as
$\int(c_{\ell,\epsilon^{\prime}}(\xi_{2})b_{\ell,\epsilon}(\xi_{1})\psi,\
G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}a_{\epsilon}(\xi_{3})\phi)\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\
,$
where $\psi$, $\phi\in{\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W}$.
By mimicking the proof of [24, Theorem X.44], we get a closed operator,
denoted by $H^{(1)}_{I,\ell,\epsilon,\epsilon^{\prime}}$, associated with the
quadratic form such that it is the unique operator in
${\mathfrak{F}}_{\ell}\otimes{\mathfrak{F}}_{W}$ such that
${\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W}\subset\
{\mathcal{D}}(H^{(1)}_{I,\ell,\epsilon,\epsilon^{\prime}})$ is a core for
$H^{(1)}_{I,\ell,\epsilon,\epsilon^{\prime}}$ and
$H^{(1)}_{I,\ell,\epsilon,\epsilon^{\prime}}=\int
G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})b^{*}_{\ell,\epsilon}(\xi_{1})c^{*}_{\ell,\epsilon^{\prime}}(\xi_{2})a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}$
as quadratic forms on
$({\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W})\times({\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W})$.
The formal operator
$\int\overline{G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})}c_{\ell,\epsilon^{\prime}}(\xi_{2})b_{\ell,\epsilon}(\xi_{1})a^{*}_{\epsilon}(\xi_{3})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}$
is similarly associated with
$(H^{(1)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}$ and
$(H^{(1)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}=\int\overline{G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})}c_{\ell,\epsilon^{\prime}}(\xi_{2})b_{\ell,\epsilon}(\xi_{1})a^{*}_{\epsilon}(\xi_{3})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}$
as quadratic forms on
$({\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W})\times({\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W})$.
Moreover,
${\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W}\subset{\mathcal{D}}((H^{(1)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*})$
is a core for $(H^{(1)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}$.
Again, there exists two closed operators
$H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}}$ and
$(H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}$ such that
${\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W}\subset{\mathcal{D}}(H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}})$,
${\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W}\subset{\mathcal{D}}((H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*})$
and ${\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W}$ is a core for
$H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}}$ and
$(H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}$ and such that
$H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}}=\int
G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})b^{*}_{\ell,\epsilon}(\xi_{1})c^{*}_{\ell,\epsilon^{\prime}}(\xi_{2})a^{*}_{\epsilon}(\xi_{3})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\
,$ $(H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}=\int
G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})a_{\epsilon}(\xi_{3})c_{\ell,\epsilon^{\prime}}(\xi_{2})b_{\ell,\epsilon}(\xi_{1})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}$
as quadratic forms on
$({\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W})\times({\mathfrak{D}}_{\ell}\otimes{\mathfrak{D}}_{W})$.
We shall still denote $H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}}$ and
$(H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}$ ($\alpha=1,2$) their
extensions to ${\mathfrak{F}}$. The set ${\mathfrak{D}}$ is then a core for
$H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}}$ and
$(H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}$
Thus
$H=H_{0}+g\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}(H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}}+(H^{(2)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*})$
is a symmetric operator defined on ${\mathfrak{D}}$.
We now want to prove that $H$ is essentially self-adjoint on ${\mathfrak{D}}$
by showing that $H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}}$ and
$(H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*}$ are relatively
$H_{0}$-bounded.
Once again, as above, for almost every $\xi_{3}\in\Sigma_{2}$, there exists
closed operators in ${\mathfrak{F}}_{L}$, denoted by
$B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})$ and
$(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}$ such that
$B^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})=-\int\overline{G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})}b_{\ell,\epsilon}(\xi_{1})c_{\ell,\epsilon^{\prime}}(\xi_{2})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\
,$ $(B^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}=\int
G^{(1)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})b^{*}_{\ell,\epsilon}(\xi_{1})c^{*}_{\ell,\epsilon^{\prime}}(\xi_{2})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\
,$ $B^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})=\int
G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})b^{*}_{\ell,\epsilon}(\xi_{1})c^{*}_{\ell,\epsilon^{\prime}}(\xi_{2})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\
,$
$(B^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}=-\int\overline{G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})}b_{\ell,\epsilon}(\xi_{1})c_{\ell,\epsilon^{\prime}}(\xi_{2})\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\
\,$
as quadratic forms on ${\mathfrak{D}}_{\ell}\times{\mathfrak{D}}_{\ell}$.
We have that
${\mathfrak{D}}_{\ell}\subset{\mathcal{D}}(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))$
(resp.
${\mathfrak{D}}_{\ell}\subset{\mathcal{D}}((B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*})$
is a core for $B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})$ (resp.
for $(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}$). We still
denote by $B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))$ and
$(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*})$ their
extensions to ${\mathfrak{F}}_{L}$.
It then follows that the operator $H_{I}$ with domain ${\mathfrak{D}}$ is
symmetric and can be written in the following form
$\begin{split}&H_{I}=\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}(H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}}+(H^{(\alpha)}_{I,\ell,\epsilon,\epsilon^{\prime}})^{*})\\\
&\\!=\\!\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int
B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})\otimes
a^{*}_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}+\\!\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}\otimes
a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\,.\end{split}$
Let $N_{\ell}$ denote the operator number of massive leptons $\ell$ in
${\mathfrak{F}}_{\ell}$, i.e.,
(2.15) $N_{\ell}=\sum_{\epsilon}\int
b_{\ell,\epsilon}^{*}(\xi_{1})b_{\ell,\epsilon}(\xi_{1})\mathrm{d}\xi_{1}\ .$
The operator $N_{\ell}$ is a positive self-adjoint operator in
${\mathfrak{F}}_{\ell}$. We still denote by $N_{\ell}$ its extension to
${\mathfrak{F}}_{L}$. The set ${\mathfrak{D}}_{L}$ is a core for $N_{\ell}$.
We then have
###### Proposition 2.4.
For a.e. $\xi_{3}\in\Sigma_{2}$,
${\mathcal{D}}(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))$,
${\mathcal{D}}((B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*})\supset{\mathcal{D}}(N_{\ell}^{\frac{1}{2}})$,
and for
$\Phi\in{\mathcal{D}}(N_{\ell}^{\frac{1}{2}})\subset{\mathfrak{F}}_{L}$ we
have
(2.16)
$\|B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})\Phi\|_{{\mathfrak{F}}_{L}}\leq\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(.,.,\xi_{3})\|_{L^{2}(\Sigma_{1}\times\Sigma_{1})}\|N_{\ell}^{\frac{1}{2}}\Phi\|_{{\mathfrak{F}}_{L}}\
,$ (2.17)
$\|(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}\Phi\|_{{\mathfrak{F}}_{L}}\leq\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(.,.,\xi_{3})\|_{L^{2}(\Sigma_{1}\times\Sigma_{1})}\|N_{\ell}^{\frac{1}{2}}\Phi\|_{{\mathfrak{F}}_{L}}\
.$
###### Proof.
The estimates (2.16) and (2.17) are examples of $N_{\tau}$ estimates (see
[16]). We give a proof for sake of completeness. We only consider
$B^{(1)}_{1,+,-}$. The other cases are quite similar.
Let $\Phi=(\Phi^{(Q)})_{Q}$ and $\Psi=(\Psi^{(Q^{\prime})})_{Q^{\prime}}$ be
two vectors in ${\mathfrak{D}}_{L}$, where we use the notations
$Q={(q_{\ell},\bar{q}_{\ell},r_{\ell},\bar{r}_{\ell})}_{\ell=1,2,3}$, and
$Q^{\prime}=(q_{\ell}^{\prime},\bar{q}_{\ell}^{\prime},r_{\ell}^{\prime},\bar{r}_{\ell}^{\prime})_{\ell=1,2,3}$.
We have
(2.18)
$\begin{split}&(\Psi^{(Q^{\prime})},B^{(1)}_{1,+,-}(\xi_{3})\Phi^{(Q)})_{{\mathfrak{F}}_{L}}=-\delta_{q^{\prime}_{1}\,q_{1}-1}\delta_{\bar{q}_{1}^{\prime}\,\bar{q}_{1}}\delta_{r^{\prime}_{1}\,r_{1}}\delta_{\bar{r}_{1}^{\prime}\,\bar{r}_{1}-1}\prod_{\ell=2}^{3}\delta_{q_{\ell}^{\prime}q_{\ell}}\delta_{\bar{q}_{\ell}^{\prime}\bar{q}_{\ell}}\delta_{r_{\ell}^{\prime}r_{\ell}}\delta_{\bar{r}_{\ell}^{\prime}\bar{r}_{\ell}}\\\
&\int_{\Sigma_{1}\times\Sigma_{1}}(\Psi^{(\tilde{Q})},b_{1,+}(\xi_{1})c_{1,-}(\xi_{2})\Phi^{(Q)})_{{\mathfrak{F}}_{L}}\overline{G^{(1)}_{1,+,-}(\xi_{1},\xi_{2},\xi_{3})}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\
.\end{split}$
Here
$\tilde{Q}=(q_{1}-1,\bar{q}_{1},r_{1},\bar{r}_{1}-1,q_{2},\bar{q}_{2},r_{2},\bar{r}_{2},q_{3},\bar{q}_{3},r_{3},\bar{r}_{3})$.
For each $Q$,
(2.19)
$B^{(1)}_{1,+,-}(\xi_{3})\Phi^{(Q)}\in{\mathfrak{F}}_{1}^{(q_{1}-1,\bar{q}_{1},r_{1},\bar{r}_{1}-1)}\otimes{\mathfrak{F}}_{2}^{(q_{2},\bar{q}_{2},r_{2},\bar{r}_{2})}\otimes{\mathfrak{F}}_{3}^{(q_{3},\bar{q}_{3},r_{3},\bar{r}_{3})}.$
By the Fubini theorem we have
$\begin{split}&\left|(\Psi^{(\tilde{Q})},B^{(1)}_{1,+,-}(\xi_{3})\Psi^{(Q)})_{{\mathfrak{F}}_{L}}\right|\\\
&=\left|\int_{\Sigma_{1}}\left(\int_{\Sigma_{1}}G^{(1)}_{1,+,-}(\xi_{1},\xi_{2},\xi_{3})c_{1,-}^{*}(\xi_{2})\Psi^{(\tilde{Q})}\mathrm{d}\xi_{2},b_{1,+}(\xi_{1})\Phi^{(Q)}\right)_{{\mathfrak{F}}_{L}}\mathrm{d}\xi_{1}\right|\
.\end{split}$
By (2.9), and the Cauchy-Schwarz inequality we get
$\begin{split}&\left|(\Psi^{(\tilde{Q})},B^{(1)}_{1,+,-}(\xi_{3})\Psi^{(Q)})_{{\mathfrak{F}}_{L}}\right|^{2}\\\
&\leq\left(\int_{\Sigma_{1}}\|b_{1,+}(\xi_{1})\Phi^{(Q)}\|\left(\int_{\Sigma_{1}}|G^{(1)}_{1,+,-}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{2}\right)^{\frac{1}{2}}\mathrm{d}\xi_{1}\right)^{2}\|\Psi^{(\tilde{Q})}\|^{2}\
.\end{split}$
By the definition of $b_{1,+}(\xi_{1})\Phi^{(Q)}$ and the Cauchy-Schwarz
inequality we get
$\begin{split}&|(\Psi^{(\tilde{Q})},B^{(1)}_{1,+,-}(\xi_{3})\Phi^{(Q)})_{{\mathfrak{F}}_{L}}|^{2}\\\
&\leq
q_{1}\left(\int_{\Sigma_{1}}\int_{\Sigma_{1}}|G^{(1)}_{1,+,-}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\right)\|\Psi^{(\tilde{Q})}\|^{2}_{{\mathfrak{F}}_{L}}\|\Phi^{(Q)}\|^{2}_{{\mathfrak{F}}_{L}}\\\
&=\left(\int_{\Sigma_{1}}\int_{\Sigma_{1}}|G^{(1)}_{1,+,-}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\right)\|\Psi^{(\tilde{Q})}\|^{2}_{{\mathfrak{F}}_{L}}\|N_{1}^{\frac{1}{2}}\Phi^{(Q)}\|^{2}_{{\mathfrak{F}}_{L}}\
.\end{split}$
By (2.19) we have
$\begin{split}|(\Psi,B^{(1)}_{1,+,-}(\xi_{3})\Phi^{(Q)})_{{\mathfrak{F}}_{L}}|^{2}\leq\|\Psi\|_{{\mathfrak{F}}_{L}}^{2}\|N_{1}^{\frac{1}{2}}\Phi^{(Q)}\|^{2}_{{\mathfrak{F}}_{L}}\int_{\Sigma_{1}\times\Sigma_{1}}|G^{(1)}_{1,+,-}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\
,\end{split}$
for every $\Psi\in{\mathfrak{D}}_{L}$. Therefore we get
$\|B^{(1)}_{1,+,-}(\xi_{3})\Phi^{(Q)}\|^{2}_{{\mathfrak{F}}_{L}}\leq\left(\int_{\Sigma_{1}\times\Sigma_{1}}|G^{(1)}_{1,+,-}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\right)\|N_{1}^{\frac{1}{2}}\Phi^{(Q)}\|^{2}_{{\mathfrak{F}}_{L}}\
,$
and by (2.19) we finally obtain
$\|B^{(1)}_{1,+,-}(\xi_{3})\Phi\|^{2}_{{\mathfrak{F}}_{L}}\leq\left(\int_{\Sigma_{1}\times\Sigma_{1}}|G^{(1)}_{1,+,-}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\right)\|N_{1}^{\frac{1}{2}}\Phi\|^{2}_{{\mathfrak{F}}_{L}}\
,$
for every $\Phi\in{\mathfrak{D}}$.
Since ${\mathfrak{D}}_{L}$ is a core for $N_{1}^{\frac{1}{2}}$ and
$B^{(1)}_{1,+,-}$ with domain ${\mathfrak{D}}_{L}$ is closable,
${\mathcal{D}}(B^{(1)}_{1,+,-}(\xi_{3}))\supset{\mathcal{D}}(N_{1}^{\frac{1}{2}})$,
and (2.16) is satisfied for every $\Phi\in{\mathcal{D}}(N_{1}^{\frac{1}{2}})$.
∎
Let
$H^{(3)}_{0,\epsilon}=\int
w^{(3)}(\xi_{3})a_{\epsilon}^{*}(\xi_{3})a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\
.$
Then $H^{(3)}_{0,\epsilon}$ is a self-adjoint operator in
${\mathfrak{F}}_{W}$, and ${\mathfrak{D}}_{W}$ is a core for
$H^{(3)}_{0,\epsilon}$.
We get
###### Proposition 2.5.
(2.20)
$\begin{split}&\|\int(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}\otimes
a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\Psi\|^{2}\\\
&\leq(\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}\frac{|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})|^{2}}{w^{(3)}(\xi_{3})}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3})\
\|(N_{\ell}+1)^{\frac{1}{2}}\otimes(H_{0,\epsilon}^{(3)})^{\frac{1}{2}}\Psi\|^{2}\end{split}$
and
(2.21) $\begin{split}&\|\int
B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})\otimes
a^{*}_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\Psi\|^{2}\\\
&\leq(\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}\frac{|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})|^{2}}{w^{(3)}(\xi_{3})}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3})\
\|(N_{\ell}+1)^{\frac{1}{2}}\otimes(H_{0,\epsilon}^{(3)})^{\frac{1}{2}}\Psi\|^{2}\\\
&+(\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3})\
(\eta\|(N_{\ell}+1)^{\frac{1}{2}}\otimes{\mathbf{1}}\
\Psi\|^{2}+\frac{1}{4\eta}\|\Psi\|^{2})\ ,\end{split}$
for every $\Psi\in{\mathcal{D}}(H_{0})$ and every $\eta>0$.
###### Proof.
Suppose that
$\Psi\in{\mathcal{D}}(N_{\ell}^{\frac{1}{2}})\hat{\otimes}{\mathcal{D}}((H_{0,\epsilon}^{(3)})^{\frac{1}{2}})$.
Let
$\Psi_{\epsilon}(\xi_{3})=w^{(3)}(\xi_{3})^{\frac{1}{2}}((N_{\ell}+1)^{\frac{1}{2}}\otimes
a_{\epsilon}(\xi_{3}))\Phi\ .$
We have
$\int_{\Sigma_{2}}\|\Psi_{\epsilon}(\xi_{3})\|^{2}\mathrm{d}\xi_{3}=\|(N_{\ell}+1)^{\frac{1}{2}}\otimes(H_{0,\epsilon}^{(3)})^{\frac{1}{2}}\Psi\|^{2}\
.$
We get
$\begin{split}&\int(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}\otimes
a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\Psi\\\
&=\int_{\Sigma_{2}}\frac{1}{(w^{(3)}(\xi_{3}))^{\frac{1}{2}}}((B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}(N_{\ell}+1)^{-\frac{1}{2}}\otimes{\mathbf{1}})\Psi_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\
.\end{split}$
Therefore
(2.22)
$\begin{split}&\|\int(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}\otimes
a_{\epsilon}(\xi_{3})\,\Psi\mathrm{d}\xi_{3}\|^{2}_{\mathfrak{F}}\\\
&\leq(\int_{\Sigma_{2}}\frac{1}{w^{(3)}(\xi_{3})}\|(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}(N_{\ell}+1)^{-\frac{1}{2}}\|_{{\mathfrak{F}}_{L}}\|\Psi_{\epsilon}(\xi_{3})\|_{{\mathfrak{F}}}\mathrm{d}\xi_{3})^{2}\\\
&\leq(\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}\frac{|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{2},\xi_{2},\xi
3)|^{2}}{w^{(3)}(\xi_{3})}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3})\|(N_{\ell}+1)^{\frac{1}{2}}\otimes(H^{(3)}_{0,\epsilon})^{\frac{1}{2}}\Psi\|_{\mathfrak{F}}^{2}\
,\end{split}$
as it follows from Proposition 2.4.
We now have
$\begin{split}&\|\int
B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})\otimes
a_{\epsilon}^{*}(\xi_{3})\,\Psi\mathrm{d}\xi_{3}\|^{2}_{\mathfrak{F}}\\\
&=\int(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})\otimes
a_{\epsilon}(\xi_{3}^{\prime})\Psi,\
B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}^{\prime})\otimes
a_{\epsilon}(\xi_{3})\,\Psi)\mathrm{d}\xi_{3}\mathrm{d}\xi_{3}^{\prime}+\int\|(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}\otimes{\mathbf{1}})\Psi\|^{2}\mathrm{d}\xi_{3}\
,\end{split}$
and
(2.23)
$\begin{split}&\int_{\Sigma_{2}\times\Sigma_{2}}(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})\otimes
a_{\epsilon}(\xi_{3}^{\prime})\Psi,B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}^{\prime})\otimes
a_{\epsilon}(\xi_{3})\Psi)\mathrm{d}\xi_{3}\mathrm{d}\xi_{3}^{\prime}\\\
&=\int_{\Sigma_{2}\times\Sigma_{2}}\frac{1}{w^{(3)}(\xi_{3})^{\frac{1}{2}}w^{(3)}(\xi_{3}^{\prime})^{\frac{1}{2}}}\Big{(}(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})(N_{\ell}+1)^{-\frac{1}{2}}\otimes{\mathbf{1}})\Psi_{\epsilon}(\xi_{3}^{\prime}),\\\
&(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}^{\prime})(N_{\ell}+1)^{-\frac{1}{2}}\otimes{\mathbf{1}})\Psi_{\epsilon}(\xi_{3})\Big{)}\mathrm{d}\xi_{3}\mathrm{d}\xi_{3}^{\prime}\\\
&\leq(\int_{\Sigma_{2}}\frac{1}{w^{(3)}(\xi_{3})^{\frac{1}{2}}}\|B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})(N_{\ell}+1)^{-\frac{1}{2}}\|_{{\mathfrak{F}}_{L}}\|\Psi_{\epsilon}(\xi_{3})\|\mathrm{d}\xi_{3})^{2}\\\
&\leq(\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}\frac{|G^{(\alpha)}(\xi_{1},\xi_{2},\xi_{3})|^{2}}{w^{(3)}(\xi_{3})}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3})\|(N_{\ell}+1)^{\frac{1}{2}}\otimes(H_{0,\epsilon}^{(3)})^{\frac{1}{2}}\Psi\|^{2}\
.\end{split}$
Furthermore
(2.24)
$\begin{split}&\int_{\Sigma_{2}}\|B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})\otimes{\mathbf{1}})\Psi\|^{2}\mathrm{d}\xi_{3}\\\
&=\int_{\Sigma_{2}}\|B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3})(N_{\ell}+1)^{-\frac{1}{2}}\otimes{\mathbf{1}})((N_{\ell}+1)^{\frac{1}{2}}\otimes{\mathbf{1}})\Psi\|^{2}\mathrm{d}\xi_{3}\\\
&\leq\left(\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\right)\,(\eta\|(N_{\ell}+1)\Psi\|^{2}+\frac{1}{4\eta}\|\Psi\|^{2})\
,\end{split}$
for every $\eta>0$.
By (2.22), (2.23), and (2.24), we finally get (2.20) and (2.21) for every
$\Psi\in{\mathcal{D}}(N_{\ell}^{\frac{1}{2}})\hat{\otimes}{\mathcal{D}}(H_{0,\epsilon}^{(3)})$.
The set
${\mathcal{D}}(N_{\ell}^{\frac{1}{2}})\hat{\otimes}{\mathcal{D}}(H_{0,\epsilon}^{(3)})$
is a core for $N_{\ell}^{\frac{1}{2}}\otimes H_{0,\epsilon}^{(3)}$ and
${\mathcal{D}}(H_{0})\subset{\mathcal{D}}(N_{\ell}^{\frac{1}{2}}\otimes
H_{0,\epsilon}^{(3)})$. It then follows that (2.20) and (2.21) are verified
for every $\Psi\in{\mathcal{D}}(H_{0})$. ∎
We now prove that $H$ is a self-adjoint operator in ${\mathfrak{F}}$ for $g$
sufficiently small.
###### Theorem 2.6.
Let $g_{1}>0$ be such that
$\frac{3g_{1}^{2}}{m_{W}}(\frac{1}{m_{1}^{2}}+1)\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}\|^{2}_{L^{2}(\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2})}<1\
.$
Then for every $g$ satisfying $g\leq g_{1}$, $H$ is a self-adjoint operator in
${\mathfrak{F}}$ with domain ${\mathcal{D}}(H)={\mathcal{D}}(H_{0})$, and
${\mathfrak{D}}$ is a core for $H$.
###### Proof.
Let $\Psi$ be in ${\mathfrak{D}}$. We have
(2.25)
$\begin{split}\|H_{I}\Psi\|^{2}\leq&12\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\Big{\\{}\left\|\int(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))^{*}\otimes
a_{\epsilon}(\xi_{3})\,\Psi\mathrm{d}\xi_{3}\right\|^{2}\\\
&+\left\|\int(B^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{3}))\otimes
a^{*}_{\epsilon}(\xi_{3})\,\Psi\mathrm{d}\xi_{3}\right\|^{2}\Big{\\}}\
.\end{split}$
Note that
$\|H_{0,\epsilon}^{(3)}\Psi\|\leq\|H_{0}^{(3)}\Psi\|\leq\|H_{0}\Psi\|\ ,$
and
$\|N_{\ell}\Psi\|\leq\frac{1}{m_{\ell}}\|H_{0,\ell}\Psi\|\leq\frac{1}{m_{1}}\|H_{0,\ell}\Psi\|\leq\frac{1}{m_{1}}\|H_{0}\Psi\|\
,$
where
(2.26) $H_{0,\ell}=\sum_{\epsilon}\int
w_{\ell}^{(1)}(\xi_{1})b_{\ell,\epsilon}^{*}(\xi_{1})b_{\ell,\epsilon}(\xi_{1})\mathrm{d}\xi_{1}+\sum_{\epsilon}\int
w_{\ell}^{(2)}(\xi_{2})c_{\ell,\epsilon}^{*}(\xi_{2})c_{\ell,\epsilon}(\xi_{2})\mathrm{d}\xi_{2}\
.$
We further note that
(2.27)
$\|(N_{\ell}+1)^{\frac{1}{2}}\otimes(H^{(3)}_{0,\epsilon})^{\frac{1}{2}}\Psi\|^{2}\leq\frac{1}{2}(\frac{1}{m_{1}^{2}}+1)\|H_{0}\Psi\|^{2}+\frac{\beta}{2m_{1}^{2}}\|H_{0}\Psi\|^{2}+(\frac{1}{2}+\frac{1}{8\beta})\|\Psi\|^{2},$
for $\beta>0$, and
(2.28)
$\eta\|((N_{\ell}+1)\otimes{\mathbf{1}})\Psi\|^{2}+\frac{1}{4\eta}\|\Psi\|^{2}\leq\frac{\eta}{m_{1}^{2}}\|H_{0}\Psi\|^{2}+\frac{\eta\beta}{m_{1}^{2}}\|H_{0}\Psi\|^{2}+\eta(1+\frac{1}{4\beta})\|\Psi\|^{2}+\frac{1}{4\eta}\|\Psi\|^{2}.$
Combining (2.25) with (2.20), (2.21), (2.27) and (2.28) we get for $\eta>0$,
$\beta>0$
(2.29) $\begin{split}&\|H_{I}\Psi\|^{2}\leq
6(\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}\|^{2})\\\
&\Big{(}\frac{1}{2m_{W}}(\frac{1}{m_{1}^{2}}+1)\|H_{0}\Psi\|^{2}+\frac{\beta}{2m_{W}m_{1}^{2}}\|H_{0}\Psi\|^{2}+\frac{1}{2m_{W}}(1+\frac{1}{4\beta})\|\Psi\|^{2}\Big{)}\\\
&+12(\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}\|^{2})(\frac{\eta}{m_{1}^{2}}(1+\beta)\|H_{0}\Psi\|^{2}+(\eta(1+\frac{1}{4\beta})+\frac{1}{4\eta})\|\Psi\|^{2}),\end{split}$
by noting
(2.30)
$\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}\frac{|G_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})|^{2}}{w^{(3)}(\xi_{3})}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\leq\frac{1}{m_{W}}\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}\|^{2}.$
By (2.29) the theorem follows from the Kato-Rellich theorem. ∎
## 3\. Main results
In the sequel, we shall make the following additional assumptions on the
kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$.
###### Hypothesis 3.1.
$(i)$ For $\alpha=1,2,\ \ell=1,2,3,\ \epsilon,\epsilon^{\prime}=\pm$,
$\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}\frac{|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})|^{2}}{|p_{2}|^{2}}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}<\infty,\quad$
$(ii)$ There exists $C>0$ such that for $\alpha=1,2,\ \ell=1,2,3,\
\epsilon,\epsilon^{\prime}=\pm$,
$\left(\int_{\Sigma_{1}\times\\{|p_{2}|\leq\sigma\\}\times\Sigma_{2}}|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\right)^{\frac{1}{2}}\leq
C\sigma^{2}.$
$(iii)$ For $\alpha=1,2,\ \ell=1,2,3,\ \epsilon,\epsilon^{\prime}=\pm$, and
$i,j=1,2,3$
$(iii.a)\quad\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}\left|[(p_{2}\cdot\nabla_{p_{2}})G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}](\xi_{1},\xi_{2},\xi_{3})\right|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}<\infty\
,$
and
$(iii.b)\quad\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}p_{2,i}^{2}\,p_{2,j}^{2}\left|\frac{\partial^{2}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}}{\partial
p_{2,i}\partial
p_{2,j}}(\xi_{1},\xi_{2},\xi_{3})\right|^{2}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}<\infty\
.$
$(iv)$ There exists $\Lambda>m_{1}$ such, that for $\alpha=1,2$, $\ell=1,2,3$,
$\epsilon,\epsilon^{\prime}=\pm$,
$G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})=0\quad\mbox{if}\quad|p_{2}|\geq\Lambda\
.$
###### Remark 3.2.
Hypothesis 3.1 (ii) is nothing but an infrared regularization of the kernels
$G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$. In order to satisfy this
hypothesis it is, for example, sufficient to suppose
$G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})=|p_{2}|^{\frac{1}{2}}\tilde{G}^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})\
,$
where $\tilde{G}^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$ is a smooth
function of $(p_{1},p_{2},p_{3})$ in the Schwartz space.
The Hypothesis 3.1 (iv), which is a sharp ultraviolet cutoff, is actually not
necessary, and can be removed at the expense of some additional technicalities
in Appendix A. However, in order to simplify the proof of Proposition 3.5, we
shall leave it.
Our first result is devoted to the existence of a ground state for $H$
together with the location of the spectrum of $H$ and of its absolutely
continuous spectrum when $g$ is sufficiently small.
###### Theorem 3.3.
Suppose that the kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$
satisfy Hypothesis 2.1 and Hypothesis 3.1 (i). Then there exists $0<g_{2}\leq
g_{1}$ such that $H$ has a unique ground state for $g\leq g_{2}$. Moreover
$\sigma(H)=\sigma_{\rm{ac}}(H)=[\inf\sigma(H),\infty)\ ,$
with $\inf\sigma(H)\leq 0$.
According to Theorem 3.3 the ground state energy $E=\inf\sigma(H)$ is a simple
eigenvalue of $H$ and our main results are concerned with a careful study of
the spectrum of $H$ above the ground state energy. The spectral theory
developed in this work is based on the conjugated operator method as described
in [23], [3] and [25]. Our choice of the conjugate operator denoted by $A$ is
the second quantized dilation generator for the neutrinos.
Let $a$ denote the following operator in $L^{2}(\Sigma_{1})$
$a=\frac{1}{2}(p_{2}\cdot i\nabla_{p_{2}}+i\nabla_{p_{2}}\cdot p_{2})\ .$
The operator $a$ is essentially self-adjoint on
$C_{0}^{\infty}({\mathbb{R}}^{3},{\mathbb{C}}^{2})$. Its second quantized
version $\mathrm{d}\Gamma(a)$ is a self-adjoint operator in
${\mathfrak{F}}_{a}(L^{2}(\Sigma_{1}))$. From the definition (2.4) of the
space ${\mathfrak{F}}_{\ell}$, the following operator in
${\mathfrak{F}}_{\ell}$
$A_{\ell}={\mathbf{1}}\otimes{\mathbf{1}}\otimes\mathrm{d}\Gamma(a)\otimes{\mathbf{1}}+{\mathbf{1}}\otimes{\mathbf{1}}\otimes{\mathbf{1}}\otimes\mathrm{d}\Gamma(a)$
is essentially self-adjoint on ${\mathfrak{D}}_{L}$.
Let now $A$ be the following operator in ${\mathfrak{F}}_{L}$
$A=A_{1}\otimes{\mathbf{1}}_{2}\otimes{\mathbf{1}}_{3}+{\mathbf{1}}_{1}\otimes
A_{2}\otimes{\mathbf{1}}_{3}+{\mathbf{1}}_{1}\otimes{\mathbf{1}}_{2}\otimes
A_{3}\ .$
Then $A$ is essentially self-adjoint on ${\mathfrak{D}}_{L}$.
We shall denote again by $A$ its extension to ${\mathfrak{F}}$. Thus $A$ is
essentially self-adjoint on ${\mathfrak{D}}$ and we still denote by $A$ its
closure.
We also set
$\langle A\rangle=(1+A^{2})^{\frac{1}{2}}\ .$
We then have
###### Theorem 3.4.
Suppose that the kernels $G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)}$
satisfy Hypothesis 2.1 and 3.1. For any $\delta>0$ satisfying $0<\delta<m_{1}$
there exists $0<g_{\delta}\leq g_{2}$ such that, for $0<g\leq g_{\delta}$,
$(i)$ The spectrum of $H$ in $(\inf\sigma(H),\,m_{1}-\delta]$ is purely
absolutely continuous.
$(ii)$ Limiting absorption principle.
For every $s>1/2$ and $\varphi$, $\psi$ in ${\mathfrak{F}}$, the limits
$\lim_{\varepsilon\rightarrow 0}(\varphi,\ \langle A\rangle^{-s}(H-\lambda\pm
i\varepsilon)\langle A\rangle^{-s}\psi)$
exist uniformly for $\lambda$ in any compact subset of
$(\inf\sigma(H),\,m_{1}-\delta]$.
$(iii)$ Pointwise decay in time.
Suppose $s\in(\frac{1}{2},1)$ and $f\in C_{0}^{\infty}({\mathbb{R}})$ with
$\mathrm{supp}f\subset(\inf\sigma(H),\,m_{1}-\delta)$. Then
$\|\langle A\rangle^{-s}\mathrm{e}^{-itH}f(H)\langle
A\rangle^{-s}\|=\mathcal{O}({t^{\frac{1}{2}-s}})\ ,$
as $t\rightarrow\infty$.
The proof of Theorem 3.4 is based on a positive commutator estimate, called
the Mourre estimate and on a regularity property of $H$ with respect to $A$
(see [23], [3] and [25]). According to [13], the main ingredient of the proof
are auxiliary operators associated with infrared cutoff Hamiltonians with
respect to the momenta of the neutrinos that we now introduce.
Let $\chi_{0}(.)$, $\chi_{\infty}(.)\in C^{\infty}({\mathbb{R}},[0,1])$ with
$\chi_{0}=1$ on $(-\infty,1]$, $\chi_{\infty}=1$ on $[2,\infty)$ and
$\chi_{0}{}^{2}+\chi_{\infty}{}^{2}=1$.
For $\sigma>0$ we set
(3.1) $\begin{split}&\chi_{\sigma}(p)=\chi_{0}(|p|/\sigma)\ ,\\\
&\chi^{\sigma}(p)=\chi_{\infty}(|p|/\sigma)\ ,\\\
&\tilde{\chi}^{\sigma}(p)=1-\chi_{\sigma}(p)\ ,\end{split}$
where $p\in{\mathbb{R}}^{3}$.
The operator $H_{I,\sigma}$ is the interaction given by (2.10), (2.11) and
(2.12) and associated with the kernels
$\tilde{\chi}^{\sigma}(p_{2})G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})$.
We then set
$H_{\sigma}:=H_{0}+gH_{I,\sigma}\ .$
Let
$\displaystyle\Sigma_{1,\sigma}=\Sigma_{1}\cap\\{(p_{2},s_{2});\
|p_{2}|<\sigma\\}\ ,$ $\displaystyle\Sigma_{1}^{\
\sigma}=\Sigma_{1}\cap\\{(p_{2},s_{2});\ |p_{2}|\geq\sigma\\}$
$\displaystyle{\mathfrak{F}}_{\ell,2,\sigma}={\mathfrak{F}}_{a}(L^{2}(\Sigma_{1,\sigma}))\otimes{\mathfrak{F}}_{a}(L^{2}(\Sigma_{1,\sigma}))\
,$ $\displaystyle{\mathfrak{F}}_{\ell,2}^{\
\sigma}={\mathfrak{F}}_{a}(L^{2}(\Sigma_{1}^{\
\sigma}))\otimes{\mathfrak{F}}_{a}(L^{2}(\Sigma_{1}^{\ \sigma}))\ ,$
$\displaystyle{\mathfrak{F}}_{\ell,2}={\mathfrak{F}}_{\ell,2,\sigma}\otimes{\mathfrak{F}}_{\ell,2}^{\
\sigma}\ ,$
$\displaystyle{\mathfrak{F}}_{\ell,1}=\bigotimes^{2}{\mathfrak{F}}_{a}(L^{2}(\Sigma_{1}))\
.$
The space ${\mathfrak{F}}_{\ell,1}$ is the Fock space for the massive leptons
$\ell$ and ${\mathfrak{F}}_{\ell,2}$ is the Fock space for the neutrinos and
antineutrinos $\ell$.
Set
$\displaystyle{\mathfrak{F}}_{\ell}^{\
\sigma}={\mathfrak{F}}_{\ell,1}\otimes{\mathfrak{F}}_{\ell,2}^{\ \sigma}\ ,$
$\displaystyle{\mathfrak{F}}_{\ell,\sigma}={\mathfrak{F}}_{\ell,2,\sigma}\ .$
We have
${\mathfrak{F}}_{\ell}\simeq{\mathfrak{F}}_{\ell}^{\
\sigma}\otimes{\mathfrak{F}}_{\ell,\sigma}\ .$
Set
$\begin{split}&{\mathfrak{F}}_{L}^{\
\sigma}=\bigotimes_{\ell=1}^{3}{\mathfrak{F}}_{\ell}^{\ \sigma}\ ,\\\
&{\mathfrak{F}}_{L,\sigma}=\bigotimes_{\ell=1}^{3}{\mathfrak{F}}_{\ell,\sigma}\
.\end{split}$
We have
${\mathfrak{F}}_{L}\simeq{\mathfrak{F}}_{L}^{\
\sigma}\otimes{\mathfrak{F}}_{L,\sigma}\ .$
Set
$\begin{split}&{\mathfrak{F}}^{\ \sigma}={\mathfrak{F}}_{L}^{\
\sigma}\otimes{\mathfrak{F}}_{W}\ ,\\\ \end{split}$
We have
${\mathfrak{F}}\simeq{\mathfrak{F}}_{L,\sigma}\otimes{\mathfrak{F}}^{\
\sigma}\ .$
Set
$\begin{split}&H_{0}^{(1)}=\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int
w_{\ell}^{(1)}(\xi_{1})\,b^{*}_{\ell,\epsilon}(\xi_{1})b_{\ell,\epsilon}(\xi_{1})\mathrm{d}\xi_{1}\
,\\\ &H_{0}^{(2)}=\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int
w_{\ell}^{(2)}(\xi_{2})\,c^{*}_{\ell,\epsilon}(\xi_{2})c_{\ell,\epsilon}(\xi_{2})\mathrm{d}\xi_{2}\
,\\\ &H_{0}^{(3)}=\sum_{\epsilon=\pm}\int
w^{(3)}(\xi_{3})a^{*}_{\epsilon}(\xi_{3})a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{3}\
,\\\ \end{split}$
and
$\begin{split}&H_{0}^{(2){\
\sigma}}=\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int_{|p_{2}|>\sigma}w_{\ell}^{(2)}(\xi_{2})\,c^{*}_{\ell,\epsilon}(\xi_{2})c_{\ell,\epsilon}(\xi_{2})\mathrm{d}\xi_{2}\
,\\\
&H_{0,\sigma}^{(2)}=\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int_{|p_{2}|\leq\sigma}w_{\ell}^{(2)}(\xi_{2})\,c^{*}_{\ell,\epsilon}(\xi_{2})c_{\ell,\epsilon}(\xi_{2})\mathrm{d}\xi_{2}\
.\end{split}$
We have on ${\mathfrak{F}}^{\ \sigma}\otimes{\mathfrak{F}}_{\sigma}$
$H_{0}^{(2)}=H_{0}^{(2)\sigma}\otimes{\mathbf{1}}_{\sigma}+{\mathbf{1}}^{\
\sigma}\otimes H_{0,\sigma}^{(2)}\ .$
Here, ${\mathbf{1}}^{\sigma}$ (resp. ${\mathbf{1}}_{\sigma}$) is the identity
operator on ${\mathfrak{F}}^{\sigma}$ (resp. ${\mathfrak{F}}_{\sigma}$).
Define
(3.2) $H^{\sigma}=H_{\sigma}|_{{\mathfrak{F}}^{\,\sigma}}\quad\mbox{and}\quad
H_{0}^{\,\sigma}=H_{0}|_{{\mathfrak{F}}^{\sigma}}\ .$
We get
$H^{\sigma}=H_{0}^{(1)}+H_{0}^{(2)\,\sigma}+H_{0}^{(3)}+gH_{I,\sigma}\quad\mbox{on}\
{\mathfrak{F}}^{\,\sigma}\ ,$
and
$H_{\sigma}=H^{\sigma}\otimes{\mathbf{1}}_{\sigma}+{\mathbf{1}}^{\,\sigma}\otimes
H_{0,\sigma}^{(2)}\quad\mbox{on}\
{\mathfrak{F}}^{\,\sigma}\otimes{\mathfrak{F}}_{\sigma}\ .$
In order to implement the conjugate operator theory we have to show that
$H^{\,\sigma}$ has a gap in its spectrum above its ground state.
We now set, for $\beta>0$ and $\eta>0$,
(3.3)
$C_{\beta\,\eta}=\left(\frac{3}{m_{W}}(1+\frac{1}{m_{1}{}^{2}})+\frac{3\beta}{m_{W}m_{1}{}^{2}}+\frac{12\,\eta}{m_{1}{}^{2}}(1+\beta)\right)^{\frac{1}{2}}\
,$
and
(3.4)
$B_{\beta\,\eta}=\left(\frac{3}{m_{W}}(1+\frac{1}{4\beta})+12(\,\eta(1+\frac{1}{4\beta})+\frac{1}{4\eta}\,)\right)^{\frac{1}{2}}\
.$
Let
(3.5)
$G=\left(G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(.,.,.)\right)_{\alpha=1,2;\ell=1,2,3;\epsilon,\epsilon^{\prime}=\pm,\epsilon\neq\epsilon^{\prime}}$
and set
(3.6)
$K(G)=\left(\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}\|^{2}_{L^{2}(\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2})}\right)^{\frac{1}{2}}\
.$
Let
(3.7)
$\tilde{C}_{\beta\eta}=C_{\beta\eta}\left(1+\frac{g_{1}K(G)C_{\beta\eta}}{1-g_{1}K(G)C_{\beta\eta}}\right),$
(3.8)
$\tilde{B}_{\beta\eta}=\Big{(}\,1+\frac{g_{1}\,K(G)C_{\beta\eta}}{1-g_{1}\,K(G)\,C_{\beta\eta}}(\,2+\frac{g_{1}K(G)B_{\beta\eta}C_{\beta\eta}}{1-g_{1}K(G)C_{\beta\eta}}\,)\,\Big{)}B_{\beta\eta}\
.$
Let
$\tilde{K}(G)=\left(\sum_{\alpha=1,2}\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int_{\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2}}\frac{|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})|^{2}}{|p_{2}|^{2}}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\right)^{\frac{1}{2}}\
.$
Let $\delta\in{\mathbb{R}}$ be such that
$0<\delta<m_{1}\ .$
We set
(3.9)
$\tilde{D}=\,\sup(\frac{4\Lambda\gamma}{2m_{1}-\delta},\,1)\,\tilde{K}(G)\,(\,2m_{1}\,\tilde{C}_{\beta\eta}+\tilde{B}_{\beta\eta}\,)\
,$
where $\Lambda>m_{1}$ has been introduced in Hypothesis 3.1(iv).
Let us define the sequence $(\sigma_{n})_{n\geq 0}$ by
$\begin{split}&\sigma_{0}=\Lambda\ ,\\\ &\sigma_{1}=m_{1}-\frac{\delta}{2}\
,\\\ &\sigma_{2}=m_{1}-\delta=\gamma\sigma_{1}\ ,\\\
&\sigma_{n+1}=\gamma\sigma_{n},\ n\geq 1\ ,\end{split}$
where $\gamma=1-\delta/(2m_{1}-\delta)$.
Let $g_{\delta}^{(1)}$ be such that
$0<g^{(1)}_{\delta}<\inf(1,g_{1},\frac{\gamma-\gamma^{2}}{3\tilde{D}})\ .$
For $0<g\leq g_{\delta}^{(1)}$ we have
$0<\gamma<(1-\frac{3g\tilde{D}}{\gamma})\ ,$
and
(3.10) $0<\sigma_{n+1}<(1-\frac{3g\tilde{D}}{\gamma})\sigma_{n},\quad n\geq 1\
.$
Set
$\begin{split}&H^{n}=H^{\sigma_{n}};\quad H_{0}^{n}=H_{0}^{\sigma_{n}},\quad
n\geq 0\,\\\ &E^{n}=\inf\sigma(H^{n})\,,\quad n\geq 0\ .\end{split}$
We then get
###### Proposition 3.5.
Suppose that the kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$
satisfy Hypothesis 2.1, Hypothesis 3.1(i) and 3.1(iv). Then there exists
$0<\tilde{g}_{\delta}\leq g^{(1)}_{\delta}$ such that, for
$g\leq\tilde{g}_{\delta}$ and $n\geq 1$, $E^{n}$ is a simple eigenvalue of
$H^{n}$ and $H^{n}$ does not have spectrum in
$(\,E^{n},\,E^{n}+(1-\frac{3g\tilde{D}}{\gamma})\sigma_{n}\,)$.
The proof of Proposition 3.5 is given in Appendix A.
We now introduce the positive commutator estimates and the regularity property
of $H$ with respect to $A$ in order to prove Theorem 3.4
The operator $A$ has to be split into two pieces depending on $\sigma$.
Let
$\displaystyle\eta_{\sigma}(p_{2})=\chi_{2\sigma}(p_{2})\ ,$
$\displaystyle\eta^{\sigma}(p_{2})=\chi^{2\sigma}(p_{2})\ ,$ $\displaystyle
a_{\sigma}=\eta_{\sigma}(p_{2})\,a\,\eta_{\sigma}(p_{2})\ ,$ $\displaystyle
a^{\sigma}=\eta^{\sigma}(p_{2})\,a\,\eta^{\sigma}(p_{2})\ .$
Since $\eta_{\sigma}^{2}+(\eta^{\sigma})^{2}=1$, and
$[\eta_{\sigma},\,[\eta_{\sigma},\,a]\,]=0=[\eta^{\sigma},\,[\eta^{\sigma},\,a]\,]$,
we obtain (see [13])
$a=a^{\sigma}+a_{\sigma}\ .$
Note that we also have
$\displaystyle a_{\sigma}=\frac{1}{2}\left(\eta_{\sigma}(p_{2})^{2}p_{2}\cdot
i\nabla p_{2}+i\nabla p_{2}.\eta_{\sigma}(p_{2})^{2}p_{2}\right)\ ,$
$\displaystyle a^{\sigma}=\frac{1}{2}\left(\eta^{\sigma}(p_{2})^{2}p_{2}\cdot
i\nabla p_{2}+i\nabla p_{2}.\eta^{\sigma}(p_{2})^{2}p_{2}\right)\ .$
The operators $a$, $a_{\sigma}$ and $a^{\sigma}$ are essentially self-adjoint
on $C_{0}^{\infty}({\mathbb{R}}^{3},\,{\mathbb{C}}^{2})$ (see [3, Proposition
4.2.3]). We still denote by $a$, $a_{\sigma}$ and $a^{\sigma}$ their closures.
If $\tilde{a}$ denotes any of the operator $a$, $a_{\sigma}$ and $a^{\sigma}$,
we have
${\mathcal{D}}(\tilde{a})=\\{\ u\in L^{2}(\Sigma_{1});\ \tilde{a}u\in
L^{2}(\Sigma_{1})\ \\}\ .$
The operators $\mathrm{d}\Gamma(a)$, $\mathrm{d}\Gamma(a^{\sigma})$,
$\mathrm{d}\Gamma(a_{\sigma})$ are self-adjoint operators in
${\mathfrak{F}}_{a}(L^{2}(\Sigma_{1}))$ and we have
$\mathrm{d}\Gamma(a)=\mathrm{d}\Gamma(a^{\sigma})+\mathrm{d}\Gamma(a_{\sigma})\
.$
By (2.4), the following operators in ${\mathfrak{F}}_{\ell}$, denoted by
$A_{\ell}^{\,\sigma}$ and $A_{\sigma\ell}$ respectively,
$A_{\ell}^{\,\sigma}={\mathbf{1}}\otimes{\mathbf{1}}\otimes\mathrm{d}\Gamma(a^{\sigma})\otimes{\mathbf{1}}+{\mathbf{1}}\otimes{\mathbf{1}}\otimes{\mathbf{1}}\otimes\mathrm{d}\Gamma(a^{\sigma})\
,$
$A_{\sigma\ell}={\mathbf{1}}\otimes{\mathbf{1}}\otimes\mathrm{d}\Gamma(a_{\sigma})\otimes{\mathbf{1}}+{\mathbf{1}}\otimes{\mathbf{1}}\otimes{\mathbf{1}}\otimes\mathrm{d}\Gamma(a_{\sigma})\
,$
are essentially self-adjoint on ${\mathfrak{D}}_{\ell}$.
Let $A^{\sigma}$ and $A_{\sigma}$ be the following two operators in
${\mathfrak{F}}_{L}$,
$A^{\sigma}=A_{1}^{\,\sigma}\otimes{\mathbf{1}}_{2}\otimes{\mathbf{1}}_{3}+{\mathbf{1}}_{1}\otimes
A_{2}^{\,\sigma}\otimes{\mathbf{1}}_{3}+{\mathbf{1}}_{1}\otimes{\mathbf{1}}_{2}\otimes
A_{3}^{\,\sigma}\ ,$ $A_{\sigma}=A_{\sigma
1}\otimes{\mathbf{1}}_{2}\otimes{\mathbf{1}}_{3}+{\mathbf{1}}_{1}\otimes
A_{\sigma
2}\otimes{\mathbf{1}}_{3}+{\mathbf{1}}_{1}\otimes{\mathbf{1}}_{2}\otimes
A_{\sigma 3}.$
The operators $A^{\sigma}$ and $A_{\sigma}$ are essentially self-adjoint on
${\mathfrak{D}}_{L}$. Still denoting by $A^{\sigma}$ and $A_{\sigma}$ their
extensions to ${\mathfrak{F}}$, $A^{\sigma}$ and $A_{\sigma}$ are essentially
self-adjoint on ${\mathfrak{D}}$ and we still denote by $A^{\sigma}$ and
$A_{\sigma}$ their closures.
We have
$A=A^{\sigma}+A_{\sigma}\ .$
The operators $a$, $a^{\sigma}$ and $a_{\sigma}$ are associated to the
following $C^{\infty}$-vector fields in ${\mathbb{R}}^{3}$ respectively,
(3.11) $\begin{split}&v(p_{2})=p_{2}\ ,\\\
&v^{\sigma}(p_{2})=\eta^{\sigma}(p_{2})^{2}p_{2}\ ,\\\
&v_{\sigma}(p_{2})=\eta_{\sigma}(p_{2})^{2}p_{2}\ .\end{split}$
Let ${\mathcal{V}}(p)$ be any of these vector fields. We have
$|{\mathcal{V}}(p)|\leq\Gamma\,|p|\ ,$
for some $\Gamma>0$ and we also have
(3.12) ${\mathcal{V}}(p)=\tilde{v}(|p|)p\ ,$
where the $\tilde{v}$’s are defined by (3.11) and (3.12), and fulfill
$|p|^{\alpha}\frac{\mathrm{d}^{\alpha}}{\mathrm{d}|p|^{\alpha}}\tilde{v}(|p|)$
bounded for $\alpha=0,1,2$.
Let $\psi_{t}(.):\,{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}^{3}$ be the
corresponding flow generated by ${\mathcal{V}}$:
$\begin{split}&\frac{\mathrm{d}}{\mathrm{d}t}\psi_{t}(p)={\mathcal{V}}(\psi_{t}(p))\
,\\\ &\psi_{0}(p)=p\ .\end{split}$
$\psi_{t}(p)$ is a $C^{\infty}$-flow and we have
(3.13)
$\mathrm{e}^{-\Gamma|t|}\,|p|\leq|\psi_{t}(p)|\leq\mathrm{e}^{\Gamma|t|}\,|p|\
.$
$\psi_{t}(p)$ induces a one-parameter group of unitary operators $U(t)$ in
$L^{2}(\Sigma_{1})\simeq L^{2}({\mathbb{R}}^{3},\,{\mathbb{C}}^{2})$ defined
by
$(U(t)f)(p)=f(\psi_{t}(p))(\det\nabla\psi_{t}(p))^{\frac{1}{2}}$
Let $\phi_{t}(.)$, $\phi^{\,\sigma}_{t}(.)$ and $\phi_{\sigma t}(.)$ be the
flows associated with the vector fields $v(.)$, $v^{\sigma}(.)$ and
$v_{\sigma}(.)$ respectively.
Let $U(t)$, $U^{\sigma}(t)$ and $U_{\sigma}(t)$ be the corresponding one-
parameter groups of unitary operators in $L^{2}(\Sigma_{1})$. The operators
$a$, $a^{\sigma}$, and $a_{\sigma}$ are the generators of $U(t)$,
$U^{\sigma}(t)$ and $U_{\sigma}(t)$ respectively, i.e.,
$\begin{split}&U(t)=\mathrm{e}^{-iat}\ ,\\\
&U^{\sigma}(t)=\mathrm{e}^{-ia^{\sigma}t}\ ,\\\
&U_{\sigma}(t)=\mathrm{e}^{-ia_{\sigma}t}\ .\end{split}$
Let
$w^{(2)}(\xi_{2})=(w^{(2)}_{\ell}(\xi_{2}))_{\ell=1,2,3}$
and
$\mathrm{d}\Gamma(w^{(2)})=\sum_{\ell=1}^{3}\sum_{\epsilon}\int
w^{(2)}_{\ell}(\xi_{2})c_{\ell,\epsilon}^{*}(\xi_{2})c_{\ell\epsilon}(\xi_{2})\mathrm{d}\xi_{2}\
.$
Let $V(t)$ be any of the one-parameter groups $U(t)$, $U^{\sigma}(t)$ and
$U_{\sigma}(t)$. We set
$V(t)w^{(2)}V(t)^{*}=(V(t)w_{\ell}^{(2)}V(t)^{*})_{\ell=1,2,3}\ ,$
and we have
$V(t)w^{(2)}V(t)^{*}=w^{(2)}(\psi_{t})\ .$
Here $\psi_{t}$ is the flow associated to $V(t)$.
This yields, for any $\varphi\in{\mathfrak{D}}$, (see [9, Lemma 2.8])
(3.14) $\begin{split}\mathrm{e}^{-iAt}H_{0}\mathrm{e}^{iAt}\varphi-
H_{0}\varphi&=(\mathrm{d}\Gamma(\mathrm{e}^{-iat}w^{(2)}\mathrm{e}^{iat})-\mathrm{d}\Gamma(w^{(2)}))\varphi\\\
&=(\mathrm{d}\Gamma(w^{(2)}\circ\phi_{t}-w^{(2)}))\varphi\ ,\end{split}$
(3.15)
$\begin{split}\mathrm{e}^{-iA^{\sigma}t}H_{0}\mathrm{e}^{iA^{\sigma}t}\varphi-
H_{0}\varphi&=(\mathrm{d}\Gamma(\mathrm{e}^{-ia^{\sigma}t}w^{(2)}\mathrm{e}^{ia^{\sigma}t})-\mathrm{d}\Gamma(w^{(2)}))\varphi\\\
&=(\mathrm{d}\Gamma(w^{(2)}\circ\phi_{t}^{\,\sigma}-w^{(2)}))\varphi\
,\end{split}$ (3.16)
$\begin{split}\mathrm{e}^{-iA_{\sigma}t}H_{0}\mathrm{e}^{iA_{\sigma}t}\varphi-
H_{0}\varphi&=(\mathrm{d}\Gamma(\mathrm{e}^{-ia_{\sigma}t}w^{(2)}\mathrm{e}^{ia_{\sigma}t})-\mathrm{d}\Gamma(w^{(2)}))\varphi\\\
&=(\mathrm{d}\Gamma(w^{(2)}\circ\phi_{\sigma t}-w^{(2)}))\varphi\
.\end{split}$
###### Proposition 3.6.
Suppose that the kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$
satisfy Hypothesis 2.1.
For every $t\in{\mathbb{R}}$ we have, for $g\leq g_{1}$,
$\begin{split}(i)&\quad\mathrm{e}^{itA}{\mathcal{D}}(H_{0})=\mathrm{e}^{itA}{\mathcal{D}}(H)\subset{\mathcal{D}}(H_{0})={\mathcal{D}}(H)\
,\\\
(ii)&\quad\mathrm{e}^{itA^{\sigma}}{\mathcal{D}}(H_{0})=\mathrm{e}^{itA^{\sigma}}{\mathcal{D}}(H)\subset{\mathcal{D}}(H_{0})={\mathcal{D}}(H)\
,\\\
(iii)&\quad\mathrm{e}^{itA_{\sigma}}{\mathcal{D}}(H_{0})=\mathrm{e}^{itA_{\sigma}}{\mathcal{D}}(H)\subset{\mathcal{D}}(H_{0})={\mathcal{D}}(H)\
.\end{split}$
###### Proof.
We only prove $i)$, since $ii)$ and $iii)$ can be proved similarly. By (3.14)
we have, for $\varphi\in{\mathfrak{D}}$,
(3.17)
$\mathrm{e}^{-itA}H_{0}\mathrm{e}^{itA}\varphi=(H_{0}^{(1)}+H_{0}^{(3)}+\mathrm{d}\Gamma(w^{(2)}\circ\phi_{t}))\varphi\
.$
It follows from (3.13) and (3.17) that
$\|H_{0}\mathrm{e}^{itA}\varphi\|\leq\mathrm{e}^{\Gamma|t|}\|H_{0}\varphi\|\
.$
This yields $i)$ because ${\mathfrak{D}}$ is a core for $H_{0}$. Moreover we
get
$\|H_{0}\mathrm{e}^{itA}(H_{0}+1)^{-1}\|\leq\mathrm{e}^{\Gamma|t|}\ .$
In view of ${\mathfrak{D}}(H_{0})={\mathfrak{D}}(H)$, the operators
$H_{0}(H+i)^{-1}$ and $H(H_{0}+i)^{-1}$ are bounded and there exists a
constant $C>0$ such that
$\|H\mathrm{e}^{itA}(H+i)^{-1}\|\leq C\mathrm{e}^{\Gamma|t|}\ .$
Similarly, we also get
$\begin{split}&\|H_{0}\mathrm{e}^{itA^{\sigma}}(H_{0}+1)^{-1}\|\leq\mathrm{e}^{\Gamma|t|}\
,\\\
&\|H_{0}\mathrm{e}^{itA_{\sigma}}(H_{0}+1)^{-1}\|\leq\mathrm{e}^{\Gamma|t|}\
,\\\ &\|H\mathrm{e}^{itA^{\sigma}}(H+i)^{-1}\|\leq C\mathrm{e}^{\Gamma|t|}\
,\\\ &\|H\mathrm{e}^{itA_{\sigma}}(H+i)^{-1}\|\leq C\mathrm{e}^{\Gamma|t|}\
.\end{split}$
∎
Let $H_{I}(G)$ be the interaction associated with the kernels
$G=(G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)})_{\alpha=1,2;\ \ell=1,2,3;\
\epsilon\neq\epsilon^{\prime}=\pm}$, where the kernels
$G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)})$ satisfy Hypothesis 2.1
We set
$V(t)G=(V(t)G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)})_{\alpha=1,2;\
\ell=1,2,3;\ \epsilon\neq\epsilon^{\prime}=\pm}$
We have for $\varphi\in{\mathfrak{D}}$ (see [9, Lemma 2.7]),
(3.18)
$\begin{split}&\mathrm{e}^{-iAt}H_{I}(G)\mathrm{e}^{iAt}\varphi=H_{I}(\mathrm{e}^{-iat}G)\varphi\
,\\\
&\mathrm{e}^{-iA^{\sigma}t}H_{I}(G)\mathrm{e}^{iA^{\sigma}t}\varphi=H_{I}(\mathrm{e}^{-ia^{\sigma}t}G)\varphi\
,\\\
&\mathrm{e}^{-iA_{\sigma}t}H_{I}(G)\mathrm{e}^{iA_{\sigma}t}\varphi=H_{I}(\mathrm{e}^{-ia_{\sigma}t}G)\varphi\
.\end{split}$
According to [3] and [25], in order to prove Theorem 3.4 we must prove that
$H$ is locally of class $C^{2}(A^{\sigma})$, $C^{2}(A_{\sigma})$ and
$C^{2}(A)$ in $(-\infty,m_{1}-\frac{\delta}{2})$ and that $A$ and $A_{\sigma}$
are locally strictly conjugate to $H$ in $(E,m_{1}-\frac{\delta}{2})$.
Recall that $H$ is locally of class $C^{2}(A)$ in
$(-\infty,m_{1}-\frac{\delta}{2})$ if, for any $\varphi\in
C_{0}^{\infty}((-\infty,m_{1}-\frac{\delta}{2}))$, $\varphi(H)$ is of class
$C^{2}(A)$, i.e.,
$t\rightarrow\mathrm{e}^{-iAt}\varphi(H)\mathrm{e}^{itA}\psi$ is twice
continuously differentiable for all $\varphi\in
C_{0}^{\infty}((-\infty,m_{1}-\frac{\delta}{2})$ and all
$\psi\in{\mathfrak{F}}$.
Thus, one of our main results is the following one
###### Theorem 3.7.
Suppose that the kernels $G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)}$
satisfy Hypothesis 2.1 and 3.1(i)-(iii).
* (a)
$H$ is locally of class $C^{2}(A)$, $C^{2}(A^{\sigma})$ and
$C^{2}(A_{\sigma})$ in $(-\infty,m_{1}-{\delta}/{2})$.
* (b)
$H^{\sigma}$ is locally of class $C^{2}(A^{\sigma})$ in
$(-\infty,m_{1}-{\delta}/{2})$.
It follows from Theorem 3.7 that $[H,\,iA]$, $[H,\,iA_{\sigma}]$,
$[H,\,iA^{\sigma}]$ and $[H^{\sigma},\,iA^{\sigma}]$ are defined as
sesquilinear forms on $\cup_{K}E_{K}(H){\mathfrak{F}}$, where the union is
taken over all the compact subsets $K$ of $(-\infty,m_{1}-\delta/2)$.
Furthermore, by Proposition 3.6, Theorem 3.7 and [13, Lemma 29], we get for
all $\varphi\in C_{0}^{\infty}((E,m_{1}-\delta/2))$ and all
$\psi\in{\mathfrak{F}}$,
(3.19)
$\begin{split}&\varphi(H)\,[H,\,iA]\,\varphi(H)\,\psi=\lim_{t\rightarrow
0}\varphi(H)\,\big{[}H,\,\frac{\mathrm{e}^{itA}-1}{t}\big{]}\,\varphi(H)\,\psi\
,\\\ &\varphi(H)\,[H,\,iA_{\sigma}]\,\varphi(H)\,\psi=\lim_{t\rightarrow
0}\varphi(H)\,\big{[}H,\,\frac{\mathrm{e}^{itA_{\sigma}}-1}{t}\big{]}\,\varphi(H)\,\psi\
,\\\ &\varphi(H)\,[H,\,iA^{\sigma}]\,\varphi(H)\,\psi=\lim_{t\rightarrow
0}\varphi(H)\,\big{[}H,\,\frac{\mathrm{e}^{itA^{\sigma}}-1}{t}\big{]}\,\varphi(H)\,\psi\
,\\\
&\varphi(H^{\sigma})\,[H^{\sigma},\,iA^{\sigma}]\,\varphi(H^{\sigma})\,\psi=\lim_{t\rightarrow
0}\varphi(H^{\sigma})\,\big{[}H^{\sigma},\,\frac{\mathrm{e}^{itA^{\sigma}}-1}{t}\big{]}\,\varphi(H^{\sigma})\,\psi\
.\\\ \end{split}$
The following proposition allows us to compute $[H,\,iA]$,
$[H,\,iA^{\sigma}]$, $[H,\,iA_{\sigma}]$ and $[H^{\sigma},\,iA^{\sigma}]$ as
sesquilinear forms. By Hypothesis 2.1 and 3.1 (iii.a), the kernels
$G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)}(\xi_{1},.,\xi_{3})$ belong to
the domains of $a$, $a^{\sigma}$, and $a_{\sigma}$.
###### Proposition 3.8.
Suppose that the kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$
satisfy Hypothesis 2.1 and 3.1 (iii.a). Then
* (a)
For all $\psi\in{\mathcal{D}}(H)$ we have
* $(i)$
$\lim_{t\rightarrow
0}\big{[}H,\frac{\mathrm{e}^{itA}-1}{t}\big{]}\psi=\big{(}\,\mathrm{d}\Gamma(w^{(2)})+gH_{I}(-iaG)\,\big{)}\psi$,
* $(ii)$
$\lim_{t\rightarrow
0}\big{[}H,\frac{\mathrm{e}^{itA^{\sigma}}-1}{t}\big{]}\psi=\big{(}\,\mathrm{d}\Gamma((\eta^{\sigma})^{2}w^{(2)})+gH_{I}(-ia^{\sigma}G)\,\big{)}\psi$,
* $(iii)$
$\lim_{t\rightarrow
0}\big{[}H,\frac{\mathrm{e}^{itA_{\sigma}}-1}{t}\big{]}\psi=\big{(}\,\mathrm{d}\Gamma((\eta_{\sigma})^{2}w^{(2)})+gH_{I}(-ia_{\sigma}G)\,\big{)}\psi$,
* $(iv)$
$\lim_{t\rightarrow
0}\big{[}H^{\sigma},\frac{\mathrm{e}^{itA^{\sigma}}-1}{t}\big{]}\psi=\big{(}\,\mathrm{d}\Gamma((\eta^{\sigma})^{2}w^{(2)})+gH_{I}(-ia^{\sigma}(\tilde{\chi}^{\sigma}(p_{2})G))\,\big{)}\psi$.
* (b)
* $(i)$
$\sup_{0<|t|\leq
1}\big{\|}\big{[}H,\frac{\mathrm{e}^{itA}-1}{t}\big{]}(H+i)^{-1}\big{\|}<\infty$,
* $(ii)$
$\sup_{0<|t|\leq
1}\big{\|}\big{[}H,\frac{\mathrm{e}^{itA^{\sigma}}-1}{t}\big{]}(H+i)^{-1}\big{\|}<\infty$,
* $(iii)$
$\sup_{0<|t|\leq
1}\big{\|}\big{[}H,\frac{\mathrm{e}^{itA_{\sigma}}-1}{t}\big{]}(H+i)^{-1}\big{\|}<\infty$,
* $(iv)$
$\sup_{0<|t|\leq
1}\big{\|}\big{[}H^{\sigma},\frac{\mathrm{e}^{itA^{\sigma}}-1}{t}\big{]}(H+i)^{-1}\big{\|}<\infty$.
###### Proof.
Part $\mathrm{(b)}$ follows from part $\mathrm{(a)}$ by the uniform
boundedness principle. For part $\mathrm{(a)}$, we only prove
$\mathrm{(a)}$(i), since other statements can be proved similarly.
By (3.13), we obtain
$\frac{1}{|t|}\big{|}w_{\ell}^{(2)}(\phi_{t}(p_{2}))-w_{\ell}^{(2)}(p_{2})\big{|}\leq\frac{1}{|t|}\big{(}\mathrm{e}^{\Gamma\,|t|}-1\big{)}w_{\ell}^{(2)}(p_{2})\
,$
for $\ell=1,2,3$.
By (3.14)-(3.16) and the Lebesgue’s Theorem we then get for all
$\psi\in{\mathcal{D}}(H_{0})$
$\begin{split}&\lim_{t\rightarrow
0}\big{[}H_{0},\frac{\mathrm{e}^{itA}-1}{t}\big{]}\psi=\lim_{t\rightarrow
0}\frac{1}{t}\big{[}\mathrm{e}^{-itA}H_{0}\mathrm{e}^{itA}-H_{0}\big{]}\psi=\mathrm{d}\Gamma(w^{(2)})\psi\
,\\\ &\lim_{t\rightarrow
0}\big{[}H_{0},\frac{\mathrm{e}^{itA^{\sigma}}-1}{t}\big{]}\psi=\lim_{t\rightarrow
0}\frac{1}{t}\big{[}\mathrm{e}^{-itA^{\sigma}}H_{0}\mathrm{e}^{itA^{\sigma}}-H_{0}\big{]}\psi=\mathrm{d}\Gamma((\eta^{\sigma})^{2}w^{(2)})\psi\
,\\\ &\lim_{t\rightarrow
0}\big{[}H_{0},\frac{\mathrm{e}^{itA_{\sigma}}-1}{t}\big{]}\psi=\lim_{t\rightarrow
0}\frac{1}{t}\big{[}\mathrm{e}^{-itA_{\sigma}}H_{0}\mathrm{e}^{itA_{\sigma}}-H_{0}\big{]}\psi=\mathrm{d}\Gamma((\eta_{\sigma})^{2}w^{(2)})\psi\
.\end{split}$
By (3.18), we obtain for all $\psi\in{\mathcal{D}}(H)$,
$\begin{split}&\lim_{t\rightarrow
0}\big{[}H_{I}(G),\frac{\mathrm{e}^{itA}\\!-1\\!}{t}\big{]}\psi=\lim_{t\rightarrow
0}\frac{1}{t}\big{[}\mathrm{e}^{-itA}H_{I}(G)\mathrm{e}^{itA}-H_{I}(G)\big{]}\psi=H_{I}(-i(a\,G))\psi,\\\
&\lim_{t\rightarrow
0}\big{[}H_{I}(G),\frac{\mathrm{e}^{itA^{\sigma}}\\!-\\!1}{t}\big{]}\psi=\lim_{t\rightarrow
0}\frac{1}{t}\big{[}\mathrm{e}^{-itA^{\sigma}}H_{I}(G)\mathrm{e}^{itA^{\sigma}}-H_{I}(G)\big{]}\psi=H_{I}(-i(a^{\sigma}G))\psi,\\\
&\lim_{t\rightarrow
0}\big{[}H_{I}(G),\frac{\mathrm{e}^{itA_{\sigma}}\\!-\\!1}{t}\big{]}\psi=\lim_{t\rightarrow
0}\frac{1}{t}\big{[}\mathrm{e}^{-itA_{\sigma}}H_{I}(G)\mathrm{e}^{itA_{\sigma}}-H_{I}(G)\big{]}\psi=H_{I}(-i(a_{\sigma}G))\psi,\\\
&\lim_{t\rightarrow
0}\big{[}H_{I}(\tilde{\chi}^{\sigma}(p_{2})G),\frac{\mathrm{e}^{itA^{\sigma}}\\!-\\!1}{t}\big{]}\psi\\\
&=\lim_{t\rightarrow
0}\frac{1}{t}\big{[}\mathrm{e}^{-itA^{\sigma}}H_{I}(\tilde{\chi}^{\sigma}(p_{2})G)\mathrm{e}^{itA^{\sigma}}-H_{I}(\tilde{\chi}^{\sigma}(p_{2})G)\big{]}\psi=H_{I}(-i(a^{\sigma}(\tilde{\chi}^{\sigma}(p_{2})G)))\psi\
.\end{split}$
This concludes the proof of Proposition 3.8. ∎
Combining (3.19) with Proposition 3.8, we finally get for every $\varphi\in
C_{0}^{\infty}((-\infty,m_{1}-\delta/2))$ and every $\psi\in{\mathfrak{F}}$
(3.20)
$\varphi(H)\big{[}H,\,iA\big{]}\varphi(H)\psi=\varphi(H)\big{[}\mathrm{d}\Gamma(w^{(2)})+gH_{I}(-i(a\,G))\big{]}\varphi(H)\psi\
,$ (3.21)
$\varphi(H)\big{[}H,\,iA^{\sigma}\big{]}\varphi(H)\psi=\varphi(H)\big{[}\mathrm{d}\Gamma((\eta^{\sigma})^{2}w^{(2)})+gH_{I}(-i(a^{\sigma}G))\big{]}\varphi(H)\psi\
,$ (3.22)
$\varphi(H)\big{[}H,\,iA_{\sigma}\big{]}\varphi(H)\psi=\varphi(H)\big{[}\mathrm{d}\Gamma((\eta_{\sigma})^{2}w^{(2)})+gH_{I}(-i(a_{\sigma}G))\big{]}\varphi(H)\psi\
,$
and
(3.23)
$\varphi(H^{\sigma})\big{[}H^{\sigma},\,iA^{\sigma}\big{]}\varphi(H^{\sigma})\psi=\varphi(H^{\sigma})\big{[}\mathrm{d}\Gamma((\eta^{\sigma})^{2}w^{(2)})+gH_{I}(-i(a^{\sigma}(\tilde{\chi}^{\sigma}G)))\big{]}\varphi(H^{\sigma})\psi\
.$
We now introduce the Mourre inequality.
Let $N$ be the smallest integer such that
$N\gamma\geq 1.$
We have, for $g\leq g^{(1)}_{\delta}$,
(3.24)
$\begin{split}&\gamma<\gamma+\frac{1}{N}(1-\frac{3g\tilde{D}}{\gamma}-\gamma)<1-\frac{3g\tilde{D}}{\gamma}\
,\\\
&\frac{\gamma}{N}\leq\gamma-\frac{1}{N}(1-\frac{3g\tilde{D}}{\gamma}-\gamma)<\gamma\
.\end{split}$
Let
$\epsilon_{\gamma}=\frac{1}{2N}(1-\frac{3g_{\delta}^{(1)}\tilde{D}}{\gamma}-\gamma)\
.$
We choose $f\in C_{0}^{\infty}({\mathbb{R}})$ such that $1\geq f\geq 0$ and
(3.25) $f(\lambda)=\left\\{\begin{array}[]{ll}1&\mbox{ if
}\lambda\in[(\gamma-\epsilon_{\gamma})^{2},\gamma+\epsilon_{\gamma}]\ ,\\\
0&\mbox{ if
}\lambda>\gamma+\frac{1}{N}(1-\frac{3g_{\delta}^{(1)}\tilde{D}}{\gamma}-\gamma)=\gamma+2\epsilon_{\gamma}\
,\\\ 0&\mbox{ if
}\lambda<(\gamma-\frac{1}{N}(1-\frac{3g_{\delta}^{(1)}\tilde{D}}{\gamma}-\gamma))^{2}=(\gamma-2\epsilon_{\gamma})^{2}\
.\end{array}\right.$
Note that $\gamma+2\epsilon_{\gamma}<1-3g\tilde{D}/\gamma$ for $g\leq
g_{\delta}^{(1)}$ and $\gamma-\epsilon_{\gamma}>\gamma/N$.
We set, for $n\geq 1$,
$f_{n}(\lambda)=f\left(\frac{\lambda}{\sigma_{n}}\right)\ .$
Let
$\begin{split}&H_{n}=H_{\sigma_{n}}\ ,\\\ &E_{n}=\inf\sigma(H_{n})\ ,\\\
&H_{0\,n}^{(2)}=H_{0\,\sigma_{n}}^{(2)}\ .\end{split}$
Let $P^{n}$ denote the ground state projection of $H^{n}$. It follows from
proposition 3.5 that, for $n\geq 1$ and $g\leq\tilde{g}_{\delta}\leq
g_{\delta}^{(1)}$,
(3.26) $f_{n}(H_{n}-E_{n})=P^{n}\otimes f_{n}(H_{0,\,n}^{(2)})\ .$
Note that
(3.27) $E_{n}=E^{n}=\inf\sigma(H^{n})\ .$
Set
$\begin{split}&a^{n}=a^{\sigma_{n}}\ ,\\\ &a_{n}=a_{\sigma_{n}}\ ,\\\
&A^{n}=A^{\sigma_{n}}\ ,\\\ &A_{n}=A_{\sigma_{n}}\ ,\\\
&{\mathfrak{F}}^{n}={\mathfrak{F}}^{\sigma_{n}}\ ,\\\
&{\mathfrak{F}}_{n}={\mathfrak{F}}_{\sigma_{n}}\ .\end{split}$
We have
$\begin{split}&{\mathfrak{F}}\simeq{\mathfrak{F}}^{n}\otimes{\mathfrak{F}}_{n}\
,\\\ &A=A^{n}+A_{n}\ .\end{split}$
We further note that
(3.28) $a^{n}\tilde{\chi}^{\sigma_{n}}(p_{2})=a^{n}\ .$
By (3.21), (3.23) and (3.28), we obtain
$[H,iA^{n}]=[H^{n},iA^{n}]\otimes{\mathbf{1}}\ ,$
as sesquilinear forms with respect to
${\mathfrak{F}}={\mathfrak{F}}^{n}\otimes{\mathfrak{F}}_{n}$.
Furthermore, it follows from the virial Theorem (see [25, Proposition 3.2] and
Proposition 6.1) that
(3.29) $P^{n}[H^{n},iA^{n}]P^{n}=0\ .$
By (3.26) and (3.29) we then get, for $g\leq\tilde{g}_{\delta}\leq
g_{\delta}^{(1)}$,
$f_{n}(H_{n}-E_{n})[H,iA^{n}]f_{n}(H_{n}-E_{n})=0\ .$
We then have
###### Proposition 3.9.
Suppose that the kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$
satisfy Hypothesis 2.1 and 3.1. Then there exists $\tilde{C}_{\delta}>0$ and
$\tilde{g}_{\delta}^{(1)}>0$ such that
$\tilde{g}_{\delta}^{(1)}\leq\tilde{g}_{\delta}$ and
$f_{n}(H_{n}-E_{n})[H,iA_{n}]f_{n}(H_{n}-E_{n})\geq\tilde{C}_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}f_{n}(H_{n}-E_{n})^{2}$
for $n\geq 1$ and $g\leq\tilde{g}_{\delta}^{(1)}$.
Let $E_{\Delta}(H-E)$ be the spectral projection for the operator $H-E$
associated with the interval $\Delta$, and let
(3.30)
$\Delta_{n}=[(\gamma-\epsilon_{\gamma})^{2}\sigma_{n},\,(\gamma+\epsilon_{\gamma})\sigma_{n}],\
n\geq 1\ .$
Note that
(3.31)
$[\sigma_{n+2},\sigma_{n+1}]\subset\left((\gamma-\epsilon_{\gamma})^{2}\sigma_{n},\,(\gamma+\epsilon_{\gamma})\sigma_{n}\right),\
n\geq 1\ .$
###### Theorem 3.10.
Suppose that the kernels $G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)}$
satisfy Hypothesis 2.1 and 3.1. Then there exists $C_{\delta}>0$ and
$\tilde{g}_{\delta}^{(2)}>0$ such that
$\tilde{g}_{\delta}^{(2)}\leq\tilde{g}_{\delta}^{(1)}$ and
$E_{\Delta_{n}}(H-E)[H,\,iA]E_{\Delta_{n}}(H-E)\geq
C_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}E_{\Delta_{n}}(H-E)\ ,$
for $n\geq 1$ and $g\leq\tilde{g}_{\delta}^{(2)}$.
## 4\. Existence of a ground state and location of the absolutely continuous
spectrum
We now prove Theorem 3.3. The scheme of the proof is quite well known (see
[5], [20]). It follows from Proposition 3.5 that $H^{n}$ has an unique ground
state, denoted by $\phi^{n}$, in ${\mathfrak{F}}^{n}$,
$H^{n}\phi^{n}=E^{n}\phi^{n},\quad\phi^{n}\in{\mathcal{D}}(H^{n}),\quad\|\phi^{n}\|=1,\quad
n\geq 1\ .$
Therefore $H_{n}$ has an unique normalized ground state in ${\mathfrak{F}}$,
given by $\tilde{\phi}_{n}=\phi^{n}\otimes\Omega_{n}$, where $\Omega_{n}$ is
the vacuum state in ${\mathfrak{F}}_{n}$,
$H_{n}\tilde{\phi}_{n}=E^{n}\tilde{\phi}_{n},\quad\tilde{\phi}_{n}\in{\mathcal{D}}(H_{n}),\quad\|\tilde{\phi}_{n}\|=1,\quad
n\geq 1\ .$
Since $\|\tilde{\phi}_{n}\|=1$, there exists a subsequence $(n_{k})_{k\geq
1}$, converging to $\infty$ such that $(\tilde{\phi}_{n_{k}})_{k\geq 1}$
converges weakly to a state $\tilde{\phi}\in{\mathfrak{F}}$. We have to prove
that $\tilde{\phi}\neq 0$. By adapting the proof of Theorem 4.1 in [2] (see
also [7]), the key point is to estimate
$\|c_{\ell,\epsilon}(\xi_{2})\tilde{\Phi}_{n}\|_{{\mathfrak{F}}}$ in order to
show that
(4.1)
$\sum_{\ell=1}^{3}\sum_{\epsilon}\int\|c_{\ell,\epsilon}(\xi_{2})\tilde{\phi}_{n}\|^{2}\mathrm{d}\xi_{2}=\mathcal{O}(g^{2})\
,$
uniformly with respect to $n$.
The estimate (4.1) is a consequence of the so-called “pull-through” formula as
it follows.
Let $H_{I_{,}n}$ denote the interaction $H_{I}$ associated with the kernels
${\mathbf{1}}_{\\{|p_{2}|\geq\sigma_{n}\\}}(p_{2})G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$.
We thus have
$\begin{split}&H_{0}c_{\ell,\epsilon}(\xi_{2})\tilde{\phi}_{n}=c_{\ell,\epsilon}(\xi_{2})H_{0}\tilde{\phi}_{n}-w_{\ell}^{(2)}(\xi_{2})c_{\ell,\epsilon}(\xi_{2})\tilde{\phi}_{n}\\\
&gH_{I,n}c_{\ell,\epsilon}(\xi_{2})\tilde{\phi}_{n}=c_{\ell,\epsilon}(\xi_{2})gH_{I,n}\tilde{\phi}_{n}+gV_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{2})\tilde{\phi}_{n}\
,\end{split}$
with
$\begin{split}V_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{2})=&g\int
G^{(1)}_{\ell,\epsilon^{\prime}\epsilon}(\xi_{2},\xi_{2},\xi_{3})b_{\ell,\epsilon^{\prime}}^{*}(\xi_{1})a_{\epsilon}(\xi_{3})\mathrm{d}\xi_{1}\,\mathrm{d}\xi_{3}\\\
&+g\int
G^{(2)}_{\ell,\epsilon^{\prime}\epsilon}(\xi_{2},\xi_{2},\xi_{3})b_{\ell,\epsilon^{\prime}}^{*}(\xi_{1})a_{\epsilon}^{*}(\xi_{3})\mathrm{d}\xi_{1}\,\mathrm{d}\xi_{3}\
.\end{split}$
This yields
(4.2)
$\left(H_{n}-E_{n}+w_{\ell}^{(2)}(\xi_{2})\right)c_{\ell,\epsilon}(\xi_{2})\tilde{\phi}_{n}=V_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{2})\tilde{\phi}_{n}\
.$
By adapting the proof of Propositions 2.4 and 2.5 we easily get
(4.3)
$\begin{split}\|V_{\ell,\epsilon,\epsilon^{\prime}}\psi\|_{{\mathfrak{F}}}&\leq\frac{g}{m_{W}{}^{\frac{1}{2}}}\left(\sum_{\alpha=1,2}\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(.,\xi_{2},.)\|_{L^{2}(\Sigma_{1}\times\Sigma_{2})}\right)\|H_{0}^{\frac{1}{2}}\psi\|\\\
&+g\,\|G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(.,\xi_{2},.)\|_{L^{2}(\Sigma_{1}\times\Sigma_{2})}\|\psi\|\
,\end{split}$
where $\psi\in{\mathcal{D}}(H_{0})$.
Let us estimate $\|H_{0}\tilde{\phi}_{n}\|$. By (2.29), (2.30), (3.3), (3.4)
and (3.6) we have
$g\|H_{I,n}\tilde{\phi}_{n}\|\leq
gK(G)(C_{\beta\eta}\|H_{0}\tilde{\phi}_{n}\|+B_{\beta\eta})$
and
$\|H_{0}\tilde{\phi}_{n}\|\leq|E_{n}|+g\|H_{I,n}\tilde{\phi}_{n}\|\ .$
Therefore
(4.4)
$\|H_{0}\tilde{\phi}_{n}\|\leq\frac{|E_{n}|}{1-g_{1}K(G)C_{\beta\eta}}+\frac{gK(G)B_{\beta\eta}}{1-g_{1}K(G)C_{\beta\eta}}\
.$
By (3.27), (A.3) and (4.4), there exists $C>0$ such that
(4.5) $\|H_{0}\tilde{\phi}_{n}\|\leq C\ ,$
uniformly in $n$ and $g\leq g_{1}$.
By (4.2), (4.3) and (4.5) we get
$\|c_{\ell,\epsilon}\tilde{\phi}_{n}\|\leq\frac{g}{|p_{2}|}\left(C^{\frac{1}{2}}\left(\sum_{\alpha=1}^{2}\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(.,\xi_{2},.)\|_{L^{2}(\Sigma_{1}\times\Sigma_{2})}\right)+\|G^{(2)}_{\ell,\epsilon,\epsilon^{\prime}}(.,\xi_{2},.)\|_{L^{2}(\Sigma_{1}\times\Sigma_{2})}\right)$
By Hypothesis 3.1(i), there exists a constant $C(G)>0$ depending on the
kernels
$G=(G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})_{\ell=1,2,3;\alpha=1,2;\epsilon\neq\epsilon^{\prime}=\pm}$
and such that
$\sum_{\ell=1}^{3}\sum_{\epsilon}\int\|c_{\ell,\epsilon}(\xi_{2})\tilde{\phi}_{n}\|^{2}\mathrm{d}\xi_{2}\leq
C(G)^{2}g^{2}\ .$
The existence of a ground state $\tilde{\phi}$ for $H$ follows by choosing $g$
sufficiently small, i.e. $g\leq g_{2}$, as in [2] and [7]. By adapting the
method developed in [19] (see [19, Corollary 3.4]), one proves that the ground
state of $H$ is unique. We omit here the details.
Statements about $\sigma(H)$ are consequences of the existence of a ground
state and follows from the existence of asymptotic Fock representations for
the CAR associated with the $c_{\ell,\epsilon}^{\sharp}(\xi_{2})$’s. For $f\in
L^{2}({\mathbb{R}}^{3},\,{\mathbb{C}}^{2})$, we define on
${\mathcal{D}}(H_{0})$ the operators
$c_{\ell,\epsilon}^{\sharp\,t}(f)=\mathrm{e}^{itH}\mathrm{e}^{-itH_{0}}c_{\ell,\epsilon}^{\sharp}(f)\mathrm{e}^{itH_{0}}\mathrm{e}^{{}_{i}tH}\
.$
By mimicking the proof given in [20, 28] one proves, under the hypothesis of
Theorem 3.3 and for $f\in C_{0}^{\infty}({\mathbb{R}}^{3}\,{\mathbb{C}}^{2})$,
that the strong limits of $c_{\ell,\epsilon}^{\sharp\,t}(f)$ when
$t\rightarrow\pm\infty$ exist for $\psi\in{\mathcal{D}}(H_{0})$,
(4.6)
$\lim_{t\rightarrow\pm\infty}c_{\ell,\epsilon}^{\sharp\,t}(f)\psi:=c_{\ell,\epsilon}^{\sharp\,\pm}(f)\psi\
.$
The operators $c_{\ell,\epsilon}^{\sharp\,\pm}(f)$ satisfy the CAR and we have
(4.7) $c_{\ell,\epsilon}^{\,\pm}(f)\tilde{\phi}=0,\quad f\in
C_{0}^{\infty}({\mathbb{R}}^{3}\,{\mathbb{C}}^{2})\,,$
where $\tilde{\phi}$ is the ground state of $H$.
It then follows from (4.6) and (4.7) that the absolutely continuous spectrum
of $H$ equals to $[\inf\sigma(H),\,\infty)$. We omit the details (see [20,
28]).
## 5\. Proof of the Mourre Inequality
We first prove Proposition 3.9. In view of Proposition 3.8(a) (iii) and
(3.22), we have, as sesquilinear forms,
(5.1)
$[H,\,iA_{\sigma}]=(1-g)\mathrm{d}\Gamma((\eta_{\sigma})^{2}w^{(2)})+g(\mathrm{d}\Gamma((\eta_{\sigma})^{2}w^{(2)})+gH_{I}(-i(a_{\sigma}G))\
.$
Let ${\mathfrak{F}}_{\ell}^{(1)}$ (respectively ${\mathfrak{F}}_{\ell}^{(2)}$)
be the Fock space for the massive leptons $\ell$ (respectively the neutrinos
and antineutrinos $\ell$).
We have
${\mathfrak{F}}_{\ell}\simeq{\mathfrak{F}}_{\ell}^{(1)}\otimes{\mathfrak{F}}_{\ell}^{(2)}\
.$
Let
${\mathfrak{F}}^{(1)}={\mathfrak{F}}_{W}\otimes(\otimes_{\ell=1}^{3}\,{\mathfrak{F}}_{\ell}^{(1)})\quad\mbox{and}\quad{\mathfrak{F}}^{(2)}=\otimes_{\ell=1}^{3}{\mathfrak{F}}_{\ell}^{(2)}\
.$
We have
(5.2) ${\mathfrak{F}}\simeq{\mathfrak{F}}^{(1)}\otimes{\mathfrak{F}}^{(2)}\ ,$
${\mathfrak{F}}^{(1)}$ is the Fock space for the massive leptons and the
bosons $W^{\pm}$, and ${\mathfrak{F}}^{(2)}$ is the Fock space for the
neutrinos and antineutrinos.
We have, as sesquilinear forms and with respect to (5.2),
(5.3)
$\begin{split}&\mathrm{d}\Gamma((\eta_{\sigma})^{2}(p_{2})w_{\ell}^{(2)})+H_{I}(-i(a_{\sigma}G))\\\
&=\sum_{\ell=1}^{3}\sum_{\epsilon}\int\eta_{\sigma}(p_{2})^{2}|p_{2}|c^{*}_{\ell,\epsilon}(\xi_{2})c_{\ell,\epsilon}(\xi_{2})\mathrm{d}\xi_{2}\\\
&+\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int|p_{2}|\left({\mathbf{1}}_{1}\otimes\eta_{\sigma}(p_{2})c^{*}_{\ell,\epsilon}(\xi_{2})+\sum_{\alpha=1,2}\frac{\mathcal{M}^{(\alpha)\,*}_{\ell,\epsilon,\epsilon^{\prime},\sigma}(\xi_{2})}{|p_{2}|}\otimes{\mathbf{1}}_{2}\right)\\\
&\left({\mathbf{1}}_{1}\otimes\eta_{\sigma}(p_{2})c_{\ell,\epsilon}(\xi_{2})+\sum_{\alpha=1,2}\frac{\mathcal{M}^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime},\sigma}(\xi_{2})}{|p_{2}|}\otimes{\mathbf{1}}_{2}\right)\mathrm{d}\xi_{2}\\\
&-\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\int\left(\sum_{\alpha=1,2}\frac{\mathcal{M}^{(\alpha)\,*}_{\ell,\epsilon,\epsilon^{\prime},\sigma}(\xi_{2})}{|p_{2}|^{\frac{1}{2}}}\otimes{\mathbf{1}}_{2}\right)\left(\sum_{\alpha=1,2}\frac{\mathcal{M}^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime},\sigma}(\xi_{2})}{|p_{2}|^{\frac{1}{2}}}\otimes{\mathbf{1}}_{2}\right)\mathrm{d}\xi_{2}\
,\end{split}$
where
$\mathcal{M}_{\ell,\epsilon,\epsilon^{\prime},\sigma}^{(\alpha)}(\xi_{2})=i\int\left(\sum_{\alpha=1,2}(a\,\eta_{\sigma}(p_{2})G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)}(\xi_{2},\xi_{2},\xi_{3}))\right)b^{*}_{\ell,\epsilon^{\prime}}(\xi_{1})a_{\epsilon^{\prime}}(\xi_{3})\mathrm{d}\xi_{1}\mathrm{d}\xi_{3}\
,$
and where ${\mathbf{1}}_{j}$ is the identity operator in
${\mathfrak{F}}^{(j)}$.
By mimicking the proofs of Proposition 2.4 and 2.5, we get, for every
$\psi\in{\mathfrak{D}}$,
$\begin{split}&\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\left(\psi,\,\int(\sum_{\alpha=1,2}\frac{\mathcal{M}_{\ell,\epsilon,\epsilon^{\prime},\sigma}^{(\alpha)\,*}(\xi_{2})}{|p_{2}|^{\frac{1}{2}}}\otimes{\mathbf{1}}_{2})(\sum_{\alpha=1,2}\frac{\mathcal{M}_{\ell,\epsilon,\epsilon^{\prime},\sigma}^{(\alpha)}(\xi_{2})}{|p_{2}|^{\frac{1}{2}}}\otimes{\mathbf{1}}_{2})\psi\,\mathrm{d}\xi_{2}\right)\\\
&=\sum_{\ell=1}^{3}\sum_{\epsilon\neq\epsilon^{\prime}}\left\|\int(\sum_{\alpha=1,2}\frac{\mathcal{M}_{\ell,\epsilon,\epsilon^{\prime},\sigma}^{\alpha}(\xi_{2})}{|p_{2}|^{\frac{1}{2}}}\otimes{\mathbf{1}}_{2})\psi\,\mathrm{d}\xi_{2}\right\|^{2}\\\
&\leq\left(\int\frac{|\sum_{\alpha=1,2}|(a\,\eta_{\sigma}(p_{2})G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)})(\xi_{2},\xi_{2},\xi_{3})|^{2}}{w^{(3)}(\xi_{3})|p_{2}|}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\right)\,\|(H_{0}^{(3)})^{\frac{1}{2}}\psi\|\
.\end{split}$
Noting that $|(a\,\eta_{\sigma})(p_{2})|\leq C$ uniformly with respect to
$\sigma$, it follows from hypothesis 2.1 and 3.1 that there exists a constant
$C(G)>0$ such that
$\int\frac{|\sum_{\alpha=1,2}(a\,\eta_{\sigma}(p_{2})G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})(\xi_{1},\xi_{2},\xi_{3})|^{2}}{w^{(3)}(\xi_{3})|p_{2}|}\mathrm{d}\xi_{1}\mathrm{d}\xi_{2}\mathrm{d}\xi_{3}\leq
C(G)\sigma\ .$
This yields
(5.4)
$-\int(\sum_{\alpha=1,2}\frac{\mathcal{M}^{(\alpha)\,*}_{\ell,\epsilon,\epsilon^{\prime},\sigma}(\xi_{2})}{|p_{2}|^{\frac{1}{2}}}\otimes{\mathbf{1}}_{2})(\sum_{\alpha=1,2}\frac{\mathcal{M}^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime},\sigma}(\xi_{2})}{|p_{2}|^{\frac{1}{2}}}\otimes{\mathbf{1}}_{2})\mathrm{d}\xi_{2}\geq-C(G)\sigma\
.$
Combining (5.1), (5.3) with (5.4), we obtain
(5.5)
$[H,\,iA_{n}]\geq(1-g)\mathrm{d}\Gamma((\eta_{\sigma_{n}})^{2}w_{\ell}^{(2)})-gC(G)\sigma_{n}\
.$
We have
(5.6) $\mathrm{d}\Gamma((\eta_{\sigma_{n}})^{2}w_{\ell}^{(2)})\geq
H_{0\,n}^{(2)}\ .$
By (3.24), (3.26) and (5.6) we get
$\begin{split}f_{n}(H_{n}-E_{n})\mathrm{d}\Gamma(\eta_{\sigma_{n}}{}^{2}w_{\ell}^{(2)})f_{n}(H_{n}-E_{n})&\geq
P_{n}\otimes f_{n}(H_{0\,n}^{(2)})\,H_{0\,n}^{(2)}\,f_{n}(H_{0\,n}^{(2)})\\\
&\geq\frac{\gamma^{2}}{N^{2}}\sigma_{n}f_{n}(H_{n}-E_{n})^{2}\ ,\end{split}$
for $g\leq g_{\delta}^{(1)}$.
This, together with (5.5), yields for $g\leq g_{\delta}^{(1)}$
$\begin{split}&f_{n}(H_{n}-E_{n})[H,\,iA_{n}]f_{n}(H_{n}-E_{n})\\\
&\geq(1-g_{\delta}^{(1)})\frac{\gamma^{2}}{N^{2}}\sigma_{n}f_{n}(H_{n}-E_{n})^{2}-g\,C(G)\,\sigma_{n}f_{n}(H_{n}-E_{n})^{2}\
.\end{split}$
Setting
$g_{\delta}^{(2)}=\inf(g_{\delta}^{(1)},\,\frac{1-g_{\delta}^{(1)}}{2\,C(G)}\frac{\gamma^{2}}{N^{2}})\
,$
we get
$f_{n}(H_{n}-E_{n})[H,\,iA_{n}]f_{n}(H_{n}-E_{n})\geq\frac{1-g_{\delta}^{(1)}}{2}\frac{\gamma^{2}}{N^{2}}\,\sigma_{n}f_{n}(H_{n}-E_{n})^{2}\
,$
for $g\leq g_{\delta}^{(2)}$.
Proposition 3.9 is proved by setting
$\tilde{g}_{\delta}^{(1)}=g_{\delta}^{(2)}$ and
$\tilde{C}_{\delta}=\frac{1-g_{\delta}^{(1)}}{2}$.
The proof of Theorem 3.10 is the consequence of the following two lemmata.
###### Lemma 5.1.
Assume that the kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$
satisfy Hypothesis 2.1 and 3.1(ii). Then there exists a constant $D>0$ such
that
$|E-E_{n}|\leq g\,D\,\sigma_{n}{}^{2}\ ,$
for $n\geq 1$ and $g\leq g^{(2)}$.
###### Proof.
Let $\phi$ (respectively $\tilde{\phi}_{n}$) be the unique normalized ground
state of $H$ (respectively $H_{n}$). We have
(5.7)
$\begin{split}&E-E_{n}\leq(\tilde{\phi}_{n},(H-H_{n})\tilde{\phi}_{n})\\\
&E_{n}-E\leq(\phi,(H_{n}-H)\phi)\ ,\end{split}$
with
(5.8) $H-H_{n}=gH_{I}(\chi_{\sigma_{n}}(p_{2})G)\ .$
Combining (2.29) and (2.30) with (3.3)-(3.6) and (5.8), we get
(5.9) $\|(H-H_{n})\tilde{\phi}_{n}\|\leq
g\,K(\chi_{\sigma_{n}}(p_{2})G)\,(C_{\beta\eta}\|H_{0}\tilde{\phi}_{n}\|+B_{\beta\eta})$
and
(5.10) $\|(H-H_{n})\phi\|\leq
g\,K(\chi_{\sigma_{n}}(p_{2})G)\,(C_{\beta\eta}\|H_{0}\phi\|+B_{\beta\eta})$
It follows from Hypothesis 3.1(ii), (4.5), (5.9) and (5.10) that there exists
a constant $D>0$ such that
$\max(\|(H-H_{n})\tilde{\phi}_{n}\|,\,\|(H-H_{n})\phi\|\leq
g\,D\,\sigma_{n}{}^{2}\ ,$
for $n\geq 1$ and $g\leq g^{(2)}$.
By (5.7), this proves Lemma 5.1. ∎
###### Lemma 5.2.
Suppose that the kernels $G_{\ell,\epsilon,\epsilon^{\prime}}^{(\alpha)}$
satisfy Hypothesis 2.1 and 3.1(ii). Then there exists a constant $C>0$ such
that
(5.11) $\|f_{n}(H-E)-f_{n}(H_{n}-E_{n})\|\leq g\,C\,\sigma_{n}\ ,$
for $n\geq 1$ and $g\leq g^{(2)}$.
###### Proof.
Let $\tilde{f}(.)$ be an almost analytic extension of $f(.)$ given by (3.25)
satisfying
(5.12) $\left|\partial_{\bar{z}}\tilde{f}(x+iy)\right|\leq Cy^{2}\ .$
Note that $\tilde{f}(x+iy)\in C_{0}^{\infty}({\mathbb{R}}^{2})$. We thus have
(5.13)
$f(s)=\int\frac{\mathrm{d}\tilde{f}(z)}{z-s},\quad\mathrm{d}\tilde{f}(z)=-\frac{1}{\pi}\frac{\partial\tilde{f}}{\partial\bar{z}}\,\mathrm{d}x\,\mathrm{d}y\
.$
Using the functional calculus based on this representation of $f(s)$, we get
(5.14)
$f_{n}(H-E)-f_{n}(H_{n}-E_{n})=\sigma_{n}\int\frac{1}{H-E-z\sigma_{n}}(H-H_{n}+E_{n}-E)\frac{1}{H_{n}-E_{n}-z\sigma_{n}}\mathrm{d}\tilde{f}(z)\
.$
Combining (2.29) and (2.30) with (3.3)-(3.6) and Hypothesis 3.1(ii), we get,
for every $\psi\in{\mathcal{D}}(H_{0})$ and for $g\leq g^{(2)}$,
(5.15) $g\|H_{I}(\chi_{\sigma_{n}}G)\psi\|\leq
2\,g\,C\,\sigma_{n}{}^{2}K(G)\,(C_{\beta\eta}\|(H_{0}+1)\psi\|+(C_{\beta\eta}+B_{\beta\eta})\|\psi\|)\
.$
This yields
(5.16) $g\|H_{I}(\chi_{\sigma_{n}}(p_{2})G)(H_{0}+1)^{-1}\|\leq
g\,C_{1}\,\sigma_{n}{}^{2}\ ,$
for some constant $C_{1}>0$ and for $g\leq g^{(2)}$.
By mimicking the proof of (A.12) we show that there exists a constant
$C_{2}>0$ such that
(5.17) $\|(H_{0}+1)(H_{n}-E_{n}-z\sigma_{n})^{-1}\|\leq
C_{2}(1+\frac{1}{|\mathrm{Im}z|\sigma_{n}})\ ,$
for $g\leq g^{(1)}$.
Combining Lemma 5.1 and (5.14) with (5.15)-(5.17) we obtain
$\|f_{n}(H-E)-f_{n}(H_{n}-E_{n})\|\leq
g\,C\,\sigma_{n}\int\frac{|\frac{\partial\tilde{f}}{\partial\bar{z}}(x+iy)|}{y^{2}}\mathrm{d}x\mathrm{d}y\
,$
for some constant $C>0$ and for $g\leq g^{(2)}$.
Using (5.12) and $\tilde{f}(x+iy)\in C_{0}^{\infty}({\mathbb{R}}^{2})$ one
concludes the proof of Lemma 5.2. ∎
We now prove Theorem 3.10.
###### Proof.
It follows from Proposition 3.9 that
$\begin{split}&f_{n}(H_{n}-E_{n})[H,\,iA]f_{n}(H_{n}-E_{n})\\\
&=f_{n}(H_{n}-E_{n})[H,\,iA_{n}]f_{n}(H_{n}-E_{n})\geq\tilde{C}_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}\,f_{n}(H_{n}-E_{n})^{2}\
,\end{split}$
for $n\geq 1$ and $g\leq\tilde{g}_{\delta}^{(1)}$.
This yields
$\begin{split}&f_{n}(H-E)[H,iA_{n}]f_{n}(H-E)\geq\tilde{C}_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}\,f_{n}(H-E)^{2}\\\
&-f_{n}(H-E)[H,\,iA](f_{n}(H_{n}-E_{n})-f_{n}(H-E))\\\
&-(f_{n}(H_{n}-E_{n})-f_{n}(H-E))[H,\,iA]f_{n}(H_{n}-E_{n})\\\
&+\tilde{C}_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}(f_{n}(H_{n}-E_{n})-f_{n}(H-E))^{2}\\\
&+\tilde{C}_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}f_{n}(H-E)(f_{n}(H_{n}-E_{n})-f_{n}(H-E))\\\
&+\tilde{C}_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}(f_{n}(H_{n}-E_{n})-f_{n}(H-E))f_{n}(H-E)\
.\end{split}$
Combining Proposition 3.8 (i) and (5.13) with (5.16) and (5.17) we show that
$[H,\,iA]f_{n}(H_{n}-E_{n})$ and $f_{n}(H-E)[H,\,iA]$ are bounded operators
uniformly with respect to $n$. This, together with Lemma 5.2, yields
(5.18)
$f_{n}(H-E)[H,\,iA]f_{n}(H-E)\geq\tilde{C}_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}f_{n}(H-E)^{2}-\tilde{C}\,g\,\sigma_{n}\
,$
for some constant $\tilde{C}>0$ and for
$g\leq\inf(g^{(2)},\,\tilde{g}_{\delta}^{(1)})$.
Multiplying both sides of (5.18) with $E_{\Delta_{n}}(H-E)$ we then get
$E_{\Delta_{n}}(H-E)[H,\,iA]E_{\Delta_{n}}(H-E)\geq\tilde{C}_{\delta}\frac{\gamma^{2}}{N^{2}}\sigma_{n}E_{\Delta_{n}}(H-E)-\tilde{C}\,g\,\sigma_{n}E_{\Delta_{n}}(H-E)\
.$
Setting
$\tilde{g}_{\delta}^{(2)}<\inf\left(\frac{\tilde{C}_{\delta}}{\tilde{C}}\frac{\gamma^{2}}{N^{2}},\,g^{(2)},\,\tilde{g}_{\delta}^{(1)}\right)\
,$
Theorem 3.10 is proved with
$C_{\delta}=\tilde{C}_{\delta}-\tilde{C}\frac{N^{2}}{\gamma^{2}}\tilde{g}_{\delta}^{(2)}>0$.
∎
## 6\. Proof of Theorem 3.7
We set
$\begin{split}&A_{t}=\frac{\mathrm{e}^{itA}-1}{t}\ ,\\\
&\mathrm{ad}_{A_{t}}\cdot=[A_{t},\,.\,]\ ,\\\
&A_{t}^{\sigma}=\frac{\mathrm{e}^{itA^{\sigma}-1}}{t}\ ,\\\
&A_{\sigma\,t}=\frac{\mathrm{e}^{itA_{\sigma}}-1}{t}\ .\end{split}$
The fact that $H$ is of class $C^{1}(A)$, $C^{1}(A^{\sigma})$ and
$C^{1}(A_{\sigma})$ in $(-\infty,\,m_{1}-\frac{\delta}{2})$ is the consequence
of the following proposition
###### Proposition 6.1.
Suppose that the kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$
satisfy Hypothesis 2.1 and 3.1(iii.a). For every $\varphi\in
C_{0}^{\infty}((-\infty,m_{1}-\frac{\delta}{2}))$ and $g\leq g_{1}$, we then
have
$\begin{split}&\sup_{0<|t|\leq 1}\|[\varphi(H),\,A_{t}]\|<\infty\,,\\\
&\sup_{0<|t|\leq 1}\|[\varphi(H),A_{t}^{\sigma}]\|<\infty\,,\\\
&\sup_{0<|t|\leq 1}\|[\varphi(H),\,A_{\sigma\,t}]\|<\infty\,,\\\
&\sup_{0<|t|\leq 1}\|[\varphi(H^{\sigma}),\,A_{t}^{\sigma}]\|<\infty\
.\end{split}$
###### Proof.
We use the representation
$\varphi(H)=\int\mathrm{d}\phi(z)(z-H)^{-1}\ ,$
where $\phi(z)$ is an almost analytic extension of $\varphi$ with
$|\partial_{\bar{z}}\phi(x+iy)|\leq
C|y|^{2}\quad\mbox{and}\quad\mathrm{d}\phi(z)=-\frac{1}{\pi}\frac{\partial}{\partial\bar{z}}\phi(z)\mathrm{d}x\mathrm{d}y\
.$
Note that $\phi(x+iy)\in C_{0}^{\infty}({\mathbb{R}}^{2})$.
We get
$\mathrm{ad}_{A_{t}}\varphi(H)=\int\mathrm{d}\phi(z)(z-H)^{-1}[A_{t},\,H](z-H)^{-1}\
.$
This yields
$\begin{split}&\|\mathrm{ad}_{A_{t}}\varphi(H)\|\\\ &\leq\sup_{0<|t|\leq
1}\|[A_{t},\,H](i-H)^{-1}\|\,\int|\mathrm{d}\phi(z)|\,\|(z-H)^{-1}\|\,\|(i-H)(z-H)^{-1}\|\
.\end{split}$
It is easy to prove that
(6.1) $\int|\mathrm{d}\phi(z)|\,\|(z-H)^{-1}\|\,\|(i-H)(z-H)^{-1}\|\leq
C\int\frac{|\mathrm{d}\phi(z)|}{|\mathrm{Im}z|^{2}}<\infty\ .$
By Proposition 3.8(b)$(i)$ and (6.1) we finally get, for $g\leq g_{1}$
$\sup_{0<|t|\leq 1}\|\mathrm{ad}_{A_{t}}\,\varphi(H)\|<\infty\ .$
In a similar way we obtain, for $g\leq g_{1}$
$\begin{split}\sup_{0<|t|\leq 1}\|[A_{t}^{\sigma},\,\varphi(H)]\|<\infty\,,\\\
\sup_{0<|t|\leq 1}\|[A_{\sigma\,t},\,\varphi(H)\|<\infty\,,\\\ \sup_{0<|t|\leq
1}\|[A_{t}^{\sigma},\,\varphi(H^{\sigma})]\|<\infty\ .\end{split}$
∎
The proof of Theorem 3.7 is the consequence of the following proposition
###### Proposition 6.2.
Suppose that the kernels $G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}$
satisfy Hypothesis 2.1 and 3.1 (i)-(iii). We then have, for $g\leq g_{1}$,
$\begin{split}&\sup_{0<|t|\leq
1}\|[A_{t},\,[A_{t},\,H]](H+i)^{-1}\|<\infty\,,\\\ &\sup_{0<|t|\leq
1}\|[A_{t}^{\sigma},[A_{t}^{\sigma},\,H](H+i)^{-1}\|<\infty\,,\\\
&\sup_{0<|t|\leq
1}\|[A_{\sigma\,t},[A_{\sigma\,t},\,H](H+i)^{-1}\|<\infty\,,\\\
&\sup_{0<|t|\leq
1}\|[A_{t}^{\sigma},[A_{t}^{\sigma},\,H^{\sigma}](H^{\sigma}+i)^{-1}\|<\infty\,,\end{split}$
###### Proof.
We have, for every $\psi\in{\mathcal{D}}(H)$,
(6.2)
$[A_{t},[A_{t},H]]\psi=\frac{1}{t^{2}}\mathrm{e}^{2itA}(\mathrm{e}^{-2itA}H\mathrm{e}^{2itA}-2\mathrm{e}^{-itA}H\mathrm{e}^{itA}+H)\psi\
.$
By (3.14) we get
(6.3)
$[A_{t},[A_{t},H_{0}]]\psi=\frac{1}{t^{2}}\mathrm{e}^{2itA}(\mathrm{d}\Gamma(w^{(2)}\circ\phi_{2t}-2w^{(2)}\circ\phi_{t}+w^{(2)}))\psi\
,$
where, for $\ell=1,2,3$,
(6.4)
$(w_{\ell}^{(2)}\circ\phi_{2t})(p_{2})-2(w_{\ell}^{(2)}\circ\phi_{t})(p_{2})+w_{\ell}^{(2)}(p_{2})=|\phi_{2t}(p_{2})|-2|\phi_{t}(p_{2})|+|p_{2}|\
.$
We further note that
(6.5)
$\frac{1}{t^{2}}\big{|}\,|\phi_{2t}(p_{2})|-2|\phi_{t}(p_{2})|+|p_{2}|\,\big{|}\leq\sup_{|s|\leq
2|t|}\left|\frac{\partial^{2}}{\partial s^{2}}|\phi_{s}(p_{2})|\,\right|\ ,$
and
(6.6) $\frac{\partial^{2}}{\partial
s^{2}}|\phi_{s}(p_{2})|=|\phi_{s}(p_{2})|\leq\mathrm{e}^{\Gamma|s|}|p_{2}|\ .$
Combining (6.3) with (6.4)-(6.6) we get
$\|[A_{t},\,[A_{t},\,H_{0}]](H_{0}+1)^{-1}\|\leq\mathrm{e}^{2\Gamma|t|}\ ,$
and
$\sup_{0<|t|\leq
1}\|[A_{t},\,[A_{t},\,H_{0}]](H_{0}+1)^{-1}\|\leq\mathrm{e}^{2\Gamma}\ .$
In a similar way we obtain
$\sup_{0<|t|\leq
1}\|[A_{t}^{\sigma},\,[A_{t}^{\sigma},H_{0}]](H_{0}+1)^{-1}\|\leq
C\mathrm{e}^{2\Gamma}\ ,$ $\sup_{0<|t|\leq
1}\|[A_{\sigma\,t},\,[A_{\sigma\,t},H_{0}]](H_{0}+1)^{-1}\|\leq
C\mathrm{e}^{2\Gamma}\ .$
Here $C$ is a positive constant.
Let us now prove that
$\sup_{0<|t|\leq 1}\|[A_{t},\,[A_{t},\,H_{I}(G)]](H+i)^{-1}\|<\infty$
By (3.18) and (6.2) we get, for every $\psi\in{\mathcal{D}}(H)$,
(6.7) $\begin{split}&[A_{t},\,[A_{t},\,H_{I}(G)]]\psi\\\
&=\sum_{\alpha=1,2}\sum_{\ell=1,2,3}\sum_{\epsilon\neq\epsilon^{\prime}}\frac{\mathrm{e}^{2itA}}{t^{2}}\Big{(}\mathrm{e}^{-2itA}H_{I}(G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\mathrm{e}^{2itA}-2\mathrm{e}^{-itA}H_{I}(G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\mathrm{e}^{itA}\\\
&+H_{I}(G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\Big{)}\psi\\\
&=\sum_{\alpha=1,2}\sum_{\ell=1,2,3}\sum_{\epsilon\neq\epsilon^{\prime}}\frac{\mathrm{e}^{2itA}}{t^{2}}\Big{(}H_{I}(G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};2t})-2H_{I}(G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};t})+H_{I}(G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};0})\Big{)}\psi\
,\end{split}$
where
$\begin{split}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};t}(\xi_{1},\xi_{2},\xi_{3})&=(D\phi_{t}(p_{2}))^{\frac{1}{2}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1};\,\phi_{t}(p_{2}),s_{2};\,\xi_{3})\\\
&=(e^{-ita}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})(\xi_{1},\xi_{2},\xi_{3})\
.\end{split}$
Combining (2.29) and (2.30) with (3.3)-(3.6) and (6.7) we get
(6.8) $\|[A_{t},\,[A_{t},\,H_{I}(G)]]\psi\|\leq
g\,K(G_{t})(C_{\beta\eta}\|(H_{0}+I)\psi\|+(C_{\beta\eta}+B_{\beta\eta})\|\psi\|)\
.$
Here $K(G_{t})>0$ and
(6.9)
$K(G_{t})^{2}=\sum_{\alpha=1,2}\sum_{\ell=1,2,3}\sum_{\epsilon\neq\epsilon^{\prime}}\frac{1}{t^{2}}\|G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};2t}-2G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};t}+G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}\|^{2}_{L^{2}(\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2})}\
.$
We further note that, for $0<|t|\leq 1$,
(6.10) $K(G_{t})\leq\sup_{0<|s|\leq
2}\Big{(}\sum_{\alpha=1,2}\sum_{\ell=1,2,3}\sum_{\epsilon\neq\epsilon^{\prime}}\left\|\frac{\partial^{2}}{\partial
s^{2}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};s}\right\|^{2}_{L^{2}(\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2})}\Big{)}^{\frac{1}{2}}\
.$
We get
(6.11) $\begin{split}&\left(\frac{\partial}{\partial
t}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};t}\right)\\\
&=\frac{3}{2}(\mathrm{e}^{-ita}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})+(\mathrm{e}^{-ita}(p_{2}\cdot\nabla_{p_{2}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}))\,,\end{split}$
and
(6.12) $\begin{split}&\left(\frac{\partial^{2}}{\partial
t^{2}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime};t}\right)\\\
&=\frac{9}{4}(\mathrm{e}^{-ita}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})+\frac{7}{2}(\mathrm{e}^{-ita}(p_{2}\cdot\nabla_{p_{2}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}))+\\!\sum_{i,j=1,2,3}\mathrm{e}^{-ita}\big{(}p_{2,i}p_{2,j}\partial^{2}_{p_{2,i}p_{2,j}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}\big{)}.\end{split}$
Recall that $\mathrm{e}^{-ita}$ is an one parameter group of unitary operators
in $L^{2}(\Sigma_{1}\times\Sigma_{1}\times\Sigma_{2})$.
Combining Hypothesis 3.1(iii.a) and (iii.b), with (6.8)-(6.12) we finally get
$\sup_{0<|t|\leq 1}\|[\,A_{t},\,[A_{t},\,H_{I}(G)]\,](H_{0}+1)^{-1}\|<\infty\
.$
In view of ${\mathcal{D}}(H)={\mathcal{D}}(H_{0})$ the operators
$H_{0}(H+i)^{-1}$ and $H(H_{0}-1)^{-1}$ are bounded and we obtain
$\sup_{0<|t|\leq 1}\|[\,A_{t},\,[A_{t},\,H_{0}]\,](H+i)^{-1}\|<\infty\ ,$
(6.13) $\sup_{0<|t|\leq
1}\|[\,A_{t},\,[A_{t},\,H_{I}(G)]\,](H+i)^{-1}\|<\infty\ .$
This yields
(6.14) $\sup_{0<|t|\leq 1}\|[\,A_{t},\,[A_{t},\,H]\,](H+i)^{-1}\|<\infty\ ,$
for $g\leq g_{1}$.
Let $V(p_{2})$ denote any of the two $C^{\infty}$-vector fields
$v^{\sigma}(p_{2})$ and $v_{\sigma}(p_{2})$ and let $\tilde{a}$ denote the
corresponding $a^{\sigma}$ and $a_{\sigma}$ operators. We get
$\begin{split}&\left(\frac{\partial^{2}}{\partial
t^{2}}(\mathrm{e}^{-i\tilde{a}t}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\right)(\xi_{1},\xi_{2},\xi_{3})\\\
&=\frac{1}{4}\left(\mathrm{e}^{-i\tilde{a}t}((\mathrm{div}V(p_{2}))^{2}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\right)(\xi_{1},\xi_{2},\xi_{3})\\\
&+\frac{1}{2}\left(\mathrm{e}^{-i\tilde{a}t}((\mathrm{div}V(p_{2}))V(p_{2})\cdot\nabla_{p_{2}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\right)(\xi_{1},\xi_{2},\xi_{3})\\\
&+\frac{1}{2}\left(\mathrm{e}^{-i\tilde{a}t}(\sum_{i,j=1}^{3}(V_{i}(p_{2})(\partial^{2}_{p_{2,i}p_{2,j}}V_{j}(p_{2})))G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\right)(\xi_{1},\xi_{2},\xi_{3})\\\
&+\frac{1}{2}\left(\mathrm{e}^{-i\tilde{a}t}(\sum_{i,j=1}^{3}V_{i}(p_{2})\frac{\partial
V_{j}}{\partial p_{2,i}}(p_{2})\frac{\partial}{\partial
p_{2,j}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\right)(\xi_{1},\xi_{2},\xi_{3})\\\
&+\frac{1}{2}\left(\mathrm{e}^{-i\tilde{a}t}(\sum_{i,j=1}^{3}V_{i}(p_{2})V_{j}(p_{2})\frac{\partial^{2}}{\partial
p_{2,i}\partial
p_{2,j}}G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}})\right)(\xi_{1},\xi_{2},\xi_{3})\
.\end{split}$
Combining the properties of the $C^{\infty}$ fields $v^{\sigma}(p_{2})$ and
$v_{\sigma}(p_{2})$ together with Hypothesis 2.1 and 3.1 we get, from (6.13)
and by mimicking the proof of (6.14),
(6.15) $\sup_{0<|t|\leq
1}\|\,[\,A_{t}^{\sigma},\,[A_{t}^{\sigma},\,H]\,](H+i)^{-1}\|<\infty\ ,$
$\sup_{0<|t|\leq
1}\|\,[\,A_{\sigma\,t},\,[A_{\sigma\,t},\,H]\,](H+i)^{-1}\|<\infty\ ,$
for $g\leq g_{1}$.
Similarly, by mimicking the proof of (6.15), we easily get, for $g\leq g_{1}$,
$\sup_{0<|t|\leq
1}\|\,[\,A_{t}^{\sigma},\,[A_{t}^{\sigma},\,H^{\sigma}]\,](H^{\sigma}+i)^{-1}\|<\infty\
.$
This concludes the proof of Proposition 6.2 ∎
We now prove Theorem 3.7.
Proof of Theorem 3.7. In view of [3, Lemma 6.2.3] (see also [13, Proposition
28]), the proof of Theorem 3.7 will follow from Proposition 6.1 and the
following estimates
(6.16) $\sup_{0<|t|\leq 1}\|\,[\,A_{t},\,[A_{t},\,\varphi(H)]\,]\,\|<\infty\
,$ (6.17) $\sup_{0<|t|\leq
1}\|\,[\,A_{t}^{\sigma},\,[A_{t}^{\sigma},\,\varphi(H)]\,]\,\|<\infty\ ,$
(6.18) $\sup_{0<|t|\leq
1}\|\,[\,A_{\sigma\,t},\,[A_{\sigma\,t},\,\varphi(H)]\,]\,\|<\infty\ ,$ (6.19)
$\sup_{0<|t|\leq
1}\|\,[\,A_{t}^{\sigma},\,[A_{t}^{\sigma},\,\varphi(H^{\sigma})]\,]\,\|<\infty\
,$
for every $\varphi\in C_{0}^{\infty}((-\infty,m_{1}-\delta/2))$ and for $g\leq
g_{1}$.
Let us prove (6.16). The inequalities (6.17)-(6.19) can be proved similarly.
To this end, let $\phi$ be an almost analytic extension of $\varphi$
satisfying
$|\partial_{\bar{z}}\phi(x+iy)|\leq C|y|^{3}\ ,$
and
$\varphi(H)=\int(z-H)^{-1}\mathrm{d}\phi(z)\
,\quad\mathrm{d}\phi(z)=-\frac{1}{\pi}\frac{\partial}{\partial\bar{z}}\phi(z)\mathrm{d}x\mathrm{d}y\
.$
It follows that
$\begin{split}&[A_{t}\,[A_{t},\,\varphi(H)]\,]=\int\Big{(}(z-H)^{-1}[A_{t}\,[A_{t},\,H]\,](z-H)^{-1}\\\
&+2(z-H)^{-1}[A_{t},\,H](z-H)^{-1}[A_{t},\,H](z-H)^{-1}\Big{)}\mathrm{d}\phi(z)\end{split}$
We note that
(6.20) $\|(H+i)(H-z)^{-1}\|\leq\frac{C}{|\mathrm{Im}z|},\quad\mbox{for
}z\in\mathrm{supp}\phi\ .$
We also have
(6.21) $\begin{split}&\sup_{0<|t|\leq
1}\|\int(z-H)^{-1}[A_{t}\,[A_{t},\,H]\,](z-H)^{-1}\mathrm{d}\phi(z)\|\\\
&\leq\sup_{0<|t|\leq
1}\int\|[A_{t}\,[A_{t},\,H]\,](H+i)^{-1}\|\,\|(H+i)(z-H)^{-1}\|\frac{|\mathrm{d}\phi(z)|}{|\mathrm{Im}z|}\\\
&\leq C\sup_{0<|t|\leq
1}\|\,\left[A_{t},\,[A_{t},H]\,\right](H+i)^{-1}\,\|\int\frac{|\mathrm{d}\phi(z)|}{|\mathrm{Im}z|^{2}}\
.\end{split}$
Therefore, combining Proposition 3.8 (b)(i) and (6.20) we obtain
(6.22) $\begin{split}&\sup_{0<|t|\leq
1}\|\int\mathrm{d}\phi(z)(H-z)^{-1}[A_{t},\,H](H-z)^{-1}[A_{t},\,H](H-z)^{-1}\|\\\
&=\sup_{0<|t|\leq 1}\|\int(H-z)^{-1}[A_{t},\,H](H+i)^{-1}(H+i)(H-z)^{-1}\\\
&\quad\quad[A_{t},\,H](H+i)^{-1}(H+i)(H-z)^{-1}\|\mathrm{d}\phi(z)\\\ &\leq
C\left(\int\frac{|\mathrm{d}\phi(z)|}{|y|^{3}}\right)\sup_{0<|t|\leq
1}\|\,[A_{t},\,H](H+i)^{-1}\|^{2}<\infty\ .\end{split}$
Inequality (6.22) together with (6.21) yields (6.16), and $H$ is locally of
class $C^{2}(A)$ on $(-\infty,\,m_{1}-\delta/2)$ for $g\leq g_{1}$.
In a similar way it follows from Proposition 3.8(b), Proposition 6.1 and
Proposition 6.2 that $H$ is locally of class $C^{2}(A^{\sigma})$ and
$C^{2}(A_{\sigma})$ in $(-\infty,m_{1}-\delta/2)$ and that $H^{\sigma}$ is
locally of class $C^{2}(A^{\sigma})$ in $(-\infty,m_{1}-\delta/2)$, for $g\leq
g_{1}$. This ends the proof of Theorem 3.7. ∎
## 7\. Proof of Theorem 3.4
By (3.31), $\cup_{n\geq
1}\left((\gamma-\epsilon_{\gamma})^{2}\sigma_{n},\,(\gamma+\epsilon_{\gamma})\sigma_{n})\right)$
is a covering by open sets of any compact subset of $(E,\,m_{1}-\delta]$ and
of the interval $(E,\,m_{1}-\delta]$ itself. Theorem 3.4 (i) and (ii) follow
from Theorems 0.1 and 0.2 in [25] and Theorems 3.7 and 3.10 above with
$g_{\delta}=\tilde{g}_{\delta}^{(2)}$, where $\tilde{g}_{\delta}^{(2)}$ is
given in Theorem 3.10. Theorem 3.4 (iii) follows from Theorem 25 in [23].
## Appendix A
In this appendix, we will prove Proposition 3.5. We apply the method developed
in [4] because every infrared cutoff Hamiltonian that one considers has a
ground state energy which is a simple eigenvalue.
Let, for $n\geq 0$,
$\begin{split}&{\mathfrak{F}}^{\sigma_{n}}={\mathfrak{F}}^{n}\,,\\\
&\Sigma_{1\,n}^{\ \,n+1}=\Sigma_{1}\cap\\{p_{2};\
\sigma_{n+1}\leq|p_{2}|<\sigma_{n}\\}\ ,\\\ &{\mathfrak{F}}_{\ell,2,n}^{\ \ \
\,n+1}={\mathfrak{F}}_{a}(L^{2}(\Sigma_{1\,n}^{\
\,n+1}))\otimes{\mathfrak{F}}_{a}(L^{2}(\Sigma_{1\,n}^{\ \,n+1}))\,,\\\
&{\mathfrak{F}}_{n}^{n+1}=\otimes_{\ell=1}^{3}\,{\mathfrak{F}}_{\ell,2,n}^{\ \
\ \,n+1}.\end{split}$
We have
${\mathfrak{F}}^{n+1}\simeq{\mathfrak{F}}^{n}\otimes{\mathfrak{F}}_{n}^{n+1}\
.$
Let $\Omega^{n}$ (respectively $\Omega_{n}^{n+1}$) be the vacuum state in
${\mathfrak{F}}^{n}$ (respectively in ${\mathfrak{F}}_{n}^{n+1}$). We now set
$H_{0\,n}^{\
\,n+1}=H_{0}^{(1)}+H_{0}^{(3)}+\sum_{\ell=1}^{3}\sum_{\epsilon=\pm}\int_{\sigma_{n+1}\leq|p_{2}|<\sigma_{n}}\\!\\!\\!w_{\ell}^{(2)}(\xi_{2})c_{\ell,\epsilon}^{*}(\xi_{2})c_{\ell,\epsilon}(\xi_{2})\mathrm{d}\xi_{2}\
.$
The operator $H_{0\,n}^{\ \,n+1}$ is a self-adjoint operator in
${\mathfrak{F}}_{n}^{n+1}$.
Let us denote by $H_{I}^{n}$ and $H_{I\,n}^{\ \,n+1}$ the interaction $H_{I}$
given by (2.10)-(2.12) but associated with the following kernels
$\tilde{\chi}^{\sigma_{n}}(p_{2})G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})\
,$
and
$(\tilde{\chi}^{\sigma_{n+1}}(p_{2})-\tilde{\chi}^{\sigma_{n}}(p_{2}))G^{(\alpha)}_{\ell,\epsilon,\epsilon^{\prime}}(\xi_{1},\xi_{2},\xi_{3})\
,$
respectively, where $\tilde{\chi}^{\sigma_{n+1}}$ is defined by (3.1).
Let for $n\geq 0$,
$\begin{split}&H_{+}^{n}=H^{n}-E^{n}\ ,\\\
&\tilde{H}_{+}^{n}=H_{+}^{n}\otimes{\mathbf{1}}_{n}^{n+1}+{\mathbf{1}}_{n}\otimes
H_{0\,n}^{\ \,n+1}\ .\end{split}$
The operators $H_{+}^{n}$ and $\tilde{H}_{+}^{n}$ are self-adjoint operators
in ${\mathfrak{F}}^{n}$ and ${\mathfrak{F}}^{n+1}$ respectively. Here
${\mathbf{1}}^{n}$ and ${\mathbf{1}}_{n}^{n+1}$ are the identity operators in
${\mathfrak{F}}^{n}$ and ${\mathfrak{F}}_{n}^{n+1}$ respectively.
Combining (2.29) and (2.30) with (3.3)-(3.6) we obtain for $n\geq 0$,
(A.1) $g\|H_{I}^{n}\psi\|\leq
gK(G)(C_{\beta\eta}\|H_{0}\psi\|+B_{\beta\eta}\|\psi\|)\ ,$
for every $\psi\in{\mathcal{D}}(H_{0}^{n})\subset{\mathfrak{F}}^{n}$.
It follows from [22, §V, Theorem 4.11] that
$H^{n}\geq-\frac{gK(G)B_{\beta\eta}}{1-g_{1}K(G)C_{\beta\eta}}\geq-\frac{g_{1}K(G)B_{\beta\eta}}{1-g_{1}K(G)C_{\beta\eta}}\
,$
and
$E^{n}\geq-\frac{gK(G)B_{\beta\eta}}{1-g_{1}K(G)C_{\beta\eta}}\ .$
We have
(A.2) $(\Omega^{n},\ H^{n}\Omega^{n})=0\ .$
Therefore
$E^{n}\leq 0\ ,$
and
(A.3) $|E^{n}|\leq\frac{gK(G)B_{\beta\eta}}{1-g_{1}K(G)C_{\beta\eta}}\ .$
Let
(A.4) $K_{n}^{n+1}(G)=K({\mathbf{1}}_{\sigma_{n+1}\leq|p_{2}|\leq
2\sigma_{n}}\,G)\ .$
Combining (2.29) and (2.30) with (3.3), (3.4) and (A.4) we obtain for $n\geq
0$
(A.5) $g\|H_{I\,n}^{\ \,n+1}\psi\|\leq
g\,K_{n}^{n+1}(G)\,(C_{\beta\eta}\|H_{0}^{n+1}\psi\|+B_{\beta\eta}\|\psi\|)\
,$
for $\psi\in{\mathcal{D}}(H_{0}^{n+1})\subset{\mathfrak{F}}^{n+1}$, where we
remind that $H_{0}^{n+1}=H_{0}|_{{\mathfrak{F}}^{\sigma_{n+1}}}$ as defined in
(3.2).
We have for every $\psi\in{\mathcal{D}}(H_{0}^{n+1})$,
(A.6)
$H_{0}^{n+1}\psi=\tilde{H}_{+}^{n}\psi+E^{n}\psi-g(H_{I}^{n}\otimes{\mathbf{1}}_{n}^{n+1})\psi\
,$
and by (A.1)
(A.7) $g\|(H_{I}^{n}\otimes{\mathbf{1}}_{n}^{n+1})\psi\|\leq
g\,K(G)\,(C_{\beta\eta}\|H_{0}^{n+1}\psi\|+B_{\beta\eta}\|\psi\|)\ .$
In view of (A.3) and (A.6) it follows from (A.7) that
(A.8) $\begin{split}&g\|(H_{I}^{n}\otimes{\mathbf{1}}_{n}^{n+1})\psi\|\\\
&\leq\frac{g\,K(G)\,C_{\beta\eta}}{1-g_{1}\,K(G)\,C_{\beta\eta}}\|\tilde{H}_{+}^{n}\psi\|+\frac{g\,K(G)\,B_{\beta\eta}}{1-g_{1}\,K(G)\,C_{\beta\eta}}\big{(}1+\frac{g\,K(G)\,B_{\beta\eta}}{1-g_{1}\,K(G)\,C_{\beta\eta}}\big{)}\|\psi\|\
.\end{split}$
By (3.7), (3.8), (A.5), (A.6), (A.8) we finally get
(A.9) $g\|H_{I\,n}^{\ \,n+1}\psi\|\leq
gK_{n}^{n+1}(G)(\tilde{C}_{\beta\eta}\|\tilde{H}_{+}^{n}\psi\|+\tilde{B}_{\beta\eta}\|\psi\|)\
.$
For $n\geq 0$, a straightforward computation yields
(A.10)
$K_{n}^{n+1}(G)\leq\sigma_{n}\tilde{K}(G)\leq\sup(\frac{4\Lambda\gamma}{2m_{1}-\delta},\
1)\,\tilde{K}(G)\frac{\sigma_{n+1}}{\gamma}\ .$
Recall that for $n\geq 0$,
(A.11) $\sigma_{n+1}<m_{1}\ .$
By (A.9), (A.10) and (A.11), we get, for $\psi\in{\mathcal{D}}(H_{0})$,
$g\,\|H_{I\,n}^{\ \,n+1}\psi\|\leq
g\,K_{n}^{n+1}(G)\,\big{(}\,\tilde{C}_{\beta\eta}\|(\tilde{H}_{+}^{n}+\sigma_{n+1})\psi\|+(\tilde{C}_{\beta\eta}\,m_{1}+\tilde{B}_{\beta\eta})\|\psi\|\,\big{)}\
,$
and for $\phi\in{\mathfrak{F}}$,
(A.12) $\begin{split}g\|H_{I\,n}^{\
\,n+1}(\tilde{H}_{+}^{n}+\sigma_{n+1})^{-1}\phi\|&\leq
g\,K_{n}^{n+1}(G)\,\big{(}\,\tilde{C}_{\beta\eta}+\frac{m_{1}\tilde{C}_{\beta\eta}+\tilde{B}_{\beta\eta}}{\sigma_{n+1}}\,\big{)}\|\phi\|\\\
&\leq\frac{g}{\gamma}\,\sup(\frac{4\Lambda\gamma}{2m_{1}-\delta},\,1)\,\tilde{K}(G)(2m_{1}\tilde{C}_{\beta\eta}+\tilde{B}_{\beta\eta})\|\phi\|\
.\end{split}$
Thus, by (A.12), the operator $H_{I\,n}^{\
\,n+1}(\tilde{H}_{+}^{n}+\sigma_{n+1})^{-1}$ is bounded and
$g\|H_{I\,n}^{\ \,n+1}(\tilde{H}_{+}^{n}+\sigma_{n+1})^{-1}\|\leq
g\frac{\tilde{D}}{\gamma}\ ,$
where $\tilde{D}$ is given by (see (3.9)
$\tilde{D}=\,\sup(\frac{4\Lambda\gamma}{2m_{1}-\delta},\,1)\,\tilde{K}(G)\,(2m_{1}\tilde{C}_{\beta\eta}+\tilde{B}_{\beta\eta}).$
This yields, for $\psi\in{\mathcal{D}}(\tilde{H}_{+}^{n})$,
$g\|H_{I\,n}^{\ \,n+1}\psi\|\leq
g\frac{\tilde{D}}{\gamma}\|(\tilde{H}_{+}^{n}+\sigma_{n+1})\psi\|\ .$
Hence it follows from [22, §V, Theorems 4.11 and 4.12] that
(A.13) $g|(H_{I\,n}^{\ \,n+1}\psi,\,\psi)|\leq
g\frac{\tilde{D}}{\gamma}(\,(\tilde{H}_{+}^{n}+\sigma_{n+1})\psi,\,\psi\,)\ .$
Let $g_{\delta}^{(2)}>0$ be such that
$g_{\delta}^{(2)}\frac{\tilde{D}}{\gamma}<1\quad\mbox{and}\quad
g_{\delta}^{(2)}\leq g_{\delta}^{(1)}\ .$
By (A.13) we get, for $g\leq g_{\delta}^{(2)}$,
(A.14) $H^{n+1}=\tilde{H}_{+}^{n}+E^{n}+gH_{I\,n}^{\ \,n+1}\geq
E^{n}-\frac{g\,\tilde{D}}{\gamma}\,\sigma_{n+1}+(1-\frac{g\,\tilde{D}}{\gamma})\tilde{H}_{+}^{n}\
.$
Because $(1-g\tilde{D}/\gamma)\tilde{H}_{+}^{n}\geq 0$ we get from (A.14)
(A.15) $E^{n+1}\geq E^{n}-\frac{g\,\tilde{D}}{\gamma}\,\sigma_{n+1},\ n\geq 0\
.$
Suppose that $\psi^{n}\in{\mathfrak{F}}^{n}$ satisfies $\|\psi^{n}\|=1$ and
for $\epsilon>0$,
(A.16) $(\psi^{n},\,H^{n}\psi^{n})\leq E^{n}+\epsilon\ .$
Let
(A.17)
$\tilde{\psi}^{n+1}=\psi^{n}\otimes\Omega_{n}^{n+1}\in{\mathfrak{F}}^{n+1}\ .$
We obtain
(A.18) $E^{n+1}\leq(\tilde{\psi}^{n+1},\,H^{n+1}\tilde{\psi}^{n+1})\leq
E^{n}+\epsilon+g(\tilde{\psi}^{n+1},\ H_{I\,n}^{\ \,n+1}\,\tilde{\psi}^{n+1})$
By (A.13), (A.16), (A.17) and (A.18) we get, for every $\epsilon>0$,
$E^{n+1}\leq
E^{n}+\epsilon(1+\frac{g\,\tilde{D}}{\gamma})+\frac{g\,\tilde{D}}{\gamma}\,\sigma_{n+1}\
,$
where $g\leq g_{\delta}^{(2)}$.
This yields
(A.19) $E^{n+1}\leq E^{n}+\frac{g\,\tilde{D}}{\gamma}\,\sigma_{n+1}\ ,$
and by (A.15), we obtain
$|E^{n}-E^{n+1}|\leq\frac{g\,\tilde{D}}{\gamma}\,\sigma_{n+1}\ .$
For $n=0$, since $\sigma_{0}=\Lambda$, remind that
$H_{0}^{\,0}=H_{0}^{n=0}=H_{0}^{\sigma_{0}}=H_{0}|_{{\mathfrak{F}}^{\Lambda}}$.
Thus, the ground state energy of $H_{0}^{\,0}$ is $0$ and it is a simple
isolated eigenvalue of $H_{0}^{\,0}$ with $\Omega^{0}$, the vacuum in
${\mathfrak{F}}^{0}$, as eigenvector. Moreover, since $\Lambda>m_{1}$,
$\inf\left(\sigma(H_{0}^{\,0}\right)\setminus\\{0\\})=m_{1}\ ,$
thus $(0,m_{1})$ belongs to the resolvent set of $H_{0}^{\,0}$.
By Hypothesis 3.1(iv) we have $H^{0}=H_{0}^{\,0}$. Hence $E^{0}=\\{0\\}$ is a
simple isolated eigenvalue of $H^{0}$ and $H^{0}=H_{+}^{\,0}$. We finally get
(A.20)
$\inf\left(\sigma(H_{+}^{\,0})-\\{0\\}\right)=m_{1}>m_{1}-\frac{\delta}{2}=\sigma_{1}\
.$
We now prove Proposition 3.5 by induction in $n\in{\mathbb{N}}^{*}$. Suppose
that $E^{n}$ is a simple isolated eigenvalue of $H^{n}$ such that
$\inf\left(\sigma(H_{+}^{n})\setminus\\{0\\}\right)\geq(1-\frac{3g\tilde{D}}{\gamma})\sigma_{n},\quad
n\geq 1\ .$
Since (3.10) gives $\sigma_{n+1}<(1-\frac{3g\tilde{D}}{\gamma})\sigma_{n}$ for
$g\leq g_{\delta}^{(2)}$, $0$ is also a simple isolated eigenvalue of
$\tilde{H}_{+}^{n}$ such that
(A.21)
$\inf\left(\sigma(\tilde{H}_{+}^{n})\setminus\\{0\\}\right)\geq\sigma_{n+1}\
.$
We must now prove that $E^{n+1}$ is a simple isolated eigenvalue of $H^{n+1}$
such that
$\inf\left(\sigma(H_{+}^{n+1})\setminus\\{0\\}\right)\geq(1-\frac{3g\tilde{D}}{\gamma})\sigma_{n+1}\
.$
Let
$\lambda^{(n+1)}=\sup_{\psi\in{\mathfrak{F}}^{n+1};\,\psi\neq 0}\ \
\inf_{(\phi,\psi)=0;\,\phi\in{\mathcal{D}}(H^{n+1});\,\|\phi\|=1}(\phi,\,H_{+}^{n+1}\phi)\
.$
By (A.14) and (A.19), we obtain, in ${\mathfrak{F}}^{n+1}$
(A.22) $\begin{split}H_{+}^{n+1}&\geq
E^{n}-E^{n+1}-\frac{g\tilde{D}}{\gamma}\sigma_{n+1}+(1-\frac{g\tilde{D}}{\gamma})\tilde{H}_{+}^{n}\\\
&\geq(1-\frac{g\tilde{D}}{\gamma})\tilde{H}_{+}^{n}-\frac{2g\tilde{D}}{\gamma}\sigma_{n+1}\
.\end{split}$
By (A.17), $\tilde{\psi}^{n+1}$ is the unique ground state of
$\tilde{H}_{+}^{n}$ and by (A.21) and (A.22), we have, for $g\leq
g_{\delta}^{(2)}$,
$\begin{split}\lambda^{(n+1)}&\geq\inf_{(\phi,\tilde{\psi}^{n+1})=0;\,\phi\in{\mathcal{D}}(H^{n+1});\,\|\phi\|=1}(\phi,H_{+}^{n+1}\phi)\\\
&\geq(1-\frac{g\tilde{D}}{\gamma})\sigma_{n+1}-\frac{2g\tilde{D}}{\gamma}\sigma_{n+1}=(1-\frac{3g\tilde{D}}{\gamma})\sigma_{n+1}>0\
.\end{split}$
This concludes the proof of Proposition 3.5 by choosing
$g_{\delta}=g_{\delta}^{(2)}$, if one proves that $H^{1}$ satisfies
Proposition 3.5. By noting that $0$ is a simple isolated eigenvalue of
$\tilde{H}_{+}^{0}$ such that
$\inf(\sigma(\tilde{H}_{+}^{0})\setminus\\{0\\})=\sigma_{1}$, we prove that
$E^{1}$ is indeed an isolated simple eigenvalue of $H^{1}$ such that
$\inf(\sigma(H_{+}^{1})\setminus\\{0\\})\geq(1-\frac{3g\tilde{D}}{\gamma})\sigma_{1}$
by mimicking the proof given above for $H_{+}^{n+1}$.
∎
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|
arxiv-papers
| 2009-04-21T13:12:53 |
2024-09-04T02:49:02.036161
|
{
"license": "Public Domain",
"authors": "J.-M. Barbaroux, J.-C. Guillot",
"submitter": "Jean-Marie Barbaroux",
"url": "https://arxiv.org/abs/0904.3171"
}
|
0904.3376
|
# Stability and chaotic behaviors of Bose-Einstein condensates in optical
lattices with two- and three-body interactions
Yan Chen Institute of Theoretical Physics, Lanzhou University, Lanzhou
$730000$, China Ke-Zhi Zhang Physics and Electronics Engineering College,
Northwest Normal University, Lanzhou 730070, China Yong Chen Corresponding
author. Email: ychen@gmail.com Institute of Theoretical Physics, Lanzhou
University, Lanzhou $730000$, China Key Laboratory for Magnetism and Magnetic
Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000,
China
###### Abstract
The stability and chaotic behaviors of Bose-Einstein condensates with two- and
three-atom interactions in optical lattices are discussed with analytical and
numerical methods. It is found that the steady-state relative population
appears tuning-fork bifurcation when the system parameters are changed to
certain critical values. In particular, the existence of three-body
interaction not only transforms the bifurcation point of the system but also
affects greatly on the macroscopic quantum self-trapping behaviors of the
system associated with the critically stable steady-state solution. In
addition, we also investigated the influence of the initial conditions, three-
body interaction and the energy bias on the macroscopic quantum self-trapping.
Finally, by applying the periodic modulation on the energy bias, we find that
the relative population oscillation exhibits a process from order to chaos,
via a series of period-doubling bifurcations.
###### pacs:
03.75.Kk, 67.85.Jk, 03.65.Ge,
## I Introduction
In Recent years, Bose-Einstein condensates (BECs) in optical lattices have
attracted enormous attention both experimentally and theoretically ce1 ; ce2 .
This is mainly because the lattice parameters and interaction strength can be
manipulated using a modern experimental technique. Making use of this,
researchers have discovered many long-predicted phenomena, for example non-
linear Landau-Zener tunneling, energetic and dynamical instability and the
strongly inhibited transport of one-dimensional BEC in optical lattices ce3 ;
ce4 ; ce5 ; ce6 ; ce7 ; ce8 ; ce9 ; ce10 . More attracting phenomena, namely,
self-trapping, was recently observed experimentally in this system ce11 . In
such an experiment, a BEC cloud with repulsive interaction initially loaded in
optical lattices was self-trapped. Many theoretical analysis was also
presented about self-trapping ce12 ; ce13 ; ce14 ; ce15 . It is well know the
macroscopic quantum self-trapping (MQST) means self-maintained population
imbalance with non-zero average value of the fractional population imbalance
which was detailed discussed ce16 ; ce17 . Marino et. al. considered that the
damping decays all different oscillations to the zero-phase mode ce18 .
Besides, macroscopic quantum fluctuations have also been discussed by taking
advantage of second-quantization approaches ce19 . However, when the trapping
potential is time dependent and the damping and finite-temperature effect can
not be neglected, chaos emerges. Abdullaev and Kraenkel studied the nonlinear
resonances and chaotic oscillation of the fractional imbalance between two
coupled BEC’s in a double-well trap with a time-dependent tunneling amplitude
for different damping ce20 . When the asymmetry of the trap potential is time-
dependent and its amplitude is so small that can be took as a perturbation,
Lee et al. studied the chaotic and frequency-locked atomic population
oscillation between two coupled BECs with a weak damping, and discovered that
the system comes to an stationary frequency-locked atomic population
oscillations from transient chaos ce21 .
It is important to note that theoretical studies of stability are mainly
focused on the effect of two-body interactions. It is clear that in low
temperature and density, where interatomic distance is much greater than the
distance scale of atom-atom interactions, two-body s-wave scattering should be
important and three-body interactions can be neglected. But, if the atom
density is higher, for example, in the case of BEC in optical lattices, three-
body interactions will play an important role ce22 . As reported in Ref. ce23
, even for a small strength of the three-body force, the region of stability
for the condensate can be extended considerably.
Therefore, the purpose of this paper is to investigate the steady-state
solution of BEC in an one-dimensional periodic optical lattice when both the
two-body and three-body interactions are taken into account. By using the
mean-field approximation and linear stability theorem, one interesting result
is found that the tuning-fork bifurcation of steady-state relative population
appears when the system parameters are changed to certain critical values. The
existence of three-body interaction not only transforms the bifurcation point
of the system but also affects greatly on the self-trapping behaviors of the
system associated with the critically stable steady-state solutions.
Additionally, we also study the effects of the initial conditions, three-body
interaction and the energy bias on the MQST. Besides, we discuss the chaos
behaviors of the system by applying the periodic modulation on the energy
bias. The result shows the relative population oscillation can undergo a
process from order to chaos, via a series of period-doubling bifurcations.
This paper is organized as follows. In Sec. II, we introduce the mean-field
description of BEC in optical lattices with two- and three-atom interactions.
In Sec. III, with linear stability theorem, we analysis the stability of
steady-state solutions. Then the influences of three-body interaction on the
macroscopic quantum self-trapping of the system are displayed In Sec. IV. In
Sec. V, by applying the periodic modulation in the energy bias, we discuss
chaotic behaviors of the system using the numerical simulation method. In the
last section, summary and conclusion of our work are presented.
## II Mean-field description of BEC in optical lattices with two- and three-
atom interactions
We focus our attention on a BEC with both two- and three-body interactions is
subjected to one dimensional (1D) optical lattices where the motion in the
perpendicular directions is confined. In the mean-field approximation , the
dynamics of BEC can be modeled by the 1D Gross-Pitaevskii (GP) equation in the
comoving frame of the lattice ce3 ; ce6 ; ce24 ; ce25 ,
$i\hbar\frac{\partial\Phi}{\partial
t}=-\frac{1}{2m}\left(\hbar\frac{\partial}{\partial
t}-ima_{l}t\right)^{2}\Phi+\upsilon_{0}\cos(2K_{l}x)\Phi+\frac{2\hbar^{2}a_{s}}{a_{\bot}^{2}m}|\Phi|^{2}\Phi+\frac{g_{2}}{3\pi^{2}a_{\bot}^{4}}|\Phi|^{4}\Phi,$
(1)
where $\Phi$ is the wave function of the condensate, $m$ is the mass of atoms,
$a_{s}$ is the two-body s-wave scattering length, $\upsilon_{0}$ is the
strength of the periodic potential, $K_{l}$ is the wave number of the laser
light which is used to generate the optical lattice, $ma_{l}$ stands for
either the inertial force in the comoving frame of an accelerating lattice or
the gravity force, $a_{\bot}=\sqrt{\hbar/(m\omega_{\bot})}$, where
$\omega_{\bot}$ is the radial frequencies of the anisotropic harmonic trap,
$g_{2}|\Phi|^{4}\Phi/(3\pi^{2}a_{\bot}^{4})$ is three-body interactions
related to the GP equation. Among Eq. (1), all the variables can be rescaled
to be dimensionless by the following system’s basic parameter $x\sim
2K_{l}x,\Phi\sim\frac{\Phi}{\sqrt{2K_{l}N}},t\sim\frac{4\hbar}{m}K_{l}^{2}t$.
we obtain the normalized 1D-GP equation in optical lattices with cubic and
quintic nonlinearities,
$i\frac{\partial\Phi}{\partial t}=-{{1}\over{2}}\left(\frac{\partial}{\partial
t}-i\alpha
t\right)^{2}\Phi+\upsilon\cos(x)\Phi+c|\Phi|^{2}\Phi+\lambda|\Phi|^{4}\Phi,$
(2)
where
$\upsilon=\frac{m\upsilon_{0}}{4\hbar^{2}K_{l}^{2}},\alpha=\frac{m^{2}}{8\hbar^{2}K_{l}^{3}}a_{l},c=\frac{Na_{s}}{K_{l}a\bot^{2}}$
is the effective two-body interaction, $N$ is the total numbers of atoms,
$\lambda=\frac{mg_{2}N^{2}}{3\pi^{2}\hbar^{2}a_{\bot}^{4}}$ is the effective
interaction among three atoms, here the three-body interaction is expected to
be positive with a value of $0<\lambda<1$.
In the neighborhood of the Brillouin Zone edge $k=1/2$, the wave function can
be approximated by ce3
$\Phi(x,t)=a(t)e^{ikx}+b(t)e^{i(k-1)x},$ (3)
where $a(t)$, $b(t)$ are the probability amplitudes of atoms in each of the
two wells respectively and $|a|^{2}+|b|^{2}=1$. By inserting such wave
functions into Eq. (2) and performing some spatial integrals, we obtain the
dynamical equations with two- and three-body interactions.
$\displaystyle i\frac{\partial a}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2}a+\frac{c}{2}\left(|b|^{2}-|a|^{2}\right)a+\lambda\left(1+2|a|^{2}|b|^{2}+2|b|^{2}\right)a+\frac{\upsilon}{2}b,$
(4) $\displaystyle i\frac{\partial b}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\gamma}{2}b-\frac{c}{2}\left(|b|^{2}-|a|^{2}\right)b+\lambda\left(1+2|a|^{2}|b|^{2}+2|b|^{2}\right)b+\frac{\upsilon}{2}a.$
(5)
Here, the level bias $\gamma(t)=\alpha t$, and $\alpha$ is the sweeping rate,
$c$ and $\lambda$ represent the nonlinear parameters, $\upsilon$ is the
coupling constant between the two condensates. We introduce the relative
population variance
$s=|b|^{2}-|a|^{2},$ (6)
with the parameters $a=|a|e^{i}\theta_{a}$, $b=|b|e^{i}\theta_{b}$,
$\theta=\theta_{b}-\theta_{a}.$ (7)
Combining Eqs. (4-7), one yields the equations of the relative population and
relative phase,
$\displaystyle\dot{s}$ $\displaystyle=$
$\displaystyle-\upsilon\sqrt{1-s^{2}}\sin\theta,$ (8)
$\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\gamma+(c+2\lambda)s+\frac{\upsilon
s}{\sqrt{1-s^{2}}}\cos\theta.$ (9)
$\dot{s}$ and $\dot{\theta}$ denote the time derivative of the relative
population and the relative phase. If we regard $s$ and $\theta$ as the
canonically conjugate variables Eqs. (8) and (9), become a pair of Hamilton’s
canonical equations with the conserved effective Hamiltonian
$H=\gamma s+\frac{1}{2}(c+2\lambda)s^{2}+\upsilon\sqrt{1-s^{2}}\cos\theta.$
(10)
In the following section, we will discuss the stability of steady-state in the
symmetric condition ($\gamma=0$) with linear stability theorem.
## III Stability analysis of the steady-state solutions
In Sec. II, we have given the dynamical equations of the system with three-
body interaction. In this section, we will discuss the stability of steady-
state in the symmetric condition. Generally, there are two ways to study the
stability of nonlinear system, the linear stability theorem and the Lyapunov
direct method. We will investigate the stability of the system with the first
method.
The steady-state solution of this system can be obtained by setting Eqs. (8)
and (9) to zero. The forms of steady-state solutions are very complicated when
the level bias $\gamma\neq 0$. For simplicity, we set $\gamma=0$, leading to
$\displaystyle\dot{s}$ $\displaystyle=$ $\displaystyle
f_{1}(s,\theta)=-\upsilon\sqrt{1-s^{2}}\sin\theta,$ (11)
$\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle
f_{2}(s,\theta)=(c+2\lambda)s+\frac{\upsilon s}{\sqrt{1-s^{2}}}\cos\theta.$
(12)
and the conserved energy
$H=\frac{1}{2}(c+2\lambda)s^{2}+\upsilon\sqrt{1-s^{2}}\cos\theta.$ (14)
Taking $\dot{s}=0$, $\dot{\theta}=0$, we get
$\displaystyle-\upsilon\sqrt{1-s^{2}}\sin\theta$ $\displaystyle=$
$\displaystyle 0,$ (15) $\displaystyle(c+2\lambda)s+\frac{\upsilon
s}{\sqrt{1-s^{2}}}\cos\theta$ $\displaystyle=$ $\displaystyle 0.$ (16)
The steady-state solutions obeyed Eqs. (14) and (15) regard as
$\displaystyle\theta_{1}$ $\displaystyle=$ $\displaystyle 2n\pi,\quad
s_{1}=0\quad\mathrm{for}\quad H=-\upsilon,$ (17) $\displaystyle\theta_{2}$
$\displaystyle=$ $\displaystyle(2n+1)\pi,\quad s_{2}=0\quad\mathrm{for}\quad
H=\upsilon,$ (18) $\theta_{3,4}=(2n+1)\pi,\quad
s_{3,4}=\pm\sqrt{1-(\frac{\upsilon}{c+2\lambda})^{2}}\quad\mathrm{for}\quad
H=\frac{(c+2\lambda)^{2}+\upsilon^{2}}{2(c+2\lambda)^{2}}.$ (19)
According to the linear stability theorem, we look for the perturbed solutions
which are near the steady-state solutions,
$s(t)=s_{i}(t)+\varepsilon_{1}(t),\qquad\theta(t)=\theta_{i}(t)+\varepsilon_{2}(t)$
(20)
where $s_{i}(t)$, $\theta_{i}(t)$ for $i=1,2,3,4$ signify the steady-state
solutions, $|\varepsilon_{1}(t)|\ll|s_{i}(t)|$ and
$|\varepsilon_{2}(t)|\ll|\theta_{i}(t)|$ which is relate to the first-order
perturbed. Inserting the above expression into Eqs. (11) and (12), we can
obtain the linear equations near to the steady-states of the nonlinear
equations as
$\dot{\varepsilon_{1}}=\left(\frac{\partial f_{1}}{\partial
s}\right)_{1}\varepsilon_{1}+\left(\frac{\partial
f_{1}}{\partial\theta}\right)_{1}\varepsilon_{2}\qquad
namely\qquad\dot{\varepsilon_{1}}=a_{11}\varepsilon_{1}+a_{12}\varepsilon_{2}$
(21) $\dot{\varepsilon_{2}}=\left(\frac{\partial f_{2}}{\partial
s}\right)_{2}\varepsilon_{1}+\left(\frac{\partial
f_{2}}{\partial\theta}\right)_{2}\varepsilon_{2}\qquad
namely\qquad\dot{\varepsilon_{2}}=a_{21}\varepsilon_{1}+a_{22}\varepsilon_{2}$
(22)
Now, we make use of the above expression to investigate the stability of the
steady-states of Eqs. (16-18).
(1)For $\theta_{1}=2n\pi,s_{1}=0,H=-\upsilon$, we can calculate the matrix
elements $a_{11}=0$, $a_{12}=-\upsilon$, $a_{21}=(c+2\lambda)+\upsilon$,
$a_{22}=0$. So, the coefficient matrix of the linearized equations (20) and
(21) becomes $A_{1}=\left[\begin{array}[]{cc}{0}&{-\upsilon}\\\
{c+2\lambda+\upsilon}&{0}\\\ \end{array}\right]$ such that the characteristic
equation writes $\det(A_{1}-\lambda
I)=\left[\begin{array}[]{cc}{0-\lambda}&{-\upsilon}\\\
{c+2\lambda+\upsilon}&{0-\lambda}\\\ \end{array}\right]=0$, which reveals that
$\lambda^{2}+\upsilon(c+2\lambda+\upsilon)=0$. We solve the equation to get
the two eigenvalues of the matrix A as
$\lambda_{1}=\sqrt{-\upsilon(c+2\lambda+\upsilon)},\lambda_{2}=-\sqrt{-\upsilon(c+2\lambda+\upsilon)}$.
In response to the forms of the eigenvalues, there exist two cases for the
stabilities:
(a) $\upsilon(c+2\lambda+\upsilon)\geq 0$, that is
$\upsilon>0\quad and\quad(c+2\lambda)\geq-\upsilon$ (23) $\upsilon<0\quad
and\quad(c+2\lambda)\leq-\upsilon$ (24)
so the two eigenvalues are both pure imaginary numbers. Thus, the stability of
the steady-state solutions $(\theta_{1},s_{1})$ corresponds to a critical case
ce26 and the dynamical bifurcations between the unstable and stable steady-
states will appear when the parameters with two- and three-body interactions
are changed.
(b) $\upsilon(c+2\lambda+\upsilon)<0$, namely
$\upsilon>0\quad and\quad(c+2\lambda)<-\upsilon$ (25) $\upsilon<0\quad
and\quad(c+2\lambda)>-\upsilon$ (26)
so the two eigenvalues are real number. It means that $\varepsilon_{1}$ and
$\varepsilon_{2}$ tend to infinity with the increase of time, and the steady-
state solutions $(\theta_{1},s_{1})$ are unstable.
(2)For $\theta_{2}=(2n+1)\pi$, $s_{2}=0$, $H=\upsilon$, the matrix elements
write as $a_{11}=0,a_{12}=-\upsilon,a_{21}=(c+2\lambda)-\upsilon,a_{22}=0$.
The corresponding eigenvalues of the matrix $A_{2}$ become
$\lambda_{1}=\sqrt{-\upsilon(\upsilon-(c+2\lambda))},\lambda_{2}=-\sqrt{-\upsilon(\upsilon-(c+2\lambda))}$.
Similarly, there are two cases of the stabilities:
(a) $\upsilon(\upsilon-(c+2\lambda))>0$, that is
$\displaystyle(c+2\lambda)>0\quad$
$\displaystyle\mathrm{and}\quad\upsilon>(c+2\lambda)$ (27)
$\displaystyle(c+2\lambda)<0\quad$ $\displaystyle\mathrm{and}\quad\upsilon>0.$
(28)
so the two eigenvalues are both pure imaginary numbers. And the stability of
the steady-state solutions $(\theta_{2},s_{2})$ of the nonlinear equations are
reviewed as critical and the dynamical bifurcations will occur.
(b) $\upsilon(\upsilon-(c+2\lambda))\leq 0$, that is
$\displaystyle\upsilon>0\quad$ $\displaystyle\mathrm{and}$
$\displaystyle\quad(c+2\lambda)\geq\upsilon$ (29)
$\displaystyle(c+2\lambda)<\upsilon\quad$ $\displaystyle\mathrm{and}$
$\displaystyle\quad\upsilon<0$ (30)
so the two eigenvalues are positive or negative real number, respectively.
$\varepsilon_{1}$, $\varepsilon_{2}$ tend to infinity as increasing the time
to infinity, and the steady-state solutions $(\theta_{2},s_{2})$ are losing
their stability.
(3)For
$\theta_{3,4}=(2n+1)\pi,s_{3,4}=\pm\sqrt{1-(\frac{\upsilon}{c+2\lambda})^{2}},H=\frac{(c+2\lambda)^{2}+\upsilon^{2}}{2(c+2\lambda)^{2}}$,
the matrix elements read $a_{11}=0$, $a_{12}=\upsilon^{2}/(c+2\lambda)$,
$a_{21}=(c+2\lambda)-(c+2\lambda)^{3}/\upsilon^{2}$, $a_{22}=0$, and the
eigenvalues $\lambda_{1}=\sqrt{\upsilon^{2}-(c+2\lambda)^{2})}$,
$\lambda_{2}=-\sqrt{\upsilon^{2}-(c+2\lambda)^{2})}$. In Eq. (18) the
population $s_{3,4}$ are both real quantities which implies
$(c+2\lambda)^{2}>\upsilon^{2}$ (31)
Figure 1: Plots of the tuning-fork bifurcation from Eqs. (17) and (18), where
$s_{2},s_{3},s_{4}$ are the steady-state solutions and the bifurcation point
is $\frac{\upsilon}{c+2\lambda}=1$
Therefore, the two eigenvalues are pure imaginary numbers. The stability of
the steady-state solutions$(\theta_{3,4},s_{3,4})$ of the nonlinear equations
are regarded as critical and the dynamical bifurcations will emerge at the
bifurcation point $(c+2\lambda)=\upsilon$, $s=0$. Obviously, the existence of
three-body interaction can change the bifurcation point of the system. It
plays a important role for stability analysis of the system, as shown in Fig.
1. For $\frac{\upsilon}{c+2\lambda}>1$, the system is in the critically stable
steady-state ($\theta_{2},s_{2}$), and for $\frac{\upsilon}{c+2\lambda}<1$,
($\theta_{2},s_{2}$) is unstable and the two steady-state solutions
($\theta_{3,4},s_{3,4}$) are critically stable. This is a typical tuning-fork
bifurcation, and the bifurcation point is $\frac{\upsilon}{c+2\lambda}=1$
According to the above analysis, we conclude that three steady-state solutions
possess different stability for different parameter regions. And it is very
interesting to arrive at the critically stable steady-state solution in
experiment which relate to the stable stationary MQST ce26 . In the following
section, we will illustrate the MQST of the non-stationary states in detail by
two different methods.
## IV The macroscopic quantum self-trapping of BEC with two- and three-atom
interactions
In this section, we investigate the macroscopic quantum self-trapping by
plotting the phase trajectories and the time evolution of the relative
population of the system.
### IV.1 The phase trajectories diagram
The macroscopic quantum self-trapping refers to the phase space trajectories
whose the relative population is not equal zero. This can be well understood
from the analysis Eqs. (8)-(10), corresponding to the critically stable
steady-state solutions discussed in sec.II. Three kinds of cases occur with
different three-body interaction parameters, as shown in Fig.2.
(1) In the case of $\upsilon=0.2,c=0.1,0<\lambda<0.05$ in the phase space ,
there are two stable points $P_{1},P_{2}$ at $s=0,\theta=\pi$ and
$s=0,\theta=0$ respectively [Fig. 2(a)], from the circumstance described by
Eqs. (22) and (26). Obviously, for the stable points $P_{1}$, $P_{2}$, the
atoms distributions are equal in the two adjacent wells, the relative
population of the trajectories around them is equal to $0$. It means that
atoms oscillate between two adjacent wells and the macroscopic quantum self-
trapping phenomenon does not emerge in this case.
(2) When parameter is set to $\upsilon=0.2$, $c=0.1$, $0.05\leq\lambda<0.15$,
two more fixed points emerge in the line $\theta=\pi$ marked by $P_{3}$,
$P_{4}$. Among them, $P_{1}$, $P_{3}$ are steady which is corresponding to
condition of Eq. (30). They are located in
$s=\pm\sqrt{1-(\frac{\upsilon}{c+2\lambda})^{2}}$, hence, $P_{4}$ is unstable
point which lies in $s=0$ and corresponds to condition of Eq.(26). As seen
from Fig. 2(b), for the stable points $P_{1},P_{3}$, the atoms distributions
are not equilibrium between two adjacent wells, and the relative population of
the trajectories around them is not equal to $0$. It indicates that atoms are
self-trapped in one well. We take it as oscillating-phase-type because the
relative population $s$ and the relative phase $\theta$ oscillate around the
fixed points.
Figure 2: Trajectories on the phase space of the system with three-body
interaction varying from $\lambda=0$ to $\lambda=0.25$(the first row).
Corresponding to in the second row we plot the energy profiles for the
relative phase $\theta=0$ (red dashed) and $\theta=\pi$ (blue solid)
(3) For $\upsilon=0.2,c=0.1$, $\lambda\geq 0.15$ , It emerges new trajectories
, i.e.the trajectories across point $P_{c}$ [Fig. 2(c)]. Only the fixed point
$P_{2}$ is stable which is relate to Eq. (22). So for these trajectories, $s$
varies with time from region of $[-1,0]$ to $[0,1]$, Apparently $\langle
s\rangle\neq 0$, atoms are self-trapped in one well. We regard it as running-
phase-type macroscopic quantum self-trapping, as described in Refs. ce27 ;
ce28 and observed in experiment ce29 .
The above changes on the topological structure of the phase space are
concerned with the change of the energy profile. When the relative phase is
zero or $\pi$, energy relying on the parameter with three-body interaction and
the average population $s$ can be derived from Eq. (10). Seeing Fig. 2 , the
transition from case(1)to case(2) corresponds to the bifurcation of the energy
profile of $\theta=\pi$: energy curve bifurcates from a single minimum to the
curve of two minima. It means the system goes from the Rabi regime into the
self-trapping regime through this bifurcation. The lowest order of energy
profile with $\theta=0$ is $-\frac{c+2\lambda}{2}$, and the energy of the
unstable point $P_{4}$ is $-\upsilon$ which is located on the maximal order of
energy profile with $\theta=\pi$. The results displayed by the phase space
trajectories conform to the case of steady-state solutions discussed in
Sec.III. The transition from case (2) to case(3) is signified by the overlap
of the two energy regions of the profile. In this condition the trajectory
stared from $s=-1$, $\theta=0$ should be confined to the lower half of phase
plane, corresponding to the running-phase-type macroscopic quantum self-
trapping.
Connecting the analysis of the steady-state solutions to the above analysis on
the energy profile, it concludes that stable behaviors of the system change
constantly with the increase of $\lambda$ and we obtain a general criterion
for the macroscopic quantum self-trapping trajectories, namely,
$H(s,\theta)<-\upsilon$. It plays a critical role to find the transition
parameters of macroscopic quantum self-tapping.
### IV.2 Numerical simulations of the MQST
Now, we focus on the dynamic behavior which dominated by Eq. (8) and (9)
without the time-dependent system parameters. We study the effect parameters
of the system on the MQST with numerical method starting form Eq. (8) and (9).
Figure 3: The time evolution of the relative population from Eqs.(8) and (9)
with initial conditions $s(0)=0$, $\theta_{0}=\pi/2$ and parameter: (a)
$c=0.1$, $\lambda=0.45$, $v=0.2$, and $\gamma=0$; (b) $c=0.1$, $\lambda=0.95$,
$v=0.2$, and $\gamma=0$; (c) $c=0.1$, $\lambda=0.45$, $v=0.8$, and $\gamma=0$;
(d) $c=0.1$, $\lambda=0.45$, $v=0.2$, and $\gamma=0.5$; (e) $c=0.1$,
$\lambda=0.95$, $v=0.2$, and $\gamma=0.5$; (f) $c=0.1$, $\lambda=0.45$,
$v=0.8$, and $\gamma=0.5$;
Figure 4: the time evolution of the relative population from Eqs. (8) and (9).
(a) initial conditions $s(0)=0.8$, $\theta_{0}=\pi$ (b) initial conditions
$s(0)=0.8$, $\theta_{0}=\pi/2$, and the other parameters $c=0.1$,
$\lambda=0.45$, $v=0.2$, and $\gamma=0$.
Choosing initial condition $s(0)=0$, $\theta(0)=\pi/2$, the time evolutions of
the relative population Fig. (3a)-(3d) show some very absorbing features. In
Fig. 3(a), the oscillations are regular and the average the relative
population $\bar{s}$ is zero for symmetric well case ($\gamma=0$) with a
special parameter, but the corresponding MQST does not appear. If we increase
$\lambda$ from 0.45 to 0.95 in Fig. 3(b), the MQST does not still appear, but
the oscillating period becomes short. Similarly, rising $\upsilon$ , we obtain
the same result as shown in Fig. 3(c).
Here, we study impacting asymmetric well case ($\gamma\neq 0$) on the MQST.
when we enhance the level bias to $\gamma=0.5$ the average the relative
population is changed to $-0.41$ in Fig.3(d). Correspondingly, the oscillating
period of $s$ is longer and the MQST emerges. Note that parameter $c$,
$\lambda$ and $\upsilon$ impact greatly on the MQST which are plotted in Fig.
3(e) and (f). In fig. 3(e), when $\lambda$ is from $0.45$ to $0.95$, the MQST
is suppressed with shorter oscillating period. Similarly, with increasing
$\upsilon$, the average relative population are changed to $-0.21$ and the
oscillating period becomes shorter again, as seen in Fig. 3(f). Thus, the
influence of parameter $c$ ,$\lambda$ ,$\upsilon$ and $\gamma$ on the MQST of
the system is very dramatic. In the case of $\gamma=0$, fixing the other
parameters and changing the initial condition from $s(0)=0,\theta(0)=\pi/2$ of
Fig.3 to $s(0)=0.8,\theta(0)=\frac{\pi}{2}$ and $s(0)=0.8,\theta(0)=\pi$, we
observe that the MQST always emerges with varying $s(0),\theta(0)$. The
oscillating period is decreased comparing to Fig.3(a)and Fig. 3(d), but the
$\bar{s}$ is increased to $-0.86,-0.72$ as shown in Fig. 4.
According to the above analysis, we can draw conclusion that when the initial
conditions $s(0)=0$, $\theta(0)=\pi/2$ are read, the parameter $c$, $\lambda$,
$\upsilon$ can impact on the MQST for asymmetric well case($\gamma\neq 0$). In
addition, in the symmetric case, the MQST does not appear and those parameters
only affect the oscillating period of the system. Besides, the initial
conditions can impact the MQST for anyone parameter set.
## V Numerical simulation of chaos by applying periodic modulation on the
lever bias
As a whole, the elementary features of chaos is that the dynamic behaviors are
unpredictable for a deterministic system. It is very sensitive for the initial
conditions and parameters of the system. So, according to these
characteristics, we can adjust the parameters to make the system get into or
get out of the chaos, in other words, we can control the regime appearing
chaos. In this section We discuss the chaotic behaviors of the system by
numerical method.
Figure 5: Dynamical phase orbits of the dimensionless variables ($s,ds/dt$)
from Eqs. (31) and (32) with parameters $\upsilon=0.001$, $c=0.1$,
$\lambda=0.45$, $\omega=0.1$, $s(0)=0$, $\theta_{0}=\pi$, and (a)
$A_{1}=0.002$, (b) $A_{1}=0.009$, (c) $A_{1}=0.04$, (d) $A_{1}=0.12$, (e)
$A_{1}=0.3$, (f)=$A_{1}=1$. Here, $A_{1}$ denotes the amplitude of the time-
dependent relative energy.
If we apply periodic modulation on the lever bias
$\gamma=A_{0}+A_{1}sin(\omega t)$, the chaos will appear in a special region,
where $A_{0},A_{1}$ stand for initial phase and amplitude respectively.
Inserting this into Eqs. (8)and (9), one derives the below dynamic equation.
$\displaystyle\dot{s}$ $\displaystyle=$
$\displaystyle-\upsilon\sqrt{1-s^{2}}\sin\theta$ (32)
$\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle
A_{0}+A_{1}\sin(\omega t)+(c+2\lambda)s+\frac{\upsilon
s}{\sqrt{1-s^{2}}}\cos\theta$ (33)
Figure 6: (a) and (b): The time evolution of the relative population of the
relative population from Eqs. (31)and (32) with the parameters
$\upsilon=0.001$, $A_{0}=0.4$, $c=0.1$, $\lambda=0.45$, $\omega=0.1$,
$s(0)=0$, $\theta(0)=\pi$, and (a) $A_{1}=0.002$, (b) $A_{1}=0.3$ (c) and (d):
The corresponding power spectrum, where the parameters in Fig. 6(c)are the
same with Fig. 6(a) and the parameters in Fig. 6(d)are the same with Fig.
6(b).
Starting from Eqs. (32), It is found that the dynamics behavior of the system
is periodic in some special parameters region and it will vary from order to
chaos with the increase of $A_{1}$ , as shown in Fig.5. With initial
conditions $s(0)=0,\theta(0)=\pi$, the phase orbit is a period-one cycle and
the corresponding oscillation is a Rabi oscillation for the set of parameters
with amplitude $A_{1}=0.002$, as in Fig. 5(a). In this case, we set the
oscillating period of the relative population $T$. When $A_{1}=0.009$, the
phase orbit becomes period-two in Fig. 5(b). It means the oscillating period
of $s$ arriving at $2T$. Then the phase orbit increases from that of period-
four to period-eight with increasing $A_{1}$ as shown in Fig. 5(c)and (d).
Fig. 5(e) and 5(f) are plotted for $A_{1}=0.3$ and $A_{1}=1$, where the phase
orbit does not show a clear periodicity which signifies the emergence of
chaos.
From the above analysis, we find that the oscillating period of the relative
population varies from a period-one limit-cycle to period-two to period-four
and then to period-eight and finally all limit-cycles tend to infinity with
$\gamma$ increasing. It exhibits a process from order to chaos, through the
period-doubling bifurcations ce26 . That is to say, for a set of fixed
parameter $\upsilon$, $c$, $\lambda$, $A_{0}$, $A_{1}$, $s(0)$, $\theta(0)$
and $\omega$, the first-order derivative of relative population transform from
the single period to multiple period and get into chaos at last with the
increase of vibration amplitude $A_{1}$.
For the aim of showing the chaotic MQST, we present the plots of the time
evolution of the relative population and corresponding plots of power spectra
from Eqs. (31) and (32) in Fig. 6. And the parameter of Fig.5(a) is accord
with Fig. 6(a) and 6(c) where the system oscillates periodically. Making use
of those parameters of Fig. 5(e), we plot Fig. 6(b)and 6(d). It shows that the
power spectrum appears confusion and the average value of the relative
population is less than zero, which implies the existence of the chaotic
behaviors .
## VI Summary and conclusion
In this paper, we study the stability and chaos of BEC with repulsive two- and
three-body interactions immersed in a one-dimensional optical lattice. The
stability of the steady-state solution are analyzed with the linear stability
theorem. The analytical results show: (1) For $\upsilon>0$ and
$c+2\lambda\geq-\upsilon$ or $\upsilon<0$ and $c+2\lambda\leq-\upsilon$, the
stability of the steady-state solution($\theta_{1}=2n\pi,s_{1}=0$) is in the
critical case. (2) For $c+2\lambda>0$ and $\upsilon>c+2\lambda$ or
$c+2\lambda<0$ and $\upsilon>0$, the steady-state
solution($\theta_{2}=(2n+1)\pi,s_{2}=0$) is the critical stability. (3) For
$(c+2\lambda)_{2}>\upsilon_{2}$, the steady-state solution
($\theta_{3,4}=(2n+1)\pi,S_{3,4}=\pm\sqrt{1-(\frac{\upsilon}{c+2\lambda})^{2}}$)
is also critically stable. When these relationship are not satisfied, the
corresponding steady-state solution are unstable. A typical tuning-fork
bifurcation of steady-state relative population appears in special parameter
region. And the existence of three-body interaction can change the bifurcation
point of the system, which is shown as Fig. 1. It plays a important role for
stability analysis of the system.
The critically stable steady-state solution indicates the existence of the
stationary MSQT. The stable behaviors of the system change constantly with the
increase of $\lambda$ and get a general criteria for the self-trapping
trajectories, $H<-\upsilon$. In addition, we also investigate the effects of
the initial conditions, a set of parameters $c,\upsilon,\lambda,\gamma$ on
MQST. It shows that $c,\upsilon,\lambda$ could affect on the MQST when
$s(0)=0,\theta_{0}=\pi$ for $\gamma\neq 0$. Particularly, the initial value
$s(0)=0,\theta_{0}=\pi$ or $s(0)=0,\theta_{0}=\pi/2$ can directly impact on
the MQST. Finally, we discuss the chaos behaviors by applying the modulation
on the energy bias ($\gamma=A_{0}+A_{1}sin\omega t$). In this case, the system
will go into chaos through the period-doubling bifurcations with the
increasing of $\lambda$, and the time evolution of the relative population and
power spectra indicate the existence of the chaos MQST. It suggests that one
can adjust the lasing detuning and intensity to change the values of the
parameters in experiments. This adjustable parameters supply the possibility
for controlling the instabilities of the system, MQST state and the chaotic
behaviors.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China
and by the Open Project of Key Laboratory for Magnetism and Magnetic Materials
of the Ministry of Education, Lanzhou University.
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|
arxiv-papers
| 2009-04-22T02:50:40 |
2024-09-04T02:49:02.057240
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yan Chen, Ke-Zhi Zhang, and Yong Chen",
"submitter": "Yong Chen",
"url": "https://arxiv.org/abs/0904.3376"
}
|
0904.3456
|
11institutetext: Physics Department and INFM - University of Milan and
European Theoretical Spectroscopy Facility
Via Celoria 16, 20133 Milano, Italy 22institutetext: Department of Physics,
University of Cagliari and SLACS-INFM/CNR Sardinian Laboratory for
Computational Materials Science
Cittadella Universitaria, I-09042 Monserrato (Ca), Italy
# Atomistic simulations of the sliding friction of graphene flakes
Federico Bonelli 11 Nicola Manini 11 Emiliano Cadelano 22 Luciano Colombo 22
(April 2, 2008)
###### Abstract
Using a tight-binding atomistic simulation, we simulate the recent atomic-
force microscopy experiments probing the slipperiness of graphene flakes made
slide against a graphite surface. Compared to previous theoretical models,
where the flake was assumed to be geometrically perfect and rigid, while the
substrate is represented by a static periodic potential, our fully-atomistic
model includes quantum mechanics with the chemistry of bond breaking and bond
formation, and the flexibility of the flake. These realistic features, include
in particular the crucial role of the flake rotation in determining the static
friction, in qualitative agreement with experimental observations.
###### pacs:
68.35.Af, 62.20.Qp, 81.05.Uw, 07.79.Sp
## 1 Introduction
The scanning tunneling microscope (STM) Binning82 , and even more the atomic-
force microscope (AFM) Binning86 , have triggered perhaps the biggest wave of
advances and discoveries ever in surface science and nanoscience. Experimental
investigations of friction on the atomic scale have become possible by virtue
of the friction force microscope (FFM). In a FFM experiment a sharp tip scans
a sample surface with atomic precision, while lateral forces are recorded with
a resolution that can reach the pN range. Since Mate et al. Mate87
investigated the nanoscale periodic frictional force map of a graphite surface
using a tungsten tip, many studies have been conduced experimentally and
theoretically. In recent works the Leiden group Dienwiebel04 ; Dienwiebel05
has probed quantitatively the well known slipperiness of graphite, responsible
for its excellent lubrication properties. Morita et al. Morita96 suggested
that in FFM experiments on layered materials, such as $\rm MoS_{2}$ or
graphite, a flake, consisting of several hundred atoms in contact with the
substrate, can attach to the tip. By controlling the relative angles of
individual nanoflakes to achieve a suitable lattice mismatch, thus
incommensurate contact Hirano90 , an almost frictionless sliding was
demonstrated for dry and wearless tip-surface contact, a phenomenon known as
superlubricity. Several experiments and calculations an have been probing the
effects of lattice mismatch on friction Maier07 ; Maier08 ; Filleter09 ;
Vanossi06 ; Vanossi07PRL ; Castelli08Lyon , showing that incommensuracy often
prevents global ingraining of large areas, thus attenuating the consequent
strongly dissipative stick-slip motion. Theoretically, atomic-scale friction
on ideal solid surfaces is often described by simple balls-and-springs models
such as the Tomlinson model Tomlinson29 , where a single atom, or a more
structured tip Tomanek91 , is dragged through a spring along a static periodic
potential energy surface. In the present work we implement a Tomlinson-like
model of a simulated FFM experiment where dissipation of a finite graphene
flake is driven along a graphite substrate, in a fully atomistic scheme based
on a tight-binding (TB) force field. Compared to similar models in the
literature Sorensen96 ; Verhoeven04 ; Fusco04 ; Filippov08 , where the flake
is assumed to be perfect and rigid and the substrate is represented by an
analytically defined static periodic potential, our fully-atomistic TB
simulation explores two realistic features, namely: (i) The flake-substrate
interaction potential is not classical and contains quantum mechanics with the
chemistry of bond breaking and bond formation. (ii) The flake is nonrigid, so
that during its advancement it can deform and relax. In Sec. 2 we introduce
the model implementation details. Section 3 reports the results obtained, in
particular for the friction dependency on the flake size, the rotation angle
relative to the substrate, and the applied load; we compare the results of the
present model to experiment and to previous calculations. The final Sec. 4
discusses the results and the advantages and drawbacks of the present model.
## 2 The model
We describe the sliding by means of a generalized Tomlinson-like model similar
to that of Ref. Verhoeven04 , but including the following features: (i) the
interaction among all carbon atoms is realized in terms of the tight-binding
scheme of Xu et al. Xu92 , and (ii) the flake is not rigid but is allowed to
deform and rotate while sliding. Interatomic forces are computed as customary
in the TB scheme Colombo05 ; the hopping parameters and the pairwise repulsive
potential term follow the scaling form given by Xu et al. Xu92 . All
interatomic interactions vanish at a cutoff distance $r_{c}=2.6$ Å. This
distance sits in between the nearest-neighbor and the next-nearest-neighbor
distances of carbon atoms of the equilibrium sp3 diamond structure, and of the
sp2 graphene plane. It is also shorter than the interlayer distance of
graphite, which is as long as 3.35 Å Zacharia04 . The present TB model has
been applied successfully to investigate several low-dimensional carbon
systems Canning97 ; Yamaguchi07 ; Cadelano09 . In particular, this
parameterization reproduces the experimental equilibrium distance $d_{\rm
graph}=1.4224$ Å of the graphitic plane. To study friction, we use a model
constituted by a graphene flake sliding over a single infinite rigid graphene
sheet. The infiniteness of the substrate is simulated by repeating
periodically a regular arrangement of $N^{\rm sub}$ atoms at positions
$\vec{r}^{\rm\,sub}_{i}$ in the $x-y$ plane. We consider a periodic
rectangular supercell, containing as many carbon atoms as necessary for the
flake not to interact with its periodic images.
Figure 1: (Color online) The definition of the angle $\phi$ measuring the
initial rotation of the flake (thick, pink lines) with respect to the
substrate honeycomb structure (thin, black lines). The pulling line is defined
by its distance $h^{\prime}$ to the center of a substrate hexagon and the
pulling angle $\theta$. To take as a reference the AB stacking ($\phi=0$), we
define $h^{\prime}$ in terms of the parameter $h=h^{\prime}+\frac{1}{2}d_{\rm
graph}$.
We have studied three different regular hexagonal flakes composed of $N^{\rm
fl}=24$, 54, and 96 atoms, respectively. Initially, the flake is rotated by an
angle $\phi$ with respect to the substrate and translated horizontally to put
its center of mass along a sliding line at a distance $h^{\prime}$ from the
center of a substrate hexagon. As illustrated in Fig. 1, this line is oriented
at an angle $\theta$ from the $\hat{x}$ direction, which is defined by being
parallel to the zig-zag direction of the honeycomb lattice. In our simulations
we drag the flake along this pulling line and usually let the flake atoms
relax in all directions ($x$, $y$ and $z$), whereas the substrate remains
completely rigid. The $\phi$ angle defines the stacking mismatch, which has a
central importance for friction. Specific values of $\theta$ such as
$0^{\circ}$ and $30^{\circ}$ define special pulling directions where the
pulled flake encounters periodic corrugations, while aperiodic corrugations
are experienced for generic $\theta$ angles. The range of interest of both
angles $\theta$ and $\phi$ runs from $0^{\circ}$ to $30^{\circ}$: outside this
range we recover equivalent geometries.
Figure 2: (Color online) The computed normal force per atom $F_{N\;\rm atom}$
as a function of the fixed rigid flake-substrate distance $d$. The curves
refer to different stackings (AA or eclipsed, AB and an incommensurate
obtained starting from AB rotated by $\phi=30^{\circ}$) for a 96-atom flake
undeformed flake. The compression curve based on the low-pressure data of Ref.
Yeoman69 is reported for comparison.
To simulate an AFM experiment we introduce a constant load $F_{N}$ pushing the
flake against the substrate along the vertical $z$ direction and simulating
the force applied by the actual tip. This load acts against the reaction
forces produced by the interaction with the substrate. These forces are
reported in Fig. 2 for a $N^{\rm fl}=96$-atom flake in several configurations,
and of course vanish beyond $r_{c}=2.6$ Å, the TB interaction cutoff. For a
distance $d\geq 0.18$ nm, the load force per atom does not exceed 10 nN (a
total load in the tens to few hundred nN for a flake composed by 10 to
$10^{2}$ atoms, corresponding to a load pressure $\simeq 4$ Mbar), which is a
value practically accessible to FFM experiments Maier07 ; Maier08 ; Filleter09
. A load per atom of 0.5 nN, withing the selected force-field model, produces
an approach distance near 0.21 nm, similar to the one obtained by assuming for
the flake-substrate system the equilibrium interlayer separation of graphite
(3.35 Å) and the very soft $c$-axis compressibility Yeoman69 , expressed as
$d\ln c/dP=-2.8\times 10^{-6}$ bar-1. Note however that we apply this
compressibility relation, also sketched in Fig. 2 for distances and pressures
that go beyond its linear-response range of validity: in the region of close
approach, $d\leq 0.22$ nm, the actual force is likely to increase more
rapidly, like in the TB model curves. We implement a Tomlinson-like dynamics
with each flake atom coupled horizontally to a rigid “support” by elastic
springs. The support is a set of ideal graphene-net points, which coincide
with the initial flake atomic positions:
$\vec{r}_{i}^{\rm\,sup}=\vec{r}_{i}^{\rm\,fl}(t=0)$. This support is then
translated rigidly parallel to the substrate. Its orientation is fixed once
and for all by the angle $\phi$. The support advances by steps of length
$\delta x$ along the direction defined by the pulling angle $\theta$ and
lateral shift $h^{\prime}$. After a few tests, we select an advancement step
$\delta x=0.0024$ nm. After each advancement step relaxation, we evaluate and
store the total spring energy and the total dragging force, defined as
follows:
$\displaystyle E^{\rm spr}$ $\displaystyle=$
$\displaystyle\frac{K}{2}\sum_{i=1}^{N^{\rm fl}}\left(r^{\rm fl}_{x\,i}-r^{\rm
sup}_{x\,i}\right)^{2}+\left(r^{\rm fl}_{y\,i}-r^{\rm
sup}_{y\,i}\right)^{2}\,;$ (1)
$\displaystyle\vec{F}_{\varparallel}^{{\rm\,spr}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N^{\rm
fl}}\vec{F}_{\varparallel\,i}^{{\rm\,spr}}\,,$ (2)
$\displaystyle\vec{F}_{\varparallel\,i}^{{\rm\,spr}}$ $\displaystyle=$
$\displaystyle-K\left(\vec{r}_{i}^{\rm\,fl}-\vec{r}^{\rm\,sup}_{i}\right)_{\varparallel}\,,$
(3)
where the $\varparallel$ symbol indicates the in-plane component. The
component $F^{{\rm\,spr}}_{\varparallel\,i}$ of
$\vec{F}_{\varparallel\,i}^{{\rm\,spr}}$ along the pulling direction equals
the force needed to make the support advance, so that the work of this force
equals $F^{{\rm\,spr}}_{\varparallel\,i}\,\delta x$, for an infinitesimal
displacement in the pulling direction.
In an AFM experiment, the scanning tip speed is typically of the order of tens
or hundreds nm/s, much slower than the fast dynamics of the flake. Assuming
that the substrate temperature is fairly low, it is appropriate to consider a
quasi-static motion of the flake as follows: after each advancement of the
support, all flake atomic positions are made relax in all directions by damped
dynamics 111 The advantage of damped dynamics with respect to a standard
energy-minimization algorithm is that at each step it relaxes smoothly and
predictably to the nearest local-minimum configuration. On the other hand,
this algorithm is computationally less efficient than, e.g., a conjugated-
gradient technique. Eventually we settle on a fairly fast converging damped
dynamics with a time step $\delta t=4$ fs. under the combined action of (i)
the TB forces, (ii) the vertical load force $F_{N}$, and (iii) the horizontal
spring forces that attract the flake atoms near the support points. When the
stationary equilibrium position is reached (defined by no force component
exceeding a threshold of $10^{-2}$ nN), the support moves one step further and
the whole relaxation procedure is repeated until the support reaches the end
of a pre-defined path. With the selected fairly small advancement step $\delta
x$, each relaxation requires typically 10 to 200 MD steps. We consider a path
of moderate length $d\simeq 1$ nm divided into approximately 400 advancement
steps. The spring elastic constant coupling the support and the flake in the
$x$-$y$ plane is a quite critical parameter of the present model. Softer
springs allow the flake a greater freedom to translate, rotate, and deform,
with better pinning to energetically favorable sites and more pronounced
stick-slip motion and higher friction. Harder springs enforce a more stiff
flake showing little or no stick-slip motion. The limit $K\to\infty$ matches
the model by Verhoeven et al. Verhoeven04 . The spring constant mimic the
combined interaction between the flake and the tip and the elastic tip
response. We suggest that a value
$K=0.5\,\frac{\rm eV}{\AA^{2}}\simeq 8.01\,\frac{\rm N}{\rm m}$ (4)
corresponding to about 10% of the stretching stiffness of a carbon-carbon bond
within a graphene layer should probably be fairly realistic. We also perform
computations with softer ($K=0.1$ eV/Å2) and harder ($K_{x}=K_{y}=2.5$ eV/Å2)
springs, as detailed in Sec. 3.
Figure 3: (Color online) The spring total lateral force component
$\vec{F}_{\varparallel}^{{\rm\,spr}}$ projected along the scan line, for a
24-atom flake dragged starting from an initial stacking AB along a pulling
angle $\theta=15^{\circ}$, and with a stacking angle $\phi=30^{\circ}$. The
spring constant is $K=0.1$ eV/Å2, the total load $F_{N}=100$ nN. The $95\%$
force level (dashed line) estimates the static friction force $F_{\rm fric}$.
In our calculations, by convention, we estimate the static friction force
$F_{\rm fric}$ along a given sliding path by the force level below which 95%
of the spring force values $F_{\varparallel}^{\rm spr}$ encountered along the
path (at the end of each relaxation, and excluding an initial transient) are
found. This definition is illustrated in Fig. 3.
Figure 4: (Color online) The dynamic-friction force is evaluated as the
energy dissipated through a forward-backward scan (solid and dashed lines
respectively). The shaded area between the two curves measures the energy
dissipated by friction. The conditions are as follows: $N^{\rm fl}=24$,
$F_{N}=100$ nN, $\theta=0^{\circ}$, $\phi=0^{\circ}$, $K=0.1$ eV/Å2.
The dynamic-friction force is defined in terms of the energy dissipated in a
forward-backward loop. Figure 4 illustrates this concept: the component of the
spring force parallel to the advancement direction shows a clear hysteretic
behavior in a forward-backward scan. The shaded area enclosed between the two
curves measures the energy $E_{\rm fric}$ dissipated by friction, and is
clearly related to the stick-slip events. The average dynamical friction force
$F_{\rm fric}^{\rm dynamic}=E_{\rm fric}/d\simeq 0.325$ nN is of course
smaller than the static friction force $F_{\rm fric}\simeq 0.9$ nN evaluated
according to the 95% protocol described above. In the following we will focus
on the static friction force $F_{\rm fric}$, which is cheaper to compute and
eventually of the same order as $F_{\rm fric}^{\rm dynamic}$. As the flake
advances along its path, it deforms and rotates around its center of mass. In
particular, to understand the evolution of the static friction force, it is
useful to track the flake instantaneous stacking angle $\phi_{A}$, which
generally differs from the fixed support angle $\phi$. At each relaxed
configuration, we calculate $\phi_{A}$ as an average over all flake atoms $i$,
of a sine projection obtained as the length of the vector product of two
vectors in the horizontal $xy$ plane: ${\bf R}_{i}^{\rm cm}$, joining the
flake center of mass to the $i$-th flake atom, and the corresponding vector
${\bf R}_{i}^{0\,\rm cm}$ computed for the unrotated $\phi=0^{\circ}$ support.
Explicitly:
$\phi_{A}=\arcsin\\!\left(\sum_{i}\frac{R_{i}^{{\rm cm}\ y}R_{i}^{0\,{\rm cm}\
x}-R_{i}^{{\rm cm}\ x}R_{i}^{0\,{\rm cm}\ y}}{N^{\rm fl}\,|{\bf R}_{i}^{{\rm
cm}}|\,|{\bf R}_{i}^{0\,{\rm cm}}|}\right).$ (5)
## 3 Results
We analyze the friction force dependence on several physical parameters,
namely the pulling angle $\theta$, stacking angle $\phi$, applied load, flake
size and position of the scan-line. Experimentally, the number of flake atoms,
estimated in the order of 100, is not well determined, while the total applied
load and the total force acting on the flake are under control in the FFM. For
ease of comparison with experiments, our discussion shall always deal with
total quantities, i.e. summed over the flake atoms.
### 3.1 Relaxation to equilibrium
Figure 5: (Color online) The relaxed configuration of the $24$-atom flake.
Static substrate atoms are clear (white), flake atoms are dark (red). Figure
6: (Color online) The relaxed configuration of the $54$-atom flake. Static
substrate atoms are clear (white), flake atoms are dark (red). Figure 7:
(Color online) The relaxed configuration of the $96$-atom flake. Static
substrate atoms are clear (white), flake atoms are dark (red).
Optimally stacked configurations are important in providing the most efficient
sticking points during a friction sliding experiment. Figs. 5, 6, and 7 report
typical such relaxed configurations for $24-$, $54-$ and $96-$atom flakes
respectively, obtained under the action of a total load of $100$ nN, and with
no spring connection to a support ($K=0$). The relaxed configuration (up to a
symmetry rotation/translation) depends only moderately on the starting
stacking, unless the initial stacking angle is strongly incommensurate. The
average vertical flake-substrate separation is 0.192 nm. The equilibrium
configuration tends to arrange the flake so as to minimize the number of flake
atoms stacked on top of a substrate atom: indeed Fig. 2 shows that the
eclipsed “AA” stacking produces the strongest repulsive force at the same
distance. Geometrically different non-optimal configurations are characterized
by typical excess total energies of 1 eV or less.
### 3.2 The stick-slip movement
Figure 8: (Color online) The component of the springs force
$F_{\varparallel}^{\rm spr}$ in the dragging direction as a function of the
support advancement distance $x_{\rm tip}$ for two different pulling
directions: $\theta=0^{\circ}$ (solid) and $\theta=15^{\circ}$ (dashed). The
simulation involves a $24$ atom flake with total applied load of 100 nN,
support stacking angle $\phi=0^{\circ}$ and spring constant $K=0.1$ eV/Å2.
We come now to the actual simulation of sliding friction: to start with, Fig.
8 displays the pulling force $F_{\varparallel}^{\rm spr}$ measured along two
sliding paths of different commensurability nature: $\theta=0^{\circ}$
(periodic) and $\theta=15^{\circ}$ (aperiodic). As expected, the regular
stick-slip pattern of the $\theta=0^{\circ}$ path is replaced by an irregular
dependency in the $\theta=15^{\circ}$ trajectory. The initial part of the
$\theta=0^{\circ}$ trajectory is not periodic because of the usual startup
transient behavior: the first 0.2 nm are omitted from the calculation of the
friction force, as discussed above.
Figure 9: (Color online) Several physical quantities plotted as functions of
the support position $x_{\rm tip}$: (a) the parallel component of the springs
lateral force $F_{\varparallel}^{\rm spr}$, (b) the flake excess energy
$E^{\rm tot}-E^{\rm spr}-E^{\rm eq}$, (c) the flake center-of-mass advancement
$x_{\rm CM}$ along the pulling direction, and (d) the flake instantaneous
rotation angle $\phi_{A}$ relative to the substrate. Two different values of
the spring constant are compared: soft, $K=0.1$ eV/Å2 (solid, squares), and
hard, $K=2.5$ eV/Å2 (dashed, triangles). These simulations involve $N^{\rm
fl}=24$, $F_{N}=100$ nN, $\theta=0^{\circ}$, and $\phi=0^{\circ}$.
The stick-slip motion and the role of the spring stiffness is understood even
better by comparing other physical quantities with $F_{\varparallel}^{\rm
spr}$. In particular, Fig. 9 displays the internal energy of the flake-
substrate interaction, the displacement of the flake center of mass along the
pulling direction, and the actual stacking angle $\phi_{A}$. We focus
initially on the solid curves, obtained in a simulation based on soft springs
with $K=0.1$ eV/Å2. After the initial transient, where the flake explores once
a configuration with a negative $\phi_{A}$, it then jumps back and forth
between two kinds of sticking configurations: the most favorable one
characterized by $\phi_{A}\simeq 4^{\circ}$, and another one, at 0.1 to 0.2 eV
higher energy, near $\phi_{A}\simeq 0^{\circ}$. These stick-slip jumps
overcome energy barriers whose heights are of order 0.2 to 0.3 eV. This energy
amplitude sets the temperature range of validity of the present zero-
temperature calculations to a few hundred Kelvin: when thermally-activated
slips through energy barriers do not have enough time to occur, i.e. for a
not-too-small tip advancement speed Riedo03 ; Gnecco03 , these slips are
unlikely and our estimates of friction should be fairly reliable. Stick-slip
events occur with correlated jumps in the spring force, flake excess energy,
flake position, and stacking angle. The very different dashed curves show that
stiff springs (with $K=2.5$ eV/Å2) produce a much stronger and more rigid
binding of the flake to the rigid support. Accordingly, such an
unrealistically rigid coupling suppresses the stick-slip behavior: both the
advancement, shown in Fig. 9(c), and the spring force, Fig. 9(a), become
smooth and jump-less. Despite the suppression of stick-slip, we observe higher
force peaks for the stiffer springs, thus indicating a higher static friction
than for the softer springs. This is due to the flake being forced to cross
high potential-energy barriers, Fig. 9(b), with little or no possibility to
avoid them by (i) shifting away from the pulling direction, (ii) deforming,
and (iii) “rotating around” ($|\phi_{A}|<0.2^{\circ}$). A similar behavior is
observed also in different geometries. The spring strength tuning the coupling
between the flake and the AFM tip plays therefore an important role in the
model calculations. We checked that up to spring constants of an intermediate
value ($K=0.5$ eV/Å2) our model still performs a stick-slip motion similar to
the one found in experiment Dienwiebel04 (where the cantilever harmonic
constants values was estimated $K\simeq 0.36$ eV /Å2), especially in the fully
commensurate $\phi=0^{\circ}$ stacking.
### 3.3 Flake-size effects
Figure 10: (Color online) The parallel component of springs lateral force,
$F_{\varparallel}^{\rm spr}$, as a function of the support position $x_{\rm
tip}$ for three flake sizes: 24-atom flake (solid), 54-atom flake atoms
(dashed), and 96-atom flake atoms (dot-dashed). The three simulations involve
a total applied load $F_{N}=100$ nN, $\theta=0^{\circ}$, $\phi=30^{\circ}$,
and soft springs constants $K=0.1$ eV/Å2.
Figure 10 compares the frictional behavior of flakes of different size,
showing that friction tends to decrease with increasing flake size. In detail,
the static friction force of the 3 flakes is $F_{\rm fric}=2.99$ nN, 1.38 nN,
and 1.24 nN for the 24, 54, and 96-atom flakes respectively. This decrease is
not surprising, since the more reactive atoms at the flake boundary tend to
bend down toward the substrate. As a result, friction is dominated by these
boundary atoms, which amount to 75% of the 24-atom flake but only 44% of the
96-atom flake. Moreover, the 96-atom flake advances continuously, and shows no
stick-slip, at variance with the 24-atom and 54-atom flakes. This difference
is due to a reduced rotational freedom of the 96-atom flake flake, due to
coupling to the support acting at a larger distance from the flake center. The
flake rotational freedom, i.e. the angular range of $\phi_{A}$ explored around
$\phi$, does represent a key issue in the friction physics of carbon flakes
sliding over a graphite surface, as pointed out by Filippov et al. Filippov08
.
In the framework of our model, with each atom tied to the moving support by an
individual spring, the flake can both shift normally to the pulling direction
and rotate around its center of mass: these degrees of freedom (and, more
weakly, the possibility of the flake to distort) affect friction in two very
different manners depending on the contact being commensurate or
incommensurate. When the flake is pulled at a commensurate stacking, e.g.
$\phi=0^{\circ}$, it encounters high potential-energy barriers thus high
friction: the combined possibility of rotations and lateral shifts allows the
flake dribble the high barriers through local changes of trajectory. The first
effect of flake shifts and rotations is then to reduce the friction of highly
commensurate contacts. In contrast, when the flake slides with an
incommensurate stacking, e.g. $\phi=15^{\circ}$, it does not encounter high-
energy barriers nor efficiently binding configurations, thus producing a low-
friction motion. However rotations and shifts allow the flake to locate deeper
energy wells (both moving apart from the pulling direction and rotating so as
to explore different stacking configurations), where the flake can stick,
eventually providing sizeable friction. This second effect is therefore to
raise the friction for incommensurate contacts and eventually destroy
superlubricity, as was observed and discussed by Filippov et al. Filippov08 .
In our simple model the single parameter $K$ tunes the flake rotational
freedom and that of shifting perpendicular to the pulling direction. However,
while its effect on the translational freedom is independent of size, the
rotational freedom does depend on the larger torque that springs of the same
stiffness exert on flake atoms more remote from the flake center, as is also
to be expected for a flake sticking to a not too sharp AFM tip.
Figure 11: (Color online) The instantaneous rotation angle $\phi_{A}$ as a
function of the support position $x_{\rm tip}$ for three values of the support
stacking angle: (a) $\phi=0^{\circ}$, (b) $\phi=15^{\circ}$, and (c)
$\phi=30^{\circ}$. Three flake sizes are considered: $N^{\rm fl}=24$ (black
solid line), $N^{\rm fl}=54$ (red dashed line) and $N^{\rm fl}=96$ (blue dot-
dashed line). Simulations are carried out with total applied load of $100$ nN,
pulling angle $\theta=0^{\circ}$ and soft springs $K=0.1$ eV/Å2.
This point shows clearly in Fig. 11, which displays the evolution of the
actual rotation angle $\phi_{A}$ along the scanline, for three different
support stacking angles and three different flake sizes: $N^{\rm fl}=24$,
$54$, and $96$. Rotational fluctuations decrease as the flake size increases.
Indeed significant systematic deviations of $\phi_{A}$ from $\phi$ are
apparent in many cases, especially $N^{\rm fl}=24$. In particular, the small
24-atom flake for $\phi=30^{\circ}$ rotates all the way to $\phi_{A}\leq
10^{\circ}$, thus displaying angular oscillations in excess of $15^{\circ}$,
for an average angle $\langle\phi_{A}\rangle\simeq 16^{\circ}$, very different
from $\phi$. When plotting the dependence of friction force $F_{\rm fric}$ on
the stacking angle, it will make more sense to use, instead of the initial
stacking angle $\phi$, the average flake rotation angle
$\langle\phi_{A}\rangle$, although even this indicator does not account
properly for rotational fluctuations. This large rotational freedom, for hard
springs, is almost completely frozen: in that case the largest rotational
fluctuation we observe is as little as $\simeq 3^{\circ}$ for $N^{\rm fl}=24$
and $\phi=15^{\circ}$.
Figure 12: (Color online) Subsequent points marking the trajectories of the
center of mass of a 24-atom flake in the $x$-$y$ plane for $\phi=30^{\circ}$
(incommensurate stacking, blue circles) and for $\phi=0^{\circ}$ (commensurate
stacking, black diamonds) at the end of each relaxation cycle. The dashed
lines represent the support scanlines ($\theta=0^{\circ}$); large circles
represent the substrate atomic positions. The simulations are the soft-spring
ones of Fig. 9.
As for lateral shifts, for soft springs $K=0.1$ eV/Å2 we observe shifts
perpendicular to the pulling line of the order of 1 Å, depicted in Fig. 12
which displays the actual path followed by the center of mass of a 24-atom
flake pulled by the support along a $\theta=0^{\circ}$ scanline. Note that,
similarly to rotational fluctuations, the incommensurate stacking
$\phi=30^{\circ}$ yields larger lateral shifts than the commensurate stacking
$\phi=0^{\circ}$. With hard springs the possibility of the flake to perform
lateral shifts is strongly reduced, so that the actual trajectory of the flake
center of mass remains very close to the support scanline. For example,
springs of $K=2.5$ eV/Å2 yield perpendicular shifts $\leq 0.2$ Å along the
same trajectory. The rotational freedom plus the lateral shifts of the flake
can lead to effectively commensurate contacts even for an incommensurate
stacking, thus explaining the deep energy valleys of the soft-spring pattern
of Fig. 9b, eventually responsible for the stick-slip motion demonstrated by
the lateral force patterns of Fig. 9a. For hard springs constants, locking
into deep energy minima does not occur, but at the same time the flake is
driven into highly repulsive geometries which it cannot dribble. This leads to
higher force peaks, and eventually to a larger static friction.
### 3.4 Angular dependence of friction
As a reference benchmark we consider a nearly rigid flake model, where atoms
are allowed to relax only in the $z$ direction, corresponding to the
$K\to\infty$ limit of the model studied until here, and comparing directly to
the model used by Verhoeven et al. Verhoeven04 . The complete suppression of
angular fluctuations should produce an extremely sharp angular dependency of
the friction force.
Figure 13: (Color online) Friction force $F_{\rm fric}$ as a function of the
fixed stacking angle $\phi$ for three different flake sizes: $24$ atoms (black
solid line), $54$ atoms (red dashed line) and $96$ atoms (blue dot-dashed
line). The simulations are carried out with pulling angle $\theta=0^{\circ}$
and total applied load of $100$ nN. Flake atoms are allowed to relax only
along $z$ direction.
Figure 13 shows the computed static friction force as a function of the fixed
stacking angle $\phi$ for different flakes. We note that friction decreases
with the $\phi$ angle, showing a maximum peak centered at $\phi=0^{\circ}$,
similar to the outcome of previous model calculation Verhoeven04 . As shown by
experiments Dienwiebel04 ; Dienwiebel05 , friction is maximum at an highly
commensurate contact ($\phi=0^{\circ}$) and decreases rapidly as the flake
rotates to incommensurate stackings. The friction peak is sharper for wider
flakes. The sharpest peak for the 96-atom flake is similar to the one
exhibited by the rigid model of Ref. Verhoeven04 . $F_{\rm fric}$ decreases by
nearly one order of magnitude from the high-friction $\phi=0^{\circ}$
commensurate angle to the low-friction $\phi\simeq 30^{\circ}$ incommensurate
one. This drop is smaller than was found by experiment Dienwiebel04 , where it
exceeded significantly one order of magnitude. Also the absolute values of
friction are systematically larger than experiment. Experiment shows friction
peak values near 0.2 nN, while the present model yields a peak value of order
10 nN, 50 times larger. This difference is even larger in the “superlubric”
region near $\phi=30^{\circ}$. These differences are to be ascribed to the
larger load, the adopted short-ranged TB parameterization, and the neglect of
thermal fluctuations, as discussed below.
Figure 14: (Color online) The friction force $F_{\rm fric}$ as a function of
the average rotation angle $\langle\phi_{A}\rangle$ for three different flake
sizes: $24$ atoms (black solid line), $54$ atoms (red dashed line) and $96$
atoms (blue points). Simulations involve a pulling angle $\theta=0^{\circ}$,
total applied load of $100$ nN and soft springs constants $K=0.1$ eV/Å2.
The rigid-flake models, studied here and in previous work Dienwiebel04 do not
look not especially realistic, since in practice a carbon flake does deform,
shift and rotate while interacting with the graphite substrate and the AFM
tip. Figure 14 reports the dependence of friction force $F_{\rm fric}$ on the
average rotation angle $\langle\phi_{A}\rangle$, for a flake whose atoms are
allowed to relax in all directions, for soft spring constants $K=0.1$ eV/Å2.
In all calculations except those of 96-atom flake we use the same angles
$\phi=0^{\circ}$, $5^{\circ}$, $10^{\circ}$, $15^{\circ}$, $20^{\circ}$,
$25^{\circ}$ and $30^{\circ}$, but the possibility of flake rotations allowed
by the soft springs produces significantly different effective average angles
$\langle\phi_{A}\rangle$, especially for the 24-atom flake. Not surprisingly,
with its vast rotational freedom, the 24-atom flake displays an almost
$\phi$-independent, constant friction. For such a small flake with soft
springs, rotations and shifts are so effective to hinder the possibility to
observe any reliable $\phi$-dependency of $F_{\rm fric}$. The 54-atom flake
and, more clearly, the 96-atom flake show average angles nearer to the support
values, with smaller-amplitude fluctuations, and therefore display a friction
curve behavior with a peak at $\langle\phi_{A}\rangle\simeq 0^{\circ}$, fairly
similar to the one obtained in the semi-rigid case and observed in experiment,
and with smaller friction at incommensurate angles. These results suggest that
when the FFM tip happens to bind to a graphene flake constituted by
substantially less than approximately $10^{2}$ atoms, no clear angular
dependency and no superlubric regimes are observed.
Figure 15: (Color online) Comparison of the friction force dependence on the
average rotation angle $\langle\phi_{A}\rangle$ for a 54-atom flake for two
values of springs strength: soft $K=0.1$ eV/Å2 (black solid line) and hard
$K=2.5$ eV/Å2 (red dashed line), plus the semi-rigid case (blue dot-dashed
line). Simulations involve a pulling angle $\theta=0^{\circ}$, total applied
load of $100$ nN.
Figure 15 summarizes the effect of increasing the rigidity of the springs on
the friction dependency on $\langle\phi_{A}\rangle$: the friction peak at a
commensurate arrangement becomes sharper and sharper as the spring rigidity
increases. At variance with the radical changes in $\langle\phi_{A}\rangle$
dependency of the 24-atom flake as a function of the spring constant, for the
54-atom flake the shift-rotational effects become comparably less important,
suggesting that for realistically large flakes in excess of one hundred atoms,
the precise value of the spring stiffness should become irrelevant, as long as
it remains in the $\lesssim 1$ eV/Å2 region.
### 3.5 Load dependency
Figure 16: (Color online) Nonlinear dependency of (a) the friction force and
(b) the friction coefficient $\mu=\frac{F_{\rm fric}}{F_{N}}$ on the total
applied load $F_{N}$, for a commensurate contact ($\phi=0^{\circ}$, black
solid line) and an incommensurate contact ($\phi=15^{\circ}$, red dashed line)
of the 24-atom flake flake. Simulations are carried out for pulling angle
$\theta=0^{\circ}$ and rigid springs, $K=2.5$ eV/Å2.
We come now to study the dependence of the friction force $F_{\rm fric}$ on
the applied load $F_{N}$, exploring a range 20 to 100 nN, matching typical
experiment values Dienwiebel04 ; Sasaki02 . Figure 16 shows the dependence of
the friction force $F_{\rm fric}$ and coefficient $\mu\equiv F_{\rm
fric}/F_{N}$ on the applied load. Hard springs are selected to reduce the
flake shift-rotational effects, in order to focus on the load dependence of
friction and simpler comparison with earlier results. Friction increases with
load, but significant deviations from the linear Coulomb law are observed,
especially for commensurate stacking $\phi=0^{\circ}$. Observe that experiment
found an even weaker dependency of the friction force on load Dienwiebel04 .
Although the data do not point clearly in the direction of a power-law
behavior $F_{\rm fric}\propto F_{N}^{\alpha}$, it is clear that if any such
law was to be estimated, it would have $\alpha<1$. This is at variance with
previous findings for a sharp undeformable tip-surface contact Fusco04 , and
with recent studies of the sliding of hydrogen-passivated carbon Mo09 . The
resulting static friction coefficient approaches the standard macroscopic
value CRC94 of graphite-graphite contact, $\mu=0.1$, while much smaller
values are found in the single-crystal FFM experiments addressed by the
present model. Regardless of the applied load, the flake-substrate distance
being smaller in the model than in real life produces larger absolute values
of friction, and this overestimation is particularly severe at small load.
### 3.6 Scanline dependency
Figure 17: (Color online) (a) Friction force $F_{\rm fric}$ as a function of
the average stacking angle $\langle\phi_{A}\rangle$ for three different
scanlines drawn in panel (b), defined by the three following initial stackings
of the support over the substrate: AB (black solid), AB with a transverse
shift $h=d_{\rm graph}/4$ (red dashed) and AB with transverse shift $h=d_{\rm
graph}/2$ (blue dot-dashed). The simulations involve a 24-atom flake, pulling
angle $\theta=0^{\circ}$, applied load $F_{N}=100$ nN and hard springs of
constants $K=2.5$ eV/Å2.
We investigate the dependence of the friction force versus stacking angle on
the actual scanline followed by the support. Changing scanline determines a
different potential profile seen by the flake, thus modifying the frictional
behavior Fusco04 . Figure 17 reports the friction angular dependency for three
equally spaced scanlines. We carry out simulations for hard springs $K=2.5$
eV/Å2 where scanline effects are the most visible, because of hindered lateral
shifts. For the $h=0$ and $h=d_{\rm graph}/2$ scanlines we find a similar
friction for all values of the stacking angle $\langle\phi_{A}\rangle$, while
the $h=d_{\rm graph}/4$ scanline shows systematically lower friction,
especially for small $\phi$. The reason is that for $\phi_{A}\simeq 0$ along
this special line each flake atom never hits any substrate atom directly on
top, therefore effectively finding a significantly lower corrugation. In
contrast along the two other scanlines, for $\phi=0^{\circ}$ one half of flake
atoms encounters periodically a substrate atom right below its trajectory,
thus finding a high corrugation. Softer springs produce a much weaker
dependence on the scanline: the flake takes advantage of its freedom to
displace laterally, thus following low-corrugation lines (such as $h=d_{\rm
graph}/4$) even when the support pulls it along some nearby parallel line.
Figure 18: Friction force $F_{\rm fric}$ as a function of the pulling angle
$\theta$. Static friction data are obtained by averaging three different
scanlines, defined by initial stacking AB, AB shifted by $d_{\rm graph}/4$ and
AB shifted by $d_{\rm graph}/2$ perpendicular to the pulling direction.
Simulations involve a 24-atom flake with load of $100$ nN, support stacking
angle $\phi=0^{\circ}$ and springs constants $K=2.5$ eV/Å2.
The scanlines of Fig. 17 involves $\theta=0^{\circ}$, i.e. a pulling along the
$x$ direction, where the flake encounters periodic repetitions of the
substrate potential. A different pulling angle affects directly this
periodicity of the problem, in general leading to a nonperiodic profile.
Figure 18 displays the friction force as a function of the pulling angle
$\theta$. Data are averaged over three different scanlines, defined by initial
stacking with $h=0$, $h=d_{\rm graph}/4$ and $h=d_{\rm graph}/2$. Like in
previous calculations Verhoeven04 , we find a minimum friction for pulling
angle $\theta=0^{\circ}$, followed by a fast growth in friction (until
$\theta=10^{\circ}$). We attribute the observed differences between our
results and those by Verhoeven et al. to the different interaction models.
## 4 Discussion and conclusion
We find fair qualitative agreement between the results obtaining by our TB
atomistic model and the existing experimental data, with a few significant
differences. Firstly, our calculations recover the stick-slip behavior of the
lateral forces, characteristic of FFM sliding experiments. In particular, we
find the correct qualitative dependence of stick-slip on the springs stiffness
characterizing the cantilever-tip-flake coupling: soft springs allow for clean
stick-slip behavior, while hard springs inhibit it. Our calculations also
reproduce correctly the friction pattern as a function of the average stacking
angle $\langle\phi_{A}\rangle$ especially as long as the rotational degree of
freedom $\phi_{A}$ is quenched. We also find that for larger flakes, the
fluctuation in $\phi_{A}$ are suppressed automatically anyway, due to the
larger torque exerted by the springs connecting the flake to the tip.
Accordingly, for flakes of sufficiently large size incommensurability produces
significantly less friction, although the friction drop is smaller than in
experiment. In the quantitative comparison between the experimental results
and our model, we find static friction force $F_{\rm fric}$ and coefficient
$\mu$ systematically at least one order of magnitude larger than experiment,
this difference being especially significant in the incommensurate
configurations where no proper superlubric regime is observed. These and other
quantitative discrepancies are to be attributed to: (i) The reduced interlayer
equilibrium distance, related to the small cut-off distance of the present TB
parameterization, which is responsible for the increased energy corrugation
experienced by our model flake with respect to real graphene on graphite. (ii)
The extra reactivity of the isolated model graphene flake with respect to a
real one, which is bond to the AFM tip and thus somewhat passivated;
accordingly, especially the atoms at the flake border, show a greater tendency
to react with substrate atoms, thus increasing friction. (iii) The neglect of
thermally-activated slips through energy barriers Riedo03 ; Gnecco03 : this
neglect generates an overestimation of friction especially where these
barriers are lower, i.e. at incommensurate stackings. Indeed the current
understanding Frenken09conf of the observed Dienwiebel04 superlubric sliding
involves thermolubricity associated to a high attempt rate for overcoming the
corrugation barriers due to the microscopic mass of the vibrating tip apex. If
the experiment of Ref. Dienwiebel04 could be repeated at the much lower
temperature of a few degree Kelvin, the observed friction values and
dependency on the $\Phi$ angle would probably look much more similar to the
one obtained in the present model.
Calculations carried out with comparably soft springs and small flakes
($N_{\rm fl}\leq 54$) show that the flake shift-rotational freedom increases
friction for incommensurate stackings (by allowing the flake to explore
deeper-bound minima) and decreases it for commensurate ones (by allowing the
flake to dribble the top barriers): the result is a substantial flattening of
the dependency of the friction static force $F_{\rm fric}$ on the stacking
angle $\phi$. Harder springs (e.g. $K=2.5$ eV/Å2) would suppress the flake
freedom to rotate and shift laterally but are incompatible with the clear
stick-slip behavior observed experimentally. These considerations confirm that
the size of the flakes showing superlubric sliding in actual FFM experiments
is large $N_{\rm fl}\geq 96$. Many discrepancies with experiment would
probably be disposed of, if a longer ranged interatomic interaction was
employed, for example a TB model based on a longer cutoff
Papaconstantopoulos98 . This way, a much weaker flake-surface interaction
would effectively correspond to comparably stronger tip-flake interaction,
thus a significant hindering of rotations and translations even with a
realistically weak tip-flake coupling $K\leq 0.5$ eV/Å2. If accurate long-
range C-C interactions up to distances of the order of 1 nm were present in
the force field, one could even include substrate relaxation to study an even
more realistic model. Such an improved model would however imply significantly
larger computational workload, especially if thermal excitations were also
included. Further research should also investigate the effect of the presence
of structural defects in the flake or in the substrate, as proposed by Guo et
al. Guo07 , and the effect of flake shape.
## Acknowledgments
We are grateful to J. Frenken and R. Buzio for useful discussion. We
acknowledge financial support by the project MIUR-PON ”CyberSar”, and by the
Italian National Research Council (CNR) through contract
ESF/EUROCORES/FANAS/AFRI.
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|
arxiv-papers
| 2009-04-22T13:05:14 |
2024-09-04T02:49:02.066573
|
{
"license": "Public Domain",
"authors": "Federico Bonelli, Nicola Manini, Emiliano Cadelano and Luciano Colombo",
"submitter": "Emiliano Cadelano",
"url": "https://arxiv.org/abs/0904.3456"
}
|
0904.3493
|
# surfaces with central convex cross-sections
Bruce Solomon Math Department, Indiana University, Bloomington IN 47405
solomon@indiana.edu mypage.iu.edu/$\sim$solomon
###### Abstract.
Say that a surface in $\,S\subset\mathbf{R}^{3}\,$ has the _central plane oval
property, or cpo_, if
* •
$S\,$ meets some affine plane transversally along an oval, and
* •
Every such transverse plane oval on $\,S\,$ has central symmetry.
We show that a complete, connected $\,C^{2}\,$ surface with cpo must either be
a cylinder over a central oval, or else _quadric_.
We apply this to deduce that a complete $\,C^{2}\,$ surface containing a
transverse plane oval but no skewloop, must be cylindrical or quadric.
###### Key words and phrases:
Quadric surface, oval, central symmetry, skewloop
## 1\. Introduction and overview
Call a set in a euclidean space _central_ if it has symmetry with respect to
reflection through a point—its _center_. Call an embedded plane loop an _oval_
if its curvature never vanishes.
Figure 1. Ovals and their centrices (see §2.3). Only the rightmost oval is
central.
If we erect a cylinder over a central oval in $\,\mathbf{R}^{3}\,$, its
transverse planar cross-sections, whenever compact, will be central ovals too.
The same goes for _quadrics_ —level-sets of a quadratic polynomials on
$\,\mathbf{R}^{3}\,$: Their transverse planar cross-sections, when compact,
are always ellipses, which are certainly central ovals.
We show here that these two kinds of examples provide the _only_ complete
$\,C^{2}\,$ surfaces in $\,\mathbf{R}^{3}\,$ whose planar ovals are all
central. We will call this the _central plane oval_ property and abbreviate it
by _cpo_ :
###### Definition 1.1 (cpo).
A $\,C^{2}$-immersed surface $\,S\subset\mathbf{R}^{3}\,$ has the _central
plane oval property_ , or _cpo_ , if
* •
$S\,$ intersects _at least one_ affine plane transversally along an oval, and
* •
_Every_ time $\,S\,$ intersects an affine plane transversally along an oval,
that oval is _central_
Given this terminology, we can state our main result as follows:
Theorem 5.2 (Main Theorem) _A complete, connected $\,C^{2}$-immersed surface
in $\,\mathbf{R}^{3}\,$ with cpo is either a cylinder, or quadric._
This result complements a fundamentally local fact about _convex_ surfaces
proven long ago by W. Blaschke in [Bl]:
###### Proposition 1.2 ([Bl, 1918]).
Suppose every plane transverse, and nearly tangent to, a smooth _convex_
surface $\,S\subset\mathbf{R}^{3}\,$ cuts $\,S\,$ along a central loop. Then
$\,S\,$ is quadric.
Though it resembles—and helped to inspire—our Main Theorem above, Blaschke’s
result seems much easier to prove, for the simple reason that convex surfaces
lie on one side of their tangent planes. By pushing such a plane slightly into
the surface, one always cuts it in a small convex loop. Blaschke merely
observed that when all such loops are _central_ , one can Taylor-expand the
surface as a graph over any tangent plane with no cubic term. This annihilates
the Pick invariant on the surface, making it quadric.
Contrastingly, our Theorem allows some, or even all of the surface, to have
negative Gauss curvature. In a negatively curved region, one _never_ finds
arbitrarily small planar ovals, and this totally blocks any direct
generalization of Blaschke’s argument—as he himself laments in [Bl].
We thus find it necessary to approach Theorem 5.2 using a global, multi-stage
argument that ultimately rests on the rotationally symmetric case. We
published the latter result in [S]:
###### Proposition 1.3 ([S]).
Let $\,M\,$ be a surface of revolution. If $\,M\,$ intersects every plane
nearly perpendicular to its axis in a central set, then $\,M\,$ is quadric.
The fundamental problem we must solve to get from this basic result to our
Main Theorem boils down to the case of a general “tube”. For suppose an
immersed surface $\,M\,$ meets some plane transversally along an oval as our
definition of cpo requires. Then some neighborhood, in $\,M\,$, of that oval
embeds into $\,\mathbf{R}^{3}\,$ as a roughly cylindrical tube with cpo. Such
tubes turn out to form the critical test case for our work. To explain
further, we need some precise language.
Let $\,I:=(-1,1)\,$ denote the open unit interval.
###### Definition 1.4 (Transversely convex tube).
Suppose $\,X:\mathbf{S}^{1}\times I\to\mathbf{R}^{3}\,$ is an embedding of the
form
$X(\theta,z):=\bigl{(}\mathbf{c}(z)+\gamma(z;\theta),\,z\bigr{)}\,,$
where $\,\mathbf{c}:I\to\mathbf{R}^{2}\,$ and
$\,\gamma:I\times\mathbf{S}^{1}\to\mathbf{R}^{2}\,$ are $\,C^{2}\,$, and for
each fixed $\,z\in I\,$, the map
$\,\gamma(z;\cdot):\mathbf{S}^{1}\to\mathbf{R}^{2}\,$ parametrizes a plane
oval having its centroid at the origin.
A transversely convex tube is any embedded annulus that, after an affine
isomorphism, can be parametrized in this way. We call $\,\mathbf{c}\,$ its
central curve. When studying a transversely convex tube, we lose no generality
by assuming it to lie in the slab $\,|z|<1\,$ as parametrized above, and we
will routinely do so without further comment.
Discarding the central curve $\,\mathbf{c}\,$ of a transversely convex tube
$\,\mathcal{T}\,$ in standard position, we get the rectification
$\,\mathcal{T}\,$, denoted $\,\mathcal{T}^{*}\,$, and given by the image of
$X^{*}(\theta,z):=\bigl{(}\gamma(z;\theta),\,z\bigr{)}$
(Figure 2). Finally, we say that $\,\mathcal{T}^{*}\,$ splits when
$\gamma(z;\theta)=r(z)\,\gamma(\theta)$
for some fixed oval $\,\gamma:\mathbf{S}^{1}\to\mathbf{R}^{2}\,$, and some
positive scaling function $\,r:I\to(0,\infty)\,$. Note that a split tube is a
surface of revolution precisely when$\,\gamma\,$ parametrizes an origin-
centered circle.
Figure 2. A transversely convex tube $\,\mathcal{T}\,$ (left) and its
rectification $\,\mathcal{T}^{*}\,$ (right).
In these terms, we reach a key analytical juncture in our work when we prove
the following technical result:
Proposition 3.10. (Splitting Lemma) _If a transversely convex tube
$\,\mathcal{T}\,$ in standard position has cpo, then its rectification
$\,\mathcal{T}^{*}\,$ splits._
Simple as this statement is, proving it was the most challenging part of our
work. Much of the effort goes toward deriving a pair of partial differential
equations satisfied by the function $\,h:\mathbf{S}^{1}\times
I\to\mathbf{R}\,$ which, for each $\,z\in I\,$, yields the support function
$\,h(z,\cdot)\,$ of the oval $\,\gamma(z;\cdot)\,$—the height-$z$ cross-
section of the rectified tube $\,\mathcal{T}^{*}\,$. These PDE’s form the
conclusion of Proposition 3.9, and we devote most of §3 to their derivation.
Our approach has a variational flavor that we sketch out at the beginning of
§3.
We then obtain our Splitting Lemma by playing these PDE’s off against each
other. Specifically, we use information gleaned from the second equation to
rewrite the first as an equation for the _square_ of $\,h\,$. We then notice a
first integral for that equation, and finally prove splitting with the help of
ODE techniques in which the second equation again plays a role.
Once we have Splitting, we return again to the first PDE from Proposition 3.9,
where we can now separate variables. This yields independent elementary ODE’s
for the horizontal and vertical behavior of our tube. Solving these, we reach
the key geometric turning point of our work: We find that the possibilities
for a tube with cpo branch in two directions:
Proposition 3.11. (Cylinder/Quadric) Suppose $\,\mathcal{T}\,$ is a
transversely convex tube with cpo. Then its rectification
$\,\mathcal{T}^{*}\,$ is either
* (i)
The cylinder over a central oval, or
* (ii)
Affinely congruent to a surface of revolution.
By Proposition 1.3, however, surfaces of revolution having cpo are already
quadric. So we now see that, insofar as tubes go, it remains only to eliminate
the rectification step. We do this in §4 by proving
Proposition 4.1. (Axis lemma) _Suppose $\,\mathcal{T}\,$ is a transversally
convex tube with cpo. Then its central curve is affine, so that
$\,\mathcal{T}\,$ is affinely congruent to its rectification
$\,\mathcal{T}^{*}\,$._
Together, the Cylinder/Quadric Proposition, Axis Lemma, and rotationally
invariant case (Proposition 1.3) combine to show that a transversely convex
tube with cpo is either cylindrical or quadric. In other words, we have a
“tubular” version of our Main Theorem:
Proposition 5.1 (Collar Theorem) A transversely convex tube with cpo is either
cylindrical or quadric.
In §5, we start with this fact, and show that it “propagates,” using an
open/closed argument, to any complete $\,C^{2}\,$ immersion with cpo. This
proves our Main Theorem 5.2, and the argument is not difficult. For as we
mentioned above, any surface $\,M\,$ with cpo contains an annular subset that
embeds in $\,\mathbf{R}^{3}\,$ as a transversely convex tube. Our Collar
Theorem now makes that tube either cylindrical or quadric. But the boundaries
of such a tube, in either case, are again transverse central ovals. So they
too have annular neighborhoods that embed as transversely convex tubes.
Roughly speaking, this pushes the boundaries of the tube a little further out
along $\,M\,$, and by completeness, the process terminates only when the tube
engulfs all of $\,M\,$.
We conclude in §6, with an application that first motivated us toward the Main
Theorem here: We extend the main result from our earlier paper with M. Ghomi
on skewloops [GS].
A _skewloop_ is a smoothly immersed loop in $\,\mathbf{R}^{3}\,$ with no pair
of distinct parallel tangent lines. In [GS], we showed that _when a complete
$C^{2}$-immersed surface in $\,\mathbf{R}^{3}\,$ has a point of positive
curvature, it contains a skewloop if and only if it is not quadric._ We
required the positive curvature assumption because our proof cited Proposition
1.2 above (Blaschke’s theorem) in an essential way. The Main Theorem here lets
us bypass that result, eliminating the positive curvature assumption in favor
of one that holds for many surfaces with _no_ positive curvature: the
existence of a single transverse planar oval. We thus obtain
Theorem 6.5. Suppose a $C^{2}$-immersed surface $\,M\subset\mathbf{R}^{3}\,$
crosses some plane transversally along an oval. Then exactly one of the
following holds:
* (i)
$S\,$ contains a skewloop.
* (ii)
$S\,$ is the cylinder over an oval.
* (iii)
$S\,$ is a non-cylindrical quadric.
For instance, this result characterizes the tube (i.e. one-sheeted)
hyperboloids as _the only negatively curved surfaces that contain a transverse
plane oval, but no skewloop_.
We now proceed from the overview above to the details of our paper, starting
with some preliminary facts about ovals.
## 2\. Oval and Centrix
Recall that by an _oval_ in the plane, we mean an embedded, _strictly_ convex
$\,C^{2}\,$ loop, and that a _central oval_ has central symmetry—symmetry with
respect to reflection through a point called its _center_.
###### Definition 2.1 (Support parametrization/support function).
A map $\,\gamma:\mathbf{S}^{1}\to\mathbf{R}^{2}\,$ _support parametrizes_ an
oval $\,\mathcal{O}\subset\mathbf{R}^{2}\,$ if and only if it satisfies
(2.1)
$\gamma^{\prime}(\theta)=\left|\gamma^{\prime}(\theta)\right|\mathbf{i}\,e^{\mathbf{i}\theta}\quad\text{for
all $\,\theta\in\mathbf{R}\,$}\,.$
Here we have identified $\,\mathbf{C}\approx\mathbf{R}^{2}\,$, and we regard
$\,2\pi$-periodic maps $\,\mathbf{R}\to\mathbf{R}^{2}\,$ as maps from
$\,\mathbf{S}^{1}\,$ to $\,\mathbf{R}^{2}\,$, in the obvious ways. We use
these identifications without further comment below.
Notice that (2.1) characterizes parametrization by the inverse of the outer
unit normal. This is a diffeomorphism $\,\mathcal{O}\to\mathbf{S}^{1}\,$ on
any $\,C^{2}\,$ oval $\,\mathcal{O}$, a fact that yields both existence and
uniqueness of the support parametrization.
By an easy exercise, the _support function_ $\,h:\mathbf{R}\to\mathbf{R}\,$,
given by
(2.2) $h(\theta):=\sup_{p\in\mathcal{O}}p\cdot e^{\mathbf{i}\theta}\,,$
determines $\,\gamma\,$ via the formula
(2.3)
$\gamma(\theta)=\left(h(\theta)+\mathbf{i}\,h^{\prime}(\theta)\right)\,e^{\mathbf{i}\theta}\,.$
Note that when we rotate an oval $\,\mathcal{O}\,$ counterclockwise through an
angle $\,\phi\,$ about the origin, (2.2) shifts its support function right by
$\,\phi\,$:
(2.4) $h(\theta)\mapsto h(\theta-\phi)\,.$
Elementary calculations using (2.3) further show that the support
parametrization makes speed and curvature reciprocal to each other:
(2.5)
$\left|\gamma^{\prime}(\theta)\right|=h(\theta)+h^{\prime\prime}(\theta)\qquad\text{and}\qquad\kappa(\theta)=\frac{1}{h(\theta)+h^{\prime\prime}(\theta)}\,.$
In particular, strict convexity of an oval ensures that its support
parametrization _immerses_ the circle into $\,\mathbf{R}^{2}\,$.
We eventually want to show that the cross-sectional ovals of a tube with cpo
are circular up to affine isomorphism—ellipses. We will do so by invoking
###### Observation 2.2.
An oval is an origin-centered ellipse if and only if its support function
$\,h\,$ satisfies
$\left(h^{2}\right)^{\prime\prime\prime}+4\left(h^{2}\right)^{\prime}=0\,.$
###### Proof.
We may parametrize any origin-centered ellipse by
$\alpha(t)=A\,e^{\mathbf{i}\,t}$
for some symmetric invertible matrix $\,A_{2\times 2}\,$. In that case, (2.2)
computes its support function as
$h(\theta)=\sup_{t}\,A\,e^{\mathbf{i}\,t}\cdot e^{\mathbf{i}\theta}\
=\sup_{t}\,e^{\mathbf{i}\,t}\cdot Ae^{\mathbf{i}\theta}\,.$
This supremum here clearly occurs when
$e^{\mathbf{i}\,t}=\frac{Ae^{\mathbf{i}\theta}}{\left|Ae^{\mathbf{i}\theta}\right|}\,,$
which instantly yields $\,h(\theta)=\left|Ae^{\mathbf{i}\theta}\right|\,$.
Familiar trig identities then make it easy to deduce
(2.6) $h^{2}(\theta)=a\cos(2\theta+b)+c>0\,,$
for some constants $\,a,b\,$ and $\,c\,$, with $\,|a|<c\,$, and the positive
solutions of $\,f^{\prime\prime\prime}+4f^{\prime}=0\,$ are precisely the
functions given by (2.6). ∎
Geometrically, (2.6) characterizes the support function of an ellipse with
major and minor axes $\,\sqrt{c\pm a}\,$.
### 2.3. The centrix.
We measure the failure of an oval to be centrally symmetric by examining the
auxilliary curve that we call its _centrix_ :
###### Definition 2.4 (Centrix).
Given an oval $\,\mathcal{O}\subset\mathbf{R}^{2}\,$ and a unit vector
$\,e^{\mathbf{i}\theta}\in\mathbf{S}^{1}\,$, there exist exactly two points on
$\,\mathcal{O}\,$ with tangent lines perpendicular to
$\,e^{\mathbf{i}\theta}\,$. We call the line segment joining these two points
the _$\theta$ -diameter_ of $\,\mathcal{O}\,$. Denoting its midpoint by
$\,\mathbf{c}(\theta)\,$, we then call the image of the resulting map
$\,\mathbf{c}:\mathbf{S}^{1}\to\mathbf{R}^{2}\,$ the _centrix_ of
$\,\mathcal{O}\,$.
Figure 3. Midpoints of diameters trace out the centrix.
###### Definition 2.5 (Even/odd).
Given the support parametrization $\,\gamma\,$ of an oval $\,\mathcal{O}\,$,
we call the maps
${\textstyle{\frac{1}{2}}}\,\left(\gamma(\theta)+\gamma(\theta+\pi)\right)\quad\text{and}\quad{\textstyle{\frac{1}{2}}}\,\left(\gamma(\theta)-\gamma(\theta+\pi)\right)\,,$
the _even_ and _odd_ parts of $\,\gamma\,$ respectively.
###### Observation 2.6.
The centrix $\,\mathbf{c}:\mathbf{S}^{1}\to\mathbf{R}^{2}\,$ of
$\,\mathcal{O}\,$ coincides with the even part of $\,\gamma\,$. It is a
constant if and only if $\,\mathcal{O}\,$ has central symmetry. In that case,
the odd part of $\,\gamma\,$ support-parametrizes the origin-centered oval
$\,\mathcal{O}-\mathbf{c}\,$.
###### Proof.
The defining condition for the support parametrization (2.1) puts the
endpoints of each $\theta$-diameter on $\,\mathcal{O}\,$ at
$\,\gamma(\theta)\,$ and $\,\gamma(\theta+\pi)\,$. It follows immediately that
the even part of $\,\gamma\,$ parametrizes the centrix$\,\mathbf{c}\,$.
When $\,\mathbf{c}(\theta)\equiv\mathbf{c}_{0}\in\mathbf{R}^{2}\,$, reflection
through $\,\mathbf{c}_{0}\,$ clearly preserves $\,\mathcal{O}\,$.
Conversely, if reflection through some point $\,\mathbf{c}_{0}\,$ preserves
$\,\mathcal{O}\,$, it—like any affine isomorphism—must preserve pairs of
parallel lines. In particular, it will swap the endpoints of each
$\theta$-diameter, preserving their midpoints. But reflection through
$\,\mathbf{c}_{0}\,$ preserves no other point. So central symmetry means
$\,\mathbf{c}(\theta)\equiv\mathbf{c}_{0}\,$.
The even and odd parts of $\,\gamma\,$ always add back to $\,\gamma\,$. So
when $\,\mathcal{O}\,$ is central, the odd part $\,\gamma^{*}\,$ clearly
parametrizes $\,\mathcal{O}-\mathbf{c}\,$, whose center of symmetry obviously
lies at the origin. In this case, we also have
$\,(\gamma^{*})^{\prime}(\theta)=\gamma^{\prime}(\theta)\,$, a multiple of
$\,\mathbf{i}\,e^{\mathbf{i}\theta}\,$. It follows that (2.1) must hold for
$\,\gamma^{*}\,$, which makes it a support parametrization. ∎
## 3\. Splitting
In this section we tackle the technical key to our Main Theorem, establishing
that cpo forces the support function of a transversely convex tube to split
along purely horizontal and vertical factors. Our Splitting Lemma 3.10 states
this precisely, and the geometric consequence that makes it interesting, our
Cylinder/Quadric Proposition 3.11, then follows fairly easily.
To prepare for the Splitting Lemma, we need calculations that stretch over a
number of pages. We hope the following descriptive plan-of-attack will help
the reader navigate them with a clear sense of our intentions.
Our strategy is to focus on the families of ovals one gets by intersecting a
transversely convex tube $\,\mathcal{T}\,$ with planes tilted slightly away
from the horizontal. Specifically, given any $\,\varepsilon\in\mathbf{R}\,$
and any unit-vector $\,\tau\in\mathbf{S}^{1}\,$, we consider the
$\varepsilon$-tilted plane given by
(3.1)
$P_{\tau,b}(\varepsilon):=\left\\{(p,z)\in\mathbf{R}^{2}\times\mathbf{R}:\
z=\varepsilon\,(p\cdot\tau)+b\right\\}.$
We call $\,\tau\,$ the tilt-direction, $\,b\,$ the $z$-intercept, and
$\,\varepsilon\,$ the slope of this plane. Fixing $\,\tau\in\mathbf{S}^{1}\,$
and $\,b\in(-1,1)\,$, we vary the slope $\,\varepsilon\,$ of this plane, and
study the resulting intersections with $\,\mathcal{T}\,$ near
$\,\varepsilon=0\,$.
Since $\,\mathcal{T}\,$ is transversely convex, it intersects _horizontal_
planes in $\,C^{2}\,$ ovals. By transversality, the cross-section
$\,P_{\tau,b}(\varepsilon)\cap\mathcal{T}\,$ remains a $\,C^{2}\,$ oval for
all sufficiently small $\,\varepsilon\,$. When we assume that
$\,\mathcal{T}\,$ has cpo, these ovals all have central symmetry too.
Our key idea is to study the _centrices_ of these cross-sections. The
preservation of central symmetry makes them all singletons, by Observation
2.6—they are independent of the variable $\,\theta\,$ along each oval.
Differentiation with respect to $\,\theta\,$ therefore yields a vanishing
condition. By taking an initial $\varepsilon$-derivative of this condition at
$\,\varepsilon=0\,$, we produce the two partial differential equations of
Proposition 3.9. As explained in our introduction, these equations lead fairly
directly to our Splitting Lemma.
We now work out the details of this program.
### 3.1. The support map of $\,\mathcal{T}\,$.
As above, we let $\,\mathcal{T}\,$ denote a transversely convex tube in
standard position. By Definition 1.4 $\,\mathcal{T}\,$ intersects the
horizontal plane at any height $\,b\in(-1,1)\,$ in an oval we shall call
$\,\mathcal{O}(b)\,$. Denote by $\,\nu:\mathcal{T}\to\mathbf{S}^{1}\,$ the map
that assigns to each point $\,p=(x,y,z)\in\mathcal{T}\,$ the (horizontal)
outer unit normal to $\,\mathcal{O}(z)\,$ at $\,p\,$. Clearly, the map
$\mathcal{T}\to\mathbf{S}^{1}\times(-1,1)\quad\text{given by}\quad
p\longmapsto\left(\nu(p),\ z(p)\right)\,.$
is a diffeomorphism, whose inverse takes the form
(3.2)
$\left(e^{\mathbf{i}\theta},\,z\right)\longmapsto\left(\Gamma(\theta,z),\,z\right)$
for some smooth map $\,\Gamma:\mathbf{S}^{1}\times(-1,1)\to\mathbf{R}^{2}\,$.
Indeed, $\,\Gamma\,$ reparametrizes $\,\mathcal{T}\,$, and for fixed
$\,b\in(-1,1)\,$, it inverts the unit normal map on $\,\mathcal{O}(b)\,$. As
mentioned following Definition 2.1, this means that $\,\Gamma(\cdot,b)\,$
support-parametrizes $\,\mathcal{O}(b)\,$, and for this reason, we call it the
_support map_ of the tube $\,\mathcal{T}\,$.
### 3.2. The height function $\zeta\,$
We now take an arbitrary intercept $\,-1<b<1\,$ and tilt direction
$\,\tau\in\mathbf{S}^{1}\,$, and regard them, for now, as fixed.
Define the cross-section
$\bar{\mathcal{O}}(b,\varepsilon):=\mathcal{T}\cap P_{\tau,b}(\varepsilon)\,,$
and its image under the projection
$\,(x,y,z)\stackrel{{\scriptstyle\pi}}{{\mapsto}}(x,y)\,$,
$\mathcal{O}(b,\varepsilon):=\pi\left(\bar{\mathcal{O}}(b,\varepsilon)\right)\,.$
We abbreviate the _horizontal_ ($\varepsilon=0$) cross-section by
$\mathcal{O}(b):=\bar{\mathcal{O}}(b,0)\,,$
and we will not hesitate to identify $\,\mathcal{O}(b)\,$ with
$\,\mathcal{O}(b,0)\,$ too, since the latter is clearly congruent to
$\,\bar{\mathcal{O}}(b,0)\,$.
As discussed above, the transverse convexity of $\,\mathcal{T}\,$ ensures that
$\,\bar{\mathcal{O}}(b,\varepsilon)\,$ is an oval for all sufficiently small
$\,\varepsilon\,$. When $\,\mathcal{T}\,$ has cpo, these tilted ovals will
clearly have central symmetry as well, but we need not assume cpo for our
immediate goal here: We want to introduce and study the “height function”
$\,\zeta(\varepsilon,\theta)\,$ that lets us parametrize
$\,\mathcal{O}(b,\varepsilon)\,$ by the map (compare (3.2))
(3.3)
$\theta\longmapsto\Bigl{(}\Gamma\left(\theta,\,\zeta(\varepsilon,\theta)\right),\
\zeta\left(\varepsilon,\theta\right)\Bigr{)}\,.$
The Implicit Function Theorem ensures the existence and $\,C^{2}\,$ smoothness
of $\,\zeta\,$. For suppose—informed by the characterization of
$\,P_{\tau,b}(\varepsilon)\,$ in (3.1)—we define a map
$\,G:\mathbf{R}\times\mathbf{S}^{1}\times(-1,1)\to\mathbf{R}\,$ via
(3.4)
$G(\varepsilon,\theta,\zeta):=\zeta-b-\varepsilon\,\tau\cdot\Gamma(\theta,\zeta)\,.$
Then $\,G\,$ inherits $\,C^{2}\,$ smoothness from $\,\Gamma\,$, and the pre-
image of $\,\mathcal{O}(b,\varepsilon)\,$ in $\,\mathbf{S}^{1}\times(-1,1)\,$
under the parametrization of $\,\mathcal{T}\,$ in (3.2) clearly solves
$G(\varepsilon,\theta,\zeta)=0\,.$
On the _horizontal_ oval $\,\mathcal{O}(b)\,$, we have $\,\zeta\equiv b\,$, so
that trivially,
$G(0,\theta,b)\equiv 0\quad\text{and}\quad\frac{\partial
G}{\partial\zeta}\left(0,\theta,b\right)=1\neq 0\quad\text{for all
$\,\theta\in\mathbf{S}^{1}\,$.}$
The Implicit Function Theorem then provides a $\,\delta>0\,$, and a
$\,C^{2}\,$ mapping
$\,\zeta:(-\delta,\delta)\times\mathbf{S}^{1}\to\mathbf{R}\,$ that satisfies
(3.5) $\zeta(0,\theta)\equiv b\quad\text{for all
$\,\theta\in\mathbf{S}^{1}\,$}\,,$
and
$G\left(\varepsilon,\theta,\zeta(\varepsilon,\theta)\right)\equiv
0\quad\text{for all $\,\theta\in\mathbf{S}^{1},\,|\varepsilon|<\delta\,$.}$
Written out using (3.4), the latter equation becomes
(3.6)
$\zeta(\varepsilon,\theta)=b+\varepsilon\,\tau\cdot\Gamma\left(\theta,\,\zeta(\varepsilon,\theta)\right),$
which shows that, as hoped, (3.3) parametrizes
$\,\mathcal{O}(b,\varepsilon)\,$.
Now observe that the projection
$\,(x,y,z)\stackrel{{\scriptstyle\pi}}{{\to}}(x,y)\,$ induces an affine
isomorphism $\,P_{\tau,b}(\varepsilon)\approx\mathbf{R}^{2}\,$. Such maps
preserve strict convexity, so that $\,\mathcal{O}(b,\varepsilon)\,$, and of
course $\,\mathcal{O}(b)\,$, are again ovals.
For future reference, we note that affine isomorphisms also preserve central
symmetry. So when $\,\mathcal{T}\,$ has cpo, the projected oval
$\,\mathcal{O}(b,\varepsilon)\,$ further inherits the central symmetry that
cpo ascribes to $\,\bar{\mathcal{O}}(b,\varepsilon)\,$.
In any case, it will suffice henceforth to study the projected ovals
$\,\mathcal{O}(b,\varepsilon)\,$ as it varies with $\,\varepsilon\,$. In view
of (3.3), we may clearly parametrize $\,\mathcal{O}(b,\varepsilon)\,$ by the
immersion
(3.7) $\theta\longmapsto\Gamma\left(\theta,\,\zeta(\varepsilon,\theta)\right)\
.$
To analyze the initial variation of the centrix of
$\,\mathcal{O}(b,\varepsilon)\,$, we will eventually requires following facts
about the derivatives of $\,\zeta\,$. The reader will easiliy confirm them by
differentiating (3.6) implicitly, and using (3.5):
###### Observation 3.3.
We have
$\frac{\partial\zeta}{\partial\varepsilon}(0,\theta)=\tau\cdot\Gamma\left(\theta,\,b\right)$
and
$\frac{\partial^{2}\zeta}{\partial\varepsilon\,\partial\theta}\left(0,\theta\right)=\tau\cdot\frac{\partial{\Gamma}}{\partial\theta}\left(\theta,b\right)\
.$
### 3.4. The support-reparametrizing map $\,\theta_{\varepsilon}\,$
Though (3.7) parametrizes $\,\mathcal{O}(b,\varepsilon)\,$, we want to study
the _centrix_ of $\,\mathcal{O}(b,\varepsilon)\,$. Observation 2.6 offers a
way to parametrize the centrix, but it derives from the _support_
parametrization of $\,\mathcal{O}(b,\varepsilon)\,$, not the one given by
(3.7). The Proposition below details the needed reparametrization, and its
final conclusion yields a crucial input to our proof of the Splitting Lemma
3.10. Notation is as above.
###### Proposition 3.5.
There exists a $\,\delta>0\,$ and a differentiable 1-parameter family of
diffeomorphisms
$\theta_{\varepsilon}:\mathbf{S}^{1}\to\mathbf{S}^{1}\quad-\delta<\varepsilon<\delta\,,$
such that the composition
$\Gamma_{\varepsilon}\circ\theta_{\varepsilon}=\Gamma\left(\theta_{\varepsilon},\,\zeta(\varepsilon,\theta_{\varepsilon})\right)$
support-parametrizes $\,\mathcal{O}(b,\varepsilon)\,$ for each
$\,\varepsilon\in(-\delta,\delta)\,$. The initial map $\,\theta_{0}\,$ is the
identity on $\,\mathbf{S}^{1}\,$, with initial $\varepsilon$-derivative given
by
$\frac{d\theta_{\varepsilon}}{d\varepsilon}\Big{|}_{\varepsilon=0}=\left(\tau\cdot\mathbf{i}e^{\mathbf{i}\theta}\right)\left(\frac{\partial\Gamma}{\partial\zeta}(\theta,b)\cdot
e^{\mathbf{i}\theta}\right)\,.$
###### Proof.
The existence of $\,\theta_{\varepsilon}\,$ is routine. For,
$\,\Gamma\left(\theta,\zeta(\varepsilon,\theta)\right)\,$ parametrizes
$\,\mathcal{O}(b,\varepsilon)\,$, and is $\,C^{2}\,$ in both $\,\theta\,$ and
$\,\varepsilon\,$. This makes the unit outer normal
$\,\nu_{\varepsilon}(\theta)\,$ on $\,\mathcal{O}(b,\varepsilon)\,$
continuously differentiable in both variables, while the strict convexity of
$\,\mathcal{O}(b,\varepsilon)\,$ ensures that $\,\nu_{\varepsilon}\,$ induces
a diffeomorphism $\,\mathbf{S}^{1}\to\mathbf{S}^{1}\,$ that varies smoothly
with $\,\varepsilon\in(-\delta,\delta)\,$. By the Inverse Function Theorem,
the inverse of this map varies smoothly in $\,\varepsilon\,$ too. As noted
after Definition 2.1, however, the inverse of the outer normal on an oval
gives its support parametrization. We therefore get the desired family of
reparametrizing maps by setting
$\,\theta_{\varepsilon}:=(\nu_{\varepsilon})^{-1}\,$ for each
$\,|\varepsilon|<\delta\,$.
Note too that by (3.5), setting $\,\varepsilon=0\,$ reduces
$\,\Gamma\left(\theta,\zeta(\varepsilon,\theta)\right)\,$ to
$\,\Gamma(\theta,b)\,$, which already support-parametrizes
$\,\mathcal{O}(b)\,$, by definition of $\,\Gamma\,$. So $\,\theta_{0}\,$ is
the trivial reparametrization—the identity map—as claimed.
It remains to verify the stated formula for
$\,\partial\theta_{\varepsilon}/\partial\varepsilon\,$ at $\,\varepsilon=0\,$.
This requires some careful calculations.
Start by observing that since
$\,\Gamma_{\varepsilon}\circ\theta_{\varepsilon}\,$ support-parametrizes
$\,\mathcal{O}(b,\varepsilon)\,$ when $\,|\varepsilon|<\delta\,$. By (2.1),
this makes its velocity at any input $\,\theta\,$ a multiple of
$\,\mathbf{i}e^{\mathbf{i}\theta}\,$. Hence
$0\equiv
e^{\mathbf{i}\theta}\cdot\frac{\partial}{\partial\theta}\left(\Gamma_{\varepsilon}\circ\theta_{\varepsilon}\right)\,.$
Use the chain rule to expand the derivative, abbreviating
$\,\theta_{\varepsilon}(\theta)\,$ as simply $\,\theta_{\varepsilon}\,$, to
rewrite this condition as
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
e^{\mathbf{i}\theta}\cdot\frac{\partial}{\partial\theta}\,\Gamma\bigl{(}\theta_{\varepsilon},\,\zeta(\varepsilon,\,\theta_{\varepsilon})\bigr{)}$
$\displaystyle=$ $\displaystyle
e^{\mathbf{i}\theta}\cdot\left[\frac{\partial\Gamma}{\partial\theta}\bigl{(}\theta_{\varepsilon},\zeta(\varepsilon,\theta_{\varepsilon})\bigr{)}+\frac{\partial\Gamma}{\partial\zeta}\bigl{(}\theta_{\varepsilon},\zeta(\varepsilon,\theta_{\varepsilon})\bigr{)}\frac{\partial\zeta}{\partial\theta}\left(\varepsilon,\theta_{\varepsilon}\right)\right]\frac{\partial\theta_{\varepsilon}}{\partial\theta}$
Since $\,\theta_{\varepsilon}\,$ is a diffeomorphism of $\,\mathbf{S}^{1}\,$,
its derivative along the circle never vanishes. So we can divide out the final
factor above and conclude that for all $\,|\varepsilon|<\delta\,$, we have
(3.8)
$\frac{\partial\zeta}{\partial\theta}\left(\varepsilon,\theta_{\varepsilon}\right)\,\frac{\partial\Gamma}{\partial\zeta}\Bigl{(}\theta_{\varepsilon},\zeta\left(\varepsilon,\theta_{\varepsilon}\right)\Bigr{)}\cdot
e^{\mathbf{i}\theta}\ =\
-\frac{\partial\Gamma}{\partial\theta}\Bigl{(}\theta_{\varepsilon},\zeta\left(\varepsilon,\theta_{\varepsilon}\right)\Bigr{)}\cdot
e^{\mathbf{i}\theta}\,.$
Regarding this as a characterization of $\,\theta_{\varepsilon}\,$, we will
differentiate implicitly with respect to $\,\varepsilon\,$, then set
$\,\varepsilon=0\,$ to verify the Proposition’s final claim. To manage the
task, we differentiate the two sides of (3.8) separately before equating them
to get our final conclusion.
Left side of (3.8): Differentiate the left-hand side of (3.8). Because
$\,\zeta(0,\theta)\equiv b\,$, all pure $\theta$-derivatives of $\,\zeta\,$
vanish at $\,\varepsilon=0\,$, and we can rewrite the sole surviving summand
using Observation 3.3:
$\displaystyle\frac{\partial}{\partial\varepsilon}\Big{|}_{\varepsilon=0}\left[\frac{\partial\zeta}{\partial\theta}\left(\varepsilon,\theta_{\varepsilon}\right)\,\frac{\partial\Gamma}{\partial\zeta}\Bigl{(}\theta_{\varepsilon},\zeta\left(\varepsilon,\theta_{\varepsilon}\right)\Bigr{)}\cdot
e^{\mathbf{i}\theta}\right]$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}\zeta}{\partial\theta\,\partial\varepsilon}\left(0,\theta\right)\,\frac{\partial\Gamma}{\partial\zeta}\left(\theta,b\right)\cdot
e^{\mathbf{i}\theta}$ $\displaystyle=$
$\displaystyle\left(\tau\cdot\frac{\partial\Gamma}{\partial\theta}\left(\theta,b\right)\right)\,\left(\frac{\partial\Gamma}{\partial\zeta}\left(\theta,b\right)\cdot
e^{\mathbf{i}\theta}\right)\,.$
Right side of (3.8): Now differentiate the right side of (3.8). Again, the
constancy of $\,\zeta(\varepsilon,\theta)\,$ at $\,\varepsilon=0\,$ eliminates
most summands, so that
(3.10)
$\displaystyle\frac{\partial}{\partial\varepsilon}\Big{|}_{\varepsilon=0}\left[-\frac{\partial\Gamma}{\partial\theta}\Bigl{(}\theta_{\varepsilon},\,\zeta\left(\varepsilon,\theta_{\varepsilon}\right)\Bigr{)}\,\cdot
e^{\mathbf{i}\theta}\right]=$
$\displaystyle-\left(\frac{\partial^{2}\Gamma}{\partial\theta^{2}}\bigl{(}\theta,b\bigr{)}\cdot
e^{\mathbf{i}\theta}\right)\frac{\partial\theta_{\varepsilon}}{\partial\varepsilon}\Big{|}_{\varepsilon=0}\
-\
\left(\frac{\partial^{2}\Gamma}{\partial\theta\,\partial\zeta}\bigl{(}\theta,b\bigr{)}\cdot
e^{\mathbf{i}\theta}\right)\frac{\partial\zeta}{\partial\varepsilon}\bigl{(}0,\theta\bigr{)}\,.$
We can now simplify this further, because $\,\Gamma(\,\cdot,b)\,$ support-
parametrizes $\,\mathcal{O}(b)\,$. This implies, via (2.1), that at the
preimage $\,(\theta,b)\,$ of any point in that oval, we have two identities:
$\frac{\partial\Gamma}{\partial\theta}\cdot e^{\mathbf{i}\theta}\equiv
0\quad\text{and}\quad\frac{\partial\Gamma}{\partial\theta}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}=\left|\frac{\partial\Gamma}{\partial\theta}\right|\,.$
The first of these lets us deduce
$\frac{\partial^{2}\Gamma}{\partial\theta\,\partial\zeta}\cdot
e^{\mathbf{i}\theta}=\frac{\partial}{\partial\zeta}\,\left(\frac{\partial\Gamma}{\partial\theta}\cdot
e^{\mathbf{i}\theta}\right)=0\,,$
which eliminates the final term on the right in (3.10).
Alternatively, if we differentiate the first of the two identities above with
respect to $\,\theta\,$, and then use the second, we get
(3.11) $\frac{\partial^{2}\Gamma}{\partial\theta^{2}}\cdot
e^{\mathbf{i}\theta}=-\frac{\partial\Gamma}{\partial\theta}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}=-\left|\frac{\partial\Gamma}{\partial\theta}\right|\,.$
This lets us rewrite the first term on the right in (3.10), collapsing the
whole equation to
(3.12)
$\frac{\partial}{\partial\varepsilon}\Big{|}_{\varepsilon=0}\left[-\frac{\partial\Gamma}{\partial\theta}\Bigl{(}\theta_{\varepsilon},\,\zeta\left(\varepsilon,\theta_{\varepsilon}\right)\Bigr{)}\,\cdot
e^{\mathbf{i}\theta}\right]=\left|\frac{\partial\Gamma}{\partial\theta}\right|\
\frac{\partial\theta_{\varepsilon}}{\partial\varepsilon}\Big{|}_{\varepsilon=0}\,.$
We now finish by setting (3.4) equal to (3.12). This exhibits the initial
$\varepsilon$-derivative of equation (3.8) as
$\left(\frac{\partial\Gamma}{\partial\theta}\cdot\tau\right)\,\left(\frac{\partial\Gamma}{\partial\zeta}\cdot
e^{\mathbf{i}\theta}\right)\ =\
\left|\frac{\partial\Gamma}{\partial\theta}\right|\
\frac{\partial\theta_{\varepsilon}}{\partial\varepsilon}\Big{|}_{\varepsilon=0}\,.$
Since this holds at the preimage $\,(\theta,b)\,$ of any point in
$\,\mathcal{O}(b,\varepsilon)\,$, and since, by (2.1) again,
$\,\partial\Gamma/\partial\theta\,$ normalizes to
$\,\mathbf{i}\,e^{\mathbf{i}\theta}\,$, this proves the last conclusion of our
Proposition. ∎
### 3.6. The symmetry obstruction.
We shall write $\,\mathbf{c}_{\varepsilon}\,$ for the centrix of
$\,\mathcal{O}(b,\varepsilon)\,$. By Observation 2.6,
$\,\mathcal{O}(b,\varepsilon)\,$ is _central_ if and only if
$\,\mathbf{c}_{\varepsilon}\,$ is _constant_ , or equivalently,
$\frac{\partial}{\partial\theta}\mathbf{c}_{\varepsilon}\equiv 0\,.$
Now observe that when $\,\mathcal{O}(b,\varepsilon)\,$ has central symmetry
for all $\,\varepsilon\,$ sufficiently near zero—as it clearly does when
$\,\mathcal{T}\,$ has cpo—we will also have
(3.13)
$\frac{\partial^{2}}{\partial\theta\,\partial\varepsilon}\Big{|}_{\varepsilon=0}\mathbf{c}_{\varepsilon}\equiv
0\,.$
The initial mixed second partial of $\,\mathbf{c}_{\varepsilon}\,$ thus forms
an _obstruction_ to cpo.
We want to show that conversely, the vanishing of this
obstruction—independently of the tilt-direction $\,\tau\,$ and the height
$\,b\,$ at which we compute it—has a strong consequence. Indeed, this
vanishing condition ultimately yields the partial differential equations of
Proposition 3.9, which in turn imply the Splitting Lemma 3.10.
To get there, we first need to rewrite the vanishing condition (3.13) in terms
of the support function of the horizontal oval $\,\mathcal{O}(b)\,$. Toward
that goal, we abbreviate
$\theta_{\varepsilon}:=\theta_{\varepsilon}(\theta)\quad\text{and}\quad{\bar{\theta}}_{\varepsilon}:=\theta_{\varepsilon}(\theta+\pi)$
for each $\,\theta\in\mathbf{S}^{1}\,$, then combine Observation 2.6 with
Proposition 3.5 to get a formula for $\,\mathbf{c}_{\varepsilon}\,$:
(3.14)
$\mathbf{c}_{\varepsilon}(\theta)=\frac{\Gamma\left(\theta_{\varepsilon},\zeta(\varepsilon,\theta_{\varepsilon})\right)+\Gamma\left({\bar{\theta}}_{\varepsilon},\zeta(\varepsilon,{\bar{\theta}}_{\varepsilon})\right)}{2}\,.$
In order to unpack (3.13), we must differentiate this formula twice: First
with respect to $\varepsilon\,$, and then with respect to $\,\theta\,$. We
record the initial $\varepsilon$-derivative as Lemma 3.7 below.
To prepare, let $\,\Gamma^{*}(\,\cdot,z)\,$ denote the _odd_ part of
$\,\Gamma(\,\cdot,z)\,$ as specified by Definition 2.5, and let
$\,\mathbf{c}(z)\,$ denote the centroid of $\,\mathcal{O}(z)\,$ for each
$\,-1<z<1\,$. In the language of Definition 1.4, $\,\mathbf{c}\,$ parametrizes
the _central curve_ of $\,\mathcal{T}\,$, while $\,\Gamma^{*}\,$ parametrizes
its _rectification_ $\,\mathcal{T}^{*}\,$.
###### Lemma 3.7.
Suppose the horizontal cross-section $\,\mathcal{O}(z)\,$ of a transversely
convex tube $\,\mathcal{T}\,$ is central about $\,(\mathbf{c}(z),z)\,$ for
each $\,-1<z<1\,$. Then for any fixed tilt-direction
$\,\tau\in\mathbf{S}^{1}\,$, we have
$\displaystyle\frac{\partial\mathbf{c}_{\varepsilon}}{\partial\varepsilon}\left(\theta,z\right)\Big{|}_{\varepsilon=0}$
$\displaystyle=$
$\displaystyle\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\Bigl{(}\frac{\
\partial\Gamma^{*}}{\partial\zeta}\cdot e^{\mathbf{i}\theta}\Bigr{)}\frac{\
\partial\Gamma^{*}}{\partial\theta}$ $\displaystyle\qquad\ +\
\bigl{(}\tau\cdot\mathbf{c}(z)\bigr{)}\,\mathbf{c}^{\prime}(z)\ +\
\bigl{(}\tau\cdot\Gamma^{*}\bigr{)}\,\frac{\
\partial\Gamma^{*}}{\partial\zeta}\,.$
We evaluate $\,\Gamma^{*}\,$ and its derivatives here at $\,(\theta,z)$
throughout.
###### Proof.
With (3.14) in view, we first compute the initial $\varepsilon$-derivative of
$\,\Gamma(\theta_{\varepsilon},\,\zeta(\varepsilon,\theta_{\varepsilon}))$.
Recall that by Proposition 3.5, $\,\theta_{0}(\theta)=\theta\,$, and
abbreviate
$\theta_{0}^{\prime}:=\frac{\partial\theta_{\varepsilon}}{\partial\varepsilon}\Big{|}_{\varepsilon=0}\,.$
A routine application of the chain rule then gives
$\displaystyle\frac{\partial}{\partial\varepsilon}\Big{|}_{\varepsilon=0}\Gamma\bigl{(}\theta_{\varepsilon},\,\zeta\left(\varepsilon,\theta_{\varepsilon}\right)\bigr{)}$
$\displaystyle=$
$\displaystyle\frac{\partial\Gamma}{\partial\theta}\left(\theta,z\right)\theta_{0}^{\prime}+\frac{\partial\Gamma}{\partial\zeta}(\theta,z)\Bigl{(}\frac{\partial{\zeta}}{\partial\varepsilon}(0,\theta)+\frac{\partial\zeta}{\partial\theta}\left(0,\theta\right)\theta^{\prime}_{0}\Bigr{)}$
$\displaystyle=$
$\displaystyle\frac{\partial\Gamma}{\partial\theta}\left(\theta,z\right)\theta_{0}^{\prime}+\frac{\partial\Gamma}{\partial\zeta}(\theta,z)\Bigl{(}\tau\cdot\Gamma\left(\theta,z\right)\Bigr{)}\,,$
where we have used equation (3.5) and Observation 3.3 to evaluate the
derivatives of $\,\zeta\,$.
We must average (LABEL:eqn:Geps) over $\,\\{\theta,\bar{\theta}\\}\,$ to get
the initial $\varepsilon$-derivative of $\,\mathbf{c}_{\varepsilon}\,$ via
(3.14). We assume $\,\Gamma(\theta,z)\,$ support-parametrizes an oval
$\,\mathcal{O}(z)\,$ having central symmetry about $\,\mathbf{c}(z)\,$ for
each $\,-1<z<1\,$, so we have
$\Gamma(\theta,z)=\mathbf{c}(z)+\Gamma^{*}(\theta,z)$
as in Observation 2.6. Here $\,\Gamma^{*}\,$ and all its $\theta$-derivatives
are odd, so that for instance
$\Gamma^{*}(\bar{\theta},z)=-\Gamma^{*}(\theta,z)\,.$
All $\theta$-derivatives of $\,\mathbf{c}(z)\,$, on the other hand, clearly
vanish. If we average (LABEL:eqn:Geps) over $\,\\{\theta,\bar{\theta}\\}\,$
with all these facts in mind, we get
$\displaystyle\frac{\partial\mathbf{c}_{\varepsilon}}{\partial\varepsilon}\left(\theta,z\right)\Big{|}_{\varepsilon=0}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\frac{\partial\Gamma}{\partial\theta}\left(\theta,z\right)\,\theta_{0}^{\prime}+\frac{\partial\Gamma}{\partial\zeta}\left(\theta,z\right)\left(\tau\cdot\Gamma\left(\theta,z\right)\right)\right.$
$\displaystyle\quad+\left.\quad\frac{\partial\Gamma}{\partial\theta}\left(\bar{\theta},z\right)\,\bar{\theta}_{0}^{\prime}+\frac{\partial\Gamma}{\partial\zeta}\left(\bar{\theta},z\right)\left(\tau\cdot\Gamma\left(\bar{\theta},z\right)\right)\right\\}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left\\{\frac{\
\partial\Gamma^{*}}{\partial\theta}\,\theta_{0}^{\prime}+\left(\mathbf{c}^{\prime}(z)+\frac{\
\partial\Gamma^{*}}{\partial\zeta}\right)\Bigl{(}\tau\cdot\left(\mathbf{c}(z)+\Gamma^{*}\right)\Bigr{)}\right.$
$\displaystyle\quad-\left.\frac{\
\partial\Gamma^{*}}{\partial\theta}\,\bar{\theta}_{0}^{\prime}+\left(\mathbf{c}^{\prime}(z)-\frac{\
\partial\Gamma^{*}}{\partial\zeta}\right)\Bigl{(}\tau\cdot\left(\mathbf{c}(z)-\Gamma^{*}\right)\Bigr{)}\right\\},$
where we now evaluate $\,\Gamma^{*}\,$ and its derivatives at $\,(\theta,z)\,$
throughout. To simplify further, note that the four mixed products involving
$\,\mathbf{c}$ and $\,\Gamma^{*}$-terms cancel in pairs, so that
$\displaystyle\frac{\partial\mathbf{c}_{\varepsilon}}{\partial\varepsilon}\left(\theta,z\right)\Big{|}_{\varepsilon=0}$
$\displaystyle=$
$\displaystyle\left(\frac{\theta_{0}^{\prime}-\bar{\theta}_{0}^{\prime}}{2}\right)\frac{\
\partial\Gamma^{*}}{\partial\theta}+\bigl{(}\tau\cdot\mathbf{c}(z)\bigr{)}\mathbf{c}^{\prime}(z)+\left(\tau\cdot\Gamma^{*}\right)\frac{\
\partial\Gamma^{*}}{\partial\zeta}$
This will give the formula we seek—we just need to prove
(3.16)
$\frac{\theta_{0}^{\prime}-\bar{\theta}_{0}^{\prime}}{2}=\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\Bigl{(}\frac{\
\partial\Gamma^{*}}{\partial\zeta}\cdot e^{\mathbf{i}\theta}\Bigr{)}\,.$
For that, we invoke Proposition 3.5. Since $\,\Gamma^{*}\,$ and
$\,e^{\mathbf{i}\theta}\,$ are both odd, that Proposition yields
$\displaystyle\theta_{0}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\Bigl{(}\mathbf{c}^{\prime}(z)\cdot
e^{\mathbf{i}\theta}+\frac{\ \partial\Gamma^{*}}{\partial\zeta}\cdot
e^{\mathbf{i}\theta}\Bigr{)}$ $\displaystyle\bar{\theta}_{0}^{\prime}$
$\displaystyle=$
$\displaystyle\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\Bigl{(}\mathbf{c}^{\prime}(z)\cdot
e^{\mathbf{i}\theta}-\frac{\ \partial\Gamma^{*}}{\partial\zeta}\cdot
e^{\mathbf{i}\theta}\Bigr{)}\,.$
Subtract the second line from the first to get (3.16), and the desired formula
follows. ∎
To finish analyzing the vanishing condition (3.13), we next need to
differentiate the result just proven with respect to $\theta$. That seems to
require a lengthy calculation, but if we work with respect to the frame
$\,\\{e^{\mathbf{i}\theta},\,\mathbf{i}\,e^{\mathbf{i}\theta}\\}\,$, a simple
observation eliminates the $\,e^{\mathbf{i}\theta}$ term entirely.
###### Observation 3.8.
Suppose the horizontal cross-section $\,\mathcal{O}(z)\,$ of a transversely
convex tube $\,\mathcal{T}\,$ is central about $\,(\mathbf{c}(z),z)\,$ for
each $\,-1<z<1\,$. Then for each tilt-direction $\,\tau\in\mathbf{S}^{1}\,$,
there exists a function $\,f_{\tau}:\mathbf{S}^{1}\times(-1,1)\to\mathbf{R}\,$
such that
$\frac{\,\partial^{2}\mathbf{c}_{\varepsilon}}{\partial\varepsilon\,\partial\theta}\bigl{(}\theta,z\bigr{)}\Big{|}_{\varepsilon=0}=f_{\tau}(\theta,z)\,\mathbf{i}\,e^{\mathbf{i}\theta}$
for all $\,(\theta,z)\in\mathbf{S}^{1}\times(-1,1)\,$.
###### Proof.
We get $\,\mathbf{c}_{\varepsilon}\,$ by symmetrizing each member in a smooth
family of support parametrizations:
$\mathbf{c}_{\varepsilon}\,(\theta)={\textstyle{\frac{1}{2}}}\left(\gamma_{\varepsilon}(\theta)+\gamma_{\varepsilon}(\theta+\pi)\right)$
Indeed, our formula (3.14) expresses $\,\mathbf{c}_{\varepsilon}\,$ in this
way. It then follows from the defining condition (2.1) for support
parametrizations, that
$\frac{\partial}{\partial\theta}\mathbf{c}_{\varepsilon}={\textstyle{\frac{1}{2}}}\bigl{(}\left|\gamma_{\varepsilon}^{\prime}(\theta)\right|+\left|\gamma_{\varepsilon}^{\prime}(\theta+\pi)\right|\bigr{)}\mathbf{i}\,e^{\mathbf{i}\theta}$
Differentiation with respect to $\,\varepsilon\,$ affects only the scalar
coefficient of $\,\mathbf{i}\,e^{\mathbf{i}\theta}\,$ here, making the desired
fact obvious. ∎
Thanks to Observation 3.8, the vanishing condition (3.13) reduces to
$\,f_{\tau}\equiv 0\,$. The two crucial PDE’s we have been aiming toward
merely interpret this simple equation and now make their appearance in the
statement of Proposition 3.9 below.
As we have explained above, Proposition 3.9 is the technical heart of this
section. It also marks our first real use of the cpo assumption: Up to now,
our results have at most assumed central symmetry for the _horizontal_ cross-
sections of $\,\mathcal{T}\,$.
To set up the statement of Proposition 3.9, recall that for each $\,|z|<1\,$,
$\,\Gamma^{*}(\,\cdot,z)\,$ support-parametrizes the horizontal cross-section
$\,\mathcal{O}(z)-\mathbf{c}(z)\,$ of the rectified tube
$\,\mathcal{T}^{*}\,$. There consequently exists a $\,C^{2}\,$ function
$h:\mathbf{S}^{1}\times[-1,1]\to\mathbf{R}$
which, for each fixed $\,|z|<1\,$, yields the support function of that oval.
We call $\,h\,$ the _transverse support function of $\,\mathcal{T}^{*}\,$_.
To simplify notation, we now adopt the convention of indicating partial
differentiation with respect to a given variable by subscripting with that
variable.
###### Proposition 3.9.
On a transversely convex tube $\,\mathcal{T}\,$ with cpo, the transverse
support function $\,h\,$ of $\,\mathcal{T}^{*}\,$ satisfies two partial
differential equations:
$\bigl{(}h_{z}\left(h+h_{\theta\theta}\right)\bigr{)}_{\theta}+\bigl{(}h_{\theta}\left(h+h_{\theta\theta}\right)\bigr{)}_{z}=0$
and
$h\,\left(h+h_{\theta\theta}\right)_{z}-\left(h+h_{\theta\theta}\right)\,h_{z}=0\,.$
###### Proof.
Differentiation with respect to $\,\theta\,$ annihilates $\,\mathbf{c}\,$ and
$\,\mathbf{c}^{\prime}\,$, and hence Lemma 3.7 combines with Observation 3.8
to give
$\displaystyle f_{\tau}$ $\displaystyle=$
$\displaystyle\mathbf{i}\,e^{\mathbf{i}\theta}\cdot\frac{\partial^{2}\mathbf{c}_{\varepsilon}}{\partial\varepsilon\,\partial\theta}\Big{|}_{\varepsilon=0}$
$\displaystyle=$
$\displaystyle\mathbf{i}\,e^{\mathbf{i}\theta}\cdot\bigl{[}\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left(\Gamma^{*}_{z}\cdot
e^{\mathbf{i}\theta}\right)\Gamma^{*}_{\theta}\bigr{]}_{\theta}+\mathbf{i}\,e^{\mathbf{i}\theta}\cdot\bigl{[}\left(\tau\cdot\Gamma^{*}\right)\Gamma^{*}_{z}\bigr{]}_{\theta}$
Since $\,\Gamma^{*}\,$ support-parametrizes $\,\mathcal{O}(z)-\mathbf{c}(z)\,$
for each $\,z\,$, however, we have
$\,\Gamma^{*}_{\theta}=\left|\Gamma^{*}_{\theta}\right|\mathbf{i}\,e^{\mathbf{i}\theta}\,$.
This is perpendicular to
$\,-e^{\mathbf{i}\theta}=\left(\mathbf{i}\,e^{\mathbf{i}\theta}\right)_{\theta}\,$,
so the product rule lets us rewrite the first term on the right above as
$\bigl{[}\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left(\Gamma^{*}_{z}\cdot
e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|\bigr{]}_{\theta}\,.$
To evaluate the second term, note that
$\,\Gamma^{*}_{\theta}\cdot\tau=\left|\Gamma^{*}_{\theta}\right|\mathbf{i}\,e^{\mathbf{i}\theta}\cdot\tau\,$,
and
$\Gamma^{*}_{z\theta}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}=\left(\Gamma^{*}_{\theta}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)_{z}=\left|\Gamma^{*}_{\theta}\right|_{z}\,.$
Taking all these facts into account, our expansion of $\,f_{\tau}\,$ becomes
$\displaystyle f_{\tau}$ $\displaystyle=$
$\displaystyle\bigl{[}\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left(\Gamma^{*}_{z}\cdot
e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|\bigr{]}_{\theta}$
$\displaystyle\qquad+\
\left|\Gamma^{*}_{\theta}\right|\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\Gamma^{*}_{z}\cdot\mathbf{i}\,e^{\mathbf{i}\,\theta}\
+\ \left(\tau\cdot\Gamma^{*}\right)\left|\Gamma^{*}_{\theta}\right|_{z}$
Now separate multiples of $\,\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\,$ from
those of $\,\tau\cdot e^{\mathbf{i}\theta}\,$, noting that
$\,\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)_{\theta}=-\left(\tau\cdot
e^{\mathbf{i}\theta}\right)\,$, and that by orthonormal expansion,
$\tau\cdot\Gamma^{*}=\left(\tau\cdot
e^{\mathbf{i}\theta}\right)\left(e^{\mathbf{i}\theta}\cdot\Gamma^{*}\right)+\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left(\mathbf{i}\,e^{\mathbf{i}\theta}\cdot\Gamma^{*}\right)\,.$
Use these facts to expand $\,f_{\tau}\,$ further, collecting multiples of
$\,\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\,$ and $\,\tau\cdot
e^{\mathbf{i}\theta}\,$, and noticing that
$\left(\Gamma^{*}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|_{z}+\left(\Gamma^{*}_{z}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|=\bigl{(}\left(\Gamma^{*}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|\bigr{)}_{z}$
to get
$\displaystyle f_{\tau}=$
$\displaystyle\left(\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\Bigl{[}\bigl{(}\left(\Gamma^{*}_{z}\cdot
e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|\bigr{)}_{\theta}+\bigl{(}\left(\Gamma^{*}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|\bigr{)}_{z}\Bigr{]}$
$\displaystyle\qquad\qquad+\left(\tau\cdot
e^{\mathbf{i}\theta}\right)\Bigl{[}\left(e^{\mathbf{i}\theta}\cdot\Gamma^{*}\right)\left|\Gamma^{*}_{\theta}\right|_{z}-\left(\Gamma^{*}_{z}\cdot
e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|\Bigr{]}\,.$
Now we invoke the central plane oval assumption, observing that _when
$\,\mathcal{T}\,$ has cpo, we must have $\,f_{\tau}\equiv 0\,$_.
Indeed, cpo endows the tilted ovals $\,\bar{\mathcal{O}}(z,\varepsilon)\,$
with central symmetry for all $\,\tau\in\mathbf{S}^{1}\,$, all $\,-1<z<1\,$,
and all sufficiently small $\,\varepsilon\,$. As noted earlier, the projected
ovals $\,\mathcal{O}(z,\varepsilon)\,$ inherit that symmetry too, since the
projection $\,(x,y,z)\to(x,y)\,$ induces an affine isomorphism from any non-
vertical plane to $\,\mathbf{R}^{2}\,$.
Observation 2.6 then makes the centrix $\,\mathbf{c}_{\varepsilon}\,$ of
$\,\mathcal{O}(z,\varepsilon)\,$ constant (i.e. independent of $\,\theta\,$)
for any tilt-direction $\,\tau\,$, any $\,|z|<1\,$ and all any sufficiently
small $\,\varepsilon\,$. The vanishing condition (3.13) therefore obtains.
Given Observation 3.8, this forces $\,f_{\tau}\equiv 0\,$ as claimed.
We may consequently set the right-hand side of (3.6) equal to zero. But the
resulting identity holds for _any_ tilt-direction
$\,\tau=:e^{\mathbf{i}\phi}\in\mathbf{S}^{1}\,$, and _the coefficients
$\,\tau\cdot e^{\mathbf{i}\theta}=\cos(\phi-\theta)\,$ and
$\,\tau\cdot\mathbf{i}\,e^{\mathbf{i}\theta}=\sin(\phi-\theta)\,$ appearing
there are clearly linearly independent functions of $\,\tau\,$._ The terms
they multiply must therefore vanish _individually_. In short, we now have
(3.18) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\Bigl{(}\left(\Gamma^{*}_{z}\cdot
e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|\Bigr{)}_{\theta}+\Bigl{(}\left(\Gamma^{*}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|\Bigr{)}_{z}$
(3.19) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(e^{\mathbf{i}\theta}\cdot\Gamma^{*}\right)\left|\Gamma^{*}_{\theta}\right|_{z}-\left(\Gamma^{*}_{z}\cdot
e^{\mathbf{i}\theta}\right)\left|\Gamma^{*}_{\theta}\right|$
For each $\,|z|<1\,$, the relationship between the support parametrization
$\,\Gamma^{*}(\,\cdot,z)\,$ of $\,\mathcal{O}(z)\,$ and its support function
$\,h(\,\cdot,z)\,$, as detailed in §2, now lets us write
$\Gamma^{*}=\left(h+\mathbf{i}\,h_{\theta}\right)\,e^{\mathbf{i}\theta}\quad\text{and}\quad\Gamma^{*}_{\theta}=\left(h+h_{\theta\theta}\right)\mathbf{i}\,e^{\mathbf{i}\theta}\,,$
from which we can immediately deduce
$\begin{array}[]{rclcccl}\Gamma^{*}\cdot
e^{\mathbf{i}\theta}&=&h\,,&&\Gamma^{*}\cdot\mathbf{i}\,e^{\mathbf{i}\theta}&=&h_{\theta}\\\
\Gamma^{*}_{z}\cdot
e^{\mathbf{i}\theta}&=&h_{z}\,,&&\left|\Gamma^{*}_{\theta}\right|&=&h+h_{\theta\theta}\,,\end{array}$
Substituting these into (3.18) and (3.19) instantly gives the differential
equations we want. ∎
We can now prove our Splitting Lemma 3.10, restated below. As above, $\,h\,$
denotes the transverse support function of $\,\mathcal{T}^{*}\,$, the
rectification of a transversely convex tube $\,\mathcal{T}\,$ with central
curve $\,\mathbf{c}\,$. Recall that we say $\,\mathcal{T}^{*}\,$ _splits_ if
we can factor its support map $\,\Gamma^{*}(z,\theta)\,$ as a product
$\,\gamma(\theta)r(z)\,$, with $\,\gamma\,$ parametrizing a fixed oval and
$\,r>0\,$.
###### Proposition 3.10 (Splitting Lemma).
If a transversely convex tube $\,\mathcal{T}\,$ in standard position has cpo,
then its rectification $\,\mathcal{T}^{*}\,$ splits.
###### Proof.
It will clearly suffice to prove that the transverse support function $\,h\,$
of $\,\Gamma^{*}\,$ factors as $\,h(z,\theta)=h(\theta)\,r(z)\,$. We know that
$\,h(z,\theta)\,$ satisfies the two differential equations of Proposition 3.9,
and we start by noticing that the second equation there forms the numerator of
a quotient-rule calculation. Specifically, it implies
$\frac{\partial}{\partial z}\left(\frac{h+h_{\theta\theta}}{h}\right)=0\,,$
from which we easily deduce
(3.20) $h_{\theta\theta}+h=q^{2}(\theta)\,h$
for some _strictly_ positive, $z$-independent function $\,q\,$ on
$\,\mathbf{S}^{1}\,$. We can assume positivity of $\,q\,$ because
$\,\mathcal{O}(z)-\mathbf{c}(z)\,$ is origin-centered and strictly convex for
each $\,z\,$, properties that, by equations (2.2) and (2.5), make both $\,h\,$
and $\,h_{\theta\theta}+h\,$ strictly positive.
In any case, since $\,q\,$ depends only on $\,\theta\,$, we see that the
support functions of the translated ovals $\,\mathcal{O}(z)-\mathbf{c}(z)\,$
all solve the same ordinary differential equation, namely (3.20). Such
equations have independent solutions, of course, so by itself, (3.20) leaves
us short of splitting. But it lets us rewrite the _first_ differential
equation of Proposition 3.9 as
(3.21)
$\bigl{(}h_{z}\,h\,q^{2}\bigr{)}_{\theta}+\bigl{(}h_{\theta}\,h\,q^{2}\bigr{)}_{z}=0\,.$
Since $\,h_{z}h\,$ and $\,h_{\theta}h\,$ are derivatives of (half) the
_squared_ support function
$H(\theta,z):=h^{2}(\theta,z)\,,$
we can the exploit $z$-independence of $\,q\,$, and use
$\,H_{z\theta}=H_{\theta z}\,$ to rewrite (3.21) in the form of a first-order
equation for $\,H_{z}\,$:
$2H_{z\theta}\,q^{2}+H_{z}\bigl{(}q^{2}\bigr{)}_{\theta}=0\,.$
Now multiply by $\,H_{z}\,$ to recognize that (3.21) actually reduces to
$\bigl{(}H_{z}^{2}q^{2}\bigr{)}_{\theta}=0\,.$
Evidently, there exists a $\theta$-independent function $\,\phi(z)\,$ such
that
$H_{z}(\theta,z)=\phi(z)\big{/}q(\theta)\,.$
Integrating with respect to $\,z\,$ then yields
$H(\theta,z)=H(\theta,0)+\frac{\Phi(z)}{q(\theta)}\,,\quad\text{where}\quad\Phi(z):=\int_{0}^{z}\phi(s)\,ds\,.$
Rewrite this as
$H(\theta,z)=H(\theta,0)\left(1+\alpha(\theta)\Phi(z)\right)\,,$
where
$\alpha(\theta):=\frac{1}{H(\theta,0)\,q(\theta)}\,.$
Since $\,H=h^{2}\,$, and, as the support function of an origin-centered oval,
$\,h(\theta,z)$ is always positive, we see that $\,1+\alpha\,\Phi>0\,$ too.
Hence
(3.22) $h(\theta,z)=h(\theta,0)\sqrt{1+\alpha(\theta)\,\Phi(z)}\,.$
The continuity of $\,\alpha\,$ guarantees it a maximum value
$\,\bar{\alpha}\,$ at some point $\,\bar{\theta}\in\mathbf{S}^{1}\,$, and
there, (3.22) yields
$\displaystyle h(\bar{\theta},z)$ $\displaystyle=$ $\displaystyle
h(\bar{\theta},0)\sqrt{1+\bar{\alpha}\,\Phi(z)}$ $\displaystyle
h_{\theta}(\bar{\theta},z)$ $\displaystyle=$ $\displaystyle
h_{\theta}(\bar{\theta},0)\sqrt{1+\bar{\alpha}\,\Phi(z)}\,.\rule{0.0pt}{17.07164pt}$
These identities show that for any fixed $\,z\,$ with $\,|z|<1\,$, the
functions $\,h(\theta,z)\,$ and
$\,h(\theta,0)\sqrt{1+\bar{\alpha}\,\Phi(z)}\,$ both obey the same initial
conditions at $\,\theta=\bar{\theta}\,$. Since both also solve (3.20),
Picard’s uniqueness theorem forces them to agree everywhere.
The Lemma consequently holds with
$r(z)=\sqrt{1+\bar{\alpha}\,\Phi(z)}\quad\text{and}\quad
h(\theta)=h(\theta,0)\,.$
∎
We now reach the main goal of this section—a geometric consequence of the
Splitting lemma:
###### Proposition 3.11.
Suppose $\,\mathcal{T}\,$ is a transversely convex tube with cpo. Then its
rectification $\,\mathcal{T}^{*}\,$ is either
* (i)
The cylinder over a central oval, or
* (ii)
Affinely congruent to a surface of revolution.
###### Proof.
We show that when $\,\mathcal{T}\,$ is a transversely convex tube in standard
position, and $\,\mathcal{T}^{*}\,$ is _not_ a cylinder, there exists a single
linear isomorphism that fixes the $z$-axis while making each horizontal cross-
section $\,\mathcal{O}(z)\,$ of $\,\mathcal{T}^{*}\,$ simultaneously circular.
This clearly implies the desired result.
We start by using the Splitting Lemma to factor the transverse support
function $\,h\,$ of $\,\mathcal{T}^{*}\,$ as
(3.23) $h(\theta,z)=r(z)\,h(\theta)\,.$
Put this factorization _back_ into the first differential equation in
Proposition 3.9 and simplify, to find that $\,r\,$ and $\,h\,$ now jointly
solve
(3.24)
$r\,r^{\prime}\bigl{(}h\,h^{\prime\prime\prime}+3\,h^{\prime}h^{\prime\prime}+4\,h\,h^{\prime}\bigr{)}=0$
on $\,\mathbf{S}^{1}\times(-1,1)\,$. We have assumed that
$\,\mathcal{T}^{*}\,$ is not cylindrical, so $\,r^{\prime}(z_{0})\neq 0\,$ for
some $\,-1<z_{0}<1\,$. Evaluating (3.24) at that height, we then deduce that
the horizontal support function $\,h(\theta)\,$ solves the following ordinary
differential equation:
$h\,h^{\prime\prime\prime}+3\,h^{\prime}h^{\prime\prime}+4\,h\,h^{\prime}=0\,.$
The reader will find it routine to verify what came as a pleasant surprise to
us: That this quadratic ODE for $\,h\,$ reduces to a linear equation—one that
could hardly be more familiar—for the _squared_ support function
$\,H(\theta):=h^{2}(\theta)\,$:
$H^{\prime\prime\prime}+4H^{\prime}=0\,.$
By Proposition 2.2, this makes $\,h(\theta)\,$ the support function of an
origin-centered _ellipse_ $\,\mathcal{O}_{0}\,$. By (3.23), _every_ horizontal
cross-section of $\,\mathcal{T}^{*}\,$ is then homothetic to
$\,\mathcal{O}_{0}\,$, and since it is origin-centered, $\,\mathcal{O}_{0}\,$
is congruent to the unit circle via some linear mapping $\,A\,$ of
$\,\mathbf{R}^{2}\,$. Extending $\,A\,$ trivially to $\,\mathbf{R}^{3}\,$, we
clearly map $\,\mathcal{T}^{*}\,$ to a surface of revolution, precisely as we
sought to prove. ∎
## 4\. Straightening the central curve
So far we have shown, using variational and analytic arguments, that when a
transversely convex tube has cpo, it rectifies to either a cylinder or—up to
affine isomorphism—a surface of revolution. We now use more elementary
arguments of a local geometric type to show that the rectification step is
actually superfluous. Specifically, we prove
###### Proposition 4.1 (Axis lemma).
Suppose $\,\mathcal{T}\,$ is a transversally convex tube with cpo. Then its
central curve is affine, so that $\,\mathcal{T}\,$ is affinely congruent to
its rectification $\,\mathcal{T}^{*}\,$.
###### Proof.
We can assume $\,\mathcal{T}\,$ lies in the standard position described by
Definition 1.4, and it clearly suffices to prove that when
$\,\mathcal{T}^{*}\,$ is either a cylinder or a surface of revolution, cpo
forces the axis of $\,\mathcal{T}\,$ itself to be a straight line. The latter
occurs if and only if the tube’s central curve
$\,\mathbf{c}:(-1,1)\to\mathbf{R}^{2}\,$ is affine (linear plus constant). We
will establish exactly that, using the following
Linearity Criterion: _A $\,C^{2}\,$ mapping $\,\mathbf{c}:I\to\mathbf{R}\,$ is
affine on an open interval $\,I\,$ if and only if it is locally _odd_ around
each input, in the sense that for all $\,b\in I\,$, we have_
(4.1)
$\mathbf{c}(b+t)-\mathbf{c}(b)=-\bigl{(}\mathbf{c}(b-t)-\mathbf{c}(b)\bigr{)}$
_for all sufficiently small $\,t\,$._
When $\,\mathbf{c}\,$ is affine, (4.1) clearly holds. To prove the converse,
it suffices to show that (4.1) implies $\,\mathbf{c}^{\prime\prime}\equiv
0\,$. But that follows instantly if we differentiate it twice, and then let
$\,t\to 0\,$.
With this criterion in hand, we proceed, treating the cylindrical and
rotationally symmetric cases separately.
Cylindrical case: When $\,\mathcal{T}^{*}\,$ is a cylinder, its horizontal
cross-section $\,\mathcal{O}(z)$ at every height $\,z\in(-1,1)\,$ translates
to a fixed central oval $\,\mathcal{O}_{0}\in\mathbf{R}^{2}\,$. Take
$\,\mathcal{O}_{0}\,$ to be centered at the origin and denote its support
parametrization by $\,\gamma\,$ to get this parametrization
$\,X:\mathbf{S}^{1}\times(-1,1)\to\mathcal{T}\,$:
(4.2) $X(t,z)=\left(\mathbf{c}(z)+\gamma(t),\,z\right)\,.$
Now consider, for any height $\,b\in(-1,1)\,$, and any angle
$\,\theta\in\mathbf{R}\,$, the $\theta$-diameter of $\,\mathcal{O}(b)$
(Definition 2.4). Since $\,\gamma\,$ support-parametrizes
$\,\mathcal{O}_{0}\,$, the endpoints of this diameter clearly lie at
$\,X(\theta,b)\,$ and $\,X(\theta+\pi,b)\,$, and the crucial point is that
_the tangent planes to $\,\mathcal{T}\,$ at these endpoints are parallel._ To
see that, compute the partial derivatives $\,X_{t}\,$ and $\,X_{z}\,$ at these
points. Since $\,\mathcal{O}_{0}\,$ is central, we have
$\,\gamma^{\prime}(\theta+\pi)=-\gamma^{\prime}(\theta)\,$, and this makes the
tangent planes parallel, since both are spanned by
$\left(\gamma^{\prime}(\theta),\,0\right)=\pm
X_{t}\quad\text{and}\quad\bigl{(}\mathbf{c}^{\prime}(b),\,1\bigr{)}=X_{z}\,.$
Now suppose, fixing the $\theta$-diameter of $\,\mathcal{O}(b)$ as axis, we
tilt the plane $\,z=b\,$ away from the horizontal with some small slope
$\,\varepsilon>0\,$ to get a new plane $\,P_{\varepsilon}(\theta)\,$. For
sufficiently small $\,\varepsilon>0\,$, the intersection
$\,\mathcal{O}(b,\theta,\varepsilon):=\mathcal{T}\cap
P_{\varepsilon}(\theta)\,$ will remain an oval—and a central oval, since
$\,\mathcal{T}\,$ has cpo.
Further, since $\,P_{\varepsilon}(\theta)\,$ contains the $\theta$-diameter of
$\,\mathcal{O}(b)$, the endpoints $\,X(\theta,b)\,$ and $\,X(\theta+\pi,b)\,$
of that diameter remain on $\,\mathcal{O}(b,\theta,\varepsilon)\,$
independently of $\,\varepsilon\,$. And since the tangent planes to
$\,\mathcal{T}\,$ at these points are parallel, and their intersections with
$\,P_{\varepsilon}(\theta)\,$ clearly form lines tangent to
$\,\mathcal{O}(b,\theta,\varepsilon)\,$ at $\,X(\theta,b)\,$ and
$\,X(\theta+\pi,b)\,$, _those tangent lines are parallel_.
The latter fact shows that the $\theta$-diameter of $\,\mathcal{O}(b)\,$
_remains_ a diameter of $\,\mathcal{O}(b,\theta,\varepsilon)\,$ independently
of $\,\varepsilon\,$, and hence that _$\,(\mathbf{c}(b),b)$ forms the center
of $\,\mathcal{O}(b,\theta,\varepsilon)\,$_, for each $\,\theta\,$ and each
sufficiently small $\,\varepsilon>0\,$. The center of
$\,\mathcal{O}(b,\theta,\varepsilon)\,$ remains fixed as we vary
$\,\varepsilon\,$.
Now observe that every point sufficiently close to $\,\mathcal{O}(b)$ on
$\,\mathcal{T}\,$ belongs $\,\mathcal{O}(b,\theta,\varepsilon)\,$ for some
$\,\theta\,$ and some small $\,\varepsilon>0\,$, so that by cpo, its
reflection through $\,(\mathbf{c}(b),b)\,$ also lies on $\,\mathcal{T}\,$. It
follows that an entire neighborhood of $\,\mathcal{O}(b)$ in $\,\mathcal{T}\,$
has reflection symmetry through $\,(\mathbf{c}(b),b)\,$. In some neighborhood
of $\,(\mathbf{c}(b),b)\,$, the central curve $\,\mathbf{c}\,$ of
$\,\mathcal{T}\,$ then inherits that same reflection symmetry. Since
$\,b\in(-1,1)\,$ was arbitrary, this clearly means that (4.1) holds for
$\,\mathbf{c}\,$, and our Linearity Criterion now straightens the central
curve, as desired.
Surface-of-revolution case: Here, each horizontal plane $\,z\equiv b\,$ cuts
the original tube $\,\mathcal{T}\,$ in a _circle_ centered at
$\,(\mathbf{c}(b),b)\,$ for each $\,b\in(-1,1)\,$. Write $\,F(b)>0\,$ for the
squared radius of this circle, and $\,(\xi(b),\eta(b)):=\mathbf{c}(b)\,$ for
the horizontal coordinates of its center. Then $\,\mathcal{T}\,$ clearly
constitutes the solution set of
(4.3) $\left(x-\xi(z)\right)^{2}+\left(y-\eta(z)\right)^{2}=F(z)\,.$
The $\,C^{2}\,$ differentiability of $\,\mathcal{T}\,$ ensures that $\,F\,$,
$\,\xi\,$ and $\,\eta\,$ are all $\,C^{2}\,$ on $\,(-1,1)\,$.
We want to show that cpo forces $\,\mathbf{c}\,$ to be affine. To do so, we
study the even and odd components of $\,\xi,\,\eta,$ and $\,F\,$ with respect
to reflection through a point, and for that we introduce the following
notation.
Suppose $\,\beta\in\mathbf{R}\,$, and let $\,f\,$ denote any function defined
on a neighborhood of $\,\beta\,$. We define the $\,\beta$-translate of $\,f\,$
by
$f_{\beta}(t):=f(\beta+t)\ .$
We also define the even and odd parts of $\,f_{\beta}\,$ respectively as
$f_{\beta}^{+}(t)=\frac{f_{\beta}(t)+f_{\beta}(-t)}{2}\ ,\qquad
f_{\beta}^{-}(t)=\frac{f_{\beta}(t)-f_{\beta}(-t)}{2}\,.$
As usual, we then have
$f_{\beta}^{+}(-t)=f_{\beta}^{+}(t)\ ,\quad
f_{\beta}^{-}(-t)=-f_{\beta}^{-}(t)$
and
$f_{\beta}(t)=f_{\beta}^{+}(t)+f_{\beta}^{-}(t)\ ,\quad
f_{\beta}(-t)=f_{\beta}^{+}(t)-f_{\beta}^{-}(t)\ .$
Now fix an arbitrary height $\,\beta\in(-1,1)\,$. Since $\,\mathcal{T}\,$ is
horizontally circular, has cpo, and lies in standard position, we can find a
small slope $\,m>0\,$, and a $z$-intercept $\,b=b(\beta)\,$ such that the
plane $\,P\,$ given by
$z=mx+b\quad\text{or}\quad x=\frac{z-b}{m}\,,$
cuts $\,\mathcal{T}\,$ is a central oval $\,\mathcal{O}\,$, depending on
$\,m\,$ and $\,\beta\,$, and centered at height $\,\beta\,$. In the
$\,(y,z)\,$ coordinate system on $\,P\,$, we get the following equation for
$\,\mathcal{O}\,$ by restricting (4.3):
(4.4)
$\left(\frac{z-b}{m}-\xi(z)\right)^{2}+\left(y-\eta(z)\right)^{2}=F(z)\,.$
Solve this for $\,y\,$ in terms of the $\beta$-centered variable
$\,t:=z-\beta\,$ to split $\,\mathcal{O}\,$ into a pair of arcs, graphs of
functions we shall call $\,y_{\pm}(t)\,$, over the symmetric interval
(4.5) $|t|<\sup\\{z-\beta\colon(x,y,z)\in\mathcal{O}\\}\,.$
Using the notation defined above, we can express these functions as
(4.6)
$y_{\pm}(t):=\eta_{\beta}(t)\pm\sqrt{F_{\beta}(t)-\left(\frac{\bar{\beta}+t}{m}-\xi_{\beta}(t)\right)^{2}}\,,$
where $\,\bar{\beta}:=\beta-b\,$.
Since the chord joining $\,(y_{+}(0),\beta)\,$ to $\,(y_{-}(0),\beta)\,$ has
height $\,\beta\,$, it clearly passes through the center of $\,\mathcal{O}\,$.
It must therefore be a diameter. But the midpoint of any diameter locates the
center of $\,\mathcal{O}\,$, so using (4.6) to average $\,y_{\pm}(0)\,$, we
can now deduce that:
_The center of $\,\mathcal{O}\,$ has coordinates $\,(\eta(\beta),\beta)\,$ in
the $\,(y,z)\,$ coordinate system on $\,P\,$._
This fact lets us express the central symmetry of $\,\mathcal{O}\,$ as the
coordinate swap
$(\eta(\beta)+s,\,\beta+t)\ \longleftrightarrow\ (\eta(\beta)-s,\,\beta-t)\,.$
When $\,t\,$ is small enough as measured by (4.5), this swap always exchanges
diametrically opposed solutions of (4.4). Write the resulting two statements
in terms of the notation introduced above to get two simultaneous identities:
$\displaystyle F^{+}_{\beta}(t)+F^{-}_{\beta}(t)$ $\displaystyle\quad=\
\left(\eta(\beta)+s-\eta^{+}_{\beta}(t)-\eta^{-}_{\beta}(t)\right)^{2}$
$\displaystyle\qquad\qquad+\left(\displaystyle{\frac{\bar{\beta}+t}{m}}-\xi_{\beta}^{+}(t)-\xi_{\beta}^{-}(t)\right)^{2}$
and
$\displaystyle F^{+}_{\beta}(t)-F^{-}_{\beta}(t)$ $\displaystyle\quad=\
\left(\eta(\beta)-s-\eta^{+}_{\beta}(t)+\eta^{-}_{\beta}(t)\right)^{2}$
$\displaystyle\qquad\qquad+\left(\displaystyle{\frac{\bar{\beta}-t}{m}}-\xi_{\beta}^{+}(t)+\xi_{\beta}^{-}(t)\right)^{2}$
Subtract (4) from (4), factor differences between corresponding squares on the
right, and divide by two, to obtain
$\displaystyle F^{-}_{\beta}(t)$ $\displaystyle\quad=\
2\left(\eta(\beta)-\eta_{\beta}^{+}(t)\right)\left(s-\eta_{\beta}^{-}(t)\right)$
$\displaystyle\qquad\qquad+\
2\left(\frac{\bar{\beta}}{m}-\xi_{\beta}^{+}(t)\right)\left(\frac{t}{m}-\xi_{\beta}^{-}(t)\right)$
The strict convexity of $\,\mathcal{O}\,$ now guarantees that the line
$\,z=\beta+t\,$ in $\,P\,$ cuts $\,\mathcal{O}\,$ in two distinct points
whenever $\,t\,$ is sufficiently small. Call the $y$-coordinates of these
points $\,\eta(\beta)+s\,$ and $\,\eta(\beta)+s^{\prime}\,$ respectively.
Equation (4) clearly remains true if we replace $\,s\,$ by $\,s^{\prime}\,$.
When we subtract the resulting $s^{\prime}$-version of (4) from the
$s$-version and simplify, however, we find that for all sufficiently small
$\,t\,$, we have
$\left(s-s^{\prime}\right)\left(\eta_{\beta}^{+}(t)-\eta(\beta)\right)=0\,.$
Since $\,s\,$ and $\,s^{\prime}\,$ are distinct for the small $\,t\,$ in
question, we evidently must have $\,\eta_{\beta}^{+}(t)\equiv\eta(\beta)\,$
for all sufficiently small $\,t\,$. By definition of $\,\eta_{\beta}^{+}\,$,
this means
$\eta(\beta+t)-\eta(\beta)=-\bigl{(}\eta(\beta-t)-\eta(\beta)\bigr{)}\,,$
so that (4.1) holds for $\,\eta\,$. But by swapping the roles of $\,x\,$ and
$\,y\,$ in the argument above, we find that in precisely the same way, it
holds for $\,\xi\,$, and hence for $\,\mathbf{c}=(\xi,\eta)\,$. Our Linearity
Criterion then makes the $\,\mathbf{c}\,$ affine, as desired. ∎
## 5\. Main theorem
By combining the Axis Lemma just proven with our Cylinder/Quadric Proposition
3.11 and the rotationally invariant case (Proposition 1.3), one immediately
deduces
###### Proposition 5.1 (Collar Theorem).
A transversely convex tube with cpo is either cylindrical or quadric.
We can strengthen this statement substantially, however, without much extra
effort:
###### Theorem 5.2 (Main Theorem).
A complete, connected $\,C^{2}$-immersed surface in $\,\mathbf{R}^{3}\,$ with
cpo is either a cylinder, or quadric.
###### Proof.
Suppose $\,F\,$ immerses a complete $\,C^{2}\,$ surface $\,M^{2}\,$ into
$\,\mathbf{R}^{3}\,$ with cpo. The latter assumption ensures, first of all,
that $\,F(M)\,$ crosses some affine plane—we take it to be the $\,z=0\,$
plane—transversally (if not exclusively) along a central oval
$\,\mathcal{O}\,$.
This being the case, define, for any two heights $\,a<0<b\,$, the open
connected component
$M_{a,b}\subset F^{-1}\left(\left\\{(x,y,z)\in\mathbf{R}^{3}\colon
a<z<b\right\\}\right)$
as the unique component containing $\,F^{-1}(\mathcal{O})\,$.
Since $\,\mathcal{O}\,$ is strictly convex and $\,F(M)\,$ is transverse to the
plane $\,z=0\,$ along $\,\mathcal{O}\,$, standard arguments from basic
differential topology show that for $\,a<0<b\,$ sufficiently near $0\,$,
* (i)
The pullback $\,F^{*}z\,$ of the height function $\,z\,$ on
$\,\mathbf{R}^{3}\,$ has no critical points in $\,M_{a,b}\,$, and
* (ii)
$F\,$ embeds $\,M_{a,b}\,$ in $\,\mathbf{R}^{3}\,$ as a transversely convex
tube.
There consequently exist _minimal_ and _maximal_ heights $\,-\infty\leq
A<0<B\leq\infty\,$ such that (i) and (ii) above both hold for every finite
$\,a<b\,$ in the closed interval $\,[A,B]\,$.
Our proof now forks in three directions, depending on whether both, neither,
or exactly one of the endpoints $\,A\,$ and $\,B\,$ are finite.
Case $\,-\infty<A<B<\infty\,$ (Ellipsoid). In this case, by (ii), the image of
$\,M_{a,b}\,$ under $\,F\,$ is a transversely convex tube for every $\,a<b\,$
in the interval $\,(A,B)\,$. This trivially extends to $\,M_{A,B}\,$, and the
resulting maximal tube clearly inherits cpo from $\,F(M)\,$. Our Collar
Theorem 5.1 then says that $\,F(M_{A,B})\,$ is either the cylinder on a
central oval, or quadric.
We can rule out the first possibility, because on a cylinder, horizontal
cross-sections are uniformly convex, and the gradient of $\,z\,$ is bounded
away from zero. But these facts, by continuity, would extend slightly beyond
$\,A\,$ and $\,B\,$, contradicting their maximality with respect to (i) and
(ii) above.
It follows that when $\,-\infty<A<B<\infty\,$, $\,F(M_{A,B})\,$ is quadric. By
affine invariance, however, we lose no generality by assuming that $\,F\,$
immerses $\,M_{A,B}\,$ as a quadric surface of revolution around the $z$-axis:
a vertical segment of an ellipsoid, cone, elliptic paraboloid, or a
hyperboloid. On all these surfaces, horizontal cross-sections in any compact
slab are uniformly convex. So the maximality of $\,A\,$ and $\,B\,$ must be
dictated by condition (i) above, not (ii). The completeness of $\,M\,$, then
ensures that $\,F^{*}z\,$ must have critical points on both boundaries of
$\,M_{A,B}\,$. But among the quadrics listed above, $\,z\,$ has multiple
critical points only on the ellipsoid, where it attains both a max and a min.
The closure of $\,F(M_{A,B})\,$ must therefore be a complete ellipsoid, which,
by continuity of $\,F\,$ and connectedness of $\,M\,$ must coincide with
$F(M)\,$.
Case $\,-A=B=\infty\,$ (Tube hyperboloid or cylinder). In this case we can
immediately from the connectedness of $\,M\,$ that $\,M_{A,B}=M\,$. Moreover,
since (ii) holds for every finite $\,a<0<b\,$, $\,F\,$ must embed
$\,M_{-r,r}\,$ in $\,\mathbf{R}^{3}\,$ as a transversely convex tube
$\,\mathcal{T}_{r}\,$ for every $\,r>0\,$. As above, $\,\mathcal{T}_{r}\,$
inherits cpo from $\,F(M)\,$, so by the Collar Theorem 5.1, $\,F\,$ maps
$\,M_{-r,r}\,$ to a cylinder over some central oval, or to a non-degenerate
quadric, for each $\,r>0\,$. Let $\,S\,$ denote the unique complete unbounded
cylinder or quadric that extends $\,F(M_{-1,1})\,$. We then clearly have
$\,F(M_{-r,r})=S\,$ in the slab $\,|z|<r\,$ for all $\,r>1\,$. But then
$\,S=F(M)\,$ in its entirety, for otherwise, $\,F(M)\,$ deviates from $\,S\,$
at some finite height $\,\rho\,$, a contradition when $\,r>|\rho|\,$. The only
smooth quadric that contains a horizontal oval and extends infinitely far both
above and below the plane $\,z=0\,$ is the tube hyperboloid. So in this case,
$\,M\,$ is either a tube hyperboloid or a cylinder.
Cases $\,|A|<B=\infty\,$ or $\,|B|<|A|=\infty\,$ (Paraboloid or convex
hyperboloid). Since the reflection $\,z\to-z\,$ is affine, these two cases are
equivalent. So we assume $\,|A|<B=\infty\,$, and arguing as in the previous
two cases, we now quickly deduce the existence of a quadric surface of
revolution $\,S\,$ such that (modulo some fixed affine isomorphism)
$\,F(M_{A,b})=S\,$ for all $\,b<\infty\,$. Further, here as in the doubly-
finite case, the maximality of $\,A\,$ must be dictated by a critical point at
height $\,A\,$. No cylinder has such a critical point, and among the quadrics,
only the elliptic paraboloid and convex hyperboloid do. Clearly then, $\,S\,$
is one of these two surfaces, and $\,F(M)=S\,$. ∎
## 6\. Application to skew loops
We originally conceived our Main Theorem 5.2 above as a tool for proving the
existence of skewloops on a class of negatively curved tubes. In this final
section we implement that idea.
###### Definition 6.1.
A skewloop is a circle differentiably immersed into $\,\mathbf{R}^{3}\,$ with
no pair of parallel tangent lines.
The existence of skewloops is not so obvious: Segre published the first
construction in 1968 [Se]. A more recent construction and application appeared
in M. Ghomi’s paper [Gh], and sparked our own interest. We coined the term
_skewloop_ in [GS], a subsequent joint paper that characterized _positively_
curved quadrics in $\,\mathbf{R}^{3}\,$ as the _only_ surfaces having a point
of positive curvature, but no skewloop:
###### Theorem 6.2 ([GS, 2002]).
A connected $\,C^{2}\,$ surface immersed in $\,\mathbf{R}^{3}\,$ with at least
one point of positive Gauss curvature admits no skewloop if and only if it is
quadric.
In particular, this identifies ellipsoids as the only _compact_ surfaces
lacking skewloops in $\,\mathbf{R}^{3}$. Its proof made strong use of
Blaschke’s result (Proposition 1.2) which, as explained in §1, applies to
_convex_ surfaces only, and is fundamentally local.
Our dependence on Blaschke’s theorem in [GS] thus compelled us to assume
positive curvature, and at that time, we could only raise the question as to
whether our skewloop-free characterization of quadrics might extend to non-
positively curved surfaces [GS, Appendix B].
S. Tabachnikov, however, took a significant and interesting step toward an
answer in [T], when he showed that—modulo genericity and $\,C^{2}\,$
assumptions that were later eliminated in [SS]— _negatively_ curved quadrics
admit no skewloops. That still left the converse question open, however: Does
lack of skewloops _characterize_ negatively curved quadrics?
We can now affirm that within a large class of surfaces, it does. To do so, we
merely combine results of the present paper with a lemma from [GS]:
###### Lemma 6.3 ([GS, Lemma 5.1]).
Suppose a $\,C^{2}\,$ embedded surface in $\,\mathbf{R}^{3}\,$ contains no
skewloop, and some affine plane cuts it transversely along an oval
$\,\mathcal{O}\,$. Then $\,\mathcal{O}\,$ is central.
Indeed, suppose $\,F:M\to\mathbf{R}^{3}\,$ immerses an open $\,C^{2}\,$
surface so that it cuts some affine plane transversally along an oval
$\,\mathcal{O}\,$. Then $\,F\,$ clearly embeds some annular neighborhood of
$\,F^{-1}(\mathcal{O})\subset M\,$ into $\,\mathbf{R}^{3}\,$ as a transversely
convex tube. Such a tube either does, or does not, have cpo, and
correspondingly, it either belongs to a central cylinder or quadric by
Proposition 5.1, or else it contains a skewloop by Lemma 6.3. We have thus
proven
###### Proposition 6.4.
Suppose a $\,C^{2}$-immersed surface $\,M\subset\mathbf{R}^{3}\,$ cuts an
affine plane transversally along an oval $\,\mathcal{O}$, but admits no
skewloop. Then some neighborhood of $\,\mathcal{O}\,$ in $\,M\,$ belongs to a
central cylinder or quadric.
If we assume completeness, we get a more elegant global statement:
###### Theorem 6.5.
Suppose a $C^{2}$-immersed surface $\,M\subset\mathbf{R}^{3}\,$ crosses some
plane transversally along an oval. Then exactly one of the following holds:
* (i)
$S\,$ contains a skewloop.
* (ii)
$S\,$ is the cylinder over an oval.
* (iii)
$S\,$ is a non-cylindrical quadric.
###### Proof.
Our hypotheses explicitly guarantee the existence of at least one oval
$\,\mathcal{O}\,$ along which $\,M\,$ cuts an affine plane transversally. But
they actually ensure that _all_ such ovals are central. For otherwise, Lemma
6.3 puts a skewloop on $\,M\,$. It follows that $\,M\,$ has cpo, and the
desired conclusion then follows from our Main Theorem 5.2 ∎
###### Corollary 6.6.
Every complete embedded negatively curved surface that meets a plane
transversely along an oval admits a skewloop, _unless_ it is affinely
congruent to the tube hyperboloid $\,x^{2}+y^{2}-z^{2}=1\,$.
###### Proof.
This follows immediately from Theorem 6.5, for among all cylinders and
quadrics having a compact cross-section, only the tube hyperboloid has
negative curvature. ∎
## Acknowledgments
Many thanks to the Technion—Israel Institute of Technology—for their
hospitality during a sabbatical in which much of this work got done, and to
the Lady Davis Foundation and Indiana University for the financial support
that made our visit there possible.
## References
* [1]
* [Bl] W. Blaschke, _Über affine Geometrie XXII: Bestimmung der Flächen mit zentrischen ebenen Schnitten_ , Gesammelte Werke. Band 4: Affine Differentialgeometrie: Differentialgeometrie der Kreis-und Kugelgruppen. Thales-Verlag, Essen, 1985.
* [Gh] M. Ghomi, _Shadows and convexity of surfaces_ , Ann. of Math. 155 281–293 (2002)
* [GS] M. Ghomi & B. Solomon, Skew loops and quadric surfaces, Comment. Math. Helv. 77 (4), 767–782 (2002)
* [Se] B. Segre, _Sulle coppie di tangenti fra ioro parallele relative ad una curve chuisa sghemba_ , Hommage au Professeur Lucien Godeaux, 141–167, Libraire Universitaire, Louvain (1968)
* [S] B. Solomon, _Symmetric cross-sections make surfaces of revolution quadric_ , Amer. Math. Monthly 116 (4), 351–355 (2009)
* [SS] J.-P. Sha & B. Solomon, _No skew branes on non-degenerate hyperquadrics_ , Math. Zeit. 257:225–229 (2007)
* [T] S. Tabachnikov, _On skew loops, skew branes, and quadratic hypersurfaces_ , Moscow Math. J. 3, 681–690 (2003)
|
arxiv-papers
| 2009-04-22T16:25:10 |
2024-09-04T02:49:02.076023
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bruce Solomon",
"submitter": "Bruce Solomon",
"url": "https://arxiv.org/abs/0904.3493"
}
|
0904.3562
|
# Radial velocity study of the CP star $\epsilon$ Ursae Majoris
N. A. Sokolov1,2
1Central Astronomical Observatory at Pulkovo, St. Petersburg 196140, Russia
2Isaac Newton Institute of Chile, Branch at St. Petersburg E-mail:
sokolov@gao.spb.ruBased on observations collected with the ELODIE spectrograph
on the 193-cm telescope at the Observatoire de Haute-Provence (CNRS), France
(Accepted 2007 November 16. Received 2007 November 16; in original form 2007
July 27)
###### Abstract
In this Letter, the radial velocity variability of the chemically peculiar
star $\epsilon$ Ursae Majoris ($\epsilon$ UMa) from the sharp cores of the
hydrogen lines is investigated. This study is based on the ELODIE archival
data obtained at different phases of the rotational cycle. The star exhibits
low-amplitude radial velocity variations with a period of $P$ = 5.0887 d. The
best Keplerian solution yields an eccentricity $e$ = 0.503 and a minimum mass
$\sim$14.7$M_{\rm Jup}$ on the hypothesis that the rotational axis of
$\epsilon$ UMa is perpendicular to the orbital plane. This result indicate
that the companion is the brown-dwarf with the projected semi-amplitude
variation of the radial velocity $K_{\rm 2}$ = 135.9 km ${\rm s}^{\rm-1}$ and
the sine of inclination times semi-major axis $a_{2}$$\sin$(i) = 0.055 au.
###### keywords:
binaries: spectroscopic – stars: chemically peculiar – stars: individual:
$\epsilon$ Ursae Majoris.
††pagerange: Radial velocity study of the CP star $\epsilon$ Ursae
Majoris–References††pubyear: 2008
## 1 Introduction
Epsilon Ursae Majoris ($\epsilon$ UMa, HD 112185, HR 4905) is the brightest
($V$ = 1.77 mag) chemically peculiar (CP) star and has been extensively
studied during the last century. Guthnick (1934) established a period of
5.d0887 d from variations in the intensity of the Ca II K line and also
noticed a periodic splitting of some lines. Struve & Hiltner (1943)
subsequently reported doubling of lines of Cr II, Fe II, V II and other
elements at certain phases. Since the overall widths of the double lines are
the same and not all lines double, they ruled out the orbital motion and
instead suggested that the phenomenon is related to rotation of the star.
Provin (1953) measured a double wave light variation with the same 5.d0887
period. The star is brightest when the Ca II K line intensity is near its
minimum and other elements are near their maximum strength.
Perhaps the most interesting and controversial aspect of $\epsilon$ UMa is its
the radial velocity variations. Abt & Snowden (1973) carried out the radial
velocity analysis of the 62 brightest northern CP stars for spectroscopic
binaries. They were used special observing and measuring techniques of the
hydrogen lines and did not find the radial velocity variations for $\epsilon$
UMa. Not surprisingly, they could not to measure the radial velocity
variations because the mean internal error per spectrum of 1.2 km ${\rm
s}^{-1}$ is near the limit at which the photographic technique is useful. On
the other hand, Morgan et al. (1978) detected the duplicity for the system
$\epsilon$ UMa with a separation of 0.053 arcsec, using speckle interferometry
technique. Although, the authors noted that the observed separation
corresponds to the diffraction limit of the 2.5-m Isaac Newton telescope.
Woszczyk & Jasiński (1980) measured the radial velocity variations of many
lines in the star and found sinusoidal variations in the radial velocities of
Fe, Cr and Ti lines with amplitudes of about 20 km ${\rm s}^{-1}$ and
attributed this to the existence of a spots of enhanced abundance of these
elements. Several surface abundance Doppler images of $\epsilon$ UMa have been
produced (Wehlau et al., 1982; Rice & Wehlau, 1990; Hatzes, 1991; Rice et al.,
1997; Holmgren & Rice, 2000). Most of the published Doppler images relate to
Fe, O, Ca, or Cr abundance. Recently, Lueftinger et al. (2003) determined for
$\epsilon$ UMa for the first time the abundance distributions of Mn, Ti, Sr,
and Mg. Attempt to determine orbital motion using these spectral lines would
be impossible. However, the hydrogen lines do not show significant rotational
effect in their radial velocity. They would be preferable for the measurement
of binary motion if the radial velocity of the hydrogen lines could be
measured with precision.
In this Letter, we present the radial velocity measurement of the hydrogen
lines in the spectrum of CP star $\epsilon$ UMa. The ELODIE observations of
this star and data reduction are described in Sect. 2. The orbital solution
derived for the satellite candidate are presented in Sect. 3. Section 4
reports discussion of our results for $\epsilon$ UMa and conclusions are
presented in Sect. 5.
## 2 Observational material and data reduction
Table 1: Measured radial velocities of $\epsilon$ UMa. All data are relative to the Solar system barycentre. File name | Exposure time | MJD | Phase | Radial velocity (km ${\rm s}^{-1}$) | |
---|---|---|---|---|---|---
| (s) | (2,450,000+) | | $H_{\delta}$ | $H_{\gamma}$ | $H_{\beta}$ | $H_{\alpha}$ | RVMean | $\sigma_{Mean}$
19960204 0022 | 168.73 | 0117.9814 | 0.821 | -9.56 | -9.19 | -9.93 | -9.47 | -9.54 | 0.31
19970218 0032 | 200.81 | 0497.9778 | 0.496 | -9.06 | -8.72 | -9.67 | -9.27 | -9.18 | 0.40
19970218 0033 | 200.37 | 0497.9819 | 0.497 | -8.90 | -8.77 | -9.51 | -9.19 | -9.09 | 0.33
20000124 0030 | 60.58 | 1568.1285 | 0.795 | -9.60 | -9.07 | -9.73 | -9.48 | -9.47 | 0.29
20000124 0031 | 90.57 | 1568.1440 | 0.798 | -9.50 | -9.16 | -9.71 | -9.63 | -9.50 | 0.24
20000126 0040 | 101.56 | 1570.1447 | 0.191 | -8.11 | -8.24 | -8.99 | -8.94 | -8.57 | 0.46
20000126 0041 | 50.58 | 1570.1499 | 0.192 | -8.29 | -8.28 | -9.07 | -8.99 | -8.66 | 0.43
20000519 0012 | 120.79 | 1683.9584 | 0.557 | -9.04 | -8.82 | -9.17 | -8.99 | -9.01 | 0.14
20000520 0009 | 120.76 | 1684.8706 | 0.737 | -9.56 | -9.02 | -9.32 | -9.13 | -9.26 | 0.24
20000521 0006 | 600.78 | 1685.8694 | 0.933 | -10.14 | -9.49 | -9.71 | -9.36 | -9.68 | 0.34
20000521 0007 | 600.44 | 1685.8784 | 0.935 | -9.85 | -9.44 | -9.63 | -9.44 | -9.59 | 0.20
20000606 0017 | 30.54 | 1701.8490 | 0.073 | -8.05 | -7.89 | -9.04 | -8.55 | -8.38 | 0.52
20000607 0007 | 30.27 | 1702.8632 | 0.272 | -8.44 | -8.33 | -8.73 | -8.44 | -8.48 | 0.17
20000608 0006 | 30.53 | 1703.8467 | 0.466 | -9.09 | -8.80 | -9.24 | -8.87 | -9.00 | 0.20
20000609 0003 | 30.97 | 1704.8443 | 0.662 | -9.34 | -8.64 | -9.35 | -8.68 | -9.00 | 0.40
20000615 1417 | 300.17 | 1710.8469 | 0.841 | -9.51 | -8.94 | -9.76 | -9.08 | -9.32 | 0.38
20000615 1420 | 50.47 | 1710.8505 | 0.842 | -9.46 | -8.74 | -9.79 | -9.30 | -9.32 | 0.44
20000615 1423 | 30.76 | 1710.8608 | 0.844 | -9.61 | -8.63 | -9.77 | -9.05 | -9.27 | 0.52
20000615 1425 | 30.26 | 1710.8634 | 0.844 | -9.68 | -8.84 | -9.67 | -9.17 | -9.34 | 0.41
20000615 1426 | 30.26 | 1710.8661 | 0.845 | -9.67 | -8.99 | -10.04 | -9.13 | -9.46 | 0.49
20000616 1418 | 50.92 | 1711.8591 | 0.040 | -8.35 | -7.88 | -9.11 | -8.85 | -8.55 | 0.55
20000616 1421 | 20.71 | 1711.8630 | 0.041 | -8.17 | -7.81 | -9.17 | -8.94 | -8.52 | 0.64
20000616 1424 | 20.65 | 1711.8670 | 0.042 | -8.13 | -8.00 | -9.33 | -8.94 | -8.60 | 0.64
20000618 0006 | 30.59 | 1713.8460 | 0.431 | -8.52 | -8.45 | -9.04 | -8.80 | -8.70 | 0.27
20000618 0007 | 30.71 | 1713.8506 | 0.432 | -8.52 | -8.40 | -8.76 | -8.74 | -8.61 | 0.17
20000618 0008 | 30.97 | 1713.8542 | 0.432 | -8.43 | -8.42 | -8.84 | -8.80 | -8.62 | 0.23
20000619 0024 | 30.49 | 1714.8368 | 0.625 | -8.85 | -8.47 | -9.47 | -8.77 | -8.89 | 0.42
20000619 0025 | 30.54 | 1714.8715 | 0.632 | -9.37 | -8.68 | -9.22 | -8.76 | -9.01 | 0.34
20000619 0026 | 30.04 | 1714.8742 | 0.633 | -9.23 | -8.96 | -9.17 | -8.83 | -9.05 | 0.19
Table 2: Best Keplerian orbital solution derived for $\epsilon$ UMa Parameter | | Value | Error
---|---|---|---
P (fixed) | (d) | 5.0887 |
T (periastron) | (JD-2450000) | 1743.932 | 0.161
e | | 0.503 | 0.063
${V}_{\rm 0}$ | (km ${\rm s}^{-1}$) | -8.920 | 0.030
$\omega$ | ($\degr$) | 260.53 | 12.91
a $\sin$(i) | (${\rm 10}^{-3}$ au) | 0.256 | 0.022
K | (km ${\rm s}^{-1}$) | 0.634 | 0.067
f(m) | (${\rm 10}^{-7}$ ${\rm M}_{\odot}$) | 0.868 | 0.227
The spectra of $\epsilon$ UMa were retrieved from the ELODIE archive (Moultaka
et al., 2004). This archive contains the complete collection of high-
resolution echelle spectra using the ELODIE fiber-fed echelle spectrograph
(Baranne et al., 1996) mounted on the 1.93-m telescope at the Haute-Provence
Observatory (France). The spectra have a resolution
($\lambda$/$\triangle$$\lambda$) of about 42000. The archived signal-to-noise
ratio was between $\sim$200 and $\sim$400 in the spectral region near
$\lambda$ 5550 Å. In addition, eight spectra obtained 15 and 16 June 2000 at
the same telescope were retrieved from the Hypercat Fits Archive
(http://leda.univ-lyon1.fr/11/spectrophotometry.html). In Table 1, each
spectrum is presented by its file name, exposure time, Julian date of the
observations and its corresponding phase. Note that the phases were computed
with the ephemeris from Table 2 (see below).
Special technique of radial velocity measurement of the hydrogen lines was
used and should be explained. Well known, that in slowly rotating late B-type
stars the hydrogen lines have broad wings and sharp cores. For $\epsilon$ UMa
the value of $v$ sin $i$ is equal 35 km ${\rm s}^{-1}$ (Lueftinger et al.,
2003). Unfortunately, the wings of the hydrogen lines can be affected by
different spectral lines. For example, in spectrum of $\epsilon$ UMa the wing
of $H_{\beta}$ line affected by many Cr lines (Žižnovský & Zverko, 1995). On
the other hand, the sharp cores of the hydrogen lines are not affected by
spectral lines and the flux is formed in the upper layers of the atmosphere.
Thus our technique was to use only the sharp cores of the hydrogen lines in
spectra of $\epsilon$ UMa. The spectra were processed using the spectral
reduction software SPE developed by S. Sergeev at Crimean Astrophysical
Observatory (CrAO). The program allows detecting the variations of the centre
gravity of the sharp cores of the hydrogen lines. After processing, all the
spectra were corrected for the motion of the Earth around the Sun. For
example, the demonstration of the positional variability of the core
$H_{\alpha}$ line is presented in Fig. 1. In Table 1, each spectrum is
presented by its radial velocities computed from the hydrogen lines, the mean
radial velocity and errors of the mean radial velocity.
Figure 1: The observed intensity profiles of the cores of $H_{\alpha}$ line
obtained on May 21, 2000 (phase = 0.93) and on June 6, 2000 (phase = 0.07)
marked by the solid and dashed lines, respectively. The velocity scale is
given with respect to the $\lambda$ = 6562.797 Å.
## 3 Orbital parameters
Orbital elements have been determined by a non-linear least-squares fitting of
the mean radial velocities from Table 1 using the program BINARY writing D.H.
Gudehus from Georgia State University
(http://www.chara.gsu.edu/$\sim$gudehus/binary.html). The solution for a
single-lined binary is modelled by up to six parameters:
1. 1.
P period
2. 2.
T time of periastron passage
3. 3.
e eccentricity
4. 4.
${V}_{\rm 0}$ system radial velocity
5. 5.
$\omega$ longitude of periastron
6. 6.
a $\sin$(i) sine of inclination times semi-major axis
The expected radial velocities are
$RV=K[\cos{(\theta+\omega)}+e\cos{\omega}]$ (1)
where $\theta$ is the angular position of the star measured from the centre of
mass at a given instant. The program also calculate the projected semi-
amplitude variation of the radial velocity:
$K=\frac{2\pi a\sin{i}}{P\sqrt{1-e^{2}}},$ (2)
though this is never used as a parameter in the solution and the mass
function:
$f(m)=\frac{M^{3}_{2}{\sin}^{3}i}{(M_{1}+M_{2})^{2}}.$ (3)
The program BINARY gives the estimated standard deviations of the orbital
parameters as well. The orbital solution of Table 2 was obtained by fitting a
Keplerian orbit to the 29 ELODIE radial velocity measurements. Note that most
of observations were obtained between JD=2451568 (January 2000) and JD=2451714
(June 2000) (see Table 1).
Experience shows that the best Keplerian fit to the data with the fixed period
P = 5.0887 d, the eccentricity e = 0.503 and the semi-amplitude K = 0.634 km
${\rm s}^{-1}$. The parameters of the best Keplerian orbital solution for
$\epsilon$ UMa are presented in Table 2. In close binary system with Bp-Ap
stars, there is evidence for a tendency toward synchronization between the
rotational and orbital motions. This effect is thought to be produced by the
tidal forces (Gerbaldi et al., 1985). The radial velocity curve is displayed
in Fig. 2 with the residuals around solution. A linear trend is not observed
in the residuals around the orbital solution that can be explained by the
absent of a second companion in a longer-period orbit. Although, the weighted
r.m.s. around the best Keplerian solution ($\sigma$(O-C)) is equal to 0.131 km
${\rm s}^{-1}$. This value is a bit large compared to the typical radial
velocity measurements from the ELODIE cross-correlation function (Naef et al.,
2003).
Figure 2: $Top:$ Phase diagram of the radial velocity measurements and
Keplerian orbital solution for $\epsilon$ UMa. $Bottom:$ Residuals around the
solution.
## 4 Discussion
The presence of the spots on the stellar surface of $\epsilon$ UMa can change
the observed spectral line profiles and induces a periodic radial velocity
signal similar to the one expected from the presence of a satellite. The
hydrogen lines analysis is one of the best tools to discriminate between
radial velocity variations due to changes in the spectral line shapes and
variations due to the real orbital motion of the star. It is obviously of
interest to compare the phase diagram of the radial velocity of $\epsilon$ UMa
derived above with the phase diagram of the radial velocity computed from
metallic lines. In this way, we selected the spectral region of the very
prominent unblended Cr II line at $\lambda$ 4558 Å.
Figure 3: Positional variations of strong photosphere lines in the spectral
region of the Cr II line at $\lambda$ 4558 Å obtained on June 15, 2000 (phase
= 0.84) and on January 26, 2000 (phase = 0.19) marked by the solid and dashed
lines, respectively. The velocity scale is given with respect to the $\lambda$
= 4558.65 Å.
Figure 3 shows the strong photosphere lines in the spectral region of the Cr
II line at $\lambda$ 4558 Å obtained at phases before and after the epoch of
periastron passage. The radial velocity shift of the Cr II $\lambda\lambda$
4554, 4558 and 4565 ÅÅ lines and the Fe II $\lambda$ 4555 Å line at different
phases is clearly seen. The Ti II line at $\lambda$ 4563 Å shows the line
doubling that appears at phase 0.84 and disappears at phase 0.19. The maximum
splitting of this line is at phase 0.04. The splitting of the lines was first
observed by Struve & Hiltner (1943) in the spectrum of $\epsilon$ UMa.
However, the authors concluded that the doubling is not caused by orbital
motion but may be due to a combination of the physical effects with Doppler
effect in a rotating star.
Wade (1997) estimated the masses for 10 magnetic CP stars using the position
of stars in the log$R_{\sun}$ – log$T_{\rm eff}$ plane. Position of $\epsilon$
UMa in the log$R_{\sun}$ – log$T_{\rm eff}$ plane seems to be among the most
evolved CP stars and gives the value of $M_{\epsilon{\rm UMa}}$ =
3.0$\pm$0.4$M_{\sun}$. According to Lueftinger et al. (2003) the radius of
$\epsilon$ UMa is equal to 4.2$\pm$0.2$R_{\sun}$ corresponding to their choice
of $v\sin i$ = 35 km ${\rm s}^{\rm-1}$ and $i$ = ${\rm 45}^{\circ}$ using the
trigonometric parallax measured by Hipparcos, $\pi$ = 40.30 mas (ESO, 1997)
and an angular diameter of 1.561 mas. The effective temperature of $\epsilon$
UMa was taken from the paper by Sokolov (1998) and is equal to 9340$\pm$530 K.
On the hypothesis that the rotational axis of $\epsilon$ UMa is perpendicular
to the orbital plane we can estimate the mass of the secondary star. For the
value of $M_{\rm 1}$ = 3.0$M_{\sun}$ and $i$ = ${\rm 45}^{\circ}$ Eq.(3) gives
$M_{\rm 2}$ = 0.014$M_{\sun}$. This result gives a value $\sim$14.7$M_{\rm
Jup}$, strongly suggesting that the companion is in the typical brown-dwarf
regime.
If we know the value $M_{\rm 1}$/$M_{\rm 2}$ then it is possible to estimate
the projected semi-amplitude variation of the radial velocity for the
companion according to formula:
$K_{2}=K_{1}\frac{M_{1}}{M_{2}},$ (4)
where $K_{1}$ is the projected semi-amplitude variation of the radial velocity
for $\epsilon$ UMa taken from Table 2. For the value of $K_{\rm 1}$ = 0.634 km
${\rm s}^{\rm-1}$ Eq.(4) gives $K_{\rm 2}$ = 135.9 km ${\rm s}^{\rm-1}$. Thus,
we can estimate the sine of inclination times semi-major axis using Eq.(2).
Computation gives the value of $a_{2}$$\sin$(i) = 0.055 au. These estimates
show that the proposed brown-dwarf is quite close to the surface of $\epsilon$
UMa at periastron. But, such close orbits are not new. For example, the
subgiant star HD 118203 have the planet with eccentric orbit ($e$ = 0.31), the
period of $P$ = 6.1335 d, and is close to its parent star ($a$ = 0.06 au) (Da
Silva et al., 2005).
Another way to interpret the radial velocity variations is the radial
pulsation of $\epsilon$ UMa. Retter et al. (2004) are analysed observations of
$\epsilon$ UMa obtained with the star tracker on the Wide Field Infrared
Explorer satellite. The authors observed that a light curve has about 2 per
cent amplitude of photometric variability. On the other hand, Molnar (1975)
has presented ultraviolet light curves for $\epsilon$ UMa from the OAO-2
satellite which indicate that the photometric variations of this star are due
to variable ultraviolet absorption effects which redistribute flux into the
visible region (see his Fig. 5). Note that no colour index changes (Provin,
1953). Certainly the radial pulsation appears unlikely given that the
rotational period is synchronized with the orbital period of $\epsilon$ UMa.
## 5 Conclusions
The archival ELODIE high-resolution echelle spectra of $\epsilon$ UMa permit
us to analyse the radial velocity variations of the sharp cores of the
hydrogen lines. This allowed determining the orbital elements of binary system
for the CP star $\epsilon$ UMa. The best Keplerian fit to the data shown that
the rotational period is synchronized with the orbital period. We are
estimated the mass of the secondary star which is equal $\sim$14.7$M_{\rm
Jup}$. This result indicate that the companion is the brown-dwarf with the
projected semi-amplitude variation of the radial velocity $K_{\rm 2}$ = 135.9
km ${\rm s}^{\rm-1}$ and the sine of inclination times semi-major axis
$a_{2}$$\sin$(i) = 0.055 au.
## Acknowledgements
The author would like to thank the referee Dr. J.B. Rice of this Letter for
his extremely helpful comments.
## References
* Abt & Snowden (1973) Abt H.A., Snowden M.S., 1973, ApJSS, 25, 137
* Baranne et al. (1996) Baranne A., Queloz D., Mayor M., Adrianzyk G., Knispel G., Kohler D., Lacroix D., Meunier J.-P., et al, 1996, A&AS, 119, 373
* Da Silva et al. (2005) Da Silva R., Udry S., Bouchy F., Mayor M., Moutou C., Pont F., Queloz D., Santos N.C., et al, 2005, (astro-ph/0510048)
* Gerbaldi et al. (1985) Gerbaldi M., Floquet M., Hauck B., 1985, A&A, 146, 341
* Guthnick (1934) Guthnick P., 1934, Sitz. Preuss. Akad. Wiss. Berlin, 30, 506
* Hatzes (1991) Hatzes A.P., 1991, MNRAS, 253, 89
* Holmgren & Rice (2000) Holmgren D.E., Rice J.B., 2000, A&A, 364, 660
* Lueftinger et al. (2003) Lueftinger T., Kuschnig R., Piskunov N.E., Weiss W.W., 2003, A&A, 406, 1033
* Molnar (1975) Molnar M.R., 1975, AJ, 80, 137
* Morgan et al. (1978) Morgan B.L., Beddoes D.R., Scaddan R.J., Dainty J.C., 1978, MNRAS, 183, 701
* Moultaka et al. (2004) Moultaka J., Ilovaisky S.A., Prugniel P., Soubiran C., 2004, PASP, 116, 693
* Naef et al. (2003) Naef D., Mayor M., Beuzit J.L., Perrier C., Queloz D., Sivan J.P., Udry S., 2003, (astro-ph/0310261)
* Provin (1953) Provin S.S., 1953, ApJ, 118, 489
* Retter et al. (2004) Retter A., Bedding T,R., Buzasi D.L., Kjeldsen H., Kiss L.L., 2004, ApJ, 601, L95
* Rice & Wehlau (1990) Rice J.B., Wehlau W.H., 1990, A&A, 233, 503
* Rice et al. (1997) Rice J.B., Wehlau W.H., Holmgren D.E., 1997, A&A, 326, 988
* Sokolov (1998) Sokolov N.A., 1998, A&AS, 130, 215
* Struve & Hiltner (1943) Struve O., Hiltner W.A, 1943, ApJ, 98, 225
* Stibbs (1950) Stibbs D.W.N., 1950, MNRAS, 110,395
* Wade (1997) Wade G.A., 1997, A&A, 325,1063
* Wehlau et al. (1982) Wehlau W., Rice J., Piskunov N., Khokhlova V., 1982, Pis’ma Astron. Zh., 8, 30
* Woszczyk & Jasiński (1980) Woszczyk A., Jasiński M., 1980, Acta Astron., 30, 331
* Žižnovský & Zverko (1995) Žižňovský J., Zverko J., 1995, Contrib. Astron. Obs. Skalnaté Pleso, 25, 39
|
arxiv-papers
| 2009-04-23T15:09:16 |
2024-09-04T02:49:02.090184
|
{
"license": "Public Domain",
"authors": "N.A. Sokolov",
"submitter": "Sokolov Nikolay",
"url": "https://arxiv.org/abs/0904.3562"
}
|
0904.3577
|
# Solving the Wheeler-DeWitt of Small Universe
Shintaro Sawayama sawayama0410@gmail.com Sawayama Cram School of Physics
Atsuhara 328, Fuji-city, Shizuoka prefecture 419-0201, Japan
###### Abstract
We can solve the Wheeler-DeWitt equation of the small universe enough to
metric becomes diagonal and take a Gaussian normal coordinate. Our previous
works are concerning to this paper. In this paper, we only write how to solve
the Wheeler-DeWitt equation of such universe. Our motivation is simple, that
is to solve the Wheeler-DeWitt equation. Even if the Wheeler-DeWitt equation
is solved, quantum gravity does not complete yet. However, this work may be
one of the first step to quantum gravity.
###### pacs:
04.60.-m, 04.60.Ds
## I Introduction
In the quantum gravity, there are many approaches, for example loop quantum
gravityAs Rov Thi or mini-superspaceHart approachs or string approachesAL .
However, quantum gravity has not completed yet. In the canonical quantum
gravity, the difficulties comes from the Wheeler-DeWitt equation. At first we
should solve the Wheeler-DeWittDe equation the quantum gravity does not
start.
In our previous work is concerning the Wheeler-DeWitt equation. Our motivation
is simple to solve the Wheeler-DeWitt equation. However, Wheeler-DeWitt is
difficult to solve. Because it is second order elliptic functional partial
differential equation with non-linear term. The elliptic differential equation
is difficult, but if we choose diagonal metric, this problem is solved. The
partial differential equation is difficult, but if we use additional
constraint equation, this problem is solved. The functional differential
equation is difficult, but we construct how to solve the functional
differential equation. The non-linear term is difficult, but we construct the
method to remove this difficulty.
We quantize diagonal metric universe by using Wheeler-DeWitt without
approximation. We use the fact metric become diagonal by coordinate
transformation and Gaussian normal coordinate. Such universe is small
universe. Or we treat the Wheeler-DeWitt equation locally. Because gravity is
quantize in very small universe. To treat such small universe is theoretical.
In section II, we simplify the Wheeler-DeWitt equation. In section III we
quantize toy model. And in this section we show the step to solve the Wheeler-
DeWitt. In section IV we quantize full quantum gravity, using the obtained
result in section III. In section V we summarize and conclude obtained result.
## II Simplification of the Wheeler-DeWitt
For simplicity we treat the only diagonal metric universes as,
$\displaystyle\begin{pmatrix}g_{00}&0&0&0\\\ 0&g_{11}&0&0\\\ 0&0&g_{22}&0\\\
0&0&0&g_{33}.\end{pmatrix}$ (1)
And in all the spacetime metrics become diagonal locally. Or we treat small
universe enough to metric become diagonal. We start from decomposition of the
Einstein Hilbert action of the above diagonal metric universe. The action of
the above universe is written by
$\displaystyle S=\int RdM=\int R[g_{\mu\mu}]dSdt.$ (2)
Here $S$ is the hyper-surface with constant time. Because, our method is
different from the usual Wheeler-DeWitt equation formalism, our obtained
Hamiltonian constraint is a different type of the Wheeler-DeWitt equation. If
we decompose this action as 3+1, then we can obtain
$\displaystyle{\cal L}=\dot{q}_{ii}P^{ii}+NH-2\sqrt{q}D^{i}N_{,i}.$ (3)
Here $N$ is the lapse functional and $N_{,i}$ is the sift vectors, and $H$ is
the Hamiltonian constraint such that
$\displaystyle H=\frac{1}{2}q_{ii}q_{jj}P^{ii}P^{jj}+{\cal R}.$ (4)
Here ${\cal R}$ is the three dimensional Ricci scalar and $P^{ii}$ is the
momentum whose commutation relation with $q_{ii}$ is not $i$, it is
$i\sqrt{q}$. In this formulation there are not appear $q_{ij}$ and $P^{ij}$
and sift vectors and momentum constraint. So we can ignore the constraint as
$[P^{ij},H]$ or $[[P^{ij},H],H]$, because we start with metric diagonal
setting. In this simple case we can ignore the diffeomorphism constraints. If
we write the Hamiltonian constraint in the operator representation, we obtain
$\displaystyle
H=\sum_{ij}\frac{1}{2}\frac{\delta^{2}}{\delta\phi_{i}\delta\phi_{j}}+{\cal
R}[q_{11},q_{22},q_{33}]=0$
$\displaystyle=\sum_{ij}\frac{1}{2}\frac{\delta^{2}}{\delta\phi_{i}\delta\phi_{j}}+\sum_{i\not=j}(\hat{\phi}_{i,jj}+\hat{\phi}_{j,i}\hat{\phi}_{i,i})e^{\hat{\phi}_{i}}=0.$
(5)
Here $\phi_{i}=\ln q_{ii}$. If we consider the $\phi$ were only depend
$t,x_{i}$, the Hamiltonian constraint becomes,
$\displaystyle
H=\sum_{ij}\frac{1}{2}\frac{\delta^{2}}{\delta\phi_{i}\delta\phi_{j}}-\Lambda=0.$
(6)
We use this Hamiltonian constraint in section III Because we only treat metric
diagonal universe, the Hamiltonian constraint has different form from the
orthodox Hamiltonian constraint. This setting is similar to mini-superspace
model.
If universe is small enough to apply Gaussian normal coordinate, the
Hamiltonian constraint become
$\displaystyle
H=\sum_{ij}\frac{1}{2}\frac{\delta^{2}}{\delta\phi_{i}\delta\phi_{j}}+\sum_{i\not=j}\hat{\phi}_{i,jj}e^{\phi_{i}}=0$
(7)
We treat this Hamiltonian constraint in section IV
The static restriction deviated from up-to-down methodSa1 is
$\displaystyle\sum_{i\not=j}\frac{\delta}{\delta\phi_{i}\delta\phi_{j}}.$ (8)
## III Simple example of Wheeler-DeWitt
Before solving Eq.(7), we solve Eq.(6) for simplicity. We consider the
following spacetime
$\displaystyle\begin{pmatrix}-N^{2}&0&0&0\\\ 0&g_{1}(t,x)&0&0\\\
0&0&g_{2}(t,y)&0\\\ 0&0&0&g_{3}(t,z)\end{pmatrix}.$ (9)
To quantize this spacetime we take steps. The step 1 is using the static
restriction we obtain special solution. The step 2 is to remove the static
restriction we obtain general solution. The we carry out step 1. The Eq.(6)
and Eq.(8) is consistent and simultaneously quantized. And the solution is
$\displaystyle
f[\phi_{1},\phi_{2},\phi_{3}]=\prod_{i}\exp(a_{i}\Lambda^{1/2}\int\delta\phi_{i})$
(10)
where
$\displaystyle a_{1}a_{2}+a_{1}a_{3}+a_{2}a_{3}=1$ (11) $\displaystyle
a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=1$ (12)
Then we carry out step 2. Using the above solution as a special solution, we
assume the state is a form
$\displaystyle|\Psi\rangle=f[\phi_{i}]g[\phi_{i}].$ (13)
And we remove the static restriction, then $g[\phi_{i}]$ should satisfy
$\displaystyle\nabla(\nabla+a)g[\phi_{i}]=0$ (14)
Here, $\nabla$ is defined by
$\displaystyle\nabla=\sum_{i}\frac{\delta}{\delta\phi_{i}}.$ (15)
And $a$ is defined by $a=\Lambda^{1/2}\sum_{i}a_{i}$ This equation can be
solved easily and solution is
$\displaystyle
e^{-a\int\delta\phi_{1}}(-2\int\delta\phi_{1}+\int\delta\phi_{2}+\int\delta\phi_{3})+e^{-a\int\delta\phi_{2}}(-2\int\delta\phi_{2}+\int\delta\phi_{1}+\int\delta\phi_{3})$
$\displaystyle+e^{-a\int\delta\phi_{3}}(-2\int\delta\phi_{3}+\int\delta\phi_{2}+\int\delta\phi_{1})$
(16)
So the quantum state of (9) is
$\displaystyle|\Psi(\phi_{i})\rangle=\prod_{i}\exp(a_{i}\Lambda^{1/2}\int\delta\phi_{i})\bigg{[}e^{-a\int\delta\phi_{1}}(-2\int\delta\phi_{1}+\int\delta\phi_{2}+\int\delta\phi_{3})$
$\displaystyle+e^{-a\int\delta\phi_{2}}(-2\int\delta\phi_{2}+\int\delta\phi_{1}+\int\delta\phi_{3})+e^{-a\int\delta\phi_{3}}(-2\int\delta\phi_{3}+\int\delta\phi_{1}+\int\delta\phi_{2})\bigg{]}$
(17)
## IV Solving the Wheeler-DeWitt of small universe
By the same method we can quantize following universe
$\displaystyle\begin{pmatrix}-N^{2}&0&0&0\\\ 0&g_{1}(t,x,y,z)&0&0\\\
0&0&g_{2}(t,x,y,z)&0\\\ 0&0&0&g_{3}(t,x,y,z)\end{pmatrix}$ (18)
Here $x,y,z$ are Gaussian normal coordinates. The step 1 is to use a static
restriction to the Eq.(7), then we obtain
$\displaystyle\sum_{i}\frac{\delta^{2}}{\delta\phi_{i}^{2}}+2\phi_{i,jj}e^{\phi_{i}}=0.$
(19)
Then we use parameter separation, i.e. the solution of the above equation is
assumed to be written as
$\displaystyle f[\phi_{i}]=f_{1}[\phi_{1}]f_{2}[\phi_{2}]f_{3}[\phi_{3}]$ (20)
Then Eq. (19) becomes
$\displaystyle\frac{\delta^{2}}{\delta\phi_{i}^{2}}+2\phi_{i,jj}e^{\phi_{i}}=0$
(21)
Or,
$\displaystyle\frac{\delta^{2}}{\delta a_{i}^{2}}+8\partial_{j}\partial^{j}\ln
a_{i}=0.$ (22)
Here $a_{i}=g_{i}^{1/2}$. Then we use a following equation
$\displaystyle-i(\ln a_{i,jj})^{1/2}\frac{\delta}{\delta
a_{i}}+i\frac{\delta}{\delta a_{i}}(\ln a_{i,jj})^{1/2}=i\delta\frac{1}{2}(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}$ (23)
Now we briefly write $\partial_{j}\partial^{j}\ln a_{i}=\ln a_{i,jj}$. Off
course it is diferent, we simplisitily use the latter. Then Eq.(22) becomes
$\displaystyle\frac{\delta^{2}}{\delta a_{i}^{2}}+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\frac{\delta}{\delta a_{i}}-2\sqrt{2}i\frac{\delta}{\delta
a_{i}}(\ln a_{i,jj})^{1/2}+8\ln a_{i,jj}=i\sqrt{2}\delta(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}$ (24)
Or
$\displaystyle\bigg{(}\frac{\delta}{\delta a_{i}}+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\bigg{)}\bigg{(}\frac{\delta}{\delta a_{i}}-2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\bigg{)}=\sqrt{2}i\delta(\ln a_{i,jj})^{-1/2}a_{i,jj}^{-1}.$
(25)
Then we take following assumptions
$\displaystyle\frac{\delta}{\delta a_{i}}+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}=g_{1}[a_{i}]$ (26) $\displaystyle\frac{\delta}{\delta
a_{i}}-2\sqrt{2}i(\ln a_{i,jj})^{1/2}=g_{2}[a_{i}].$ (27)
Here,
$\displaystyle g_{1}[a_{i}]g_{2}[a_{i}]=i\delta\sqrt{2}(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}$ (28)
Then second partial functional derivative become ordinal functional derivative
and the equation become following
$\displaystyle\frac{\delta f^{1/2}[a_{i}]}{\delta a_{i}}+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}f^{1/2}[a_{i}]=g_{1}[a_{i}]f^{1/2}[a_{i}]$ (29)
$\displaystyle\frac{\delta f^{1/2}[a_{i}]}{\delta a_{i}}-2\sqrt{2}i(\ln
a_{i,jj})^{1/2}f^{1/2}[a_{i}]=g_{2}[a_{i}]f^{1/2}[a_{i}].$ (30)
From this equation we obtain
$\displaystyle\frac{1}{2}\ln f[a_{i}]=\int g_{1}[a_{i}]+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\delta a_{i}$ (31) $\displaystyle\frac{1}{2}\ln f[a_{i}]=\int
g_{2}[a_{i}]-2\sqrt{2}i(\ln a_{i,jj})^{1/2}\delta a_{i}$ (32)
We can obtain the solution
$\displaystyle f[a_{i}]=\exp\bigg{(}2\int g_{1}[a_{i}]+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\delta a_{i}\bigg{)}$ (33) $\displaystyle
f[a_{i}]=\exp\bigg{(}2\int g_{2}[a_{i}]-2\sqrt{2}i(\ln a_{i,jj})^{1/2}\delta
a_{i}\bigg{)}$ (34)
Because $f[a_{i}]$ is same
$\displaystyle g_{1}[a_{i}]=g_{2}[a_{i}]-4\sqrt{2}(\ln a_{i,jj})^{1/2}.$ (35)
Inserting this equation to Eq.(28), we obtain
$\displaystyle g_{2}[a_{i}]^{2}-4\sqrt{2}(\ln
a_{i,jj})^{1/2}g_{2}[a_{i}]=i\delta\sqrt{2}(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}.$ (36)
Exchangeng this equation, we obtain
$\displaystyle g_{2}[a_{i}]^{2}-4\sqrt{2}(\ln
a_{i,jj})^{1/2}g_{2}[a_{i}]-i\delta\sqrt{2}(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}=0.$ (37)
By solving this second order function equation, we obtain
$\displaystyle g_{2}[a_{i}]=2\sqrt{2}(\ln a_{i,jj})^{1/2}\pm\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{-1/2}a_{i,jj}^{-1}}.$ (38)
Inserting this equation to Eq.(34), we obtain
$\displaystyle f[a_{i}]=\exp\bigg{(}\int\pm 2\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{-1/2}a_{i,jj}^{-1}}\delta
a_{i}\bigg{)}.$ (39)
So the quantization of the spacetime as (18) is
$\displaystyle f[a_{1},a_{2},a_{3}]=\prod_{i}\exp\bigg{(}\int\pm 2\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{1/2}a_{i,jj}^{-1}}\delta
a_{i}\bigg{)}.$ (40)
This is the special solution of quantization of the spacetime (18).
Then we carry out step 2, we remove the static restriction and we assume the
state is the form of the
$\displaystyle|\Psi\rangle=f[a_{i}]h[a_{i}]$ (41)
Then we obtain
$\displaystyle\nabla(\nabla+h^{\prime}[a_{i}])h[a_{i}]=0.$ (42)
Here,
$\displaystyle h^{\prime}[a_{i}]=\sum_{i}\pm\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{1/2}a_{i,jj}^{-1}}$ (43)
The functional partial derivative
$\displaystyle(\nabla+h^{\prime}[a_{i}])h[a_{i}]=0$ (44)
is solved analytically. The solution is
$\displaystyle
h[a_{i}]=\exp\bigg{(}\int^{a_{1}}h^{\prime}[a_{1}^{\prime},-a_{1}+a_{2}+a_{1}^{\prime},-a_{1}+a_{3}+a_{1}^{\prime}]\delta
a_{1}^{\prime}\bigg{)}(-2\int\delta a_{1}+\int\delta a_{2}+\int\delta a_{3})$
$\displaystyle+\exp\bigg{(}\int^{a_{2}}h^{\prime}[a_{1}-a_{2}+a_{2}^{\prime},a_{2}^{\prime},-a_{2}+a_{3}+a_{2}^{\prime}]\delta
a_{2}^{\prime}\bigg{)}(-2\int\delta a_{2}+\int\delta a_{1}+\int\delta a_{3})$
$\displaystyle+\exp\bigg{(}\int^{a_{3}}h^{\prime}[-a_{3}+a_{1}+a_{3}^{\prime},-a_{3}+a_{2}+a_{3}^{\prime},a_{3}^{\prime}]\delta
a_{3}^{\prime}\bigg{)}(-2\int\delta a_{3}+\int\delta a_{1}+\int\delta a_{2})$
(45)
So the quantum state of spacetime (18) is
$\displaystyle|\Psi\rangle=\prod_{i}\exp\bigg{(}\int\pm 2\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{-1/2}a_{i,jj}^{-1}}\delta
a_{i}\bigg{)}$
$\displaystyle\times\bigg{[}\exp\bigg{(}\int^{a_{1}}h^{\prime}[a_{1}^{\prime},-a_{1}+a_{2}+a_{1}^{\prime},-a_{1}+a_{3}+a_{1}^{\prime}]\delta
a_{1}^{\prime}\bigg{)}(-2\int\delta a_{1}+\int\delta a_{2}+\int\delta a_{3})$
$\displaystyle+\exp\bigg{(}\int^{a_{2}}h^{\prime}[a_{1}-a_{2}+a_{2}^{\prime},a_{2}^{\prime},-a_{2}+a_{3}+a_{2}^{\prime}]\delta
a_{2}^{\prime}\bigg{)}(-2\int\delta a_{2}+\int\delta a_{1}+\int\delta a_{3})$
$\displaystyle+\exp\bigg{(}\int^{a_{3}}h^{\prime}[-a_{3}+a_{1}+a_{3}^{\prime},-a_{3}+a_{2}+a_{3}^{\prime},a_{3}^{\prime}]\delta
a_{3}^{\prime}\bigg{)}(-2\int\delta a_{3}+\int\delta a_{1}+\int\delta
a_{2})\bigg{]}$ (46)
This is the main result of our paper.
## V Conclusion and Discussions
We quantized diagonal metric space with Gaussian normal coordinate. By solving
the Wheeler-DeWitt equation, we know the form of the solution. We quantize two
universe. The common feature is similarity of step 2. The step 2 is all ways
solved. So the important point is to find the special solution.
There are many further work. Once we should calculate the inner product and
norm. If the normalization is end the state is used to calculate the averaged
value. Or we should discuss problem of the norm. This open issue is very
important one. And we should search the initial singularity or the black hole
singularity. Our work does not end.
## References
* (1) C.J.Isham and A.Ashtekar Class. Quant. Grav. 9 1433 (1992)
* (2) C.Rovelli Quantum Gravity; Cambridge monographs on mathematical physics (2004)
* (3) T.Thiemann gr-qc/0110034 (2001)
* (4) A.Linde hep-th/0503195 (2005)
* (5) S.Sawayama gr-qc/0604007 (2006)
* (6) B.S.DeWitt Phys. Rev. 160 1113 (1967)
* (7) J.J.Halliwell and J.B.Hartle Phys. Rev. D 43 1170 (1991)
|
arxiv-papers
| 2009-04-23T00:04:12 |
2024-09-04T02:49:02.095537
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shintaro Sawayama",
"submitter": "Shintaro Sawayama",
"url": "https://arxiv.org/abs/0904.3577"
}
|
0904.3670
|
CAS-KITPC/ITP-106
KU-TP 031
Topological Black Holes in Hořava-Lifshitz Gravity
Rong-Gen Caia,***e-mail address: cairg@itp.ac.cn, Li-Ming Caob,†††e-mail
address: caolm@apctp.org, Nobuyoshi Ohtac,‡‡‡e-mail address:
ohtan@phys.kindai.ac.jp
a Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences,
P.O. Box 2735, Beijing 100190, China
Kavli Institute for Theoretical Physics China (KITPC), Chinese Academy of
Sciences,
P.O. Box 2735, Beijing 100190, China
bAsia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea
cDepartment of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
Abstract
We find topological (charged) black holes whose horizon has an arbitrary
constant scalar curvature $2k$ in Hořava-Lifshitz theory. Without loss of
generality, one may take $k=1,0$ and $-1$. The black hole solution is
asymptotically AdS with a nonstandard asymptotic behavior. Using the
Hamiltonian approach, we define a finite mass associated with the solution. We
discuss the thermodynamics of the topological black holes and find that the
black hole entropy has a logarithmic term in addition to an area term. We find
a duality in Hawking temperature between topological black holes in Hořava-
Lifshitz theory and Einstein’s general relativity: the temperature behaviors
of black holes with $k=1,0$ and $-1$ in Hořava-Lifshitz theory are
respectively dual to those of topological black holes with $k=-1,0$ and $1$ in
Einstein’s general relativity. The topological black holes in Hořava-Lifshitz
theory are thermodynamically stable.
## 1 Introduction
Recently a field theory model for a UV complete theory of gravity was proposed
by Hořava [1], which is a non-relativistic renormalisable theory of gravity
and reduces to Einstein’s general relativity at large scales. This theory is
named Hořava-Lifshitz theory in the literature since at the UV fixed point of
the theory space and time have different scalings. Since then much attention
has been attracted to this gravity theory [2, 3, 4, 5, 6, 7, 8, 9], including
its implications in cosmology [3, 4, 5, 7, 8, 9]. In [7] the authors find some
static spherically symmetric black hole solutions in Hořava-Lifshitz theory.
In the $(3+1)$-dimensional ADM formalism, where the metric can be written as
$ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$ (1.1)
and for a spacelike hypersurface with a fixed time, its extrinsic curvature
$K_{ij}$ is
$K_{ij}=\frac{1}{2N}(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i}),$ (1.2)
where a dot denotes a derivative with respect to $t$ and covariant derivatives
defined with respect to the spatial metric $g_{ij}$, the action of Hořava-
Lifshitz theory is [1]
$\displaystyle I$ $\displaystyle=$ $\displaystyle\int
dtd^{3}x\sqrt{g}N\left(\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})+\frac{\kappa^{2}\mu^{2}(\Lambda
R-3\Lambda^{2})}{8(1-3\lambda)}+\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}\right.$
(1.3)
$\displaystyle\left.-\frac{\kappa^{2}\mu^{2}}{8}R_{ij}R^{ij}+\frac{\kappa^{2}\mu}{2\omega^{2}}\epsilon^{ijk}R_{il}\nabla_{j}R^{l}_{\
k}-\frac{\kappa^{2}}{2\omega^{4}}C_{ij}C^{ij}\right),$
where $\kappa^{2}$, $\lambda$, $\mu$, $\omega$ and $\Lambda$ are constant
parameters and the Cotten tensor, $C_{ij}$, is defined by
$C^{ij}=\epsilon^{ikl}\nabla_{k}\left(R^{j}_{\
l}-\frac{1}{4}R\delta^{j}_{l}\right)=\epsilon^{ikl}\nabla_{k}R^{j}_{\
l}-\frac{1}{4}\epsilon^{ikj}\partial_{k}R.$ (1.4)
In (1.3), the first two terms are the kinetic terms, while the others give the
potential of the theory in the so-called “detailed-balance” form.
Comparing the action to that of general relativity, one can see that the speed
of light, Newton’s constant and the cosmological constant are
$c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda}{1-3\lambda}},\ \
G=\frac{\kappa^{2}c}{32\pi},\ \ \tilde{\Lambda}=\frac{3}{2}\Lambda,$ (1.5)
respectively. Let us notice that when $\lambda=1$, the first three terms in
(1.3) could be reduced to the usual ones of Einstein’s general relativity.
However, in Hořava-Lifshitz theory, $\lambda$ is a dynamical coupling
constant, susceptible to quantum correction [1]. In addition, we see from
(1.5) that when $\lambda>1/3$, the cosmological constant $\Lambda$ must be
negative. However, the cosmological constant can be positive if we make an
analytic continuation $\mu\to i\mu,w^{2}\to-iw^{2}$ [7]. In this paper, we
consider the former case with negative cosmological constant.
For later convenience, we rewrite the action (1.3) as follows [7]:
$\displaystyle I$ $\displaystyle=$ $\displaystyle\int dtd^{3}x({\cal
L}_{0}+{\cal L}_{1}),$ (1.6) $\displaystyle{\cal L}_{0}$ $\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})+\frac{\kappa^{2}\mu^{2}(\Lambda
R-3\Lambda^{2})}{8(1-3\lambda)}\right\\},$ $\displaystyle{\cal L}_{1}$
$\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\right\\}.$
The equations of motion for the action are given in [5, 7], but they are very
lengthy and we will not reproduce them here.
In this note we are interested in black hole solutions in the action (1.6).
Considering the static, spherically symmetric solutions with the metric ansatz
$ds^{2}=-N^{2}(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (1.7)
Without the term ${\cal L}_{1}$, the solution is the just the (A)dS
Schwarzschild black hole solution with metric functions [7]
$N^{2}(r)=f(r)=1-\frac{\Lambda}{2}r^{2}-\frac{m}{r}.$ (1.8)
With the term ${\cal L}_{1}$, a general static, spherically symmetric black
hole solution with an arbitrary $\lambda$ is also found in [7], but the
solution is elusive. We discuss the general solution in the appendix. Of
particular interest is the case with $\lambda=1$, on which we focus in the
following. The solution is then given by
$N^{2}=f=1+x^{2}-\alpha\sqrt{x},$ (1.9)
where $x=\sqrt{-\Lambda}r$ and $\alpha$ is an integration constant. This
solution is asymptotically $AdS_{4}$ and has a singularity at $x=0$ if
$\alpha\neq 0$. The singularity could be covered by black hole horizon at
$x_{+}$, the largest root of the equation $f=0$ if $\alpha>0$. The Hawking
temperature of the black hole horizon is easily given by [7]
$T=\frac{3x_{+}^{2}-1}{8\pi x_{+}}\sqrt{-\Lambda}.$ (1.10)
Note that here we have corrected a typo in [7]. One can see from (1.10) that
there exists an extremal limit, $x_{+}=1/\sqrt{3}$, where the temperature
vanishes. Another remarkable point one can see by comparing the solution (1.9)
and (1.8) is that general relativity is not always recovered at large distance
[7]. In addition, one may naively expect that the mass of the black hole
solution (1.9) is divergent due to the square root term.
The black hole solution (1.9) is obtained from the action (1.6) in the
detailed balance [1]. The authors in [7] also considered black hole solution
in Hořava-Lifshitz theory without the condition of the detailed balance,
namely in the theory given by
${\cal L}={\cal L}_{0}+(1-\epsilon^{2}){\cal L}_{1},$ (1.11)
where $\epsilon$ is a constant. In this theory, the black hole solution they
found turns to be
$N^{2}=f=1+\frac{x^{2}}{1-\epsilon^{2}}-\frac{\sqrt{\alpha^{2}(1-\epsilon^{2})x+\epsilon^{2}x^{4}}}{1-\epsilon^{2}}.$
(1.12)
In the large distance limit, the solution reduces to
$f=1+\frac{x^{2}}{1+\epsilon}-\frac{\alpha^{2}}{2\epsilon x}+{\cal
O}(x^{-4}).$ (1.13)
The authors in [7] suggest that the solution has a finite mass for non-
vanishing $\epsilon$, while it becomes divergent as $\epsilon=0$. In the
latter case, the solution goes back to the one (1.9). Furthermore, when
$\epsilon=1$, the solution becomes the (A)dS Schwarzschild black hole solution
(1.8).
In this note we are going to discuss thermodynamics of the black hole
solutions (1.9) and (1.12), which have not been studied in [7]. Since the
solutions (1.9) and (1.12) are asymptotically AdS, we will generalize those
solutions to the case of topological black holes with any constant scalar
curvature horizon [10, 11, 12, 13]. We will also discuss the topological
charged black holes in Hořava-Lifshitz theory by including Maxwell field.
## 2 Topological black holes and thermodynamics
In this section we first generalize the spherically symmetric black hole
solution (1.9) to the topological black hole case with arbitrary constant
scalar curvature horizon. The black hole solution is of the metric ansatz
$ds^{2}=-\tilde{N}^{2}(r)f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{k}^{2},$
(2.1)
where $d\Omega_{k}^{2}$ denotes the line element for an 2-dimensional Einstein
space with constant scalar curvature $2k$. Without loss of generality, one may
take $k=0$, $\pm 1$, respectively. Following [7], substituting the metric
(2.1) into (1.6), we find
$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Lambda\Omega_{k}}{8(1-3\lambda)}\int
dtdr\tilde{N}\left\\{-3\Lambda r^{2}-2(f-k)-2r(f-k)^{\prime}\right.$ (2.2)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+\frac{(\lambda-1)f^{\prime
2}}{2\Lambda}+\frac{(2\lambda-1)(f-k)^{2}}{\Lambda
r^{2}}-\frac{2\lambda(f-k)}{\Lambda r}f^{\prime}\right\\},$
where a prime denotes the derivative with respect to $r$ and $\Omega_{k}$ is
the volume of the 2-dimensional Einstein space. Again, we consider the
solution in the case of $\lambda=1$. In that case, we have
$I=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\int
dtdx\tilde{N}\left(x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}\right)^{\prime}.$ (2.3)
Note that here $x=\sqrt{-\Lambda}r$ and a prime becomes the derivative with
respect to $x$. From the action, we obtain the equations of motion
$\displaystyle 0=\tilde{N}^{\prime},$ $\displaystyle
0=(x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x})^{\prime}.$ (2.4)
From the first equation, we have $\tilde{N}=N_{0}$, a constant. One can set
$N_{0}=1$ by rescaling the time coordinate $t$. From the second one, one can
obtain $x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}=c_{0}$, here $c_{0}$ is an
integration constant. Solving this yields
$f(r)=k+x^{2}-\sqrt{c_{0}x}.$ (2.5)
Note that $c_{0}$ should be positive here. When $k=1$, the solution reduces to
the one given by [7]. Thus we generalize the solution in [7] to the case of
topological black holes with arbitrary $k$. In addition, let us stress here
that although we have obtained the black hole solution through the
minisuperspace approach, it has been checked that the solution (2.5) with
$N_{0}=1$ indeed satisfies the equations of motion given in [7].
A remarkable property of black holes is that they are associated with
thermodynamics. Now we are going to discuss thermodynamics of the black hole
solution (2.5), which has not yet been discussed. Comparing to the AdS
Schwarzschild black hole solution, one may naively expect that the mass of the
solution (2.5) is divergent and one could not define a finite mass for this
solution. However, this conclusion is not true. In fact, such non-standard
asymptotic behavior also appears for the black hole solutions in the so-called
dimensionally continued gravity [14, 13]. For the dimensionally continued
black hole solutions, a finite mass can be obtained by using the Hamiltonian
approach. We find that this approach also works for the Hořava-Lifshitz
theory. Note that the action (2.3) can be written as
$I=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}(t_{2}-t_{1})\int
dx\tilde{N}\left(x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}\right)^{\prime}+B,$ (2.6)
where $B$ is a surface term, which must be chosen so that the action has an
extremum under variations of the fields with appropriate boundary conditions.
One demands that the fields approach the classical solutions at infinity.
Varying the action (2.6), one finds the boundary term
$\delta B=-(t_{2}-t_{1})N_{0}\delta M.$ (2.7)
The boundary term $B$ is the conserved charge associated to the “improper
gauge transformations” produced by time evolution [15]. Here $M$ and $N_{0}$
are a conjugate pair. Therefore when one varies $M$, $N_{0}$ must be fixed.
Thus the boundary term should be in the form
$B=-(t_{2}-t_{1})N_{0}M+B_{0},$ (2.8)
where $B_{0}$ is an arbitrary constant, which should be fixed by some physical
consideration; for example, mass vanishes when black hole horizon goes to
zero. For details, see [14]. According to this Hamiltonian approach, we get
the mass of the solution (2.5) as
$M=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}c_{0}.$ (2.9)
Note that here $\Lambda$ is negative, therefore the black hole mass is always
positive because we have already set $c_{0}>0$. One can easily obtain the
Hawking temperature of the black hole, either by directly calculating the
surface gravity at the horizon, or by requiring the absence of the conical
singularity at the horizon of the Euclidean black hole. Both methods give the
same result
$T=\frac{3x_{+}^{2}-k}{8\pi x_{+}}\sqrt{-\Lambda}.$ (2.10)
The next step is to get the entropy associated with the topological black
hole. In Einstein’s general relativity, entropy of black hole is always given
by one quarter of black hole horizon area. But in higher derivative gravities,
in general, the area formula breaks down. Here we will obtain the black hole
entropy by using the first law of black hole thermodynamics with assumption
that as a thermodynamical system [11, 12, 13, 14], the first law always keeps
valid: $dM=TdS$. Integrating this relation yields
$S\equiv\int T^{-1}dM+S_{0}=\int T^{-1}\frac{dM}{dx_{+}}dx_{+}+S_{0},$ (2.11)
where $S_{0}$ is an integration constant, which should be fixed by physical
consideration. Through (2.11), we obtain
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{4}\left(x_{+}^{2}+2k\ln
x_{+}\right)+S_{0},$ (2.12) $\displaystyle=$
$\displaystyle\frac{c^{3}}{4G}\left(A-\frac{k\Omega_{k}}{\Lambda}\ln\frac{A}{A_{0}}\right),$
where the Newton’s constant and speed of light are given in (1.5),
$A=\Omega_{k}r_{+}^{2}$ is the black hole horizon area, and $A_{0}$ is a
constant of dimension of length squared. The leading term is just one quarter
of horizon area in units of $c=G=1$, which should be the contribution from the
${\cal L}_{0}$ term. The second term is a logarithmic function, therefore we
cannot fix the integration constant $S_{0}$ or $A_{0}$, unfortunately, by some
physical consideration, for example, black hole entropy should vanish when
black hole horizon goes to zero. The integration constant $S_{0}$ could be
fixed by counting micro degrees of freedom in some quantum theory of gravity
like string theory. An interesting fact is that such a term often appears in
the quantum correction of black hole entropy. In addition, when $k=0$, namely,
for black hole with Ricci flat horizon, the logarithmic term disappears. Thus,
the area formula of black hole entropy is recovered in this case. It might be
a universal result that the area formula still holds for Ricci flat black
holes in higher derivative gravity theories [11, 12, 13].
Two additional points are worth stressing here. One is on the temperature
(2.10). For $k=1$, as pointed out in [7], there is an extremum at
$x_{+}=1/\sqrt{3}$, where the temperature vanishes, and it corresponds to an
extremal black hole. For $k=0$, the temperature
$T=3x_{+}\sqrt{-\Lambda}/8\pi$. In these two cases, the temperature always
monotonically increases as the horizon $x_{+}$ grows. For $k=-1$, the inverse
temperature starts from zero at $x_{+}=0$, monotonically increases and reaches
a maximal value, $\beta=1/T=4\pi/\sqrt{-3\Lambda}$ at $x_{+}=1/\sqrt{3}$, then
monotonically decreases as $x_{+}$ grows. It is interesting to compare these
temperature behaviors of the topological black holes in Hořava-Lifshitz theory
with those for topological black holes in Einstein’s general relativity (the
latter could be obtained by replacing $1$ by $k$ in (1.8)). The temperature
for the topological black holes in Einstein’s general relativity is
$T_{\rm TSch}=\frac{\sqrt{-\Lambda}}{8\pi x_{+}}(3x_{+}^{2}+2k).$ (2.13)
We see that except for the coefficient difference in front of the horizon
curvature constant $k$, there is a duality relation in these two temperatures:
the temperature behaviors of black holes in Hořova-Lifshitz theory in the
cases of $k=1$,$0$ and $-1$, are dual to the cases of $k=-1$, $0$ and $1$ in
Einstein’s general relativity, respectively. Note that for topological black
holes in Einstein’s general relativity [10], in the cases of $k=0$ and $k=-1$,
the black holes are always thermodynamically stable, while in the case of
$k=1$, the small black hole with $x_{+}<\sqrt{2/3}$ is thermodynamically
unstable and it becomes thermodynamically stable for large horizon radius
$x_{+}>\sqrt{2/3}$.
However, a close check tells us that in the case of $k=-1$, there exists a
minimal horizon at $x_{+}=1$ for the topological black hole in Hořava-Lifshitz
theory, which can be seen from the metric function $f(r)$ in (2.5), namely for
the case of $c_{0}=0$. This is just the massless black hole in AdS space. Thus
in the range $x_{+}\in[1,\infty)$, the temperature of the topological black
hole is also a monotonically increasing function of $x_{+}$. Thus the unstable
phase for the topological black hole with $k=-1$ in Hořava-Lifshitz theory
does not appear, and the black hole is always thermodynamically stable.
To see this more clearly, let us calculate heat capacity of black hole,
defined as $C=dM/dT$. The heat capacity of the black hole in Hořava-Lifshitz
gravity is
$C=\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{2}\frac{(3x_{+}^{2}-k)(x_{+}^{2}+k)}{3x_{+}^{2}+k}.$
(2.14)
We see that for the cases $k=1$ and $k=0$, the heat capacity is always
positive, which implies that the black hole is local thermodynamically stable,
while in the case of $k=-1$, if $x_{+}>1$, it is also positive. For
comparison, we give the heat capacity for the topological AdS black hole in
Einstein’s general relativity
$C_{\rm
TSch}=\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{2}\frac{3x_{+}^{2}+2k}{3x_{+}^{2}-2k}x_{+}^{2}.$
(2.15)
When $k=0$ and $-1$, it is always positive while when $k=1$, it is negative
for $x_{+}^{2}<2/3$, positive for $x_{+}^{2}>2/3$ and diverges at
$x_{+}^{2}=2/3$.
Another interesting question is whether there exists the Hawking-Page phase
transition associated with the black holes in Hořava-Lifshitz gravity. It is
well known that there is a Hawking-Page transition for static, spherically
symmetric AdS-Schwarzschild black hole (the case of $k=1$) between a large AdS
black hole and thermal gas in AdS space [16]. On the other hand, for the cases
of $k=0$ and $k=-1$ topological black hole in Einstein’s general relativity,
the Hawking-Page phase transition does not exist. To discuss the Hawking-Page
transition, one has to calculate the Euclidean action or free energy of the
black hole. The Euclidean action has a relation to the free energy by $I=\beta
F$, here $\beta$ is the inverse temperature of the black hole. By definition,
the free energy $F$ is given by $F=M-TS$. By using (2.9), (2.10) and (2.12),
we find
$F=\frac{\kappa^{2}\mu^{2}\Omega_{k}\sqrt{-\Lambda}}{32x_{+}}\left(-x_{+}^{4}+5kx_{+}^{2}+2k^{2}-6kx_{+}^{2}\ln
x_{+}+2k^{2}\ln x_{+}\right)-TS_{0}.$ (2.16)
Due to the uncertainty of $S_{0}$, we cannot determine the signature of the
free energy. However, if one can neglect the term $S_{0}$, we see the free
energy is negative for large enough horizon radius, which means that large
black holes in Hořava-Lifshitz gravity is thermodynamically stable globally.
Now we turn to the case without the detailed balance condition, namely
$\epsilon^{2}\neq 0$. Replacing (2.3) we have
$I=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\int
dtdx\tilde{N}\left(x^{3}-2x(f-k)+(1-\epsilon^{2})\frac{(f-k)^{2}}{x}\right)^{\prime}.$
(2.17)
In this case, one has the solution
$\displaystyle\tilde{N}=N_{0},$ $\displaystyle
f(r)=k+\frac{x^{2}}{1-\epsilon^{2}}-\frac{\sqrt{\epsilon^{2}x^{4}+(1-\epsilon^{2})c_{0}x}}{1-\epsilon^{2}}.$
(2.18)
Again, $c_{0}$ is an integration constant and $N_{0}$ could be set to one.
Similar to the case of $\epsilon^{2}=0$, we find the mass of the solution is
$M=\frac{\kappa^{2}\mu^{2}\Omega_{k}\sqrt{-\Lambda}}{16}c_{0},$ (2.19)
and $c_{0}$ can be expressed in terms of black hole horizon radius $x_{+}$,
$c_{0}=\frac{x_{+}^{4}+2kx_{+}+(1-\epsilon^{2})k^{2}}{x_{+}}.$ (2.20)
The Hawking temperature of the black hole is found to be
$T=\frac{\sqrt{-\Lambda}}{8\pi}\frac{3x_{+}^{4}+2kx_{+}^{2}-(1-\epsilon^{2})k^{2}}{x_{+}(x_{+}^{2}+(1-\epsilon^{2})k)}.$
(2.21)
With the mass and temperature, we obtain the entropy of the black hole
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{4}\left(x_{+}^{2}+2k(1-\epsilon^{2})\ln
x_{+}\right)+S_{0},$ (2.22) $\displaystyle=$
$\displaystyle\frac{c^{3}}{4G}\left(A-(1-\epsilon^{2})\frac{k\Omega_{k}}{\Lambda}\ln\frac{A}{A_{0}}\right).$
When $\epsilon^{2}=0$, it goes back to (2.12), while it reduces to the well-
known area formula for $\epsilon^{2}=1$, as expected, since in that case, the
effect of higher derivative terms disappears.
Now let us discuss the behavior of the temperature (2.21).
(i) When $k=0$, the temperature is independent of $\epsilon^{2}$, given by
$T=\frac{3\sqrt{-\Lambda}}{8\pi}x_{+}.$ (2.23)
Clearly it is a monotonically increasing function of $x_{+}$
(ii) When $k=-1$ and $\epsilon^{2}<1$, an extremal black hole with $T=0$ is
obtained at $x_{+}^{2}=(1+\sqrt{1+3(1-\epsilon^{2})})/3$. While to keep the
denominator in (2.21) positive, one has to have $x_{+}^{2}>(1-\epsilon^{2})$,
which is always smaller than $(1+\sqrt{1+3(1-\epsilon^{2})})/3$. This
indicates that there does exist an extremal black hole in this case with the
minimal horizon radius $x_{+\rm min}^{2}=(1+\sqrt{1+3(1-\epsilon^{2})})/3$.
When $\epsilon^{2}>1$, according to (2), the minimal horizon radius is
$x_{+}^{2}=1+\epsilon$. In both cases of $\epsilon^{2}>1$ and $<1$, the
temperature of the black hole is a monotonically increasing function of
$x_{+}$ in the physical regime.
(iii) When $k=1$, let us first consider the case of $\epsilon^{2}<1$. A
vanishing temperature happens at $x^{2}_{+\rm
min}=(-1+\sqrt{1+3(1-\epsilon^{2})})/3$. When $\epsilon^{2}>1$, there does not
exist an extremal black hole, but keep the temperature positive, a physical
horizon radius must obey $x_{+}^{2}>\epsilon^{2}-1$. As the case of $k=-1$
with any $\epsilon^{2}$, the temperature of the black hole is a monotonically
increasing function of $x_{+}$ in the physical regime, again.
In summary, the case with $\epsilon^{2}\neq 0$ is similar to the case with
$\epsilon^{2}=0$, the Hawking temperature of the black holes with any $k$ is
always a monotonically increasing function of horizon radius $x_{+}$ in the
physical regime. This implies that the topological black holes in Hořava-
Lifshitz theory are thermodynamically stable. Note that when $\epsilon^{2}=1$,
the situation is reduced to the case of the well-known topological AdS
Schwarzschild black holes [10].
## 3 Topological charged black holes
In this section we consider the charged generalization of the topological
black hole found in Sec. 2. To give a universal result, we assume
$\epsilon^{2}\neq 0$. Following [14, 13], the Hamiltonian action for the
Maxwell field can be written as
$I_{\rm em}=\int dtd^{3}x\left[p^{i}\dot{A}_{i}-\frac{1}{2}N\left(\alpha
g^{-1/2}p^{i}p_{i}+\frac{g^{1/2}}{2\alpha}F_{ij}F^{ij}\right)+\varphi
p^{i},_{i}\right]+B_{\rm em},$ (3.1)
where $p^{i}$ is the momentum conjugate of the spatial components of the
Maxwell field $A_{i}$, $\varphi=A_{0}$, $B_{\rm em}$ is a boundary term, $N$
is the lapse function, and $\alpha$ is a parameter to be fixed shortly.
Considering the static topological black hole solution with the metric ansatz
(2.1), the action (3.1) is reduced to
$I_{\rm em}=\frac{\Omega_{k}}{\alpha}\int
dtdr\left(-\frac{1}{2}\tilde{N}r^{2}p^{2}+\varphi(r^{2}p)^{\prime}\right)+B_{\rm
em},$ (3.2)
where $p=\alpha p^{r}/r^{2}\gamma^{1/2}$ and $\gamma$ is the determinate of
the 2-dimensional Einstein space $d\Omega^{2}_{k}$. Note that here the
solution without magnetic charge $F_{ij}=0$ has been assumed. To be consistent
with (2.17), we set $x=\sqrt{-\Lambda}r$, The action (3.2) then becomes
$I_{\rm em}=\frac{\Omega_{k}}{\alpha\sqrt{-\Lambda}}\int
dtdx\left(-\frac{1}{2}\tilde{N}x^{2}\tilde{p}^{2}+\varphi(x^{2}\tilde{p})^{\prime}\right)+B_{\rm
em},$ (3.3)
where a prime denotes derivative with respective to $x$ and
$\tilde{p}=p/\sqrt{-\Lambda}$. Now we set
$\alpha^{-1}=-\frac{\kappa^{2}\mu^{2}\Lambda}{16}.$ (3.4)
Combining (2.17) and (3.4), we have
$I=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\int
dtdx\left(\tilde{N}(U^{\prime}-\frac{1}{2}x^{2}\tilde{p}^{2})+\varphi(x^{2}\tilde{p})^{\prime}\right)+B,$
(3.5)
where
$U=x^{3}-2x(f-k)+(1-\epsilon^{2})\frac{(f-k)^{2}}{x}.$
From the action (3.5) we obtain the equations of motion
$\displaystyle U^{\prime}=\frac{1}{2}x^{2}\tilde{p}^{2},\ \ \
(x^{2}\tilde{p})^{\prime}=0,$ (3.6)
$\displaystyle\varphi^{\prime}=-\tilde{N}\tilde{p},\ \ \
\tilde{N}^{\prime}=0,$
which have the solution
$\displaystyle\tilde{N}=N_{0},\ \ \ \ \varphi=\frac{N_{0}q}{x}+\varphi_{0},$
$\displaystyle\tilde{p}=\frac{q}{x^{2}},\ \ \ \ U=-\frac{q^{2}}{2x}+c_{0}.$
(3.7)
Here $N_{0}$, $\varphi_{0}$, $c_{0}$ and $q$ are integration constants, their
physical meanings are clear. Physical electric charge and mass of the solution
are
$Q=\frac{\kappa^{2}\mu^{2}\Omega_{k}\sqrt{-\Lambda}}{16}q,\ \ \
M=\frac{\kappa^{2}\mu^{2}\Omega_{k}\sqrt{-\Lambda}}{16}c_{0},$ (3.8)
respectively, and the metric function $f$ is given by
$f(r)=k+\frac{x^{2}}{1-\epsilon^{2}}-\frac{\sqrt{\epsilon^{2}x^{4}+(1-\epsilon^{2})(c_{0}x-q^{2}/2)}}{1-\epsilon^{2}},$
(3.9)
while $\tilde{N}=N_{0}$ could be set to one. Taking the limit $\epsilon\to 1$,
the solution is reduced to
$f(r)=k+\frac{x^{2}}{2}-\frac{c_{0}}{2x}+\frac{q^{2}}{4x^{2}},$ (3.10)
as expected, it is just the AdS Reissner-Nordström black hole solution. The
Hawking temperature of the black hole is
$T=\frac{\sqrt{-\Lambda}(3x_{+}^{4}+2kx_{+}^{2}-(1-\epsilon^{2})k^{2}-q^{2}/2)}{8\pi
x_{+}(x_{+}^{2}+(1-\epsilon^{2})k)}.$ (3.11)
Putting the temperature (3.11) and mass (3.8) into the first law of black hole
thermodynamics, it is easy to check that one reproduces the entropy (2.22),
the charge $q$ does not appear explicitly in the expression of black hole
entropy in terms of horizon radius. This is consistent with the fact that
black hole entropy is a function of horizon geometry. The behavior of the
temperature can be analyzed as the case without the electric charge, but we do
not repeat here. Instead we only point out that due to the appearance of the
electric charge, extremal black holes with vanishing temperature always exist
within reasonable parameter regime.
## 4 Conclusion
In this paper we found topological (charge) black hole solutions with
arbitrary constant scalar curvature horizon in Hořava-Lifshitz theory,
generalizing the static, spherically symmetric black hole solutions in [7].
Although there is a square root term in the metric function $f(r)$, we can
define a finite mass associated with the black hole solution by use of the
Hamiltonian approach. We have calculated the Hawking temperature of the black
hole and the black hole entropy by using the first law of black hole
thermodynamics, and found that, except for the well-known horizon area term,
the black hole entropy has a logarithmic term. Such a logarithmic term often
occurs on the occasion of considering quantum corrections to black hole
entropy. In our entropy expression, there is a undetermined constant $S_{0}$.
To fix the constant entropy $S_{0}$, one has to invoke quantum theory of
gravity.
We find that the temperature behavior of the topological black holes in
Hořava-Lifshitz theory is very interesting. Indeed there is a duality for
temperature between topological black holes in Hořava-Lifshitz theory and
topological black holes in Einstein’s general relativity. The temperatures of
topological black holes with $k=1$, $0$ and $-1$ in Hořava-Lifshitz theory are
dual to those of black holes with $k=-1$, $0$ and $1$ in Einstein’s general
relativity, respectively.
In this paper we have only considered thermodynamics of topological black
holes in Hořava-Lifshitz theory with $\lambda=1$. It is of great interest to
see whether one can find a way to study thermodynamics for the general
topological black holes in the theory with $\lambda\neq 1$.
## Acknowledgments
This work was supported partially by grants from NSFC, China (No. 10821504 and
No. 10525060), a grant from the Chinese Academy of Sciences with
No.KJCX3-SYW-N2, the Grant-in-Aid for Scientific Research Fund of the JSPS No.
20540283, and the Japan-U.K. Research Cooperative Program.
## Appendix A Appendix: Topological black holes for general $\lambda$
Here we briefly discuss topological black hole solution with a general
$\lambda$. In terms of the new function $F$ defined by
$\displaystyle F(r)=k-\Lambda r^{2}-f(r),$ (A.1)
the action (2.2) takes the form
$\displaystyle I=\frac{\kappa^{2}\mu^{2}\Omega_{k}}{8(1-3\lambda)}\int
dtdr\tilde{N}\left\\{\frac{(\lambda-1)}{2}F^{\prime
2}-\frac{2\lambda}{r}FF^{\prime}+\frac{(2\lambda-1)}{r^{2}}F^{2}\right\\}.$
(A.2)
The equations of motion are then
$\displaystyle
0=\left(\frac{2\lambda}{r}F-(\lambda-1)F^{\prime}\right)\tilde{N}^{\prime}+(\lambda-1)\left(\frac{2}{r^{2}}F-F^{\prime\prime}\right)\tilde{N},$
(A.3) $\displaystyle 0=(\lambda-1)r^{2}F^{\prime 2}-4\lambda
rFF^{\prime}+2(2\lambda-1)F^{2}.$ (A.4)
The latter is easily solved to give [7]
$\displaystyle F(r)=\alpha
r^{\frac{2\lambda\pm\sqrt{2(3\lambda-1)}}{\lambda-1}},$ (A.5)
and then the first gives
$\displaystyle\tilde{N}=\beta r^{-\frac{1+3\lambda\pm
2\sqrt{2(3\lambda-1)}}{\lambda-1}},$ (A.6)
where $\alpha$ and $\beta$ are both integration constants. When $\alpha=0$ or
$F=0$, Eq. (A.3) does not restrict $\tilde{N}$. Note that the exponent of Eq.
(A.5) for the negative branch is always less than 2 for positive $\lambda$,
and thus the $r^{2}$ term in the metric function (A.1) dominates at large
distances. The other branch gives a power larger than 2. We are interested in
the solutions with asymptotic AdS behavior. In that case, we should look at
the negative branch with constant $\tilde{N}$. It follows from Eq. (A.3) that
either $\lambda=1$ or $F^{\prime\prime}=\frac{2}{r^{2}}F$. The latter leads to
$F\sim r^{2}$ or $1/r$; the first one does not satisfy (A.4), and the second
solution requires $\lambda=1/3$, which may be of some interest [1], but the
action (1.3) appears singular. So we discuss $\lambda=1$ case mainly in this
paper.
## References
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|
arxiv-papers
| 2009-04-23T12:59:17 |
2024-09-04T02:49:02.101803
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rong-Gen Cai, Li-Ming Cao, Nobuyoshi Ohta",
"submitter": "Rong-Gen Cai",
"url": "https://arxiv.org/abs/0904.3670"
}
|
0904.3706
|
# Recent Results from the MINOS experiment
Milind V. Diwan
###### Abstract
MINOS is an accelerator neutrino oscillation experiment at Fermilab. An
intense high energy neutrino beam is produced at Fermilab and sent to a near
detector on the Fermilab site and also to a 5 kTon far detector 735 km away in
the Soudan mine in northern Minnesota. The experiment has now had several
years of running with millions of events in the near detector and hundreds of
events recorded in the far detector. I will report on the recent results from
this experiment which include precise measurement of $|\Delta m^{2}_{32}|$,
analysis of neutral current data to limit the component of sterile neutrinos,
and the search for $\nu_{\mu}\to\nu_{e}$ conversion. The focus will be on the
analysis of data for $\nu_{\mu}\to\nu_{e}$ conversion. Using data from an
exposure of $3.14\times 10^{20}$ protons on target, we have selected electron
type events in both the near and the far detector. The near detector is used
to measure the background which is extrapolated to the far detector. We have
found 35 events in the signal region with a background expectation of $27\pm
5(stat)\pm 2(syst)$. Using this observation we set a $90\%$ C.L. limit of
$\sin^{2}2\theta_{13}<0.29$ for $\delta_{cp}=0$ and normal mass hierarchy.
Further analysis is under way to reduce backgrounds and improve sensitivity.
Physics Department, Brookhaven National Laboratory
Upton, NY 11973, USA
E-mail: diwan@bnl.gov
April 20, 2009
## 1 Introduction
In the current picture of neutrino oscillations, three flavors of neutrinos
are related to three mass states by the Pontecorvo-Maki-Nakagawa-Sakata mixing
matrix ?,?). The mixing can be described by two $\Delta m^{2}$ parameters
($|\Delta m^{2}_{32}|$, $|\Delta m^{2}_{21}|$), three mixing angles
($\theta_{23}$, $\theta_{12}$, and $\theta_{13}$) and a CP violating phase
($\delta_{cp}$)?). The oscillation phenomena naturally falls into two domains:
the atmospheric neutrino oscillations, and Solar neutrino oscillations ?). The
atmospheric neutrino oscillations are well-described by
$\nu_{\mu}\rightarrow\nu_{\tau}$ oscillations, with parameters
$\sin^{2}2\theta_{23}>0.92$ and $1.9\times 10^{-3}<|\Delta
m_{32}^{2}|<3.0\times 10^{-3}~{}{\rm eV}^{2}$ at 90$\%$ C.L ?). The K2K
experiment and MINOS have confirmed the atmospheric neutrino oscillations with
accelerator beams ?,?) Solar $\nu_{e}\rightarrow\nu_{\mu,\tau}$ oscillations
are described by $\sin^{2}2\theta_{12}=0.86^{+0.03}_{-0.04}$ and $\Delta
m_{21}^{2}=8.0^{+0.4}_{-0.3}\times 10^{-5}~{}{\rm eV}^{2}$ are also consistent
with multiple observations ?,?), and are confirmed by disappearance of reactor
$\bar{\nu}_{e}$?). As yet, very little is known about either $\theta_{13}$ or
$\delta_{cp}$, although lack of observed disappearance of reactor
$\bar{\nu}_{e}$ over a few km baseline?) has shown that $\theta_{13}$ must be
small: $\sin^{2}2\theta_{13}<0.19$ at $90\%$ C.L. Furthermore, the sign of
$|\Delta m^{2}_{32}|$ (or the ordering of the mass eigenstates) is unknown.
The sign of $\Delta m^{2}_{12}$ is known using the strong matter effects that
must be considered when analyzing neutrinos from the Sun. Determination of the
unknowns in neutrino mixing needs further experiments in the oscillation range
of $|\Delta m^{2}_{32}|$ for conversion of muon and electron type neutrinos
into each other.
For our present discussion, it is useful to exhibit an approximate analytic
formula for the oscillation of $\nu_{\mu}\to\nu_{e}$ for 3-generation mixing
obtained with the simplifying assumption of constant matter density ?,?).
Assuming a constant matter density, the oscillation of
$\nu_{\mu}\rightarrow\nu_{e}$ in the Earth for 3-generation mixing is
described approximately by Equation LABEL:qe1. In this equation $\alpha=\Delta
m^{2}_{21}/\Delta m^{2}_{31}$, $\Delta=\Delta m^{2}_{31}L/4E$,
$\hat{A}=2VE/\Delta m^{2}_{31}$, $V=\sqrt{2}G_{F}n_{e}$. $n_{e}$ is the
density of electrons in the Earth. Recall that $\Delta m^{2}_{31}=\Delta
m^{2}_{32}+\Delta m^{2}_{21}$. Also notice that $\hat{A}\Delta$, which has
absolute value of $LG_{F}n_{e}/\sqrt{2}$, is sensitive to the sign of $\Delta
m^{2}_{31}$.
$\displaystyle P(\nu_{\mu}\rightarrow\nu_{e})$ $\displaystyle\approx$
$\displaystyle\sin^{2}\theta_{23}{\sin^{2}2\theta_{13}\over(\hat{A}-1)^{2}}\sin^{2}((\hat{A}-1)\Delta)$
$\displaystyle+\alpha{\sin\delta_{CP}\cos\theta_{13}\sin 2\theta_{12}\sin
2\theta_{13}\sin
2\theta_{23}\over\hat{A}(1-\hat{A})}\sin(\Delta)\sin(\hat{A}\Delta)\sin((1-\hat{A})\Delta)$
$\displaystyle+\alpha{\cos\delta_{CP}\cos\theta_{13}\sin 2\theta_{12}\sin
2\theta_{13}\sin
2\theta_{23}\over\hat{A}(1-\hat{A})}\cos(\Delta)\sin(\hat{A}\Delta)\sin((1-\hat{A})\Delta)$
$\displaystyle+\alpha^{2}{\cos^{2}\theta_{23}\sin^{2}2\theta_{12}\over\hat{A}^{2}}\sin^{2}(\hat{A}\Delta)$
For anti-neutrinos, the second term in Equation LABEL:qe1 has the opposite
sign. It proportional to the CP violating quantity $\sin\delta_{CP}$. An
accelerator experiment using high energy neutrinos, a sufficiently long
baseline, and the ability to detect $\nu_{\mu}\to\nu_{e}$ conversion with low
backgrounds and high statistics, has sensitivity to all four terms in Equation
LABEL:qe1. The first term dominates the sensitivity to the unknown parameter
$\theta_{13}$. MINOS is the first high energy accelerator experiment so far to
have sensitivity to more than the first term in Equation LABEL:qe1. In the
following we describe an analysis of the MINOS data to reduce backgrounds to
allow the observation of $\nu_{\mu}\to\nu_{e}$.
## 2 MINOS beam and detector
The MINOS detectors ?) and the NuMI beam line ?) are described elsewhere. In
brief, NuMI is a conventional two-horn-focused neutrino beam with a 675 m long
decay tunnel. The neutrino beam goes through the Earth to upper Minnesota over
a distance of 735 km to the far detector. The horn current and position of the
hadron production target relative to the horns can be configured to produce
different $\nu_{\mu}$ energy spectra. In figure 1 we show the spectrum of all
$\nu_{\mu}$ events in the fiducial volume for the horn-on and horn-off
configurations. Several beam configurations with different mean energies have
been used for studying backgrounds and systematics: in particular, the horn-
off configuration has been particularly useful for the $\nu_{e}$ search. The
high energy, $\sim 5-10GeV$, obtained by moving the production target, as well
as intermediate energy configurations have been used for the analysis of the
beam systematics for the muon neutrino disappearance data. Most of the physics
data has been in the low energy (horn-on) configuration in which the peak of
the spectrum is $\sim 3GeV$. In the low energy configuration, $92.9\%$ of the
flux is $\nu_{\mu}$, $5.8\%$ is $\bar{\nu}_{\mu}$ and $1.3\%$ is the
$\nu_{e}/\bar{\nu}_{e}$ contamination.
Figure 1: (in color) Spectrum of all muon neutrino events in the MINOS near
detector fiducial volume for the horn on and horn off configurations.
MINOS consists of two detectors: a 0.98 kt Near Detector (ND) 1.04 km from the
NuMI target; and a 5.4 kt Far Detector (FD) 735 km from the target. Both are
segmented, magnetized calorimeters that permit particle tracking, optimized
for neutrino energy range of $1<E_{\nu}<50{\rm GeV}$. The curvature of muons
produced in $\nu_{\mu}+\mbox{Fe}\rightarrow\mu^{-}+X$ interactions
aaaApproximately 5% of the neutrino interactions occur in aluminum and
scintillator. is used for energy determination of muons that exit the detector
and to distinguish the $\nu_{\mu}$ component of the beam from the
contamination. The energy of muons contained in the detector is measured by
their range. The muon and shower energies are added to obtain the
reconstructed muon neutrino energy ($E_{reco}=E_{\mu}+E_{h}$) with a
resolution given by $\Delta p_{\mu}/p_{\mu}\approx 10\%$ and $\Delta
E_{h}/E_{h}\approx 56\%/\sqrt{E_{h}}$. For electron neutrino detection the
relevant parameters of the detector concern the calorimetric segmentation.
Both the near and far detectors have identical segmentation of 1 inch (1.44
radiation length) steel and 1 cm thick plastic scintillator. Transversely the
scintillator is in strips of 4.1 cm width, corresponding to Moliere radius of
3.7 for electromagnetic showers. The scintillator is read by wavelength
shifting fibers into multianode PMTs. The scintillator strips range in length
from a maximum of 8 meters in the far detector down to $\sim$ 1 m in the near
detector. The light yield is on the average $\sim$6 photo-electrons for a
minimum ionizing particles. Although the near and far detectors have identical
granularity, there are important differences: the light yield, the type of PMT
used (Hamamtsu M16 in the far, and Hamamatsu M64 in the near), the cross talk
between channels in the PMTs, and multiplexing of scintillator strips onto the
PMT pixels ?). These differences are carefully calibrated using cosmic rays
and simulated in the Monte Carlo programs to limit the near and far
differences. After electron particle identification and selection cuts as
described below, the energy resolution for $\nu_{e}$ events is $\sim
30\%/\sqrt{E}$.
## 3 Data Reduction
This note describes results from data recorded between May 2005, and July
2007. Over this period, a total of $3.36\times 10^{20}$ protons on target
(POT) were accumulated. A $1.27\times 10^{20}$ POT subset of this exposure
(hereafter referred to as Run I) forms the data set from Ref ?). In Run I and
for most of the new running period (Run II), the beam line was configured to
enhance $\nu_{\mu}$ production with energies 1-5 GeV (the low-energy
configuration). An exposure of $0.15\times 10^{20}$ POT was accumulated with
the beam line configured to enhance the $\nu_{\mu}$ energy spectrum at 5-10
GeV (the high-energy configuration). The Run II data were collected with a
replacement target of identical construction due to failure of the motion
system of the first target. The new target was found to be displaced
longitudinally $\sim$1 cm relative to the first target, resulting in a 30 MeV
shift in the neutrino spectrum. This effect is incorporated in the Monte Carlo
simulation, and the Run I and Run II data sets are analyzed separately to
account for this shift.
The data reduction has several components: cuts are first applied to remove
data from periods of bad detector and beam conditions. After event
reconstruction, preselection cuts select events that are enriched in the types
of events that are under analysis: negative or positive muons, electrons or
neutral currents. These cuts are performed as identically as possible for the
near and far detectors. Differences are accounted for in the Monte Carlo.
After the preselection, particle identification cuts are applied to extract a
pure sample of the events under consideration. The muon and neutral current
analysis has been described in detail in previous publications ?,?).
For the electron analysis the selected sample of data (the low energy horn on
configuration) corresponds to $3.14\times 10^{20}$ protons on target for the
far detector. The near detector data was sampled uniformly and scaled to
correspond to $10^{19}$ protons on target. Reconstructed events were chosen
within the well calibrated parts of the detectors corresponding to fiducial
masses of 29 ton and 4 kton for the near and far detectors, respectively,
within the 10 $\mu$sec beam pulse gate to reject cosmic ray events. After
these cuts cosmics contribute $<0.5$ event background in the final sample. The
$\nu_{e}$ preselection cuts selected an initial sample of events with single
electromagnetic showers according to the reconstruction algorithm and rejected
events with any tracks longer than 25 planes. A second cut examined planes
with track-like hits, and eliminated events with more than 16 planes with such
hits. After the precuts, the event sample is composed of $\sim 30\%$
$\nu_{\mu}$ CC events in which the muon is too short to be rejected, $\sim
65\%$ NC events, and about $\sim 5\%$ $\nu_{e}$ events from the beam
contamination.
## 4 Selection of Electron Neutrinos
After preselection, further rejection of neutral current, and muon charged
current events is needed. To achieve this rejection we use the short compact
nature of the electromagnetic showers compared to the diffuse nature of
hadronic showers. We have developed two software algorithms to examine each
candidate event and classify it as a potential electron neutrino signal or
background.
Selection ANN We use the pattern of energy deposition, after eliminating hits
with less than 2 photo-electrons, to characterize each event by several
parameters. The artificial neural network (ANN) algorithm combines 11 such
reconstructed quantities that exhibit signal and background separation. Some
of these quantities are the maximum energy fraction in 4 planes, fraction of
energy in a 3 strip wide road, the RMS of transverse energy deposition, etc.
The output from the ANN is between 0 (background like) and 1 (signal-like).
With a cut at 0.7, the efficiency for the signal, after preselection, is
expected to be approximately 41%; the expected neutral current and
$\nu_{\mu}$CC rejection efficiency is $\sim 92.3\%$ and $\geq 99.4\%$,
respectively.
Selection LEM The second discrimination technique is a novel approach called
Library Event Matching (LEM) selection in which each event candidate is
compared to a large library of simulated $\nu_{e}$-CC and NC events. The best
50 library matches are found for each candidate event, by considering the
probability that two different energy deposition patterns in the detector
originated from the same neutrino interaction. This computationally intensive
technique can be carried out because the size of the events is generally
small, and all the strip information can be used. Three variables are
constructed: the fraction of these matches that are $\nu_{e}$-CC events, the
mean hadronic $y$ of the best matches, and the mean fractional charge $q$
matched within those best matches. A likelihood is then formed from these
variables as a function of energy. With a cut at 0.65, the efficiency for the
signal, after preselection, is expected to be approximately 46%; the expected
neutral current and $\nu_{\mu}$CC rejection efficiency is $\sim 92.9\%$ and
$\geq 99.3\%$, respectively.
The two selection algorithms rely on very different techniques, provide
different signal to background ratios, and are sensitive to different
systematic uncertainties. Assuming the signal is at the Chooz limit, LEM has
the potential to achieve a better signal to background ratio (1:3) compared to
ANN (1:4), but it is more sensitive to systematic uncertainties on the
relative energy calibrations in the near and far detectors. Both algorithms
select predominantly NC events and higher y, $\nu_{\mu}$ CC events. The
background consists mainly of deep inelastic scattering events, with nearly
half of the background showers containing a single $\pi^{0}$. In the analysis
reported here, the ANN selected sample is used to derive the final results,
but the LEM selection is examined as a cross check.
## 5 Calculation of Backgrounds
The rate and spectrum of events selected as electron like in the near detector
are used to predict the number of background events expected in the far
detector. Figure 2 shows the distribution of ANN PID and LEM PID for the near
detector data. The plots also show the prediction from the Monte Carlo which
deviates from the data. This level of deviation is within the systematic
errors due to cross section and hadronic shower modeling uncertainties. The
Monte Carlo is based on past data ?) with much lower statistics in the MINOS
energy region, and therefore while the Monte Carlo can be used for
understanding ratios, and relative changes, the MINOS near data itself must be
used to determine the normalization of the background in the far detector.
Figure 2: (in color) The ANN PID distribution for the near detector data
(left). The LEM PID distribution for the near detector data (right). The plots
show the chosen cuts for selection of $\nu_{e}$-like events.
The near detector background spectrum has three different components, NC
events, $\nu_{\mu}$-CC events and beam $\nu_{e}$ events. At lowest order the
far detector background calculation is the near detector event rate (5524
events per $10^{19}$ POT for ANN) multiplied by the energy averaged far/near
ratio of the neutrino flux $\sim 1.3\times 10^{-6}$ and the ratio of the
fiducial masses $4{\rm kton}/29{\rm ton}$. However, the $\nu_{\mu}$ charged
current component of the background is affected by oscillations, and therefore
a more sophisticated calculation is needed. For such a calculation we need to
separate the background components and extrapolate them separately to the far
detector. The far detector will also have a small component from $\nu_{\tau}$
events which will be calculated by Monte Carlo.
The components of the background are determined using the horn off data sample
recorded in the ND. Applying the $\nu_{e}$ selection to data taken with the
focusing horns turned off (horn-off) provides a neutral current enriched
sample. The higher mean energy spectrum (see figure 1) of the horn-off sample
allows almost complete rejection of the $\nu_{\mu}$ charged current events
because the muons tend to be longer. These data are used in conjunction with
the standard low energy beam configuration data (horn-on) to extract the
individual NC and CC-$\nu_{\mu}$ components of the samples as a function of
reconstructed energy. The Monte Carlo is used to calculate the ratios $r_{NC}$
and $r_{CC}$, which are the ratios of the horn-off to horn-on configurations
for NC or CC events, respectively. An additional input from the Monte Carlo is
the small contamination of $\nu_{e}$ events in the beam. With this information
a calculation is performed for every energy bin to extract the CC and NC
composition in both horn on and horn off spectra. Figure 3 shows the final
result for the ANN selection. Integrating over the energy spectrum, the ND
background is 57$\pm$5% NC, 32$\pm$7% $\nu_{\mu}$-CC and 11$\pm$3% intrinsic
beam $\nu_{e}$-CC events. The errors on the NC and $\nu_{\mu}$-CC components
arise from the statistics of the horn-off data and systematics on the ratios;
the uncertainty on the beam $\nu_{e}$-CC includes systematic errors from the
beam flux, cross-section and selection efficiency for electrons.
Figure 3: (in color) Separation of the types of backgrounds in the near
detector data. A calculation is performed using the horn-off data which is
enriched in NC events because of the higher energy neutrino spectrum (see
text). Large fraction of the error is due to the statistics of the horn-off
sample. This is for the ANN selection, results are similar for the alternate
selection (LEM).
After decomposing the Near Detector energy spectrum into its background
components, each background spectrum is multiplied by the ratio of the Far to
Near ratio from the MC simulation for each component to provide a prediction
of the FD spectrum for that component. The far/near ratios are shown in figure
4 for the ANN selection. The MC simulations take into account differences in
the spectrum of events at the ND and FD due to the beam line geometry as well
as possible differences in detector calibrations and topological response.
Oscillations are included when predicting the $\nu_{\mu}$-CC component. The
smaller $\nu_{\tau}$-CC and beam $\nu_{e}$-CC components are calculated by
Monte Carlo using the expected energy spectrum in the FD. All background
components are then added together and summed over the energy range to provide
the total predicted background in the Far Detector. The detailed modeling of
all far/near differences change the background prediction from the lowest
order by $\sim$10%. We expect a total background of 26.6 events for the ANN
selection, of which 18.2 are NC, 5.1 are $\nu_{\mu}$-CC, 2.2 are beam
$\nu_{e}$ and 1.1 are $\nu_{\tau}$ for $3.14\times 10^{20}$ POT bbbUsing
$\Delta m^{2}_{32}$=$2.43\times 10^{-3}{\rm eV^{2}}$,
$\sin^{2}2\theta_{23}=1.0$, and $\sin^{2}2\theta_{13}=0.$. With LEM, we expect
21.4 background events, with 14.8 NC, 2.9 $\nu_{\mu}$-CC, 1.1 beam $\nu_{e}$
and 2.7 $\nu_{\tau}$.
Figure 4: (in color) Monte Carlo calculation of the far/near ratios for NC and
CC background components. The calculation includes effects of beamline
geometry (including the $1/r^{2}$ loss), fiducial mass, difference in spetra,
detector calibrations, and differences in the analysis efficiencies. Plots are
similar for the alternate selection (LEM).
The effects of systematic errors were evaluated by generating modified MC
samples, and quantifying the change in the number of predicted background
events in the Far Detector using Far to Near ratios from the modified samples
relative to the unmodified case. Many uncertainties, including those that
affect neutrino interaction physics, shower hadronization, intranuclear re-
scattering, and absolute energy scale errors affect the events in both
detectors in a similar manner and largely cancel. Other effects give rise to
Far/Near differences such as relative event rate normalization, calibration
errors, reconstruction differences between the detectors and low level
modeling of each detector. The individual systematic errors are added in
quadrature along with the systematic error arising from the decomposition of
the background sources in the ND to give an overall systematic error of 7.3%
on the number of background events selected with the ANN selection. The LEM
selection is more sensitive to uncertainties in the PMT gains, relative energy
calibration and crosstalk. The total systematic error on the number of
background events selected by the LEM technique is 12.0%.
### 5.1 Examination of events outside the signal region and other checks
Three main checks are performed by utilizing an independent data set obtained
from $\nu_{\mu}$ charged current events in which the muon is removed in
software and the remaining hadronic shower is analyzed as if it is a complete
neutrino event. This procedure is carried out on data and MC and the $\nu_{e}$
selections are applied to both. The discrepancy between muon-removed data and
MC simulation is similar to that found in the standard sample as a function of
reconstructed energy and of many different reconstructed shower topology
variables used in the selections.
In the first check, the muon-removed data is used to obtain the relative
contributions of the background components present in the ND data spectrum.
The number of selected NC events in the MC simulation of the standard sample
is scaled in each energy bin by the ratio of the number of events in the muon-
removed data to the muon-removed simulation. Once the number of NC events is
determined, the number of $\nu_{\mu}$-CC events selected in each reconstructed
energy bin in the data is determined using
$N_{CC}=N^{data}_{total}-N_{NC}-N_{\nu_{e}}$, in which the number of beam
$\nu_{e}$ events are obtained from the MC. The background components as
derived from the muon-removed sample agree well with those obtained from the
horn-off method.
In the second check, we treat the muon removed data from the near and far
detectors as if they are real events and perform a complete analysis. From the
near data we create a prediction for the far detector and count the number in
the far detector. Using this procedure we predicted $29\pm 5(stat)\pm 2(syst)$
events for the ANN selection and observed 39 events. For the LEM selection the
prediction was $17\pm 4(stat)\pm 2(stat)$ and the observation was 25. The
observed excess in this sample, which contains no electron signal events, was
a cause of concern, however, upon examination of the full distribution with
and without the particle ID cut, it was considered likely to be a statistical
fluctuation. This issue will be explored with the larger data sample being
acquired at this time.
In the third check, we estimate the efficiency for selecting $\nu_{e}$-CC
events. We use the sample of muon removed events and embed a simulated
electron of the same momentum as the removed muon. Test beam measurements
indicate that electrons are well simulated in the MINOS detectors. Data from
the test beam ?) was analyzed using the same selection cuts and agrees with
Monte Carlo within 2.6% for ANN and 2.2% for LEM. sxs Comparisons between
muon-removed data and simulated samples of events with embedded electrons
indicate that the selection efficiency of $\nu_{e}$ signal events is well
modeled by the MC. The algorithms focus on the EM core of the shower and are
not affected by hadronic shower modeling discrepancies. The difference between
the data and the MC is used as a correction to the signal selection efficiency
and it is -0.3% for ANN and -5.3% for LEM. The selection efficiency of the ANN
selection is calculated to be 41.4$\pm$1.4% and for LEM is 45.2$\pm$1.5%.
The prediction of the backgrounds in the FD and the systematic uncertainties
on that prediction were established before examining the data in the FD. Some
additional checks were performed before opening the signal region. The number
of events passing the preselection cuts, but failing the $\nu_{e}$-CC
selection cuts were compared to the expectation. In the FD data, 146 events
were observed below the ANN selection cut, with an expectation of $132\pm
12{\rm(stat)}\pm 8{\rm(syst)}$. The events below the LEM cut totaled 176
events compared to an expectation of $157\pm 13{\rm(stat)}\pm 3{\rm(syst)}$.
Both observations deviate from the background prediction by approximately
$1\sigma$ assuming no signal events in this part of the data.
## 6 Results for $\nu_{\mu}\to\nu_{e}$
After examining the sideband data sets, we proceeded to count the number of
events passing the predetermined selection cut. We observe $35$ events in the
FD when using the $\nu_{e}$ selection based on the ANN algorithm, with a
background expectation of $27\pm 5{\rm(stat.)}\pm 2{\rm(syst.)}$. With LEM
(the secondary selection) we observe $28$, with a background expectation of
$22\pm 5{\rm(stat.)}\pm 3{\rm(syst.)}$. The distributions are shown in Fig. 5
and 6.
Figure 5: (in color) Distribution of far detector events for the ANN PID. Left
shows the ANN PID distribution. Right shows the energy distribution after the
PID cut. The plots below show the data minus the background prediction with
the expected distribution of the signal if all the excess is interpreted as
signal.
Figure 6: (in color) Distribution of far detector events for the LEM PID. Left
shows the LEM PID distribution. Right shows the energy distribution after the
PID cut. The plots below show the data minus the background prediction with
the expected distribution of the signal if all the excess is interpreted as
signal.
Figure 7 shows the 90% confidence level interval in the $\sin^{2}2\theta_{13}$
and $\delta_{CP}$ plane for each mass hierarchy using our observation for ANN
PID. To set this limit we have used only the total observed number of events;
detailed fitting of the data distributions was not performed for this result.
We use the current best fit value of $|\Delta m^{2}_{32}|$=2.43$\times
10^{-3}$ ${\rm eV^{2}}$ and $\sin^{2}\left(2\theta_{23}\right)$=1.0 for this
calculation. Fluctuations (Poisson) and systematic effects (Gaussian) are
incorporated via the Feldman-Cousins approach ?). The oscillation probability
is computed using a full 3-flavor neutrino mixing framework that includes
matter effects.
Figure 7: The 90% confidence interval in the
$\sin^{2}\left(2\theta_{13}\right)$ and $\delta_{CP}$ plane using the ANN PID
results. Black lines show the best fit to our data in both the normal
hierarchy (solid) and inverted hierarchy (dotted). Blue (red) lines show the
90 C.L. boundaries for the normal (inverted) hierarchy.
## 7 Updates to MINOS measurement of $\nu_{\mu}$ disappearance
MINOS has recently reported updated measurements of $\nu_{\mu}$ disappearance
?) on the same data set that was described above (the high energy spectrum
data was also included). We observed 848 $\nu_{\mu}$-CC events in the far
detector across the energy range of 0 to 120 GeV compared to the expectation
of $1065\pm 60(syst)$. The observed spectrum is shown in figure 8. The same
figure also shows the confidence interval in the $\sin^{2}2\theta_{23}$ versus
$|\Delta m^{2}_{32}|$ plane. We obtain $|\Delta m^{2}_{32}|=2.43\pm 0.13\times
10^{-3}eV^{2}$ at 68 % C.L. and the mixing angle of
$\sin^{2}2\theta_{23}>0.90$ at 90% C.L. At present time the measurement of
$|\Delta m^{2}_{32}|$ is dominated by MINOS and the measurement of the mixing
angle is dominated by Super-Kamiokande.
Figure 8: (in color) Measurement of MINOS for the disappearance of muon
neutrinos.
While the disappearance of $\nu_{\mu}$ from the atmosphere and the NuMI/MINOS
beam experiment is largely explained by 3 generation neutrino mixing with
$\nu_{\mu}\to\nu_{\tau}$ as the mechanism, any small admixture of sterile
neutrinos is still an experimental issue. MINOS performed a search for
disappearance of active neutrinos using neutral current interactions ?). The
final spectrum of neutral current events and the prediction based on near
detector data is shown in figure 9. No anomalous depletion in the
reconstructed energy spectrum is observed. Assuming oscillations occur at a
single mass-squared splitting, a fit to the neutral- and charged-current
energy spectra limits the fraction of $\nu_{\mu}$ oscillating to a sterile
neutrino to be below 0.68 at 90% confidence level. Electron neutrinos can
constitute a background to the neutral current analysis, therefore any
possible contribution from $\nu_{e}$ appearance at the current experimental
bound leads to a less stringent limit.
Figure 9: (in color) Measurement of Neutral Current spectrum in MINOS.
## 8 Conclusions
In summary, we report the first results of a search for $\nu_{e}$ appearance
in the MINOS experiment. The observed rate of events in the Far Detector after
$\nu_{e}$ selection for $3.14\times 10^{20}{\rm\,POT}$ is consistent with the
background expectation within 1.5 standard deviations. For this data set,
assuming $|\Delta m^{2}_{32}|$=2.43$\times 10^{-3}$ ${\rm eV^{2}}$,
$\sin^{2}\left(2\theta_{23}\right)$=1.0, and $\delta_{CP}=0$, we set an upper
limit of $\sin^{2}(2\theta_{13})<0.29$ at 90% C.L. for the normal hierarchy
and $\sin^{2}(2\theta_{13})<0.42$ for the inverted hierarchy.
## 9 Acknowledgements
This was prepared for the proceedings of the XIII International Workshop on
Neutrino Telescopes at the Istituto Veneto di Scienze, Lettere ed Arti in
Venice held on March 10-13, 2009. The presentation was on behalf of the MINOS
collaboration. This work was supported by the US Department of Energy under
contract number DE-AC02-98CH10886.
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|
arxiv-papers
| 2009-04-23T19:15:27 |
2024-09-04T02:49:02.109581
|
{
"license": "Public Domain",
"authors": "Milind V. Diwan",
"submitter": "Milind Vaman Diwan",
"url": "https://arxiv.org/abs/0904.3706"
}
|
0904.3797
|
# Internet Traffic Periodicities and Oscillations: A Brief Review
Reginald D. Smith Bouchet-Franklin Research Institute, PO Box 10051,
Rochester, NY 14610 rsmith@bouchet-franklin.org
(April 23, 2009)
###### Abstract
Internet traffic displays many persistent periodicities (oscillations) on a
large range of time scales. This paper describes the measurement methodology
to detect Internet traffic periodicities and also describes the main
periodicities in Internet traffic.
###### keywords:
internet traffic, packets, FFT, wavelets, periodicities
††journal: Computer Networks
## 1 Introduction
Internet traffic has exploded in the last fifteen years as an area of intense
theoretical and experimental research. As the largest engineered
infrastructure and information system in human history, the Internet’s
staggering size and complexity are reinforced by its decentralized and self-
organizing structure. Using packets of encapsulated data and a commonly agreed
protocol suite, the Internet has far outgrown its origins as ARPANET whose
traffic has demanded new models and ways of thinking to understand and
predict.
Amongst the earliest discoveries were the researches of Leland and Wilson [1]
who identified the non-Poisson nature of Internet traffic. This was followed
by the seminal paper of Leland, Taqqu, Willinger, and Wilson [2] which proved
that Internet packet interarrival times are both self-similar and portray
long-range dependence. Though self-similarity is present at all time scales,
it is most well-defined when traffic is stationary, an assumption that can
only last a few hours at the most. The lack of stationarity on long time
scales is due to one of the most widely known periodicities (or oscillations)
in Internet traffic, the diurnal cycle with 12 and 24 hour peaks.
Internet periodicities are not new and have been well-studied since the
earliest days of large-scale measurements of packet traffic, however, they
rarely receive primary attention in discussions of traffic and are often
mentioned only as an aside or a footnote. Gradually, however, they are gaining
more attention. This new area of research has been dubbed _network
spectroscopy_ [3] or _Internet spectroscopy_. In this paper, they will take
front and center as the most important periodicities, as well as the
techniques to measure them, are described.
## 2 Detection Methodologies
Identifying periodicities in Internet traffic is, in general, not markedly
different from standard spectral analysis of any time series. The same
cautions apply with sampling rates and the Nyquist theorem to determine the
highest identifiable frequency as well as to be aware of possible aliasing. In
addition, the sampling period is important due to the large ranges of
magnitudes the periods of Internet periodicities occupy.
The standard method is covered in [4, 5]. A continuous time series is
collected and binned with a sampling rate $p$ where the number of packets
arriving every $p$ interval seconds are counted. Next, to remove the DC
component of the signal, every time step has the mean of the entire time
series subtracted from it. Next you calculate the autocovariance (ACVF) of the
adjusted time series. where for a time series of $N$ sampling periods (total
sampling time $pN$) the ACVF, $c$ at lag, $k$ is defined as
$c(k)=\sum^{N-k-1}_{t=0}(X(t)-\overline{X})(X(t+k)-\overline{X})$ (1)
with a typical lag range chosen of $0<k<N/2$. Finally, a Fourier transform is
taken of the ACVF with maximum lag $M$ and the periodogram created from the
absolute value (amplitude) of the Fourier series
$P(f)=\left|\sum^{M-1}_{k=0}c(k)e^{-i2\pi fk}\right|$ (2)
A resulting periodogram (see figure 1) has several typical features. First,
low frequency $1/f$ noise can be present, again testifying to the self-similar
nature of the traffic. This can sometimes obscure low-frequency periodicities
in the data. Second, are any periodicities, their harmonics, and occasionally
even small peaks perhaps representing nonlinear mixing of a sort between two
periodicities, often with periods of different orders of magnitude.
### 2.1 Wavelet methods
Given the nonstationary nature of Internet traffic and the frequent presence
of transients, methods based on the Fourier transform can only given an
incomplete view of the periodic dynamics of Internet traffic. In particular,
especially for rapidly changing periodicities such as those caused by RTT of
flows, periodicities may only be temporary before shifting, disappearing, or
being displaced. Wavelet methods have been developed in great theoretical and
practical detail in the last several decades to allow for the analysis of a
signal’s periodic nature on multiple times scales. Wavelet techniques will not
be covered here in detail though there are many good references [6, 7, 8, 9].
The continuous wavelet function on the signal $x(t)$, here an Internet traffic
trace, is given for a mother wavelet, $\psi$ with $a$ representing a
stretching coefficient (scale) and $b$ represents a translation coefficient
(time)
$T(a,b)=\frac{1}{\sqrt{a}}\int^{\infty}_{-\infty}x(t)\psi^{*}\left(\frac{t-b}{a}\right)dt$
(3)
In figure 1 alongside the FFT of the signal is a contour plot generated by
plotting $T(a,b)$ using the Morlet mother wavelet over 12 octaves. One of the
key advantages of wavelets is seeing the periodic variation over time. The
y-axis represents the period of the signal represented and the x-axis is the
time of the traffic trace in seconds. A first feature is the continuous strong
periodicity at 30 seconds as a result of the update packets. A second and more
intriguing feature are the inverse triangular ‘bursts’ of high frequency
traffic with an average period close to one hour. These are update packets
generated by route flapping, which are damped for a maximum period of one hour
according to the most common presets for route flapping damping. The packets
with the most pernicious flapping routers announcing withdrawals were removed
in the third figure where the hourly oscillation largely disappears.
Figure 1: A FFT periodogram and two wavelet contour plots of the ACVF of BGP
update packet traffic on node rrc00 on the RIPE Routing Information Service
(RIS) BGP update traffic on February 1, 2009 [11]. In the FFT plot, the strong
peaks are due to the 30s BGP KEEPALIVE update packet messages with subsequent
harmonics. The contour plot is based on a continuous wavelet transform using
the Morlet wavelet for a 20,000s (5.5 hour) trace starting at 0000 GMT. The
strong periodicity at 30s is evident, as well as the periodic bursts of high
frequency traffic below it due to route flapping and the one hour periodic
maximum suppression by route flap damping. The peaks correspond to the minimum
penalty for the flapping route while the troughs correspond to the maximum
penalty. In the third figure, the contour plot is recreated with the signal
omitting packets that announce a withdrawal of one of the top 5 (most likely
flapping) withdrawn IP addresses. The high frequency flapping is still present
but not the coordinated hourly damping as in the second figure.
## 3 Traffic oscillations/periodicities
There are a plethora of traffic periodicities that represent oscillations in
traffic over periods of many orders of magnitude from milliseconds to weeks.
Broido, et. al. [10] believe there are thousands of periodic processes in the
Internet. The sheer range of the periods of the periodicities means that many
times, only certain periodicities appear in packet arrival time series due
either to the sampling rate or sampling duration. This is one of the reasons
why a comprehensive description of all Internet periodicities has rarely been
done.
Internet periodicities have origins which broadly correspond to two general
causes: first, there are protocol or data transmission driven periodicities.
These range on the time scale from microseconds to seconds, or in rare cases,
hours. These periodicties can again be broken down into two smaller groups,
periodicities driven by packet data transmission on the link layer and
periodicities driven by protocol behavior on the transport layer.
Second are application driven periodicities. Their periods range on the time
scale of minutes to hours to weeks, and quite possibly longer. These are all
generated from activities at the application layer, either by automated
applications such as BGP or DNS or user driven applications via HTTP or other
user application protocols.
The major known periodicities are summarized in figure 2 and will be described
in detail in the next two subsections.
Figure 2: A rough breakdown of the major periodicities in Internet traffic
showing the responsible protocols and their period in seconds. The
periodicities span over 12 orders of magnitude and different protocol layers
tend to operate on different time scales.
### 3.1 Key Periodicities: Link and Transport Layer
A key link level periodicity due to the throughput of packet transmission [12,
5] of a link and can be deduced from the equation:
$f=\frac{T}{s}$ (4)
Where $T$ is the average throughput of the link and $s$ is the average packet
size at the link level. The base frequency is the rate of packet emission
across the link at the optimum throughput and packet size. The base frequency
for data transmission is given by
$f_{max}=\frac{B}{MTU}$ (5)
where $B$ is the bandwidth of the link and the packet size is the MTU packet
size. Therefore for 1 Gigabit, 100 Mbps, and 10 Mbps Ethernet links with MTU
sizes of 1500 bytes, the theoretical optimal base frequencies are 83.3 kHz,
8.3 kHz, and 833 Hz respectively. Other technologies have their own specific
periods such as SONET frames identified with periods of 125$\mu$s[3].
These are among the most difficult traffic to identify due to the need for
high sampling rates of packet traffic. At a minimum, a microsecond sampling
rate is usually necessary to make sure you can identify all link-layer
periodicities. It is rare that both link layer and other periodicities are
displayed together since the massive memory overhead of recording the
timestamp of almost every packet is necessary.
The link layer periodicities are receiving much of the attention in the
research, however, due to their possible use in inferring bottlenecks and
malicious traffic. The main practical applications being researched are
inferring network path characteristics such as bandwidth, digital
fingerprinting of link transmissions, and detecting malicious attack traffic
by changes in the frequency domain of the transmission signal. [13, 14] use
analysis of the distribution of packet interarrival times to infer congestion
and bottlenecks on network paths upstream. In [5, 4, 15, 16, 17, 18] various
measures of packet arrival distributions, particularly in the frequency
domain, are being tested to recognize and analyze distributed denial of
service or other malicious attacks against computer networks. Inspecting the
frequency domain of a signal can also reveal the fingerprints of the various
link level technologies used along the route of the signal as is done in [10,
19].
The transport layer also produces its own periodicities. In particular, both
TCP and ICMP often times operate bidirectional flows with the interarrival of
ACK packets corresponding to the RTT between the source and destination [10,
15, 20], often in the range of 10 ms to 1 s. Instead of just frequency peaks
there usually are wide bands corresponding to the dominant RTT in the TCP or
ICMP traffic measured. According to most equations of TCP throughput such as
that by Semke et. al. [21] the throughput of TCP depends inversely on the RTT
so that the TCP RTT periodicities often can give a relative estimate of
throughput of the flows producing them and the distribution of RTT for flows
in the traffic trace. Exact estimates are difficult though since packet loss
and maximum segment size are usually unknown. ICMP, though a connectionless
protocol also has echo replies which can also appear as periodicities if they
are persistent through time.
### 3.2 Key Periodicities: Application Layer
Once you rise to periods above one second, application layer periodicities
dominate the spectrum. These come from a variety of sources including software
settings and human activity. At the low end are the 30s and sometimes 60s
periodicities in BGP traffic. The 30s oscillation, shown in figure 1, is the
most common set time for routers to advertise their presence and continuing
function to neighboring routers using KEEPALIVE BGP updates. These are the
strongest periodicities present in BGP traffic. Large-scale topological
perturbations such as BGP storms can also produce transient periodicities in
traffic such as large-scale route flapping which is shown in figure 1.
UDP traffic periodicities are rarely consistent and large-scale and are
generally generated by DNS, the largest application using UDP. Claffy et. al.
identified periodicities of DNS updates transmitted with periods of 75
minutes, 1 hour, and 24 hours due to default settings in Windows 2000 and XP
DNS software[3]. They warn that such software settings could possibly cause
problems in Internet traffic if they lead to harmful periods of traffic
oscillations and congestion. Large numbers of usually source and software
specific UDP periodicities were also identified by Brondman [22].
User traffic driven periodicities were the first known and most easily
recognized. The first discovered and most well-known periodicity is the 24
hour diurnal cycle and its companion cycle of 12 hours. These cycles have been
known for decades and reported as early as 1980 and again in 1991 as well as
in many subsequent studies[23, 24, 25, 26, 27, 28, 29]. This obviously refers
to the 24 hour work-day and its 12-hour second harmonic as well as activity
from around the globe. The other major periodicity from human behavior is the
week with a period of 7 days [25, 26, 30] and a second harmonic at 3.5 days
and barely perceptible third harmonic at 2.3 days. There are reports as well
of seasonal variations in traffic over months [12], but mostly these have not
been firmly characterized. Long period oscillations have been linked to
possible causes of congestion and other network behavior related to network
monitoring [27, 28]. One note is that user traffic driven periodicities tend
to appear in protocols that are directly used by most end users. The
periodicities appear TCP/IP not UDP/IP and are mainly attributable to activity
with the HTTP and SMTP protocols. They also often do not appear in networks
with low traffic or research aims such as the now defunct 6Bone IPv6 test
network.
## 4 Discussion
These periodicities range in roughly 12 orders of magnitude. However, they
share one particular characteristic. Namely, the longer the period of the
periodicity, the less likely it is to betray variations in period or phase
over time. For example, the diurnal and weekly periodicities have their roots
in human activity and are based on the Earth’s rotation and the seven week
social convention. These do not vary appreciably over long-time periods and
since they help drive human behavior which drives traffic, these could be
considered the most permanent of all periodicities and this is partially why
these were the earliest known. The BGP KEEPALIVE updates and DNS updates are
based on commonly agreed software settings. These also do not vary appreciably
and only change by user preference. However, the transport and link layer
periodicities are much more variable. The RTT of TCP or ICMP varies depending
on the topological distance and congestion between two points. Hardly, stable
variables. Assuming the bandwidth of the link layers is steady, the average
packet size, which depends on both the maximum transmission unit (MTU)
software settings can cause large variability to be seen in actual network
traffic. Understanding the range of these periodicities is more important than
memorizing a distance frequency value since it is always different depending
on the time and place of measurement. Internet periodicities will likely play
a large role in full characterization and simulation of Internet traffic.
Hopefully further work will put them in their rightful place as fundamental
phenomena of data traffic.
## References
* [1] WE Leland, & DV Wilson, High time-resolution measurement and analysis of LAN traffic: Implications for LAN interconnection, Proceedings IEEE lNFOCOM ’91, (1991) 1360-1366.
* [2] WE Leland, MS Taqqu, W Willinger, & DV Wilson, On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Transactions on Networking, 2 1, (1994) -151.
* [3] A Broido, E Nemeth, & KC Claffy, Spectroscopy of DNS Update Traffic, ACM SIGMETRICS 2003, 31 (2003) 320-321.
* [4] A Hussain, J Heidemann, & C Papadopoulos, A framework for classifying denial of service attacks in Proceedings of the ACM SIGCOMM’2003. Karlsruhe, Germany, August 2003, (2003) 99-110.
* [5] X He, C Papadopoulos, J Heidemann, U Mitra, U Riaz, U & A Hussain, Remote detection of bottleneck links using spectral and statistical methods, Computer Networks, 53 (2009) 279-298.
* [6] DB Percival & AT Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, New York, 2000.
* [7] Y Nievergelt, Wavelets Made Easy Springer, Berlin, 1999\.
* [8] G Kaiser, A Friendly Guide to Wavelets Springer, Berlin, 1994.
* [9] P Addison, The Illustrated Wavelet Transform Handbook, CRC Press, Boca Raton, 2002.
* [10] A Broido, R King, E Nemeth, KC Claffy, Radon Spectroscopy of Packet Delay, in Proceedings of the IEEE High-Speed Networking Workshop 2003. San Diego, CA (2003).
* [11] RIPE Network Co-ordination Centre - Routing Information Service (RIS) http://www.ripe.net/ris/
* [12] X He, C Papadopoulos, J Heidemann, & A Hussain, Spectral Characteristics of Saturated Links, University of Southern California Technical Report, USC-CSD-TR-827 (2004).
* [13] D Katabi & C Blake, Inferring Congestion Sharing and Path Characteristics from Packet Interarrival Times, MIT Technical Report, MIT-LCSTR-828 (2001).
* [14] X He, C Papadopoulos, J Heidemann, U Mitra, U Riaz, U & A Hussain, Spectral Analysis of Bottleneck Traffic, University of Southern California Technical Report, USC/CS Technical Report 05-853 (2005).
* [15] CM Cheng, HT Kung, & KS Tan, Use of spectral analysis in defense against DoS attacks, in Proceedings of IEEE GLOBECOM ’02, 3 (2002) 2143-2148.
* [16] Y Chen & K Hwang, Collaborative detection and filtering of shrew DDoS attacks using spectral analysis, Journal of Parallel and Distributed Computing, 66 (2006) 1137-1151.
* [17] A Hussain, J Heidemann, & C Papadopolous, Identification of Repeated Attacks Using Network Traffic Forensics, USC/ISI Technical Report ISI-TR-2003-577b (2004).
* [18] L Li & G Lee, DDoS Attack Detection and Wavelets, Telecommunication Systems, 28 (2005) 435-451.
* [19] M Coates, A Hero, R Nowak, & B Yu, Internet Tomography, IEEE Signal Processing Magazine, 19 no. 3 (2002) 47-65.
* [20] A Broido, Invariance of Internet RTT spectrum, in Proceedings of ISMA Conference, October 2002 (2002).
* [21] M Mathis, J Semke, & J Madhavi, The Macroscopic Behavior of the TCP Congestion Avoidance Algorithm, ACM SIGCOMM Computer Communication Review, 27 no. 3 (1997) 67-82.
* [22] Grondman, I, Identifying short-term periodicities in Internet traffic, BSc. Thesis, University of Twente (2006).
* [23] JF Shoch & JA Hupp, Measured performance of an Ethernet local network, Communications of the ACM, 23 (1980) 711-721.
* [24] A Lakhina, K Papagiannaki, ME Crovella, C Diot, E Kolaczyk, & N Taft, Structural analysis of network traffic flows, ACM SIGMETRICS Performance Evaluation Review, 32 (2004) 61-72.
* [25] K Papagiannaki, N Taft, Z Zhang, & C Diot, Long-term forecasting of Internet backbone traffic, IEEE Transactions on Neural Networks, 16 (2005) 1110-1124.
* [26] M Roughan, A Greenberg, C Kalmanek, M Rumsewicz, J Yates, & Y Zhang, Experience in measuring backbone traffic variability: models, metrics, measurements and meaning, Proceedings of the 2nd ACM SIGCOMM Workshop on Internet Measurement, (2002) 91-92.
* [27] A Mukherjee, On The Dynamics and Significance of Low Frequency Components of Internet Load, University of Pennsylvania Technical Reports, MS-CIS-92-83 (1992).
* [28] P Owezarski & N Larrieu, Internet Traffic Characterization - An Analysis of Traffic Oscillations, in High Speed Networks and Multimedia Communications edited by MM Freire, P Lorenz, & M Lee, Springer, Berlin, 2004, 96
* [29] HJ Fowler & WE Leland, Local area network characteristics, with implications for broadbandnetwork congestion management, IEEE Journal on Selected Areas in Communications, 19 (1991) 1139-1149.
* [30] M Burgess, H Haugerud, S Straumsnes, & T Reitan, Measuring system normality, ACM Transactions on Computer Systems, 20 2, (2002) 125-160.
|
arxiv-papers
| 2009-04-24T12:35:01 |
2024-09-04T02:49:02.117852
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Reginald D. Smith",
"submitter": "Reginald Smith",
"url": "https://arxiv.org/abs/0904.3797"
}
|
0904.3855
|
# Exploring Progressions: A Collection of Problems
Konstantine Zelator
Department of Mathematics
and Computer Science
Rhode Island College
600 Mount Pleasant Avenue
Providence, RI 02908
USA
## 1 Introduction
In this work, we study the subject of arithmetic, geometric, mixed, and
harmonic progressions. Some of the material found in Sections 2,3,4, and 5,
can be found in standard precalculus texts. For example, refer to the books in
[1] and [2]. A substantial portion of the material in those sections cannot be
found in such books. In Section 6, we present 21 problems, with detailed
solutions. These are interesting, unusual problems not commonly found in
mathematics texts, and most of them are quite challenging. The material of
this paper is aimed at mathematics educators as well as math specialists with
a keen interest in progressions.
## 2 Progressions
In this paper we will study arithmetic and geometric progressions, as well as
mixed progressions. All three kinds of progressions are examples of sequences.
Almost every student who has studied mathematics, at least through a first
calculus course, has come across the concept of sequences. Such a student has
usually seen some examples of sequences so the reader of this book has quite
likely at least some informal understanding of what the term sequence means.
We start with a formal definition of the term sequence.
Definition 1:
1. (a)
A finite sequence of $k$ elements, ($k$ a fixed positive integer) and whose
terms are real numbers, is a mapping $f$ from the set $\\{1,2,\ldots,k\\}$
(the set containing the first $k$ positive integers) to the set of real
numbers $\mathbb{R}$. Such a sequence is usually denoted by
$a_{1},\ldots,a_{n},\ldots,a_{k}$. If $n$ is a positive integer between $1$
and $k$, the $n$th term ${\boldmath a}_{\boldmath n}$, is simply the value of
the function $f$ at $n$; $a_{n}=f(n)$.
2. (b)
An infinite sequence whose terms are real numbers, is a mapping $f$ from the
set of positive integers or natural numbers to the set of real numbers
$\mathbb{R}$, we write $F:\mathbb{N}\rightarrow\mathbb{R}$; $f(n)=a_{n}$.
Such a sequence is usually denoted by $a_{1},a_{2},\ldots a_{n},\ldots$ . The
term $a_{n}$ is called the $n$th term of the sequence and it is simply the
value of the function at $n$.
Remark 1: Unlike sets, for which the order in which their elements do not
matter, in a sequence the order in which the elements are listed does matter
and makes a particular sequence unique. For example, the sequences $1,\ 8,\
10,$ and $8,\ 10,\ 1$ are regarded as different sequences. In the first case
we have a function $f$ from $\\{1,2,3\\}$ to $\mathbb{R}$ defined as follows:
$f:=\\{1,2,3\\}\rightarrow\mathbb{R};\ f(1)=1=a_{1},\ f(2)=8=a_{2}$, and
$f(3)=10=a_{3}$. In the second case, we have a function
$g:\\{1,2,3\\}\rightarrow\mathbb{R};\ g(1)=b_{1}=8,\ g(2)=b_{2}=10$, and
$g(3)=b_{3}=1$.
Only if two sequences are equal as functions, are they regarded one and the
same sequence.
## 3 Arithmetic Progressions
Definition 2: A sequence $a_{1},a_{2},\ldots,a_{n},\ldots$ with at least two
terms, is called an arithmetic progression, if, and only if there exists a
(fixed) real number $d$ such that $a_{n+1}=a_{n}+d$, for every natural number
$n$, if the sequence is infinite. If the sequence if finite with $k$ terms,
then $a_{n+1}=a_{n}+d$ for $n=1,\ldots,k-1$. The real number $d$ is called the
difference of the arithmetic progression.
Remark 2: What the above definition really says, is that starting with the
second term $a_{2}$, each term of the sequence is equal to the sum of the
previous term plus the fixed number $d$.
Definition 3: An arithmetic progression is said to be increasing if the real
number $d$ (in Definition 2) is positive, and decreasing if the real number
$d$ is negative, and constant if $d=0$.
Remark 3: Obviously, if $d>0$, each term will be greater than the previous
term, while if $d<0$, each term will be smaller than the previous one.
Theorem 1: Let $a_{1},a_{2},\ldots,a_{n},\ldots$ be an arithmetic progression
with difference $d,m$ and $n$ any natural numbers with $m<n$. The following
hold true:
1. (i)
$a_{n}=a_{1}+(n-1)d$
2. (ii)
$a_{n}=a_{n-m}+md$
3. (iii)
$a_{m+1}+a_{n-m}=a_{1}+a_{n}$
Proof:
1. (i)
We may proceed by mathematical induction. The statement obviously holds for
$n=1$ since $a_{1}=a_{1}+(1-1)d$; $a_{1}=a_{1}+0$, which is true. Next we show
that if the statement holds for some natural number $t$, then this assumption
implies that the statement must also hold for $(t+1)$. Indeed, if the
statement holds for $n=t$, then we have $a_{t}=a_{1}+(t-1)d$, but we also know
that $a_{t+1}=a_{t}+d$, since $a_{t}$ and $a_{t+1}$ are successive terms of
the given arithmetic progression. Thus, $a_{t}=a_{1}+(t-1)d\Rightarrow
a_{t}+d=a_{1}(t-1)d+d\Rightarrow a_{t}+d=a_{1}+d\cdot t\Rightarrow
a_{t+1}=a_{1}+d\cdot t$; $a_{t+1}=a_{1}+d\cdot[(t+1)-1]$, which proves that
the statement also holds for $n=t+1$. The induction process is complete.
2. (ii)
By part (i) we have established that $a_{n}=a_{1}+(n-1)d$, for every natural
number $n$. So that
$\begin{array}[]{rcl}a_{n}&=&a_{1}+(n-1)d-md+md;\\\ \\\
a_{n}&=&a_{1}+[(n-m)-1]d+md.\end{array}$
Again, by part (i) we know that $a_{n-m}=a_{1}+[(n-m)-1]d$. Combining this
with the last equation we obtain, $a_{n}=a_{n-m}+md$, and the proof is
complete.
3. (iii)
By part (i) we know that $a_{m+1}=a_{1}+[(m+1)-1]d\Rightarrow
a_{m+1}=a_{1}+md$; and by part (ii), we have already established that
$a_{n}=a_{n-m}+md$. Hence, $a_{m+1}+a_{n-m}=a_{1}+md+a_{n-m}=a_{1}+a_{n}$, and
the proof is complete. $\square$
Remark 4: Note that what Theorem 1(iii) really says is that in an arithmetic
progression $a_{1},\ldots,a_{n}$ with $a_{1}$ being the first term and $a_{n}$
being the $n$th or last term; if we pick two in between terms $a_{m+1}$ and
$a_{n-m}$ which are “equidistant” from the first and last term respectively
($a_{m+1}$ is $m$ places or spaces to the right of $a_{1}$ while $a_{n-m}$ is
$m$ spaces or places to the left of $a_{n}$), the sum of $a_{m+1}$ and
$a_{n-m}$ remains fixed: it is always equal to $(a_{1}+a_{n})$, no matter what
the value of $m$ is ($m$ can take values from $1$ to $(n-1)$). For example, if
$a_{1},a_{2},a_{3},a_{4},a_{5}$ is an arithmetic progression we must have
$a_{1}+a_{5}=a_{2}+a_{4}=a_{3}+a_{3}=2a_{3}$. Note that $(a_{2}+a_{4})$
corresponds to $m=1$, while $(a_{3}+a_{3})$ corresponds to $m=2$, but also
$a_{4}+a_{2}$ corresponds to $m=3$ and $a_{5}+a_{1}$ corresponds to $m=4$.
Likewise, if $b_{1},b_{2},b_{3},b_{4},b_{5},b_{6}$ are the successive terms of
an arithmetic progression we must have $b_{1}+b_{6}=b_{2}+b_{5}=b_{3}+b_{4}$.
The following theorem establishes two equivalent formulas for the sum of the
first $n$ terms of an arithmetic progression.
Theorem 2: Let $a_{1},a_{2},\ldots,a_{n},\ldots,$ be an arithmetic progression
with difference $d$.
1. (i)
The sum of the first (successive) $n$ terms $a_{1},\ldots,a_{n}$, is equal to
the real number $\left({\displaystyle\frac{a_{1}+a_{n}}{2}}\right)\cdot n$; we
write
$a_{1}+a_{2}+\cdots+a_{n}={\displaystyle\sum^{n}_{i=1}}a_{i}={\displaystyle\frac{n\cdot(a_{1}+a_{n})}{2}}$.
2. (ii)
${\displaystyle\sum^{n}_{i=1}}a_{i}=\left({\displaystyle\frac{a_{1}+[a_{1}+(n-1)d]}{2}}\right)\cdot
n$.
Proof:
1. (i)
We proceed by mathematical induction. For $n=1$ the statement is obviously
true since
$a_{1}={\displaystyle\frac{1\cdot(a_{1}+a_{1})}{2}}={\displaystyle\frac{2a_{1}}{2}}$
. Assume the statement to hold true for some $n=k\geq 1$. We will show that
whenever the statement holds true for some value $k$ of $n,\ k\geq 1$, it must
also hold true for $n=k+1$. Indeed, assume
$a_{1}+\cdots+a_{k}={\displaystyle\frac{k\cdot(a_{1}+a_{k})}{2}}$; add
$a_{k+1}$ to both sides to obtain
$\begin{array}[]{rcl}a_{1}+\cdots+a_{k}+a_{k+1}&=&{\displaystyle\frac{k\cdot
a_{1}+a_{k}}{2}}+a_{k+1}\\\ &&\Rightarrow a_{1}+\cdots+a_{k}+a_{k+1}\\\ \\\
&=&{\displaystyle\frac{ka_{1}+ka_{k}+2a_{k+1}}{2}}\end{array}$ $None$
But since the given sequence is an arithmetic progression by Theorem 1(i), we
must have $a_{k+1}=a_{1}+kd$ where $d$ is the difference. Substituting back in
equation (1) for $a_{k+1}$ we obtain,
$\begin{array}[]{rcl}a_{1}+\cdots+a_{k}+a_{k+1}&=&{\displaystyle\frac{ka_{1}+ka_{k}+(a_{1}+kd)+a_{k+1}}{2}}\\\
\\\ \Rightarrow
a_{1}+\cdots+a_{k}+a_{k+1}&=&{\displaystyle\frac{(k+1)a_{1}+k(a_{k}+d)+a_{k+1}}{2}}\end{array}$
$None$
We also have $a_{k+1}=a_{k}+d$, since $a_{k}$ and $a_{k+1}$ are successive
terms. Replacing $a_{k}+d$ by $a_{k+1}$ in equation (2) we now have
$a_{1}+\cdots+a_{k}+a_{k+1}={\displaystyle\frac{(k+1)a_{1}+ka_{k+1}+a_{k+1}}{2}}={\displaystyle\frac{(k+1)a_{1}+(k+1)a_{k+1}}{2}}=(k+1)\cdot{\displaystyle\frac{(a_{1}+a_{k+1})}{2}}$,
and the proof is complete. The statement also holds for $n=k+1$. $\square$
2. (ii)
This is an immediate consequence of part (i). Since
${\displaystyle\sum^{n}_{i=1}}a_{i}={\displaystyle\frac{n(a_{1}+a_{n})}{2}}$
and $a_{n}=a_{1}+(n-1)d$ (by Theorem 1(i)) we have,
${\displaystyle\sum^{n}_{i=1}}a_{i}=n\left({\displaystyle\frac{a_{1}+[a_{1}+(n-1)d]}{2}}\right),$
and we are done. $\square$
Example 1:
1. (i)
The sequence of positive integers $1,2,3,\ldots,n,\ldots,$ is an infinite
sequence which is an arithmetic progression with first term $a_{1}=1$,
difference $d=1$, and the $n$th term $a_{n}=n$. According to Theorem 2(i) the
sum of the first $n$ terms can be easily found:
$1+2+\ldots+n={\displaystyle\frac{n\cdot(1+n)}{2}}$.
2. (ii)
The sequence of the even positive integers $2,4,6,8,\ldots,2n,\ldots$ has
first term $a_{1}=2$, difference $d=2$, and the $n$th term $a_{n}=2n$.
According to Theorem 2(i),
$2+4+\cdots+2n={\displaystyle\frac{n\cdot(2+2n)}{2}}={\displaystyle\frac{n\cdot
2\cdot(n+1)}{2}}=n\cdot(n+1)$.
3. (iii)
The sequence of the odd natural numbers $1,3,5,\ldots,(2n-1),\ldots$, is an
arithmetic progression with first term $a_{1}=1$, difference $d=2$, and $n$th
term $a_{n}=2n-1$. According to Theorem 2(i) we have
$1+3+\cdots+(2n-1)=n\cdot\left({\displaystyle\frac{1+(2n-1)}{2}}\right)={\displaystyle\frac{n\cdot(2n)}{2}}=n^{2}$.
4. (iv)
The sequence of all natural numbers which are multiples of $3\ :\ 3,6,9,12,$
$\ldots,3n,\ldots$ is an arithmetic progression with first term $a_{1}=3$,
difference $d=3$ and $n$th term $a_{n}=3n$. We have
$3+6+\cdots+3n={\displaystyle\frac{n\cdot(3+3n)}{2}}={\displaystyle\frac{3n(n+1)}{2}}$.
Observe that this sum can also be found from (i) by observing that
$3+6+\cdots+3n=3\cdot(1+2+\cdots+n)={\displaystyle\frac{3\cdot n(n+1)}{2}}$.
If we had to find the sum of all natural numbers which are multiples of $3$,
starting with $3$ and ending with $33$; we know that $a_{1}=3$ and that
$a_{n}=33$. We must find the value of $n$. Indeed, $a_{n}=a_{1}+(n-1)\cdot d$;
and since $d=3$, we have $33=3+(n-1)\cdot 3\Rightarrow 33=3\cdot[1+(n-1)]$;
$11=n$. Thus,
$3+6+\cdots+30+33={\displaystyle\frac{11\cdot(3+33)}{2}}={\displaystyle\frac{11\cdot
36}{2}}=11\cdot 18=198$.
Example 2: Given an arithmetic progression
$a_{1},\ldots,a_{m},\ldots,a_{n},\ldots$, and natural numbers $m,n$ with
$2\leq m<n$, one can always find the sum $a_{m}+a_{m+1}+\cdots+a_{n-1}+a_{n}$;
that is, the sum of the $[(n-m)+1]$ terms starting with $a_{m}$ and ending
with $a_{n}$. If we know the values of $a_{m}$ and $a_{n}$ then we do not need
to know the value of the difference. Indeed, the finite sequence
$a_{m},a_{m+1},\ldots,a_{n-1},a_{n}$ is a finite arithmetic progression with
first term $a_{m}$, last term $a_{n}$, (and difference $d$); and it contains
exactly $[(n-m)+1]$ terms. According to Theorem 2(i) we must have
$a_{m}+a_{m+1}+\cdots+a_{n-1}+a_{n}=\frac{(n-m+1)\cdot[a_{m}+a_{n}]}{2}$.
If, on the other hand, we only know the values of the first term $a_{1}$ and
difference $d$ ( and the values of $m$ and $n$), we can apply Theorem 2(ii) by
observing that
$\begin{array}[]{rcl}a_{m}+a_{m+1}+\cdots+a_{n-1}+a_{n}&=&\underset{\underset{n\
{\rm terms}}{{\rm sum\ of\ the\
first}}}{\left(\underbrace{a_{1}+a_{2}+\ldots+a_{n}}\right)}\\\
&&-\underset{\underset{(m-1)\ {\rm terms}}{{\rm sum\ of\ the\
first}}}{\left(\underbrace{a_{1}+\ldots+a_{m-1}}\right)}\\\ \\\ {\rm by\ Th.\
2(ii)}&=&\left(\frac{2a_{1}+(n-1)d}{2}\right)\cdot n\\\
&&-\left(\frac{2a_{1}+(m-2)d}{2}\right)\cdot(m-1)\\\ \\\
&=&\frac{2[n-(m-1)]a_{1}+[n\cdot(n-1)-\cdot(m-2)\cdot(m-1)]d}{2}\\\ \\\
&=&\frac{2(n-m+1)a_{1}+[n(n-1)-(m-2)(m-1)]d}{2}\end{array}$
Example 3:
1. (a)
Find the sum of all multiples of $7$, starting with $49$ and ending with
$133$. Both $49$ and $133$ are terms of the infinite arithmetic progression
with first term $a_{1}=7$, and difference $d=7$. If $a_{m}=49$, then
$49=a_{1}+(m-1)d;\ 49=7+(m-1)\cdot 7\Rightarrow\frac{49}{7}=m;\ m=7$.
Likewise, if $a_{n}=$ then $133=a_{1}+(n-1)d;\ 133=7+(n-1)7\Rightarrow 19=n$.
According to Example 2, the sum we are looking for is given by
$a_{7}+a_{8}+\ldots+a_{18}+a_{19}=\frac{(19-7+1)(a_{7}+a_{19})}{2}=\frac{13\cdot(49+133)}{2}=\frac{13\cdot
182}{2}=(13)\cdot(91)=1183$.
2. (b)
For the arithmetic progression with first term $a_{1}=11$ and difference
$d=5$, find the sum of its terms starting with $a_{5}$ and ending with
$a_{13}$.
We are looking for the sum $a_{5}+a_{6}+\ldots+a_{12}+a_{13}$; in the usual
notation $m=5$ and $n=13$. According to Example 2, since we know the first
term $a_{1}=11$ and the difference $d=5$ we may use the formula we developed
there:
$\begin{array}[]{rcl}a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}&=&\frac{2(n-m+1)a_{1}+[n(n-1)-(m-2)(m-1)]d}{2};\\\
\\\ a_{5}+a_{6}+\ldots+a_{12}+a_{13}&=&\frac{2\cdot(13-5+1)\cdot
11+[13\cdot(13-1)-(5-2)(5-1)]5}{2}\\\ \\\ &=&\frac{2\cdot 9\cdot
11+[(13)(12)-(3)(4)]5}{2}=\frac{198+(156-12)\cdot 5}{2}\\\ \\\
&=&\frac{198+720}{2}=\frac{918}{2}=459\end{array}$
The following Theorem is simple in both its statement and proof but it serves
as an effective tool to check whether three real numbers are successive terms
of an arithmetic progression.
Theorem 3: Let $a,b,c$ be real numbers with $a<b<c$.
1. (i)
The three numbers $a,b$, and $c$ are successive of an arithmetic progression
if, and only if, $2b=a+c$ or equivalently $b=\frac{a+c}{2}$.
2. (ii)
Any arithmetic progression containing $a,b,c$ as successive terms must have
the same difference $d$, namely $d=b-a=c-b$
Proof:
1. (i)
Suppose that $a,b$, and $c$ are successive terms of an arithmetic progression;
then by definition we have $b=a+d$ and $c=b+d$, where $d$ is the difference.
So that $d=b-a=c-b$; from $b-a=c-b$ we obtain $2b=a+c$ or $b=\frac{a+c}{2}$.
Conversely, if $2b=a+c$, then $b-a=c-b$; so by setting $d=b-a=c-b$, it
immediately follows that $b=a+d$ and $c=b+d$ which proves that the real
numbers $a,b,c$ are successive terms of an arithmetic progression with
difference $d$.
2. (ii)
This has already been shown in part (i), namely that $d=b-a=c-b$. Thus, any
arithmetic progression containing the real numbers $a,b,c$ as successive terms
must have difference $d=b-a=c-b$.
Remark 5: According to Theorem 3, the middle term $b$ is the average of $a$
and $c$. This is generalized in Theorem 4 below. But, first we have the
following definition.
Definition 4: Let $a_{1},a_{2},\ldots,a_{n}$ be a list (or sequence) of $n$
real numbers($n$ a positive integer). The arithmetic mean or average of the
given list, is the real number $\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$.
Theorem 4: Let $m$ and $n$ be natural numbers with $m<n$. Suppose that the
real numbers $a_{m},a_{m+1},\ldots,a_{n-1},a_{n}$ are the $(n-m+1)$ successive
terms of an arithmetic progression (here, as in the usual notation, $a_{k}$
stands for the $k$th term of an arithmetic progression whose first term is
$a_{1}$ and difference is $d$).
1. (i)
If the natural number $(n-m+1)$ is odd, then the arithmetic mean or average of
the reals $a_{m},a_{m+1},\ldots,a_{n-1},a_{n}$ is the term
$a_{(\frac{m+n}{2})}$. In other words,
$a_{(\frac{m+n}{2})}=\frac{a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}}{n-m+1}$. (Note
that since $(n-m+1)$ is odd, it follows that $n-m$ must be even, and thus so
must be $n+m$; and hence $\frac{m+n}{2}$ must be a natural number).
2. (ii)
If the natural number is even, then the arithmetic mean of the reals
$a_{m},a_{m+1},\ldots,a_{n-1},a_{n}$ must be the average of the two middle
terms $a_{(\frac{n+m-1}{2})}$ and $a_{(\frac{n+m+1}{2})}$.
In other words
$\frac{a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}}{n-m+1}=\frac{a_{(\frac{n+m-1}{2})}+a_{(\frac{n+m+1}{2})}}{2}$.
Remark 6: To clearly see the workings of Theorem 4, let’s look at two
examples; first suppose $m=3$ and $n=7$. Then $n-m+1=7-3+1=5$; so if
$a_{3},a_{4},a_{5},a_{6},a_{7}$ are successive terms of an arithmetic
progression, clearly $a_{5}$ is the middle term. But since the five terms are
equally spaced or equidistant from one another (because each term is equal to
the sum of the previous terms plus a fixed number, the difference $d$), it
makes sense that $a_{5}$ would also turn out to be the average of the five
terms.
If, on the other hand, the natural number $n-m+1$ is even; as in the case of
$m=3$ and $n=8$. Then we have two middle numbers: $a_{5}$ and $a_{6}$.
Proof (of Theorem 4):
1. (i)
Since $n-m+1$ is odd, it follows $n-m$ is even; and thus $n+m$ is also even.
Now, if we look at the integers $m,m+1,\ldots,n-1,n$ we will see that since
$m+n$ odd, there is a middle number among them, namely the natural number
$\frac{m+n}{2}$. Consequently among the terms
$a_{m},a_{m+1},\ldots,a_{n-1},a_{n}$, the term $a_{(\frac{m+n}{2})}$ is the
middle term. Next we perform two calculations. First we compute
$a_{(\frac{m+n}{2})}$ in terms of $m,n$ the first term $a_{1}$ and the
difference $d$. According to Theorem 1(i), we have,
$a_{(\frac{m+n}{2})}=a_{1}+\left(\frac{m+n}{2}-1\right)d=a_{1}+\left(\frac{m+n-2}{2}\right)d.$
Now let us compute the sum $\frac{a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}}{n-m+1}$.
First assume $m\geq 2$; so that $2\leq m<n$. Observe that
$\begin{array}[]{rl}&a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}\\\ \\\
=&\underset{{\rm sum\ of\ the\ first}\ n\ {\rm
terms}}{\left({\underbrace{a_{1}+a_{2}+\ldots+a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}}}\right)}\\\
\\\ &-\underset{\underset{{\rm note\ that}\ m-1\geq 1,\ {\rm since}\ m\geq
2}{{\rm sum\ of\ the\ first}\ (m-1)\ {\rm
terms}}}{(\underbrace{a_{1}+\ldots+a_{m-1}})}\end{array}$
Apply Theorem 2(ii), we have,
$a_{1}+a_{2}+\ldots+a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}=\frac{n[2a_{1}+(n-1)d]}{2}$
and
$a_{1}+\ldots+a_{m-1}=\frac{(m-1)[2a_{1}+(m-2)d]}{2}.$
Putting everything together we have
$\begin{array}[]{rl}&a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}\\\ \\\
=&(a_{1}+a_{2}+\ldots+a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n})\\\ \\\
&-(a_{1}+\ldots+a_{m-1})=\frac{n[2a_{1}+(n-1)d]}{2}\\\ \\\
&-\frac{(m-1)[2a_{1}+(m-2)d]}{2}\\\ \\\
=&\frac{2(n-m+1)a_{1}+[n(n-1)-(m-1)(m-2)]d}{2}.\end{array}$
Thus,
$\begin{array}[]{rcl}&&\frac{a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}}{n-m+1}\\\ \\\
&=&\frac{2(n-m+1)a_{1}+[n(n-1)-(m-1)(m-2)]d}{2(n-m+1)}\\\ \\\
&=&a_{1}+\frac{[n(n-1)-(m-1)(m-2)]d}{2(n-m+1)}\\\ \\\
&=&a_{1}+\frac{[n^{2}-m^{2}-n+3m-2]d}{2(n-m+1)}\\\ \\\
&=&a_{1}+\frac{[(n-m)(n+m)+(n+m)-2(n-m)-2]d}{2(n-m+1)}\\\ \\\
&=&a_{1}+\frac{[(n-m)(n+m)+(n+m)-2(n-m+1)]d}{2(n-m+1)}\\\ \\\
&=&a_{1}+\frac{[(n+m)(n-m+1)-2(n-m+1)]d}{2(n-m+1)}\\\ \\\
&=&a_{1}+\frac{(n-m+1)(n+m-2)d}{2(n-m+1)}=a_{1}+\frac{(n+m-2)d}{2},\end{array}$
which is equal to the term $a_{(\frac{m+n}{2})}$ as we have already shown.
What about the case $m=1$? If $m=1$, then $n-m+1=n$ and $a_{m}=a_{1}$. In that
case, we have the sum $\frac{a_{1}+a_{2}+\ldots+a_{n-1}+a_{n}}{n}=$ (by
Theorem 2(ii)) $\frac{n\cdot[2a_{1}+(n-1)d]}{2n}$; but the middle term
$a_{(\frac{m+n}{2})}$ is now $a_{(\frac{n+1}{2})}$ since $m=1$; but
$a_{(\frac{n+1}{2})}=a_{1}+(\frac{1+n-2}{2})d\Rightarrow
a_{(\frac{n+1}{2})}=a_{1}+(\frac{n-1}{2})d$; compare this answer with what we
just found right above, namely
$\frac{n\cdot[2a_{1}+(n-1)d]}{2n}=\frac{2a_{1}+(n-1)d}{2}=a_{1}+(\frac{n-1}{2})d,$
they are the same. The proof is complete.
2. (ii)
This is left as an exercise to the student. (See Exercise 23).
Definition 5: A sequence $a_{1},a_{2},\ldots,a_{n},\ldots$ (finite or
infinite) is called a harmonic progression, if, and only if, the corresponding
sequence of the reciprocal terms:
$b_{1}=\frac{1}{a_{1}},\ \
b_{2}=\frac{1}{a_{2}},\ldots,b_{n}=\frac{1}{a_{n}},\ldots,$
is an arithmetic progression.
Example 4: The reader can easily verify that the following three sequences are
harmonic progressions:
1. (a)
$\frac{1}{1},\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n},\ldots$
2. (b)
$\frac{1}{2},\frac{1}{4},\frac{1}{6},\ldots,\frac{1}{2n},\ldots$
3. (c)
$\frac{1}{9},\frac{1}{16},\frac{1}{23},\ldots,\frac{1}{7n+2},\ldots$
## 4 Geometric Progressions
Definition 6: A sequence $a_{1},a_{2},\ldots,a_{n},\ldots$ (finite or
infinite) is called a geometric progression, if there exists a (fixed) real
number $r$ such that $a_{n+1}=r\cdot a_{n}$, for every natural number $n$ (if
the progression is finite with $k$ terms $a_{1},\ldots,a_{k}$; with $k\geq 2$,
then $a_{n+1}=r\cdot a_{n}$, for all $n=1,2,\ldots,k-1$). The real number $r$
is called the ratio of the geometric progression. The first term of the
arithmetic progression is usually denoted by $a$, we write $a_{1}=a$.
Theorem 5: Let $a=a_{1},a_{2},\ldots,a_{n},\ldots$ be a geometric progression
with first term $a$ and ratio $r$.
1. (i)
$a_{n}=a\cdot r^{n-1}$, for every natural number $n$.
2. (ii)
$a_{1}+\ldots+a_{n}={\displaystyle\sum^{n}_{i=1}}a_{i}=\frac{a_{n}\cdot
r-a}{r-1}=\frac{a(r^{n}-1)}{r-1}$, for every natural number $n$, if $r\neq 1$;
if on the other hand $r=1$, then the sum of the first $n$ terms of the
geometric progression is equal to $n\cdot a$.
Proof:
1. (i)
By induction: the statement is true for $n=1$, since $a_{1}=a\cdot
r^{\circ}=a$. Assume the statement to hold true for $n=k$; for some natural
number $k$. We will show that this assumption implies the statement to be also
true for $n=(k+1)$. Indeed, since the statement is true for $n=k$, we have
$a_{k}=a\cdot r^{k-1}\Rightarrow r\cdot a_{k}=r\cdot a\cdot r^{k-1}=a\cdot
r^{k}$; but $k=(k+1-1)$ and $r\cdot a_{k}=a_{k+1}$, by the definition of a
geometric progression. Hence, $a_{k+1}=a\cdot r^{(k+1)-1}$, and so the
statement also holds true for $n=k$.
2. (ii)
Most students probably have seen in precalculus the identity
$r^{n}-1=(r-1)(r^{n-1}+\ldots+1)$ to hold true for all natural numbers $n$ and
all reals $r$. For example, when $n=2,\ r^{2}-1=(r-1)(r+1)$; when $n=3$,
$r^{3}-1=(r-1)(r^{2}+r+1)$.
We use induction to actually prove it. Note that the statement $n=1$ simply
takes the form, $r-1=r-1$ so it holds true; while for $n=2$ the statement
becomes $r^{2}-1=(r-1)(r+1)$, which is again true. Now assume the statement to
hold for some $n=k,\ k\geq 2$ a natural number. So we are assuming that the
statement $r^{k}-1=(r-1)(r^{k-1}+\ldots+r+1)$. Multiply both sides by $r$:
$\begin{array}[]{rl}&r\cdot(r^{k}-1)=r\cdot(r-1)\cdot(r^{k-1}+\ldots+r+1)\\\
\\\ \Rightarrow&r^{k+1}-r=(r-1)\cdot(r^{k}+r^{k-1}+\ldots+r^{2}+r);\\\ \\\
&r^{k+1}-r=(r-1)\cdot(r^{k}+r^{k-1}+\ldots r^{2}+r+1-1)\\\ \\\
\Rightarrow&r^{k+1}-r=(r-1)\cdot(r^{k}+r^{k-1}+\ldots+r^{2}+r+1)\\\
&+(r-1)\cdot(-1)\\\ \\\
\Rightarrow&r^{k+1}-r=(r-1)\cdot(r^{k}+r^{k-1}+\ldots+r^{2}+r+1)-r+1\\\ \\\
\Rightarrow&r^{k+1}-1=(r-1)\cdot(r^{(k+1)-1}+r^{(k+1)-2}+\ldots+r^{2}+r+1),\end{array}$
which proves that the statement also holds true for $n=k+1$. The induction
process is complete. We have shown that
$r^{n}-1=(r-1)(r^{n-1}+r^{n-2}+\ldots+r+1)$ holds true for every real number
$r$ and every natural $n$. If $r\neq 1$, then $r-1\neq 0$, and so
$\frac{r^{n}-1}{r-1}=r^{n-1}+r^{n-2}+\ldots+r+1$. Multiply both sides by the
first term $a$ we obtain
$\begin{array}[]{rcl}{\displaystyle\frac{a\cdot(r^{n}-1)}{r-1}}&=&ar^{n-1}+ar^{n-2}+\ldots
ar+a\\\ \\\ &=&a_{n}+a_{n-1}+\ldots+a_{2}+a_{1}.\end{array}$
Since by part (i) we know that $a_{i}=a\cdot r^{i-1}$, for $i=1,2,\ldots,n$;
if on the other hand $r=1$, then the geometric progression is obviously the
constant sequence, $a,a,\ldots,a,\ldots\ ;\ \ a_{n}=a$ for every natural
number $n$. In that case $a_{1}+\ldots+a_{n}=\underset{n\ {\rm
times}}{\underbrace{a+\ldots+a}}=na$. The proof is complete. $\square$
Remark 7: We make some observation about the different types of geometric
progressions that might occur according to the different types of values of
the ratio $r$.
1. (i)
If $a=0$, then regardless of the value of the ratio $r$, one obtains the zero
sequence $0,0,0,\ldots,0,\ldots$ .
2. (ii)
If $r=1$, then for any choice of the first term $a$, the geometric progression
is the constant sequence, $a,a,\ldots,a,\ldots$ .
3. (iii)
If the first term $a$ is positive and $r>1$ one obtains a geometric
progression of positive terms, and which is increasing and which eventually
exceed any real number (as we will see in Theorem 8, given a positive real
number $M$, there is a term $a_{n}$ that exceeds M; in the language of
calculus, we say that it approaches positive infinity). For example:
$a=\frac{1}{2}$, and $r=2$; we have the geometric progression
$a_{1}=a=\frac{1}{2},\ a_{2}=\frac{1}{2}\cdot 2=1,a_{3}=\frac{1}{2}\cdot
2^{2}=2;$
The sequence is,
$\frac{1}{2},1,2,2^{2},2^{3},2^{4},\ldots,\underset{a_{n}}{\underbrace{\frac{1}{2}\cdot
2^{n-1}}}$.
4. (iv)
When $a>0$ and $0<r<1$, the geometric progression is decreasing and in the
language of calculus, it approaches zero (it has limit value zero).
For example: $a=4,\ r=\frac{1}{3}$.
We have $a_{1}=a=4,\ a_{2}=4\cdot\frac{1}{3},\
a_{3}=a\cdot\left(\frac{1}{3}\right)^{2},\
a_{4}=4\cdot\left(\frac{1}{3}\right)^{3};$$4,\ \frac{4}{3},\
\frac{4}{9},\ldots,\frac{4}{3^{n-1}}\ n$th term, $\ldots$ .
5. (v)
For $a>0$ and $-1<r<0$, the geometric sequence alternates (which means that if
we pick any term, the succeeding term will have opposite sign). Still, in this
case, such a sequence approaches zero (has limit value zero).
For example: $a=9,\ r=-\frac{1}{2}$.
$a_{1}=a=9,\ a_{2}=9\cdot\left(-\frac{1}{2}\right)=-\frac{9}{2},\
a_{3}=9\cdot\left(\cdot\frac{1}{2}\right)^{2}=\frac{9}{4},\ldots$
$9,\ -\frac{9}{2},\ \frac{9}{2^{2}},\ \frac{-9}{2^{3}},\ldots,\underset{n{\rm
th\
term}}{\underbrace{9\cdot\left(-\frac{1}{2}\right)^{n-1}=\frac{9\cdot(-1)^{n-1}}{2^{n-1}}}}\\\
$
6. (vi)
For $a>0$ and $r=-1$, we have a geometric progression that oscillates:
$a,-a,a,-a,\ldots,a_{n}=(-1)^{n-1},\ldots$ .
7. (vii)
For $a>0$ and $r<-1$, the geometric progression has negative terms only, it is
decreasing, and in the language of calculus we say that approaches negative
infinity.
For example: $a=3,r=-2$
$\begin{array}[]{rcl}a_{1}&=&a=3,\ a_{2}=3\cdot(-2)=-6,\\\ \\\
a_{3}&=&3\cdot(-2)^{2}=12,\ldots 3,\ -6,\ 12,\ldots,\\\ \\\
&&\displaystyle{\underbrace{3\cdot(-2)^{n-1}=3\cdot
2^{n-1}\cdot(-1)^{n-1}}_{n{\rm th\ term}}},\ldots\end{array}$
8. (viii)
What happens when the first term $a$ is negative? A similar analysis holds
(see Exercise 24).
Theorem 6: Let $a=a_{1},a_{2},\ldots,a_{n},\ldots$ be a geometric progression
with ratio $r$.
1. (i)
If $m$ and $n$ are any natural numbers such that $m<n,\ a_{n}=a_{n-m}\cdot
r^{m}$.
2. (ii)
If $m$ and $n$ are any natural numbers such that $m<n$, then $a_{m+1}\cdot
a_{n-m}=a_{1}\cdot a_{n}$.
3. (iii)
For any natural number $n$,
$\left(\overset{n}{\underset{i=1}{\Pi}}a_{i}\right)^{2}=(a_{1}\cdot
a_{2}\ldots a_{n})^{2}=(a_{1}\cdot a_{n})^{n}$, where
$\overset{n}{\underset{i=1}{\Pi}}a_{i}$ denotes the product of the first $n$
terms $a_{1},a_{2},\ldots,a_{n}$.
Proof:
1. (i)
By Theorem 5(i) we have $a_{n}=a\cdot r^{n-1}$ and $a_{n-m}=a\cdot r^{n-m-1}$;
thus $a_{n-m}\cdot r^{m}=a\cdot r^{n-m-1}\cdot r^{m}=a\cdot r^{n-1}=a_{n}$,
and we are done. $\square$
2. (ii)
Again by Theorem 5(i) we have,
$a_{m+1}=a\cdot r^{m},\ a_{n-m}=a\cdot r^{n-m-1},\ {\rm and}\ a_{n}=a\cdot
r^{n-1}$
so that $a_{m+1}\cdot a_{n-m}=a\cdot r^{m}\cdot a\cdot r^{n-m-1}=a^{2}\cdot
r^{n-1}$ and $a_{1}\cdot a_{n}=a\cdot(a\cdot r^{n-1})=a^{2}\cdot r^{n-1}$.
Therefore, $a_{m+1}\cdot a_{n-m}=a_{1}\cdot a_{n}$.
3. (iii)
We could prove this part by using mathematical induction. Instead, an
alternative proof can be offered by making use of the fact that the sum of the
first $n$ natural integers is equal to $\frac{n\cdot(n+1)}{2}$;
$1+2+\ldots+n=\frac{n(n+1)}{2}$; we have already seen this in Example 1(i).
(Go back and review this example if necessary; $1,2,\ldots,n$ are the
consecutive first $n$ terms of the infinite arithmetic progression with first
term $1$ and difference $1$). This fact can be applied neatly here:
$\begin{array}[]{rcl}a_{1}\cdot a_{2}\ldots a_{i}\ldots a_{n}&=&\ {\rm(by\
Theorem\ 1(i))}\\\ \\\ &=&a\cdot(a\cdot r)\ldots(a\cdot r^{i-1})\ldots(a\cdot
r^{n-1})\\\ \\\ &=&\underset{n\ {\rm times}}{(\underbrace{a\cdot a\ldots
a})}\cdot r^{[1+2+\ldots+(i-1)+\ldots+(n-1)]}\end{array}$
The sum $[1+2+\ldots+(i-1)+\ldots+(n-1)]$ is simply the sum of the first
$(n-1)$ natural numbers, if $n\geq 2$. According to Example 1(i) we have,
$1+2+\ldots+(i-1)+\ldots+(n-1)=\frac{(n-1)\cdot[(n-1)+1]}{2}=\frac{(n-1)\cdot
n}{2}.$
Hence, $a_{1}\cdot a_{2}\ldots a_{i}\ldots a_{n}=\underset{n\ {\rm
times}}{(\underbrace{a\cdot a\ldots a})}\cdot
r^{[1+2+\ldots+(i-1)+\ldots+(n-1)]}=a^{n}\cdot
r^{\frac{(n-1)n}{2}}\Rightarrow(a_{1}\cdot a_{2}\ldots a_{i}\ldots
a_{n})^{2}=a^{2n}\cdot r^{(n-1)\cdot n}$. On the other hand, $(a_{1}\cdot
a_{n})^{n}=[a\cdot(a\cdot r^{n-1})]^{n}=[a^{2}\cdot r^{n-1}]^{n}=a^{2n}\cdot
r^{n(n-1)}=(a_{1}\cdot a_{2}\ldots a_{i}\ldots a_{n})^{2}$; we are done.
$\square$
Definition 7: Let $a_{1},a_{2},\ldots,a_{n}$ be positive real numbers. The
positive real number $\sqrt[n]{a_{1}a_{2}\ldots a_{n}}$ is called the
geometric mean of the numbers $a_{1},a_{2},\ldots a_{n}$.
We saw in Theorem 3 that if three real numbers $a,b,c$ are consecutive terms
of an arithmetic progression, the middle term $b$ must be equal to the
arithmetic mean of $a$ and $c$. The same is true for the geometric mean if the
positive reals $a,b,c$ are consecutive terms in a geometric progression. We
have the following theorem.
Theorem 7: If the positive real numbers $a,b,c$, are consecutive terms of a
geometric progression, then the geometric mean of $a$ and $c$ must equal $b$.
Also, any geometric progression containing $a,b,c$ as consecutive terms, must
have the same ratio $r$, namely $r=\frac{b}{a}=\frac{c}{b}$. Moreover, the
condition $b^{2}=ac$ is the necessary and sufficient condition for the three
reals $a,b,c$ to be consecutive terms in a geometric progression.
Proof: If $a,b,c$ are consecutive terms in a geometric progression, then
$b=ar$ and $c=b\cdot r$; and since both $a$ and $b$ are positive and thus
nonzero, we must have $r=\frac{b}{a}={c}{b}\Rightarrow b^{2}=ac\Rightarrow
b=\sqrt{ac}$ which proves that $b$ is the geometric mean of $a$ and $c$.
Conversely, if the condition $b^{2}=ac$ is satisfied (which is equivalent to
$b=\sqrt{ac}$, since $b$ is positive), then since $a$ and $b$ are positive and
thus nonzero, infer that $\frac{b}{a}=\frac{c}{b}$; thus if we set
$r=\frac{b}{a}=\frac{c}{b}$, it is now clear that $a,b,c$ are consecutive
terms of a geometric progression whose ratio is uniquely determined in terms
of the given reals $a,b,c$ and any other geometric progression containing
$a,b,c$ as consecutive terms must have the same ratio $r$. $\square$
For the theorem to follow we will need what is called Bernoulli’s Inequality:
for every real number $a\geq-1$, and every natural number $n$, $(a+1)^{n}\geq
1+na.$
Let $a\geq-1$; Bernoulli’s Inequality can be easily proved by induction:
clearly the statement holds true for $n=1$ since $1+a\geq 1+a$ (the equal sign
holds). Assume the statement to hold true for some $n=k\geq 1:(a+1)^{k}\geq
1+ka$; since $a+1\geq 0$ we can multiply both sides of this inequality by
$a+1$ without affecting its orientation:
$\begin{array}[]{rcl}(a+1)^{k+1}&\geq&(a+1)(1+ka)\Rightarrow\\\ \\\
(a+1)^{k+1}&\geq&a+ka^{2}+1+ka;\\\ \\\ (a+1)^{k+1}&\geq&1+(k+1)a+ka^{2}\geq
1+(k+1)a,\end{array}$
since $ka^{2}\geq 0$ (because $a^{2}\geq 0$ and $k$ is a natural number). The
induction process is complete.
Theorem 8:
1. (i)
If $r>1$ and $M$ is a real number, then there exists a natural number $N$ such
that $r^{n}>M$, for every natural number $n$. For parts (ii), (iii), (iv) and
(v), let $a_{1}=a,a_{2},\ldots,a_{n},\ldots,$ be an infinite geometric
progression with first term $a$ and ratio $r$.
2. (ii)
Suppose $r>1$ and $a>0$. If $M$ is a real number, then there is a natural
number $N$ such that $a_{n}>M$, for every natural number $n\geq N$.
3. (iii)
Suppose $r>1$ and $a<0$. If $M$ is a real number, then there is a natural
number $N$ such that $a_{n}<M$, for every natural number $n\geq N$.
4. (iv)
Suppose $|r|<1$, and $r\neq 0$. If $\epsilon>0$ is a positive real number,
then there is a natural number $N$ such that $|a_{n}|<\epsilon$, for every
natural number $n\geq N$.
5. (v)
Suppose $|r|<1$ and let $S_{n}=a_{1}+a_{2}+\ldots+a_{n}$. If $\epsilon>0$ is a
positive real number, then there exists a natural number $N$ such that
$\left|S_{n}-\frac{a}{1-r}\right|<\epsilon$, for every natural number $n\geq
N$.
Proof:
1. (i)
We can write $r=(r-1)+1$; let $a=r-1$, since $r>1$, $a$ must be a positive
real. According to the Bernoulli Inequality we have, $r^{n}=(a+1)^{n}\geq
1+na$; thus, in order to ensure that $r^{n}>M$, it is sufficient to have
$1+na>M\Leftrightarrow na>M-1\Leftrightarrow n>{\displaystyle\frac{M-1}{a}}$
(the last step is justified since $a>0$). Now, if
$\left[\left[\,{\displaystyle\frac{|M-1|}{a}}\,\right]\right]$ stands for the
integer part of the positive real number ${\displaystyle\frac{|M-1|}{a}}$ we
have by definition, $\left[\left[{\displaystyle\frac{|M-1|}{a}}\
\right]\right]\ \leq\
{\displaystyle\frac{|M-1|}{a}}<\left[\left[\,{\displaystyle\frac{|M-1|}{a}}\,\right]\right]+1$.
Thus, if we choose
$N=\left[\left[{\displaystyle\frac{|M-1|}{a}}\right]\right]+1$, it is clear
that $N>{\displaystyle\frac{|M-1|}{a}}\geq{\displaystyle\frac{M-1}{a}}$ so
that for every natural number $n\geq N$, we will have
$n>{\displaystyle\frac{M-1}{a}}$, and subsequently we will have (since $a>0$),
$na>M-1\Rightarrow na+1>M$. But $(1+a)^{n}\geq 1+na$ (Bernoulli), so that
$r^{n}=(1+a)^{n}\geq 1+na>M$; $r^{n}>M$, for every $n\geq N$. We are done.
$\square$
2. (ii)
By part (i), there exists a natural number $N$ such that
$r^{n}>\frac{M}{a}\cdot r$, for every natural number $n\geq N$ (apply part (i)
with $\frac{M}{a}\cdot r$ replacing $M$). Since both $r$ and $a$ are positive,
so is $\frac{a}{r}$; multiplying both sides of the above inequality by
$\frac{a}{r}$ we obtain $\frac{a}{r}\cdot
r^{n}>\frac{a}{r}\cdot\frac{M}{a}\cdot r\Rightarrow a\cdot r^{n-1}>M$. But
$a\cdot r^{n-1}$ is the $n$th term $a_{n}$ of the geometric progression. Hence
$a_{n}>M$, for every natural number $n\geq N$. $\square$
3. (iii)
Apply part (ii) to the opposite geometric progression:
$-a_{1},-a_{2},\ldots$,$-a_{n},\ldots$ , where $a_{n}$ is the $n$th term of
the original geometric progression (that has $a_{1}=a<0$ and $r>1$, it is also
easy to see that the opposite sequence is itself a geometric progression with
the same ratio $r>1$ and opposite first term $-a$). According to part (ii)
there exists a natural number $N$ such that $-a_{n}>-M$, for every natural
number $n\geq N$. Thus $-(-a_{n})<-(-M)\Rightarrow a_{n}<M$, for every $n\geq
N$. $\square$
4. (iv)
Since $|r|<1$, assuming $r\neq 0$ it follows that $\frac{1}{|r|}>1$. Let
$M=\frac{|a|}{\epsilon\cdot|r|}$. According to part (i), there exists a
natural number $N$ such that
$\left(\frac{1}{|r|}\right)^{n}>M=\frac{|a|}{\epsilon|r|}$ (just apply part
(i) with $r$ replaced by $\frac{1}{|r|}$ and $M$ replaced by
$\frac{|a|}{\epsilon\cdot|r|}$ for every natural number $n\geq N$. Thus
$\frac{1}{|r|^{n}}>\frac{|a|}{\epsilon\cdot|r|}$; multiply both sides by
$|r|^{n}\cdot\epsilon$ to obtain
$\frac{|r|^{n}\cdot\epsilon}{|r|^{n}}>\frac{|a|\cdot|r|^{n}\cdot\epsilon}{\epsilon\cdot|r|}\Rightarrow|a|\cdot|r|^{n-1}<\epsilon$;
but $|a|\cdot|r|^{n-1}=|ar^{-n}|=|a_{n}|$, the absolute value of the $n$th
term of the geometric progression; $|a_{n}|<\epsilon$, for every natural
number $n\geq N$. Finally if $r=0$, then $a_{n}=0$ for $n\geq 2$, and so
$|a_{n}|=0<\epsilon$ for all $n\geq 2$. $\square$
5. (v)
By Theorem 5(ii) we know that,
$S_{n}=a_{1}+a_{2}+\ldots+a_{n}=a+ar+\ldots+ar^{n-1}=\frac{a(r^{n}-1)}{r-1}$
We have
$S_{n}-\frac{a}{1-r}=\frac{a(r^{n}-1)}{r-1}+\frac{a}{r-1}=\frac{ar^{n}-a+a}{r-1}=\frac{ar^{n}}{r-1}$.
Consequently,
$\left|S_{n}-\frac{a}{1-r}\right|=\left|\frac{ar^{n}}{r-1}\right|=|r|^{n}\cdot\left|\frac{a}{r-1}\right|$.
Assume $r\neq 0$. Since $|r|<1$, we can apply the already proven part (iv),
using the positive real number $\frac{\epsilon\cdot|r-1|}{|r|}$ in place of
$\epsilon$: there exists a natural number $N$ such that
$|a_{n}|<\frac{\epsilon\cdot|r-1|}{|r|}$, for every natural number $n\geq N$.
But $a_{n}=a\cdot r^{n-1}$ so that,
$|a|\cdot|r|^{n-1}<\frac{\epsilon\cdot|r-1|}{|r|}\Rightarrow$
$\Rightarrow$ (multiplying both sides by $|r|$)
$|a||r|^{n}<\epsilon\cdot|r-1|\Rightarrow$
$\Rightarrow$ (dividing both sides by $|r-1|$) $\frac{|a|\
|r|^{n}}{|r-1|}<\epsilon$.
And since
$\left|S_{n}-\frac{a}{1-r}\right|=|r|^{n}\cdot\left|\frac{a}{r-1}\right|$ we
conclude that, $\left|S_{n}-\frac{a}{1-r}\right|<\epsilon$. The proof will be
complete by considering the case $r=0$: if $r=0$, then $a_{n}=0$, for all
$n\geq 2$. And thus $S_{n}=\frac{a(r^{n}-1)}{r-1}=\frac{-a}{-1}=a$, for all
natural numbers $n$. Hence,
$\left|S_{n}-\frac{a}{1-r}\right|=\left|a-\frac{a}{1}\right|=|a-a|=0<\epsilon$,
for all natural numbers $n$. $\square$
Remark 6: As the student familiar with, will recognize, part (iv) of Theorem 8
establishes the fact that the limit value of the sequence whose $n$th term is
$a_{n}=a\cdot r^{n-1}$ and under the assumption $|r|<1$, is equal to zero. In
the language of calculus, when $|r|<1$, the geometric progression approaches
zero. Also, part (v), establishes the sequence of (partial) sums whose $n$th
term is $S_{n}$, approaches the real number $\frac{a}{1-r}$, under the
assumption $|r|<1$. In the language of calculus we say that the infinite
series $a+ar+ar^{2}+\ldots+ar^{n-1}+\ldots$ converges to $\frac{a}{1-r}$.
## 5 Mixed Progressions
The reader of this book who has also studied calculus, may have come across
the sum,
$1+2x+3x^{2}+\ldots+(n+1)x^{n}.$
There are $(n+1)$ terms in this sum; the $i$th term is equal to $i\cdot
x^{i-1}$, where $i$ is a natural number between $1$ and $(n+1)$. Note that if
$a_{i}=i\cdot x^{i-1},\ b_{i}=i$, and $c_{i}=x^{i-1}$, we have
$a_{i}=b_{i}\cdot c_{i}$; what is more, $b_{i}$ is the $i$th term of an
arithmetic progression (that has both first term and difference equal to $1$);
and $c_{i}$ is the $i$th term of a geometric progression (with first term
$c=1$ and ratio $r=x$). Thus the term $a_{i}$ is the product of the $i$th term
of an arithmetic progression with the $i$th term of a geometric progression;
then we say that $a_{i}$ is the $i$th term of a mixed progression. We have the
following definition.
Definition 8: Let $b_{1},b_{2},\ldots,b_{n},\ldots$ be an arithmetic
progression; and $c_{1},c_{2},$ $\ldots,c_{n},\ldots$ be a geometric
progression. The sequence $a_{1},a_{2},\ldots,a_{n},\ldots,$ where
$a_{n}=b_{n}\cdot c_{n}$, for every natural number $n$, is called a mixed
progression. (Of course, if both the arithmetic and geometric progressions are
finite sequences with the same number of terms, so it will be with the mixed
progression.)
Back to our example. With a little bit of ingenuity, we can compute this sum;
that is, find a closed form expression for it, in terms of $x$ and $n$.
Indeed, we can write the given sum in the form,
$\begin{array}[]{ll}\underset{(n+1)\ {\rm
terms}}{\left(\underbrace{1+x+x^{2}+\ldots+x^{n-1}+x^{n}}\right)}+\underset{n\
{\rm terms}}{\left(\underbrace{x+x^{2}+\ldots+x^{n-1}+x^{n}}\right)}\\\ \\\
+\underset{(n-1)\ {\rm
terms}}{\left(\underbrace{x^{2}+x^{3}+\ldots+x^{n-1}+x^{n}}\right)}+\ldots+\underset{2\
{\rm terms}}{\left(\underbrace{x^{n-1}+x^{n}}\right)}+\underset{{\rm one\
term}}{\underbrace{x^{n}}}\end{array}.$
In other words we have written the original sum
$1+2x+3x^{2}+\ldots+(n+1)x^{n}$ as a sum of $(n+1)$ sums, each containing one
term less than the previous one.
Now according to Theorem 5(ii),
$1+x+x^{2}+\ldots+x^{n-1}+x^{n}=\frac{x^{n+1}-1}{x-1}\ {\rm\left(\right.}{\rm
assuming}\ x\neq 1\ {\rm\left.\right)},$
since this is the sum of the first $(n+1)$ terms of a geometric progression
with first term $1$ and ratio $x$.
Next, consider
$\begin{array}[]{rcl}x+x^{2}+\ldots+x^{n-1}+x^{n}&=&(1+x+x^{2}+\ldots+n^{n-1}+x^{n})-1\\\
&=&\frac{x^{n+1}-1}{x-1}-\left(\frac{x^{i}-1}{x-1}\right)\end{array}.$
Continuing this way we have,
$\begin{array}[]{rcl}x^{2}+\ldots+x^{n-1}+x^{n}&=&(1+x+x^{2}+\ldots+x^{n}-1+x^{n})-(x+1)\\\
\\\
&=&{\displaystyle\frac{x^{n+1}-1}{x-1}}-\left({\displaystyle\frac{x^{2}-1}{x-1}}\right).\end{array}$
On the $i$th level,
$\begin{array}[]{rcl}x^{i}+\ldots+x^{n-1}+x^{n}&=&(1+x+\ldots+x^{i-1}+x^{i}+\ldots+x^{n-1}+x^{n})-\\\
\\\
-(1+x+\ldots+x^{i-1})&=&{\displaystyle\frac{x^{n+1}-1}{x-1}}-\left({\displaystyle\frac{x^{i}-1}{x-1}}\right).\end{array}$
Let us list all of these sums:
$\begin{array}[]{crcl}(1)&1+x+x^{2}+\ldots+x^{n-1}+x^{n}&=&\frac{x^{n+1}-1}{x-1}\\\
\\\
(2)&x+x^{2}+\ldots+x^{n-1}+x^{n}&=&\frac{x^{n+1}-1}{x-1}-\left(\frac{x-1}{x-1}\right)\\\
\\\
(3)&x^{2}+\ldots+x^{n-1}+x^{n}&=&\frac{x^{n+1}-1}{x-1}-\left(\frac{x^{2}-1}{x-1}\right)\\\
\vdots&&&\\\
(i)&x^{i}+\ldots+x^{n-1}+x^{n}&=&\frac{x^{n+1}-1}{x-1}-\left(\frac{x^{i}-1}{x-1}\right)\\\
\vdots&&&\\\
(n)&x^{n-1}+x^{n}&=&\frac{x^{n+1}-1}{x-1}-\left(\frac{x^{n-1}-1}{x-1}\right)\\\
\\\
(n+1)&x^{n}&=&\frac{x^{n+1}-1}{x-1}-\left(\frac{x^{n}-1}{x-1}\right),\end{array}$
with $x\neq 1$.
If we add the $(n+1)$ equations or identities (they hold true for all reals
except for $x=1$), the sum of the $(n+1)$ left-hand sides is simply the
original sum $1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x$? Thus, if we add up the
$(n+1)$ equations member-wise we obtain,
$\begin{array}[]{rl}&1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}\\\
=&(n+1)\cdot\left(\frac{x^{n+1}-1}{x-1}\right)+\frac{n-(x+x^{2}+\ldots+x^{i}+\ldots+x^{n-1}+x^{n})}{x-1}\\\
=&(n+1)\cdot\left(\frac{x^{n+1}-1}{x-1}\right)+\frac{(n+1)-(1+x+x^{2}+\ldots+x^{n})}{x-1}\\\
\Rightarrow&1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}\\\
=&(n+1)\cdot\left(\frac{x^{n+1}-1}{x-1}\right)+\frac{(n+1)-\left(\frac{x^{n+1}-1}{x-1}\right)}{x-1};\\\
&1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}\\\
=&(n+1)\cdot\left(\frac{x^{n+1}-1}{x-1}\right)+\frac{(n+1)(x-1)-(x^{n+1}-1)}{(x-1)^{2}};\\\
&1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}\\\
=&\frac{(n+1)(x^{n+1}-1)(x-1)+(n+1)(x-1)-(x^{n+1}-1)}{(x-1)^{2}};\\\
&1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}\\\
=&\frac{(n+1)(x-1)\cdot[(x^{n+1}-1)+1]-(x^{n+1}-1)}{(x-1)^{2}};\\\
&1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}\\\ =&\frac{(n+1)(x-1)\cdot
x^{n+1}-(x^{n+1}-1)}{(x-1)^{2}};\\\ &1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}\\\
=&\frac{(n+1)x^{n+2}-(n+1)x^{n+1}-x^{n+1}+1}{(x-1)^{2}};\\\
&1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}\\\
=&\framebox{$\frac{(n+1)x^{n+2}-(n+2)x^{n+1}+1}{(x-1)^{2}}$}\end{array}$
for every natural number $n$.
For $x=1$, the above derived formula is not valid. However, for $x=1$;
$1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}=1+2+3+\ldots+n+(n+1)=\frac{(n+1)(n+2)}{2}$
(the sum of the first $(n+1)$ terms of an arithmetic progression with first
term $a_{1}=1$ and difference $d=1$.
The following theorem gives a formula for the sum of the first $n$ terms of a
mixed progression.
Theorem 9: Let $b_{1},b_{2},\ldots,b_{n},\ldots$ , be an arithmetic
progression with first term $b_{1}$ and difference $d$; and
$c_{1},c_{2},\ldots,c_{n},\ldots$ , be a geometric progression with first term
$c_{1}=c$ and ratio $r\neq 1$. Let $a_{1},a_{2},\ldots,a_{n},\ldots$ , the
corresponding mixed progression, that is the sequence whose $n$th term $a_{n}$
is given by $a_{n}=b_{n}\cdot c_{n}$, for every natural number $n$.
1. (i)
$a_{n}=\left[b_{1}+(n-1)\cdot d\right]\cdot c\cdot r^{n-1}$, for every natural
number $n$.
2. (ii)
For every natural number $n$, $a_{n+1}-r\cdot a_{n}=d\cdot c_{n+1}$.
3. (iii)
If $S_{n}=a_{1}+a_{2}+\ldots+a_{n}$ (sum of the first $n$ terms of the mixed
progression), then
$\begin{array}[]{rcl}S_{n}&=&\frac{a_{n}\cdot
r-a_{1}}{r-1}+\frac{d\cdot\tau\cdot c\cdot(1-r^{n-1})}{(r-1)^{2}};\\\ \\\
S_{n}&=&\frac{a_{n}\cdot r-a_{1}}{r-1}+\frac{d\cdot
r\cdot(c-c_{n})}{(r-1)^{2}}\end{array}$
(recall $c_{n}=c\cdot r^{n-1}$).
Proof:
1. (i)
This is immediate, since by Theorem 1(i), $b_{n}=b_{1}+(n-1)\cdot d$ and by
Theorem 5(i), $c_{n}=c\cdot r^{n-1}$, and so $a_{n}=b_{n}\cdot
c_{n}=[b_{1}+(n-1)d]\cdot c\cdot r^{r-1}$.
2. (ii)
We have $a_{n+1}=b_{n+1}\cdot c_{n+1},\ a_{n}=b_{n}c_{n},\ b_{n+1}=d+b_{n}$.
Thus, $a_{n+1}-r\cdot a_{n}=c_{n+1}\cdot(d+b_{n})-r\cdot b_{n}\cdot
c_{n}=d\cdot c_{n+1}+c_{n+1}b_{n}-rb_{n}c_{n}=d\cdot
c_{n+1}+b_{n}\cdot\underset{0}{(\underbrace{c_{n+1}-rc_{n}})}=dc_{n+1}$, since
$c_{n+1}=rc_{n}$ by virtue of the fact that $c_{n}$ and $c_{n+1}$ are
consecutive terms of a geometric progression with ratio $r$. End of proof.
$\square$
3. (iii)
We proceed by mathematical induction. The statement is true for $n=1$ because
$S_{1}=a_{1}$ and $\frac{a_{1}r-a_{i}}{r-1}+\frac{d\cdot
r\cdot(c-c_{1})}{(r-1)^{2}}=\frac{a_{1}(r-1)}{r-1}+0=a_{1}=S_{1}$. Assume the
statement to hold for $n=k$: (for some natural number $k\geq 1;\
S_{k}=\frac{a_{k}\cdot r-a_{1}}{r-1}+\frac{d\cdot
r\cdot(c-c_{k})}{(r-1)^{2}}$. We have $S_{k+1}=S_{k}+a_{k+1}=\frac{a_{k}\cdot
r-a_{1}}{r-1}+\frac{d\cdot
r\cdot(c-c_{k})}{(r-1)^{2}}+a_{k+1}=\frac{a_{k}\cdot r-a_{1}+a_{k+1}\cdot
r-a_{k+1}}{r-1}+\frac{d\cdot r\cdot(c-c_{k})}{(r-1)^{2}}(1)$. But by part (ii)
we know that $a_{k+1}-ra_{k}=d\cdot c_{k+1}$. Thus, by (1) we now have,
$\begin{array}[]{lrcl}&S_{k+1}&=&{\displaystyle\frac{a_{k+1}\cdot
r-a_{1}}{r-1}-\frac{d\cdot c_{k+1}}{r-1}+\frac{d\cdot
r\cdot(c-c_{k})}{(r-1)^{2}}}\\\
\Rightarrow&S_{k+1}&=&{\displaystyle\frac{a_{k+1}\cdot
r-a_{1}}{r-1}+\frac{-(r-1)\cdot d\cdot c_{k+1}+d\cdot
r\cdot(c-c_{k})}{(r-1)^{2}}};\\\ &S_{k+1}&=&{\displaystyle\frac{a_{k+1}\cdot
r-a_{1}}{r-1}+\frac{d\cdot
r\cdot(c-c_{k+1})+d\cdot\overset{0}{(\overbrace{c_{k+1}-r\cdot
c_{k}})}}{(r-1)^{2}}}.\end{array}$
But $c_{k+1}-r\cdot c_{k}=0$ (since $c_{k+1}=r\cdot c_{k}$) because $c_{k}$
and $c_{k+1}$ consecutive terms of a geometric progression with ratio $r$.
Hence, we obtain $S_{k+1}=\frac{a_{k+1}\cdot r-a_{1}}{r-1}+\frac{d\cdot
r\cdot(c-c_{k+1})}{(r-1)^{2}}$; the induction is complete.
The example with which we opened this section is one of a mixed progression.
We dealt with the sum $1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}$. This is the
sum of the first $(n+1)$ terms of a mixed progression whose $n$th term is
$a_{n}=n\cdot x^{n-1}$; in the notation of Theorem 9, $b_{n}=n,\ d=1,\
c_{n}=x^{n-1}$, and $r=x$ (we assume $x\neq 1$).
According to Theorem 9(iii)
$\begin{array}[]{rcl}S_{n}&=&1+2x+3x^{2}+\ldots+nx^{n-1}=\frac{(nx^{n-1})\cdot
x-1}{x-1}+\frac{x\cdot(1-x^{n-1})}{(x-1)^{2}}\\\ \\\
&=&\frac{nx^{n}-1}{x-1}+\frac{x-x^{n}}{(x-1)^{2}}=\frac{(nx^{n}-1)(x-1)}{(x-1)^{2}}+\frac{x-x^{n}}{(x-1)^{2}}\\\
\\\
&=&\frac{nx^{n+1}-nx^{n}-x+1+x-x^{n}}{(x-1)^{2}}=\frac{nx^{n+1}-(n+1)x^{n}+1}{(x-1)^{2}};\end{array}$
Thus, if we replace $n$ by $(n+1)$ we obtain,
$S_{n+1}=1+2x+3x^{2}+\ldots+nx^{n-1}+(n+1)x^{n}=\frac{(n+1)x^{n+2}-(n+2)x^{n+1}+1}{(x-1)^{2}}$,
and this is the formula we obtained earlier.
Definition 9: Let $a_{1},\ldots,a_{n}$ be nonzero real numbers. The real
number $\frac{n}{\frac{1}{a_{1}}+\ldots+\frac{1}{a_{n}}}$, is called the
harmonic mean of the real numbers $a_{1},\ldots,a_{n}$.
Remark 7: Note that since
$\frac{n}{\frac{1}{a_{1}}+\ldots+\frac{1}{a_{n}}}=\frac{1}{(\frac{1}{a_{1}}+\ldots+\frac{1}{a_{n}})/n}$,
the harmonic mean of the reals $a_{1},\ldots,a_{n}$, is really the reciprocal
of the mean of the reciprocal real numbers
$\frac{1}{a_{1}},\ldots,\frac{1}{a_{n}}$.
We close this section by establishing an interesting, significant and deep
inequality, that has many applications in mathematics and has been used to
prove a number of other theorems. Given $n$ positive real numbers
$a_{1},\ldots,a_{n}$ one can always designate three positive reals to the
given set $\\{a_{1},\ldots,a_{n}\\}$: the arithmetic mean denoted by A.M., the
geometric mean denoted by G.M., and the harmonic mean H.M. The arithmetic-
geometric-harmonic mean inequality asserts that A.M. $\geq$ G.M. $\geq$ H.M.
(To the reader: Do an experiment; pick a set of three positive reals; then a
set of four positive reals; for each set compute the A.M., G.M., and H.M.
values; you will see that the inequality holds; if you are in disbelief do it
again with another sample of positive real numbers.)
The proof we will offer for the arithmetic-geometric-harmonic inequality is
indeed short. To do so, we need a preliminary result: we have already proved
(in the proof of Theorem 5(i)) the identity
$r^{n}-1=(r-1)(r^{n-1}+r^{n-2}+\ldots+r+1)$, which holds true for all real
numbers $r$ and all natural numbers $n$. Moreover, if $r\neq 1$, we have
$\frac{r^{n-1}}{r-1}=r^{n-1}+r^{n-2}+\ldots+r+1$
If we set $r=\frac{b}{a}$, with $b\neq a$, in the above equation and we
multiply both sides by $a^{n}$ we obtain,
$\frac{b^{n}-a^{n}}{b-a}=b^{n-1}+b^{n-2}\cdot a+b^{n-3}\cdot
a^{2}+\ldots+b^{2}\cdot a^{n=-3}+b\cdot a^{n-2}+a^{n-1}$
Now, if $b>a>0$ and in the above equation we replace $b$ by $a$, the resulting
right-hand side will be smaller. In other words, in view of $b>a>0$ we have,
$\begin{array}[]{c}(1)\\\ (2)\\\ (3)\\\ \vdots\\\ (n-2)\\\ (n-1)\\\
(n)\end{array}\left\\{\begin{array}[]{l}b^{n-1}>a^{n-1}\\\ b^{n-2}\cdot
a>a^{n-2}\cdot a^{1}=a^{n-1}\\\ b^{n-3}\cdot a^{2}>a^{n-3}\cdot
a^{2}=a^{n-1}\\\ \vdots\\\ b^{2}\cdot a^{n-3}\cdot a^{2}>a^{2}\cdot
a^{n-3}j=a^{n-1}\\\ b\cdot a^{n-2}>a\cdot a^{n-2}=a^{n-1}\\\
a^{n-1}=a^{n-1}\end{array}\right\\}\begin{array}[]{ll}\Rightarrow&{\rm add\
memberwise}\\\ \\\ &b^{n-1}+b^{n-2}\cdot a+b^{n-3}\cdot a^{2}+\ldots\\\
+&b^{2}a^{n-3}+b\cdot a^{n-2}+a^{n-1}\\\ >&n\cdot a^{n-1}\end{array}$
Hence, the identity above, for $b>a>0$, implies the inequality
$\frac{b^{n}-a^{n}}{b-a}>na^{n-1}$; multiplying both sides by $b-a>0$ we
arrive at
$\begin{array}[]{rl}&b^{n}-a^{n}>(b-a)na^{n-1}\\\ \\\
\Rightarrow&b^{n}>nba^{n-1}-na^{n}+a^{n};\\\ \\\
&b^{n}>nba^{n-1}-(n-1)a^{n}.\end{array}$
Finally, by replacing $n$ by $(n+1)$ in the last inequality we obtain,
$b^{n+1}>(n+1)ba^{n}-na^{n+1}$, for every natural number $n$ and any real
numbers such that $b>a>0$
We are now ready to prove the last theorem of this chapter.
Theorem 10: Let $n$ be a natural number and $a_{1},\ldots,a_{n}$ positive real
numbers. Then,
$\begin{array}[]{rcccl}\underset{{\rm
A.M.}}{\underbrace{\frac{a_{1}\ldots+a_{n}}{n}}}&\geq&\underset{{\rm
G.M.}}{\underbrace{\sqrt[n]{a_{1}\ldots a_{n}}}}&\geq&\underset{{\rm
H.M.}}{\underbrace{\frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}}}}\end{array}$
Proof: Before we proceed with the proof, we mention here that if one equal
sign holds the other must also hold, and that can only happen when all $n$
numbers $a_{1},\ldots,a_{n}$ are equal. We will not prove this here, but the
reader may want to verify this in the cases $n=2$ and $n=3$. We will proceed
by using mathematical induction to first prove that,
$\frac{a_{1}+\ldots+a_{n}}{n}\geq\sqrt[n]{a_{1}\ldots a_{n}}$, for every
natural number $n$ and all positive reals $a_{1},\ldots,a_{n}$. Even though
this trivially holds true for $n=1$, we will use as our starting or base
value, $n=2$. So we first prove that
$\frac{a_{1}+a_{2}}{2}\geq\sqrt{a_{1}a_{2}}$ holds true for any two positive
reals. Since $a_{1}$ and $a_{2}$ are both positive, the square roots
$\sqrt{a_{1}}$ and $\sqrt{a_{2}}$ are both positive real numbers and
$a_{1}=(\sqrt{a_{1}})^{2},\ a_{2}=(\sqrt{a_{2}})^{2}$. Clearly,
$\begin{array}[]{rl}&(\sqrt{a_{1}}-\sqrt{a_{2}})^{2}\geq 0\\\ \\\
\Rightarrow&(\sqrt{a_{1}})^{2}-2(\sqrt{a_{1}})(\sqrt{a_{2}})+(\sqrt{a_{2}})^{2}\geq
0\\\ \\\ \Rightarrow&a_{1}-2\sqrt{a_{1}a_{2}}+a_{2}\geq 0\\\ \\\
\Rightarrow&a_{1}+a_{2}\geq 2\cdot\sqrt{a_{1}a_{2}}\\\ \\\
\Rightarrow&\frac{a_{1}+a_{2}}{2}\geq\sqrt{a_{1}a_{2}},\end{array}$
so the statement holds true for $n=2$.
The Inductive Step: Assume the statement to hold true for some natural number
$n=k\geq 2$; and show that this assumption implies that the statement must
also hold true for $n=k+1$. So assume,
$\begin{array}[]{rlll}&\frac{a_{1}+\ldots+a_{k}}{k}&\geq&\sqrt[k]{a_{1}\ldots
a_{k}}\\\ \\\ \Rightarrow&a_{1}+\ldots+a_{k}&\geq&k\cdot\sqrt[k]{a_{1}\ldots
a_{k}}\end{array}$
Now we apply the inequality we proved earlier:
$b^{k+1}>(k+1)\cdot b\cdot a^{k}-k\cdot a^{k+1};$
If we take $b=\sqrt[k+1]{a_{k+1}}$, where $a_{k+1}$ is a positive real and
$a=\sqrt[k(k+1)]{a_{1}\ldots a_{k}}$ we now have,
$\begin{array}[]{rcl}\left(\sqrt[k+1]{a_{k+1}}\right)^{k+1}&>&(k+1)\cdot\sqrt[k+1]{a_{k+1}}\cdot\left(\sqrt[k(k+1)]{a_{1}\ldots
a_{k}}\right)^{k}-k\cdot\left(\sqrt[k(k+1)]{a_{1}\ldots a_{k}}\right)^{k+1}\\\
\\\
&\Rightarrow&a_{k+1}>(k+1)\cdot\sqrt[k+1]{a_{k+1}}\cdot\sqrt[k+1]{a_{1}\ldots
a_{k}}-k\cdot\sqrt[k]{a_{1}\ldots a_{k}}\\\ \\\
&\Rightarrow&a_{k+1}+k\cdot\sqrt[k]{a_{1}\ldots
a_{k}}>(k+1)\cdot\sqrt[k+1]{a_{1}\ldots a_{k}\cdot a_{k+1}}\end{array}$
But from the inductive step we know that $a_{1}+\ldots+a_{k}\geq
k\cdot\sqrt[k]{a_{1}\ldots a_{k}}$; hence we have,
$\begin{array}[]{rcl}a_{k+1}+(a_{1}+\ldots+a_{k})&\geq&a_{k+1}+k\cdot\sqrt[k]{a_{1}\ldots
a_{k}}\geq(k+1)\cdot\sqrt[k+1]{a_{1}\ldots a_{k}\cdot a_{k+1}}\\\ \\\
&\Rightarrow&a_{1}+\ldots+a_{k}+a_{k+1}\geq(k+1)\sqrt[k+1]{a_{1}\ldots
a_{k}\cdot a_{k+1}},\end{array}$
and the induction is complete.
Now that we have established the arithmetic-geometric mean inequality, we
prove the geometric-harmonic inequality. Indeed, if $n$ is a natural number
and $a_{1},\ldots,a_{n}$ are positive reals, then so are the real numbers
$\frac{1}{a_{1}},\ldots,\frac{1}{a_{n}}$. By applying the already proven
arithmetic-geometric mean inequality we infer that,
$\frac{\frac{1}{a_{1}}+\ldots+\frac{1}{a_{n}}}{n}\geq\sqrt[n]{\frac{1}{a_{1}}\ldots\frac{1}{a_{n}}}$
Multiplying both sides by the product
$\left(\frac{n}{\frac{1}{a_{1}}+\ldots+\frac{1}{a_{n}}}\right)\cdot\sqrt[n]{a_{1}\ldots
a_{n}}$, we arrive at the desired result:
$\sqrt[n]{a_{1}\ldots
a_{n}}\geq\frac{n}{\frac{1}{a_{1}}+\ldots+\frac{1}{a_{n}}}.$
This concludes the proof of the theorem. $\square$
## 6 A collection of 21 problems
1. P1.
Determine the difference of each arithmetic progression whose first term is
$\frac{1}{5}$; and with subsequent terms (but not necessarily consecutive) the
rational numbers $\frac{1}{4},\ \frac{1}{3},\ \frac{1}{2}$.
Solution: Let $k,m,n$ be natural numbers with $k<m<n$ such that
$a_{k}=\frac{1}{4},\ a_{m}=\frac{1}{3},$ and $a_{n}=\frac{1}{2}$. And, of
course, $a_{1}=\frac{1}{5}$ is the first term;
$a_{1}=\frac{1}{5},\ldots,a_{k}=\frac{1}{4},\ldots,a_{m}=\frac{1}{3},\ldots,a_{n}=\frac{1}{2},\ldots$
. By Theorem 1(i) we must have,
$\left.\begin{array}[]{l}\frac{1}{4}=a_{k}=\frac{1}{5}+(k-1)d\\\ \\\
\frac{1}{3}=a_{m}=\frac{1}{5}+(m-1)d\\\ \\\
\frac{1}{2}=a_{n}+\frac{1}{5}+(n-1)d\end{array}\right\\}$; where $d$ is the
difference of the arithmetic progression.
Obviously, $d\neq 0$; the three equations yield,
$\left.\begin{array}[]{l}(k-1)d=\frac{1}{4}-\frac{1}{5}=\frac{1}{20}\\\ \\\
(m-1)d=\frac{1}{3}-\frac{1}{5}=\frac{2}{15}\\\ \\\
(n-1)d=\frac{1}{2}-\frac{1}{5}=\frac{3}{10}\end{array}\right\\}$ (1) Also, it
is clear that $1<k$; (2) so that $1<k<m<n$. (3)
Dividing (1) with (2) member-wise gives
$\frac{k-1}{m-1}=\frac{3}{8},\ \Rightarrow 8(k-1)=3(m-1)$ (4)
Dividing (2) with (3) member-wise implies
$\frac{m-1}{n-1}=\frac{4}{9}\Rightarrow 9(m-1)=4(n-1)$ (5)
Dividing (1) with (3) member-wise produces
$\frac{k-1}{n-1}=\frac{1}{6}\Rightarrow 6(k-1)=n-1$ (6)
According to Equation (4), 3 must be a divisor of $k-1$ and $8$ must be a
divisor of $m-1$; if we put $k-1=3t;\ k=3t+1$, where $t$ is a natural number
(since $k>1$), then (4) implies $8t=m-1\Rightarrow m=8t+1$
Going to equation (5) and substituting for $m-1=8t$, we obtain,
$18t=n-1\Rightarrow n=18t+1.$
Checking equation (6) we see that $6(3t)=18t$, which is true for all
nonnegative integer values of $t$. In conclusion we have the following
formulas for $k,\ m,$ and $n$:
$k=3t+1,\ m=8t+1,\ n=18t+1;\ t\in{\mathbb{N}};\ t=1,2,\ldots$
We can now calculate $d$ in terms of $t$ from any of the equations (1), (2),
or (3):
From (1), $(k-1)d=\frac{1}{20}\Rightarrow 3t\cdot d=\frac{1}{20}\Rightarrow$
$d=\frac{1}{60t}$. We see that this problem has infinitely many solutions:
there are infinitely many (infinite) arithmetic progressions that satisfy the
conditions of the problem. For each positive integer of value of $t$, a new
such arithmetic progression is determined. For example, for $t=1$ we have
$d=\frac{1}{60},\ k=4,\ m=9,\ n=19$. We have the progression,
$a_{1}=\frac{1}{5},\ldots,a_{4}=\frac{1}{4},\ldots\ldots,a_{9}=\frac{1}{3},\ldots\ldots,a_{19}=\frac{1}{2},\ldots$
2. P2.
Determine the arithmetic progressions (by finding the first term $a_{1}$ and
difference $d$) whose first term is $a_{1}=5$, whose difference $d$ is an
integer, and which contains the numbers $57$ and $113$ among their terms.
Solution: We have $a_{1}=5,\ a_{m}=57,\ a_{n}=113$ for some natural numbers
$m$ and $n$ with $1<m<n$. We have $57=5+(m-1)d$ and $113=5+(n-1)d$;
$(m-1)d=52$ and $(n-1)d=108$; the last two conditions say that $d$ is a common
divisor of $52$ and $108$; thus $d=1,2,\ {\rm or}\ 4$ are the only possible
values. A quick computation shows that for $d=1$, we have $m=53$, and $n=109$;
for $d=2$, we have $m=27$ and $n=55$; and for $d=4,\ m=14$ and $n=28$. In
conclusion there are exactly three arithmetic progressions satisfying the
conditions of this exercise; they have first term $a_{1}=5$ and their
differences $d$ are $d=1,2,$ and $4$ respectively.
3. P3.
Find the sum of all three-digit natural numbers $k$ which are such that the
remainder of the divisions of $k$ with $18$ and of $k$ with $30$, is equal to
$7$.
Solution: Any natural number divisible by both $18$ and $30$, must be
divisible by their least common multiple which is $90$. Thus if $k$ is any
natural number satisfying the condition of the exercise, then the number $k-7$
must be divisible by both $18$ and $90$ and therefore $k-7$ must be divisible
by $90$; so that $k-7=90t$, for some nonnegative integer $t$; thus the three-
digit numbers of the form $k=90t+7$ are precisely the numbers we seek to find.
These numbers are terms in an infinite arithmetic progression whose first term
is $a_{1}=7$ and whose difference is $d=70:\ a_{1}=7,\ a_{2}=7+90,\
a_{3}=7+2\cdot(90),\ldots,a_{t+1}=7+90t,\ldots$ .
A quick check shows that the first such three-digit number in the above
arithmetic progression is $a_{3}=7+90(2)=187$ (obtained by setting $t=2$) and
the last such three-digit number in the above progression is
$a_{12}=7+90(11)=997$ (obtained by putting $t=11$ in the formula
$a_{t+1}=7+90t$). Thus, we seek to find the sum,
$a_{3}+a_{4}+\ldots+a_{11}+a_{12}$. We can use either of the two formulas
developed in Example 2 (after example 1 which in turn is located below the
proof of Theorem 2).
Since we know the first and last terms of the sum at hand, namely $a_{3}$, it
is easier to use the first formula in Example 2:
$\begin{array}[]{rcl}a_{m}+a_{m+1}+\ldots+a_{n-1}+a_{n}&=&\frac{(n-m+1)(a_{m}+a_{n})}{2}\end{array}$
In our case $m=3,\ n=12,\ a_{m}=a_{3}=187$, and $a_{n}=a_{12}=997$. Thus
$\begin{array}[]{rcl}a_{3}+a_{4}+\ldots+a_{11}+a_{12}&=&\frac{(12-3+1)\cdot(187+997)}{2}\\\
\\\ &=&\frac{10}{2}\cdot(1184)=5\cdot(1184)=5920.\end{array}$
4. P4.
Let $a_{1},a_{2},\ldots,a_{n},\ldots$, be an arithmetic progression with first
term $a_{1}$ and positive difference $d$; and $M$ a natural number, such that
$a_{1}\leq M$. Show that the number of terms of the arithmetic progression
that do not exceed $M$, is equal to
$\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]+1$, where
$\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]$ stands for the integer
part of the real number $\frac{M-a_{1}}{d}$.
Solution: If, among the terms of the arithmetic progression, $a_{n}$ is the
largest term which does not exceed $M$, then $a_{n}\leq M$ and $a_{\ell}>M$,
for all natural number $\ell$ greater than $n$; $\ell=n+1,n+2,\ldots$ . But
$a_{n}=a_{1}+(n-1)d$; so that $a_{1}+(n-1)d\leq M\Rightarrow(n-1)d\leq
M-a_{1}\Rightarrow n-1\leq\frac{M-a_{1}}{d}$ since $d>0$. Since, by
definition, $\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]$ is the
greatest integer not exceeding $\frac{M-a_{1}}{d}$ and since $n-1$ does not
exceed $\frac{M-a_{1}}{d}$, we conclude that
$n-1\leq\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]\Rightarrow
n\leq\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]+1$. But $n$ is a
natural number, that is, a positive integer, and so must be the integer
$N=\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]+1$ Since $a_{n}$ was
assumed to be the largest term such that $a_{n}\leq M$, it follows that $n$
must equal $N$; because the term $a_{N}$ is actually the largest term not
exceeding $M$ (note that if $n<N$, then $a_{n}<a_{N}$, since the progression
is increasing in view of the fact that $d>0$). Indeed, if
$N=\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]+1$, then by the
definition of the integer part of a real number we must have
$N-1\leq\frac{M-a_{1}}{d}<N$. Multiplying by $d>0$ yields $d(N-1)\leq
M-a_{1}\Rightarrow a_{1}+d(N-1)\leq M\Rightarrow a_{N}\leq M$.
In conclusion we see that the terms $a_{1},\ldots,a_{N}$ are precisely the
terms not exceeding $\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]+1$;
therefore there are exactly
$\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]+1$ terms not exceeding $M$.
5. P5.
Apply the previous problem P4 to find the value of the sum of all natural
numbers $k$ not exceeding $1,000$, and which are such that the remainder of
the division of $k^{2}$ with $17$ is equal to $9$.
Solution: First, we divide those numbers $k$ into two disjoint classes or
groups. If $q$ is the quotient of the division of $k^{2}$ with $17$, and with
remainder $9$, we must have,
$k^{2}=17q+9\Leftrightarrow(k-3)(k+3)=17q,$
but $17$ is a prime number and as such it must divide at least one of the two
factors $k-3$ and $k+3$; but it cannot divide both. Why? Because for any value
of the natural number $k$, it is easy to see that the greatest common divisor
of $k-3$ and $k+3$ is either equal to $1,2$, or $6$. Thus, we must have either
$k-3=17n$ or $k+3=17m$; either $k=17n+3$ or
$\begin{array}[]{rcl}k=17m-3&=&17(m-1)+14\\\ &=&17\cdot\ell+14\end{array}$
(here we have set $m-1=\ell$). The number $n$ is a nonnegative integer and the
number $\ell$ is also a nonnegative integer. So the two disjoint classes of
the natural numbers $k$ are,
$\begin{array}[]{rrcl}&k&=&3,20,37,54,\ldots\\\ \\\ {\rm
and}&k&=&14,31,48,65,\ldots\end{array}$
Next, we find how many numbers $k$ in each class do not exceed $M=10,000$.
Here, we are dealing with two arithmetic progressions: the first being
$3,20,37,54,\ldots,$ having first term $a_{1}=3$ and difference $d=17$. The
second arithmetic progression has first term $b_{1}=14$ and the same
difference $d=17$.
According to the previous practice problem, P4, there are exactly
$N_{1}=\left[\\!\left[\frac{M-a_{1}}{d}\right]\\!\right]+1=\left[\\!\left[\frac{1000-3}{17}\right]\\!\right]+1=\left[\\!\left[\frac{997}{17}\right]\\!\right]+1=58+1=59$
terms of the first arithmetic progression not exceeding $1000$ (also, recall
from Chapter 6 that $\left[\\!\left[\frac{997}{17}\right]\\!\right]$ is really
none other than the quotient of the division of $997$ with $17$).
Again, applying problem P4 to the second arithmetic progression, we see that
there are
$N_{2}=\left[\\!\left[\frac{M-b_{1}}{d}\right]\\!\right]+1=\left[\\!\left[\frac{1000-14}{17}\right]\\!\right]+1=\left[\\!\left[\frac{986}{17}\right]\\!\right]+1=58+1=59$.
Finally, we must find the two sums:
$\begin{array}[]{rcl}S_{N_{1}}&=&a_{1}+\ldots+a_{N_{1}}=\frac{N_{1}\cdot(a_{1}+a_{N_{1}})}{2}=\frac{N_{1}\cdot\left[2a_{1}+(N_{1}-1)d\right]}{2}\\\
\\\ &=&\frac{59\cdot\left[2(3)+(59-1)\cdot
17\right]}{2}=\frac{59\cdot\left[6+(58)(17)\right]}{2}\end{array}$
and
$\begin{array}[]{rcl}S_{N_{2}}&=&b_{1}\ldots+b_{N_{2}}=\frac{N_{2}\cdot\left[2b_{1}+(N_{2}-1)d\right]}{2}\\\
\\\
&=&\frac{59\cdot\left[2(14)+(59-1)17\right]}{2}=\frac{59\cdot\left[28+(58)(17)\right]}{2}\end{array}$
Hence,
$\begin{array}[]{rcl}S_{N_{1}}+S_{N_{2}}&=&\frac{59\cdot\left[6+28+2(58)(17)\right]}{2}\\\
\\\
&=&\frac{59\left[34+1972\right]}{2}=\frac{59\cdot(2006)}{2}=59\cdot(1003)=59,177.\end{array}$
6. P6.
If $S_{n},\ S_{2n},\ S_{3n}$, are the sums of the first $n,\ 2n,\ 3n$ terms of
an arithmetic progression, find the relation or equation between the three
sums.
Solution: We have $S_{n}=\frac{n\cdot\left[a_{1}+(n-1)d\right]}{2}$,
$S_{2n}=\frac{2n\cdot\left[a_{1}+(2n-1)d\right]}{2}$, and
$S_{3n}=\frac{3n\cdot\left[a_{1}+(3n-1)d\right]}{2}$.
We can write
$\begin{array}[]{rcl}S_{2n}&=&\frac{2n\cdot\left[2a_{1}+2(n-1)d+(d-a_{1})\right]}{2}\
{\rm and}\\\ \\\
S_{3n}&=&\frac{3n\cdot\left[3a_{1}+3(n-1)d+(2d-2a_{1})\right]}{2}.\end{array}$
So that,
$S_{2n}=\frac{2n\cdot
2\cdot\left[a_{1}+(n-1)d\right]}{2}+\frac{2n\cdot(d-a_{1})}{2}$ (1)
and
$S_{3n}=\frac{3n\cdot 3\cdot\left[a_{1}+(n-1)d\right]}{2}+\frac{3n\cdot
2\cdot(d-a_{1})}{2}$ (2)
To eliminate the product $n\cdot(d-a_{1})$ in equations (1) and (2) just
consider $3S_{2n}-S_{3n}$: equations (1) and (2) imply,
$\begin{array}[]{rcl}3S_{2n}-S_{3n}&=&\frac{3\cdot 2n\cdot
2\cdot\left[a_{1}+(n-1)d\right]}{2}-\frac{3n\cdot
3\cdot\left[a_{1}+(n-1)d\right]}{2}\\\ \\\
&&+\underset{0}{\underbrace{\frac{3\cdot 2n\cdot(d-a_{1})}{2}-\frac{3n\cdot
2\cdot(d-a_{1})}{2}}}\\\ \\\ \Rightarrow
3S_{2n}-S_{3n}&=&\frac{3n\cdot\left[a_{1}+(n-1)d\right]}{2}\end{array}$
but $S_{n}=\frac{n\cdot\left[a_{1}+(n-1)d\right]}{2}$; hence the last equation
yields
$\begin{array}[]{rl}&3S_{2n}-S_{3n}=3\cdot S_{n}\\\ \\\
\Rightarrow&\framebox{$3S_{2n}=3S_{n}+S_{3n}$};\\\ \\\ {\rm
or}&3(S_{2n}-S_{n})=S_{3n}\end{array}$
7. P7.
If the first term of an arithmetic progression is equal to some real number
$a$, and the sum of the first $m$ terms is equal to zero, show that the sum of
the next $n$ terms must equal to $\frac{a\cdot m(m+n)}{1-m}$; here, we assume
that $m$ and $n$ are natural numbers with $m>1$
Solution: We have
$a_{1}+\ldots+a_{m}=0=\frac{m\cdot\left[2a_{1}+d(m-1)\right]}{2}\Rightarrow$
(since $m>1$) $2a_{1}+d(m-1)=0\Rightarrow
d=\frac{-2a_{1}}{m-1}=\frac{2a_{1}}{1-m}=\frac{2a}{1-m}$. Consider the sum of
the next $n$ terms
$\begin{array}[]{rcl}a_{m+1}+\ldots+a_{m+n}&=&\frac{n\cdot(a_{m+1}+a_{m+n})}{2};\\\
\\\
a_{m+1}+\ldots+a_{m+n}&=&\frac{n\cdot\left[(a_{1}+md)+(a_{1}+(m+n-1)d)\right]}{2};\\\
\\\
a_{m+1}+\ldots+a_{m+n}&=&\frac{n\cdot\left[2a_{1}+(2m+n-1)d\right]}{2}\end{array}$
Now substitute for $d=\frac{2a}{1-m}$: (and of course, $a=a_{1}$)
$\begin{array}[]{rcl}a_{m+1}+\ldots+a_{m+n}&=&\frac{n[2a+(2m+n-1)\cdot\frac{2a}{1-m}]}{2};\\\
\\\ a_{m+1}+\ldots+a_{m+n}&=&\frac{n\cdot 2a[(1-m)+(2m+n-1)]}{2(1-m)};\\\ \\\
a_{m+1}+\ldots+a_{m+n}&=&\frac{2an[1-m+2m+n-1]}{2(1-m)}=\frac{a\cdot
n\cdot(m+n)}{1-m}\end{array}$
8. P8.
Suppose that the sum of the $m$ first terms of an arithmetic progression is
$n$; and thesum of the first $n$ terms is equal to $m$. Furthermore, suppose
that the first term is $\alpha$ and the difference is $\beta$, where $\alpha$
and $\beta$ are given real numbers. Also, assume $m\neq n$ and $\beta\neq 0$.
1. (a)
Find the sum of the first $(m+n)$ in terms of the constants $\alpha$ and
$\beta$ only.
2. (b)
Express the integer $mn$ and the difference $(m-n)$ in terms of $\alpha$ and
$\beta$.
3. (c)
Drop the assumption that $m\neq n$, and suppose that both $\alpha$ and $\beta$
are integers. Describe all such arithmetic progressions.
Solution:
1. (a)
We have $a_{1}+\ldots+a_{m}=n$ and $a_{1}+\ldots+a_{n}=m$;
$\frac{m\cdot[2\alpha+(m-1)\beta]}{2}=n\ \ {\rm and}\ \
\frac{n\cdot[2\alpha+(n-1)\beta]}{2}=m,$
since $a_{1}=\alpha$ and $d=\beta$.
Subtracting the second equation from the first one to obtain,
$\begin{array}[]{rcl}2\alpha\cdot(m-n)&+&\beta\cdot[m(m-1)-n(n-1)]=2n-2m;\\\
2\alpha\cdot(m-n)&+&\beta\cdot[(m^{2}-n^{2})-(m-n)]+2(m-n)=0;\\\
2\alpha\cdot(m-n)&+&\beta\cdot[(m-n)(m+n)-(m-n)]+2(m-n)=0;\\\
2\alpha\cdot(m-n)&+&\beta\cdot(m-n)\cdot[m+n-1]+2(m-n)=0;\end{array}$
$(m-n)\cdot[2\alpha+\beta(m+n-1)+2]=0$; but $m-n\neq 0$, since $m\neq n$ by
the hypothesis of the problem. Thus,
$\begin{array}[]{ll}&2\alpha+\beta\cdot(m+n-1)+2=0\Rightarrow\beta(m+n-1)=-2(1+a)\\\
\\\ \Rightarrow&m+n-1=\frac{-2(1+a)}{\beta}\Rightarrow
m+n=1-\frac{2(1+\alpha)}{\beta}=\frac{\beta-2\alpha-2}{\beta}.\end{array}$
Now, we compute the sum
$a_{1}+\ldots+a_{m+n}=\frac{(m+n)\cdot[2\alpha+(m+n-1)\beta]}{2}$
$\begin{array}[]{rl}\Rightarrow&a_{1}+\ldots+a_{m+n}=\frac{\left(\frac{\beta-2\alpha-2}{\beta}\right)\cdot\left[2\alpha\left(\frac{\beta-2\alpha-2}{\beta}\right)\cdot\beta\right]}{2};\\\
\\\
&a_{1}+\ldots+a_{m+n}=\framebox{$\frac{\left(\beta-2\alpha-2\right)\cdot\left(\beta-2\right)}{2\beta}$}\end{array}$
2. (b)
If we multiply the equations $\frac{m\cdot[2\alpha+(m-1)\beta]}{2}=n$ and
$\frac{n\cdot[2\alpha+(n-1)\beta]}{2}=m$ member-wise we obtain, $\frac{m\cdot
n\cdot[2\alpha+(n-1)\beta][2\alpha+(m-1)\beta]}{4}=mn$ and since $mn\neq 0$,
we arrive at
$\begin{array}[]{rl}&[2\alpha+(n-1)\beta]\cdot[2\alpha+(m-1)\beta]=4\\\ \\\
\Rightarrow&4\alpha^{2}+2\alpha\beta\cdot(m-1+n-1)+(n-1)(m-1)\beta^{2}=4\\\
\\\
\Rightarrow&4\alpha^{2}+2\alpha\beta\cdot(m+n)-4\alpha\beta+nm\beta^{2}-(n+m)\beta^{2}+\beta^{2}=4;\\\
\\\
&(2\alpha-\beta)^{2}+(m+n)\cdot(2\alpha\beta-\beta^{2})+nm\beta^{2}=4.\end{array}$
Now let us substitute for $m+n=\frac{\beta-2\alpha-2}{\beta}$ (from part (a))
in the last equation above; we have,
$\begin{array}[]{rl}&(2\alpha-\beta)^{2}+\left(\frac{\beta-2\alpha-2}{\beta}\right)\cdot\beta\cdot(2\alpha-\beta)+nm\beta^{2}=4\\\
\\\
\Rightarrow&(2\alpha-\beta)^{2}+(\beta-2\alpha-2)(2\alpha-\beta)+nm\beta^{2}=4\\\
\\\
\Rightarrow&4\alpha^{2}-4\alpha\beta+\beta^{2}+2\alpha\beta-\beta^{2}-4\alpha^{2}+4\alpha\beta-4\alpha+2\beta+nm\beta^{2}=4\\\
\\\ \Rightarrow&nm\beta^{2}+2\alpha\beta-4\alpha+2\beta=4\Rightarrow
nm\beta^{2}=4-2\alpha\beta+4\alpha-2\beta\\\ \\\
\Rightarrow&\framebox{$nm=\frac{2\cdot(2-\alpha\beta+2\alpha-\beta)}{\beta^{2}}$}\end{array}$
Finally, from the identity $(m-n)^{2}=(m+n)^{2}-4nm$, it follows that
$\begin{array}[]{rl}&(m-n)^{2}=\left(\frac{\beta-2\alpha-2}{\beta}\right)^{2}-\frac{8(2-\alpha\beta+2\alpha-\beta)}{\beta^{2}}\\\
\\\
\Rightarrow&(m-n)^{2}=\frac{\beta^{2}+4\alpha^{2}+4-4\alpha\beta-4\beta+8\alpha-16+8\alpha\beta-16\alpha+8\beta}{\beta^{2}}\\\
\\\
&(m-n)^{2}=\frac{\beta^{2}+4\alpha^{2}-12+4\alpha\beta+4\beta-8\alpha}{\beta^{2}};\\\
\\\
&|m-n|=\frac{\sqrt{\beta^{2}+4\alpha^{2}-12+4\alpha\beta+4\beta-8\alpha}}{|\beta|}\\\
&=\frac{\sqrt{(2\alpha+\beta)^{2}-12+4\beta-8\alpha}}{|\beta|};\\\ \\\
&\framebox{$m-n=\pm\frac{\sqrt{(2\alpha+\beta)^{2}-12+4\beta-8\alpha}}{|\beta|}$}\end{array}$
the choice of the sign depending on whether $m>n$ or $m<n$ respectively. Also
note, that a necessary condition that must hold here is
$(2\alpha+\beta)^{2}-12+4\beta-8\alpha>0.$
3. (c)
Now consider $\dfrac{m[2\alpha+(m-1)\beta]}{2}=n$ and
$\dfrac{n[2\alpha+(n-1)\beta]}{2}=m$, with $\alpha$ and $\beta$ being
integers. There are four cases.
Case 1: Suppose that $m$ and $n$ are odd. Then we see that $m\mid n$ and
$n\mid m$, which implies $m=n$ (since $m,n$ are positive integers; if they are
divisors of each other, they must be equal). We obtain,
$2\alpha+(n-1)\beta=2\Leftrightarrow
n=\dfrac{\beta+2-2\alpha}{\beta}=1+\dfrac{2(1-\alpha)}{\beta};\ \beta\mid
2(1-\alpha).$
If $\beta$ is odd, it must be a divisor of $1-\alpha$. Put
$1-\alpha=\beta\rho$ and so $n=1+2\rho$, with $\rho$ being a positive integer.
So, the solution is
$m=n=1+2\rho,\ \ \alpha=1-\beta\rho,\ \ \rho\in\mathbb{Z}^{+},\ \
\beta\in\mathbb{Z}$
If $\beta$ is even, set $\beta=2B$. We obtain $1-\alpha-B\rho$, for some odd
integer $\rho\geq 1$. The solution is
$m=n=1+\rho,\ \ \alpha=1-B\rho,\ \ \beta=2B,\ \ \rho\ {\rm an\ odd\ positive\
integer}$.
Case 2: Suppose that $m$ is even, $n$ is odd; put $m=2k$. We obtain
$k\left[2\alpha+(2k-1)\beta\right]=n\ {\rm and}\
n\left[2\alpha+(n-1)\beta\right]=4k.$
Since $n$ is odd, $n$ must be a divisor of $k$ and since $k$ is also a divisor
of $n$, we conclude that since $n$ and $k$ are positive, we must have $n=k$.
So, $2\alpha+(2n-1)\beta=1$ and $2\alpha+(n-1)\beta=4$. From which we obtain
$n\beta=-3\Leftrightarrow(n=1\ {\rm and}\ \beta=-3)$ or $(n=3\ {\rm and}\
\beta=01$).
The solution is
$\begin{array}[]{rl}&n=1,\ \beta=-3,\ m=2,\ \alpha=2\\\ {\rm or}&n=3,\
\beta=-1,\ m=6,\ \alpha=3\end{array}$
Case 3: $m$ odd and $n$ even. This is exactly analogous to the previous case.
One obtains the solutions (just switch $m$ and $n$)
$\begin{array}[]{l}m=1,\ \beta=-3,\ n=2,\ \alpha=2\\\ m=3,\ \beta=-1,\ n=6,\
\alpha=3\end{array}$
Case 4: Assume $m$ and $n$ to be both even. Set
$m=2^{e}_{m_{1}},n=2^{f}_{n_{1}}$, where $e,f$ are positive integers and
$m_{1},n_{1}$ are odd positive integers. Since $n-1$ and $m-1$ are odd, by
inspection we see that $\beta$ must be even. We have,
$\left\\{\begin{array}[]{rl}&2^{e}\cdot
m_{1}\cdot\left[2\alpha+\left(2^{3}_{m_{1}}-1\right)\cdot\beta\right]=2^{f+1}\cdot
n_{1}\\\ \\\ {\rm and}&2^{f}\cdot
n_{1}\cdot\left[2\alpha+\left[2\alpha+\left(2^{f}_{n_{1}}-1\right)\cdot\beta\right]\right]=2^{e+1}\cdot
m_{1}.\end{array}\right.$
We see that the left-hand side of the first equation is divisible by a power
of 2 which is at least $2^{e+1}$; and the left-hand side of the equation is
divisible by at least $2^{f+1}$.
This then implies that $e+1\leq f+1$ and $f+1\leq e+1$. Hence $e=f$.
Consequently,
$\begin{array}[]{rcll}m_{1}\left[2\alpha+\left(2^{e}_{m_{1}}-1\right)\beta\right]&=&2_{n_{1}}&{\rm
and}\\\ \
n_{1}\left[2\alpha+\left(2^{e}_{n_{1}}-1\right)\beta\right]&=&2m_{1}&\end{array}$
Let $\beta=2k$. By cancelling the factor 2 from both sides of the two
equations, we infer that $m_{1}$ is a divisor of $n_{1}$ and $n_{1}$ a divisor
of $m_{1}$. Thus $m_{1}=n_{1}$.
The solution is
$\begin{array}[]{l}\alpha=1-\left(2^{e}\cdot n_{1}-1\right)k\\\ \\\
\beta=2k\\\ \\\ m=2^{e}_{n_{1}}=n\end{array}$ ,
where $k$ is an arbitrary integer, $e$ is a positive integer, and $n_{1}$ can
be any odd positive integer.
9. P9.
Prove that if the real numbers $\alpha,\beta,\gamma,\delta$ are successive
terms of a harmonic progression, then
$3(\beta-\alpha)(\delta-\gamma)=(\gamma-\beta)(\delta-\alpha).$
Solution: Since $\alpha,\beta,\gamma,\delta$ are members of a harmonic
progression they must all be nonzero; $\alpha\beta\gamma\delta\neq 0$. Thus
$3(\beta-\alpha)(\delta-\gamma)=(\gamma-\beta)(\delta-\alpha)$
is equivalent to
$\frac{3(\beta-\alpha)(\delta-\gamma)}{\alpha\beta\gamma\delta}=\frac{(\gamma-\beta)(\delta-\alpha)}{\alpha\beta\gamma\delta}$
$\begin{array}[]{rl}\Leftrightarrow&3\cdot\left(\frac{\beta-\alpha}{\beta\alpha}\right)\cdot\left(\frac{\delta-\gamma}{\delta\gamma}\right)=\left(\frac{\gamma-\beta}{\gamma\beta}\right)\cdot\left(\frac{\delta-\alpha}{\alpha\delta}\right)\\\
\\\
\Leftrightarrow&3\cdot\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)\cdot\left(\frac{1}{\gamma}-\frac{1}{\delta}\right)=\left(\frac{1}{\beta}-\frac{1}{\gamma}\right)\cdot\left(\frac{1}{\alpha}-\frac{1}{\delta}\right)\end{array}$
By definition, since $\alpha,\beta,\gamma,\delta$ are consecutive terms of a
harmonic progression; the numbers
$\frac{1}{\alpha},\frac{1}{\beta},\frac{1}{\gamma},\frac{1}{\delta}$ must be
successive terms of an arithmetic progression with difference $d$; and
$\frac{1}{\alpha}-\frac{1}{\beta}=-d,\ \frac{1}{\gamma}-\frac{1}{\delta}=-d$,
$\frac{1}{\beta}-\frac{1}{\gamma}=-d$, and
$\frac{1}{\alpha}-\frac{1}{\delta}=-3d$ (since
$\frac{1}{\delta}=\frac{1}{\gamma}+d=\frac{1}{\beta}+2d=\frac{1}{\alpha}+3d$).
Thus the above statement we want to prove is equivalent to
$3\cdot(-3)\cdot(-d)=(-d)\cdot(-3d)\Leftrightarrow 3d^{2}=3d^{2}$
which is true.
10. P10.
Suppose that $m$ and $n$ are fixed natural numbers such that the $m$th term
$a_{m}$ in a harmonic progression is equal to $n$; and the $n$th term $a_{n}$
is equal to $m$. We assume $m\neq n$.
1. (a)
Find the $(m+n)$th term $a_{m+n}$ in terms of $m$ and $n$ .
2. (b)
Determine the general $k$th term $a_{k}$ in terms of $k,m$, and $n$.
Solution:
1. (a)
Both $\frac{1}{a_{m}}$ and are the $m$th and $n$th terms respectively of an
arithmetic progression with first term $\frac{1}{a_{1}}$ and difference $d$;
so that $\frac{1}{a_{m}}=\frac{1}{a_{1}}+(m-1)d$ and
$\frac{1}{a_{n}}=\frac{1}{a_{1}}+(n-1)d$. Subtracting the second equation from
the first and using the fact that $a_{m}=n$ and $a_{n}=m$ we obtain,
$\frac{1}{n}-\frac{1}{m}=(m-n)d\Rightarrow\frac{m-n}{nm}=(m-n)d$; but $m-n\neq
0$; cancelling the factor $(m-n)$ from both sides, gives $\frac{1}{mn}=d$.
Thus from the first equation ,
$\frac{1}{n}=\frac{1}{a_{1}}+(m-1)\cdot\frac{1}{mn}\Rightarrow\frac{1}{n}-\frac{(m-1)}{mn}=\frac{1}{a_{1}}\Rightarrow\frac{m-(m-1)}{mn}=\frac{1}{a_{1}};\
\frac{1}{mn}=\frac{1}{a_{1}}\Rightarrow\framebox{$a_{1}=mn$}$. Therefore,
$\frac{1}{a_{m+n}}=\frac{1}{a_{1}}+(m+n-1)d\Rightarrow\frac{1}{a_{m+n}}=\frac{1}{mn}+\frac{m+n-1}{mn}\Rightarrow\framebox{${a}_{m+n}=\frac{mn}{m+n}$}$.
2. (b)
We have
$\frac{1}{a_{k}}=\frac{1}{a_{1}}+(k-1)d\Rightarrow\frac{1}{a_{k}}=\frac{1}{mn}+\frac{(k-1)}{mn}=\frac{k}{mn}\Rightarrow\framebox{$a_{k}=\frac{mn}{k}$}$
.
11. P11.
Use mathematical induction to prove that if $a_{1},a_{2},\ldots,a_{n}$, with
$n\geq 3$, are the first $n$ terms of a harmonic progression, then
$(n-1)a_{1}a_{n}=a_{1}a_{2}+a_{2}a_{3}+\ldots+a_{n-1}a_{n}$.
Solution: For $n=3$ the statement is
$2a_{1}a_{3}=a_{1}a_{2}+a_{2}a_{3}\Leftrightarrow
a_{2}\cdot(a_{1}+a_{3})=2a_{1}a_{3}$; but $a_{1},a_{2},a_{3}$ are all nonzero
since they are the first three terms of a harmonic progression. Thus, the last
equation is equivalent to
$\frac{2}{a_{2}}=\frac{a_{1}+a_{d}}{a_{1}a_{3}}\Leftrightarrow\frac{2}{a_{2}}=\frac{1}{a_{3}}+\frac{1}{a_{1}}$
which is true, because $\frac{1}{a_{1}},\frac{1}{a_{2}},\frac{1}{a_{3}}$ are
the first three terms of a harmonic expression.
The inductive step: prove that whenever the statement holds true for some
natural number $n=k\geq 3$, then it must also hold true for $n=k+1$. So we
assume $(k-1)a_{1}a_{k}=a_{1}a_{2}+a_{2}a_{3}+\ldots+a_{k-1}a_{k}$. Add
$a_{k}a_{k+1}$ to both sides to obtain,
$(k-1)a_{1}a_{k}+a_{k}a_{k+1}=a_{1}a_{2}+a_{2}a_{3}+\ldots+a_{k-1}a_{k}+a_{k}a_{k+1}$
(1)
If we can show that the left-hand side of (1) is equal to $ka_{1}a_{k+1}$, the
induction process will be complete. So we need to show that
$(k-1)a_{1}a_{k}+a_{k}a_{k+1}=k\cdot a_{1}\cdot a_{k+1}$ (2)
(dividing both sides of the equation by $a_{1}\cdot a_{k}\cdot a_{k+1}\neq 0$)
$\Leftrightarrow\frac{(k-1)}{a_{k+1}}+\frac{1}{a_{1}}=\frac{k}{a_{k}}.$ (3)
To prove (3), we can use the fact that $\frac{1}{a_{k+1}}$ and
$\frac{1}{a_{k}}$ are the $(k+1)$th and $k$th terms of an arithmetic
progression with first term $\frac{1}{a_{1}}$ and ratio $d$:
$\frac{1}{a_{k+1}}=\frac{1}{a_{1}}+k\cdot d$ and
$\frac{1}{a_{k}}=\frac{1}{a_{1}}+(k-1)d$; so that,
$\frac{k-1}{a_{k+1}}=\frac{k-1}{a_{1}}+(k-1)kd$ and
$\frac{k}{a_{k}}=\frac{k}{a_{1}}+k(k-1)d$. Subtracting the second equation
from the first yields,
$\frac{k-1}{a_{k+1}}-\frac{k}{a_{k}}=\frac{(k-1)-k}{a_{1}}\Rightarrow\frac{k-1}{a_{k+1}}+\frac{1}{a_{1}}=\frac{k}{a_{k}}$
which establishes (3) and thus equation (2). The induction is complete since
we have show (by combining (1) and (3)).
$k\cdot a_{1}a_{k+1}=a_{1}a_{2}+a_{2}a_{3}+\ldots+a_{k-1}a_{k}+a_{k}a_{k+1},$
the statement also holds for $n=k+1$.
12. P12.
Find the necessary and sufficient condition that three natural numbers $m,n$,
and $k$ must satisfy, in order that the positive real numbers
$\sqrt{m},\sqrt{n},\sqrt{k}$ be consecutive terms of a geometric progression.
Solution: According to Theorem 7, the three positive reals will be consecutive
terms of an arithmetic progression if, and only if,
$(\sqrt{n})^{2}=\sqrt{m}\sqrt{k}\Leftrightarrow n=\sqrt{mk}\Leftrightarrow$
(since both $n$ and $mk$ are positive) $n^{2}=mk$. Thus, the necessary and
sufficient condition is that the product of $m$ and $k$ be equal to the square
of $n$.
13. P13.
Show that if $\alpha,\beta,\gamma$ are successive terms of an arithmetic
progression, $\beta,\gamma,\delta$ are consecutive terms of a geometric
progression, and $\gamma,\delta,\epsilon$ are the successive terms of a
harmonic progression, then either the numbers $\alpha,\gamma,\epsilon$ or the
numbers $\epsilon,\gamma,\alpha$ must be the consecutive terms of a geometric
progression.
Solution: Since $\frac{1}{\gamma},\frac{1}{\delta},\frac{1}{\epsilon}$ are by
definition successive terms of an arithmetic progression and the same holds
true for $\alpha,\beta,\gamma$, Theorem 3 tells us that we must have
$2\beta=\alpha+\gamma$ (1) and
$\frac{2}{\delta}=\frac{1}{\gamma}+\frac{1}{\epsilon}$ (2). And by Theorem 7,
we must also have $\gamma^{2}=\beta\delta$ (3). (Note that $\gamma,\delta$,
and $\epsilon$ must be nonzero and thus so must be $\beta$.)
Equation (2) implies $\delta=\frac{2\gamma\epsilon}{\gamma+\epsilon}$ and
equation (1) implies $\beta=\frac{\alpha+\gamma}{2}$. Substituting for $\beta$
and $\delta$ in equation (3) we now have
$\begin{array}[]{rl}&\gamma^{2}=\left(\frac{\alpha+\gamma}{2}\right)\cdot\left(\frac{2\gamma\epsilon}{\gamma+\epsilon}\right)\\\
\\\
\Rightarrow&\gamma^{2}\cdot(\gamma+\epsilon)=(\alpha+\gamma)\cdot\gamma\epsilon\Rightarrow\gamma^{3}+\gamma^{2}\epsilon=\alpha\gamma\epsilon+\gamma^{2}\epsilon\\\
\\\
\Rightarrow&\gamma^{3}-\alpha\gamma\epsilon=0\Rightarrow\gamma(\gamma^{2}-\alpha\epsilon)=0\end{array}$
and since $\gamma\neq 0$ we conclude
$\gamma^{2}-\alpha\epsilon=0\Rightarrow\gamma^{2}=\alpha\epsilon$, which, in
accordance with Theorem 7, proves that either $\alpha,\gamma,\epsilon$; or
$\epsilon,\gamma,\alpha$ are consecutive terms in a geometric progression.
14. P14.
Prove that if $\alpha$ is the arithmetic mean of the numbers $\beta$ and
$\gamma$; and $\beta$, nonzero, the geometric mean of $\alpha$ and $\gamma$,
then $\gamma$ must be the harmonic mean of $\alpha$ and $\beta$. (Note: the
assumption $\beta\neq 0$, together with the fact that $\beta$ is the geometric
mean of $\alpha$ and $\gamma$, does imply that both $\alpha$ and $\gamma$ must
be nonzero as well.)
Solution: From the problems assumptions we must have $2\alpha=\beta+\gamma$
and $\beta^{2}=\alpha\gamma$; $\beta^{2}=\alpha\gamma\Rightarrow
2\beta^{2}=2\alpha\gamma$; substituting for $2\alpha=\beta+\gamma$ in the last
equation produces
$\begin{array}[]{rl}&2\beta^{2}=(\beta+\gamma)\gamma\Rightarrow
2\beta^{2}=\beta\gamma+\gamma^{2}\\\ \\\
\Rightarrow&2\beta^{2}-\gamma^{2}-\beta\gamma=0\Rightarrow(\beta^{2}-\gamma^{2})+(\beta^{2}-\beta\gamma)=0\\\
\\\
\Rightarrow&(\beta-\gamma)(\beta+\gamma)+\beta\cdot(\beta-\gamma)=0\Rightarrow(\beta-\gamma)\cdot(2\beta+\gamma)=0.\end{array}$
If $\beta-\gamma\neq 0$, then the last equation implies
$2\beta+\gamma=0\Rightarrow\gamma=-2\beta$; and thus from $2a=\beta+\gamma$ we
obtain $2\alpha=\beta-2\beta$; $2\alpha=-\beta$; $\alpha=-\beta/2$. Now
compute, $\frac{2}{\gamma}=\frac{2}{-2\beta}=-\frac{1}{\beta}$, since
$\beta\neq 0$; and
$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{1}{-\frac{\beta}{2}}+\frac{1}{\beta}=-\frac{2}{\beta}+\frac{1}{\beta}=-\frac{1}{\beta}$.
Therefore $\frac{2}{\gamma}=\frac{1}{\alpha}+\frac{1}{\beta}$, which proves
that $\gamma$ is the harmonic mean of $\alpha$ and $\beta$. Finally, by going
back to the equation $(\beta-\gamma)(2\beta+\gamma)=0$ we consider the other
possibility, namely $\beta-\gamma=0$; $\beta=\gamma$ (note that $\beta-\gamma$
and $2\beta+\gamma$ cannot both be zero for this would imply $\beta=0$,
violating the problem’s assumption that $\beta\neq 0$). Since $\beta=\gamma$
and $2\alpha=\beta+\gamma$, we conclude $\alpha=\beta=\gamma$. And then
trivially, $\frac{2}{\gamma}=\frac{1}{\alpha}+\frac{1}{\beta}$, so we are
done.
15. P15.
We partition the set of natural numbers in disjoint classes or groups as
follows: $\\{1\\},\\{2,3\\},\\{4,5,6\\},\\{7,8,9,10\\},\ldots$; the $n$th
class contains $n$ consecutive positive integers starting with
$\frac{n\cdot(n-1)}{2}+1$. Find the sum of the members of the $n$th class.
Solution: First let us make clear why the first member of $n$th class is the
number $\frac{n(n-1)}{2}+1$; observe that the $n$th class is preceded by
$(n-1)$ classes; so since the $k$th class, $1\leq k\leq n-1$, contains exactly
$k$ consecutive integers, then there precisely $(1+2+\ldots+k+\ldots+(n-1))$
consecutive natural numbers preceding the $n$th class; but the sum
$1+2+\ldots+(n-2)+(n-1)$ is the sum of the first $(n-1)$ terms of the infinite
arithmetic progression that has first term $a_{1}=1$ difference $d=1$, hence
$\begin{array}[]{rcl}1+2+\ldots+(n-1)&=&a_{1}+a_{2}+\ldots+a_{n-1}=\frac{(n-1)\cdot(a_{1}+a_{n-1})}{2}\\\
\\\ &=&\frac{(n-1)(1+(n-1))}{2}=\frac{(n-1)\cdot n}{2}.\end{array}$
This explains why the $n$th class starts with the natural number
$\frac{n(n-1)}{2}+1$; the members of the $n$th class are the numbers
$\frac{n(n-1)}{2}+1,\ \frac{n(n-1)}{2}+2,\ldots,\frac{n(n-1)}{2}+n$. These $n$
numbers form a finite arithmetic progression with first term
$\underset{a}{\underbrace{\frac{n(n-1)}{2}+1}}$ and difference $d=1$. Hence
their sum is equal to
$\begin{array}[]{rcl}\frac{n\cdot[2a+(n-1)d]}{2}&=&\frac{n\cdot\left[2\left(\frac{n(n-1)}{2}+1\right)+(n-1)\right]}{2}\\\
\\\
&=&\frac{n\cdot[n(n-1)+2+n-1]}{2}=\frac{n\cdot[n^{2}-n+2+n-1]}{2}=\framebox{$\frac{n\cdot(n^{2}+1)}{2}$}\end{array}$
16. P16.
We divide 8,000 objects into $(n+1)$ groups of which the first $n$ of them
contain $5,8,11,14,\ldots,[5+3\cdot(n-1)]$ objects respectively; and the
$(n+1)$th group contains fewer than $(5+3n)$ objects; find the value of the
natural number $n$ and the number of objects that the $(n+1)$th group
contains.
Solution: The total number of objects that first $n$ groups contain is equal
to, $S_{n}=5+8+11+14+\ldots+[5+3(n-1)]$; this sum, $S_{n}$, is the sum of the
first $n$ terms of the infinite arithmetic progression with first term
$a_{1}=5$ and difference $d=3$; so that its $n$th term is $a_{n}=5+3(n-1)$.
According to Theorem 2,
$S_{n}=\frac{n\cdot[a_{1}+a_{n}]}{2}=\frac{n\cdot[5+5+3(n-1)]}{2}=\frac{n\cdot[5+5+3n-3]}{2}=\frac{n\cdot(7+3n)}{2}$.
Thus, the $(n+1)$th group must contain, $8,000-\frac{n(7+3n)}{2}$ objects. By
assumption, the $(n+1)$th group contains fewer than $(5+3n)$ objects. Also
$8,000-\frac{n(7+3n)}{2}$ must be a nonnegative integer, since it represents
the number of objects in a set (the $(n+1)$th class; theoretically this number
may be zero). So we have two simultaneous inequalities to deal with:
$0\leq 8,000-\frac{n(7+3n)}{2}\Leftrightarrow\frac{n(7+3n)}{2}\leq 8,000;\ \
n(7+3n)\leq 16,000.$
And (the other inequality)
$\begin{array}[]{rcl}8,000-\frac{n(7+3n)}{2}&<&5+3n\Leftrightarrow
16,000-n(7+3n)<10+6n\Leftrightarrow 16,000\\\ &<&3n^{2}+13n+10\Leftrightarrow
16,000<(3n+10)(n+1).\end{array}$
So we have the following system of two simultaneous inequalities
$\left.\begin{array}[]{rc}&n(7+3n)\leq 16,000\\\ \\\ {\rm
and}&16,000<(3n+10)(n+1)\end{array}\right\\}\begin{array}[]{c}(1)\\\ \\\
(2)\end{array}$
Consider (1): At least one of the factors $n$ and $7+3n$ must be less than or
equal to $\sqrt{16,000}$; for if both were greater than $\sqrt{16,000}$ then
their product would exceed $\sqrt{16,000}\cdot\sqrt{16,000}=16,000$,
contradicting inequality (1); and since $n<7+3n$, it is now clear that the
natural number $n$ cannot exceed
$\sqrt{16,000}:n\leq\sqrt{16,000}\Leftrightarrow n\leq\sqrt{16\cdot 10^{3}};\
n\leq 4\cdot\sqrt{10^{2}\cdot 10};\ n\leq 4\cdot 10\cdot\sqrt{10}=40\sqrt{10}$
so $40\sqrt{10}$ is a necessary upper bound for $n$. The closest positive
integer to $40\sqrt{10}$, but less than $40\sqrt{10}$ is the number $126$; but
actually, an upper bound for $n$ must be much less than $126$ in view of the
factor $7+3n$. If we consider (1), we have $3n^{2}+7n-16,000\leq 0$ (3)
The two roots of the quadratic equation $3x^{2}+7x-16,000=0$ are the real
numbers
$r_{1}=\frac{-7+\sqrt{(7)^{2}-4(3)(-16,000)}}{6}=\frac{-7+\sqrt{192,049}}{6}=\approx$${\rm
approximately}\ 71.872326$; and
$r_{2}=\frac{-7-\sqrt{192,049}}{6}\approx-74.20566$.
Now, it is well known from precalculus that if $r_{1}$ and $r_{2}$ are the two
roots of the quadratic polynomial $ax^{2}+bx+c$, then
$ax^{2}+bx+c=a\cdot(x-r_{1})(x-r_{2})$, for all real numbers $x$. In our case
$3x^{2}+7x-16,000=3\cdot(x-r_{1})(x-r_{2})$, where $r_{1}$ and $r_{2}$ are the
above calculated real numbers. Thus, in order for the natural number $n$ to
satisfy the inequality (3), $3n^{2}+7n-16,000\leq 0$; it must satisfy
$3(n-r_{1})(n-r_{2})\leq 0$; but this will only be true if, and only if,
$r_{1}\leq n\leq r_{2}$; $-74.20566\leq n\leq 71.872326$; but $n$ is a natural
number; thus $1\leq n\leq 71$; this upper bound for $n$ is much lower than the
upper bound of the upper bound $126$ that we estimated more crudely earlier.
Now consider inequality (2): it must hold true simultaneously with (1); which
means we have,
$\left.\begin{array}[]{rl}&16,000<(3n+10)\cdot(n+1)\\\ \\\ {\rm and}&1\leq
n\leq 71\end{array}\right\\}$
If we take the highest value possible for $n$; namely $n=71$, we see that
$(3n+10)(n+1)=(3\cdot(71)+10)\cdot(72)=(223)(72)=16,052$ which exceeds the
number $16,000$, as desired. But, if we take the next smaller value, $n=70$,
we have $(3n+10)(n+1)=(220)(71)=15,620$ which falls below $16,000$. Thus, this
problem has a unique solution, $n=71$. The total number of objects in the
first $n$ groups (or 71 groups) is then equal to,
$\frac{n\cdot(7+3n)}{2}=\frac{(7)\cdot(7+3(7))}{2}=\frac{(71)\cdot(220)}{2}=(71)\cdot(110)-7,810.$
Thus, the $(n+1)$th or $72$nd group contains, $8,000-7,810=\framebox{190}$
objects; note that $190$ is indeed less that $5n+3=5(71)+3=358$.
17. P17.
1. (a)
Show that the real numbers $\frac{\sqrt{2}+1}{\sqrt{2}-1},\
\frac{1}{2-\sqrt{2}},\ \frac{1}{2}$, can be three consecutive terms of a
geometric progression. Find the ratio $r$ of any geometric progression that
contains these three numbers as consecutive terms.
2. (b)
Find the value of the infinite sum of the terms of the (infinite) geometric
progression whose first three terms are the numbers
$\frac{\sqrt{2}+1}{\sqrt{2}-1},\ \frac{1}{2-\sqrt{2}},\ \frac{1}{2};\
\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)+\left(\frac{1}{2-\sqrt{2}}\right)+\frac{1}{2}+\ldots$
.
Solution:
1. (a)
Apply Theorem 7: the three numbers will be consecutive terms of a geometric
progression if, and only if,
$\left({\displaystyle\frac{1}{2-\sqrt{2}}}\right)^{2}={\displaystyle\frac{(\sqrt{2}+1)}{(\sqrt{2}-1)}}\cdot\frac{1}{2}$
(1)
Compute the left-hand side:
$\begin{array}[]{rcl}{\displaystyle\frac{1}{(2-\sqrt{2})^{2}}}&=&{\displaystyle\frac{1}{4-4\sqrt{2}+2}}={\displaystyle\frac{1}{6-4\sqrt{2}}}\\\
\\\
&=&{\displaystyle\frac{1}{2(3-2\sqrt{2})}}={\displaystyle\frac{3+2\sqrt{2}}{2\cdot(3-2\sqrt{2})(3+2\sqrt{2})}}\\\
\\\
&=&{\displaystyle\frac{3+2\sqrt{2}}{2\cdot[9-8]}}={\displaystyle\frac{3+2\sqrt{2}}{3}}.\end{array}$
Now we simplify the right-hand side:
$\begin{array}[]{rcl}\left({\displaystyle\frac{\sqrt{2}+1}{\sqrt{2}-1}}\right)\cdot{\displaystyle\frac{1}{2}}&=&{\displaystyle\frac{1}{2}\cdot\frac{(\sqrt{2}+1)^{2}}{(\sqrt{2}-1)(\sqrt{2}+1)}}\\\
\\\
&=&{\displaystyle\frac{1}{2}\cdot\frac{(2+2\sqrt{2}+1)}{(2-1)}=\frac{3+\sqrt{2}}{2}}\end{array}$
so the two sides of (1) are indeed equal; (1) is a true statement. Thus, the
three numbers can be three consecutive terms in a geometric progression. To
find $r$, consider $\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}\right)\cdot
r=\frac{1}{2-\sqrt{2}}$; and also $\left(\frac{1}{2-\sqrt{2}}\right)\cdot
r=\frac{1}{2}$; from either of these two equations we can get the value of
$r$; if we use the second equation we have, $r=\frac{2-\sqrt{2}}{2}$.
2. (b)
Since $|r|=\left|\frac{2-\sqrt{2}}{2}\right|=\frac{2-\sqrt{2}}{2}<1$,
according to Remark 6, the sum $a+ar+ar^{2}+\ldots+ar^{n-1}+\ldots$ converges
to $\frac{a}{1-r}$; in our case $a=\frac{\sqrt{2}+1}{\sqrt{2}-1}$ and
$r=\frac{2-\sqrt{2}}{2}$. Thus the value of the infinite sum is equal to
$\begin{array}[]{rcl}{\displaystyle\frac{a}{1-r}}&=&{\displaystyle\frac{\frac{\sqrt{2}+1}{\sqrt{2}-1}}{1-\left(\frac{2-\sqrt{2}}{2}\right)}}={\displaystyle\frac{\frac{\sqrt{2}+1}{\sqrt{2}-1}}{\frac{2-\left(2-\sqrt{2}\right)}{2}}}\\\
\\\
&=&{\displaystyle\frac{2(\sqrt{2}+1)}{\sqrt{2}(\sqrt{2}-1)}=\frac{2(\sqrt{2}+1)\cdot(\sqrt{2}+1)\cdot\sqrt{2}}{\sqrt{2}\cdot\sqrt{2}\cdot(\sqrt{2}-1)(\sqrt{2}+1)}}\\\
\\\
&=&{\displaystyle\frac{2\sqrt{2}\cdot(\sqrt{2}+1)^{2}}{2\cdot(2-1)}=\sqrt{2}\cdot(2+2\sqrt{2}+1)=\sqrt{2}\cdot(3+2\sqrt{2})}\\\
\\\
&=&3\sqrt{2}+2\cdot\sqrt{2}\cdot\sqrt{2}=3\sqrt{2}+4=\framebox{$4+3\sqrt{2}$}\end{array}$
3. P18.
(For student who had Calculus.) If $|\rho|<1$ and $|\beta\rho|<1$, calculate
the infinite sum,
$S=\underset{1{\rm st}}{\underbrace{\alpha\rho}}+\underset{2{\rm
nd}}{(\underbrace{\alpha+\alpha\beta})}\rho^{2}+\ldots+\underset{n{\rm th\
term}}{(\underbrace{\alpha+\alpha\beta+\ldots+\alpha\beta^{n-1}})}\rho^{n}+\ldots\
.$
Solution: First we calculate the $n$th term which itself is a sum of $n$
terms:
$(\alpha+\alpha\beta+\ldots+\alpha\beta^{n-1})\cdot\rho^{n}=\alpha\cdot\rho^{n}\cdot(1+\beta+\ldots+\beta^{n-1})=\alpha\cdot\rho^{n}\cdot\left(\frac{\beta^{n}-1}{\beta-1}\right)$
by Theorem 5(ii). Now we have,
$\begin{array}[]{rcl}S&=&\alpha\rho+(\alpha+\alpha\beta)\rho^{2}+\ldots+\alpha\cdot\rho^{n}\cdot\left(\frac{\beta^{n}-1}{\beta-1}\right)+\ldots\\\
\\\
S&=&\alpha\rho\left(\frac{\beta-1}{\beta-1}\right)+\alpha\rho^{2}\cdot\left(\frac{\beta^{2}-1}{\beta-1}\right)\\\
\\\
&&+\ldots+\alpha\cdot\rho^{n}\cdot\left(\frac{\beta^{n}-1}{\beta-1}\right)+\ldots\end{array}$
Note that $S={\displaystyle\lim_{n\rightarrow\infty}}S_{n}$, where
$\begin{array}[]{rcl}S_{n}&=&\alpha\rho\cdot\left(\frac{\beta-1}{\beta-1}\right)+\alpha\rho^{n}\cdot\left(\frac{\beta^{2}-1}{\beta-1}\right)+\ldots+\alpha\cdot\rho^{n}\cdot\left(\frac{\beta^{n}-1}{\beta-1}\right);\\\
\\\
S_{n}&=&\left(\frac{\alpha\rho}{\beta-1}\right)\left[(\beta-1)+\rho(\beta^{2}-1)+\ldots+\rho^{n-1}\cdot(\beta^{n}-1)\right]\\\
\\\
S_{n}&=&\left(\frac{\alpha\rho}{\beta-1}\right)\left[\beta\cdot[1+(\rho\beta)+\ldots+(\rho\beta)^{n-1}]-(1+\rho+\ldots+\rho^{n-1})\right]\\\
\\\
S_{n}&=&\left(\frac{\alpha\rho}{\beta-1}\right)\cdot\left[\beta\cdot\frac{[(\rho\beta)^{n}-1]}{\rho\beta-1}-\left(\frac{\rho^{n}-1}{\rho-1}\right)\right]\end{array}$
Now, as $n\rightarrow\infty$, in virtue of $|\rho\beta|<1$ and $|\rho|<1$ we
have,
${\displaystyle\lim_{n\rightarrow\infty}}\frac{[(\rho\beta)^{n}-1]}{\rho\beta-1}=\frac{-1}{\rho\beta-1}=\frac{1}{1-\rho\beta}$
and
${\displaystyle\lim_{n\rightarrow\infty}}\frac{\rho^{n}-1}{\rho-1}=\frac{1}{1-\rho}$.
Hence,
$\begin{array}[]{rcl}S&=&{\displaystyle\lim_{n\rightarrow\infty}}S_{n}=\left(\frac{\alpha\rho}{\beta-1}\right)\cdot\left[\beta\cdot\left(\frac{1}{1-\rho\beta}\right)-\left(\frac{1}{1-\rho}\right)\right];\\\
\\\
S&=&\left(\frac{\alpha\rho}{\beta-1}\right)\cdot\left[\frac{\beta(1-\rho)-(1-\rho\beta)}{(1-\rho\beta)\cdot(1-\rho)}\right]=\frac{\alpha\rho\cdot(\beta-1)}{(\beta-1)\cdot(1-\rho\beta)(1-\rho)};\end{array}$
$\begin{array}[]{c}S=\frac{\alpha\rho}{(1-\rho\beta)\cdot(1-\rho)}\end{array}$
18. P19.
Let $m,n$ and $\ell$ be distinct natural numbers; and
$a_{1},\ldots,a_{k},\ldots$, an infinite arithmetic progression with first
nonzero term $a_{1}$ and difference $d$.
1. (a)
Find the necessary conditions that $n,\ell$, and $m$ must satisfy in order
that,
$\underset{\underset{{\rm first}\ m\ {\rm terms}}{\rm sum\ of\
the}}{\underbrace{a_{1}+a_{2}+\ldots+a_{m}}}=\underset{\underset{{\rm next}\
n\ {\rm terms}}{\rm sum\ of\
the}}{\underbrace{a_{m+1}+\ldots+a_{m+n}}}=\underset{\underset{{\rm next}\
\ell\ {\rm terms}}{\rm sum\ of\ the}}{\underbrace{a_{m+1}+\ldots+a_{m+\ell}}}$
2. (b)
If the three sums in part (a) are equal, what must be the relationship between
$a_{1}$ and $d$?
3. (c)
Give numerical examples.
Solution:
1. (a)
We have two simultaneous equations,
$\left.\begin{array}[]{rl}&a_{1}+a_{2}+\ldots+a_{m}=a_{m+1}+\ldots+a_{m+n}\\\
{\rm and}\\\
&a_{m+1}+\ldots+a_{m+n}=a_{m+1}+\ldots+a_{m+\ell}\end{array}\right\\}$ (1)
According to Theorem 2 we have,
$\begin{array}[]{rrcl}&a_{1}+a_{2}+\ldots+a_{m}&=&\frac{m\cdot[2a_{1}+(m-1)d]}{2};\\\
\\\ &a_{m+1}+\ldots+a_{m+n}&=&\frac{n\cdot[a_{m+1}+a_{m+n}]}{2}\\\ \\\
&&=&\frac{n\cdot[(a_{1}+md)+(a_{1}+(m+n-1)d)]}{2}\\\ \\\
&&=&\frac{n\cdot[2a_{1}+(2m+n-1)d]}{2};\\\ {\rm and}&&&\\\
&a_{m+1}+\ldots+a_{m+\ell}&=&\frac{\ell\cdot[2a_{1}+(2m+\ell-1)d]}{2}\end{array}$
Now let us use the first equation in (1):
$\begin{array}[]{rcl}\frac{m\cdot[2a_{1}+(m-1)d]}{2}&=&\frac{n\cdot[2a_{1}+(2m+n-1)d]}{2};\\\
\\\ 2ma_{1}+m(m-1)d&=&2na_{1}+n\cdot(2m+n-1)d;\\\ \\\
2a_{1}\cdot(m-n)&=&[n\cdot(2m+n-1)-m(m-1)]d;\\\ \\\
2a_{1}\cdot(m-n)&=&[2nm+n^{2}-m^{2}+m-n]d;\end{array}$
According to hypothesis $a_{1}\neq 0$ and $m-n\neq 0$; so the right-hand side
must also be nonzero and,
$d=\frac{2a_{1}\cdot(m-n)}{2nm+n^{2}-m^{2}+m-n}$ (2)
Now use the second equation in (1):
$\begin{array}[]{rrcl}&\frac{n\cdot[2a_{1}+(2m+n-1)d]}{2}&=&\frac{\ell\cdot[2a_{1}+(2m+\ell-1)d]}{2}\\\
\\\ \Leftrightarrow&2na_{1}+n(2m+n-1)d&=&2\ell a_{1}+\ell(2m+\ell-1)d\\\ \\\
\Leftrightarrow&2a_{1}\cdot(n-\ell)&=&[\ell(2m+\ell-1)-n(2m+n-1)]d\\\ \\\
\Leftrightarrow&2a_{1}\cdot(n-\ell)&=&[2m\cdot(\ell-n)+(\ell^{2}-n^{2})-(\ell-n)]d\\\
\\\
\Leftrightarrow&2a_{1}\cdot(n-\ell)&=&[2m\cdot(\ell-n)+(\ell-n)(\ell+n)-(\ell-n)]d\\\
\\\
\Leftrightarrow&2a_{1}\cdot(n-\ell)&=&(\ell-n)\cdot[2m+\ell+n-1]d;\end{array}$
and since $n-\ell\neq$, we obtain $-2a_{1}=(2m+\ell_{n}-1)d$;
$d=\frac{-2a_{1}}{2m+\ell+n-1}$ (3)
(Again, in virtue of $a_{1}\neq 0$, the product $(2m+\ell+n-1)d$ must also be
nonzero, so $2m+\ell+n-1\neq 0$, which is true anyway since, obviously,
$2m+\ell+n$ is a natural number greater than 1).
Combining Equations (2) and (3) and cancelling out the factor $2a_{1}\neq 0$
from both sides we obtain,
$\frac{m-n}{2nm+n^{2}-m^{2}+m-n}=\frac{-1}{2m+\ell+n-1}$
Cross multiplying we now have,
$\begin{array}[]{cl}&(m-n)\cdot(2m+\ell+n-1)\\\ \\\
=&(-1)\cdot(2nm+n^{2}-m^{2}+m-n);\\\ \\\
&2m^{2}+m\ell+mn-m-2mn-n\ell-n^{2}+n\\\ \\\ =&-2mn-n^{2}+m^{2}-m+n;\\\ \\\
&m^{2}+m\ell-n\ell+mn=0.\end{array}$
We can solve for $n$ in terms of $m$ and $\ell$ (or for $\ell$ in terms of $m$
and $n$) we have,
$n\cdot(\ell-m)=m\cdot(m+\ell)\Rightarrow\framebox{$n=\frac{m\cdot(m+\ell)}{\ell-m}$},\
{\rm since}\ \ell-m\neq 0.$
Also, we must have $\ell>m$, in view of the fact that $n$ is a natural number
and hence positive (also note that these two conditions easily imply $n>m$ as
well). But, there is more: The natural number $\ell-m$ must be a divisor of
the product $m\cdot(m+\ell)$. Thus, the conditions are:
1. (A)
$\ell>m$
2. (B)
$(\ell-m)$ is a divisor of $m\cdot(m+\ell)$ and
3. (C)
$n=\frac{m\cdot(m+\ell)}{\ell-m}$
2. (b)
As we have already seen $d$ and $a_{1}$ must satisfy both conditions (2) and
(3). However, under conditions (A), (B), and (C), the two conditions (2) and
(3) are, in fact, equivalent, as we have already seen; so
$d=\frac{-2a_{1}}{2m+\ell+n-1}$ (condition (3)) will suffice.
3. (c)
Note that in condition (C), if we choose $m$ and $\ell$ such $\ell-m$ is
positive and $(\ell-m)$ is a divisor of $m$, then clearly the number
$n=\frac{m\cdot(m+\ell)}{\ell-m}$, will be a natural number. If we set
$\ell-m=t$, then $m+\ell=t+2m$, so that
$n=\frac{m\cdot(t+2m)}{t}=m+\frac{2m^{2}}{t}.$
So if we take $t$ to be a divisor of $m$, this will be sufficient for
$\frac{2m^{2}}{t}$ to be a positive integer. Indeed, set $m=M\cdot t$, then
$n=M\cdot t+\frac{2M^{2}t^{2}}{t}=M\cdot t+2M^{2}\cdot t=t\cdot M\cdot(1+2M)$.
Also, in condition (3) , if we set $a_{1}=a$, then (since $\ell=m+t=Mt+t$)
$\begin{array}[]{rcl}d&=&\frac{-2a}{2M\cdot t+(Mt+t)+Mt+2M^{2}t-1};\\\ \\\
d&=&\frac{-2a}{4Mt+t+2M^{2}t-1}.\end{array}$ (4)
Thus, the formulas $\ell=Mt+t,\ n=Mt+2M^{2}\cdot t$ and (4) will generate, for
each pair of values of the natural numbers $M$ and $t$, an arithmetic
progression that satisfies the conditions of the problem; for any nonzero
value of the first term $a$.
Numerical Example: If we take $t=3$ and $M=4$, we then have $m=M\cdot t=3\cdot
4=12;\ n=t\cdot M\cdot(1+2M)=12\cdot(1+8)=108$, and $\ell=m+t=12+3=15$. And,
$d=\frac{-2a}{2m+\ell+n-1}=\frac{-2a}{24+15+108-1}=\frac{-2a}{146}=\frac{-a}{73}.$
Now let us compute
$\begin{array}[]{rcl}a_{1}+\ldots+a_{m}&=&{\displaystyle\frac{m\cdot[2a+(m-1)d]}{2}=\frac{12\cdot\left[2a+11\cdot\left(\frac{-a}{73}\right)\right]}{2}}\\\
\\\ &=&{\displaystyle\frac{12\cdot[146a-11a]}{2\cdot
73}=\frac{6\cdot(135a)}{73}}={\displaystyle\frac{810a}{73}}.\end{array}$
Next,
$\begin{array}[]{rl}&a_{m+1}+\ldots+a_{m+n^{\prime}}\\\ \\\
=&\frac{n\cdot[2a+(m+n-1)d]}{2}\\\ \\\
=&\frac{108\cdot\left[2a+(24+108-1)\cdot\left(\frac{-a}{73}\right)\right]}{2}\\\
\\\ =&\frac{108}{2}\cdot\frac{[146a-131a]}{73}\\\ \\\
=&\frac{(54)(15a)}{73}=\frac{810a}{73}\end{array}$
and
$\begin{array}[]{cl}&a_{m+1}+\ldots+a_{m+\ell}\\\ \\\
=&{\displaystyle\frac{\ell\cdot[2a+(2m+\ell-1)d]}{2}}\\\ \\\
=&{\displaystyle\frac{15\cdot\left[2a+(24+15-1)\cdot\left(\frac{-a}{73}\right)\right]}{2}}\\\
\\\ =&{\displaystyle\frac{15}{2}\cdot\frac{[146a-38a]}{73}}\\\ \\\
=&{\displaystyle\frac{15}{2}\cdot\frac{(108)a}{73}=\frac{(15)(54a)}{73}}\\\
\\\ =&{\displaystyle\frac{810a}{73}}.\end{array}$
Thus, all three sums are equal to $\frac{810a}{73}$.
19. P20.
If the real numbers $a,b,c$ are consecutive terms of an arithmetic progression
and $a^{2},b^{2},c^{2}$ are consecutive terms of a harmonic progression, what
conditions must the numbers $a,b,c$ satisfy? Describe all such numbers
$a,b,c$.
Solution: By hypothesis, we have
$2b=a+c\ {\rm and}\ \frac{2}{b^{2}}=\frac{1}{a^{2}}+\frac{1}{c^{2}}$
so $a,b,c$ must all be nonzero real numbers. The second equation is equivalent
to $b^{2}=\frac{2a^{2}c^{2}}{a^{2}+c^{2}}$ and $abc\neq 0$; so that,
$b^{2}(a^{2}+c^{2})=2a^{2}c^{2}\Leftrightarrow
b^{2}\cdot[(a+c)^{2}-2ac]=2a^{2}c^{2}$. Now substitute for $a+c=2b$:
$\begin{array}[]{rl}&b^{2}\cdot[(2b)^{2}-2ac]=2a^{2}c^{2}\\\ \\\
\Leftrightarrow&4b^{4}-2acb^{2}-2a^{2}c^{2}=0;\\\ \\\
&2b^{4}-acb^{2}-a^{2}c^{2}=0\end{array}$
At this stage we could apply the quadratic formula since $b^{2}$ is a root to
the equation $2x^{2}-acx-a^{2}c^{2}=0$; but the above equation can actually be
factored. Indeed,
$\begin{array}[]{rcl}b^{4}-acb^{2}+b^{4}-a^{2}c^{2}&=&0;\\\ \\\
b^{2}(b^{2}-ac)+(b^{2})^{2}-(ac)^{2}&=&0;\end{array}$
$\begin{array}[]{rcl}b^{2}\cdot(b^{2}-ac)+(b^{2}-ac)(b^{2}+ac)&=&0;\\\
(b^{2}-ac)\cdot(2b^{2}+ac)&=&0\end{array}$ (1)
According to Equation (1), we must have $b^{2}-ac=0$; or alternatively
$2b^{2}+ac=0$. Consider the first possibility, $b^{2}-ac=0$. Then, by going
back to equation $\frac{2}{b^{2}}=\frac{1}{a^{2}}+\frac{1}{c^{2}}$ we obtain
$\frac{2}{ac}=\frac{1}{a^{2}}+\frac{1}{c^{2}}\Leftrightarrow\frac{2a^{2}c^{2}}{ac}=a^{2}+c^{2}\Leftrightarrow
2ac=a^{2}+c^{2}$; $a^{2}+c^{2}-2ac=0\Leftrightarrow(a-c)^{2}=0$; $a=c$ and
thus $2b=a+c$ implies $b=a=c$.
Next, consider the second possibility in Equation (1):
$2b^{2}+ac=0\Leftrightarrow 2b^{2}=-ac$; which clearly imply that one of $a$
and $c$ must be positive, the other negative. Once more going back to
$\begin{array}[]{rl}&\frac{2}{b^{2}}=\frac{1}{a^{2}}+\frac{1}{c^{2}};\
\frac{4}{2b^{2}}=\frac{1}{a^{2}}+\frac{1}{c^{2}}\\\ \\\
\Leftrightarrow&\frac{4}{-ac}=\frac{c^{2}+a^{2}}{a^{2}c^{2}}\\\ \\\
\Leftrightarrow&-4ac=c^{2}+a^{2}$; $a^{2}+4ac+c^{2}=0\end{array}$ (2)
Let $t=\frac{a}{c};\ a=c\cdot t$ then Equation (2) yields (since $ac\neq 0$),
$t^{2}+4t+1=0$ (3)
Applying the quadratic formula to Equation (3), we now have
$\begin{array}[]{l}t={\displaystyle\frac{-4\pm\sqrt{16-4}}{2}=\frac{-4\pm
2\sqrt{3}}{2}};\\\ \\\ t=-2\pm\sqrt{3};\end{array}$
note that both numbers $-2+\sqrt{3}$ and $-2-\sqrt{3}$ are negative and hence
both acceptable as solutions, since we know that $a$ and $c$ have opposite
sign, which means that $t=\frac{a}{c}$ must be negative. So we must have
either $a=(-2+\sqrt{3})c$; or alternatively $a=-(2+\sqrt{3})\cdot c$. Now, we
find $b$ in terms of $c$. From $2b^{2}=-ac$; $b^{2}=-\frac{ac}{2}$; note that
the last equation says that either the numbers $-\frac{a}{2},b,c$ are the
successive terms of a geometric progression; or the numbers $-a,b,\frac{c}{2}$
(or any of the other two possible permutations: $a,b,-\frac{c}{2}$,
$\frac{a}{2},b,-c$; and four more that are obtained by switching $a$ with
$c$). So, if $a=(-2+\sqrt{3})c$, then from $2b=a+c;\
b=\frac{a+c}{2}=\frac{(-2+\sqrt{3})c+c}{2}=\frac{(\sqrt{3}-1)c}{2}$. And if
$a=-(2+\sqrt{3})c,\
b=\frac{a+c}{2}=\frac{-(2+\sqrt{3})c+c}{2}=\frac{-(1+\sqrt{3})c}{2}$. So, in
conclusion we summarize as follows:
Any three real numbers $a,b,c$ such that $a,b,c$ are consecutive terms of an
arithmetic progression and $a^{2},b^{2},c^{2}$ the successive terms of a
harmonic progression must fall in exactly one of three classes:
1. (1)
$a=b=c;\ c$ can be any nonzero real number
2. (2)
$a=(-2+\sqrt{3})\cdot c,\ b=\frac{(\sqrt{3}-1)c}{2};\ c$ can be any positive
real;
3. (3)
$a=(2+\sqrt{3})c,\ b=\frac{-(1+\sqrt{3})}{2}c;\ c$ can be any positive real.
20. P21.
Prove that if the positive real numbers $\alpha,\beta,\gamma$ are consecutive
members of a geometric progression, then $\alpha^{k}+\gamma^{k}\geq
2\beta^{k}$, for every natural number $k$.
Solution: Given any natural number $k$, we can apply the arithmetic-geometric
mean inequality of Theorem 10, with $n=2$, and $a_{1}=\alpha^{k},\
a_{2}=\gamma^{k}$, in the notation of that theorem:
$\frac{\alpha^{k}+\gamma^{k}}{2}\geq\sqrt{\alpha^{k}\cdot\gamma^{k}}=\sqrt{(\alpha\gamma)^{k}}.$
But since $\alpha,\beta,\gamma$ are consecutive terms of a geometric
progression, we must also have $\beta^{2}=\alpha\gamma$. Thus the above
inequality implies,
$\begin{array}[]{rccl}&\frac{\alpha^{k}+\gamma^{k}}{2}&\geq&\sqrt{(\beta^{2})^{k}};\\\
&\frac{\alpha^{k}+\gamma^{k}}{2}&\geq&\sqrt{(\beta^{k})^{2}}\\\
\Rightarrow&\frac{\alpha^{k}+\gamma^{k}}{2}&\geq&\beta^{k}\\\
\Rightarrow&\alpha^{k}+\gamma^{k}&\geq&2\beta^{k},\end{array}$
and the proof is complete.
## 7 Unsolved problems
1. 1.
Show that if the sequence $a_{1},a_{2},\ldots,a_{n},\ldots$ , is an arithmetic
progression, so is the sequence $c\cdot a_{1},c\cdot a_{2},\ldots,c\cdot
a_{n},\ldots$ , where $c$ is a constant.
2. 2.
Determine the difference of each arithmetic progression which has first term
$a_{1}=6$ and contains the numbers $62$ and $104$ as its terms.
3. 3.
Show that the irrational numbers $\sqrt{2},\ \sqrt{3},\ \sqrt{5}$ cannot be
terms of an arithmetic progression.
4. 4.
If $a_{1},a_{2},\ldots,a_{n},\ldots$ is an arithmetic progression and
$a_{k}=\alpha,\ a_{m}=\beta,\ a_{\ell}=\gamma$, show that the natural numbers
$k,m,\ell$ and the real numbers $\alpha,\beta,\gamma$, must satisfy the
condition
$\alpha\cdot(m-\ell)+\beta\cdot(\ell-k)+\gamma\cdot(k-m)=0.$
Hint: Use the usual formula $a_{n}=a_{1}+(n-1)d$, for $n=k,m,\ell$, to obtain
three equations; subtract the first two and then the last two (or the first
and the third) to eliminate $a_{1}$; then eliminate the difference $d$ (or
solve for $d$ in each of the resulting equations).
5. 5.
If the numbers $\alpha,\beta,\gamma$ are successive terms of an arithmetic
progression, then the same holds true for the numbers
$\alpha^{2}\cdot(\beta+\gamma),\ \beta^{2}\cdot(\gamma+\alpha),\
\gamma^{2}\cdot(\alpha+\beta)$.
6. 6.
If $S_{k}$ denotes the sum of the first $k$ terms of the arithmetic
progression with first term $k$ and difference $d=2k-1$, find the sum
$S_{1}+S_{2}+\ldots+S_{k}$.
7. 7.
We divide the odd natural numbers into groups or classes as follows:
$\\{1\\},\\{3,5\\},\\{7,9,11\\},\ldots$ ; the $n$th group contains $n$ odd
numbers starting with $(n\cdot(n-1)+1)$ (verify this). Find the sum of the
members of the $n$th group.
8. 8.
We divide the even natural numbers into groups as follows:
$\\{2\\},\\{4,6\\},$ $\\{8,10,12\\},\ldots$ ; the $n$th group contains $n$
even numbers starting with $(n(n-1)+2)$. Find the ’sum of the members of the
$n$th group.
9. 9.
Let $n_{1},n_{2},\ldots,n_{k}$ be $k$ natural numbers such that
$n_{1}<n_{2}<\ldots<n_{k}$; if the real numbers,
$a_{n_{1}},a_{n_{2}},\ldots,a_{n_{k}}$, are members of an arithmetic
progression (so that the number $a_{n_{i}}$ is precisely the $n_{i}$th term in
the progression; $i=1,2,\ldots,k)$, show that the real numbers:
$\frac{a_{n_{k}}-a_{n_{1}}}{a_{n_{2}}-a_{n_{1}}},\
\frac{a_{n_{k}}-a_{n_{2}}}{a_{n_{2}}-a_{n_{1}}}\
,\ldots,\frac{a_{n_{k}}-a_{n_{k-1}}}{a_{n_{2}}-a_{n_{1}}},$
are all rational numbers.
10. 10.
Let $m$ and $n$ be natural numbers. If in an arithmetic progression
$a_{1},a_{2},\ldots,a_{k},\ldots$; the term $a_{m}$ is equal to $\frac{1}{n}$;
$a_{m}=\frac{1}{n}$, and the term $a_{n}$ is equal to $\frac{1}{m};\
a_{n}=\frac{1}{m}$, prove the following three statements.
1. (a)
The first term $a_{1}$ is equal to the difference $d$.
2. (b)
If $t$ is any natural number, then $a_{t\cdot(mn)}=t$; in other words, the
terms $a_{mn},a_{2mn},a_{3mn},\ldots$ , are respectively equal to the natural
numbers $1,2,3,\ldots$ .
3. (c)
If $S_{t\cdot(mn)}$ ($t$ a natural number) denote the sum of the first
$(t\cdot m\cdot n)$ terms of the arithmetic progression, then
$S_{t\cdot(mn)}=\frac{1}{2}\cdot(mn+1)\cdot t$. In other words,
$S_{mn}=\frac{1}{2}(mn+1)$, $S_{2mn}=\frac{1}{2}\cdot(mn+1)\cdot 2,\
S_{3mn}=\frac{1}{2}\cdot(mn+1)\cdot 3,\ldots$ .
11. 11.
If the distinct real numbers $a,b,c$ are consecutive terms of a harmonic
progression show that
1. (a)
$\frac{2}{b}=\frac{1}{b-a}+\frac{1}{b-c}$ and
2. (b)
$\frac{b+a}{b-a}+\frac{b+c}{b-c}=2$
12. 12.
If the distinct reals $\alpha,\beta,\gamma$ are consecutive terms of a
harmonic progressionthen the same is true for the numbers
$\alpha,\alpha-\gamma,\alpha-\beta$.
13. 13.
Let $a=a_{1},a_{2},a_{3},\ldots,a_{n},\ldots$ , be a geometric progression and
$k,\ell,m$natural numbers. If $a_{k}=\beta,\ a_{\ell}=\gamma,\ a_{m}=\delta$,
show that $\beta^{\ell-m}\cdot\gamma^{m-k}\cdot\delta^{k-\ell}=1$.
14. 14.
Suppose that $n$ and $k$ are natural numbers such that $n>k+1$; and
$a_{1}=a,a_{2},\ldots,a_{t},\ldots$ a geometric progression, with positive
ratio $r\neq 1$, and positivefirst term $a$. If $A$ is the value of the sum of
the first $k$ terms of the progression and $B$ is the value of the last $k$
terms among the $n$ first terms, express the ratio $r$ in terms of $A$ and $B$
only; and also the first term $a$ in terms of $A$ and $B$.
15. 15.
Find the sum
$\left(a-\frac{1}{a}\right)^{2}+\left(a^{2}-\frac{1}{a^{2}}\right)^{2}+\ldots\left(a^{n}-\frac{a}{a^{n}}\right)^{2}$.
16. 16.
Find the infinite sum
$\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots\right)+\left(\frac{1}{5}+\frac{1}{5^{2}}+\frac{1}{5^{3}}+\ldots\right)$$+\left(\frac{1}{9}+\frac{1}{9^{2}}+\frac{1}{9^{3}}+\ldots\right)+\ldots+\underset{k{\rm
th\
sum}}{\left(\underbrace{\frac{1}{(2k+1)}+\frac{1}{(2k+1)^{2}}+\frac{1}{(2k+1)^{3}}+\ldots}\right)}+\ldots$
.
17. 17.
Find the infinite sum
$\frac{2}{7}+\frac{4}{7^{2}}+\frac{2}{7^{3}}+\frac{4}{7^{4}}+\frac{2}{7^{5}}+\frac{4}{7^{6}}+\ldots$
.
18. 18.
If the numbers $\alpha,\beta,\gamma$ are consecutive terms of an arithmetic
progression and the nonzero numbers $\beta,\gamma,\delta$ are consecutive
terms of a harmonic progression, show that
$\frac{\alpha}{\beta}=\frac{\gamma}{\delta}$.
19. 19.
Suppose that the positive reals $\alpha,\beta,\gamma$ are successive terms of
an arithmetic progression and let $x$ be the geometric mean of $\alpha$ and
$\beta$; and let $y$ be the geometric mean of $\beta$ and $\gamma$. Prove that
$x^{2},\beta^{2},y^{2}$ are successive terms of an arithmetic progression.
Give two numerical examples.
20. 20.
Show that if the nonzero real numbers $a,b,c$ are consecutive terms of a
harmonic progression, then the numbers $a-\frac{b}{2},\ \frac{b}{2},\
c-\frac{b}{2}$, must be consecutive terms of a geometric progression. Give two
numerical examples.
21. 21.
Compute the following sums:
1. (i)
$\frac{1}{2}+\frac{2}{2^{2}}+\ldots+\frac{n}{2^{n}}$
2. (ii)
$1+\frac{3}{2}+\frac{5}{4}+\ldots+\frac{2n-1}{2^{n-1}}$
22. 22.
Suppose that the sequence $a_{1},a_{2},\ldots,a_{n},\ldots$ satisfies
$a_{n+1}=(a_{n}+\lambda)\cdot\omega$, where $\lambda$ and $\omega$ are fixed
real numbers with $\omega\neq 1$.
1. (i)
Use mathematical induction to prove that for every natural number,
$a_{n}=a_{1}\cdot\omega^{n-1}+\lambda\cdot\left(\frac{\omega^{n}-\omega}{\omega-1}\right)$.
2. (ii)
Use your answer in part (i) to show that,
$\begin{array}[]{rcl}S_{n}&=&a_{1}+a_{2}+\ldots+a_{n}\\\
&=&a_{1}\cdot{\left(\displaystyle\frac{\omega^{n}-1}{\omega-1}\right)+\lambda\cdot\left(\frac{\omega^{n+1}-n\cdot\omega^{2}+(n-1)\omega}{(\omega-1)^{2}}\right)}.\end{array}$
($*$) Such a sequence is called a semi-mixed progression.
23. 23.
Prove part (ii) of Theorem 4.
24. 24.
Work out part (viii) of Remark 5.
25. 25.
Prove the analogue of Theorem 4 for geometric progressions: if the $(n-m+1)$
positive real numbers $a_{m},a_{m+1},\ldots,a_{n-1},a_{n}$ are successive
terms of a geometric progression, then
1. (i)
If the natural number $(n-m+1)$ is odd, then the geometric mean of the
$(n-m+1)$ terms is simply the middle number $a_{(\frac{m+n}{2})}$.
2. (ii)
If the natural number $(n-m+1)$ is even, then the geometric mean of the
$(n-m+1)$ terms must be the geometric mean of the two middle terms
$a_{(\frac{n+m-1}{2})}$ and $a_{(\frac{n+m+1}{2})}$.
## References
* [1] Robert Blitzer, Precalculus, Third Edition, Pearson Prentiss Hall, 2007, 1053 pp. See pages 936-958.
* [2] Michael Sullivan, Precalculus, Eighth Edition, Pearson Prentiss Hall, 2008, 894 pp. See pages 791-801.
|
arxiv-papers
| 2009-04-24T12:10:36 |
2024-09-04T02:49:02.126031
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Konstantine Zelator",
"submitter": "Konstantine Zelator",
"url": "https://arxiv.org/abs/0904.3855"
}
|
0904.3857
|
# Microscopic origin of magnetism and magnetic interactions in ferropnictides
M.D. Johannes, I.I. Mazin Code 6393, Naval Research Laboratory, Washington,
D.C. 20375
(Printed on )
###### Abstract
One year after their initial discovery, two schools of thought have
crystallized regarding the electronic structure and magnetic properties of
ferropnictide systems. One postulates that these are itinerant weakly
correlated metallic systems that become magnetic by virtue of spin-Peierls
type transition due to near-nesting between the hole and the electron Fermi
surface pockets. The other argues these materials are strongly or at least
moderately correlated, the electrons are considerably localized and close to a
Mott-Hubbard transition, with the local magnetic moments interacting via
short-range superexchange. In this paper we argue that neither picture is
fully correct. The systems are moderately correlated, but with correlations
driven by Hund’s rule coupling rather than by the on-site Hubbard repulsion.
The iron moments are largely local, driven by Hund’s intra-atomic exchange.
Superexchange is not operative and the interactions between the Fe moments are
considerably long-range and driven mostly by one-electron energies of all
occupied states.
###### pacs:
Pacs Numbers:
Ferropnictides are still attracting widespread attention from researchers both
inside and outside the field of superconductivity. There is now a nearly
universal agreement that magnetism and, specifically, proximity to an
antiferromagnetic “stripe-order” transition plays a major role in the physics
of these compounds. There is also growing evidence that the magnetic
properties and correlation effects in this system are not controlled by the
Hubbard $U$ as in cuprates (spectroscopy tells us that Hubbard correlations
are weak and the effective $U$ is on the order of 1 eVPES , smaller than the
bandwidth; first principles calculations of $U$ support thisU ). On the other
hand, the multiband character of Fe bands and the large intra-atomic (Hund’s)
exchange coupling in Fe suggest that the Hund’s $J$ may play the main role in
the magnetism..
As opposed to the Hubbard $U,$ the Hund’s $J$ is generally well accounted for
in density-functional calculations (where it is called the Stoner $I$).)
Indeed, the local density approximation (LDA) correctly predicts the
particular antiferromagnetic and structural ground state of undoped
ferropnictides, in striking contrast to the cuprates. In view of this, it is
instrumental to trace down the origin of magnetism $within$ LDA, and to
disentangle the nature of magnetic interactions captured by this approach. It
is highly likely that the physics uncovered by density functional theory will
reflect the actual physics of these systems. Given the heated (but largely
devoid of solid facts) discussion of whether antiferromagnetism in pnictides
is due to Fermi surface nesting or to second neighbor superexchange (See Ref.
MS for a review), a clear understanding of, at least, the message that LDA
calculations send seems highly necessary.
In this paper, we analyze the magnetic interactions and demonstrate that
neither of the above two views (often presented as an axiomatic dilemma) is
correct. The magnetism appears due to $local$ Hund’s rule coupling, while the
particular ground state is selected by itinerant, essentially one-electron
interactions, of which only a small part is played by the Fermi surface
nesting. We will also explain why conventional “Anderson-Kanamori”
superexchange is not operative here, and will show some striking examples
where calculations and experiment contradict both local superexchange and
spin-Peierls pictures, yet are perfectly understandable on the basis of one-
electron energy balance.
We start with a qualitative analysis. The Hund’s rule coupling energy in
density functional theory is expressed as $E_{H}=-Im^{2}/4,$ where $m$ is the
magnetization of an Fe ion, and $I\approx 0.8$ eV is the Stoner factor for Fe
(0.9 eV in GGA). Depending on the material, the self-consistent magnetic
moment on Fe appears to be between 1.5-2 $\mu_{B}$ in LDA and 1.8-2.5 in GGA.
The corresponding energy gain even in LDA is 0.5 eV, which is remarkably
large. In other words, every individual Fe wants to be strongly magnetic and
the advantage of spin polarization should lead to a magnetic ground state at
the mean field level, unless an unusually large kinetic energy penalty exists.
However, this is exactly the case for the formation of a ferromagnetic
configuration. To create a magnetization $m$ on Fe, one needs to move
approximately 1.15 (to account for the relative share of Fe-d orbitals at the
Fermi level) spin-minority electrons into unoccupied spin-majority states,
incurring an energy loss of $\approx(1.15m)^{2}/N_{\uparrow}(E_{F}).$ The
density of states (DOS) per Fe, $N_{\uparrow}(E_{F}),$ varies between 1 and
1.5 eV${}^{-1},$ depending on the system, creating an energy loss for $m=1.5$
$\mu_{B}$ of 0.5-0.8 eV. This cost is about as large as the Hund’s rule energy
gain estimated above. This shows that the system is on the verge of a
ferromagnetic instability, but nothing more.
In low-DOS metals, magnetization without a large cost in kinetic energy is
possible if some type of antiferromagnetic arrangement is formed (cf. metal Cr
and Mn). For a broad band metal, this narrows the conductivity band, but as
long as the exchange splitting is smaller than the bandwidth, the cost is
small. Because in ferropnictides the calculated bandwidth is 5-6 eV and the
exchange splitting $mI$ is at most 2 eV, this mechanism should be very
favorable.
It is interesting to consider how the system determines which particular AFM
arrangement is most profitable from the point of view of the one-electron
energy (note that LDA calculations can be forced to converge to nearly any AFM
pattern, but not to a ferromagnetic state). If the resulting magnetization is
small, the answer is obvious: the second derivative of the total energy with
respect to magnetization is defined by the noninteracting susceptibility at
the AF wave vector Q, $\partial^{2}E/\partial
m^{2}=-\chi_{0}^{-1}(\mathbf{Q)}$ (with the small caveat that an actual spin
density wave is not a single harmonic, but includes all wave vectors Q+G,
where G is a reciprocal lattice vector). The imaginary part of $\chi_{0}$ is
directly related to Fermi surface nesting, being defined, in the constant
matrix elements approximation, as
$\sum_{ij}\int\delta(\varepsilon_{\mathbf{k}i})\delta(\varepsilon_{\mathbf{k+Q,}i})d\mathbf{k,}$
while the (actually relevant) real part collects information from all states
and may or may not have any relation to the nesting conditions (for a detailed
discussion see Ref. CDW ).
Geometrical nesting, as a property of the Fermi surface, becomes even more
disconnected from a real instability in the strongly nonlinear regime,
$m\gtrsim 1$ $\mu_{B}$, which is the case for ferropnictides. Monitoring the
evolution of the electronic bands with increasing spin polarizationFSM , one
observes that at $m\sim 1$ $\mu_{B}$ the resulting bands can in no way be
described as anticrossing downfolded nonmagnetic bands with partial gapping of
the Fermi surface. Rather, the entire Fe d band is fully restructured.
Although the lowest-energy AFM state wave vector indeed coincides with the
quasi-nesting wave vector in some cases, it is not always true, as exemplified
by the case of FeTe that we discuss later.
It should be noted that while quasi-nesting is not particularly relevant for
the long-range ordering in the undoped crystals, it does define the low-energy
excitations in non-magnetic phases and these can perfectly well mediate
superconductivity.
Figure 1: (color online) Top-down view of the a) checkerboard b) stripe and c)
doublestripe magnetic patterns for a single FeAs or FeTe layer. The light
colored sites have majority up spin and the darker sites have majority down
spin.
Having established a general framework, we now address specific examples.
First, we investigate checkerboard, stripe, and double-stripe magnetic
structures (See Fig. 1) and show that the stripe order is lower in energy than
either the checkerboard or the double-stripe structure for the 122 systems,
but not for FeTe. We use BaFe2As2 as an example, but the results for LaFeAsO
are very similar. Our calculations were performed using an all-electron, full-
potential LAPW package WIEN2k, in the Generalized Gradient Approximation,
similar to Ref. PRB . All structures were fully relaxed (except where stated
otherwise) using the Vienna Ab-Initio Simulation Program (VASP) vasp , with
the PAW formulation PAW and also using GGA. In Table 1 we show the magnetic
stabilization energies of the three different antiferromagnetic structures.
Table 1: Stabilization energies for various magnetic configurations in the 122 and FeTe systems. All energies are per Fe atom. | checkerboard | stripe | double stripe
---|---|---|---
BaFe2As2 | 16 meV | 94 meV | 0.6 meV
FeTe | – | 207 meV | 230 meV
In Fig. 2a,b,c, we show the DOSs for BaFe2As2 in each of the three magnetic
configurations along with the nonmagnetic DOS. Compared to the nonmagnetic
DOS, we see that the checkerboard pattern has a very similar spectrum at and
near the Fermi energy and gains one-electron energy by shifting spectral
weight from the region between -0.5 and -1.0 downward to the region between
-1.0 and -2.0. The doublestripe pattern actually incurs an energy penalty at
and just below EF, but gains energy by shifting weight downward from between
-0.2 and -0.7 to between -1.0 and -2.0. The ground state configuration, in
contrast to the other two, gains energy all the way from EF to -0.9 by
shifting weight downward. This is accomplished through the opening of a large
pseudogap (this terminology has no connection with the pseudogap in cuprates
and simply signifies a depression in one-electron DOS around the Fermi level).
Though all three magnetic configurations are stable with respect to a
nonmagnetic state, it is visibly the case that the stripe ordering has the
greatest one electron energy advantage. This is reflected in the much larger
gain in total energy (See Table 1).
Figure 2: (color online) The densities of states for BaFe2As2 in the non-
magnetic configuration in comparison to a) checkerboard magnetic pattern b)
stripe (ground state) magnetic pattern and c) double stripe magnetic pattern
Let us now compare the results with the same calculations for FeTe. As
indicated in a number of papers, FeTe is always formed with an excess Fe, so
the fact that experiment gives the double stripe structure as the low-
temperature ground state Li should be taken $cum$ $grano$ $salis.$ However,
as Table 1 shows, it is definitely the stoichiometric ground state in density
functional calculations, and this is the only thing that matters for our
analysis note1 . We note here that we do not fully relax the FeTe structure,
but only relax the internal positions. As before these relaxations are done
separately for magnetic and nonmagnetic cases.
Figure 3: (color online) The densities of states for FeTe in the non-magnetic
configuration in comparison to a) stripe magnetic pattern and b) double stripe
magnetic pattern.
Table 1 indicates that for FeTe, as opposed to BaFe2As${}_{2},$ the energies
of the single and double stripe phases are relatively close. This suggests
that the crude method of determining the ground state by looking at the DOS
may not work here, as the DOSs for the two AFM phases will probably be
similar. Indeed, this is what we see in Fig. 3a,b where a large downward shift
of spectral weight is visible for both patterns. Interestingly, the
nonmagnetic Fermi surfaces in FeTe are extremely similar to those in the 122
and 1111 materials, whereas nonmagnetic DOS and the magnetic electronic
structure are quite different. This reinforces that FS nesting, which would be
nearly identical for BaFe2As2 and FeTe, is not driving the magnetic order.
Our calculations also provide a strong argument against superexchange. Looking
at the patterns in Fig. 1, it is easy to see that for both the stripe and
double stripe patterns, the first neighbor exchange, $J_{1}$, does not
contribute, due to equal numbers of aligned and anti-aligned spins. The second
neighbor exchange, $J_{2}$, would have to be stronger than $J_{1}/2$ in order
for the stripe pattern to be energetically favorable over the checkerboard
pattern in a superexchange picture. It has often been argued that this
situation is not unreasonable since the Fe-As-Fe paths available for $J_{1}$
and $J_{2}$ are similar. For double stripe order in the FeTe system, however,
both $J_{1}$ and $J_{2}$ cancel, leaving only $J_{3}$ to establish the
ordering. Considering the remarkably strong stability (compared to non-
magnetic) calculated for double stripe order (See Table 1), this is hard to
rationalize. Furthermore, the energy term for stripe is $J_{2}$ \- $J_{3}$
(compared to $J_{3}$ alone for double stripe). For double stripe to stabilize,
$J_{3}$ could be no smaller than $J_{2}/2$, but the ”similar hopping paths”
justification used for $J_{1}$ and $J_{2}$ and is not available: the third
neighbor exchange path is more than twice as long as the second neighbor one
and involves As-As hopping across a distance of a full lattice constant. Thus,
the existence and stability of the double stripe order severely strains the
credibility of the superexchange picture. This is, in fact, to be expected
since superexchange is not efficient when the bandwidth is much larger than
the energy cost of flipping an electron’s spin, which is precisely the case
here.
This does not, however, mean that one cannot map the dependence of the total
energy onto a suitable short-range exchange model. In fact, it is hard to
imagine a case in which this would not be possible. Yet, in carrying out this
procedure for ferropnictide systems, one should be aware of the following
caveats:
(1) There is no microscopic justification (as for instance in the Hubbard
model) for introducing any $J-t$ (or $J_{1}-J_{2}-t)$ Hamiltonian.
(2) There is no guarantee that this kind of mapping can be stopped at first or
second neighbors. In fact, accurate calculations show that at least some of
the exchange parameters in these mappings decay as $1/R^{3},$ just as in metal
ironY-A .
(3) The resulting exchange parameters strongly and qualitatively depend on the
long-range order established in the system. In particular, the parameters that
can be used to describe the ordered state cannot be used to describe the spin
fluctuations, and vice versa. (See Ref. Y-A and references therein.)
(4) In the absence of superexchange, there is no reason to believe that the
total energy can be mapped onto a Heisenberg model,
$\sum_{ij}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}.$ In fact, direct
calculations show that at least one biquadratic term needs to be added to map
the total energy onto the mean-field Hamiltonian, $\sum
K(\mathbf{S}_{i}\cdot\mathbf{S}_{i+\mathbf{1}})^{2},$ where $K\sim J$. Ole
Figure 4: The Fermi surfaces of stripe-ordered BaFe2As2. Top panel shows the
’reverse distortion’ in which the Fe-Fe distance is lengthened along the like
spin direction and shortened along the unlike spin direction. The bottom
panels shows the fully relaxed calculation which reproduces the experimentally
observed distortion (to within a few percent).
We now switch our attention to the structural transition observed
simultaneoulsy with the magnetic one in the 122 systems. Density functional
calculations very accurately reproduce the experimentally observed distortion
in which Fe ions along the stripe direction are closer to one another than Fe
atoms belong to adjacent stripes yildirim ; jesche . We investigated whether
the structural distortion, like the magnetic ordering, can be understood in
terms of one electron energies by calculating the DOS for a variety of small
changes in the $a$ and $b$ lattice constants. In contrast to changing the
magnetic pattern, changing the structural distortion has very little effect on
the DOS away from the Fermi energy. There were no large shifts of spectral
weight to lower energies, though small shifts of the order of 0.05 eV did
occur and these were within 0.5 eV of the Fermi energy (for comparison see the
heavy restructuring of the DOS in Figs. 2 and 3). The distortion can therefore
be treated as a linear perturbation with a one-electron energy lowering
observable at (or very near) the Fermi energy. Specifically, we find that the
lowest energy structure corresponds to the smallest Fermi surface area. As an
example, in Fig. 4 we show the Fermi surface in the magnetic Brillouin zone of
the fully relaxed (lowest energy) structure and a ’reverse distortion’ in
which the distances between like and unlike spins are reversed from the
correct configuration. The change in the size of the Fermi surface is clearly
visible. We were unable to engineer a further minimization of the Fermi
surface with any choice of in-plane distortions other than the optimal energy
one.
In conclusion, we have shown that the relevant physics with respect to the
magnetic ordering and structural distortion in the ferropnictides lies in the
one-electron energies. Our results resolve the superficially binary choice
between superexchange and Fermi surface nesting in favor of a third mechanism
that is neither fully localized nor fully itinerant. One-electron energy is
gained throughout an energy range of at least 1 eV below EF and the ground
state is determined by which magnetic pattern most effectively exploits a
downshift in spectral weight, not by fermiology. On the other hand, the Fermi
surface itself is the operative feature for determination of the structural
distortion. The energy minimum for an in-plane distortion corresponds to a
simultaneous minimization of the Fermi surface area.
We thank H. Eschrig, K. Koepernik and T. Yildirim for useful and engaging
discussions related to this work. We acknowledge funding from the Office of
Naval Research.
## References
* (1) T. Kroll, S. Bonhommeau, T. Kachel, H.A. Duerr, J. Werner, G. Behr, A.Koitzsch, R. Huebel, S. Leger, R. Schoenfelder, A. Ariffin, R. Manzke, F.M.F. de Groot, J. Fink, H. Eschrig, B. Buechner, M. Knupfer, Phys. Rev. B 78, 220502 (2008); F. Bondino, E. Magnano, M. Malvestuto, F. Parmigiani, M. A. McGuire, A. S. Sefat, B. C. Sales, R. Jin, D. Mandrus, E. W. Plummer, D. J. Singh, N. Mannella; Phys. Rev. Lett. 101, 267001 (2008); E. Z. Kurmaev, R. G. Wilks, A. Moewes, N. A. Skorikov, Yu. A. Izyumov, L. D. Finkelstein, R. H. Li, X. H. Chen, Phys. Rev. B 78, 220503(R) (2008); V. I. Anisimov, E. Z. Kurmaev, A. Moewes, I. A. Izyumov, Physica C (Special Issue), to be published; T.D. Devereaux et al, to be published.
* (2) V. I. Anisimov, Dm. M. Korotin, M. A. Korotin, A. V. Kozhevnikov, J. Kuneš, A. O. Shorikov, S. L. Skornyakov, S. V. Streltsov, J. Phys.: Condens. Matter 21, 075602 (2009); T. Miyake, L. Pourovskii, V. Vildosola, S. Biermann, A. Georges, J. Phys. Soc. Jpn. 77, Supplement C, 99 (2008).
* (3) I.I. Mazin and J. Schmalian, Physica C (Special Issue), to be published.
* (4) M.D. Johannes and I.I. Mazin, Phys. Rev. B77, 16535 (2008).
* (5) M.A. Korotin, S.V. Streltsov, A.O. Shorikov, and V.I. Anisimov, JETP 107, 649 (2008).
* (6) K.D. Belashchenko, V.P. Antropov, Phys. Rev. B78, 212505 (2008); T. Yildirim, Physica C (Special Issue), to be published.
* (7) A. N. Yaresko, G.-Q. Liu, V. N. Antonov, O.K. Andersen, arXiv:0810.4469.
* (8) I. I. Mazin, M. D. Johannes, L. Boeri, K. Koepernik, D. J. Singh, Phys. Rev. B78, 085104 (2008).
* (9) G. Kresse, J. Furthmuller, Phys. Rev. B 54, 169 (1996).
* (10) P. E. Blochl, Phys. Rev. B 50, 953 (1994).
* (11) S. Li, C. de la Cruz, Q. Huang, Y. Chen, J.W. Lynn, J. Hu, Y-L. Huang, F-C. Hsu, K-W. Yeh, M-K. Wu, P. Dai, Phys. Rev. B 79, 054503 (2009).
* (12) We are interested in how the double stripe structure manifests in stoichiometric FeTe within LDA. We cannot make meaningful comparisons with experiment and we therefore do not fully relax the FeTe structure, but only relax the internal positions. As before, relaxations are done separately for magnetic and nonmagnetic cases. We also neglect the checkerboard case as being of only secondary interest for this system.
* (13) T. Yildirim, arXiv:0805.2888 (2008).
* (14) A. Jesche, N. Caroca-Canales, H. Rosner, H. Borrmann, A. Ormeci, D. Kasinathan, K. Kaneko, H. H. Klauss, H. Luetkens, R. Khasanov, A. Amato, A. Hoser, C. Krellner, C. Geibel, Phys. Rev. B 78, 180504R (2008)
|
arxiv-papers
| 2009-04-24T12:21:52 |
2024-09-04T02:49:02.141573
|
{
"license": "Public Domain",
"authors": "M. D. Johannes, Igor Mazin",
"submitter": "Michelle Johannes",
"url": "https://arxiv.org/abs/0904.3857"
}
|
0904.3899
|
# Modified Gravitational Equations on Braneworld with Lorentz Invariant
Violation
Arianto(1,2,3) arianto@upi.edu Freddy P. Zen(1,2) fpzen@fi.itb.ac.id Bobby E.
Gunara(1,2) bobby@fi.itb.ac.id (1)Theoretical Physics Lab., THEPI Devision,
and
(2)Indonesia Center for Theoretical and Mathematical Physics (ICTMP)
Faculty of Mathematics and Natural Sciences,
Institut Teknologi Bandung,
Jl. Ganesha 10 Bandung 40132, INDONESIA.
(3)Department of Physics, Udayana University
Jl. Kampus Bukit Jimbaran Kuta-Bali 80361, INDONESIA.
###### Abstract
The modified gravitational equations to describe a four-dimensional braneworld
in the case with the Lorentz invariant violation in a bulk spacetime is
presented. It contains a trace part of the brane energy-momentum tensor and
the coefficients of all terms describe the Lorentz violation effects from the
bulk spacetime. As an application, we apply this formalism to study cosmology.
In respect to standard effective Friedmann equations on the brane, Lorentz
invariance violation in the bulk causes a modification of this equations that
can lead to significant physical consequences. In particular, the effective
Friedmann equation on the brane explicitly depends on the equation of state of
the brane matter and the Lorentz violating parameters. We show that the
components of five-dimensional Weyl curvature are related to the matter on
brane even at low energies. We also find that the constraints on the theory
parameters are depend on the equation of state of the energy components of the
brane matter. Finally, the stability of the model depend on the specific
choices of initial conditions and the parameters $\beta_{i}$.
###### pacs:
98.80.Cq, 98.80.Hw
## I Introduction
There has been a growing appreciation of the importance of the violations of
Lorentz invariance recently. The intriguing possibility of the Lorentz
violation is that an unknown physics at high-energy scales could lead to a
spontaneous breaking of Lorentz invariance by giving an expectation value to
certain non Standard Model fields that carry Lorentz indices, such as vectors,
tensors, and gradients of scalar fields Kostelecky:1988zi . A relativistic
theory of gravity where gravity is mediated by a tensor, a vector, and a
scalar field, thus called TeVeS gravitational theory Bekenstein:2004ne ,
provides modified Newtonian dynamics (MOND) and Newtonian limits in the weak
field nonrelativistic limit. TeVeS could also explain the large-scale
structure formation of the Universe without recurring to cold dark matter
Skordis:2005xk , which is composed of very massive slowly moving and weakly
interacting particles. On the other hand, the Einstein–Aether theory
Jacobson:2000xp is a vector-tensor theory, and TeVeS can be written as a
vector-tensor theory which is the extension of the Einstein–Aether theory
Zlosnik:2006sb . In the case of generalized Einstein–Aether theory
Zlosnik:2006zu , the effect of a general class of such theories on the solar
system has been considered in Ref. Bonvin:2007ap . On small scales the
Einstein-Aether vector field will in general lead to a renormalization of the
local Newton Constant Carroll:2004ai . Moreover, as has been shown by authors
in Ref. Li:2007vz , the Einstein–Aether theory may lead to significant
modifications of the power spectrum of tensor perturbation. The existence of
vector fields in a scalar-vector-tensor theory of gravity also leads to its
applications in modern cosmology and it might explain inflationary scenarios
Lim:2004js ; Kanno:2006ty ; Watanabe:2009ct ; Avelino:2009wj and accelerated
expansion of the universe Zlosnik:2006zu ; Tartaglia:2007mh . Based on a
dynamical vector field coupled to the gravitation and scalar fields, we have
studied to some extent the cosmological implications of a scalar-vector-tensor
theory of gravity :2007xt . The models also allow crossing of phantom divide
line Nozari:2008ff .
Motivated by string theory and its extension M-theory, the standard model
particles may be confined on a hypersurface, called brane, embedded in a
higher dimensional space, called bulk. Only gravity and other exotic matter
such as the dilaton can propagate in the bulk Horava:1995qa . The braneworld
models have been shown to be extremely rich in phenomena leading to
modifications of General Relativity (GR) at both low and high energies
Maartens:2003tw . In the context of gravity and cosmology, models proposed by
Randall and Sundrum (RS) Randall:1999ee ; Randall:1999vf have attracted much
attention, where four-dimensional gravity can be recovered at low energy
despite the infinite size of the extra dimension. In RS II model
Randall:1999vf , a positive tension brane is embedded in five-dimensional
anti-de Sitter (AdS) spacetime. To study gravity on the brane, it is useful to
derive the effective four-dimensional Einstein equation on the brane firstly
developed by Shiromizu, Maeda, and Sasaki (SMS) Shiromizu:1999wj . There are
two very important results that arise from the effective four-dimensional
Einstein equations on the brane. The first one is quadratic energy-momentum
tensor, $\pi_{\mu\nu}$, which is relevant in high energy and the second one is
the projected Weyl tensor, $E_{\mu\nu}$, on the brane which is responsible for
carrying on the brane the contribution of the bulk gravitational field. In the
RS II models, this term supplies an additional matter-like effect to the
brane. Thus, its contribution to the four-dimensional effective theory is of
crucial importance as it is non-negligible already even in low energy limit.
Then, the Friedmann equations on the brane, governing the cosmological
evolution of the brane, are non conventional in that the Hubble parameter
depends quadratically on the energy density instead of linearly as in standard
cosmology, and one radiation like term, usually referred to as a dark
radiation term in the homogeneous and isotropic background spacetime. This
dark radiation modifies the expansion of the background universe in the same
way as an usual radiation Ida:1999ui ; Kraus:1999it ; Mukohyama:1999qx ;
Ichiki:2002eh .
Recently, a braneworld scenario with bulk broken Lorentz invariance has been
developed, where a family of static self-tuning braneworld solutions was found
Koroteev:2009xd . In a different approach braneworld model, a bulk vector
field with a non-vanishing vacuum expectation value, allowing for the
spontaneous breaking of the Lorentz symmetry. The breaking of Lorentz
invariance the loss of this symmetry is transmitted to the gravitational
sector of the model. By assuming that the vacuum expectation value of the
component of the vector field normal to the brane vanishes, it found that
Lorentz invariance on the brane can be made exact via the dynamics of the
graviton, vector field, and the geometry of the extrinsic curvature of the
surface of the brane. As a consequence of the exact reproduction of Lorentz
symmetry on the brane, a condition for the matching of the observed
cosmological constant in four dimensions is found Bertolami:2006bf . The
notion of Lorentz violation in four dimensions is extended to a five-
dimensional braneworld scenario resulting the time variation in the
gravitational coupling and cosmological constant. There exist also a relation
between the maximal velocity in the bulk and the speed of light on the brane
Ahmadi:2006cr . Various Lorentz violating effects within the context of the
braneworld scenario have also been studied in Refs. Csaki:2000dm ;
Stoica:2001qe ; Libanov:2005yf ; Nozari:2008rg ; Farakos:2009ui .
In this paper we address the issue of cosmological evolution on a brane in a
theory of gravity whose action includes, in addition to the familiar Einstein
term, a Lorentz violating vector field contribution. We generalize the
gravitational effects of the vector fields in four dimensions Jacobson:2000xp
; Kostelecky:2003fs to include five dimensional braneworld gravity. In
particular, we put a vector $n^{a}$ in the direction of the extra dimension
such that the existence of the brane defines a preferred direction in the
bulk.
This paper is organized as follows. In Section II, we derive the four-
dimensional effective Einstein equations on the brane in the case with the
Lorentz invariant violation in a bulk spacetime. With non-ignoring of the
Lorentz violation effects, this equation is modified by the trace of the brane
energy–momentum tensor. Thus the relation between the projected Weyl tensor
and the brane matter may be understood. In Section III, we study the
cosmological implications of the modified four-dimensional effective Einstein
equations on the brane. In general, the effective four-dimensional Einstein
equations on the brane cannot be solved without knowing $E_{\mu\nu}$, because
it could have a non-trivial component of an anisotropic stress Maartens:2000fg
. However, it is possible to know some features of this tensor by using
constraint equations on the brane obtained by the four-dimensional Bianchi
identity. In the background spacetime, the four-dimensional equations are
sufficient to show that $E_{\mu\nu}$ induces the radiation fluid on the brane.
We will take this strategy to determine the Friedmann equation on the brane.
Interestingly, the Friedmann equation is found to depend on the equation of
state of the matter explicitly, and the Lorentz violation parameters. In
Section IV, we discuss a low energy limit of the theory. Remarkable, the
parameters of the theory can be determined by equation of state of the brane
matter. Section V is devoted to the conclusions.
## II Modified SMS Effective Equation on the brane
In this section, we derive the $4$-dimensional effective gravitational
equations in a $Z_{2}$-symmetric braneworld using the geometrical projection
approach. For this purpose, we first write the $5$-dimensional field equations
in the form of the evolution equations along the extra dimension and the
constraint equations.
The action we consider consists of the vector field $n^{a}$ minimally coupled
to gravity:
$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-\tilde{g}}\left({\cal R}-2\Lambda\right)+\int
d^{5}x\sqrt{-\tilde{g}}{\cal L}_{n}+\int d^{4}x\sqrt{-g}(-\sigma+{\cal
L}_{m})\ .$ (1)
Here, $\mathcal{R}$, $\kappa$, $\Lambda$, and $\tilde{g}$ are the scalar
curvature, the gravitational constant in $5$-dimensions, the bulk cosmological
constant, and the determinant of $5$-dimensional metric, respectively. ${\cal
L}_{m}$ and ${\cal L}_{n}$ are the Lagrangian density for the matter fields on
the brane and the vector field Lagrangian, respectively. A metric $g$ is the
induced metric on the brane while $\sigma$ denotes the brane tension. Note
that we have assumed no coupling between the matter fields and the vector
field in the action (1). Therefore, the brane observer does not feel the
present of the preferred frame.
We write the coordinate system for the bulk spacetime in the form
$ds^{2}=g_{ab}dx^{a}dx^{b}=dy^{2}+g_{\mu\nu}(y,x)dx^{\mu}dx^{\nu}\ ,$ (2)
and we may assume that the position of the brane is $y=0$ in this coordinate
system so that the induced metric on the brane is
$g_{\mu\nu}(x)=\tilde{g}_{\mu\nu}(y=0,x)$. We also assume a $Z_{2}$-symmetry
across the brane and the extrinsic curvature is defined as
$K_{\mu\nu}=-g_{\mu\nu,y}/2$.
The vector field Lagrangian, ${\cal L}_{n}$, is given by
$\displaystyle{\cal L}_{n}$ $\displaystyle=$
$\displaystyle-\beta_{1}\nabla^{a}n^{b}\nabla_{a}n_{b}-\beta_{2}\left(\nabla_{a}n^{a}\right)^{2}-\beta_{3}\nabla^{a}n^{b}\nabla_{b}n_{a}+\lambda(n^{a}n_{a}-1)\
,$ (3)
where $\beta_{i}$ are constant parameters and $\lambda$ is a Lagrangian
multiplier. In this setup, we assume that $n^{a}$ is a vector field along the
extra dimension and the preferred frame is selected by the constrained vector
field $n^{a}$ which violates Lorentz symmetry. We take $n^{a}$ as the
dimensionless vector. Hence, each $\beta_{i}$ has dimension of $(mass)^{3}$.
In other words, $\beta_{i}^{1/3}$ gives the mass scale of symmetry breakdown
in the bulk. Following the usual braneworld scenarios our spacetime is
orthogonal to the extra dimension. Then one can introduce the normal unit
vector $n^{a}$ which is orthogonal to the hypersurfaces at $y=const$. as
$n^{a}=\delta_{y}^{a}$. In particular, there is a background solution that
5-vector takes on a vacuum expectation value with components $(0,0,0,0,1)$,
thus allowing for the spontaneous breaking of the Lorentz symmetry.
Varying the action (1) with respect to the metric, $\lambda$, and $n^{a}$,
respectively, we have the field equations
$\displaystyle{}^{(5)}G_{ab}$ $\displaystyle=$ $\displaystyle-\Lambda
g_{ab}+\kappa^{2}(T_{ab}+{\cal
T}_{ab})+\kappa^{2}\delta_{a}^{\mu}\delta_{b}^{\nu}S_{\mu\nu}\delta(y)\ ,$ (4)
$\displaystyle g_{ab}n^{a}n^{b}$ $\displaystyle=$ $\displaystyle 1\ ,$ (5)
$\displaystyle\nabla_{a}J^{ab}$ $\displaystyle=$ $\displaystyle\lambda n^{b}\
,$ (6)
where current tensor $J^{a}{}_{c}$ is given by
$J^{a}{}_{b}=-\beta_{1}\nabla^{a}n_{b}-\beta_{2}\delta^{a}_{b}\nabla_{c}n^{c}-\beta_{3}\nabla_{b}n^{a}\
,$ (7)
and $S_{\mu\nu}=-\sigma g_{\mu\nu}+\tau_{\mu\nu}$ is the energy momentum
tensor on the brane, where $\tau_{\mu\nu}$ is the energy momentum tensor of
the brane matter other than the tension. $T_{ab}$ is the energy–momentum
tensor of the vector field. To be as general as possible, we also have
included a bulk energy–momentum tensor in (4), denoted by ${\cal T}_{ab}$.
Using the extrinsic curvature, the components of the left hand side of
Einstein equations (4) are
$\displaystyle{}^{(5)}G^{y}{}_{y}$ $\displaystyle=$ $\displaystyle-{1\over
2}R+{1\over 2}K^{2}-{1\over
2}K^{\alpha\beta}K_{\alpha\beta}=-\Lambda+\kappa^{2}T^{y}{}_{y}+\kappa^{2}{\cal
T}^{y}{}_{y}\ ,$ (8) $\displaystyle{}^{(5)}G^{y}{}_{\mu}$ $\displaystyle=$
$\displaystyle-
D_{\alpha}K_{\mu}{}^{\alpha}+D_{\mu}K=\kappa^{2}(T^{y}{}_{\mu}+{\cal
T}^{y}{}_{\mu})\ ,$ (9) $\displaystyle{}^{(5)}G^{\mu}{}_{\nu}$
$\displaystyle=$ $\displaystyle
G^{\mu}{}_{\nu}+(K^{\mu}{}_{\nu}-\delta^{\mu}_{\nu}K)_{,y}+{1\over
2}\delta^{\mu}_{\nu}(K^{2}+K^{\alpha\beta}K_{\alpha\beta})$ (10)
$\displaystyle=$
$\displaystyle-\Lambda\delta^{\mu}_{\nu}+\kappa^{2}(T^{\mu}{}_{\nu}+{\cal
T}^{\mu}{}_{\nu})+\kappa^{2}S^{\mu}{}_{\nu}\delta(y)\ ,$
where $G^{\mu}{}_{\nu}$ is the $4$-dimensional Einstein tensor and the
covariant derivatives $D_{\mu}$ is calculated with respect to the four-
dimensional metric $g_{\mu\nu}$. The components of the energy momentum tensor
of the vector field are given by
$\displaystyle T^{y}{}_{y}$ $\displaystyle=$
$\displaystyle\beta_{2}K^{2}+(\beta_{1}+\beta_{3})K^{\alpha\beta}K_{\alpha\beta}\
,$ (11) $\displaystyle T^{y}{}_{\mu}$ $\displaystyle=$ $\displaystyle 0\ ,$
(12) $\displaystyle T^{\mu}{}_{\nu}$ $\displaystyle=$ $\displaystyle
2(\beta_{1}+\beta_{3})K^{\mu}{}_{\nu}K+\beta_{2}\delta^{\mu}{}_{\nu}K^{2}-\delta^{\mu}_{\nu}(\beta_{1}+\beta_{3})K^{\alpha\beta}K_{\alpha\beta}-2(\beta_{1}+\beta_{3})K^{\mu}{}_{\nu,y}-2\beta_{2}\delta^{\mu}_{\nu}K_{,y}\
.$ (13)
Combining Eqs. (8) with (10) and using (11) and (13), we have
$\displaystyle-{1\over 3}\left(R^{\mu}{}_{\nu}-{1\over
4}\delta^{\mu}_{\nu}R\right)$ $\displaystyle=$ $\displaystyle{1\over
6}\delta^{\mu}_{\nu}\Lambda+{(1-\alpha_{0})\over
12}\delta^{\mu}_{\nu}K^{2}-{(1+\alpha_{1})\over
3}\left(KK^{\mu}{}_{\nu}-{3\over
4}\delta^{\mu}_{\nu}K_{\alpha\beta}K^{\alpha\beta}\right)$ (14)
$\displaystyle+{(1+\alpha_{1})\over 3}K^{\mu}{}_{\nu,y}-{(1-\alpha_{0})\over
3}\delta^{\mu}_{\nu}K_{,y}-{\kappa^{2}\over 3}\left({\cal
T}^{\mu}{}_{\nu}-{1\over 2}\delta^{\mu}_{\nu}{\cal T}^{y}{}_{y}\right)\ ,$
where we have defined
$\displaystyle\alpha_{0}=2\kappa^{2}\beta_{2},\quad\alpha_{1}=2\kappa^{2}(\beta_{1}+\beta_{3})\
.$ (15)
The trace of equation (14) yields
$\displaystyle(3-4\alpha_{0}-\alpha_{1})K_{,y}=2\Lambda-(\alpha_{0}+\alpha_{1})K^{2}+3(1+\alpha_{1})K_{\alpha\beta}K^{\alpha\beta}-{\kappa^{2}\over
3}\left({\cal T}^{\mu}{}_{\mu}-2{\cal T}^{y}{}_{y}\right).$ (16)
Substituting Eqs. (14) and (16) into the following components of the Weyl
tensor
$\displaystyle C_{y\mu y\nu}$ $\displaystyle=$ $\displaystyle-{1\over
3}\left(R_{\mu\nu}-{1\over 4}g_{\mu\nu}R\right)+{1\over
3}\left(KK_{\mu\nu}-{1\over 4}g_{\mu\nu}K^{2}\right)+{1\over
3}\left(K_{\mu}{}^{\alpha}K_{\alpha\nu}+{3\over
4}g_{\mu\nu}K_{\alpha\beta}K^{\alpha\beta}\right)$ (17) $\displaystyle+{2\over
3}\left(K_{\mu\nu,y}-{1\over 4}g_{\mu\nu}K_{,y}\right)\ ,$
we have
$\displaystyle\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}C_{y\mu y\nu}$
$\displaystyle=$ $\displaystyle{1\over 2}\Lambda
g_{\mu\nu}-\frac{3\alpha_{0}+(2+\alpha_{0})\alpha_{1}}{4(3+\alpha_{1})}g_{\mu\nu}K^{2}-\frac{(1+\alpha_{1})\alpha_{1}}{(3+\alpha_{1})}KK_{\mu\nu}+\frac{(1+\alpha_{1})(3+2\alpha_{1})}{(3+\alpha_{1})}K_{\mu}{}^{\lambda}K_{\lambda\nu}$
(18)
$\displaystyle+\frac{3(1+\alpha_{1})(4+\alpha_{1})}{4(3+\alpha_{1})}g_{\mu\nu}K_{\alpha\beta}K^{\alpha\beta}+(1+\alpha_{1})K_{\mu\nu,y}-(1-\alpha_{0})g_{\mu\nu}K_{,y}$
$\displaystyle+{\kappa^{2}\over 2}g_{\mu\nu}{\cal
T}^{y}{}_{y}-{\kappa^{2}\over 3}\left({\cal T}_{\mu\nu}+{1\over
2}g_{\mu\nu}{\cal T}^{\alpha}{}_{\alpha}\right)\ .$
Here, we have defined that the term ${\cal T}^{\alpha}{}_{\alpha}$ is the
trace defined with respect to the four-dimensional metric $g$, and not the
full trace defined with respect to $\tilde{g}$. Equation (10) can be expressed
as
$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle-\Lambda
g_{\mu\nu}+(1+\alpha_{1})KK_{\mu\nu}-{(1-\alpha_{0})\over
2}g_{\mu\nu}K^{2}-2(1+\alpha_{1})K_{\mu}{}^{\alpha}K_{\alpha\nu}-{(1+\alpha_{1})\over
2}g_{\mu\nu}K_{\alpha\beta}K^{\alpha\beta}$ (19)
$\displaystyle-(1+\alpha_{1})K_{\mu\nu,y}+(1-\alpha_{0})g_{\mu\nu}K_{,y}+\kappa^{2}{\cal
T}_{\mu\nu}\ .$
Using Eq. (18), Eq. (19) is expressed as
$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\Lambda
g_{\mu\nu}-\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}E_{\mu\nu}-\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}(K_{\mu}{}^{\alpha}K_{\alpha\nu}-KK_{\mu\nu})-\frac{6+4\alpha_{1}-(3+\alpha_{1})\alpha_{0}}{4(3+\alpha_{1})}g_{\mu\nu}K^{2}$
(20)
$\displaystyle+\frac{(1+\alpha_{1})(6+\alpha_{1})}{4(3+\alpha_{1})}g_{\mu\nu}K_{\alpha\beta}K^{\alpha\beta}+{\kappa^{2}\over
2}g_{\mu\nu}{\cal T}^{y}{}_{y}+{2\kappa^{2}\over 3}\left({\cal
T}_{\mu\nu}-{1\over 4}g_{\mu\nu}{\cal T}^{\alpha}{}_{\alpha}\right)\ ,$
where the projected Weyl tensor is $E_{\mu\nu}=C_{y\mu y\nu}|_{y=0}$. Note
that the coefficient of the four-dimensional Einstein tensor (20) is modified
by factor $(3+\alpha_{1})$. Here, we take $\alpha_{1}\neq-3$. The case
$\alpha_{1}=-3$ provides a relation between the extrinsic curvature and the
projected Weyl tensor. To eliminate the extrinsic curvature, we use the
junction conditions. It can be obtained by collecting together the terms in
field equations which contain a $\delta$-function. From Eqs. (10) and (13), we
then obtain
$\displaystyle\left[K^{\mu}{}_{\nu}-\delta^{\mu}_{\nu}K\right]|_{y=0}$
$\displaystyle=$ $\displaystyle{\kappa^{2}\over
2(1+\alpha_{1})}\left(S^{\mu}{}_{\nu}+\alpha_{2}\delta^{\mu}_{\nu}S\right)\ ,$
(21)
where
$\displaystyle\alpha_{2}=\frac{\alpha_{0}+\alpha_{1}}{3-4\alpha_{0}-\alpha_{1}}\
.$ (22)
For convenient we will take $\alpha_{1}\neq-3$ and $\alpha_{1}\neq-1$ in order
to avoid unreal singularities in Eqs. (20) and (21). Substituting (21) into
(20), we finally obtain the modified effective SMS equation on the brane as
$\displaystyle G_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\Lambda_{b}g_{\mu\nu}+8\pi
G\left(\tau_{\mu\nu}+{\alpha_{1}\over
12}g_{\mu\nu}\tau\right)+\kappa^{4}\pi_{\mu\nu}-\widetilde{E}_{\mu\nu}+F_{\mu\nu}\
,$ (23)
where we have defined the quantities
$\displaystyle\Lambda_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda+\frac{\kappa^{4}}{4(3-4\alpha_{0}-\alpha_{1})}\sigma^{2},$
(24) $\displaystyle 8\pi G$ $\displaystyle=$
$\displaystyle\frac{3\kappa^{4}}{2(3+\alpha_{1})(3-4\alpha_{0}-\alpha_{1})}\sigma,$
(25) $\displaystyle\pi_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{3}{4(3+\alpha_{1})(1+\alpha_{1})}\left[{(1-2\alpha_{0}-\alpha_{1})\over(3-4\alpha_{0}-\alpha_{1})}\tau\tau_{\mu\nu}-\tau_{\mu}{}^{\alpha}\tau_{\alpha\nu}+{(6+\alpha_{1})\over
12}g_{\mu\nu}\tau_{\alpha\beta}\tau^{\alpha\beta}\right.$ (26)
$\displaystyle\left.-{2(3-\alpha_{1})-(9+\alpha_{1})\alpha_{0}\over
12(3-4\alpha_{0}-\alpha_{1})}g_{\mu\nu}\tau^{2}\right]\ ,$
$\displaystyle\widetilde{E}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}E_{\mu\nu}\ ,$ (27)
and the bulk energy-momentum tensor projected on the brane is given by
$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left[{\kappa^{2}\over 2}g_{\mu\nu}{\cal
T}^{y}{}_{y}+{2\kappa^{2}\over 3}\left({\cal T}_{\mu\nu}-{1\over
4}g_{\mu\nu}{\cal T}^{\alpha}{}_{\alpha}\right)\right]_{y=0}\ .$ (28)
There are four features in the effective Einstein equations (23). The first
one is the presence of the bulk energy-momentum tensor. This term allows
exotic matter such as the dilaton can propagate in the bulk. The second
departure from the standard four-dimensional Einstein equation arises from the
presence of the Weyl tensor which is undetermined on the brane. The third is a
quadratic in the brane energy-momentum tensor. The last one is a linear in
addition to the brane energy-momentum tensor. It is our main result. This
trace part of the brane energy-momentum tensor is measured by local observers
at the brane and vanishes when
$\alpha_{1}=2\kappa^{2}(\beta_{1}+\beta_{3})=0$.
Equation (9) and the junction conditions (21) imply
$\displaystyle
D_{\mu}\tau^{\mu}{}_{\nu}+\alpha_{2}D_{\nu}\tau-(1+4\alpha_{2})D_{\nu}\sigma=-2(1+\alpha_{1}){\cal
T}^{y}{}_{\nu}\ .$ (29)
This equation tell us that the energy momentum tensor $\tau_{\mu\nu}$ is not
conserved on the brane. Taking the divergence of the four–dimensional
effective equations and using four–dimensional Bianchi identity, we obtain the
constraint equations for $E_{\mu\nu}$ as
$\displaystyle D_{\mu}\widetilde{E}^{\mu}{}_{\nu}$ $\displaystyle=$
$\displaystyle-D_{\nu}\Lambda_{b}+8\pi
G\left(D_{\mu}\tau^{\mu}{}_{\nu}+{\alpha_{1}\over
12}D_{\nu}\tau\right)+\kappa^{4}D_{\mu}\pi^{\mu}{}_{\nu}$ (30)
$\displaystyle+{\kappa^{2}\over 2}D_{\nu}{\cal T}^{y}{}_{y}+{2\kappa^{2}\over
3}\left(D_{\mu}{\cal T}_{\mu\nu}-{1\over 4}D_{\nu}{\cal
T}^{\alpha}{}_{\alpha}\right).$
Equations (29) and (30) indicate a time variation of the brane tension, the
cosmological constant, and the gravitational constant in general.
In the following section, we study analytically the cosmological consequences
of Eqs. (23), (29) and (30). Here, for simplicity, we consider constant
$\sigma$, because there are no theoretical observational arguments for the
evolution of $\sigma$ in time. For cosmology on the brane, we suppose here
that we can ignore the bulk matter, $F_{\mu\nu}=0$. The bulk matter is
important to get a well–behaved geometry in the bulk. We also assume that the
bulk cosmological constant is truly constant. Then, Eqs. (29) and (30) become
$\displaystyle D_{\mu}\tau^{\mu}{}_{\nu}=-\alpha_{2}D_{\nu}\tau\ ,$ (31)
$\displaystyle D_{\mu}\widetilde{E}^{\mu}{}_{\nu}=-8\pi
G\left(\alpha_{2}-{\alpha_{1}\over
12}\right)D_{\nu}\tau+\kappa^{4}D_{\mu}\pi^{\mu}{}_{\nu}\ .$ (32)
Note that the projected Weyl tensor is affected by the energy–momentum tensor
on the brane even at low energies. Thus, the model is quite different from the
conventional braneworld even at low energies.
## III Braneworld cosmology
The projected Weyl tensor in the modified Einstein equation (23) is a priori
undetermined. This comes from the five-dimensional nature of the theory and
the fact that the system of equations is not closed on the brane. This tensor
mediates some information from the bulk to the brane. In this section, we will
try to solve Einstein equation to study the cosmology braneworld from equation
(23), by assuming that there is no cosmological constant on the brane and the
constant vacuum energy. Although these assumptions are usual in braneworld
scenario, we will show, which is the main result of present paper, the
effective Friedmann equations is modified by the effect of Lorentz violation,
and the components of the projected Weyl tensor are related to the matter on
the brane. We then discuss the method to obtain the components of the
projected Weyl tensor from the brane data. For cosmological applications, we
consider a metric of the form
$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}\ ,$ (33)
where $x^{i}$ are the three ordinary spatial coordinates and $a$ is the scale
factor. The Hubble parameter $H$ on the brane, describing the cosmological
dynamics of the Universe, is defined as $H=\dot{a}/a$. For simplicity, we
ignore the bulk matter for the cosmology on the brane. Hereafter, we will
consider only the matter on the brane. For further discussions on the
gravitational field equations in the braneworld model with Lorentz violation
and their cosmological applications see Ahmadi:2006cr . We restrict the
energy-momentum tensor on the brane of the form
$\displaystyle\tau_{\mu\nu}=(\rho,Pa^{2}\delta_{ij})\ ,$ (34)
where $\rho$ is the energy density and $P$ the pressure. We will assume that
the equation of state relating $\rho$ and $P$ has the form $P=\omega\rho$,
where $\omega$ is constant. Similarly, the projected Weyl tensor is of the
form
$\displaystyle E_{\mu\nu}=(\rho_{d},P_{d}a^{2}\delta_{ij})\ .$ (35)
The traceless property of $E_{\mu\nu}$ implies: $-\rho_{d}+3P_{d}=0$. We will
be interested in the relation between the components of the projected Weyl
tensor and the brane energy-momentum tensor. The components of the quadratic
in the energy-momentum tensor (26) are given by
$\displaystyle\pi_{00}$ $\displaystyle=$
$\displaystyle\frac{1+3\alpha_{3}}{4(3+\alpha_{1})(1+\alpha_{1})^{2}}\rho^{2}\
,$ (36) $\displaystyle\pi_{ij}$ $\displaystyle=$
$\displaystyle\frac{1+2\omega-3\alpha_{4}}{4(3+\alpha_{1})(1+\alpha_{1})^{2}}\rho^{2}a^{2}\delta_{ij}\
,$ (37)
where
$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{12(3-4\alpha_{0}-\alpha_{1})}\\{[7-9(2+\omega)\omega-3(1+\omega)^{2}(2-\alpha_{1})\alpha_{1}]\alpha_{0}-[17-3\omega(8+3\omega)$
$\displaystyle+2(1-12\omega-3\omega^{2}+(1+3\omega^{2})\alpha_{1}^{2})]\alpha_{1}\\}\
,$ $\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{12(3-4\alpha_{0}-\alpha_{1})}\\{[-1-2\omega+15\omega^{2}+3(1+\omega)^{2}(6+\alpha_{1})\alpha_{1}]\alpha_{0}-[15+32\omega$
(38)
$\displaystyle-15\omega^{2}-(1+3\omega^{2})\alpha_{1}^{2}-2(1+9\omega^{2})\alpha_{1}]\alpha_{1}\\}\
.$
Substituting metric (33) and tensors (34), (35) and (36), (37) in the
effective Einstein equations (23), one finds
$\displaystyle 3H^{2}=8\pi
G\left[1+\frac{(1-3\omega)\alpha_{1}}{12}\right]\rho+\frac{\kappa^{4}(1+3\alpha_{3})}{4(1+\alpha_{1})^{2}(3+\alpha_{1})}\rho^{2}-\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}\rho_{d}\
,$ (39) $\displaystyle-2\dot{H}-3H^{2}=8\pi
G\left[\omega-\frac{(1-3\omega)\alpha_{1})}{12}\right]\rho+\frac{\kappa^{4}(1+2\omega-3\alpha_{4})}{4(1+\alpha_{1})^{2}(3+\alpha_{1})}\rho^{2}-\frac{(1+\alpha_{1})}{(3+\alpha_{1})}\rho_{d}\
.$ (40)
Obviously, these equations are quite different from the usual braneworld
equations due to the effect of bulk Lorentz violation. From Eq. (31) and the
constraint equation for the projected Weyl tensor (32), we have
$\displaystyle[1+(1-\omega)\alpha_{2}]\dot{\rho}+3H\rho(1+\omega)=0\ ,$ (41)
and
$\displaystyle\dot{\rho}_{d}+4H\rho_{d}$ $\displaystyle=$
$\displaystyle\frac{8\pi
G(1+\omega)(1-3\omega)(3+\alpha_{1})(3-4\alpha_{0}-\alpha_{1})}{3(1+\alpha_{1})^{3}}\left[\alpha_{2}-\frac{(1+\alpha_{1})^{2}\alpha_{1}}{12(1-\omega\alpha_{1}-(1+\omega)\alpha_{0})}\right]H\rho$
(42)
$\displaystyle-\frac{\kappa^{4}(1+\omega)\alpha_{5}}{4(1+\alpha_{1})^{3}(1+(1-3\omega)\alpha_{2})}H\rho^{2},$
where
$\displaystyle\alpha_{5}$ $\displaystyle=$
$\displaystyle\frac{(1+\alpha_{1})}{2(3-4\alpha_{0}-\alpha_{1})}\\{12(1+\omega)^{2}\alpha_{0}^{2}\alpha_{1}+[9(1+3\omega^{2})+(1+3\omega^{2})\alpha_{1}^{2}-2(1+12\omega-9\omega^{2})\alpha_{1}]\alpha_{1}$
(43)
$\displaystyle+[3(1-3\omega^{2})-2(7+30\omega-9\omega^{2})\alpha_{1}+(7+6\omega+15\omega^{2})\alpha_{1}^{2}]\alpha_{0}\\}\
.$
For $\omega\neq-1$, equation (41) is solved to yield
$\displaystyle\rho=a^{-\frac{3(1+\omega)}{1+(1-\omega)\alpha_{2}}}\ .$ (44)
Here, we have absorbed a constant factor into the scale factor by rescaling
it. Equation (42) can be integrated. We find
$\displaystyle{\rho}_{d}$ $\displaystyle=$ $\displaystyle-{3C\over
a^{4}}+\frac{8\pi
G(1+\omega)(3+\alpha_{1})(3-4\alpha_{0}-\alpha_{1})^{2}}{9(1+\alpha_{1})^{4}}\left\\{[1+(1-3\omega)\alpha_{2}]\alpha_{2}-\frac{\alpha_{1}(1+\alpha_{1})^{2}}{4(3-4\alpha_{0}-\alpha_{1})}\right\\}a^{-\frac{3(1+\omega)}{1+(1-\omega)\alpha_{2}}}$
(45)
$\displaystyle-\frac{\kappa^{4}(1+\omega)\alpha_{5}}{8(1+\alpha_{1})^{3}[2(1-3\omega)\alpha_{2}-(1+3\omega)]}a^{-\frac{6(1+\omega)}{1+(1-\omega)\alpha_{2}}}\
,$
where $C$ is a constant of integration. This effect of the bulk acts as
radiation fluid, hence it is called as dark radiation. Substituting Eq. (45)
into Eq. (39), we obtain the effective Friedmann equation
$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle{8\pi G_{eff}\over
3}\rho+A\rho^{2}+{\bar{C}\over a^{4}}\ ,$ (46)
where
$\displaystyle G_{eff}$ $\displaystyle=$
$\displaystyle\left\\{1-\frac{[1-\omega\alpha_{1}-(1+\omega)\alpha_{0}][(2+3\omega-(2+\alpha_{1})\alpha_{1})\alpha_{1}+3(1+\omega)\alpha_{0}]}{3(1+\alpha_{1})^{3}}\right\\}G\
,$ (47) $\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{\kappa^{4}}{12(3+\alpha_{1})(1+\alpha_{1})^{2}}\left[1+3\alpha_{3}-\frac{3(1+\omega)\alpha_{5}}{2[(1+3\omega)-2(1-3\omega)\alpha_{2}]}\right],$
(48) $\displaystyle\bar{C}$ $\displaystyle=$
$\displaystyle\frac{3(1+\alpha_{1})}{(3+\alpha_{1})}C\ .$ (49)
Note that the effective Newton constant depends on the Lorentz violating
parameters and the equations of state. It is different from the conventional
braneworld cosmology in five-dimensional case even at low energy. If the
effects of Lorentz violations are ignored, $\beta_{i}=0$, we have $G_{eff}=G$,
$A=\kappa^{4}/36$ and $\bar{C}=C$. In the alternative theory of gravity
including Brans–Dicke theory, the effective Newton constant need not be
constant in time. Observational bounds on $\dot{G}/G$ then constrain the
theory. In our case, we have the relation (47), hence the Newton constant is
always constant.
If the effective cosmological constant is included, the Friedmann equation
(46) becomes
$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle{1\over
3}\Lambda_{b}+{8\pi G_{eff}\over 3}\rho+A\rho^{2}+{\bar{C}\over a^{4}}\ ,$
(50)
where the relation between the vacuum energy and the effective cosmological
constant on a brane is given by Eq. (24). It is different from the usual four-
dimensional theory. In the RS braneworld, the vacuum energy in the brane is
not directly related to the cosmological constant on the brane in the
effective Einstein equation as in Eq. (24). In the RS braneworld, there should
be a cancellation between the four-dimensional and five-dimensional
contribution of the vacuum energy in order to have a vanishing cosmological
constant on the brane. This requires a fine-tuning for the parameters in the
action. In the present model the RS type relation is given by
$\displaystyle\sigma=\frac{6}{\kappa^{2}l}\left(1-{4\over 3}\alpha_{0}-{1\over
3}\alpha_{1}\right)^{1/2}\ ,$ (51)
and
$\displaystyle 8\pi G=\frac{3\kappa^{2}}{l(3+\alpha_{1})(1+\alpha_{1})^{1/2}}\
.$ (52)
Here, the bulk cosmological constant is defined as
$\Lambda=-6/\kappa^{2}l^{2}$, where $l$ is the scale of the bulk curvature
radius.
## IV Low Energy Constraint on $\beta_{i}$
For a well-defined theory, the constraints on the theory parameters
$\beta_{i}$ are given by Lim:2004js (see also Carroll:2004ai ):
1. 1.
Subluminal propagation of spin-0 field:
$(\beta_{1}+\beta_{2}+\beta_{3})/\beta_{1}\leq 1$,
2. 2.
Positivity of Hamiltonian: $\beta_{1}>0$,
3. 3.
Non-tachyonic propagation of spin-0 field:
$(\beta_{1}+\beta_{2}+\beta_{3})/\beta_{1}\geq 0$,
4. 4.
Subluminal propagation of spin-2 field: $\beta_{1}+\beta_{3}\leq 0$.
All these conditions together imply $(\beta_{1}+\beta_{2}+\beta_{3})\geq 0$
and $\beta_{2}\geq 0$.
At low energies, we can neglect the quadratic term of the Friedmann equation
(46). Then we have
$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle{8\pi G_{eff}\over
3}\rho+{\bar{C}\over a^{4}}\ .$ (53)
Here, we have assumed $3A/8\pi G_{eff}<<1$. Therefore, one can set $A\approx
0$ without loss of generality. Solving Eq. (48) one finds
$\displaystyle\alpha_{1}=\frac{1-\alpha_{0}(1+\omega)}{\omega},~{}~{}\text{or}~{}~{}\alpha_{1}=\frac{2(1+3\omega)-3\alpha_{0}(1+\omega)^{2}}{1+3\omega^{2}}\
,$ (54)
where $\alpha_{0}$ and $\alpha_{1}$ is given by Eq. (15). In other word, the
effect of Lorentz violation in the bulk is dependent on the equation of state
of the energy components of the Universe. Remarkable, the first solution (54)
yields $G_{eff}=G$. In this case, using the above constraints we find
1. 1.
For $\omega<-1$,
$\displaystyle\alpha_{0}>\frac{1+3\omega}{1+\omega},\qquad\alpha_{1}<-3\ ,$
(55)
and
$\displaystyle 1<\alpha_{0}<\frac{1+3\omega}{1+\omega},\qquad-3<\alpha_{1}<-1\
.$ (56)
2. 2.
For $-1<\omega<0$,
$\displaystyle 1<\alpha_{0}\leq\frac{1}{1+\omega},\qquad-1<\alpha_{1}\leq 0\
.$ (57)
3. 3.
For $\omega>0$,
$\displaystyle\frac{1}{1+\omega}\leq\alpha_{0}<1,\qquad-1<\alpha_{1}\leq 0\ .$
(58)
The above constraints give the correction in the coefficient of the dark
radiation. The second solution (54) gives the constraints:
1. 1.
For $\omega<-1$,
$\displaystyle\alpha_{0}>\frac{5+6\omega+9\omega^{2}}{3(1+\omega)^{2}},\qquad\alpha_{1}<-3\
,$ (59)
and
$\displaystyle
1<\alpha_{0}<\frac{5+6\omega+9\omega^{2}}{3(1+\omega)^{2}},\qquad-3<\alpha_{1}<-1\
.$ (60)
2. 2.
For $-1<\omega\leq-1/3$,
$\displaystyle\alpha_{0}>\frac{5+6\omega+9\omega^{2}}{3(1+\omega)^{2}},\qquad\alpha_{1}<-3\
,$ (61)
and
$\displaystyle
1<\alpha_{0}<\frac{5+6\omega+9\omega^{2}}{3(1+\omega)^{2}},\qquad-3<\alpha_{1}<-1\
.$ (62)
3. 3.
For $\omega\geq-1/3$,
$\displaystyle\frac{2(1+3\omega)}{3(1+\omega)^{2}}\leq\alpha_{0}<1,\qquad-1<\alpha_{1}\leq
0\ .$ (63)
These constraints give the corrections both in the effective Newton constant
and the dark radiation.
## V Conclusions
In the present paper, we have considered a five-dimensional braneworld model
with bulk Lorentz invariance violation, and derived the effective four-
dimensional Einstein equations on the brane. The main result of this paper is
the existence of the trace part of the brane energy-momentum tensor in the
modified Einstein equations on the brane, which is a modification of the SMS
effective equation Shiromizu:1999wj . Thus, the divergence of the projected
Weyl tensor is modified. Therefore, due to Lorentz violating effect, we have
obtained an expression for the projected Weyl tensor as a function of the
source on the brane. It becomes clear that the bulk effect can be determined
by matter localized on the brane even at low energies. As an application, we
have used the modified SMS effective equation to determine the Friedmann
equation on the brane. We have showed the effective Newton constant that
relates geometry to the matter density in Friedmann equation is dependent on
the equation of state of the energy component of the Universe, and the Lorentz
violating parameters. Note that if the brane was isotropic and homogeneous,
the matter part would have the additional property, $D^{\mu}\pi_{\mu\nu}=0$.
However, due to effect of Lorentz violation in the bulk, the effect of matter
still appears in Eq. (32). Thus, the brane matter will deform the bulk
geometry. In other word, the back-reaction of this to the brane will modify
the effective Friedmann equation even at low energies. It is interesting to
understand the low energy description of this braneworld model. The low energy
perturbation scheme proposed in Kanno:2006ty is a major achievement as it
allows for the derivation of the effective theory on the brane and for the
full comprehension of the Weyl tensor contribution to the effective theory. We
leave this issue for future studies.
Finally, we also find that the effect of Lorentz violation in the bulk is
dependent on the equation of state of the energy components of the brane
matter. This model also provides a convenient framework within which one may
study dark energy.
###### Acknowledgements.
Arianto wishes to acknowledge all members of the Theoretical Physics
Laboratory, the THEPI Divison of the Faculty of Mathematics and Natural
Sciences, ITB, for the warmest hospitality. This work was supported by Hibah
Kompetensi DIKTI, 2009\.
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|
arxiv-papers
| 2009-04-24T16:07:36 |
2024-09-04T02:49:02.148051
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Arianto, Freddy P. Zen, Bobby E. Gunara",
"submitter": "Arianto Arianto",
"url": "https://arxiv.org/abs/0904.3899"
}
|
0904.3927
|
# A Critique of “Solving the P/NP Problem Under Intrinsic Uncertainty”
Andrew Keenan Richardson Cole Arthur Brown
Although whether P equals NP is an important, open problem in computer
science, and although Jaeger’s 2008 [1] paper, “Solving the P/NP Problem Under
Intrinsic Uncertainty” presents an attempt at tackling the problem by
discussing the possibility that all computation is uncertain to some degree,
there are a number of logical oversights present in that paper which preclude
it from serious consideration toward having resolved P-versus-NP. There are
several differences between the model of computation presented in Jaeger’s
paper [1] and the standard model, as well as several bold assumptions that are
not well supported in Jaeger’s paper [1] or in the literature. In addition, we
find several omissions of rigorous proof that ultimately weaken this paper to
a point where it cannot be considered a candidate solution to the P-versus-NP
problem.
## 1 Overview
Jaeger [1] presents a paper which attempts to show that P is not equal to NP.
This follows from a novel model of computation which has intrinsic uncertainty
in its computation of all problems. However, we will show that this model
cannot be used to solve the same class of decisions that a Turing machine can.
### 1.1 Summary of the Paper in Question
Jaeger [1] in his paper: Solving the P/NP Problem Under Intrinsic Uncertainty,
attempts to resolve P-versus-NP through unique, unconventional means, by
sketching a computer science analogue to the Heisenberg Uncertainty Principle.
He begins by establishing a supposedly Turing equivalent computing machine
largely based off of the Turing machine that contains a tape of binary storage
cells of any length, infinite or finite. Rather than assuming the machine is
preprogrammed to accept certain inputs, the model described has a single tape
in which both the input and the program code are randomly placed. This allows
him to introduce the uncertainty that the paper is based on. He argues that
since multiple, unique programs can perform the same actions at different
speeds, program size should be considered when analyzing program complexity.
He continues that his concept of “intrinsic uncertainty” implies that it is
impossible to distinguish whether, given a segment of the input, a part of
this segment is program code or input.
Jaeger [1] likens the relationship of the parts of this segment to the
relationship described in Heisenberg’s Uncertainty Principle – that you can
not determine whether a part is code or input, rather you can compute the
answer given both options and then compute the “certainty” that each
computation is correct. There is a problem with this, he explains, because
according to his model every computation done on the machine has uncertainty,
including his certainty calculations. To solve this problem, he proposes what
he calls “self-computation”, or applying his Turing-machine-based machine to
itself to compute a confidence value for its output, thus increasing the
confidence value of its output. As a result of this self-computation he claims
that the confidence value of a computation is directly proportional to the
ratio of code length to input length. Through use of the sigmoid function, he
claims one can calculate the entropy, or the probability that this confidence
value is smaller than, larger than or equal to 0.5.
To relate this all to the P/NP problem, Jaeger [1] constructs three lemmas.
His first lemma says that a Turing machine simulating a computable decision in
NP to an arbitrary precision is itself computable regardless of the precision.
Lemma three states that this Turing machine is also in NP for some given
uncertainties. Lemma two allows him to state an uncertainty threshold for
which the Turing machine is not in P, thus proving that there exists some
function in NP but not in P.
### 1.2 P = NP Problem
The problem known as P=NP is arguably the biggest open problem in computer
science [2]. The Clay Mathematics Institute has offered a $1,000,000 prize for
anyone who provides a proof one way or another. This incentive has attracted
quite a few attempts by amateurs as well as professionals, since the problem
at first seems deceptively simple. The basic question asked by this problem is
this: if an answer to a yes or no problem can be verified in polynomial time,
can the answer also be computed in polynomial time? The answer is commonly
assumed to be ‘no’, although no proof has yet emerged.
One way to prove that P is not equal to NP is to give a counter example of a
problem that is in NP but proveably not in P. This is more difficult than it
sounds because it is difficult to prove that there are no algorithms in P
which compute the answer. Jaeger’s paper [1] attempts to prove that there is
some inrinsic information necessary for having the answer to a problem, and
that this intrinsic information can only be computed in nonpolynomial time.
### 1.3 Turing Machines
Jaeger 2008 [1] uses the model of a Turing machine, a theoretical model of
computing yes or no answers. The model of a Turing machine has several parts
[3]: a finite set of states including the initial state and a subset of final
or accepting states, a finite set of tape alphabet symbols composed of the
blank symbol and a set of input symbols, and a transition function which
computes the new state, the alphabet symbol to be printed, and the direction
to move (right or left), based on the current state and the current tape
symbol. Informally there are two parts in a Turing machine: a finite set of
states in which the current state is arrived at deterministically, and an
infinite tape which may start with a finite number of non-blank symbols
printed on it.
## 2 Challenging the Arguments Made
We will show that the model presented in this paper is flawed because it is
not equivalent to a Turing machine. Moreover, the self-computation algorithm
presented here has several major shortcomings which prevent it from resolving
that uncertainty.
### 2.1 Theoretical Differences with Turing Machines
The most fundamental flaw to the argument presented in the paper under
consideration is that rather than using a theoretically established model like
a Turing machine, it presents a model in which the “code” of the model,
presumably corresponding to the set of states and the transition function, is
indistinguishable from the “input” to the code, presumably corresponding to
the initial content of the tape. Although there are numerous very obvious
theoretical and practical ways of distinguishing “code” from “input”, they are
not clearly marked in this model, causing the intrinsic uncertainty upon which
the core argument is based.
“Let us assume in the following that we have a Turing-complete machine
architecture…”, reads paragraph 2, pg 5. The paper seems to rely on this model
being Turing-complete, a term which is never clearly defined but which we
assume to mean Turing equilavent. However, this is clearly not the case, since
the elements of the 7-tuple defining a Turing machine are clearly
identifiable, and the same can be said for the initial state of the tape. This
method of arbitrarily introducing uncertainty to the computation of a Turing
machine does not produce a Turing equivalent model, as it cannot be used to
simulate a Turing machine, nor can it compute the same class of decisions that
a Turing machine can. Because the output of this model is always uncertain,
there are no problems in NP that can be reduced to it. It is similar to a
model that flips a coin and upon seeing a heads, returns an arbitrary answer.
### 2.2 Lack of Rigor in Analysis
The definition of the machine the proof revolves around is seemingly flawed.
The machine which is supposed to be equivalent to a Turing machine in
computational power detailed has a tape that contains both its program code
and input, with both being randomly distributed throughout the tape. The
machine reads the tape which is partitioned into two subsets, $S_{1}$ and
$S_{2}$. We are lead to assume that these subsets form the direct sum of the
original input, but this is never verified. As a result, one could infer that
perhaps there is some intersection between $S_{1}$ and $S_{2}$. This is an
example of the unrigorous definitions throughout the paper. These subsets are
never fully defined - no methods for finding them are ever given. The machine
evaluates the tape by computing the results as if $S_{1}$ were the code and
$S_{2}$ the input and vice-versa, then calculates confidence values for both
of those computations to help decide which configuration is correct. This has
some problems, namely that the method for partitioning the input into two
distinct blocks is non-existent. On page 6, the partition is described as
being N bits long, but no further information is given. Given the fact that
the input and code are randomly intermixed, one could go as far as to say that
its impossible to partition the input with any precision - both the code and
input string are represented in binary and are thus indistinguishable. This
means that not only would the machine have to evaluate for all possible
$S_{1}$-$S_{2}$ dichotomies, but also for all possible partitions of the
input. This additional factor is left completely unaccounted for in the
remainder of the paper.
There is another error on page 6 in the uncertainty section. The claim that,
”we can accept the interpretation whose program code encompasses the larger
number of bits as more likely,” is never justified. Such a statement is
invalid - in many cases, for instance any program dealing with databases, the
size of the input vastly exceeds the size of the program code.
One mistake that appears quite often in Jaeger 2008 [1] is the constant
changing of terms. In section 2, page 5, the machine used throughout the paper
is described as a ”Turing-complete” machine based on the Turing machine. Just
two pages later on page 7, the same machine is referred to as a Turing
machine. This inconsistency is detrimental to the reader’s understanding as
well as the validity of the paper itself. The paper also mentions an ”outside
program” that performs the interpreting and execution aspects of the
previously defined computing machine. While the existence of such a ”program”
is provable, the author has omitted any kind of proof that such a program
exists.
### 2.3 The Self Computation Algorithm
The bulk of this article describes an algorithm to be used for calculation of
certainty measurements to be reported alongside the actual findings of a
Turing machine. The method of self-computation proposed is hugely complex,
riddled with nonstandard notation and very confusing; the authors of this
paper were unable to make much sense of it. Self-computation, as described in
the paper, involves executing the uncertain Turing machine on a copy of a
similar machine running arbitrary ”code”. The chance that the machine could
execute a copy of itself and gain higher precision is unlikely and left
completely unproven. The insinuation is that as you run the machine on itself,
the bitlength of the ”code” portion gets longer and longer, increasing the
ratio of code to data, which is directly proportional to certainty.
While this method may appear useful for calculating uncertainty, it is
unlikely that this will work because it assumes, among other things, that bit
length is a useful metric for telling code and data apart, that this is a
necessary determination in a Turing machine, that Turing equivalent models can
report a certainty measurement, and that no other algorithms exist for
computing this function. As we have shown these things to be untrue, the
internal merit of the self computation algorithm is irrelevant.
### 2.4 There is No Proof of Nonpolynomiality for Certainty Calculation
One problem with the argumunt presented in Jaeger 2008 [1] (Lemmas 1 and 3) is
that it is nowhere proven that the method of self-computation presented in
there is the only method of determining the certainty of a result. Even
granting that there is a problem of determining certainty of answers produced
by Turing machines, and given that this method of self-computation is a
reasonable way of determining certainty, and given that a certainty threshold
can be set such that computing certainty by this method cannot be done in
polynomial time, all doubtful claims, nothing in Jaeger’s paper [1] attempts
to prove that there is no other method of computing this certainty
measurement. It could very well be that there is a polynomial time algorithm
for computing this measurement that the author simply did not think of.
Indeed, it is unclear from the paper under consideration why it should be
possible to set a threshold ”dynamically” so that it requires NP time to reach
that level of certainty. Even using the algorithm outlined there, it is not
clearly explained why this would require exponential time rather than
polynomial time. This trick of requiring that an exponential amount of
computation go into producing the answer is only possible as a use of the
certainty value which is produced by the modified Turing machine. As the
output of a standard Turing machine is binary, this type of trick, which
requires using the extra non-binary output, would not normally be possible. It
is as if one constructed an algorithm to run in exponential time by requiring
that it print an exponential number of characters.
## 3 Conclusion
Overall, we have shown several things which should each individually render
the argument described in Jaeger’s paper [1] impotent. We have shown that a
model of computation that requires a measure of intrinsic uncertainty cannot
be reduceable to a Turing machine nor can it reliably compute any NP-complete
problem. We have also shown that the algorithm proposed for calculating
uncertainty relies on faulty or unproven assumptions. Finally, we note that
while one algorithm is presented here, which may very well have an exponential
runtime, there is no attempt to prove that there cannot be a polynomial-time
algorithm to compute the same. Any of these things strikes a fatal blow to the
argument outlined by Jaeger [1]. As such, we do not find this compelling
support for P not equal to NP.
## 4 Acknowledgements
This work was done as a project in the Spring 2009 CSC 200H course at the
University of Rochester. We thank the professor, Lane A. Hemaspaandra, and the
TA, Adam Sadilek, for their comments and advice. Any opinions, errors, or
omissions are the sole responsibility of the authors.
## References
* [1] Stefan Jaeger. Solving the P/NP Problem under Intrinsic Uncertainty. 2008\.
* [2] Michael Sipser. The History and Status of the P Versus NP Question. Proceedings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, pages 603–618, May 1992.
* [3] Michael Sipser. Introduction to the Teory of Computation, Second Edition. Course Technology, Camridge, Massachusetts, 2005.
|
arxiv-papers
| 2009-04-24T19:32:10 |
2024-09-04T02:49:02.155236
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrew Keenan Richardson, Cole Arthur Brown",
"submitter": "Cole Brown",
"url": "https://arxiv.org/abs/0904.3927"
}
|
0904.3954
|
# On supremum of bounded quantum observable††thanks: This project is supported
by Natural Science Found of China (10771191 and 10471124).
Liu Weihua, Wu Junde
Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
E-mail: wjd@zju.edu.cn
Abstract. In this paper, we present a new necessary and sufficient condition
for which the supremum $A\vee B$ exists with respect to the logic order
$\preceq$. Moreover, we give out a new and much simpler representation of
$A\vee B$ with respect to $\preceq$, our results have nice physical meanings.
Keywords: Quantum observable, logic order, supremum.
PACS numbers: 02.10-v, 02.30.Tb, 03.65.Ta.
## 1 Introduction
There some basic notations: $H$ is a complex Hilbert space, $S(H)$ is the set
of all bounded linear self-adjoint operators on $H$, $S^{+}(H)$ is the set of
all positive operators in $S(H)$, $P(H)$ is the set of all orthogonal
projection operators on $H$, ${\cal B}(\mathbb{R})$ is the set of all Borel
subsets of real number set $\mathbb{R}$. Each element in $P(H)$ is said to be
a quantum event on $H$. Each element in $S(H)$ is said to be a bounded quantum
observable on $H$. For $A\in S(H)$, let $R(A)$ be the range of $A$,
$\overline{R(A)}$ be the closure of $R(A)$, $P_{A}$ be the orthogonal
projection on $\overline{R(A)}$, $P^{A}$ be the spectral measure of $A$,
null$(A)$ be the null space of $A$, and $N_{A}$ be the orthogonal projection
on null$(A)$.
Let $A,B\in S(H)$. If for each $x\in H$, $[Ax,x]\leq[Bx,x]$, then we say that
$A\leq B$. Equivalently, there exists a $C\in S^{+}(H)$ such that $A+C=B$.
$\leq$ is a partial order on $S(H)$. The physical meaning of $A\leq B$ is that
the expectation of $A$ is not greater than the expectation of $B$ for each
state of the system. So the order $\leq$ is said to be a numerical order of
$S(H)$. But $(S(H),\leq)$ is not a lattice. Nevertheless, as a well known
theorem due to Kadison, $(S(\mathbb{H}),\leq)$ is an anti-lattice, that is,
for any two elements $A$ and $B$ in $S(\mathbb{H})$, the infimum $A\wedge B$
of $A$ and $B$ exists with respect to $\leq$ iff $A$ and $B$ are comparable
with respect to $\leq$ ([1]).
In 2006, Gudder introduced a new order $\preceq$ on $S(H)$: if there exists a
$C\in S(H)$ such that $AC=0$ and $A+C=B$, then we say that $A\preceq B$ ([2]).
Equivalently, $A\preceq B$ iff for each $\Delta\in{\cal B}(\mathbb{R})$ with
$0\notin\Delta$, $P^{A}(\Delta)\leq P^{B}(\Delta)$ ([2]). The physical meaning
of $A\preceq B$ is that for each $\Delta\in{\cal B}(\mathbb{R})$ with
$0\notin\Delta$, the quantum event $P^{A}(\Delta)$ implies the quantum event
$P^{B}(\Delta)$. Thus, the order $\preceq$ is said to be a logic order of
$S(H)$ ([2]). In [2], it is proved that $(S(H),\preceq)$ is not a lattice
since the supremum of arbitrary $A$ and $B$ may not exist in general. In [3],
it is proved that the infimum $A\wedge B$ of $A$ and $B$ with respect to
$\preceq$ always exists. In [4, 5], the representation theorems of the infimum
$A\wedge B$ of $A$ and $B$ with respect to $\preceq$ were obtained. In more
recent, Xu and Du and Fang in [6] discussed the existence of the supremum
$A\vee B$ of $A$ and $B$ with respect to $\preceq$ by the technique of
operator block. Moreover, they gave out a sufficient and necessary conditions
for the existence of $A\vee B$ with respect to $\preceq$. Nevertheless, their
conditions are difficult to be checked since the conditions depend on an
operator $W$, but $W$ is not easy to get. Moreover, their proof is so much
algebraic that we can not understand its physical meaning.
In this paper, we present a new necessary and sufficient condition for which
$A\vee B$ exists with respect to $\preceq$ in a totally different form.
furthermore, we give out a new and much simpler representation of $A\vee B$
with respect to $\preceq$, our results have nice physical meanings.
Lemma 1.1 [2]. Let $A,B\in S(H)$. If $A\preceq B$, then $A=BP_{A}$.
Lemma 1.2 [2]. If $P,Q\in P(H)$, then $P\leq Q$ iff $P\preceq Q$, and $P$ and
$Q$ have the same infimum $P\wedge Q$ and the supremum $P\vee Q$ with respect
to the orders $\leq$ and $\preceq$, we denote them by $P\wedge Q$ and $P\vee
Q$, respectively.
Lemma 1.3 [7]. Let $A,B\in S(H)$. Then $P^{A}(\\{0\\})=N(A)$,
$P_{A}=P^{A}(R\backslash\\{0\\})$, $P_{A}+N(A)=I$, $P_{A}\vee
P_{B}=I-N(A)\wedge N(B)$.
## 2 Some elementary lammas
Let $A,B\in S(H)$ and they have the following forms:
$A=\int\limits_{-M}^{M}\lambda dA_{\lambda}$
and
$B=\int\limits_{-M}^{M}\lambda dB_{\lambda},$
where $\\{A_{\lambda}\\}_{\lambda\in\mathbb{R}}$ and
$\\{B_{\lambda}\\}_{\lambda\in\mathbb{R}}$ be the identity resolutions of $A$
and $B$ ([7]), respectively, and $M=\max(\|A\|,\|B\|)$.
If $A$ has an upper bound $F$ in $S(H)$ with respect to $\preceq$, then it
follows from Lemma 1.1 that $A=FP_{A}$. Note that $A\in S(H)$, so
$FP_{A}=P_{A}F$ and thus $AF=FA$. Let $F$ have the following form:
$F=\int\limits_{-G}^{G}\lambda dF_{\lambda},$
where $\\{F_{\lambda}\\}_{\lambda\in\mathbb{R}}$ is the identity resolution of
$F$ and $G=\max(\|F\|,M)$. Then we have
$A=FP_{A}=(\int\limits_{-G}^{G}\lambda
dF_{\lambda})P_{A}=\int\limits_{-G}^{G}\lambda d(F_{\lambda}P_{A}).$
Lemma 2.1. Let $A\in S(H)$ and $F\in S(H)$ be an upper bound of $A$ with
respect to $\preceq$. Then for each $\Delta\in{\cal B}(\mathbb{R})$, we have
$P^{A}(\Delta)=\left\\{\begin{array}[]{ccc}P^{F}(\Delta)P_{A},&&0\not\in\Delta\\\
N(A),&&\Delta=\\{0\\}\\\
P^{F}(\Delta\backslash\\{0\\})P_{A}+N(A).&&0\in\Delta\\\ \end{array}\right.$
Proof. We just need to check $P^{A}(\Delta)=P^{F}(\Delta)P_{A}$ when
$0\not\in\Delta$, the rest is trivial. Note that if we restrict on the
subspace $P_{A}(H)=\overline{R(A)}$, since $AF=FA$, then
$\\{F_{\lambda}P_{A}\\}_{\lambda\in\mathbb{R}}$ is the identity resolution of
$F|_{P_{A}(H)}$ ([7]). Let $f$ be the characteristic function of $\Delta$.
Then the following equality proves the conclusion:
$P^{A}(\Delta)=f(A)=f(FP_{A})=\int\limits_{-G}^{G}f(\lambda)d(F_{\lambda}P_{A})=\int\limits_{\lambda\in\Delta}d(F_{\lambda}P_{A})=P^{F}(\Delta)P_{A}.$
It follows from Lemma 2.1 immediately:
Lemma 2.2. Let $A,B\in S(H)$ and $F\in S(H)$ be an upper bound of $A$ and $B$
with respect to $\preceq$. Then for any two Borel subsets $\Delta_{1}$ and
$\Delta_{2}$ of $\mathbb{R}$, if $\Delta_{1}\cap\Delta_{2}=\emptyset$,
$0\notin\Delta_{1}$, $0\notin\Delta_{2}$, we have
$P^{A}(\Delta_{1})P^{B}({\Delta_{2}})=P^{F}(\Delta_{1})P_{A}P^{F}(\Delta_{2})P_{B}=P_{A}P^{F}(\Delta_{1})P^{F}(\Delta_{2})P_{B}=\theta.$
Lemma 2.3. Let $A,B\in S(H)$ and have the following property: For each pair
$\Delta_{1},\Delta_{2}\in{\cal B}(\mathbb{R})$, whenever
$\Delta_{1}\cap\Delta_{2}=\emptyset$ and $0\not\in\Delta_{1}$,
$0\not\in\Delta_{2}$, we have $P^{A}(\Delta_{1})P^{B}({\Delta_{2}})=\theta$,
then the following mapping $E:{\cal B}(\mathbb{R})\rightarrow P(H)$ defines a
spectral measure:
$E(\Delta)=\left\\{\begin{array}[]{ccc}P^{A}(\Delta)\vee
P^{B}(\Delta),&&0\not\in\Delta\\\ N(A)\wedge N(B)=I-P_{A}\vee
P_{B},&&\Delta=\\{0\\}\\\ P^{A}(\Delta\backslash\\{0\\})\vee
P^{B}(\Delta\backslash\\{0\\})+N(A)\wedge N(B).&&0\in\Delta\\\
\end{array}\right.$
Proof. First, we show that for each $\Delta\in{\cal B}(\mathbb{R})$,
$E(\Delta)\in P(H)$. It is sufficient to check the case of $0\in\Delta$. Since
$P^{A}(\Delta\backslash\\{0\\})\vee P^{B}(\Delta\backslash\\{0\\})\leq
P^{A}(R\backslash\\{0\\})\vee P^{B}(R\backslash\\{0\\})=P_{A}\vee P_{B}$, so
it follows from Lemma 1.3 that $P^{A}(\Delta\backslash\\{0\\})\vee
P^{B}(\Delta\backslash\\{0\\})+N(A)\wedge N(B)\in P(H)$ and the conclusion is
hold.
Second, we have
$E(\emptyset)=P^{A}(\emptyset)\vee P^{B}(\emptyset)=\theta\vee\theta=\theta,$
$E(R)=P^{A}(R\backslash\\{0\\})\vee P^{B}(R\backslash\\{0\\})+N(A)\wedge N(B)$
$=P_{A}\vee P_{B}+N(A)\wedge N(B)=I.$
Third, if $\Delta_{1}\cap\Delta_{2}=\emptyset$, there are two cases:
(i). $0$ doesn’t belong to any one of $\Delta_{1}$ and $\Delta_{2}$. It
follows from the definition of $E$ that
$E(\Delta_{1})E(\Delta_{2})=(P^{A}(\Delta_{1})\vee
P^{B}(\Delta_{1}))(P^{A}(\Delta_{2})\vee P^{B}(\Delta_{2})).$ Note that
$P^{B}(\Delta_{1})P^{A}(\Delta_{2})=\theta$ by the conditions of the lemma and
$P^{B}(\Delta_{1})P^{B}(\Delta_{2})=\theta$, we have
$P^{B}(\Delta_{1})(P^{A}(\Delta_{2})\vee P^{B}(\Delta_{2}))=\theta$,
similarly, we have also $P^{A}(\Delta_{1})(P^{A}(\Delta_{2})\vee
P^{B}(\Delta_{2}))=\theta$, thus,
$E(\Delta_{1})E(\Delta_{2})=\theta.$
Furthermore, we have
$\begin{array}[]{rcl}E(\Delta_{1}\cup\Delta_{2})&=&P^{A}(\Delta_{1}\cup\Delta_{2})\vee
P^{B}(\Delta_{1}\cup\Delta_{2})\\\ &=&P^{A}(\Delta_{1})\vee
P^{A}(\Delta_{2})\vee P^{B}(\Delta_{1})\vee P^{B}(\Delta_{2})\\\
&=&(P^{A}(\Delta_{1})\vee P^{B}(\Delta_{1}))\vee(P^{A}(\Delta_{2})\vee
P^{B}(\Delta_{2}))\\\ &=&(P^{A}(\Delta_{1})\vee
P^{B}(\Delta_{1}))+(P^{A}(\Delta_{2})\vee P^{B}(\Delta_{2}))\\\
&=&E(\Delta_{1})+E(\Delta_{2}).\end{array}$
That is, in this case, we proved that
$E(\Delta_{1})E(\Delta_{2})=\theta,$
$E(\Delta_{1}\cup\Delta_{2})=E(\Delta_{1})+E(\Delta_{2}).$
(ii). $0$ belongs to one of $\Delta_{1}$ and $\Delta_{2}$. Without of losing
generality, we suppose that $0\in\Delta_{1}$, since
$\Delta_{1}\cap\Delta_{2}=\emptyset$, so $0\notin\Delta_{2}$, thus we have
$\begin{array}[]{rcl}E(\Delta_{1})E(\Delta_{2})&=&(P^{A}(\Delta_{1}\backslash\\{0\\})\vee
P^{B}(\Delta_{1}\backslash\\{0\\})+N(B)\wedge N(A))(P^{A}(\Delta_{2})\vee
P^{B}(\Delta_{2}))\\\ &=&(P^{A}(\Delta_{1}\backslash\\{0\\})\vee
P^{B}(\Delta_{1}\backslash\\{0\\}))(P^{A}(\Delta_{2})\vee
P^{B}(\Delta_{2}))=\theta,\\\ \end{array}$
$\begin{array}[]{rcl}E(\Delta_{1}\cup\Delta_{2})&=&P^{A}(\Delta_{1}\backslash\\{0\\}\cup\Delta_{2})\vee
P^{B}(\Delta_{1}\backslash\\{0\\}\cup\Delta_{2})+(N(B)\wedge N(A))\\\
&=&(P^{A}(\Delta_{1}\backslash\\{0\\})\vee
P^{B}(\Delta_{1}\backslash\\{0\\})+(N(B)\wedge N(A)))+(P^{A}(\Delta_{2})\vee
P^{B}(\Delta_{2}))\\\ &=&(P^{A}(\Delta_{1}\backslash\\{0\\})\vee
P^{B}(\Delta_{1}\backslash\\{0\\})+(N(A)\wedge N(B)))+(P^{A}(\Delta_{2})\vee
P^{B}(\Delta_{2}))\\\ &=&E(\Delta_{1})+E(\Delta_{2}).\\\ \end{array}$
Thus, it follows from (i) and (ii) that whenever
$\Delta_{1}\cap\Delta_{2}=\emptyset$, we have
$E(\Delta_{1})E(\Delta_{2})=\theta,$
$E(\Delta_{1}\cup\Delta_{2})=E(\Delta_{1})+E(\Delta_{2}).$
Final, if $\\{\Delta_{n}\\}_{n=1}^{\infty}$ is a sequence of pairwise disjoint
Borel sets in ${\cal B}(\mathbb{R})$, then it is easy to prove that
$E(\bigcup\limits_{n=1}^{\infty}\Delta_{n})=\sum\limits_{n=1}^{\infty}E(\Delta_{n}).$
Thus, the lemma is proved.
## 3 Main results and proofs
Theorem 3.1. Let $A,B\in S(H)$ and have the following property: For each pair
$\Delta_{1},\Delta_{2}\in{\cal B}(\mathbb{R})$, whenever
$\Delta_{1}\cap\Delta_{2}=\emptyset$ and $0\not\in\Delta_{1}$,
$0\not\in\Delta_{2}$, we have $P^{A}(\Delta_{1})P^{B}({\Delta_{2}})=\theta$.
Then the supremum $A\vee B$ of $A$ and $B$ exists with respect to the logic
order $\preceq$.
Proof. By Lemma 2.3, $E(\cdot)$ is a spectral measure and so it can generate a
bounded quantum observable $K$ and $K$ can be represented by
$K=\int\limits_{-M}^{M}\lambda dE_{\lambda}$, where
$\\{E_{\lambda}\\}=E(-\infty,\lambda]$, $\lambda\in\mathbb{R}$ and
$M=\max(\|A\|,\|B\|)$. Moreover, for each $\Delta\in{\cal B}(\mathbb{R})$,
$P^{K}(\Delta)=E(\Delta)$ ([7]). We confirm that $K$ is the supremum $A\vee B$
of $A$ and $B$ with respect to $\preceq$. In fact, for each $\Delta\in{\cal
B}(\mathbb{R})$ with $0\notin\Delta$, by the definition of $E$ we knew that
$P^{K}(\Delta)=E(\Delta)=P^{A}(\Delta)\vee P^{B}(\Delta)\geq P^{A}(\Delta)$,
$P^{K}(\Delta)=E(\Delta)=P^{A}(\Delta)\vee P^{B}(\Delta)\geq P^{B}(\Delta)$.
So it following from the equivalent properties of $\preceq$ that $A\preceq K$,
$B\preceq K$ ([2]). If $K^{\prime}$ is another upper bound of $A$ and $B$ with
respect to $\preceq$, then for each $\Delta\in{\cal B}(\mathbb{R})$ with
$0\notin\Delta$, we have $P^{A}(\Delta)\leq P^{K^{\prime}}(\Delta)$,
$P^{B}(\Delta)\leq P^{K^{\prime}}(\Delta)$ ([2]), so $P^{A}(\Delta)\vee
P^{B}(\Delta)=E(\Delta)=P^{K}(\Delta)\leq P^{K^{\prime}}(\Delta)$, thus we
have $K\preceq K^{\prime}$ and $K$ is the supremum of $A$ and $B$ with respect
to $\preceq$ is proved.
It follows from Lemma 2.2 and theorem 3.1 that we have the following theorem
immediately:
Theorem 3.2. Let $A,B\in S(H)$. Then the supremum $A\vee B$ of $A$ and $B$
exists with respect to the logic order $\preceq$ iff for each pair
$\Delta_{1},\Delta_{2}\in{\cal B}(\mathbb{R})$, whenever
$\Delta_{1}\cap\Delta_{2}=\emptyset$ and $0\not\in\Delta_{1}$,
$0\not\in\Delta_{2}$, we have $P^{A}(\Delta_{1})P^{B}({\Delta_{2}})=\theta$.
Moreover, in this case, we have the following nice representation:
$A\vee B=\int\limits_{-M}^{M}\lambda dE_{\lambda},$
where $\\{E_{\lambda}\\}=E(-\infty,\lambda]$, $\lambda\in\mathbb{R}$ and
$M=\max(\|A\|,\|B\|)$.
Remark 3.3. Let $A,B\in S(H)$. Note that for each $\Delta\in{\cal
B}(\mathbb{R})$, $P^{A}(\Delta)$ is interpreted as the quantum event that the
quantum observable $A$ has a value in $\Delta$ ([2]), and the conditions:
$\Delta_{1}\cap\Delta_{2}=\emptyset$, $0\not\in\Delta_{1}$,
$0\not\in\Delta_{2}$ must have $P^{A}(\Delta_{1})P^{B}({\Delta_{2}})=\theta$
told us that the quantum events $P^{A}(\Delta_{1})$ and $P^{B}(\Delta_{2})$
can not happened at the same time, so, the physical meanings of the supremum
$A\vee B$ exists with respect to $\preceq$ iff for each pair
$\Delta_{1},\Delta_{2}\in{\cal B}(\mathbb{R})$, whenever
$\Delta_{1}\cap\Delta_{2}=\emptyset$ and $0\not\in\Delta_{1}$,
$0\not\in\Delta_{2}$, the quantum observable $A$ takes value in $\Delta_{1}$
and the quantum observable $B$ takes value in $\Delta_{2}$ can not happen at
the same time.
References
[1]. Kadison, R. Order properties of bounded self-adjoint operators. _Proc.
Amer. Math. Soc_. 34: 505-510, (1951)
[2]. Gudder S. An Order for quantum observables. _Math Slovaca_. 56: 573-589,
(2006)
[3]. Pulmannova S, Vincekova E. Remarks on the order for quantum observables.
_Math Slovaca_. 57: 589-600, (2007)
[4]. Liu Weihua, Wu Junde. A representation theorem of infimum of bounded
quantum observables. _J Math Physi_. 49: 073521-073525, (2008)
[5]. Du Hongke, Dou Yanni. A spectral representation of infimum of self-
adjoint operators in the logic order. Acta Math. Sinica. To appear
[6]. Xu Xiaoming, Du Hongke, Fang Xiaochun. An explicit expression of supremum
of bounded quantum observables. _J Math Physi_. 50: 033502-033509, (2009)
[7]. Kadison. R. V., Ringrose J. R. Fundamentals of the Theory of Operator
Algebra. Springer-Verlag, New York, (1983)
|
arxiv-papers
| 2009-04-25T00:54:38 |
2024-09-04T02:49:02.161803
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Liu Weihua, Wu Junde",
"submitter": "Junde Wu",
"url": "https://arxiv.org/abs/0904.3954"
}
|
0904.4094
|
# On the Upper Bounds of MDS Codes
Jiansheng Yang111Supported by Shanghai Leading Academic Discipline Project
Project Number S30104. Email: yjsyjs@staff.shu.edu.cn , Yunying Zhang
Department of Mathematics, Shanghai University
Shanghai 200444, China
###### Abstract
Let $M_{q}(k)$ be the maximum length of MDS codes with parameters $q,k$. In
this paper, the properties of $M_{q}(k)$ are studied, and some new upper
bounds of $M_{q}(k)$ are obtained. Especially we obtain that $M_{q}(q-1)\leq
q+2(q\equiv 4(mod~{}6)),~{}M_{q}(q-2)\leq q+1(q\equiv
4(mod~{}6)),~{}M_{q}(k)\leq q+k-3~{}(q=36(5s+1),~{}s\in N$ and $k=6,7).$
Keywords: MDS codes; Hamming distance; codes equivalence; weight distribution
## 1 Introduction
Let $C$ be an $(n,q^{k},d)$ code, if $d=n-k+1$, then $C$ is called a maximum
distance separable (MDS) code. MDS codes are at the heart of combinatorics and
finite geometries. In their book [9] MacWilliams and Sloane describe MDS codes
as one of the most fascinating chapters in all of coding theory. These codes
can be linear or non-linear. Very little is known about non-linear
$(n,q^{k},n-k+1)$ MDS codes. R.H.Bruck , H.J.Ryser, R.Silverman and A.A.Bruen
had proved some results on MDS codes [4,5,10] early. Recently, T.L.Alderson
studies MDS codes extension. And he has obtained some important
results[1,2,3].
In this paper, we assume that $A$=$\\{0,1,2,\cdots,q-1\\}$ is a additive
group(not necessary cyclic group). Denote the maximum number $n$ of an
$(n,q^{k},n-k+1)$ MDS code over $A$ by $M_{q}(k)$. If $A$ is a field, then
denote the maximum number $n$ of a linear $(n,q^{k},n-k+1)$ MDS code over $A$
by $m_{q}(k)$. The Main Conjecture of $m_{q}(k)$ is the following.
$m_{q}(k)=\left\\{{\begin{array}[]{*{20}c}{q+2}&{for\;k=3\;and\;k=q-1\;both\;q\;even,}\\\
{q+1}&{in\;all\;other\;cases.}\\\ \end{array}}\right.$
The Main Conjecture has been proved in some cases, [7] gave us a good
summarize. For $M_{q}(k)$, it is well known that $M_{q}(k)\leq q+k-1$. In [5],
A.A.Bruen and R.Silverman proved that:
Theorem 1.1 [5] (1) If $C$ is a $(q+k-1;k)$-MDS code with $k\geq 3$ and $q>2$,
then $4$ divides $q$.
(2) If $C$ is a $(q+k-1;k)$-MDS code with $k>3$ and $q>2$, then $36$ divides
$q$.
In [2], T.L.Alderson proved that:
Theorem 1.2 [2] (1) If 36 does not divide $q$ and $k\geq 4$, then a
$q-ary~{}(n,k)$-MDS code satisfies $n\leq q+k-3$.
(2) If $q>2$ and $q\equiv 2~{}mod~{}4$ then no q-ary $(q+1;3)$-MDS codes
exist.
In [11], Wang proved that
Theorem 1.3 [11] $M_{q}(q-1)\leq q+1$ for $q$ is odd.
In this paper, we use the generalized weight enumerator(see below) and
combinatorial methods to study $M_{q}(k)$. Some new upper bounds of $M_{q}(k)$
are obtained. Especially we obtain that $M_{q}(q-1)\leq q+2(q\equiv
4(mod~{}6)),~{}M_{q}(q-2)\leq q+1(q\equiv 4(mod~{}6)),~{}M_{q}(k)\leq
q+k-3~{}(q=36(5s+1),~{}s\in N$ and $k=6,7).$
For the convenient, we introduce some notations and results as following.
Let $C$ and $D$ be two codes of length $n$ over $A$. If there exist $n$
permutations $\pi_{1},\cdots,\pi_{n}$ of the $q$ elements and a permutation
$\sigma$ of the $n$ coordinate positions such that $(u_{1},\cdots,u_{n})\in C$
iff $(\pi_{1}(u_{\sigma{(1)}}),\cdots,\pi_{n}(u_{\sigma{(n)}}))\in D$, then we
call $C$ is equivalent to $D$. If $C$ is equivalent to $D$, then $C$ and $D$
have the same Hamming distance. It is clear, if $c_{0}\in A^{n}$,
$D=c_{0}+C=\\{c_{0}+\alpha|\alpha\in C\\}$ is equivalent to $C$. Then we may
always assume that code $C$ contains the zero element
$\textbf{0}=(0,0,\cdots,0,0)$.
The generalized weight enumerator which is introduced by M.El-Khamy and
R.J.McEliece[6] is called the partition weight enumerator(PWE) . Suppose the
coordinate set $N=\\{1,2,\cdots,n\\}$ is partitioned into $p$ disjoint subsets
$N_{1},\cdots,N_{p}$, with $|N_{i}|=n_{i}$ , for $i=1,\cdots,p$. Denoting this
partition by $\mathcal{T}$, the $\mathcal{T}$-weight profile of an $v\in
A^{n}$ is defined as
$\mathcal{W}_{\mathcal{T}}(v)=(\omega_{1},\cdots,\omega_{p})$, where
$\omega_{i}$ is the Hamming weight of $v$ restricted to $N_{i}$. Given a code
$C$ of length $n$, and an $(n_{1},\cdots,n_{p})$ partition $\mathcal{T}$ of
the $n$ coordinates of $C$, the $\mathcal{T}$-weight enumerator of $C$ is
defined as following.
$A^{\mathcal{T}}(\omega_{1},\cdots,\omega_{p})=|\\{c\in
C:\mathcal{W}_{\mathcal{T}}(c)=(\omega_{1},\cdots,\omega_{p})\\}|.$
Theorem 1.4 [6] For an $(n,q^{k},d)$ MDS code $C$ which contains the zero
element, the $p$-partition weight enumerator is given by
$A^{\mathcal{T}}(\omega_{1},\cdots,\omega_{p})=E(\omega)\frac{{n_{1}\choose\omega_{1}}{n_{2}\choose\omega_{2}}\cdots{n_{p}\choose\omega_{p}}}{{n\choose\omega}}$
where $\omega=\sum_{i=1}^{n}\omega_{i}$, $E(\omega)=|\\{c\in
C:\mathcal{W}(c)=\omega\\}|$ and $\mathcal{W}(c)$ is the Hamming weight of
$C$.
Remark: In the proof, [6] assume that $A$ is a field, however, the proof is
true for any $A$(It only need the MDS codes have the zero element). Therefore,
the formula holds for non-linear MDS codes which contain the zero element.
For an $(n,q^{k},d)$ MDS code over $A$, the weight distribution is known as
$E(\omega)=(q-1){n\choose\omega}\sum_{j=0}^{\omega-d}(-1)^{j}{\omega-1\choose
j}q^{\omega-d-j}$
where $\omega\geq d$ [9], and we can know that the formula holds for MDS
codes(which contain the zero element) not only for linear MDS codes[8].
For any $\alpha=(a_{1},\cdots,a_{n})\in A^{n}$, define the support of $\alpha$
by $Supp\alpha=\\{i|a_{i}\neq 0,~{}1\leq i\leq n\\}$, and
$\overline{Supp}\alpha=\\{j|a_{j}=0,~{}1\leq j\leq n\\}$.
## 2 New Upper Bounds for MDS Codes
Theorem 2.1 If $q\equiv 4(mod~{}6)$, then $M_{q}(q-1)\leq q+2$.
Proof: Suppose $C$ is an $(q+3,q^{q-1},5)(q~{}is~{}even)$ MDS code which
contains the zero element. The partition $\mathcal{T}$ is given as following.
$\mathcal{T}=\mathcal{T}_{1}\cup\mathcal{T}_{2},~{}\mathcal{T}_{1}=\\{1,2,3\\},~{}\mathcal{T}_{2}=\\{4,5,\cdots,q+3\\}.$
$n_{1}=|\mathcal{T}_{1}|=3,~{}n_{2}=|\mathcal{T}_{2}|=q,~{}\omega_{1}=2,~{}\omega_{2}=3.$
From Theorem 1.4, we have
$A^{\mathcal{T}}(2,3)=E(5)\frac{{3\choose 2}{q\choose 3}}{{q+3\choose
5}}=\frac{3q(q-1)^{2}(q-2)}{6}.$
where $E(5)=(q-1){q+3\choose 5}$. For $(x,y)$, there are altogether
$(q-1)^{2}$ pairs $(x,y)$ with $x,y\in S$ where $S=\\{1,2,\cdots,q-1\\}$. Thus
there exists $(a,b)\in S$ such that
$|C_{a,b,0}|\geq\frac{3q(q-1)^{2}(q-2)}{6\times 3(q-1)^{2}}=\frac{q(q-2)}{6}.$
where
$C_{a,b,0}=\\{(a_{1},a_{2},a_{3},\cdots,a_{q+3})\in
C|a_{1}=a,a_{2}=b,a_{3}=0,a_{k}\in A,k=4,5,\cdots,q+3\\}.$
Since $C$ is an MDS code, w.l.g. we may assume $a=1,b=1$. Then
$C_{1,1,0}=\\{(1,1,0,a_{4},\cdots,a_{q+3})\in C|a_{k}\in
A,k=4,5,\cdots,q+3\\}.$
Let
$C_{i}=C_{1,1,0,i}=\\{(1,1,0,a_{4},\cdots,a_{q+3})\in C|a_{i}\neq
0\\}~{}(i\in\\{4,5,\cdots,q+3\\}).$
We will prove that $|C_{1,1,0}|=\frac{q(q-2)}{6}$ when $q$ is even. If this is
not true, we have $|C_{1,1,0}|>\frac{q(q-2)}{6}$, then there exists $i$,
w.l.g. assume $i$=4, such that
$|C_{4}|\geq\frac{|C_{1,1,0}|{3\choose 1}}{{q\choose
1}}>\frac{3q(q-2)}{6q}=\frac{q-2}{2}.$
Assume
$\alpha=(1,1,0,a_{4},\cdots,a_{q+3}),~{}\beta=(1,1,0,b_{4},\cdots,b_{q+3})\in
C_{4}$, we have $a_{4}\neq 0,~{}b_{4}\neq 0$. If $i\in Supp\alpha\cap
Supp\beta~{}(5\leq i\leq q+3)$, then we have $d(\alpha,\beta)\leq 4$, a
contradiction. Thus we have $Supp\alpha\cap Supp\beta=\\{1,2,4\\}$. Let
$\alpha_{1},\alpha_{2},\cdots,\alpha_{t}\in C_{4}$. Since $Supp\alpha_{i}\cap
Supp\alpha_{j}=\\{1,2,4\\}$ for all $i\neq j$ and $\omega(\alpha)=5$, we have
$\cup_{i=1}^{t}|Supp\alpha_{i}|=2t+3$. This implies $2t+3\leq q+2$, i.e.
$t\leq\frac{q-1}{2}$. Since $q$ is even, we have $t\leq\frac{q-2}{2}$.
Hence$|C_{4}|\leq\frac{q-2}{2}~{}(q~{}is~{}even)$, a contradiction.
By this, we have
$|C_{1,1,0}|=\frac{q(q-2)}{6}~{}(q~{}is~{}even).$
Thus $\frac{q(q-2)}{6}$ must be an integer, however, if $q\equiv 4(mod~{}6)$,
$\frac{q(q-2)}{6}$ is not an integer. Therefore, if $q\equiv 4(mod~{}6)$, then
$M_{q}(q-1)\leq q+2$. $\Box$
Theorem 2.2 If $q$ is even and $(l+2)!$ does not divide
$(q+l-1)\cdots(q+1)q(q-2)$ where $l\geq 1$, then $M_{q}(q-2)\leq q+l$.
Proof: Suppose $C$ is an $(q+l+1,q^{q-2},l+4)(q~{}is~{}even)$ MDS code which
contains the zero element. The partition $\mathcal{T}$ is given as following.
$\mathcal{T}=\mathcal{T}_{1}\cup\mathcal{T}_{2},~{}\mathcal{T}_{1}=\\{1,2\\},~{}\mathcal{T}_{2}=\\{3,4,\cdots,q+l+1\\}.$
$n_{1}=|\mathcal{T}_{1}|=2,~{}n_{2}=|\mathcal{T}_{2}|=q+l-1,~{}\omega_{1}=2,~{}\omega_{2}=l+2.$
From Theorem 1.4, we have
$A^{\mathcal{T}}(2,l+2)=E(l+4)\frac{{2\choose 2}{q+l-1\choose
l+2}}{{q+l+1\choose l+4}}={q+l-1\choose l+2}(q-1).$
where $E(l+4)=(q-1){q+l+1\choose l+4}$. For $(x,y)$, there are altogether
$(q-1)^{2}$ pairs $(x,y)$ with $x,y\in S$ where $S=\\{1,2,\cdots,q-1\\}$. Thus
there exists $(a,b)\in S$ such that
$|C_{a,b}|\geq\frac{{q+l-1\choose l+2}(q-1)}{(q-1)^{2}}=\frac{{q+l-1\choose
l+2}}{q-1}.$
where
$C_{a,b}=\\{(a_{1},a_{2},a_{3},\cdots,a_{q+l+1})|a_{1}=a,a_{2}=b,a_{k}\in
A,k=3,4,\cdots,q+l+1\\}.$
Since $C$ is an MDS code, we may assume $a=1,b=1$. Then
$C_{1,1}=\\{(1,1,a_{3},\cdots,a_{q+l+1})|a_{k}\in A,k=3,4,\cdots,q+l+1\\}.$
Let
$B_{\underbrace{i,j,\cdots,k}_{l}}=C_{1,1,\underbrace{i,j,\cdots,k}_{l}}=\\{(1,1,a_{3},\cdots,a_{q+l+1})|a_{k}\in
A,k=3,4,\cdots,q+l+1~{}and~{}a_{i},a_{j},\cdots,a_{k}\neq 0\\},$ where
$i,j,\cdots,k$ are the $l$ distinct numbers of $\\{3,4,\cdots,q+l+1\\}.$
We claim that $|C_{1,1}|=\frac{{q+l-1\choose l+2}}{q-1}$ when $q$ is even. If
this is not true, since$|C_{1,1}|\geq\frac{{q+l-1\choose l+2}}{q-1}$, we
have$|C_{1,1}|>\frac{{q+l-1\choose l+2}}{q-1}$ and there exist $i,j,\cdots,k$,
w.l.g. assume the $l$ numbers are $3,4,\cdots,l+2$, such that
$|B_{3,4,\cdots,l+2}|\geq\frac{|C_{1,1}|{l+2\choose l}}{{q+l-1\choose
l}}>\frac{q-2}{2}.$
Assume
$\alpha=(1,1,a_{3},\cdots,a_{q+l+1}),~{}\beta=(1,1,b_{3},\cdots,b_{q+l+1})\in
B_{3,4,\cdots,l+2}$, we have $a_{r}\neq 0,~{}b_{s}\neq 0,3\leq r,s\leq l+2$.
If $i\in Supp\alpha\cap Supp\beta~{}(l+3\leq i\leq q+l+1)$, since
$|Supp\alpha|=|Supp\beta|=l+4$, then we have $d(\alpha,\beta)\leq l+3$, a
contradiction. Thus we have $Supp\alpha\cap Supp\beta=\\{1,2,\cdots,l+2\\}$.
Let $\alpha_{1},\alpha_{2},\cdots,\alpha_{t}\in B_{3,4,\cdots,l+2}$. Since
$Supp\alpha_{i}\cap Supp\alpha_{j}=\\{1,2,\cdots,l+2\\}$ for all $i\neq j$ and
$\omega(\alpha)=l+4$, we have $\cup_{i=1}^{t}|Supp\alpha_{i}|=2t+l+2$. This
implies $2t+l+2\leq q+l+1$, i.e. $t\leq\frac{q-1}{2}$. Since $q$ is even, we
have $t\leq\frac{q-2}{2}$.
Hence$|B_{3,4,\cdots,l+2}|\leq\frac{q-2}{2}~{}(q~{}is~{}even)$, a
contradiction.
By this, we have
$|C_{1,1}|=\frac{{q+l-1\choose
l+2}}{q-1}=\frac{(q+l-1)\cdots(q+1)q(q-2)}{(l+2)!}~{}(q~{}is~{}even).$
Thus $\frac{(q+l-1)\cdots(q+1)q(q-2)}{(l+2)!}$ must be an integer. Therefore,
if $q$ is even and $(l+2)!$ does not divide $(q+l-1)\cdots(q+1)q(q-2)$ where
$l\geq 1$, then $M_{q}(q-2)\leq q+l$. $\Box$
By calculating, we can get the following.
Corollary 2.2.1 $M_{q}(q-2)\leq q+1~{}(q\equiv 4(mod~{}6)).$
Corollary 2.2.2 $M_{q}(q-2)\leq q+3~{}(q\equiv 6~{}or~{}26(mod~{}30)).$
Corollary 2.2.3 $M_{q}(q-2)\leq q+5~{}(q\equiv 8~{}or~{}36(mod~{}42)).$
Theorem 2.3 If $q$ is even and $(k-1)!$ does not divide
$(q+k-4)\cdots(q+1)q(q-2)$ where $k\geq 4$, then $M_{q}(k)\leq q+k-3$.
Proof: Suppose $C$ is an $(q+k-2,q^{k},q-1)(q~{}is~{}even)$ MDS code which
contains the zero element. The partition $\mathcal{T}$ is given as following.
$\mathcal{T}=\mathcal{T}_{1}\cup\mathcal{T}_{2},~{}\mathcal{T}_{1}=\\{1,2\\},~{}\mathcal{T}_{2}=\\{3,4,\cdots,q+k-2\\}.$
$n_{1}=|\mathcal{T}_{1}|=2,~{}n_{2}=|\mathcal{T}_{2}|=q+k-4,~{}\omega_{1}=2,~{}\omega_{2}=q-3.$
From Theorem 1.4, we have
$A^{\mathcal{T}}(2,q-3)=E(q-1)\frac{{2\choose 2}{q+k-4\choose
q-3}}{{q+k-2\choose q-1}}={q+k-4\choose q-3}(q-1).$
where $E(q-1)=(q-1){q+k-2\choose q-1}$. For $(x,y)$, there are altogether
$(q-1)^{2}$ pairs $(x,y)$ with $x,y\in S$ where $S=\\{1,2,\cdots,q-1\\}$. Thus
there exists $(a,b)\in S$ such that
$|C_{a,b}|\geq\frac{{q+k-4\choose q-3}(q-1)}{(q-1)^{2}}=\frac{{q+k-4\choose
q-3}}{q-1}.$ (1)
where
$C_{a,b}=\\{(a_{1},a_{2},a_{3},\cdots,a_{q+k-2})|a_{1}=a,a_{2}=b,a_{m}\in
A,m=3,4,\cdots,q+k-2\\}.$
Since $C$ is an MDS code, we may assume $a=1,b=1$. Then
$C_{1,1}=\\{(1,1,a_{3},\cdots,a_{q+k-2})|a_{m}\in A,m=3,4,\cdots,q+k-2\\}.$
Let
$B_{\underbrace{i,j,\cdots,r}_{k-3}}=C_{1,1,\underbrace{i,j,\cdots,r}_{k-3}}=\\{(1,1,a_{3},\cdots,a_{q+k-2})|a_{m}\in
A,~{}m=3,4,\cdots,q+k-2~{}and~{}a_{i}=a_{j}=\cdots=a_{r}=0\\},$ where
$i,j,\cdots,r$ are the $k-3$ distinct numbers of $\\{3,4,\cdots,q+k-2\\}.$
We will prove that $|C_{1,1}|=\frac{{q+k-4\choose q-3}}{q-1}$ when $q$ is
even. If this is not true, we have $|C_{1,1}|>\frac{{q+k-4\choose q-3}}{q-1}$
and there exist $i,j,\cdots,r$, w.l.g. assume the $k-3$ numbers are
$3,4,\cdots,k-1$, such that
$|B_{3,4,\cdots,k-1}|\geq\frac{|C_{1,1}|{k-1\choose k-3}}{{q+k-4\choose
k-3}}>\frac{q-2}{2}.$
Assume
$\alpha=(1,1,a_{3},\cdots,a_{q+k-2}),~{}\beta=(1,1,b_{3},\cdots,b_{q+k-2})\in
B_{3,4,\cdots,k-1}$, we have $a_{r}=0,~{}b_{s}=0,3\leq r,s\leq k-1$. If
$i\in\overline{Supp}\alpha\cap\overline{Supp}\beta~{}(k\leq i\leq q+k-2)$,
then we have $d(\alpha,\beta)\leq q-2$, a contradiction. Thus we have
$\overline{Supp}\alpha\cap\overline{Supp}\beta=\\{3,4,\cdots,k-1\\}$. Let
$\alpha_{1},\alpha_{2},\cdots,\alpha_{t}\in B_{3,4,\cdots,k-1}$. Since
$\overline{Supp}\alpha_{i}\cap\overline{Supp}\alpha_{j}=\\{3,4,\cdots,k-1\\}$
for all $i\neq j$ and $\omega(\alpha)=q-1$, we have
$\cup_{i=1}^{t}|Supp\alpha_{i}|=2t+k-3$. This implies $2t+k-3\leq q+k-4$, i.e.
$t\leq\frac{q-1}{2}$. Since $q$ is even, we have $t\leq\frac{q-2}{2}$.
Hence$|B_{3,4,\cdots,k-1}|\leq\frac{q-2}{2}~{}(q~{}is~{}even)$, a
contradiction.
By this,we have
$|C_{1,1}|=\frac{{q+k-4\choose
q-3}}{q-1}=\frac{(q+k-4)\cdots(q+1)q(q-2)}{(k-1)!}~{}(q~{}is~{}even)$
Thus $\frac{(q+k-4)\cdots(q+1)q(q-2)}{(k-1)!}$ must be an integer. Therefore,
if $q$ is even and $(k-1)!$ does not divide $(q+k-4)\cdots(q+1)q(q-2)$ where
$k\geq 4$, then $M_{q}(k)\leq q+k-3$. $\Box$
By the Theorem 2.3, we have the following.
Corollary 2.3.1 $M_{q}(k)\leq q+k-3~{}(q=36(5s+1),~{}s\in N$ and $k=6,7).$
## 3 Conclusion
In this paper, we use the generalized weight enumerator and combinatorial
methods to study $M_{q}(k)$ which denote the maximum number $n$ of an
$(n,q^{k},n-k+1)$ MDS code. Compared to Theorem 1.1, Theorem 1.2 and Theorem
1.3, we obtain some new upper bounds of $M_{q}(k)$. Especially we obtain that
$M_{q}(q-1)\leq q+2(q\equiv 4(mod~{}6)),~{}M_{q}(q-2)\leq q+1(q\equiv
4(mod~{}6)),~{}M_{q}(k)\leq q+k-3~{}(q=36(5s+1),~{}s\in N$ and $k=6,7).$
## References
* [1] T.L. Alderson, “On MDS codes and Bruen-Silverman codes,” Ph.D. Thesis, University of Western Ontario, 2002.
* [2] T.L.Alderson, “Extending MDS codes,” Annals of Combinatorics, vol. 9, pp. 125-135, 2005.
* [3] T. L. Alderson and A. A. Bruen, “Codes from cubic curves and their extensions,” the electronic journal of combinatorics, vol. 15, 2008.
* [4] R.H.Bruck and H.J.Ryser, “The nonexistence of certain finite projective planes,” Canad J.Math, vol. 1, pp. 88-93, 1949\.
* [5] A.A.Bruen and R.Silverman, “On the nonexistence of certain MDS codes and projective planes,” Math.Z, vol. 183, pp. 171-175, 1983.
* [6] M.El-Khamy and R.J.McEliece, “The Partition Weight Enumerator of MDS Codes and its Applications,” Information Theory, vol. 9, pp. 926-930, 2005.
* [7] J. W. P. Hirschfeld, “The Main Conjecture for MDS Codes,” Lecture Notes In Computer Science; Proceedings of the 5th IMA Conference on Cryptography and Coding, Springer-Verlag London, UK, vol. 1025, pp. 44-52, 1995\.
* [8] Ludo M.G.M. and Tolhuizen, “On Maximum Distance Separable codes over alphabets of arbitrary size,” Information Theory, vol. 7, pp. 431, 1994.
* [9] F.J.MacWilliams and N.J.A.Slane, “Theory of Error-Correcting Codes,” North-Holland,Amsterdam, pp. 317-329, 1977.
* [10] R.Silverman, “A metrization for power-sets with applications to combinatorial analysis,” Canad.J.Math, vol. 12, pp. 158-176, 1960.
* [11] D.X.Wang, “The MDS codes of small dimension and small weight,” Ms.Thesis, University of Shanghai, 2008.
|
arxiv-papers
| 2009-04-27T06:47:15 |
2024-09-04T02:49:02.169001
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiansheng Yang, Yunying Zhang",
"submitter": "Yang Jiansheng",
"url": "https://arxiv.org/abs/0904.4094"
}
|
0904.4143
|
# Maximizing the probability of attaining a target prior to extinction
Debasish Chatterjee Automatic Control Laboratory
ETL I19, ETH Zürich
Physikstrasse 3
8092 Zürich
Switzerland chatterjee@control.ee.ethz.ch http://control.ee.ethz.ch/~chatterd
, Eugenio Cinquemani INRIA-Grenoble - Rhône-Alpes
655 avenue de l’Europe
Montbonnot
38334 Saint Ismier cedex
France Eugenio.Cinquemani@inria.fr http://ibis.inrialpes.fr/article941.html
and John Lygeros Automatic Control Laboratory
ETL I22, ETH Zürich
Physikstrasse 3
8092 Zürich
Switzerland lygeros@control.ee.ethz.ch http://control.ee.ethz.ch/~lygeros
###### Abstract.
We present a dynamic programming-based solution to the problem of maximizing
the probability of attaining a target set before hitting a cemetery set for a
discrete-time Markov control process. Under mild hypotheses we establish that
there exists a deterministic stationary policy that achieves the maximum value
of this probability. We demonstrate how the maximization of this probability
can be computed through the maximization of an expected total reward until the
first hitting time to either the target or the cemetery set. Martingale
characterizations of thrifty, equalizing, and optimal policies in the context
of our problem are also established.
###### Key words and phrases:
dynamic programming, probability maximization, Markov control processes
###### 2000 Mathematics Subject Classification:
Primary: 90C39, 90C40; Secondary: 93E20
This research was partially supported by the Swiss National Science Foundation
under grant 200021-122072.
## 1\. Introduction
There are two basic categories of discrete-time controlled Markov processes
that deal with random temporal horizons. The first is the well-known optimal
stopping problem [Dynkin, 1963], in which the random horizon arises from some
dynamic optimization protocol based on the past history of the process. The
random ‘stopping time’ thus generated is regarded as a decision variable. This
problem arises in, among other areas, stochastic analysis, mathematical
statistics, mathematical finance, and financial engineering; see the
comprehensive monograph [Peskir and Shiryaev, 2006] for details and further
references. The second is relatively less common, and is characterized by the
fact that the random horizon arises as a result of an endogenous event of the
stochastic process, e.g., the process hitting a particular subset of the
state-space, variations in the process paths crossing a certain threshold.
This problem arises in, among others, optimization of target-level criteria
[Dubins and Savage, 1976; Bouakiz and Kebir, 1995], optimal control of
retirement investment funds [Boda et al., 2004], minimization of ruin
probabilities in insurance funds [Schmidli, 2008], ‘satisfaction of needs’
problems in economics [Simon, 1957], risk minimizing stopping problems
[Ohtsubo, 2003], attainability problems under stochastic perturbations
[Digaĭlova and Kurzhanskiĭ, 2004], and optimal control of Markov control
processes up to an exit time [Borkar, 1991].
The problem treated in this article falls under the second category above. In
broad strokes, we consider a discrete-time Markov control process with Borel
state and action spaces. We assume that there is a certain target set located
inside a safe region, the latter being a subset of the state-space. The
problem is to maximize the probability of attaining the target set before
exiting the safe set (or equivalently, hitting the cemetery set or unsafe
region). This ‘reach a good set while avoiding a bad set’ formulation arises
in, e.g., air traffic control, where aircraft try to reach their destination
while avoiding collision with other aircraft or the ground despite uncertain
weather conditions. It also arises in portfolio optimization, where it is
desired to reach a target level of wealth without falling below a certain
baseline capital with high probability. Finally, it forms the core of the
computation of safe sets for hybrid systems where the ‘good’ and the ‘bad’
sets represent states from which a discrete transition into the unsafe set is
possible [Gao et al., 2007; Tomlin et al., 2000]. Special cases of this
problem have been investigated in, e.g., [Watkins and Lygeros, 2003; Prandini
and Hu, 2006] in the context of air traffic applications, [Abate et al., 2008;
Prajna et al., 2007] in the context of probabilistic safety, [Boda et al.,
2004] in the context of maximizing the probability of attaining a preassigned
comfort level of retirement investment funds.
It is clear from the description of our problem in the preceding paragraph
that there are two random times involved, namely, the hitting times of the
target and the cemetery sets. In this article we formulate our problem as the
maximization of an expected total reward accumulated up to the minimum of
these two hitting times. As such, this formulation falls under the broad
framework of optimal control of Markov control processes up to an exit time,
which has a long and rich history. It has mostly been studied as the
minimization of an expected total cost until the first time that the state
enters a given target set, see e.g., [Borkar, 1991, Chapter II], [Hernández-
Lerma and Lasserre, 1999, Chapter 8], and the references therein. In
particular, if a unit cost is incurred as long as the state is outside the
target set, then the problem of minimizing the cost accumulated until the
state enters the target is known variously as the pursuit problem [Eaton and
Zadeh, 1962], transient programming [Whittle, 1983], the first passage problem
[Derman, 1970; Kushner, 1971], the stochastic shortest path problem
[Bertsekas, 2007], and control up to an exit time [Borkar, 1988, 1991; Kesten
and Spitzer, 1975]. Here we exploit certain additional structures of our
problem in the dynamic programming equations that we derive leading to methods
fine-tuned to the particular problem at hand.
Our main results center around the assertion that there exists a deterministic
stationary policy that maximizes the probability of hitting the target set
before the cemetery set. This maximal probability as a function of the initial
state is the optimal value function for our problem. We obtain a Bellman
equation for our problem which is solved by the optimal value function.
Furthermore, we provide martingale-theoretic conditions characterizing
‘thrifty’, ‘equalizing’, and optimal policies via methods derived from [Dubins
and Savage, 1976; Karatzas and Sudderth, 2009]; see also [Zhu and Guo, 2006]
and the references therein for martingale characterization of average
optimality. The principal techniques employed in this article are similar to
the ones in [Chatterjee et al., 2008], where the authors studied optimal
control of a Markov control process up its first entry time to a safe set. In
[Chatterjee et al., 2008] we developed a recovery strategy to enter a given
target set from its exterior while minimizing a discounted cost. The problem
was posed as one of minimizing the sum of a discounted cost-per-stage function
$c$ up to the first entry time $\tau$ to a target set, namely, minimize
$\mathsf{E}^{\pi}_{x}\bigl{[}\sum_{t=0}^{\tau-1}\alpha^{t}c(x_{t},a_{t})\bigr{]}$
over a class of admissible policies $\pi$, where $\alpha\in\;]0,1[$ is a
discount factor. Here we extend this approach to problems with two sets, a
target and a cemetery, and the case of $\alpha=1$.
This article unfolds as follows. The main results are stated in §2. In §2.1 we
define the general setting of the problem, namely, Markov control processes on
Polish spaces, their transition kernels, and the admissible control
strategies. In §2.2 we present our main Theorem (2.10) which guarantees the
existence of a deterministic stationary policy that leads to the maximal
probability of hitting the target set while avoiding the specified dangerous
set, and also provides a Bellman equation that the value function must
satisfy. In §2.3 we look at a martingale characterization of the optimal
control problem; thrifty and equalizing policies are defined in the context of
our problem, and we establish necessary and sufficient conditions for
optimality in terms of thrifty and equalizing policies in Theorem (2.17). We
discuss related reward-per-stage functions and their relationships to our
problem and treat several examples in §3. Proofs of the main results appear in
§4. The article concludes in §5 with a discussion of future work.
## 2\. Main Results
Our main results are stated in this section after some preliminary definitions
and conventions.
### 2.1. Preliminaries
We employ the following standard notations. Let $\mathbb{N}$ denote the
natural numbers $\\{1,2,\ldots\\}$ and $\mathbb{N}_{0}$ denote the nonnegative
integers $\\{0\\}\cup\mathbb{N}$. Let $\boldsymbol{1}_{A}(\cdot)$ be the usual
indicator function of a set $A$, i.e., $\boldsymbol{1}_{A}(\xi)=1$ if $\xi\in
A$ and $0$ otherwise. For real numbers $a$ and $b$ let $a\wedge
b\mathrel{\mathop{:}\\!\\!=}\min\\{a,b\\}$. A function
$f:X\longrightarrow\mathbb{R}$ restricted to $A\subseteq X$ is depicted as
$f|_{A}$.
Given a nonempty Borel set $X$ (i.e., a Borel subset of a Polish space), its
Borel $\sigma$-algebra is denoted by $\mathfrak{B}\\!\left(X\right)$. By
convention, when referring to sets or functions, “measurable” means “Borel-
measurable.” If $X$ and $Y$ are nonempty Borel spaces, a _stochastic kernel_
on $X$ given $Y$ is a function $Q(\cdot|\cdot)$ such that $Q(\cdot|y)$ is a
probability measure on $X$ for each fixed $y\in Y$, and $Q(B|\cdot)$ is a
measurable function on $Y$ for each fixed $B\in\mathfrak{B}\\!\left(X\right)$.
We briefly recall some standard definitions below, see, e.g., [Hernández-Lerma
and Lasserre, 1996] for further details. A _Markov control model_ is a five-
tuple
((2.1)) $\bigl{(}X,A,\\{A(x)\mid x\in X\\},Q,r\bigr{)}$
consisting of a nonempty Borel space $X$ called the _state-space_ , a nonempty
Borel space $A$ called the _control_ or _action set_ , a family $\\{A(x)\mid
x\in X\\}$ of nonempty measurable subsets $A(x)$ of $A$, where $A(x)$ denotes
the set of _feasible controls_ or _actions_ when the system is in state $x\in
X$ and with the property that the set
$\mathbb{K}\mathrel{\mathop{:}\\!\\!=}\bigl{\\{}(x,a)\big{|}x\in X,a\in
A(x)\bigr{\\}}$ of feasible state-action pairs is a measurable subset of
$X\times A$, a stochastic kernel $Q$ on $X$ given $\mathbb{K}$ called the
_transition law_ , and a measurable function
$r:\mathbb{K}\longrightarrow\mathbb{R}$ called the _reward-per-stage
function_.
###### (2.2) Assumption.
The set $\mathbb{K}$ of feasible state-action pairs contains the graph of a
measurable function from $X$ to $A$. $\diamondsuit$
Consider the Markov model ((2.1)), and for each $i=0,1,\ldots,$ define the
space $H_{i}$ of _admissible histories_ up to time $i$ as
$H_{0}\mathrel{\mathop{:}\\!\\!=}X$ and
$H_{i}\mathrel{\mathop{:}\\!\\!=}\mathbb{K}^{i}\times X=\mathbb{K}\times
H_{i-1},i\in\mathbb{N}$. A generic element $h_{i}$ of $H_{i}$, which is called
an admissible $i$-history, or simply $i$-history, is a vector of the form
$h_{i}=(x_{0},a_{0},\ldots,x_{i-1},a_{i-1},x_{i})$, with
$(x_{j},a_{j})\in\mathbb{K}$ for $j=0,\ldots,i-1$, and $x_{i}\in X$. Hereafter
we let the $\sigma$-algebra generated by the history $h_{i}$ be denoted by
$\mathfrak{F}_{i}$, $i\in\mathbb{N}_{0}$.
Recall that a _policy_ is a sequence $\pi=(\pi_{i})_{i\in\mathbb{N}_{0}}$ of
stochastic kernels $\pi_{i}$ on the control set $A$ given $H_{i}$ satisfying
the constraint $\pi_{i}(A(x_{i})|h_{i})=1\;\;\forall\,h_{i}\in
H_{i},i\in\mathbb{N}_{0}$. The set of all policies is denoted by $\Pi$. Let
$(\Omega,\mathfrak{F})$ be the measurable space consisting of the (canonical)
sample space $\Omega\mathrel{\mathop{:}\\!\\!=}\overline{H}_{\infty}=(X\times
A)^{\infty}$ and let $\mathfrak{F}$ be the corresponding product
$\sigma$-algebra. The elements of $\Omega$ are sequences of the form
$\omega=(x_{0},a_{0},x_{1},a_{1},\ldots)$ with $x_{i}\in X$ and $a_{i}\in A$
for all $i\in\mathbb{N}_{0}$; the projections $x_{i}$ and $a_{i}$ from
$\Omega$ to the sets $X$ and $A$ are called _state_ and _control_ (or
_action_) variables, respectively.
Let $\pi=(\pi_{i})_{i\in\mathbb{N}_{0}}$ be an arbitrary control policy, and
let $\nu$ be an arbitrary probability measure on $X$, referred to as the
initial distribution. By a theorem of Ionescu-Tulcea [Rao and Swift, 2006,
Chapter 3, §4, Theorem 5], there exists a unique probability measure
$\mathsf{P}_{\nu}^{\pi}$ on $(\Omega,\mathfrak{F})$ supported on $H^{\infty}$,
such that for all $B\in\mathfrak{B}\\!\left(X\right)$,
$C\in\mathfrak{B}\\!\left(A\right)$, $h_{i}\in H_{i}$, $i\in\mathbb{N}_{0}$,
we have $\mathsf{P}_{\nu}^{\pi}(x_{0}\in B)=\nu(B)$,
((2.3)a) $\displaystyle\mathsf{P}_{\nu}^{\pi}\bigl{(}a_{i}\in
C\,\big{|}\,h_{i}\bigr{)}$
$\displaystyle=\pi_{i}\bigl{(}C\,\big{|}\,h_{i}\bigr{)}$ ((2.3)b)
$\displaystyle\mathsf{P}_{\nu}^{\pi}\bigl{(}x_{i+1}\in
B\,\big{|}\,h_{i},a_{i}\bigr{)}$
$\displaystyle=Q\bigl{(}B\,\big{|}\,x_{i},a_{i}\bigr{)}.$
###### (2.4) Definition.
The stochastic process
$\bigl{(}\Omega,\mathfrak{F},\mathsf{P}_{\nu}^{\pi},(x_{i})_{i\in\mathbb{N}_{0}}\bigr{)}$
is called a discrete-time _Markov control process_. $\Diamond$
We note that the Markov control process in Definition (2.4) is not necessarily
Markovian in the usual sense due to the dependence on the entire history
$h_{i}$ in ((2.3)a); however, it is well-known [Hernández-Lerma and Lasserre,
1996, Proposition 2.3.5] that if $(\pi_{i})_{i\in\mathbb{N}_{0}}$ is
restricted to a suitable subclass of policies, then
$(x_{i})_{i\in\mathbb{N}_{0}}$ is a Markov process.
Let $\Phi$ denote the set of stochastic kernels $\varphi$ on $A$ given $X$
such that $\varphi(A(x)|x)=1$ for all $x\in X$, and let $\mathbb{F}$ denote
the set of all measurable functions $f:X\longrightarrow A$ satisfying $f(x)\in
A(x)$ for all $x\in X$. The functions in $\mathbb{F}$ are called _measurable
selectors_ of the set-valued mapping $X\ni x\longmapsto A(x)\subseteq A$.
Recall that a policy $\pi=(\pi_{i})_{i\in\mathbb{N}_{0}}\in\Pi$ is said to be
_randomized Markov_ if there exists a sequence
$(\varphi_{i})_{i\in\mathbb{N}_{0}}$ of stochastic kernels
$\varphi_{i}\in\Phi$ such that
$\pi_{i}(\cdot|h_{i})=\varphi_{i}(\cdot|x_{i})\;\;\forall\,h_{i}\in
H_{i},\;i\in\mathbb{N}_{0}$; _deterministic Markov_ if there exists a sequence
$(f_{i})_{i\in\mathbb{N}_{0}}$ of functions $f_{i}\in\mathbb{F}$ such that
$\pi_{i}(\cdot|h_{i})=\delta_{f(x_{i})}(\cdot)$; _deterministic stationary_ if
there exists a function $f\in\mathbb{F}$ such that
$\pi_{i}(\cdot|h_{i})=\delta_{f(x_{i})}(\cdot)$. As usual let $\Pi$,
$\Pi_{RM}$, $\Pi_{DM}$, and $\Pi_{DS}$ denote the set of all randomized
history-dependent, randomized Markov, deterministic Markov, and deterministic
stationary policies, respectively. The transition kernel $Q$ in ((2.3)b) under
a policy
$\pi\mathrel{\mathop{:}\\!\\!=}(\varphi_{i})_{i\in\mathbb{N}_{0}}\in\Pi_{RM}$
is given by $\bigl{(}Q(\cdot|\cdot,\varphi_{i})\bigr{)}_{i\in\mathbb{N}_{0}}$,
which is defined as the transition kernel $\mathfrak{B}\\!\left(X\right)\times
X\ni(B,x)\longmapsto
Q(B|x,\varphi_{i}(x))\mathrel{\mathop{:}\\!\\!=}\int_{A(x)}\varphi_{i}(\mathrm{d}a|x)Q(B|x,a)$.
Occasionally we suppress the dependence of $\varphi_{i}$ on $x$ and write
$Q(B|x,\varphi_{i})$ in place of $Q(B|x,\varphi_{i}(x))$, and
$r(x_{j},\varphi_{j})\mathrel{\mathop{:}\\!\\!=}\int_{A(x_{j})}\varphi_{j}(\mathrm{d}a|x_{j})r(x_{j},a)$.
We simply write $f^{\infty}$ for a policy $(f,f,\ldots)\in\Pi_{DS}$.
### 2.2. Maximizing the Probability of Hitting a Target before a Cemetery Set
Let $O$ and $K$ be two nonempty measurable subsets of $X$ with $O\subsetneqq
K$. Let
((2.5))
$\displaystyle\tau\mathrel{\mathop{:}\\!\\!=}\inf\bigl{\\{}t\in\mathbb{N}_{0}\;\big{|}\;x_{t}\in
O\bigr{\\}}\quad\text{and}\quad\tau^{\prime}\mathrel{\mathop{:}\\!\\!=}\inf\bigl{\\{}t\in\mathbb{N}_{0}\;\big{|}\;x_{t}\in
X\smallsetminus K\bigr{\\}}$
be the first hitting times of the above sets.111As usual we set the infimum
over an empty set to be $\infty$. These random times are stopping times with
respect to the filtration $(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$. Suppose
that the objective is to maximize the probability that the state hits the set
$O$ before exiting the set $K$; in symbols the objective is to attain
((2.6))
$V^{\star}(x)\mathrel{\mathop{:}\\!\\!=}\sup_{\pi\in\Pi}V(\pi,x)\mathrel{\mathop{:}\\!\\!=}\sup_{\pi\in\Pi}\mathsf{P}^{\pi}_{x}\bigl{(}\tau<\tau^{\prime},\tau<\infty\bigr{)},$
where the $\sup$ is taken over a class $\Pi$ of admissible policies.
###### (2.7).
_Admissible policies._ It is clear at once that the class of admissible
policies for the problem ((2.6)) is different from the classes considered in
§2.1. Indeed, since the process is killed at the stopping time
$\tau\wedge\tau^{\prime}$, it follows that the class of admissible policies
should also be truncated at the stage $\tau\wedge\tau^{\prime}-1$. For a given
stage $t\in\mathbb{N}_{0}$ we define the $t$-th policy element $\pi_{t}$ only
on the set $\\{t<\tau\wedge\tau^{\prime}\\}$. Note that with this definition
$\pi_{t}$ becomes a $\mathfrak{F}_{t\wedge\tau\wedge\tau^{\prime}}$-measurable
randomized control. It is also immediate from the definitions of $\tau$ and
$\tau^{\prime}$ that if the initial condition $x\in O\cup(X\smallsetminus K)$,
then the set of admissible policies is empty in the sense that there is
nothing to do by definition. Indeed, in this case $\tau\wedge\tau^{\prime}=0$
and no control is needed. We are thus interested only in $x\in K\smallsetminus
O$, for otherwise the problem is trivial. In other words, the domain of
$\pi_{t}$ is contained in the ‘spatial’ region
$\bigl{\\{}(x,a)\in\mathbb{K}\,\big{|}\,x\in K\smallsetminus O,a\in
A(x)\bigr{\\}}$. Equivalently, in view of the definitions of the ‘temporal’
elements $\tau$ and $\tau^{\prime}$, $\pi_{t}$ is well-defined on the set
$\\{t<\tau\wedge\tau^{\prime}\\}$. We re-define
$\mathbb{K}\mathrel{\mathop{:}\\!\\!=}\bigl{\\{}(x,a)\in\mathbb{K}\,\big{|}\,x\in
K\smallsetminus O,a\in A(x)\bigr{\\}}$, and also let $\mathbb{F}$ to be the
set of measurable selectors of the set-valued map $K\smallsetminus O\ni
x\longmapsto A(x)\subseteq A$.
_Throughout this subsection we shall denote by $\Pi_{M}$ the class of Markov
policies such that if $(\pi_{t})_{t\in\mathbb{N}_{0}}\in\Pi_{M}$, then
$\pi_{t}$ is defined on $\mathbb{K}$ for each $t$._
###### (2.8).
Recall that a transition kernel $Q$ on a measurable space $X$ given another
measurable space $Y$ is said to be _strongly Feller_ if the mapping
$y\longmapsto\int_{X}g(x)Q(\mathrm{d}x|y)$ is continuous and bounded for every
measurable and bounded function $g:X\longrightarrow\mathbb{R}$. A function
$g:\mathbb{K}\longrightarrow\mathbb{R}$ is _upper semicontinuous_ (u.s.c.) if
for every sequence $(x_{j},a_{j})_{j\in\mathbb{N}}\subseteq\mathbb{K}$
converging to $(x,a)\in\mathbb{K}$, we have
$\limsup_{j\rightarrow\infty}g(x_{j},a_{j})\leqslant g(x,a)$; or,
equivalently, if for every $r\in\mathbb{R}$, the set
$\bigl{\\{}(x,a)\in\mathbb{K}\,\big{|}\,g(x,a)\geqslant r\bigr{\\}}$ is closed
in $\mathbb{K}$. A set-valued map
$\Psi:X\longrightarrow\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\to{\;}Y$ between
topological spaces is _upper hemicontinuous at a point $x$_ if for every
neighborhood $U$ of $\Psi(x)$ there exists a neighborhood $V$ of $x$ such that
$z\in V$ implies that $\Psi(z)\subseteq U$; $\Psi$ is _upper hemicontinuous_
if it is upper hemicontinuous at every $x$ in its domain. If $X$ is equipped
with a $\sigma$-algebra $\Sigma$, then the set-valued map $\Psi$ is called
_weakly measurable_ if $\Psi^{\ell}(G)\in\Sigma$ for every open $G\subseteq
Y$, where $\Psi^{\ell}$ is the lower inverse of $\Psi$, defined by
$\Psi^{\ell}(A)\mathrel{\mathop{:}\\!\\!=}\\{x\in X\mid\Psi(x)\cap
A\neq\emptyset\\}$. See, e.g., [Aliprantis and Border, 2006, Chapters 17-18]
for further details on set-valued maps.222What we call “set-valued maps” are
“correspondences” in [Aliprantis and Border, 2006]. Whenever $B\subseteq X$ is
a nonempty measurable set and we are concerned with any set-valued map $B\ni
x\longmapsto A(x)\subseteq A$, we let $B$ be equipped with the trace of
$\mathfrak{B}\\!\left(X\right)$ on $B$. Let
$b\mathfrak{B}\\!\left(X\right)^{+}$ denote the convex cone of nonnegative,
bounded, and measurable real-valued functions on $X$, and we define
$\bar{B}\mathrel{\mathop{:}\\!\\!=}\bigl{\\{}g\in\boldsymbol{L}_{\infty}(X)\,\big{|}\,g|_{X\smallsetminus
K}=0,\left\lVert g\right\rVert_{\boldsymbol{L}_{\infty}{(X)}}\leqslant
1\bigr{\\}}$.
###### (2.9) Assumption.
In addition to Assumption (2.2), we stipulate that
1. (i)
the set-valued map $K\smallsetminus O\ni x\longmapsto A(x)\subseteq A$ is
compact-valued, upper hemicontinuous, and weakly measurable;
2. (ii)
the transition kernel $Q$ on $X$ given $\mathbb{K}$ is strongly Feller, i.e.,
the mapping $\mathbb{K}\ni(x,a)\longmapsto\int_{X}Q(\mathrm{d}y|x,a)g(y)$ is
continuous and bounded for all bounded and measurable functions
$g:X\longrightarrow\mathbb{R}$.$\diamondsuit$
The following theorem gives basic existence results for the problem ((2.6)); a
proof is presented in §4.1.
###### (2.10) Theorem.
Suppose that Assumption ((2.9)) holds, and that $\tau\wedge\tau^{\prime}$ is
finite for every policy in $\Pi_{M}$. Then:
1. (i)
The value function $V^{\star}$ is a pointwise bounded and measurable solution
to the _Bellman equation_ in $\psi$:
((2.11)) $\psi(x)=\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\max_{a\in
A(x)}\int_{X}Q(\mathrm{d}y|x,a)\boldsymbol{1}_{K}(y)\psi(y)\quad\forall\,x\in
X.$
Moreover, $V^{\star}$ is minimal in $\bar{B}\cap
b\mathfrak{B}\\!\left(X\right)^{+}$.
2. (ii)
There exists a measurable selector $f_{\star}\in\mathbb{F}$ such that
$f_{\star}(x)\in A(x)$ attains the maximum in ((2.11)) for each $x\in
K\smallsetminus O$, which satisfies
((2.12)) $V^{\star}(x)=\begin{cases}1&\text{if }x\in O,\\\
\displaystyle{\int_{K}Q(\mathrm{d}y|x,f_{\star})\,V^{\star}(y)}&\text{if }x\in
K\smallsetminus O,\\\ 0&\text{otherwise},\end{cases}$
where $V^{\star}$ is as defined in ((3.1)). Moreover, the deterministic
stationary policy $f_{\star}^{\infty}$ is optimal. Conversely, if
$f_{\star}^{\infty}$ is optimal, then it satisfies ((2.12)).
###### (2.13).
As a matter of notation we shall henceforth represent the functional equation
((2.12)) with the less formal version:
((2.14)) $V^{\star}(x)=\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}y|x,f_{\star})\,V^{\star}(y)\quad\forall\,x\in X.$
Note that the measure $Q(\cdot|x,f_{\star})$ is not well-defined for $x\in
O\cup(X\smallsetminus K)$ for $f\in\mathbb{F}$ in view of the definition in
paragraph (2.7). As such the integral
$\int_{K}Q(\mathrm{d}y|x,f_{\star})\,V^{\star}(y)$ is undefined for $x\in
O\cup(X\smallsetminus K)$. However, to preserve the form of ((2.11)) and
simplify notation, we shall stick to the representation ((2.14)) by defining
any object that is written as an integral of a bounded measurable function
with respect to the measure $Q(\cdot|x,f)$ to be $0$ whenever $x\in
O\cup(X\smallsetminus K)$ and $f\in\mathbb{F}$.
### 2.3. A Martingale Characterization
_We now return to the more general class of all possible policies (not just
Markovian), denoted by $\Pi$._
Fix an initial state $x\in X$ and a policy $\pi\in\Pi$. For each
$n\in\mathbb{N}$ we define the random variable
$W_{n}(\pi,x)\mathrel{\mathop{:}\\!\\!=}\sum_{t=0}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})$.
Let us consider the process $(\zeta_{n})_{n\in\mathbb{N}_{0}}$ defined by
((2.15)) $\displaystyle\zeta_{0}$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}V^{\star}(x),$
$\displaystyle\zeta_{n}$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}W_{n}(\pi,x)+\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}\cdot
V^{\star})(x_{n\wedge\tau\wedge\tau^{\prime}}),\;\;n\in\mathbb{N}.$
We follow the basic framework of [Karatzas and Sudderth, 2009].
###### (2.16) Definition.
A policy $\pi\in\Pi$ is called _thrifty at $x\in X$_ if
$V^{\star}(x)=\Lambda^{\pi}(x)$, and _equalizing at $x\in X$_ if
$\Lambda^{\pi}(x)=V(\pi,x)$. The action $a_{n}$, defined on
$\\{\tau\wedge\tau^{\prime}>n\\}$, is said to _conserve $V^{\star}$ at
$x_{n}$_ if $\boldsymbol{1}_{O}(x_{n})+\boldsymbol{1}_{K\smallsetminus
O}(x_{n})\int_{K}Q(\mathrm{d}y|x_{n},a_{n})V^{\star}(y)=V^{\star}(x_{n})$.
$\Diamond$
Connections between equalizing, thrifty, and optimal policies for our problem
((2.6)) are established by the following
###### (2.17) Theorem.
A policy $\pi\in\Pi$ is
* $\circ$
equalizing at $x\in X$ if and only if
$\lim_{n\to\infty}\mathsf{E}^{\pi}_{x}\bigl{[}\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{n\wedge\tau\wedge\tau^{\prime}})\bigr{]}=0;$
* $\circ$
optimal at $x\in X$ if and only if $\pi$ is both thrifty and equalizing.
A connection between thrifty policies, the process
$(\zeta_{n})_{n\in\mathbb{N}_{0}}$ defined in ((2.15)), and actions conserving
the optimal value function $V^{\star}$ is established by the following
###### (2.18) Theorem.
For a given policy $\pi\in\Pi$ and an initial state $x\in X$ the following are
equivalent:
1. (i)
$\pi$ is trifty at $x$;
2. (ii)
$(\zeta_{n})_{n\in\mathbb{N}_{0}}$ is a
$(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$ -martingale under
$\mathsf{P}^{\pi}_{x}$;
3. (iii)
$\mathsf{P}^{\pi}_{x}$-almost everywhere on $\\{\tau\wedge\tau^{\prime}>n\\}$
the action $a_{n}$ conserves $V^{\star}$.
It is possible to make a connection, relying purely on martingale-theoretic
arguments, between the process $(\zeta_{n})_{n\in\mathbb{N}_{0}}$ and the
value function corresponding to an optimal policy. This is the content of the
following theorem, which may be viewed as a partial converse to Theorem
(2.18).
###### (2.19) Theorem.
Suppose that either one of the stopping times $\tau$ and $\tau^{\prime}$
defined in ((2.5)) is finite for every policy in $\Pi$. Let $V^{\prime}$ be a
nonnegative measurable function such that $V^{\prime}|_{O}=1$,
$V^{\prime}|_{X\smallsetminus K}=0$, and bounded above by $1$ elsewhere. For a
policy $\pi\in\Pi$ define the process
$(\zeta^{\prime}_{n})_{n\in\mathbb{N}_{0}}$ as
((2.20)) $\displaystyle\zeta^{\prime}_{0}$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}V^{\prime}(x),$
$\displaystyle\zeta^{\prime}_{n}$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}W_{n}(\pi,x)+\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}\cdot
V^{\prime})(x_{n\wedge\tau\wedge\tau^{\prime}}),\;\;n\in\mathbb{N},$
where $W_{n}(\pi,x)$ is as in ((2.15)). If for some policy $\pi^{\star}\in\Pi$
the process $(\zeta^{\prime}_{n})_{n\in\mathbb{N}_{0}}$ is a
$(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$ -martingale under
$\mathsf{P}^{\pi^{\star}}_{x}$, then
$V^{\prime}(x)=\mathsf{P}^{\pi^{\star}}_{x}\bigl{(}\tau<\tau^{\prime},\tau<\infty\bigr{)}$.
Proofs of the above results are presented in §4.2.
## 3\. Discussion and Examples
Let us look at the stopped process
$(x_{t\wedge{(n-1)}\wedge\tau\wedge\tau^{\prime}})_{t\in\mathbb{N}_{0}}$. It
is clear that in this case $V_{n}(\pi,x)=1$ whenever $x\in O$ and
$V_{n}(\pi,x)=0$ whenever $x\in X\smallsetminus K$ for all policies in
$\Pi_{M}$; otherwise for $x\in K\smallsetminus O$ we have
$\displaystyle V_{n}(\pi,x)$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}\mathsf{P}^{\pi}_{x}\bigl{(}\tau<\tau^{\prime},\tau<n\bigr{)}$
$\displaystyle=\mathsf{P}^{\pi}_{x}\bigl{(}x_{1\wedge\tau\wedge\tau^{\prime}}\in
O\bigr{)}+\mathsf{P}^{\pi}_{x}\bigl{(}x_{1\wedge\tau\wedge\tau^{\prime}}\in
K\smallsetminus O,x_{2\wedge\tau\wedge\tau^{\prime}}\in O\bigr{)}$
$\displaystyle\quad+\ldots+\mathsf{P}^{\pi}_{x}\bigl{(}x_{1\wedge\tau\wedge\tau^{\prime}},\ldots,x_{(n-2)\wedge\tau\wedge\tau^{\prime}}\in
K\smallsetminus O,x_{(n-1)\wedge\tau\wedge\tau^{\prime}}\in O\bigr{)}.$
Since the $k$-th term on the right-hand side is
$\mathsf{E}^{\pi}_{x}\bigl{[}\prod_{t=1}^{k-1}\boldsymbol{1}_{K\smallsetminus
O}(x_{t\wedge\tau\wedge\tau^{\prime}})\boldsymbol{1}_{O}(x_{k\wedge\tau\wedge\tau^{\prime}})\bigr{]}$,
it follows that
$\displaystyle V_{n}(\pi,x)$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=1}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\Biggl{(}\prod_{i=0}^{t-1}\boldsymbol{1}_{K\smallsetminus
O}(x_{i})\Biggr{)}\boldsymbol{1}_{O}(x_{t})\Biggr{]}$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=1}^{(n-1)\wedge\tau}\boldsymbol{1}_{O}(x_{t\wedge\tau^{\prime}})\Biggr{]}=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=1}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}.$
We note that $V_{n}(\pi,x)=0$ whenever $x\in O\cup(X\smallsetminus K)$. A
policy that maximizes $V_{n}(\pi,x)$ is defined only on the set
$K\smallsetminus O$, and it is left undefined elsewhere. Once the process
exits $K\smallsetminus O$ or the stage reaches $n-1$, the task of our control
policy is over. Such a deterministic stationary policy (which exists, as
demonstrated below) with a measurable selector $f\in\mathbb{F}$ should be
represented as
$f^{\tau\wedge\tau^{\prime}}\mathrel{\mathop{:}\\!\\!=}\underset{\tau\wedge\tau^{\prime}\text{
times}}{\underbrace{(f,f,\ldots,f)}}$ since it is applied only for the first
$\tau\wedge\tau^{\prime}$ stages; however, for notational brevity we simply
write $f^{\infty}$ hereafter.
Quite clearly, letting $n\to\infty$, the monotone convergence theorem gives
$\displaystyle V(\pi,x)$
$\displaystyle=\lim_{n\to\infty}V_{n}(\pi,x)=\mathsf{P}^{\pi}_{x}\bigl{(}\tau<\tau^{\prime},\tau<\infty\bigr{)}$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=1}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=1}^{\tau}\boldsymbol{1}_{O}(x_{t\wedge\tau\wedge\tau^{\prime}})\Biggr{]}.$
We note that by definition, the random sum inside the expectation on the
right-hand side of the last equality above is the limit of partial (finite)
sums, and this ensures that the term inside the expectation is defined on the
event $\\{\tau\wedge\tau^{\prime}<\infty\\}$. By definition note that
((3.1))
$V^{\star}(x)=\sup_{\pi\in\Pi_{M}}V(\pi,x)=\sup_{\pi\in\Pi_{M}}\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=1}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}.$
Consider again the value-iteration functions defined by
((3.2))
$\begin{cases}v_{0}(x)\mathrel{\mathop{:}\\!\\!=}\boldsymbol{1}_{O}(x)\\\
v_{n}(x)\mathrel{\mathop{:}\\!\\!=}\displaystyle{\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\max_{a\in
A(x)}\int_{X}Q(\mathrm{d}y|x,a)\boldsymbol{1}_{K}(y)v_{n-1}(y)}\end{cases}$
for $x\in X$ and $n\in\mathbb{N}$. The function $v_{n}$ is clearly
identifiable with the optimal value function for the problem of maximizing
$\mathsf{P}^{\pi}_{x}\bigl{(}\tau<\tau^{\prime},\tau<n\bigr{)}$ of the process
stopped at the $(n-1)$-th stage, $n\in\mathbb{N}$. To get an intuitive idea,
fix a deterministic Markov policy $\pi^{\prime}=(f_{t})_{t\in\mathbb{N}_{0}}$
and take the first iterate $v_{0}$. (Of course the assumption underlying the
notation $(f_{t})_{t\in\mathbb{N}_{0}}$ is that $f_{t}$ is defined on
$\\{t<\tau\wedge\tau^{\prime}\\}$.) It is immediately clear that the reward at
the first step is $1$ if and only if $x\in K$ and $0$ otherwise, and that is
precisely $v_{0}$ irrespective of the policy. For the second iterate note the
reward under the policy $\pi^{\prime}$ is
$\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus O}(x)Q(O|x,f_{0}(x))$.
This is because the reward is $1$ if $x\in O$ and the process terminates at
the first stage, or $x\in K\smallsetminus O$ and the reward at the second
stage is the probability of hitting $O$ at the second stage. Of course there
is no reward if $x\in X\smallsetminus K$. Similarly, for the third iterate the
reward is $\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}\xi_{1}|x,f_{0}(x))\bigl{(}\boldsymbol{1}_{O}(\xi_{1})+\boldsymbol{1}_{K\smallsetminus
O}(\xi_{1})Q(O|\xi_{1},f_{1}(\xi_{1}))\bigr{)}$. Note that only those sample
paths that stay in $K\smallsetminus O$ at the first step contribute to the
reward at the second stage, only those sample paths that stay in
$K\smallsetminus O$ for the first and the second stages contribute to the
reward at the third stage, and so on.
### 3.1. A general setting and various special cases
Our problem ((2.6)) can be viewed as a special case of a more general setting.
To wit, consider a nonnegative upper semicontinuous reward-per-stage function
$r:\mathbb{K}\longrightarrow\mathbb{R}_{\geqslant 0}$ and the problem of
maximizing the total reward up to (and including) the hitting time
$\tau\wedge\tau^{\prime}$, i.e., maximize
$\mathsf{E}^{\pi}_{x}\bigl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}r(x_{t},a_{t})\bigr{]}$
over a class of policies. This corresponds to maximization of the reward until
exit from the set $K\smallsetminus O$. The value-iteration functions
$(v^{\prime}_{n})_{n\in\mathbb{N}_{0}}$ corresponding to this problem can be
written down readily: for $x\in X$ and $n\in\mathbb{N}$ let
$\displaystyle v^{\prime}_{0}(x)$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}\sup_{a\in
A(x)}r(x,a)\boldsymbol{1}_{O\cup(X\smallsetminus K)}(x),$ $\displaystyle
v^{\prime}_{n}(x)$ $\displaystyle\mathrel{\mathop{:}\\!\\!=}\sup_{a\in
A(x)}\biggl{[}r(x,a)\boldsymbol{1}_{O\cup(X\smallsetminus
K)}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{X}Q(\mathrm{d}y|x,a)v^{\prime}_{n-1}(y)\biggr{]}.$
Our problem ((2.6)) corresponds to the case of $r(x,a)=\boldsymbol{1}_{O}(x)$.
Modulo the additional technical complications involving integrability of the
value-iteration functions at each stage and the total reward corresponding to
initial conditions being well-defined real numbers, the analysis of this more
general problem can be carried out in exactly the same way as we do below for
the problem ((2.6)). While the above more general problem treats both the
target set $O$ and the cemetery state $X\smallsetminus K$ equally, the bias
towards the target set $O$ is provided in our problem ((2.6)) by the special
structure of the reward $r(x,a)=\boldsymbol{1}_{O}(x)$.
From the general framework it is not difficult to arrive at reward-per-stage
functions that are meaningful in the context of reachability, avoidance, and
safety. For the sake of simplicity, till the end of this subsubsection we
suppose that for all initial conditions and admissible policies $\pi\in\Pi$
the stopping times $\tau$ and $\tau^{\prime}$ are finite
$\mathsf{P}^{\pi}_{x}$-almost surely. With this assumption in place, let us
look at some examples:
* $\circ$
Consider a discounted version of our problem ((2.6)), namely, let
$V^{(1)}(\pi,x)\mathrel{\mathop{:}\\!\\!=}\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\alpha^{t}\boldsymbol{1}_{O}(x_{t})\Biggr{]},$
where $\alpha\in\;]0,1[$ is a constant discount factor. From the definitions
of $\tau$ and $\tau^{\prime}$ it follows that
$\sum_{t=0}^{\tau\wedge\tau^{\prime}}\alpha^{t}\boldsymbol{1}_{O}(x_{t})=\alpha^{\tau}\boldsymbol{1}_{\\{\tau<\tau^{\prime}\\}}$,
and in view of the range of $\alpha$ it follows that maximization of $V^{(1)}$
over admissible policies leads to small values of $\tau$ on the set
$\\{\tau<\tau^{\prime}\\}$ on an average, but it is silent about the values of
$\tau$ on $\\{\tau>\tau^{\prime}\\}$.
To get a more quantitative idea of the role that the discount factor $\alpha$
plays, let $\tilde{\tau}$ be a random variable independent of the Markov
control process defined in Definition (2.4),333The random variable
$\tilde{\tau}$ can be defined in a standard way by enlarging the probability
space. with distribution function
$\mathsf{P}(\tilde{\tau}=n)=(1-\alpha)\alpha^{n}$ for all
$n\in\mathbb{N}_{0}$. In a standard way we construct the product probability
measure $\mathsf{P}^{\pi}\otimes\mathsf{P}$ and denote the expectation with
respect to this measure as $\mathsf{E}^{\pi,\tilde{\tau}}_{x}[\cdot]$. We can
write
$V^{(1)}(\pi,x)=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\infty}\alpha^{t}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}\Biggr{]}=(1-\alpha)^{-1}\mathsf{E}^{\pi,\tilde{\tau}}_{x}\bigl{[}\boldsymbol{1}_{O}(x_{\tilde{\tau}})\boldsymbol{1}_{\\{\tilde{\tau}\leqslant\tau\wedge\tau^{\prime}\\}}\bigr{]}.$
In view of the definitions of $\tau$ and $\tau^{\prime}$ we get
$V^{(1)}(\pi,x)=(1-\alpha)^{-1}\mathsf{E}^{\pi,\tilde{\tau}}_{x}\bigl{[}\boldsymbol{1}_{\\{\tilde{\tau}=\tau,\tau<\tau^{\prime}\\}}\bigr{]}$.
This alternative characterization shows that maximization of $V^{(1)}$ over
admissible policies leads to smaller values of $\tau$ compared to
$\tau^{\prime}$; moreover, the random variable $\tilde{\tau}$ gives a
quantitative idea of how the choice of $\alpha$ determines the outcome since
$\tilde{\tau}$ is a geometric random variable with parameter $(1-\alpha)$.
Choosing a small $\alpha$ implies smaller $\tilde{\tau}$ with higher
probability and may appear to be profitable; however, in certain problems it
is possible that the set $O$ may be reachable at $\tilde{\tau}$ with small
probability and the corresponding event of interest
$\\{\tilde{\tau}=\tau,\tau<\tau^{\prime}\\}$ may be relatively small for a
given initial condition $x$. Moreover, the factor $(1-\alpha)^{-1}$ is small
for small values of $\alpha$, and contributes to this phenomenon.
A second quantitative view of the role of $\alpha$ is offered by the fact that
$V^{(1)}(\pi,x)=\mathsf{E}^{\pi,\tilde{\tau}}_{x}\bigl{[}\sum_{t=0}^{\tilde{\tau}\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\bigr{]}$.
Indeed, we have
$\displaystyle\mathsf{E}^{\pi,\tilde{\tau}}_{x}$
$\displaystyle\Biggl{[}\sum_{t=0}^{\tilde{\tau}\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}=\mathsf{E}^{\pi,\tilde{\tau}}_{x}\Biggl{[}\sum_{t=0}^{\tilde{\tau}}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}\Biggr{]}$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{n=0}^{\infty}\alpha^{n}(1-\alpha)\sum_{t=0}^{n}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}\Biggr{]}$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{n=0}^{\infty}\sum_{t=0}^{n}\alpha^{n}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}-\sum_{n=0}^{\infty}\sum_{t=0}^{n}\alpha^{n+1}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}\Biggr{]}$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\infty}\sum_{n=t}^{\infty}\alpha^{n}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}-\sum_{t=0}^{\infty}\sum_{n=t}^{\infty}\alpha^{n+1}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}\Biggr{]}$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\infty}\frac{\alpha^{t}}{1-\alpha}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}-\sum_{t=0}^{\infty}\frac{\alpha^{t+1}}{1-\alpha}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}\Biggr{]}$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\infty}\alpha^{t}\boldsymbol{1}_{O}(x_{t})\boldsymbol{1}_{\\{t\leqslant\tau\wedge\tau^{\prime}\\}}\Biggr{]}=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\alpha^{t}\boldsymbol{1}_{O}(x_{t})\Biggr{]}=V^{(1)}(\pi,x).$
In this setting we do not have the $(1-\alpha)^{-1}$ factor outside the
expectation as in the second version of $V^{(1)}$ above, and it demonstrates
that maximizing $V^{(1)}(\pi,x)$ over admissible policies leads to maximizing
the probability of the event $\\{\tau<\tilde{\tau}\wedge\tau^{\prime}\\}$,
where $\alpha$ controls the values of $\tilde{\tau}$ as before.
* $\circ$
Consider the reward-per-stage function
$r(x,a)=\boldsymbol{1}_{O}(x)-\boldsymbol{1}_{X\smallsetminus O}(x)$. Under
integrability assumption on $\tau\wedge\tau^{\prime}$ under all admissible
policies, we have
$\displaystyle V^{(2)}(\pi,x)$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\bigl{(}\boldsymbol{1}_{O}(x_{t})-\boldsymbol{1}_{X\smallsetminus
O}(x_{t})\bigr{)}\Biggr{]}$
$\displaystyle=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\bigl{(}\boldsymbol{1}_{O}(x_{t})-\boldsymbol{1}_{K\smallsetminus
O}(x_{t})-\boldsymbol{1}_{X\smallsetminus K}(x_{t})\bigr{)}\Biggr{]}$
$\displaystyle=\mathsf{P}^{\pi}_{x}(\tau<\tau^{\prime})-\mathsf{P}^{\pi}_{x}(\tau^{\prime}<\tau)-\mathsf{E}^{\pi}_{x}[\tau\wedge\tau^{\prime}].$
Clearly, maximization of $V^{(2)}$ over admissible policies leads to both the
maximal enlargement of the set $\\{\tau<\tau^{\prime}\\}$ and minimization of
the hitting time $\tau$ on this set.
* $\circ$
Consider $r(x,a)=\boldsymbol{1}_{O}(x)-\boldsymbol{1}_{X\smallsetminus K}(x)$.
This leads to the expected total reward until escape from $K\smallsetminus O$
as
$V^{(3)}(\pi,x)\mathrel{\mathop{:}\\!\\!=}\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\bigl{(}\boldsymbol{1}_{O}(x_{t})-\boldsymbol{1}_{X\smallsetminus
K}(x_{t})\bigr{)}\Biggr{]}=\mathsf{P}^{\pi}_{x}(\tau<\tau^{\prime})-\mathsf{P}^{\pi}_{x}(\tau^{\prime}<\tau).$
Since
$\mathsf{P}^{\pi}_{x}(\tau<\tau^{\prime})+\mathsf{P}^{\pi}_{x}(\tau^{\prime}<\tau)=1$,
maximization of $V^{(3)}$ over admissible policies maximizes the probability
of the event $\\{\tau<\tau^{\prime}\\}$. Thus, maximizing $V^{(3)}(\pi,x)$
over $\pi\in\Pi$ is a different formulation of the objective of our problem
((2.6)). The above analysis also shows that the same objective results if we
take the reward-per-stage function to be
$\boldsymbol{1}_{O}(x)-\gamma\boldsymbol{1}_{X\smallsetminus K}(x)$ for any
$\gamma\geqslant 0$.
* $\circ$
Suppose that $\tau\wedge\tau^{\prime}$ is integrable for all admissible
policies and consider the reward-per-stage
$r(x,a)=\boldsymbol{1}_{K\smallsetminus O}(x)$. Let
$V^{(4)}(\pi,x)\mathrel{\mathop{:}\\!\\!=}\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{K\smallsetminus
O}(x_{t})\Biggr{]}.$
Maximization of $V^{(4)}$ over admissible policies leads to large values of
$\tau\wedge\tau^{\prime}$ on an average. This is a form of safety problem, the
state stays inside $K\smallsetminus O$ for as long as possible on an average.
* $\circ$
Suppose that $\tau\wedge\tau^{\prime}$ is integrable for all admissible
policies and consider
$r(x,a)=\gamma\boldsymbol{1}_{O}(x)-\boldsymbol{1}_{K\smallsetminus O}(x)$ for
$\gamma\geqslant 1$. Consider
$V^{(5)}(\pi,x)\mathrel{\mathop{:}\\!\\!=}\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\bigl{(}\gamma\boldsymbol{1}_{O}(x_{t})-\boldsymbol{1}_{K\smallsetminus
O}(x_{t})\bigr{)}\Biggr{]},$
we see that
$V^{(5)}(\pi,x)=\gamma\mathsf{P}^{\pi}_{x}(\tau<\tau^{\prime})-\mathsf{E}^{\pi}_{x}[\tau\wedge\tau^{\prime}]$.
We see that maximization of $V^{(5)}$ over admissible policies leads to a
balance between maximizing the probability that the state hits the set $O$
before getting out of $K$ and exiting $K$ quickly. This is because it is more
profitable to exit from $K$ and get a zero reward than incur negative reward
by prolonging the duration of stay in $K\smallsetminus O$. The factor $\gamma$
decides the priorities of the two alternatives. It is trivially clear that
$\gamma=1$ leads to rapid exit from $K$ if the initial condition is in
$K\smallsetminus O$.
Not all the reward-per-stage functions mentioned above can be handled in our
present framework. In particular, we make the crucial assumption that the
reward-per-stage function is nonnegative, which does not hold in some of the
cases above. However, under appropriate growth-rate conditions on the reward-
per-stage function, the nonnegativity assumption can be dispensed with.
In classical finite or infinite-horizon optimal control problems a translation
of the (fixed) reward-per-stage function would not change the solution to the
problem. However, translations of the reward-per-stage function in random-
horizon problems may lead to drastically different policies. We give two
examples:
* $\circ$
Consider the reward-per-stage functions
$r^{\prime}(x,a)=\boldsymbol{1}_{O}(x)-\boldsymbol{1}_{X\smallsetminus K}(x)$
and
$r^{\prime\prime}(x,a)=2\cdot\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)$; in this case we translate $r^{\prime}$ on $X$ by $1$, i.e.,
$r^{\prime\prime}=r^{\prime}+1$. On the one hand, maximizing
$\mathsf{E}^{\pi}_{x}\bigl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}r^{\prime}(x_{t},a_{t})\bigr{]}$
yields a policy that $\mathsf{P}^{\pi}_{x}(\tau<\tau^{\prime})$ as we have
seen before (this is $V^{(3)}$ above). On the other hand, maximizing
$\mathsf{E}^{\pi}_{x}\bigl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}r^{\prime\prime}(x_{t},a_{t})\bigr{]}$
yields a policy that tries to keep the state in $K\smallsetminus O$ for as
long as possible, and at each stage accrue a reward of $1$, which is certainly
better than jumping to $O$ and accruing a reward of $2$ at most.
* $\circ$
Consider
$r^{\prime}(x,a)=\boldsymbol{1}_{O}(x)-\boldsymbol{1}_{X\smallsetminus K}(x)$
and
$r^{\prime\prime}(x,a)=-\boldsymbol{1}_{O}(x)-3\cdot\boldsymbol{1}_{X\smallsetminus
K}(x)$; in this case we translate $r^{\prime}$ by $-2$ only on its support
$O\cup(X\smallsetminus K)$. We have noted above that maximizing
$\mathsf{E}^{\pi}_{x}\bigl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}r^{\prime}(x_{t},a_{t})\bigr{]}$
yields a policy that maximizes $\mathsf{P}^{\pi}_{x}(\tau<\tau^{\prime})$.
However, maximizing
$\mathsf{E}^{\pi}_{x}\bigl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}r^{\prime\prime}(x_{t},a_{t})\bigr{]}$
yields a policy that tries to keep the state in $K\smallsetminus O$ for the
longest possible duration to avoid incurring negative reward.
### 3.2. Further examples
For one-dimensional stochastic processes initialized somewhere between two
different levels $a$ and $b$, problems such as calculating the probability of
hitting the level $a$ before the level $b$ are fairly common, e.g., in random
walks, Brownian motion, and diffusions, see, e.g., [Levin et al., 2009,
Chapters 2-3], [Revuz and Yor, 1999]. It is possible to obtain explicit
expressions of these probabilities in a handful of cases.
Let us consider a controlled Markov chain $(x_{t})_{t\in\mathbb{N}_{0}}$ with
a finite state-space $X=\\{1,2,\ldots,m\\}$ and a transition probability
matrix $Q=[q_{ij}(a)]_{m\times m}$, where $a$ is the action or control
variable. Let $O\subsetneqq X$, $K\subsetneqq X$ be subsets of $X$ with
$O\subsetneqq K$. Since $X$ is finite, Assumption (2.9) is satisfied. Consider
the problem ((2.6)) in the context of this Markov chain
$(x_{t})_{t\in\mathbb{N}_{0}}$ initialized at some $i_{0}\in K\smallsetminus
O$. By Theorem (2.10) the optimal value function $V^{\star}$ must satisfy the
equation
$\displaystyle V^{\star}(i)$
$\displaystyle=\boldsymbol{1}_{O}(i)+\boldsymbol{1}_{K\smallsetminus
O}(i)\max_{a\in A(i)}\sum_{j\in K}q_{ij}(a)V^{\star}(j)$
$\displaystyle=\boldsymbol{1}_{O}(i)+\boldsymbol{1}_{K\smallsetminus
O}(i)\max_{a\in A(i)}\Biggl{(}\sum_{j\in O}q_{ij}(a)+\sum_{j\in
K\smallsetminus O}q_{ij}(a)V^{\star}(j)\Biggr{)}$
for all $i\in X$. If the control actions are finite in number, searching for a
maximizer over an enumerated list all control actions corresponding to each of
the states may be possible if the state and action spaces are not too large.
However, the memory requirement for storing such enumerated lists clearly
increases exponentially with the dimension of the state and action spaces if
the Markov chain is extracted by a discretization procedure based on a grid on
the state-space of a discrete-time Markov process evolving, for example, on a
subset of Euclidean space. As an alternative, it is possible to search for a
maximizer from a parametrized family of functions (vectors) by applying well-
known suboptimal control strategies [Bertsekas, 2007, Chapter 6], [Bertsekas
and Tsitsiklis, 1996; Powell, 2007]. Note that in the case of an uncontrolled
Markov chain the equation above reduces to
$V^{\star}(i)=\boldsymbol{1}_{O}(i)+\boldsymbol{1}_{K\smallsetminus
O}(i)\bigl{(}\sum_{j\in O}q_{ij}+\sum_{j\in K\smallsetminus
O}q_{ij}V^{\star}(j)\bigr{)}$, and can be solved as a linear equation on
$K\smallsetminus O$ for the vector $V^{\star}|_{K\smallsetminus O}$. Thus,
solving for $V^{\star}$ yields a method of calculating the probability of
hitting $O$ before hitting $X\smallsetminus K$ in uncontrolled Markov chains,
and can serve as a verification tool [Kwiatkowska et al., 2007].
In certain cases of uncountable state-space Markov chains the policies and
value functions corresponding to maximization of
$\mathsf{P}^{\pi}_{x}\bigl{(}\tau<\tau^{\prime},\tau<n\bigr{)}$ can be
explicitly calculated for small values of $n$. As an illustration, consider a
scalar linear controlled system
((3.3)) $x_{t+1}=x_{t}+a_{t}+w_{t},\quad x_{0}=x,\;\;t\in\mathbb{N}_{0}.$
Here $x_{t}\in\mathbb{R}$ is the state of the system at time $t$, $a_{t}$ is
the action or control at time $t$ taking values in $[-1,1]$, and
$(w_{t})_{t\in\mathbb{N}_{0}}$ is a sequence of independent and identically
distributed (i.i.d) standard normal random variables treated as noise inputs
to the system. Let us suppose that our target set is $O=\;]-1,1[$, safe set is
$K=[-3,3]$, and let us find a greedy policy for our problem, i.e., a policy
that maximizes
$\mathsf{P}^{\pi}_{x}\bigl{(}\tau<\tau^{\prime},\tau<2\bigr{)}$.
The greedy policy tries to maximize
$\mathsf{P}_{x}\bigl{(}x_{1}\in\;]-1,1[\bigr{)}=\mathsf{P}_{x}\bigl{(}x+a+w\in\;]-1,1[\bigr{)}=\mathfrak{N}(1-x-a)-\mathfrak{N}(-1-x-a)=:G(x,a)$,
where $\mathfrak{N}$ is the cumulative distribution function of the standard
normal random variable. The function $G$ can be expressed in terms of the
complementary error function444Recall that the complementary error function is
defined as
$\operatorname{erfc}(r)\mathrel{\mathop{:}\\!\\!=}\frac{2}{\sqrt{\pi}}\int_{r}^{\infty}\mathrm{e}^{-t^{2}}\mathrm{d}t=1-\operatorname{erf}(r)$,
where $\operatorname{erf}(\cdot)$ is the standard error function. as
$G(x,a)=\frac{1}{2}\Bigl{(}\operatorname{erfc}\bigl{(}-\frac{1}{\sqrt{2}}(1-x-a)\bigr{)}-\operatorname{erfc}\bigl{(}-\frac{1}{\sqrt{2}}(-1-x-a)\bigr{)}\Bigr{)}$,
and $\operatorname{arg\,max}_{a\in[-1,1]}G(x,a)$ can be solved in closed form.
Indeed, $\tfrac{\partial G}{\partial
a}(x,a)=\frac{1}{\sqrt{2\pi}}\bigl{(}\mathrm{e}^{-\frac{1}{2}(x+a+1)^{2}}-\mathrm{e}^{-\frac{1}{2}(x+a-1)^{2}}\bigr{)}=0$
gives $a^{\star}=f_{\star}(x)=-x$ as the unconstrained optimizer. Since
$a\in[-1,1]$, we have the constrained maximizer as
$f_{\star}(x)=-\operatorname{sat}(x)$, where $\operatorname{sat}(\cdot)$ is
the standard saturation function.555Recall that the standard saturation
function is defined as $\operatorname{sat}(r)$ equals $r$ if
$\left\lvert{r}\right\rvert<1$, $1$ if $r\geqslant 1$ and $-1$ otherwise. In
other words, we get a bang-bang controller since $x-\operatorname{sat}(x)\neq
0$ on the interior of $K\smallsetminus O$. It is easy to discern the maximizer
from the accompanying figure. The corresponding maximal probability is found
by substituting the above optimizer back into the dynamic programming
equation, and this yields
$V_{1}^{\star}(x)=\boldsymbol{1}_{O}(x)+\frac{1}{2}\boldsymbol{1}_{K\smallsetminus
O}(x)\Bigl{(}\operatorname{erf}\bigl{(}\frac{1}{\sqrt{2}}(x-\operatorname{sat}(x)+1)\bigr{)}-\operatorname{erf}\bigl{(}\frac{1}{\sqrt{2}}(x-\operatorname{sat}(x)-1)\bigr{)}\Bigr{)}$.
For $n=3$ it turns out that we can no longer compute the optimizer
corresponding to the first stage in closed form; the optimizer for the second
stage is, of course, $f_{\star}(x)=-\operatorname{sat}(x)$ calculated above.
It is also evident from the accompanying figure that even in this simple
example there will arise nontrivial issues with nonconvexity for $n\geqslant
3$.
### 3.3. Uniqueness of optimal policies
So far in our discussion we have not addressed the issue of uniqueness of the
optimal policy in our problem ((2.6)). (Theorem (2.10) shows that an optimal
policy exists, so the uniqueness question is meaningful.) It becomes clear
from considerations of the geometry of the sets $O$ and $K$ in simple examples
that the optimal controller $f_{\star}$ in Theorem (2.10)_(ii)_ is nonunique
in general. For instance, consider the linear system considered in ((3.3))
above with initial condition $x_{0}=0$, and let $O=\;]-2,-1[\;\cup\;]1,2[$ and
$K=[-3,3]$. Since the noise is symmetric about the origin, from symmetry
considerations it immediately follows that the optimal controller $f_{\star}$
is nonunique at the origin. Note that $f_{\star}$ is, of course, defined on
$K\smallsetminus O$.
### 3.4. Relation to a probabilistic safety problem
Let us digress a little and consider the following probabilistic safety
problem: maximize the probability that the state remains inside a safe set
$C\subseteq X$ for $n$ stages, beginning from an initial condition $x\in C$.
This, as mentioned earlier, is the probabilistic safety problem addressed in
[Abate et al., 2008]. Of course the probability of staying inside $C$ for the
first $n$ stages is given by
$\mathsf{P}^{\pi}_{x}\bigl{(}\bigcap_{t=0}^{n-1}\\{x_{t}\in
C\\}\bigr{)}=\mathsf{E}^{\pi}_{x}\bigl{[}\prod_{t=0}^{n-1}\boldsymbol{1}_{\\{x_{t}\in
C\\}}\bigr{]}$. If $\sigma$ is the first exit time from $C$, then
$\mathsf{P}^{\pi}_{x}\bigl{(}\bigcap_{t=0}^{n-1}\\{x_{t}\in
C\\}\bigr{)}=\mathsf{E}^{\pi}_{x}\bigl{[}\prod_{t=0}^{(\sigma-1)\wedge(n-1)}\boldsymbol{1}_{\\{x_{t}\in
C\\}}\bigr{]}$. Therefore, in this particular problem there is no difference
between the maximal values of
$\mathsf{E}^{\pi}_{x}\bigl{[}\prod_{t=0}^{n-1}\boldsymbol{1}_{\\{x_{t}\in
C\\}}\bigr{]}$ or
$\mathsf{E}^{\pi}_{x}\bigl{[}\prod_{t=0}^{(\sigma-1)\wedge(n-1)}\boldsymbol{1}_{\\{x_{t}\in
C\\}}\bigr{]}$. However, the policies arising from the two different
maximizations are quite unlike each other. Indeed, whereas the former yields a
deterministic Markov policy [Abate et al., 2008] whose every element is
defined on all of $X$, the stopping time version yields a deterministic Markov
policy whose $t$-th element $\pi_{t}$ is defined on the set $\\{t<\sigma\wedge
n\\}$, just as discussed in paragraph (2.7). On the one hand note that the
reward in the former case is not affected by further application of the
control actions once the state has exited the safe set $C$; the policy
resulting from this formulation, however, dictates that the control actions
are carried out until (and including) the $(n-2)$-th stage nonetheless. On the
other hand, the reward in the latter stopping time version saturates at the
stage the state leaves $C$ and future control actions are not defined.
It is interesting to note that the Bellman equation developed for
probabilistic safety and reachability in [Abate et al., 2008] may be obtained
as a special case of ((2.11)) in Theorem (2.10) above. This comes as no
surprise. The problem of maximizing the probability of staying inside a
(measurable) safe set $C\subseteq X$ for $N$ steps is given by the
maximization of
$\mathsf{E}^{\pi}_{x}\bigl{[}\prod_{t=0}^{\sigma\wedge(N-1)}\boldsymbol{1}_{C}(x_{t})\bigr{]}$,
where $\sigma$ is the first time to exit $C$ and this clearly translates to
minimizing $\mathsf{P}^{\pi}_{x}(\tau<N)$. In our setting, if we let $K$ be
the entire state-space $X$, $C=X\smallsetminus O$, and $\tau$ the first time
to hit the set $O$, then our problem ((2.6)) is precisely that in [Abate et
al., 2008] with the exception of maximization in place of minimization. It
must be mentioned however, that the analysis carried out in [Abate et al.,
2008] relies on the approach in [Bertsekas and Shreve, 1978] and is purely
analytical; the strong Feller assumption on the transition kernel in our
formulation plays no role there.
## 4\. Proofs
This section collects the proofs of the various results in §2.
### 4.1. Proof of Theorem ((2.10))
We recall a few standard results about set-valued maps first, followed by
sequence of lemmas before getting to the proof of Theorem (2.10). The various
definitions in paragraphs (2.7), (2.8), and (2.13) will be employed without
further reference. Just as in §2.2, for the purposes of this subsection, we
let $\Pi_{M}$ denote the set of admissible Markov policies such that $\pi_{t}$
is defined on $\mathbb{K}$ whenever
$(\pi_{t})_{t\in\mathbb{N}_{0}}\in\Pi_{M}$.
###### (4.1) Proposition ([Aliprantis and Border, 2006, Lemma 17.30]).
Let $\Psi:X\longrightarrow\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\to{\;}Y$ be an upper
hemicontinuous set-valued map between topological spaces with nonempty compact
values, and let $f:\mathop{Graph}(\Psi)\longrightarrow\mathbb{R}$ be upper
semicontinuous.666Recall that $\mathop{Graph}(\Psi)$ is the set
$\bigl{\\{}(x,\Psi(x))\,\big{|}\,x\in X\bigr{\\}}\subseteq X\times Y$, the
graph of the set-valued map $\Psi$. Define the function
$m:X\longrightarrow\mathbb{R}$ by
$m(x)\mathrel{\mathop{:}\\!\\!=}\max_{y\in\Psi(x)}f(x,y)$. Then the function
$m$ is upper semicontinuous.
###### (4.2) Proposition ([Aliprantis and Border, 2006, Theorem 18.19]).
Let $X$ be a separable metrizable space and $(S,\Sigma)$ a measurable space.
Let $\Psi:S\longrightarrow\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\to{\;}X$ be a weakly
measurable correspondence with nonempty compact values, and suppose $f:S\times
X\longrightarrow\mathbb{R}$ is a Carathéodory function.777Recall that a
Carathéodory function $f:S\times X\longrightarrow Y$ is a mapping that is
measurable in the first variable and continuous in the second, where
$(S,\Sigma)$ is a measurable space and $X,Y$ are topological spaces. In
particular, if $X$ is a separable and metrizable space, and $Y$ is a
metrizable space, every Carathéodory function $f:S\times X\longrightarrow Y$
is jointly measurable [Aliprantis and Border, 2006, Lemma 4.51]; this is
clearly true in the Carathéodory functions we consider. Let us also define the
function $m:S\longrightarrow\mathbb{R}$ by
$m(s)\mathrel{\mathop{:}\\!\\!=}\max_{x\in\Psi(s)}f(s,x)$, and the
correspondence $\mu:S\longrightarrow\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\to{\;}X$ of
maximizers by
$\mu(s)\mathrel{\mathop{:}\\!\\!=}\bigl{\\{}x\in\Psi(s)\,\big{|}\,f(s,x)=m(s)\bigr{\\}}$.
Then the argmax correspondence $\mu$ is measurable and admits a measurable
selector.
###### (4.3) Definition.
For $u\in b\mathfrak{B}\\!\left(X\right)^{+}\cap\bar{B}$ we define the mapping
$Tu$
((4.4)) $X\ni x\longmapsto
Tu(x)\mathrel{\mathop{:}\\!\\!=}\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\sup_{a\in A(x)}\int_{K}Q(\mathrm{d}y|x,a)u(y)\in\mathbb{R}_{\geqslant
0}.$
The operator $T$ is called the _dynamic programming operator_ corresponding to
the problem ((2.6)). $\Diamond$
###### (4.5) Lemma.
Suppose that Assumption ((2.9)) holds. Then the dynamic programming operator
$T$ defined in ((4.4)) takes $b\mathfrak{B}\\!\left(X\right)^{+}\cap\bar{B}$
into itself. Moreover, there exists a measurable selector $f\in\mathbb{F}$
such that
((4.6)) $Tu(x)=\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}y|x,f)u(y)\quad\forall\,x\in X.$
###### Proof.
Fix $u\in b\mathfrak{B}\\!\left(X\right)^{+}\cap\bar{B}$. Since the transition
kernel $Q$ is strongly Feller on $\mathbb{K}$, the mapping
$\mathbb{K}\ni(x,a)\longmapsto
S(x,a)\mathrel{\mathop{:}\\!\\!=}\int_{X}Q(\mathrm{d}y|x,a)\boldsymbol{1}_{K}(y)u(y)\in\mathbb{R}_{\geqslant
0}$
is continuous on $\mathbb{K}$. Also, $S(x,a)$ is bounded whenever $u$ is, a
bound of $S$ being the essential supremum norm of $u$. Therefore, since $A(x)$
is compact for each $x\in X$, the function
$S^{\star}(x)\mathrel{\mathop{:}\\!\\!=}\sup_{a\in A(x)}S(x,a)$ is well-
defined on $K\smallsetminus O$, i.e., the sup is attained on $A(x)$ for $x\in
K\smallsetminus O$. We also note that since $K\smallsetminus O$ is a
measurable set, by Assumption (2.9)
* $\circ$
the correspondence $K\smallsetminus O\ni x\longmapsto A(x)\subseteq A$ is
upper hemicontinuous, and since $S$ is continuous on $\mathbb{K}$, the map
$K\smallsetminus O\ni x\longmapsto
S^{\star}(x)\mathrel{\mathop{:}\\!\\!=}\max_{a\in
A(x)}S(x,a)\in\mathbb{R}_{\geqslant 0}$ is an u.s.c. function by Proposition
(4.1);
* $\circ$
the correspondence $K\smallsetminus O\ni x\longmapsto A(x)\subseteq A$ is
weakly measurable, and since $S$ is continuous on $\mathbb{K}$ (and therefore
is a Carathéodory function), there exists a measurable selector
$f\in\mathbb{F}$ such that $S^{\star}(x)=S(x,f(x))$ for all $x\in
K\smallsetminus O$ by Proposition (4.2).
It follows at once that $X\ni x\longmapsto
Tu(x)=\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}y|x,f(x))u(y)\in\mathbb{R}_{\geqslant 0}$ is a member
of the set $b\mathfrak{B}\\!\left(X\right)^{+}$, and the assertion follows. ∎
###### (4.7) Lemma.
Suppose that hypotheses of Theorem ((2.10)) hold. If $u\in
b\mathfrak{B}\\!\left(X\right)^{+}\cap\bar{B}$ satisfies the inequality
$u\leqslant Tu$ pointwise on $X$, then also $u\leqslant V^{\star}$ pointwise
on $X$, where $T$ is the dynamic programming operator in ((4.4)).
###### Proof.
By definition of $T$ it is clear that we only need to examine the validity of
the assertion on $K\smallsetminus O$. Suppose that $u\in
b\mathfrak{B}\\!\left(X\right)^{+}\cap\bar{B}$ satisfies the inequality
$u\leqslant Tu$ pointwise on $X$. By Lemma (4.5) we know that there exists
$f\in\mathbb{F}$ satisfying
$Tu(x)=\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}y|x,f)u(y)\quad\forall\,x\in K\smallsetminus O.$
A straightforward calculation shows that if $u\leqslant Tu$ then $Tu\leqslant
T\circ Tu$ on $K\smallsetminus O$. Fix $x\in K\smallsetminus O$. Applying the
inequality $u\leqslant Tu$ repeatedly we have
$\displaystyle u(x)$
$\displaystyle\leqslant\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}\xi_{1}|x,f)u(\xi_{1})$
$\displaystyle\leqslant\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}\xi_{1}|x,f)\biggl{[}\boldsymbol{1}_{O}(\xi_{1})+\boldsymbol{1}_{K\smallsetminus
O}(\xi_{1})\int_{K}Q(\mathrm{d}\xi_{2}|\xi_{1},f)u(\xi_{2})\biggr{]}$
$\displaystyle\cdots$
and after $n$ steps
$\displaystyle u(x)$
$\displaystyle\leqslant\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}\xi_{1}|x,f)\biggl{[}\boldsymbol{1}_{O}(\xi_{1})+\ldots$
$\displaystyle\qquad\qquad\ldots+\boldsymbol{1}_{K\smallsetminus
O}(\xi_{n-2})\int_{K}Q(\mathrm{d}\xi_{n-1}|\xi_{n-2},f)\biggl{[}\boldsymbol{1}_{O}(\xi_{n-1})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\boldsymbol{1}_{K\smallsetminus
O}(\xi_{n-1})\int_{K}Q(\mathrm{d}\xi_{n}|\xi_{n-1},f)u(\xi_{n})\biggr{]}\cdots\biggr{]}$
$\displaystyle=\Biggl{(}\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}\xi_{1}|x,f)\biggl{[}\boldsymbol{1}_{O}(\xi_{1})+\ldots$
$\displaystyle\qquad\qquad\ldots+\boldsymbol{1}_{K\smallsetminus
O}(\xi_{n-2})\int_{O}Q(\mathrm{d}\xi_{n-1}|\xi_{n-2},f)\biggr{]}\Biggr{)}$
$\displaystyle\quad+\Biggl{(}\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K\smallsetminus O}Q(\mathrm{d}\xi_{1}|x,f)\int_{K\smallsetminus
O}Q(\mathrm{d}\xi_{2}|\xi_{1},f)\cdots\int_{K}Q(\mathrm{d}\xi_{n}|\xi_{n-1},f)u(\xi_{n})\Biggr{)}.$
We claim that the right-hand side of the last equality above is
$\mathsf{E}^{f^{\infty}}_{x}\Biggl{[}\sum_{t=0}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}+\mathsf{E}^{f^{\infty}}_{x}\Bigl{[}\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}\cdot
u)(x_{n\wedge\tau\wedge\tau^{\prime}})\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}<\infty\\}}\Bigr{]},$
where $\boldsymbol{1}_{K}\cdot
u(\xi)\mathrel{\mathop{:}\\!\\!=}\boldsymbol{1}_{K}(\xi)u(\xi)$ for $\xi\in
X$. To see this note that the first term is clear by definition. The second
term above is due to the fact that only those trajectories that stay in
$K\smallsetminus O$ for $n$ steps (i.e., from stage $0$ through stage $n-1$)
contribute to the integrand that features $u$, and this accounts for the
factor $\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})$. Since
$\\{\tau\wedge\tau^{\prime}<\infty\\}$ is a full measure set, the factor
$\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}<\infty\\}}$ does not change the
value of the integral. Taking the limit of the first term above as
$n\to\infty$, the monotone convergence theorem gives
$\lim_{n\to\infty}\mathsf{E}^{f^{\infty}}_{x}\Biggl{[}\sum_{t=0}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}=\mathsf{E}^{f^{\infty}}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}=V(f^{\infty},x)\leqslant
V^{\star}(x),$
where the last inequality follows from the definition of $V^{\star}$. Since
$u$ is bounded and nonnegative, taking the limit of the second term above as
$n\to\infty$, the dominated convergence theorem gives
$\displaystyle\lim_{n\to\infty}$
$\displaystyle\mathsf{E}^{f^{\infty}}_{x}\Bigl{[}\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}\cdot
u)(x_{n\wedge\tau\wedge\tau^{\prime}})\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}<\infty\\}}\Bigr{]}$
$\displaystyle=\mathsf{E}^{f^{\infty}}_{x}\Bigl{[}\boldsymbol{1}_{K\smallsetminus
O}(x_{\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}\cdot
u)(x_{\tau\wedge\tau^{\prime}})\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}<\infty\\}}\Bigr{]}$
$\displaystyle=0$
since $\boldsymbol{1}_{K\smallsetminus O}(x_{\tau\wedge\tau^{\prime}})=0$ on
the set $\\{\tau\wedge\tau^{\prime}<\infty\\}$ by definition of the stopping
times $\tau$ and $\tau^{\prime}$. Substituting back we see that $u(x)\leqslant
V^{\star}(x)$, and the assertion follows since $x\in K\smallsetminus O$ is
arbitrary. ∎
###### (4.8) Lemma.
Suppose that Assumption ((2.9)) holds. Then the value iteration functions
$(v_{n})_{n\in\mathbb{N}_{0}}$ defined in ((3.2)) satisfy $v_{n}\uparrow
V^{\star}$, and the function $V^{\star}$ satisfies the Bellman equation
((2.11)).
###### Proof.
From the definition of the value-iteration functions
$(v_{n})_{n\in\mathbb{N}_{0}}$ in ((3.2)) we see that
$(v_{n})_{n\in\mathbb{N}_{0}}$ is a monotone increasing sequence bounded above
by $\boldsymbol{1}_{X}$. Therefore there exists a measurable function
$v^{\star}:X\longrightarrow[0,1]$ such that $v_{n}\uparrow v^{\star}$
pointwise on $X$. By definition of $v_{n}$ we have
$\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}\leqslant\sup_{\pi\in\Pi_{M}}\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}=v_{n}(x),$
and the monotone convergence theorem shows that
$v^{\star}(x)=\lim_{n\to\infty}v_{n}(x)\geqslant\lim_{n\to\infty}\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}=\mathsf{E}^{\pi}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}=V(\pi,x).$
Taking the supremum over $\pi\in\Pi_{M}$ on the right-hand side shows that
$v^{\star}\geqslant V^{\star}$ pointwise on $X$. Note that $v_{n}|_{O}=1$ and
$v_{n}|_{X\smallsetminus K}=0$ for all $n$; therefore $v^{\star}|_{O}=1$ and
$v^{\star}|_{X\smallsetminus K}=0$.
Let us define the maps
$\displaystyle\mathbb{K}\ni(x,a)\longmapsto T^{\prime}v_{n}(x,a)$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}\int_{K}Q(\mathrm{d}y|x,a)v_{n}(y)\in[0,1],$
$\displaystyle\mathbb{K}\ni(x,a)\longmapsto T^{\prime}v^{\star}(x,a)$
$\displaystyle\mathrel{\mathop{:}\\!\\!=}\int_{K}Q(\mathrm{d}y|x,a)v^{\star}(y)\in[0,1].$
We note that the transition kernel $Q$ is strongly Feller by Assumption (2.9),
and therefore $T^{\prime}v_{n},n\in\mathbb{N}_{0}$ and $T^{\prime}v^{\star}$
are continuous functions on $\mathbb{K}$. Moreover, for all
$n\in\mathbb{N}_{0}$ we define
((4.9)) $\displaystyle T^{\prime}v_{n}(x,a)$
$\displaystyle=T^{\prime}v^{\star}(x,a)=1\quad\text{for $x\in O$ and $a\in
A(x)$},$ $\displaystyle T^{\prime}v_{n}(x,a)$
$\displaystyle=T^{\prime}v^{\star}(x,a)=0\quad\text{for $x\in X\smallsetminus
K$ and $a\in A(x)$},$
Since $v_{n}\uparrow v^{\star}$ pointwise on $X$, it follows from the
definitions above and the monotone convergence theorem that for all $x\in X$
and $a\in A(x)$
((4.10)) $T^{\prime}v_{n}(x,a)\boldsymbol{1}_{K\smallsetminus O}(x)\uparrow
T^{\prime}v^{\star}(x,a)\boldsymbol{1}_{K\smallsetminus O}(x).$
Fix $x\in K\smallsetminus O$. Since $T^{\prime}v_{n}$ and
$T^{\prime}v^{\star}$ are continuous functions on $\mathbb{K}$, for each
$n\in\mathbb{N}_{0}$ both $\sup_{a\in A(x)}T^{\prime}v_{n}(x,a)$ and
$\sup_{a\in A(x)}T^{\prime}v^{\star}(x,a)$ are attained on $A(x)$. From the
definition of $(v_{n})_{n\in\mathbb{N}_{0}}$ in ((3.2)) we have $\max_{a\in
A(x)}T^{\prime}v_{n}(x,a)\leqslant\max_{a\in A(x)}T^{\prime}v^{\star}(x,a)$
for all $n\in\mathbb{N}_{0}$. Also, $\bigl{(}\max_{a\in
A(x)}T^{\prime}v_{n}(x,a)\bigr{)}_{n\in\mathbb{N}_{0}}$ is a nondecreasing
sequence of numbers bounded above by $1$, and therefore it attains a limit. If
this limit is strictly less than $\max_{a\in A(x)}T^{\prime}v^{\star}(x,a)$,
standard easy arguments may be invoked to show that the sequence of continuous
functions $\bigl{(}T^{\prime}v_{n}(x,\cdot)\bigr{)}_{n\in\mathbb{N}_{0}}$
cannot converge pointwise to $T^{\prime}v^{\star}(x,\cdot)$ on $A(x)$, which
contradicts ((4.10)). It follows that whenever $x\in K\smallsetminus O$,
$\displaystyle v^{\star}(x)$
$\displaystyle=\lim_{n\to\infty}v_{n}(x)=\lim_{n\to\infty}Tv_{n-1}(x)$
$\displaystyle=\lim_{n\to\infty}\max_{a\in
A(x)}T^{\prime}v_{n-1}(x,a)=\max_{a\in A(x)}T^{\prime}v^{\star}(x,a)$
$\displaystyle=Tv^{\star}(x).$
Together with ((4.9)) this shows that $v^{\star}$ satisfies the Bellman
equation ((2.11)) pointwise on $X$, i.e., $v^{\star}=Tv^{\star}$. We have
already seen above that $v^{\star}\geqslant V^{\star}$ pointwise on $X$. Since
$v^{\star}=Tv^{\star}$, the reverse inequality follows from Lemma (4.7).
Therefore, we conclude that $v^{\star}=V^{\star}$ identically on $X$. ∎
###### (4.11) Lemma.
Let $f^{\infty}$ be a deterministic stationary policy. Then we have
((4.12)) $V(f^{\infty},x)=\begin{cases}1&\text{if }x\in O,\\\
\displaystyle{\int_{K}Q(\mathrm{d}y|x,f)V(f^{\infty},y)}&\text{if }x\in
K\smallsetminus O,\\\ 0&\text{otherwise}.\end{cases}$
###### Proof.
For $x\in O\cup(X\smallsetminus K)$ the assertions are trivial. Fix $x\in
K\smallsetminus O$. From the definition of $V(f^{\infty},x)$ we have
$\displaystyle V(f^{\infty},x)$
$\displaystyle=\mathsf{E}^{f^{\infty}}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\,\Bigg{|}\,x_{0}=x\Biggr{]}$
$\displaystyle=\mathsf{E}^{f^{\infty}}\Biggl{[}\boldsymbol{1}_{O}(x_{0})\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}=0\\}}+\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}>0\\}}\sum_{t=1}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\,\Bigg{|}\,x_{0}=x\Biggr{]}$
$\displaystyle=\boldsymbol{1}_{O}(x)+\mathsf{E}^{f^{\infty}}\Biggl{[}\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}>0\\}}\sum_{t=1}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\,\Bigg{|}\,x_{0}=x\Biggr{]}.$
Since $\\{\tau\wedge\tau^{\prime}>0\\}=\\{x_{0}\in K\smallsetminus O\\}$ and
this event is $\mathfrak{F}_{0}$-measurable,
$\mathsf{E}^{f^{\infty}}\Biggl{[}\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}>0\\}}\sum_{t=1}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\,\Bigg{|}\,x_{0}=x\Biggr{]}=\boldsymbol{1}_{K\smallsetminus
O}(x)\mathsf{E}^{f^{\infty}}\Biggl{[}\sum_{t=1}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\,\Bigg{|}\,x_{0}=x\Biggr{]}.$
Therefore,
$\displaystyle V(f^{\infty},x)$
$\displaystyle=\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\mathsf{E}^{f^{\infty}}\Biggl{[}\sum_{t=1}^{\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\,\Bigg{|}\,x_{0}=x\Biggr{]}$
$\displaystyle=\boldsymbol{1}_{O}(x)+\boldsymbol{1}_{K\smallsetminus
O}(x)\mathsf{E}^{f^{\infty}}\Biggl{[}\sum_{t=1}^{\tau}\boldsymbol{1}_{O}(x_{t\wedge\tau\wedge\tau^{\prime}})\,\Bigg{|}\,x_{0}=x\Biggr{]}.$
Considering the fact that $V(f^{\infty},x)=0$ for $x\in X\smallsetminus K$ by
definition, the Markov property shows that the second term on the right-hand
side above equals
$\displaystyle\boldsymbol{1}_{K\smallsetminus O}(x)$
$\displaystyle\mathsf{E}^{f^{\infty}}\Biggl{[}\mathsf{E}^{f^{\infty}}\Biggl{[}\sum_{t=1}^{\tau}\boldsymbol{1}_{O}(x_{t\wedge\tau\wedge\tau^{\prime}})\,\bigg{|}\,x_{1\wedge\tau\wedge\tau^{\prime}}\Biggr{]}\,\Bigg{|}\,x_{0}=x\Biggr{]}$
$\displaystyle=\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}y|x,f)\mathsf{E}^{f^{\infty}}\Biggl{[}\sum_{t=1}^{\tau}\boldsymbol{1}_{O}(x_{t\wedge\tau\wedge\tau^{\prime}})\,\Bigg{|}\,x_{1\wedge\tau\wedge\tau^{\prime}}=y\Biggr{]}$
$\displaystyle=\boldsymbol{1}_{K\smallsetminus
O}(x)\int_{K}Q(\mathrm{d}y|x,f)V(f^{\infty},y).$
Collecting the above equations we obtain ((4.12)), and this completes the
proof. ∎
We are now ready for the proof of the first main result.
###### Proof of Theorem ((2.10)).
(i) Note that by definition $V^{\star}$ is nonnegative. The fact that
$V^{\star}$ satisfies the Bellman equation follows from Lemma (4.8). In view
of the definition of $\bar{B}$ in Theorem (2.10) and Lemma (4.8) we conclude
that $V^{\star}$ is minimal in $b\mathfrak{B}\\!\left(X\right)^{+}\cap\bar{B}$
because $u=Tu$ pointwise on $K\smallsetminus O$ implies that $u\leqslant
V^{\star}$ pointwise on $K\smallsetminus O$ for any $u\in
b\mathfrak{B}\\!\left(X\right)^{+}\cap\bar{B}$.
(ii) Lemma (4.5) guarantees the existence of a selector
$f_{\star}\in\mathbb{F}$ such that ((2.12)) holds. Iterating the equality
((2.12)) (or ((2.14))) it follows as in the proof of Lemma (4.7) that for
$x\in X$,
$V^{\star}(x)=\mathsf{E}^{f_{\star}^{\infty}}_{x}\Biggl{[}\sum_{t=0}^{(n-1)\wedge\tau\wedge\tau^{\prime}}\boldsymbol{1}_{O}(x_{t})\Biggr{]}+\mathsf{E}^{f_{\star}^{\infty}}_{x}\bigl{[}\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{n\wedge\tau\wedge\tau^{\prime}})\bigr{]}.$
Taking limits as $n\to\infty$ on the right, the monotone and dominated
convergence theorems give $V^{\star}(x)=V(f_{\star}^{\infty},x)$. Since $x$ is
arbitrary, $V^{\star}(\cdot)=V(f_{\star}^{\infty},\cdot)$ on $K\smallsetminus
O$ and that $f_{\star}^{\infty}$ is an optimal policy. Conversely, by Lemma
(4.11) it follows that under the stationary deterministic strategy
$f_{\star}^{\infty}$ we have ((4.12)) with $f_{\star}$ in place of $f$, which
is identical to ((2.12)).∎
### 4.2. Proofs of the results in §2.3
For the purposes of this subsection we let $\Pi$ denote the set of admissible
policies such that $\pi_{t}$ is defined on $\mathbb{K}$ whenever
$(\pi_{t})_{t\in\mathbb{N}_{0}}\in\Pi$.
###### (4.13) Lemma.
For every policy $\pi\in\Pi$ and initial state $x\in X$ the processes
$(\zeta_{n})_{n\in\mathbb{N}_{0}}$ and
$\bigl{(}\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}\cdot
V^{\star})(x_{n\wedge\tau\wedge\tau^{\prime}})\bigr{)}_{n\in\mathbb{N}_{0}}$
are both nonnegative $(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$\-
supermartingales under $\mathsf{P}^{\pi}_{x}$.
###### Proof.
It is clear that both processes are nonnegative and
$(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$-adapted. Fix $n\in\mathbb{N}$, an
initial state $x\in X$, a policy $\pi\in\Pi$, and on the event
$\\{\tau\wedge\tau^{\prime}>n\\}$ fix a history
$h_{n}=\bigl{(}x,a_{0},x_{1},a_{1},\ldots,x_{n-1},a_{n-1},x_{n}\bigr{)}$. Let
$a_{n}\mathrel{\mathop{:}\\!\\!=}\pi_{n}(h_{n})$ on
$\\{\tau\wedge\tau^{\prime}>n\\}$. Then
$\displaystyle\zeta_{n+1}$
$\displaystyle=W_{n+1}(\pi,x)+\boldsymbol{1}_{K\smallsetminus
O}(x_{n\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{(n+1)\wedge\tau\wedge\tau^{\prime}})$
$\displaystyle=W_{n}(\pi,x)+\boldsymbol{1}_{O}(x_{n\wedge\tau\wedge\tau^{\prime}})\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}=n\\}}+\boldsymbol{1}_{K\smallsetminus
O}(x_{n\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{(n+1)\wedge\tau\wedge\tau^{\prime}})$
$\displaystyle=W_{n}(\pi,x)+\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}=n\\}}\boldsymbol{1}_{O}(x_{n\wedge\tau\wedge\tau^{\prime}})+\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}>n\\}}(\boldsymbol{1}_{K}V^{\star})(x_{(n+1)\wedge\tau\wedge\tau^{\prime}}).$
Since $\\{x_{n\wedge\tau\wedge\tau^{\prime}}\in
O\\}\subseteq\\{\tau\wedge\tau^{\prime}=n\\}$, we have
$\displaystyle\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}=n\\}}$
$\displaystyle\boldsymbol{1}_{O}(x_{n\wedge\tau\wedge\tau^{\prime}})+\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}>n\\}}(\boldsymbol{1}_{K}V^{\star})(x_{(n+1)\wedge\tau\wedge\tau^{\prime}})$
$\displaystyle=\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}\geqslant
n\\}}\bigl{(}\boldsymbol{1}_{O}(x_{n\wedge\tau\wedge\tau^{\prime}})+\boldsymbol{1}_{K\smallsetminus
O}(x_{n\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{(n+1)\wedge\tau\wedge\tau^{\prime}})\bigr{)}.$
Since $\\{\tau\wedge\tau^{\prime}\geqslant
n\\}=\\{\tau\wedge\tau^{\prime}>n-1\\}=\\{x_{(n-1)\wedge\tau\wedge\tau^{\prime}}\in
K\smallsetminus O\\}$, it follows that
$\displaystyle\zeta_{n+1}=W_{n}(\pi,x)$
$\displaystyle+\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})\cdot$
$\displaystyle\bigl{(}\boldsymbol{1}_{O}(x_{n\wedge\tau\wedge\tau^{\prime}})+\boldsymbol{1}_{K\smallsetminus
O}(x_{n\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{(n+1)\wedge\tau\wedge\tau^{\prime}})\bigr{)}.$
Therefore, keeping in mind the definition of $a_{n}$ above,
$\displaystyle\mathsf{E}^{\pi}_{x}\bigl{[}\zeta_{n+1}\,\big{|}\,\mathfrak{F}_{n\wedge\tau\wedge\tau^{\prime}}\bigr{]}$
$\displaystyle=W_{n}(\pi,x)+\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})T^{\prime}V^{\star}(x_{n\wedge\tau\wedge\tau^{\prime}},a_{n})$
((4.14)) $\displaystyle\leqslant W_{n}(\pi,x)+\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})V^{\star}(x_{n\wedge\tau\wedge\tau^{\prime}})$
$\displaystyle=\zeta_{n},$
where the inequality holds $\mathsf{P}^{\pi}_{x}$-almost surely. Therefore,
the process $(\zeta_{n})_{n\in\mathbb{N}_{0}}$ is a nonnegative
$(\mathfrak{F}_{n\wedge\tau\wedge\tau^{\prime}})_{n\in\mathbb{N}_{0}}$\-
supermartingale, and hence also a $(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$\-
supermartingale. Considering that the sequence
$\bigl{(}W_{n}(\pi,x)\bigr{)}_{n\in\mathbb{N}_{0}}$ is nondecreasing, from the
definitions in ((2.15)) and the fact that the process
$(\zeta_{n})_{n\in\mathbb{N}_{0}}$ is a
$(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$\- supermartingale we see that the
process $\bigl{(}\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{n\wedge\tau\wedge\tau^{\prime}})\bigr{)}_{n\in\mathbb{N}_{0}}$
is also a $(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$\- supermartingale under
$\mathsf{P}^{\pi}_{x}$. ∎
###### Proof of Theorem ((2.17)).
Lemma (4.13) confirms that both of the two adapted processes
$(\zeta_{n})_{n\in\mathbb{N}_{0}}$ and
$\bigl{(}\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{n\wedge\tau\wedge\tau^{\prime}})\bigr{)}_{n\in\mathbb{N}}$
converge almost surely and are nonincreasing in expectation, both under
$\mathsf{P}^{\pi}_{x}$. Let
$\Lambda^{\pi}(x)\mathrel{\mathop{:}\\!\\!=}\lim_{n\to\infty}\mathsf{E}^{\pi}_{x}[\zeta_{n}]$.
We then have
$\displaystyle V^{\star}(x)$
$\displaystyle=\mathsf{E}^{\pi}_{x}[\zeta_{0}]\geqslant\lim_{n\to\infty}\mathsf{E}^{\pi}_{x}[\zeta_{n}]$
((4.15))
$\displaystyle=\lim_{n\to\infty}\Bigl{(}\mathsf{E}^{\pi}_{x}\bigl{[}W_{n}(\pi,x)\bigr{]}+\mathsf{E}^{\pi}_{x}\bigl{[}\boldsymbol{1}_{K\smallsetminus
O}(x_{(n-1)\wedge\tau\wedge\tau^{\prime}})(\boldsymbol{1}_{K}V^{\star})(x_{n\wedge\tau\wedge\tau^{\prime}})\bigr{]}\Bigr{)}$
$\displaystyle\geqslant V(\pi,x).$
The assertion is now an immediate consequence of (4.2).∎
###### Proof of Theorem ((2.18)).
Suppose that (i) holds. Since $\mathsf{E}^{\pi}_{x}[\zeta_{n}]$ is
nonincreasing with $n$ it follows that
$\mathsf{E}^{\pi}_{x}[\zeta_{n+1}]=\mathsf{E}^{\pi}_{x}[\zeta_{n}]=\ldots=\mathsf{E}^{\pi}_{x}[\zeta_{0}]=V^{\star}(x)$
for every $n\in\mathbb{N}$. Therefore, equality must hold
$\mathsf{P}^{\pi}_{x}$-almost surely in (4.2), and (ii) follows.
Suppose that (ii) holds. Then equality holds in (4.2) almost surely under
$\mathsf{P}^{\pi}_{x}$, and therefore $\mathsf{P}^{\pi}_{x}$-almost everywhere
on the set $\\{x_{n\wedge\tau\wedge\tau^{\prime}}\in K\smallsetminus
O\\}=\\{\tau\wedge\tau^{\prime}>n\\}$ we have
$T^{\prime}V^{\star}(x_{n},a_{n})=V^{\star}(x_{n})$, and (iii) follows.
Suppose that (iii) holds. Then taking expectations in (4.2) we arrive at
$\mathsf{E}^{\pi}_{x}[\zeta_{n+1}]=\mathsf{E}^{\pi}_{x}[\zeta_{n}]=\ldots=\mathsf{E}^{\pi}_{x}[\zeta_{0}]=V^{\star}(x)$.
As a result we have $\Lambda^{\pi}(x)=V^{\star}(x)$, and (i) follows.∎
###### Proof of Theorem ((2.19)).
It follows readily from the definition of the stopping times $\tau$ and
$\tau^{\prime}$ that the process $(\zeta^{\prime}_{n})_{n\in\mathbb{N}_{0}}$
defined in ((2.20)) is a bounded process, and by assumption it is a
$(\mathfrak{F}_{n})_{n\in\mathbb{N}_{0}}$ -martingale under
$\mathsf{P}^{\pi^{\star}}_{x}$. Doob’s Optional Sampling Theorem [Rao and
Swift, 2006, Theorem 2, p. 422] applied to
$(\zeta^{\prime}_{n})_{n\in\mathbb{N}_{0}}$ at the stopping time
$\tau\wedge\tau^{\prime}$ gives us
$\mathsf{E}^{\pi^{\star}}_{x}\bigl{[}\zeta^{\prime}_{\tau\wedge\tau^{\prime}}\bigr{]}=\mathsf{E}^{\pi^{\star}}_{x}\bigl{[}\zeta^{\prime}_{0}\bigr{]}=V^{\prime}(x),$
where the last equality follows from the definition of $\zeta^{\prime}_{0}$.
From ((2.15)) we get
$\displaystyle\mathsf{E}^{\pi^{\star}}_{x}\bigl{[}\zeta^{\prime}_{\tau\wedge\tau^{\prime}}\bigr{]}$
$\displaystyle=\mathsf{E}^{\pi^{\star}}_{x}\Bigl{[}W_{\tau\wedge\tau^{\prime}-1}(\pi^{\star},x)+\boldsymbol{1}_{K\smallsetminus
O}(x_{\tau\wedge\tau^{\prime}-1})\bigl{(}\boldsymbol{1}_{K}\cdot
V^{\prime}\bigr{)}(x_{\tau\wedge\tau^{\prime}})\Bigr{]}$
$\displaystyle=\mathsf{E}^{\pi^{\star}}_{x}\Biggl{[}\sum_{t=0}^{\tau\wedge\tau^{\prime}-1}\boldsymbol{1}_{O}(x_{t})+\boldsymbol{1}_{K\smallsetminus
O}(x_{\tau\wedge\tau^{\prime}-1})\bigl{(}\boldsymbol{1}_{K}\cdot
V^{\prime}\bigr{)}(x_{\tau\wedge\tau^{\prime}})\Biggr{]}$
$\displaystyle=\mathsf{E}^{\pi^{\star}}_{x}\Bigl{[}\boldsymbol{1}_{K\smallsetminus
O}(x_{\tau\wedge\tau^{\prime}-1})\bigl{(}\boldsymbol{1}_{K}\cdot
V^{\prime}\bigr{)}(x_{\tau\wedge\tau^{\prime}})\Bigr{]}.$
By definition of $\tau$ and $\tau^{\prime}$, $\boldsymbol{1}_{K\smallsetminus
O}(x_{\tau\wedge\tau^{\prime}-1})$ equals $1$ on
$\\{\tau\wedge\tau^{\prime}<\infty\\}$, and by our hypotheses the set
$\\{\tau\wedge\tau^{\prime}<\infty\\}$ is a
$\mathsf{P}^{\pi^{\star}}_{x}$-full-measure set. Continuing from the last
equality above we arrive at
$\displaystyle\mathsf{E}^{\pi^{\star}}_{x}\bigl{[}\zeta^{\prime}_{\tau\wedge\tau^{\prime}}\bigr{]}$
$\displaystyle=\mathsf{E}^{\pi^{\star}}_{x}\Bigl{[}\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}<\infty\\}}\bigl{(}\boldsymbol{1}_{K}\cdot
V^{\prime}\bigr{)}(x_{\tau\wedge\tau^{\prime}})\Bigr{]}$
$\displaystyle=\mathsf{E}^{\pi^{\star}}_{x}\bigl{[}\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}<\infty\\}}\bigl{(}\boldsymbol{1}_{\\{\tau<\tau^{\prime}\\}}\boldsymbol{1}_{K}(x_{\tau})V^{\prime}(x_{\tau})+\boldsymbol{1}_{\\{\tau>\tau^{\prime}\\}}\boldsymbol{1}_{K}(x_{\tau^{\prime}})V^{\prime}(x_{\tau^{\prime}})\bigr{)}\bigr{]}$
((4.16))
$\displaystyle=\mathsf{E}^{\pi^{\star}}_{x}\bigl{[}\boldsymbol{1}_{\\{\tau\wedge\tau^{\prime}<\infty\\}}\boldsymbol{1}_{\\{\tau<\tau^{\prime}\\}}\bigr{]}$
$\displaystyle=\mathsf{P}^{\pi^{\star}}_{x}\bigl{(}\tau<\tau^{\prime},\tau<\infty\bigr{)},$
where the equality in ((4.16)) follows from the assumptions on $V^{\prime}$
and the definitions of $\tau$ and $\tau^{\prime}$. Collecting the equations
above we get
$V^{\prime}(x)=\mathsf{P}^{\pi^{\star}}_{x}\bigl{(}\tau<\tau^{\prime},\tau<\infty\bigr{)}$
as asserted.∎
It is of interest to note that the hypotheses of Theorem (2.19) requires at
least one of the stopping times $\tau$ or $\tau^{\prime}$ to be finite. Let us
examine the case of $\tau\wedge\tau^{\prime}$ being $\infty$ on a set of
positive probability. Following the proof of Theorem (2.19), we see that in
this case we have to agree on the value of
$V^{\prime}(x_{\tau\wedge\tau^{\prime}})$ on
$\\{\tau\wedge\tau^{\prime}=\infty\\}$. If
$\lim_{n\to\infty}V^{\prime}_{n}(\pi^{\star},x)$ exists, then we can always
let $V^{\prime}(x_{\tau\wedge\tau^{\prime}})$ take this value on the set
$\\{\tau\wedge\tau^{\prime}=\infty\\}$. However, the context of the problem
offers another alternative, namely, to set
$V^{\prime}(x_{\tau\wedge\tau^{\prime}})=0$ on
$\\{\tau\wedge\tau^{\prime}=\infty\\}$. This is because if $x_{t}\in
K\smallsetminus O$ for all $t\in\mathbb{N}_{0}$, then the value of
$x_{\tau\wedge\tau^{\prime}}$ is of no consequence at all.
## 5\. Conclusions and Future Work
The purpose of this article was to present a dynamic programming based
solution to the problem of maximizing the probability of attaining a target
set before hitting a cemetery set, and furnish an alternative martingale
characterization of optimality in terms of thrifty and equalizing policies.
Several related problems of interest were sketched in §3.1. Some of these
problems do not admit an immediate solution in the dynamic programming
framework we established here because of our central assumption that the cost-
per-stage function is nonnegative. This issue deserves further investigation.
The results in this article also provide clear indications to the possibility
of developing verification tools for probabilistic computation tree logic
[Kwiatkowska et al., 2007] in terms of dynamic programming operators. This
matter is under investigation and will be reported in [Ramponi et al., 2009].
Implementation of the dynamic-programming algorithm in this article is
challenging due to integration over subsets of the state-space, and suboptimal
policies are needed. In this context development of a possible connection with
‘greedy-time-optimal’ policies [Meyn, 2008, Chapters 4, 7], originally
proposed as a tractable alternative to optimal policies in demand-driven
large-scale production systems, is being sought.
## Acknowledgement
The authors thank Onésimo Hernández-Lerma for helpful suggestions and pointers
to references, and Sean Summers for posing the problem.
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|
arxiv-papers
| 2009-04-27T19:33:01 |
2024-09-04T02:49:02.175699
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Debasish Chatterjee, Eugenio Cinquemani and John Lygeros",
"submitter": "Debasish Chatterjee",
"url": "https://arxiv.org/abs/0904.4143"
}
|
0904.4145
|
Hölder continuity of solutions to the Monge-Ampère equations
on compact Kähler manifolds
PHAM HOANG HIEP
ABSTRACT. We study Hölder continuity of solutions to the Monge-Ampère
equations on compact Kähler manifolds. In [DNS] the authors have shown that
the measure $\omega_{u}^{n}$ is moderate if $u$ is Hölder continuous. We prove
a theorem which is a partial converse to this result. 2000 Mathematics Subject
Classification: Primary 32W20, Secondary 32Q15. Key words and phrases: Hölder
continuity, Complex Monge-Ampère operator, $\omega$-plurisubharmonic
functions, compact Kähler manifolds.
1\. Introduction Let $X$ be a compact $n$-dimensional Käler manifold equipped
with a fundamental form $\omega$ satisfying $\int\limits_{X}\omega^{n}=1$. An
upper semicontinuous function $\varphi:\ X\to[-\infty,+\infty)$ is called
$\omega$-plurisubharmonic ($\omega$-psh) if $\varphi\in L^{1}(X)$ and
$\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$. By PSH$(X,\omega)$ (resp.
PSH${}^{-}(X,\omega)$) we denote the set of $\omega$-psh (resp. negative
$\omega$-psh) functions on $X$. The complex Monge-Ampère equation
$\omega_{u}^{n}=f\omega^{n}$ was solved for smooth positive $f$ in the
fundamental work of S. T. Yau (see [Yau]). Later S. Kolodziej showed that
there exists a continuous solution if $f\in L^{p}(\omega^{n})$, $f\geq 0$,
$p>1$ (see [Ko2]). Recently in [Ko5] he proved that this solution is Hölder
continuous in this case (see also [EGZ] for the case $X=\mathbb{C}P^{n}$). In
Corollary 1.2 in [DNS] the authors have shown that the measure
$\omega_{u}^{n}$ is moderate if $u$ is Hölder continuous. The main result is
the following theorem which give a partial answer to the converse problem:
Theorem A. Let $\mu$ be a non-negative Radon measure on $X$ such that
$\mu(B(z,r))\leq Ar^{2n-2+\alpha},$
for all $B(z,r)\subset X$ ($A,\alpha>0$ are constants). Then for every $f\in
L^{p}(d\mu)$ with $p>1$, $\int\limits_{X}fd\mu=1$, there exists a Hölder
continuous $\omega$-psh function $u$ such that $\omega_{u}^{n}=fd\mu$. The
following results are simple applications of Theorem A: Corollary B. Let
$\varphi\in$PSH$(X,\omega)$ be a Hölder continuous function. Then for every
$f\in L^{p}(\omega_{\varphi}\wedge\omega^{n-1})$ with $p>1$,
$\int\limits_{X}f\omega_{\varphi}\wedge\omega^{n-1}=1$, there exists a Hölder
continuous $\omega$-psh function $u$ such that
$\omega_{u}^{n}=f\omega_{\varphi}\wedge\omega^{n-1}$. Corollary C. Let $S$ be
a $C^{1}$ smooth real hypersurface in $X$ and $V_{S}$ be the volume measure on
$S$. Then for every $f\in L^{p}(dV_{S})$ with $p>1$,
$\int\limits_{X}fdV_{S}=1$, there exists a Hölder continuous $\omega$-psh
function $u$ such that $\omega_{u}^{n}=fdV_{S}$. Acknowledgments. The author
is grateful to Slawomir Dinew and Nguyen Quang Dieu for valuable comments. The
author is also indebted to the referee for his useful comments that helped to
improve the paper.
2\. Preliminaries First we recall some elements of pluripotential theory that
will be used throughout the paper. Details can be found in [BT1-2], [Ce1-2],
[CK], [CGZ], [De1-3], [Di1-3], [GZ1-2], [H], [Ko1-5], [KoTi], [Si1-2],
[Ze1-2]. 2.1. In [Ko2] Kołodziej introduced the capacity $C_{X}$ on $X$ by
$C_{X}(E):=\sup\\{\int\limits_{E}\omega_{\varphi}^{n}:\
\varphi\in\text{PSH}(X,\omega),\ -1\leq\varphi\leq 0\\}$
for all Borel sets $E\subset X$. 2.2. In [GZ1] Guedj and Zeriahi introduced
the Alexander capacity $T_{X}$ on $X$ by
$T_{X}(E)=e^{-\sup\limits_{X}V_{E,X}^{*}}$
for all Borel sets $E\subset X$. Here $V_{E,X}^{*}$ is the global extremal
$\omega$-psh function for E defined as the smallest upper semicontinuous
majorant of $V_{E,X}$ i.e,
$V_{E,X}(z)=\sup\\{\varphi(z):\ \varphi\in\text{PSH}(X,\omega),\ \varphi\leq
0\ \text{on}\ E\\}.$
2.3. The following definition was introduced in [EGZ]: A probability measure
$\mu$ on $X$ is said to satisfy the condition $\Cal{H}(\alpha,A)$
($\alpha,A>0$) if
$\mu(K)\leq AC_{X}(K)^{1+\alpha},$
for any Borel subset $K$ of $X$. A probability measure $\mu$ on $X$ is said to
satisfy the condition $\Cal{H}(\infty)$ if for any $\alpha>0$ there exist
$A(\alpha)>0$ dependent on $\alpha$ such that
$\mu(K)\leq A(\alpha)C_{X}(K)^{1+\alpha},$
for any Borel subset $K$ of $X$. 2.4. The following definition was introduced
in [DS]: A measure $\mu$ is said to be moderate if for any open set $U\subset
X$, any compact set $K\subset\subset U$ and any compact family $\Cal{F}$ of
plurisubharmonic functions on $U$, there are constants $\alpha>0$ such that
$\sup\\{\int\limits_{K}e^{-\alpha\varphi}d\mu:\ \varphi\in\Cal{F}\\}<+\infty.$
2.5. The following class of $\omega$-psh functions was investigated by Guedj
and Zeriahi in [GZ2]:
$\Cal{E}(X,\omega)=\\{\varphi\in\text{PSH}(X,\omega):\
\lim\limits_{j\to\infty}\int\limits_{\\{\varphi>-j\\}}\omega_{\max(\varphi,-j)}^{n}=\int\limits_{X}\omega^{n}=1\\}.$
Let us also define
$\Cal{E}^{-}(X,\omega)=\Cal{E}(X,\omega)\cap\text{PSH}^{-}(X,\omega).$
We refer to [GZ2] for the properties of the class $\Cal{E}(X,\omega)$. 2.6.
$S$ is called a $C^{1}$ smooth real hypersurface in $X$ if for all $z\in X$
there exists a neighborhood $U$ of $z$ and $\chi\in C^{1}(U)$ such that $S\cap
U=\\{z\in U:\ \chi(z)=0\\}$ and $D\chi(z)\not=0$ for all $z\in S\cap U$. Next
we state a well-known result needed for our work. 2.7. Proposition. Let $\mu$
be a non-negative Radon measure on $X$ such that $\mu(B(z,r))\leq
Ar^{2n-2+\alpha}$ for all $B(z,r)\subset X$ ($A,\ \alpha>0$ are constants).
Then $\mu\in\Cal{H}(\infty)$. Proof. By Theorem 7.2 in [Ze2] and Proposition
7.1 in [GZ1] we can find $\epsilon,C>0$ such that
$\mu(K)\leq
Ah^{2n-2+\alpha}(K)\leq\frac{AC}{\alpha}T_{X}(K)^{\epsilon\alpha}\leq\frac{ACe}{\alpha}e^{-\frac{\epsilon\alpha}{C_{X}(K)^{\frac{1}{n}}}},$
for all Borel subsets $K$ of $X$, where $h^{2n-2+\alpha}$ is the Hausdorff
content of dimension $2n-2+\alpha$. This implies that $\mu\in\Cal{H}(\infty)$.
3\. Stability of the solutions The stability estimate of solutions to the
Monge-Ampère equations on compact Kähler manifolds was obtained by Kolodziej
([Ko2]). Recently, in [DZ] S. Dinew and Z. Zhang proved a stronger version of
this estimate. We will show a generalization of the stability theorem by S.
Kolodziej. As a first step we have the following proposition. This proof
follows ideas of the proof of Theorem 2.5 in [DH]. We include a proof for the
reader’s convenience. 3.1. Proposition. Let
$\varphi,\psi\in\Cal{E}^{-}(X,\omega)$ be such that
$\omega_{\varphi}^{n}\in\Cal{H}(\alpha,A)$. Then there exist constants
$t\in\mathbb{R}$ and $C(\alpha,A)\geq 0$ such that
$\int\limits_{\\{|\varphi-\psi-t|>a\\}}(\omega_{\varphi}^{n}+\omega_{\psi}^{n})\leq
C(\alpha,A)a^{n+1},$
here
$a=[\int\limits_{X}||\omega_{\varphi}^{n}-\omega_{\psi}^{n}||]^{\frac{1}{2n+3+\frac{n+1}{1+\alpha}}}$.
Proof. Since
$\int\limits_{\\{|\varphi-\psi-t|>a\\}}(\omega_{\varphi}^{n}+\omega_{\psi}^{n})\leq
2$, it suffices to consider the case when $a$ is small. Set
$\epsilon=\frac{1}{2}\inf\\{\int\limits_{\\{|\varphi-\psi-t|>a\\}}\omega_{\varphi}^{n}:\
t\in\mathbb{R}\\}$
Hence
$\int\limits_{\\{|\varphi-\psi-t|\leq a\\}}\omega_{\varphi}^{n}\leq
1-2\epsilon$
for all $t\in\mathbb{R}$. Set
$t_{0}=\sup\\{t\in{\mathbb{R}}:\
\int\limits_{\\{\varphi<\psi+t+a\\}}\omega_{\varphi}^{n}\leq 1-\epsilon\\}$
Replacing $\psi$ by $\psi+t_{0}$ we can assume that $t_{0}=0$. Then
$\int\limits_{\\{\varphi<\psi+a\\}}\omega_{\varphi}^{n}\leq 1-\epsilon$ and
$\int\limits_{\\{\varphi\leq\psi+a\\}}\omega_{\varphi}^{n}\geq 1-\epsilon$.
Hence
$\displaystyle\int\limits_{\\{\psi<\varphi+a\\}}\omega_{\varphi}^{n}=1-\int\limits_{\\{\varphi+a\leq\psi\\}}\omega_{\varphi}^{n}=1-\int\limits_{\\{\varphi\leq\psi+a\\}}\omega_{\varphi}^{n}$
$\displaystyle+\int\limits_{\\{\psi-a<\varphi\leq\psi+a\\}}\omega_{\varphi}^{n}\leq
1-\epsilon.$
Since $\int\limits_{\\{|\varphi-\psi|\leq a\\}}\omega_{\varphi}^{n}\leq 1$ we
can choose $s\in[-a+a^{n+2},a-a^{n+2}]$ satisfying
$\int\limits_{\\{|\varphi-\psi-s|<a^{n+2}\\}}\omega_{\varphi}^{n}\leq
2{a^{n+1}}.$
Replacing $\psi$ by $\psi+s$ we can assume that $s=0$. One easily obtains the
following inequalities
$\int\limits_{\\{\varphi<\psi+a^{n+2}\\}}\omega_{\varphi}^{n}\leq 1-\epsilon,\
\int\limits_{\\{\psi<\varphi+a^{n+2}\\}}\omega_{\varphi}^{n}\leq 1-\epsilon,\
\int\limits_{\\{|\varphi-\psi|<a^{n+2}\\}}\omega_{\varphi}^{n}\leq 2a^{n+1}.$
$None$
By [GZ2] we can find $\rho\in\Cal{E}(X,\omega)$, such that
$\omega_{\rho}^{n}=\frac{1}{1-\epsilon}1_{\\{\varphi<\psi\\}}\omega_{\varphi}^{n}+c1_{\\{\varphi\geq\psi\\}}\omega_{\varphi}^{n}\
\text{and}\ \sup\limits_{X}\rho=0,$ $None$
($c\geq 0$ is chosen so that the measure has total mass $1$). For simplicity
of notation we set $\beta=\frac{n+1}{1+\alpha}$. Set
$U=\\{(1-a^{n+2+\beta})\varphi<(1-a^{n+2+\beta})\psi+a^{n+2+\beta}\rho\\}\subset\\{\varphi<\psi\\}.$
From Theorem 2.1 in [Di3] and (2) we get
$\omega_{\varphi}^{n-1}\wedge\omega_{(1-a^{n+2+\beta})\psi+a^{n+2+\beta}\rho}\geq(1-a^{n+2+\beta})\omega_{\varphi}^{n-1}\wedge\omega_{\psi}+\frac{a^{n+2+\beta}}{(1-\epsilon)^{\frac{1}{n}}}\omega_{\varphi}^{n},$
$None$
on $U$. From Theorem 2.3 in [Di3], Lemma 2.6 in [DH] and (3) we obtain
$\displaystyle(1-a^{n+2+\beta})\int\limits_{U}\omega_{\varphi}^{n-1}\wedge\omega_{\psi}+\frac{a^{n+2+\beta}}{(1-\epsilon)^{\frac{1}{n}}}\int\limits_{U}\omega_{\varphi}^{n}$
$\displaystyle\leq\int\limits_{U}\omega_{(1-a^{n+2+\beta})\psi+a^{n+2+\beta}\rho}\wedge\omega_{\varphi}^{n-1}$
$\displaystyle\leq\int\limits_{U}\omega_{(1-a^{n+2+\beta})\varphi}\wedge\omega_{\varphi}^{n-1}=(1-a^{n+2+\beta})\int\limits_{U}\omega_{\varphi}^{n}+a^{n+2+\beta}\int\limits_{U}\omega\wedge\omega_{\varphi}^{n-1}$
$\displaystyle\leq(1-a^{n+2+\beta})(\int\limits_{U}\omega_{\varphi}^{n-1}\wedge\omega_{\psi}+2a^{2n+3+\beta})+a^{n+2+\beta}\int\limits_{U}\omega\wedge\omega_{\varphi}^{n-1}.$
Hence
$\frac{1}{(1-\epsilon)^{\frac{1}{n}}}\int\limits_{U}\omega_{\varphi}^{n}\leq
2a^{n+1}+\int\limits_{U}\omega\wedge\omega_{\varphi}^{n-1}.$ $None$
From Proposition 3.6 in [GZ1] and (4) we get
$\displaystyle(5)$
$\displaystyle\frac{1}{(1-\epsilon)^{\frac{1}{n}}}[\int\limits_{\\{\varphi\leq\psi-a^{n+2}\\}}\omega_{\varphi}^{n}-C_{1}(\alpha,A)a^{n+1}]$
$\displaystyle\leq\frac{1}{(1-\epsilon)^{\frac{1}{n}}}[\int\limits_{\\{\varphi\leq\psi-a^{n+2}\\}}\omega_{\varphi}^{n}-A[C_{X}(\\{\rho\leq-\frac{1}{2a^{\beta}}\\})]^{1+\alpha}]$
$\displaystyle\leq\frac{1}{(1-\epsilon)^{\frac{1}{n}}}[\int\limits_{\\{\varphi\leq\psi-a^{n+2}\\}}\omega_{\varphi}^{n}-\int\limits_{\\{\rho\leq-\frac{1}{2a^{\beta}}\\}}\omega_{\varphi}^{n}]$
$\displaystyle\leq\frac{1}{(1-\epsilon)^{\frac{1}{n}}}\int\limits_{U}\omega_{\varphi}^{n}$
$\displaystyle\leq 2a^{n+1}+\int\limits_{U}\omega\wedge\omega_{\varphi}^{n-1}$
$\displaystyle\leq
2a^{n+1}+\int\limits_{\\{\varphi<\psi\\}}\omega\wedge\omega_{\varphi}^{n-1},$
Similarly to $\rho$ we define $\vartheta\in\Cal{E}(X,\omega)$, such that
$\omega_{\vartheta}^{n}=\frac{1}{1-\epsilon}1_{\\{\varphi<\psi\\}}\omega_{\varphi}^{n}+l1_{\\{\psi\geq\varphi\\}}\omega_{\varphi}^{n}\
\text{and}\ \sup\limits_{X}\vartheta=0,$
($l$ plays the same role as $c$ above). Set
$V=\\{(1-a^{n+2+\beta})\psi<(1-a^{n+2+\beta})\varphi+a^{n+2+\beta}\vartheta\\}\subset\\{\psi<\varphi\\}.$
We get
$\frac{1}{(1-\epsilon)^{\frac{1}{n}}}[\int\limits_{\\{\psi\leq\varphi-a^{n+2}\\}}\omega_{\varphi}^{n}-C_{1}(\alpha,A)a^{n+1}]\leq
2a^{n+1}+\int\limits_{\\{\psi<\varphi\\}}\omega\wedge\omega_{\varphi}^{n-1}.$
$None$
From (1), (5) and (6) we obtain
$\displaystyle\frac{1}{(1-\epsilon)^{\frac{1}{n}}}[1-2a^{n+1}-2C_{1}(\alpha,A)a^{n+1}]$
$\displaystyle\leq\frac{1}{(1-\epsilon)^{\frac{1}{n}}}[\int\limits_{\\{|\varphi-\psi|\geq
a^{n+1}\\}}\omega_{\varphi}^{n}-2C_{1}(\alpha,A)a^{1+\alpha}]$
$\displaystyle\leq 4a^{n+1}+1.$
Hence
$\epsilon\leq 1-[\frac{1-2(C_{1}(\alpha,A)+1)a^{n+1}}{4a^{n+1}+1}]^{n}\leq
C_{2}(\alpha,A)a^{n+1}.$
This implies that there exists $t\in\mathbb{R}$ satisfying
$\int\limits_{\\{|\varphi-\psi-t|>a\\}}\omega_{\varphi}^{n}\leq
2C_{2}(\alpha,A)a^{n+1}.$
Finally we have
$\displaystyle\int\limits_{\\{|\varphi-\psi-t|>a\\}}(\omega_{\varphi}^{n}+\omega_{\psi}^{n})$
$\displaystyle=2\int\limits_{\\{|\varphi-\psi-t|>a\\}}\omega_{\varphi}^{n}+\int\limits_{\\{|\varphi-\psi-t|>a\\}}(\omega_{\psi}^{n}-\omega_{\varphi}^{n})$
$\displaystyle\leq 2C_{2}(\alpha,A)a^{n+1}+a^{2n+3+\beta}\leq
C(\alpha,A)a^{n+1}.$
The second step in proving our stability therem is the the following
3.2. Proposition. Let $\varphi,\psi\in\Cal{E}^{-}(X,\omega)$ be such that
$\omega_{\varphi}^{n},\omega_{\psi}^{n}\in\Cal{H}(\alpha,A)$. Then there exist
constants $t\in\mathbb{R}$ and $C(\alpha,A)\geq 0$ such that
$C_{X}(\\{|\varphi-\psi-t|>a\\})\leq C(\alpha,A)a,$
here
$a=[\int\limits_{X}||\omega_{\varphi}^{n}-\omega_{\psi}^{n}||]^{\frac{1}{2n+3+\frac{n+1}{1+\alpha}}}$.
Proof. Since $C_{X}(\\{|\varphi-\psi-t|>a\\})\leq C_{X}(X)=1$, it suffices to
consider the case when $a$ is small. Without loss of generality we can assume
that $\sup\limits_{X}\varphi=\sup\limits_{X}\psi=0$. By Remark 2.5 in [EGZ]
there exists $M(\alpha,A)>0$ such that
$||\varphi||_{L^{\infty}(X)}<M(\alpha,A)$,
$||\psi||_{L^{\infty}(X)}<M(\alpha,A)$. By Proposition 3.1 we can find $t>0$
such that
$\int\limits_{\\{|\varphi-\psi-t|>a\\}}(\omega_{\varphi}^{n}+\omega_{\psi}^{n})\leq
C_{1}(\alpha,A)a^{n+1}.$
We consider the case $a<\min(1,\frac{1}{C_{1}(\alpha,A)})$. Since
$\int\limits_{\\{|\varphi-\psi-t|>a\\}}(\omega_{\varphi}^{n}+\omega_{\psi}^{n})<1$
we get $\\{|\varphi-\psi-t|>a\\}\not=X$. This implies that
$|t|\leq\sup\limits_{X}|\varphi-\psi|+1\leq M(\alpha,A)+1$. Replacing $\psi$
by $\psi+t$ we can assume that $t=0$ and
$||\psi||_{L^{\infty}(X)}<2M(\alpha,A)+1$. Using Lemma 2.3 in [EGZ] for
$s=\frac{a}{2}$, $t=\frac{a}{2(2M(\alpha,A)+1)}$ we get
$\displaystyle C_{X}(\\{\varphi-\psi<-a\\})$ $\displaystyle\leq
C_{X}(\\{\varphi-\psi<-\frac{a}{2}-\frac{a}{2(2M(\alpha,A)+1)}\\})$
$\displaystyle\leq\frac{2^{n}(2M(\alpha,A)+1)^{n}}{a^{n}}\int\limits_{\\{\varphi-\psi<-a\\}}\omega_{\varphi}^{n}$
$\displaystyle\leq 2^{n}(2M(\alpha,A)+1)^{n}C_{1}(\alpha,A)a.$
Similarly we get
$C_{X}(\\{\psi-\varphi<-a\\})\leq 2^{n}(2M(\alpha,A)+1)^{n}C_{1}(\alpha,A)a.$
Combination of these inequalities yields
$C_{X}(\\{|\varphi-\psi|>a\\})\leq C(\alpha,A)a.$
Now we prove the promised generalization of Kolodziej stability theorem
(Theorem 1.1 in [Ko5]). 3.3. Theorem. Let
$\varphi,\psi\in\Cal{E}^{-}(X,\omega)$ be such that
$\sup\limits_{X}\varphi=\sup\limits_{X}\psi=0$ and
$\omega_{\varphi}^{n},\omega_{\psi}^{n}\in\Cal{H}(\alpha,A)$. Then there
exists $C(\alpha,A)>0$ such that
$\sup\limits_{X}|\varphi-\psi|\leq
C(\alpha,A)[\int\limits_{X}||\omega_{\varphi}^{n}-\omega_{\psi}^{n}||]^{\frac{\min(1,\frac{\alpha}{n})}{2n+3+\frac{n+1}{1+\alpha}}}.$
Proof. Set
$a=[\int\limits_{X}||\omega_{\varphi}^{n}-\omega_{\psi}^{n}||]^{\frac{1}{2n+3+\frac{n+1}{1+\alpha}}}.$
By Proposition 3.2 there exists $C_{1}(\alpha,A)>0$ and $t\in\mathbb{R}$ such
that $|t|\leq M(\alpha,A)+1$ and
$C_{X}(\\{|\varphi-\psi-t|>a\\})\leq C_{1}(\alpha,A)a.$
Moreover, by Proposition 2.6 in [EGZ] there exists $C_{2}(\alpha,A)>0$ such
that
$\displaystyle\sup\limits_{X}|\varphi-\psi-t|$ $\displaystyle\leq
2a+C_{2}(\alpha,A)[C_{X}(\\{|\varphi-\psi-t|>a\\})]^{\frac{\alpha}{n}}$
$\displaystyle\leq 2a+C_{2}(\alpha,A)[C_{1}(\alpha,A)a]^{\frac{\alpha}{n}}$
$\displaystyle\leq C_{3}(\alpha,A)a^{\min(1,\frac{\alpha}{n})}.$
Moreover, since $\sup\limits_{X}\varphi=\sup\limits_{X}\psi=0$ we obtain
$|t|\leq C_{3}(\alpha,A)a^{\min(1,\frac{\alpha}{n})}$. Combination of these
inequalities yields
$\sup\limits_{X}|\varphi-\psi|\leq\sup\limits_{X}|\varphi-\psi-t|+|t|\leq
2C_{3}(\alpha,A)a^{\min(1,\frac{\alpha}{n})}=C(\alpha,A)[\int\limits_{X}||\omega_{\varphi}^{n}-\omega_{\psi}^{n}||]^{\frac{\min(1,\frac{\alpha}{n})}{2n+3+\frac{n+1}{1+\alpha}}}.$
3.4. Corollary. Let $\mu$ be a non-negative Radon measure on $X$ such that
$\mu(B(z,r))\leq Ar^{2n-2+\alpha}$ for all $B(z,r)\subset X$ ($A,\ \alpha>0$
are constants). Given $p>1,M>0,\epsilon>0$ and $f,g\in L^{p}(d\mu)$ with
$||f||_{L^{p}(d\mu)},||g||_{L^{p}(d\mu)}\leq M$ and
$\int\limits_{X}fd\mu=\int\limits_{X}gd\mu=1$. Assume that
$\varphi,\psi\in\Cal{E}^{-}(X,\omega)$ satisfy $\omega_{\varphi}^{n}=fd\mu$,
$\omega_{\psi}^{n}=gd\mu$ and $\sup\limits_{X}\varphi=\sup\limits_{X}\psi=0$.
Then there exists $C(\alpha,A,M,\epsilon)>0$ such that
$\sup\limits_{X}|\varphi-\psi|\leq
C(\alpha,A,M,\epsilon)[\int\limits_{X}|f-g|d\mu]^{\frac{1}{2n+3+\epsilon}}.$
Proof. By Hölder inequality we have
$\int\limits_{K}fd\mu\leq||f||_{L^{p}(d\mu)}[\mu(K)]^{1-\frac{1}{p}}\leq
M[\mu(K)]^{1-\frac{1}{p}},$
$\int\limits_{K}gd\mu\leq||g||_{L^{p}(d\mu)}[\mu(K)]^{1-\frac{1}{p}}\leq
M[\mu(K)]^{1-\frac{1}{p}},$
for any Borel subset $K$ of $X$. By Proposition 2.7 we get
$fd\mu,gd\mu\in\Cal{H}(\infty)$. Using Theorem 3.3 we can find
$C(\alpha,A,M,\epsilon)>0$ such that
$\sup\limits_{X}|\varphi-\psi|\leq
C(\alpha,A,M,\epsilon)[\int\limits_{X}|f-g|d\mu]^{\frac{1}{2n+3+\epsilon}}.$
4\. Local estimates in Potential theory Let $\Omega$ be a bounded domain in
$\mathbb{R}^{n}$ ($n\geq 2$). By SH$(\Omega)$ (resp SH${}^{-}(\Omega)$) we
denote the set of subharmonic (resp. negative subharmonic) functions on
$\Omega$. For each $u\in SH(\Omega)$ and $\delta>0$ we denote
$\tilde{u}_{\delta}(x)=\frac{1}{c_{n}\delta^{n}}\int\limits_{B_{\delta}}u(x+y)dV_{n}(y),$
$u_{\delta}(x)=\sup\limits_{y\in B_{\delta}}u(x+y),$
for $x\in\Omega_{\delta}=\\{x\in\Omega:d(x,\partial\Omega)>\delta\\}$. Here
$B_{\delta}=\\{x\in\mathbb{R}^{n}:\
|x|=(x_{1}^{2}+...+x_{n}^{2})^{\frac{1}{2}}<\delta\\}$ and $c_{n}$ is the
volume of the unit ball $B_{1}$. We state some results which will be used in
our main theorems. 4.1. Theorem. Let $\mu$ be a non-negative Radon measure on
$\Omega$ such that $\mu(B(z,r))\leq Ar^{n-2+\alpha}$ for all $B(z,r)\subset
D\subset\subset\Omega$ ($A,\alpha>0$ are constants). Then for $K\subset\subset
D$ and $\epsilon>0$ there exists $C(\alpha,A,K,\epsilon)$ such that
$\int\limits_{K}[\tilde{u}_{\delta}-u]d\mu\leq
C(\alpha,A,K,\epsilon)\int\limits_{\bar{D}}\Delta u\
\delta^{\frac{\alpha-\epsilon}{1+\alpha}},$
for all $u\in\text{SH}(\Omega)$, where $\Delta$ is the Laplace operator.
Proof. Since the change of radii of the balls does not affect the statement we
can assume that $\Omega=B_{4}$, $D=B_{3}$, $K=B_{1}$ and $u$ is smooth on
$B_{4}$. By [Hö] we have
$u(x)=\int\limits_{B_{2}}G(x,z)\Delta u(z)+h(x),$
where $G(x,y)$ is the fundamental solution of Laplace equation and $h$ is
harmonic in $B_{2}$. By Fubini theorem we have
$\begin{aligned}
\int\limits_{B_{1}}[\tilde{u}_{\delta}(x)-u(x)]d\mu(x)=&\int\limits_{B_{1}}\frac{1}{c_{n}\delta^{n}}\int\limits_{B_{\delta}}[u(x+y)-u(x)]dV_{n}(y)d\mu(x)\\\
&\frac{1}{c_{n}\delta^{n}}\int\limits_{B_{1}}\int\limits_{B_{\delta}}\int\limits_{B_{2}}[G(x+y,z)-G(x,z)]\Delta
u(z)dV_{n}(y)d\mu(x)\\\ &=\int\limits_{B_{2}}\Delta
u(z)\frac{1}{c_{n}\delta^{n}}\int\limits_{B_{\delta}}dV_{n}(y)\int\limits_{B_{1}}[G(x+y,z)-G(x,z)]d\mu(x)\end{aligned}.$
Set
$F(y,z)=\int\limits_{B_{1}}[G(x+y,z)-G(x,z)]d\mu(x).$
It is enough to prove that $F(y,z)\leq
C(\alpha,A,s)\delta^{\frac{\alpha-\epsilon}{1+\alpha}}$ for all $y\in
B_{\delta},z\in B_{2}$. We consider two cases: Case 1: $n=2$. For $y\in
B_{\delta},z\in B_{2}$, $\delta<\frac{1}{2}$, we have
$\displaystyle F(y,z)$
$\displaystyle=\int\limits_{B_{1}}[\ln|x+y-z|-\ln|x-z|]d\mu(x)$
$\displaystyle=\int\limits_{B_{1}\cap\\{|x-z|\geq|y|^{\frac{1}{1+\alpha}}\\}}\ln|1+\frac{y}{x-z}|d\mu(x)+\int\limits_{B_{1}\cap\\{|x-z|<|y|^{\frac{1}{1+\alpha}}\\}}\ln|1+\frac{y}{x-z}|d\mu(x)$
$\displaystyle\leq\int\limits_{B_{1}\cap\\{|x-z|\geq|y|^{\frac{1}{1+\alpha}}\\}}\ln(1+|y|^{\frac{\alpha}{1+\alpha}})d\mu(x)+\ln
4\int\limits_{B_{1}\cap\\{|x-z|<|y|^{\frac{1}{1+\alpha}}\\}}d\mu$
$\displaystyle\
+\int\limits_{B_{1}\cap\\{|x-z|<|y|^{\frac{1}{1+\alpha}}\\}}\ln\frac{1}{|x-z|}d\mu(x)$
$\displaystyle\leq|y|^{\frac{\alpha}{1+\alpha}}\mu(B_{1})+A|y|^{\frac{\alpha}{1+\alpha}}\ln
4+|y|^{\frac{\alpha-\epsilon}{1+\alpha}}\int\limits_{\\{|x-z|<|y|^{\frac{1}{1+\alpha}}\\}}\frac{1}{|x-z|^{\alpha-\epsilon}}\ln\frac{1}{|x-z|}d\mu(x)$
$\displaystyle\leq A(1+\ln
4)|y|^{\frac{\alpha}{1+\alpha}}+|y|^{\frac{\alpha-\epsilon}{1+\alpha}}C_{1}(\alpha,\epsilon)\int\limits_{\\{|x-z|<1\\}}\frac{d\mu(x)}{|x-z|^{\alpha-\frac{\epsilon}{2}}}$
$\displaystyle\leq A(1+\ln
4)|y|^{\frac{\alpha}{1+\alpha}}+C_{1}(\alpha,\epsilon)|y|^{\frac{\alpha-\epsilon}{1+\alpha}}\sum\limits_{j=0}^{\infty}\int\limits_{\\{2^{-j-1}\leq|x-z|<2^{-j}\\}}\frac{d\mu(x)}{|x-z|^{\alpha-\frac{\epsilon}{2}}}$
$\displaystyle\leq A(1+\ln
4)|y|^{\frac{\alpha}{1+\alpha}}+C_{1}(\alpha,\epsilon)|y|^{\frac{\alpha-\epsilon}{1+\alpha}}A\sum\limits_{j=0}^{\infty}2^{(j+1)(\alpha-\frac{\epsilon}{2})-j\alpha}$
$\displaystyle\leq
C(\alpha,A,\epsilon)|y|^{\frac{\alpha-\epsilon}{1+\alpha}}\leq
C(\alpha,A,\epsilon)\delta^{\frac{\alpha-\epsilon}{1+\alpha}}.$
Case 2: $n\geq 3$. Similarly for $y\in B_{\delta},z\in B_{2}$,
$\delta<\frac{1}{2}$, we have
$\displaystyle F(y,z)$
$\displaystyle=\int\limits_{B_{1}}[-\frac{1}{|x+y-z|^{n-2}}+\frac{1}{|x-z|^{n-2}}]d\mu(x)$
$\displaystyle=\int\limits_{B_{1}\cap\\{|x-z|\geq|y|^{\frac{1}{1+\alpha}}\\}}\frac{|x+y-z|^{n-2}-|x-z|^{n-2}}{|x+y-z|^{n-2}|x-z|^{n-2}}d\mu(x)+\int\limits_{\\{|x-z|<|y|^{\frac{1}{1+\alpha}}\\}}\frac{d\mu(x)}{|x-z|^{n-2}}$
$\displaystyle\leq
C_{2}(\alpha)|y|^{\frac{\alpha}{1+\alpha}}\int\limits_{B_{1}\cap\\{|x-z|\geq|y|^{\frac{1}{1+\alpha}}\\}}d\mu(x)+|y|^{\frac{\alpha-\epsilon}{1+\alpha}}\int\limits_{\\{|x-z|<|y|^{\frac{1}{1+\alpha}}\\}}\frac{d\mu(x)}{|x-z|^{n-2+\alpha-\epsilon}}$
$\displaystyle\leq
AC_{2}(\alpha)|y|^{\frac{\alpha}{1+\alpha}}+|y|^{\frac{\alpha-\epsilon}{1+\alpha}}\int\limits_{\\{|x-z|<1\\}}\frac{d\mu(x)}{|x-z|^{n-2+\alpha-\epsilon}}$
$\displaystyle\leq
C(\alpha,A,\epsilon)|y|^{\frac{\alpha-\epsilon}{1+\alpha}}\leq
C(\alpha,A,\epsilon)\delta^{\frac{\alpha-\epsilon}{1+\alpha}},$
4.2. Theorem. Let $\mu$ be a non-negative Radon measure on $\Omega$ such that
$\mu(B(z,r))\leq Ar^{n-2+\alpha}$ for all $B(z,r)\subset
D\subset\subset\Omega$ ($A,\alpha>0$ are constants). Then for $K\subset\subset
D$ and $\epsilon>0$ there exists $C(\alpha,A,K,\epsilon)$ such that
$\int\limits_{K}[u_{\delta}-u]d\mu\leq
C(\alpha,A,K,\epsilon)||u||_{L^{\infty}(\Omega)}\
\delta^{\frac{\alpha-\epsilon}{2(1+\alpha)}},$
for all $u\in\text{SH}\cap L^{\infty}(\Omega)$. We need a well-known lemma:
4.3. Lemma. Let $u\in\text{SH}\cap L^{\infty}(\Omega)$. Then
$|\tilde{u}_{\delta}(x)-\tilde{u}_{\delta}(y)|\leq\frac{||u||_{L^{\infty}(\Omega)}|x-y|}{\delta},$
for all $x,y\in\Omega_{\delta}$. Proof of Theorem 4.2. By Lemma 4.3 we have
$u_{\delta}(x)=\sup\limits_{y\in B_{\delta}}u(x+y)\leq\sup\limits_{y\in
B_{\delta}}\tilde{u}_{\delta^{\frac{1}{2}}}(x+y)\leq\tilde{u}_{\delta^{\frac{1}{2}}}(x)+\delta^{\frac{1}{2}}||u||_{L^{\infty}(\Omega)}.$
By Theorem 4.1 we get
$\displaystyle\int\limits_{K}[u_{\delta}-u]d\mu$
$\displaystyle\leq\int\limits_{K}[\tilde{u}_{\delta^{\frac{1}{2}}}-u]d\mu+||u||_{L^{\infty}(\Omega)}\mu(K)\delta^{\frac{1}{2}}$
$\displaystyle\leq C(\alpha,A,K,\epsilon)||u||_{L^{\infty}(\Omega)}\
\delta^{\frac{\alpha-\epsilon}{2(1+\alpha)}}.$
Next we state a well-known result is a direct consequence of the Jensen
formula (see [AG]) 4.4. Proposition. Let $u\in\text{SH}(B_{2})$ be such that
$|u(x)-u(y)|\leq A|x-y|^{\alpha}$ for all $x,y\in B_{2}$. Then there exists
$C(\alpha,A)>0$ such that
$\int\limits_{B(x,r)}\Delta u\leq C(\alpha,A)r^{n-2+\alpha},$
for all $B(x,r)\subset B_{1}$.
5\. Main results Proof of Theorem A. We use the same scheme as the proof of
Theorem 2.1 in [Ko5]. From Corollary 3.4 and from Theorem 4.2 we can replace
$\omega^{n}$ by $d\mu$. This implies that $u$ is Hölder continuous with the
Hölder exponent dependent on $\alpha$, $A$, $p$, $X$ and
$||f||_{L^{p}(d\mu)}$. Proof of Corollary B. It follows from Proposition 4.4
and Theorem A. Proof of Corollary C. Direct application of Theorem A.
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Department of Mathematics University of Education (Dai hoc Su Pham Ha Noi)
CauGiay, Hanoi, Vietnam E-mail: phhiep-vnyahoo.com
|
arxiv-papers
| 2009-04-27T15:54:19 |
2024-09-04T02:49:02.187523
|
{
"license": "Public Domain",
"authors": "Pham Hoang Hiep",
"submitter": "Pham Hiep hoang",
"url": "https://arxiv.org/abs/0904.4145"
}
|
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